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recently , the both schools of takemura and takayama have developed a quite interesting minimization method called holonomic gradient descent method(hgd ) . it utilizes grbner basis in the ring of differential operator with rational coefficients . grbner basis in the differential operators plays a central role in deriving some differential equations called a pfaffian system for optimization . hgd works by a mixed use of pfaffian system and an iterative optimization method . it has been successfully applied to several maximum likelihood estimation ( mle ) problems , which have been intractable in the past . for example , hgd solve numerically the mle problems for the von mises - fisher distribution and the fisher - bingham distribution on the sphere ( see , sei et al.(2013 ) and nakayama et al.(2011 ) ) . furthermore , the method has also been applied to the evaluation of the exact distribution function of the largest root of a wishart matrix , and it is still rapidly expanding the area of applications(see , hashiguchi et al.(2013 ) ) . on the other hand , in statistical models , it is not rare that parameters are constrained and therefore the mle problem with constraints has been surely one of fundamental topics in statistics . in this paper , we develop hgd for mle problems with constraints , which we call the constrained holonomic gradient descent(chgd ) . the key of chgd is to separate the process into ( a ) updating of new parameter values by newton - raphson method with penalty function and ( b ) solving a pfaffian system . we consider the following the constrained optimization problem . @xmath2 where @xmath3 , @xmath4 and @xmath5 are all assumed to be continuously differentiable function . @xmath6 is an equality constraint function and @xmath7 is an inequality constraint function . in this paper , the objective function @xmath8 is assumed to be holonomic . we call the interior region defined by the constraint functions _ the feasible region_. a penalty function method replaces a constrained optimization problem by a series of unconstrained problems . it is performed by adding a term to the objective function that consists of a penalty parameter @xmath9 and a measure of violation of the constraints . in our simulation , we use _ the exact penalty function method_. the definition of the exact penalty function is given by ( see yabe ( 2006 ) ) . @xmath10 assume that we seek the minimum of a holonomic function @xmath8 and the point @xmath11 which gives the minimum @xmath8 . in hgd , we use the iterative method together with a pfaffian system . in this paper , we use the the newton - raphson iterative minimization method in which the renewal rule of the search point is given by @xmath12 where @xmath13 and @xmath14 is the hessian of @xmath8 at @xmath15 . hgd is based on the theory of the grbner basis . in the following , we refer to the relation of a numerical method and the grbner basis . let @xmath16 be the differential ring written as @xmath17 \langle \partial_1, .. ,\partial_n \rangle \nonumber\end{aligned}\ ] ] where @xmath18 $ ] are the rational coefficients of differential operators . suppose that @xmath19 is a left ideal of @xmath16 , @xmath20 $ ] is a field and @xmath21\langle \partial_1, .. ,\partial_n \rangle \in i$ ] . if an arbitrary function @xmath22 satisfies @xmath23 for all @xmath24 , then @xmath22 is a solution of @xmath25 . that is @xmath26 when @xmath22 satisfies equation ( [ eq_h ] ) , @xmath22 is called _ holonomic function_. let @xmath27 $ ] , with @xmath28 be a standard basis in the quotient vector space @xmath29 which is a finite dimensional vector spaces . let @xmath30 be the grbner basis of @xmath25 . the rank of arbitrary differential operations can be reduced by normalization by @xmath30 . assume that @xmath31 holds . for a solution @xmath22 of @xmath25 put @xmath32 . then , it holds that ( see , e.g.,nakayama et al.(2011 ) ) @xmath33 where @xmath34 is a @xmath35 matrix with @xmath36 as a @xmath37 element @xmath38_{j } , \ \ i=1, ... ,n,\ \ j=1 ... ,t\end{aligned}\ ] ] this proves the assertion . the above differential equations are called _ pfaffian differential equations _ or _ pfaffian system _ of @xmath25 . so we can calculate the gradient of @xmath39 by using pfaffian differential equations . then , @xmath40 and @xmath41 are also given by pfaffian differential equations . ( see hibi et al.(2012 ) ) let @xmath42 be the normal form of @xmath43 by @xmath30 and @xmath44 be the normal form of @xmath45 by @xmath30 . then we have , @xmath46 where @xmath47 denotes the first entry of a vector @xmath48 . for hgd , we first give an ideal @xmath49 for holonomic function @xmath8 and calculate the grbner basis @xmath30 of @xmath25 and then the standard basis @xmath50 are given by @xmath30 . the coefficient matrix @xmath34 for pfaffian system is led by this standard basis , and @xmath41 and @xmath40 are calculated from @xmath50 by starting from a initial point @xmath51 through the pfaffian equations . after these , we can compute automatically the optimum solution by a mixed use of then newton - raphson method . the algorithm is given by below . * set @xmath52 and take an initial point @xmath53 and evaluate @xmath54 . * evaluate @xmath40 and @xmath55 from @xmath39 and calculate the newton direction , @xmath56 * update a search point by @xmath57 . * evaluate @xmath58 by solving pfaffian equations numerically . * set @xmath59 and calculate @xmath58 and goes to step.2 and repeat until convergence . the key step of the above algorithm is step 4 . we can not evaluate @xmath58 by inputting @xmath60 in the function @xmath8 since the hgd treats the case that @xmath8 is difficult to calculate numerically . instead , we only need calculate @xmath61 and @xmath62 numerically for a given initial value @xmath51 . now , we propose the method in which we add constraint conditions to hgd and call it the constrained holonomic gradient descent method(chgd ) . for treating constraints we use the penalty function and add it to objective function and make a new objective function and can treat it as the unconstrained optimization problem . we use hgd for evaluation of gradients and hessian and use the exact penalty function method for constraints . the value of updating a search point can be obtained as the product of directional vector and step size . the step size @xmath63 is chosen so that the following armijo condition is satisfied . in fact we chose @xmath63 such that @xmath64 where @xmath65 and @xmath66 is the approximation of @xmath67 given by . @xmath68 the initial value of @xmath63 is set @xmath69 and then @xmath63 is made smaller iteratively until @xmath63 satisfies equation ( [ eq_s ] ) , or @xmath70 . in our algorithm , holonomic gradient descent plays a role to calculate the gradient vectors and then the penalty function plays a role to control the step size iteratively . we apply chgd for mle for von mises distribution(vm ) . the process of applying for hgd is shown in nakayama et al.(2011 ) . the density function of vm is given by @xmath71 . the parameters of vm , @xmath72 and @xmath73 , show concentration and mean of angle data @xmath74 respectively . we set the parameters for mle @xmath75 and @xmath76 . now we solve the constrained optimization problem given by . @xmath77 let @xmath74 be sample data . let @xmath78 be sample size . then , @xmath79 and @xmath80 . in our simulation , we set the vm s parameter @xmath81 of which the true value @xmath82 and the initial value @xmath83 . we tried the 2 patterns of constraints . both of the case worked under the same condition except constraints . in figure 1 , the constraint is @xmath84 . in figure 2 , the constraint is @xmath85 . figures 1,2 are the drawing of the trace of the search point . the result of simulation , the convergence point of hgd is @xmath86 . in figure 1 , the convergence point of chgd is @xmath87 . in figure 2 , the convergence point of chgd is @xmath88 . in the chgd , the search direction is almost same as the hgd , because the direction is decided by the hgd s algorithm . while , the constraints play the role to judge the search point is within the feasible region or not and decide the step size . chgd is the effective method for optimization with constraints . however , whenever chgd increases the cost of runtimes than hgd regardless of whether the solution is in the feasible region or not . the following table shows the runtimes when the optimization solution is within the feasible region . in table [ tb1 ] , all numbers are the means of 500 times trials . the optimization problem is equation ( [ optvm ] ) . sample data is drawn from the vm with @xmath89 . the third column of table [ tb1 ] is the result with only newton - raphson method which optimize @xmath8 directly , not use pfaffian system . thus , we see that hgd and chgd is faster than newton - raphson method . we see that the runtimes of chgd is longer than hgd in general , where the both of solutions are almost the same value when the solution is inside the feasible region . sometimes the process finishes early by constraints , when the solution is outside the feasible region . although , we need consider the cost of calculation of chgd . 99 hashiguchi , h. , numata , y. , takayama , n. , takemura , a. ( 2013 ) . _ `` the holonomic gradient method for the distribution function of the largest root of a wishart matrix''_. journal of multiva , riate analysis 117 ( 2031 ) 296 - 312 nakayama , h. , nishiyama , k. , noro , m. , ohara , k. , sei , t. , takayama , n. , takemura , a. ( 2011 ) . _ `` holonomic gradient descent and its application to the fisher bingham integral''_. advances in applied mathematics , 47(3 ) , 639 - 658 . yabe , h. ( 2006 ) . _ `` introduction and application of optimization problem(japanese)''_. surikougakusha publisher . cox , d. a. , little , j. , oshea , d. ( 2007 ) . _ `` ideals , varieties , and algorithms : an introduction to computational algebraic geometry and commutative algebra ( vol . 10)''_. springer verlag .
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recently , the school of takemura and takayama have developed a quite interesting minimization method called _ holonomic gradient descent method _
( hgd ) .
it works by a mixed use of pfaffian differential equation satisfied by an objective holonomic function and an iterative optimization method .
they successfully applied the method to several maximum likelihood estimation ( mle ) problems , which have been intractable in the past . on the other hand , in statistical models , it is not rare that parameters are constrained and therefore the mle with constraints has been surely one of fundamental topics in statistics . in this paper we develop hgd with constraints for mle . * holonomic decent minimization method for restricted maximum likelihood estimation * + rieko sakurai@xmath0 , and toshio sakata @xmath1 + @xmath0 _ graduate school of medicine , kurume university 67 asahimachi , kurume 830 - 0011 , japan _
+ @xmath1 _ faculty of design human science , kyushu university , 4 - 9 - 1 shiobaru minami - ku , fukuoka 815 - 8540 , japan _ + email : a213gm009s@std.kurume-u.ac.jp _ key words : holonomic gradinet descent method , newton - raphson method with penalty function , von mises - fisher distribution _
| 3,230 | 388 |
recent progress in quantum communication technology has confirmed that the biggest challenge in using quantum methods of communication is to provide scalable methods for building large - scale quantum networks @xcite . the problems arising in this area are related to physical realizations of such networks , as well as to designing new protocols that exploit new possibilities offered by the principles of quantum mechanics in long - distance communication . one of the interesting problems arising in the area of quantum internetworking protocols is the development of methods which can be used to detect errors that occur in large - scale quantum networks . a natural approach for developing such methods is to construct them on the basis of the methods developed for classical networks @xcite . the main contribution of this paper is the development of a method for exploring quantum networks by mobile agents which operate on the basis of information stored in quantum registers . we construct a model based on a quantum walk on cycle which can be applied to analyse the scenario of exploring quantum networks with a faulty sense of direction . one should note that the presented model allows studying the situations where all nodes in the network are connected . the reason for this is that a move can result in the shift of the token from the current position to any other position in the network . thus we do not restrict ourselves to a cycle topology . this paper is organized as follows . in the remaining part of this section we provide a motivation for the considered scenario and recall a classical scenario described by magnus - derek game . in section [ sec : quantum - magnus - derek ] we introduce a quantum the scenario of quantum network exploration with a distracted sense of direction . in section [ sec : application - quantum ] we analyse the behaviour of quantum mobile agents operating with various classes of strategies and describe non - adaptive and adaptive quantum strategies which can be employed by the players . finally , in section [ sec : final ] we summarize the presented work and provide some concluding remarks . as quantum networks consist of a large number of independent parties @xcite it is crucial to understand how the errors , that occur during the computation on nodes , influence their behaviour . such errors may arise , in the first place , due to the erroneous work of particular nodes . therefore it is important to develop the methods that allow the exploration of quantum networks and the detection of malfunctioning nodes . one of the methods used to tackle this problem in classical networks is the application of mobile agents , _ i.e. _ autonomous computer programs which move between hosts in a network . this method has been studied extensively in the context of intrusion detection @xcite , but it is also used as a convincing programming paradigm in other areas of software engineering @xcite . on the other hand , recent results concerning the exploration of quantum graphs suggest that by using the rules of quantum mechanics it is possible to solve search problems @xcite or rapidly detect errors in graphs @xcite . in this paper we aim to combine both methods mentioned above . we focus on a model of mobile agents used to explore a quantum network . for the purpose of modelling such agents we introduce and study the quantum version of the magnus - derek game @xcite . this combinatorial game , introduced in @xcite , provides a model for describing a mobile agent acting in a communication network . the magnus - derek game was introduced in @xcite and analysed further in @xcite and @xcite . the game is played by two players : derek ( from _ direction _ or _ distraction _ ) and magnus ( from _ magnitude _ or _ maximization _ ) , who operate by moving a token on a round table ( cycle ) with @xmath0 nodes @xmath1 . initially the token is placed in the position @xmath2 . in each round ( step ) magnus decides about the number @xmath3 of positions for the token to move and derek decides about the direction : clockwise ( @xmath4 or @xmath2 ) or counter - clockwise ( @xmath5 or @xmath6 ) . magnus aims to maximize the number of nodes visited during the game , while derek aims to minimize this value . derek represents a distraction in the sense of direction . for example , a sequence of moves @xmath7 allowing magnus to visit three nodes , can be changed to @xmath8 due to the influence of derek represented by the @xmath4 and @xmath5 signs . the possibility of providing biased information about the direction prevents magnus permanently from visiting some nodes . in the classical scenario one can introduce a function @xmath9 which , for a given number of nodes @xmath0 , gives the cardinality of the set of positions visited by the token when both players play optimally @xcite . it can be shown that this function is well defined and @xmath10 with @xmath11 being the smallest odd prime factor of @xmath0 . by @xmath12 we denote the number of moves required to visit the optimal number of nodes . in the case @xmath13 , the number of moves is optimal and equals @xmath14 . et al . _ proved @xcite that if @xmath15 is a positive integer not equal to a power of @xmath16 , then there exists a strategy allowing magnus to visit at least @xmath9 nodes using at most @xmath17 moves . we distinguish two main types of regimes adaptive and non - adaptive . in the adaptive regime , both players are able to choose their moves during the execution of the game . in the non - adaptive regime , magnus announces the sequence of moves he aims to perform . in particular , if the game is executed in the non - adaptive regime , derek can calculate his sequence of moves before the game . in the classical case the problem of finding the optimal strategy for derek is @xmath18-hard @xcite and is equivalent to the partition problem @xcite . let us now assume that the players operate by encoding their positions on a cycle in an @xmath0-dimensional pure quantum states . thus the position of the token is encoded in a state @xmath19 . at the @xmath20-th step of the game magnus decides to move @xmath21 and derek decides to move in direction @xmath22 . one can easily express the classical game by applying the notation of quantum states . the evolution of the system during the move described above is given by a unitary matrix of the form @xmath23 where @xmath24 . clearly , as the above permutation operators express only the classical subset of the possible moves , by using it one can not expect to gain with respect to the classical scenario . in particular , the operators @xmath25 as introduced above do not allow the preparation of a move by using the information encoded in a superposition . in order to exploit the possibilities offered by quantum mechanics in the magnus - derek scheme , we can use a quantum walk controlled by two registers . to achieve this we need to offer the players a larger state space . we introduce a quantum scheme by defining the following quantum version of the magnus - derek game . 1 . the state of the system is described by a vector of the form @xmath26 2 . the initial state of the system reads @xmath27 . 3 . at each step the players can choose their strategy , possibly using unitary gates . 1 . magnus operates on his register with any unitary gate @xmath28 resulting in a operation of the form @xmath29 performed on the full system . 2 . derek operates on his register with any unitary gate @xmath30 . if his actions are position - independent the operation performed on the full system takes the form @xmath31 . however , in section [ sec : position_control ] we also allow position - controlled actions , resulting in the operator of the form @xmath32 . [ game:3b ] 4 . the change of the token position , resulting from the players moves , is described by the shift operator @xmath33 where the addition and the subtraction is in the appropriate ring @xmath34 . the single move in the game defined according to the above description is given by the position - independent operator @xmath35 taking this into account the state of the system after the execution of @xmath36 moves reads @xmath37 where each matrix @xmath38 depends on the move of each party . the distribution of the position on the cycle after @xmath36 moves is described by a reduced density matrix @xmath39 which represents the state of the token register after tracing - out the subsystems used to process the strategies . here @xmath40 represents the operation of tracing - out the subsystems used by magnus and derek to encode their strategies . the key part of this procedure is how the players choose their strategies . the selection of the method influences the efficiency of the exploration . below we study the possible methods and show how they influence the behaviour of the quantum version of the magnus - derek game . clearly , by using the unitary gates magnus and derek are able to prepare the superpositions of base states . for this reason , one needs to provide the notion of node visiting suitable for analysing quantum superpositions of states . therefore , we introduce the notion of _ visiting _ and _ attaining _ a position . we say that the position @xmath41 is visited in @xmath42 steps , if for some step @xmath43 the probability of measuring the position register in the state @xmath44 is 1 , _ i.e._@xmath45 in order to introduce the notion of attaining we use the concepts of measured quantum walk @xcite and concurrent hitting time . a @xmath44-measured quantum walk from a discrete - time quantum walk starting in a state @xmath46 is a process defined by iteratively first measuring with the two projectors @xmath47 and @xmath48 . if @xmath49 is measured the process is stopped , otherwise a step operator is applied and the iteration is continued . a quantum random walk has a @xmath50 concurrent @xmath51 hitting - time if the @xmath44-measured walk from this walk and initial state @xmath52 has a probability @xmath53 of stopping at a time @xmath54 . we say that the position @xmath41 is attained in @xmath42 steps , if @xmath44-measured exploration walk has a @xmath55 concurrent @xmath56 hitting time , i.e. the exploration walk with initial state @xmath46 has a probability of stopping at a time @xmath57 equal to @xmath6 . with the help of these definitions , one can introduce the concepts of _ visiting strategy _ and _ attaining strategy_. [ def : visiting - strategy ] if for the given sequence of moves performed by magnus , there exists @xmath42 such that each position on the cycle is visited in @xmath42 steps , then we call such sequence of moves a _ visiting strategy_. [ def : attaining - strategy ] if for the given sequence of moves performed by magnus , each position on the cycle is attained , then we call such sequence of moves an _ attaining strategy_. the quantum scheme introduced in the previous section extends the space of strategies which can be used by both players . as there is a significant difference in situations where @xmath13 and @xmath58 , we will consider these cases separately . we start by considering the case @xmath13 . in this situation we have two possible alternatives . in the first one magnus uses the quantum version of the optimal classical strategy and derek while derek performs any possible quantum moves . in the second scenario both players are able to explore all possible quantum moves . let us first consider the quantum scheme executed by magnus with the use of the classical optimal strategy . as in the classical case derek is not able to prevent magnus from visiting all the nodes , it is natural to ask if he can achieve any advantage using unitary moves . if the number of nodes is equal to @xmath59 , for some integer @xmath36 , the optimal strategy for magnus can be computed at the beginning of the game . this strategy _ i.e. _ a sequence of magnitudes is given as ( see lemma 2 in @xcite ) @xmath60 where @xmath61 denotes the repetition of the moves starting from the beginning of the sequence until the move preceding the @xmath61 and excluding it . the first few sequences resulting from eq . ( [ eqn : classical - optimal-2k ] ) are presented in table [ tab : magnus2-moves - examples ] . .optimal moves to be performed by magnus when the number of nodes is equal to @xmath59 . magnus is able to visit all @xmath0 positions in @xmath62 moves by using this strategy . [ cols="<,<",options="header " , ] by using this strategy in the classical case , magnus is able to visit all nodes using @xmath62 moves and derek is not able to prevent him from doing this . moreover , the bound for the number of moves required to visit all the nodes in the classical case is tight . let us now assume that magnus is using quantum moves constructed for the classical optimal strategy , but derek can use arbitrary quantum moves . for example , if @xmath63 magnus optimal strategy is realized by the following sequence of unitary gates @xmath64 first of all , as the moves performed by magnus allow him the sampling of the space of positions using @xmath65 steps , it can be easily seen that derek is not able to prevent magnus from attaining all nodes using @xmath12 moves . on the other hand , derek is able to prevent magnus from visiting all nodes . he can achieve this using the strategy given as follows . [ str:2k - strategy - h - id ] for steps @xmath66 perform the following gate @xmath67 where @xmath68 denotes the hadamard gate . the probabilities of finding a token at each position for the scheme with magnus using the optimal strategy and derek using strategy [ str:2k - strategy - h - id ] is presented in fig . [ fig : qpos-2k - strategy - h - id ] . clearly , magnus is able to attain all the nodes is in @xmath69 steps . however , derek can prevent him from visiting all nodes in @xmath69 steps . this is expressed in the following . [ tw : vs - optimal ] let us take @xmath70 . then , there exists a strategy for derek preventing magnus from visiting all nodes in @xmath69 steps . moreover , there is no strategy for derek that enables him to prevent magnus from attaining all nodes in @xmath69 steps . _ proof._the first part follows from the construction of the strategy [ str:2k - strategy - h - id ] . in fact , any strategy of this form , not necessarily using hadamard gate , will prevent magnus from visiting all nodes . the second part follows from the construction of the magnus strategy . let s assume that there is a strategy that allows derek to prevent magnus from attaining a position @xmath41 i.e. this position is not attained . then a @xmath44-measured walk has no @xmath71 concurrent hitting time . thus , there is a non - zero probability that the process will not stop in @xmath0 steps . this means that at each step @xmath42 there is a non - zero amplitude for some state @xmath72 with @xmath73 _ i.e. _ the state will not get measured by effect @xmath74 . the sequence of directions resulting from the above @xmath75 used by derek in a classical version of the game would give him a strategy forbidding a visit in position @xmath41 . it is a contradiction of the properties of magnus classical strategy . @xmath76 the above proposition can be easily extended as , by using a quantum strategy with only one hadamard gate , derek can prevent magnus from visiting more than two nodes . this result shows that by using quantum moves against the classical strategy , derek is not able to exclude additional positions . however , he gains in comparison to the classical case as he is able to introduce more distraction in terms of the reliability of the exploration . in the situation @xmath77 , the quantum strategies used by derek to distract the sense of directions can depend on the type of information which is available to him . without the possibility to perform position - controlled operations he can only use classical information about history of choices of magnus unitaries that gives him an estimate of the current state . on the other hand , if he is able to decide about his move using the current position , the resulting strategy is more robust . in the classical case the adaptive strategy allows derek to use the knowledge about magnus move to choose a step according to the position of a token in the moment of the decision . in the quantum case , when a superposition of positions is possible and no measurement is allowed , derek s decision can not depend on the position of the token . instead , derek can maintain only information about the history and the current state of the walk in order to choose the optimal move . in this section we provide such quantum adaptive strategy for derek under the principles of the game introduced in section [ sec : quantum - magnus - derek ] , _ i.e. _ without using controlled operations , which can be used by derek to execute his move . using the presented strategy derek can reduce the number of visited positions to 2 ( or even one in the case of odd @xmath0 ) at the cost of increasing the number of attained positions . the main result of this section can be stated as follows . [ tw : pm1 ] in the case when @xmath78 contains in its decomposition two distinct odd prime numbers @xmath11 and @xmath79 there exists a strategy for derek that allows him to assert that : 1 . only the starting position ( and the symmetric one in the case of even @xmath0 ) will be visited , 2 . the total number of attained positions during the walk will be at most @xmath80m = n-(n / q - n / pq)$ ] , assuming that magnus uses only permutation operators . one should note that for @xmath81 magnus can not apply the provided strategy . moreover , the strategy could be applied recursively by excluding subsequent pairs of least odd prime divisors in order to slightly improve this result not all multiplications of @xmath82 need to be attained and the number of attained positions would be at most @xmath83 , for @xmath84 . in order to prove this , we provide a method for constructing a strategy for derek , which allows him to obtain the desired result . we show that the provided strategy guarantees that the amplitudes of a state , at every step corresponding to a fixed set of positions , will be equal to zero and , as a consequence , there is zero probability of measuring any such positions during the walk _ i.e. _ none of them is attained . the first requirement for derek is the choice of the set of _ restricted positions _ , _ i.e. _ positions which will be protected from being visited or attained by magnus . restricted positions have to be distributed on the cycle in a regular way . more precisely , we have the following . [ fact : structure ] a set of restricted positions which can be chosen by derek in order to construct a strategy in proposition [ tw : pm1 ] is a subset of @xmath85 where @xmath11 is a divisor of @xmath0 . _ _ to show that the set of restricted position has to be of this form it is sufficient to prove that the intervals between subsequent restricted positions have to be equal . let us assume that this is not the case and consider three subsequent restricted positions . if the distance between two of them is even , magnus , after visiting the position in the middle , would be able to visit one of the restricted positions . if both distances are odd , but different , then the sum of them is even and by repeating the reasoning track we obtain that magnus is able to visit at least one of the restricted positions . @xmath76 after choosing the set @xmath86 and for a given position on the cycle , derek can choose his move independently from the magnus choice . when we assign the positions with possible directions according to particular magnitudes it turns out that some of the positions are not distinguishable from derek s point of view . let us call two positions _ symmetric _ if their distance to the nearest restricted position in the direction indicated by the coin register is identical . this allows us to state the following . [ fact : symmetric ] considering two symmetric positions the sets of directions that can be chosen by derek in order to avoid visiting restricted positions are identical for every magnus call . one can note that the relation of being symmetric is invariant under the action of the step operator . two facts stated above allow derek to restrict the choice of moves in such manner that he is able prevent magnus from visiting the set of restricted positions . however , the most important part of derek s strategy is steering the state of the system into a superposition of symmetric states . such a state guarantees the possibility to perform a strategy in which none of the states will be visited ( only attained ) . when such a superposition is achievable from the beginning , derek achieves the result similar to the classical case ( equal number of restricted positions ) assuming that none of the states is visited . on the other hand , when he needs to adopt to the standard situation when the starting state is a base state with one particular position , then the number of states that are attained is greater than the number of the positions visited in the classical scenario . [ fact : distance ] if a superposition of two symmetric states has been created from a base state , the beginning position must be equally distant from two closest positions from every set of the restricted positions . _ if this were not the case , the resulting states in a superposition would not be equally distant from restricted positions and , as the result , not symmetric . @xmath76 the above stated facts allow the formulation of the proof of proposition [ tw : pm1 ] . _ proof of proposition [ tw : pm1 ] . _ as a consequence of fact [ fact : distance ] , the starting point has to belong to every restricted set . using fact [ fact : structure ] , magnus can design a strategy that allows him to visit all positions from an arbitrarily fixed set @xmath86 ( by calling appropriate multiplications of @xmath11 ) even when restricted to the permutation operators . thus derek has to choose two prime divisors of @xmath0 and decide which will be used as the restricted positions set , according to the magnus first move . the optimal choice is to use two smallest factors . in this case the optimal strategy for magnus would be to call @xmath87 and visit all positions that are multiplications of @xmath82 , and then switch to @xmath11 . for this reason derek uses the restricted set which is identical as in the case of restricting @xmath88 excluding all the positions numbered with common multiplications of @xmath11 and @xmath79 . from fact [ fact : symmetric ] it follows that each strategy excluding a given set of positions allows the exclusion of the same set while operating on a superposition . taking into account the above considerations , we define the strategy for derek , which fulfills the requirements of proposition [ tw : pm1 ] . [ str : adaptive - no - control ] for any classical strategy used by magnus , derek has to perform the following steps : * step :* apply the hadamard gate . if magnus chooses a magnitude equal to @xmath89 , for some @xmath36 , apply @xmath90 and repeat this step . + if magnus chooses other magnitude go to * step 3*. if magnus chooses a multiplication of @xmath11 ( respectively @xmath79 ) , set the restricted positions to be @xmath91 ( respectively @xmath92 ) . apply the classical strategy @xcite . having strategy [ str : adaptive - no - control ] , while magnus applies the magnitude equal to @xmath89 and derek performs step 2 , no positions restricted in terms of prop . [ tw : pm1 ] . will be attained . starting from the moment that magnus chooses some other magnitude derek applies unitaries that correspond to the classical strategy . this ensures that none of the restricted positions will be attained ( otherwise magnus would be able to visit more than @xmath93 positions in the classical @xmath94 case , see proof of prop . [ tw : vs - optimal ] ) . @xmath76 an example of state evolution in the game executed using strategy [ str : adaptive - no - control ] is presented in fig . [ fig : nocontrol ] . the starting position is the only one that is visited . [ fig : nocontrol ] as it was shown in the previous section a strategy allowed for derek in the scenario introduced at the beginning of this paper is not sufficient to maintain the number of restricted positions characteristic for classical strategy and limiting magnus only to attain positions . however , the notion of adaptive strategies for derek can be transferred into quantum scenario . in order to let derek use position information in his strategy we have to modify the model introduced in section [ sec : quantum - magnus - derek ] . we do this by replacing the local operators @xmath95 available to derek with the position - controlled operators of the form @xmath96 . having such operators at his disposal , derek is able to apply a different strategy to each part of the state separately . let us consider magnus - derek game on @xmath94 , @xmath97 , @xmath98 positions with @xmath11 being the least prime divisor of @xmath0 . when the set of operators available for derek includes the operators of the form @xmath99 where @xmath36 is an arbitrary position and @xmath100 is an arbitrary local unitary operation then the maximum number of attained positions for magnus is equal to @xmath101 ( as in the classical case ) and the total number of visited positions is at most 2 ( respectively 1 if @xmath0 is an odd number ) . _ proof._in the simplest case derek leads to a superposition of two states . in this case he needs only to ensure that the superposition will not vanish . an example of such strategy is presented in fig . [ fig : controlled - one - hadamard ] . [ str : adaptive - controled ] for any magnus strategy based on permutation operators , when @xmath102 and @xmath11 is a prime number , derek has to perform the following steps : * step :* apply the hadamard gate . if @xmath0 is even do nothing as long as magnus move is equal to @xmath103 . if @xmath0 is odd go to * step 3*. find a set of @xmath104 equally distant positions that is disjoint with already visited positions . apply classical strategy to both parts of the state using position controlled operators . the strategy is based on the classical one that is proven to be optimal . if there would be a strategy for magnus that allows him to attain additional position , there would be also an analogous classical strategy ( see proof of prop . [ tw : vs - optimal ] ) . @xmath76 one can also consider a modification of the above strategy with the additional ability of operating on the superposition of more than two base states . the main restriction on the strategy executed by derek is , in this case , the equality of amplitudes @xmath105 for every position @xmath41 and current magnus call @xmath106 . when the condition is satisfied derek is able to set an arbitrary direction in every position of the cycle using the @xmath107 and @xmath108 operators . the example is shown in the fig . [ fig : pos_controled ] . after the second step the state of the token is a superposition of at least three states . as the consequence , the probabilities are more distributed over the cycle . the presented game provides a model for studying the exploration of quantum networks . the model presented in this paper is based on a quantum walk on a cycle . despite its simplicity , the presented model can be used to describe complex networks and study the behaviour of mobile agents acting in such network . one should note that in the case of the magnus - derek game the main objective is to optimize the number of nodes visited during the game . the actual goal of visiting depends on the computation which is required to take place at the nodes . we have shown that by extending the space of possible moves , both players can significantly change the parameters of the exploration . in particular , if magnus uses the sequence of moves optimal for the classical case , derek is able to prevent him from visiting all nodes . we have assumed that in the quantum scenario not only the number of attained positions is at stake but also the number of positions that are visited by magnus . we have considered a modification of a classical strategy that enables both players to preform their tasks efficiently . this analysis provides an interesting insight into the difficulty of achieving quantum - oriented goals . we have also shown that without a proper model of adaptiveness , it is not possible for derek to obtain the results analogous to the classical case ( the number of restricted positions is lower or the no - visiting condition is validated ) . performing a strategy optimized in order to reduce the number of visited slots requires a trade - off with the total number of attained positions . with additional control resources the total number of attained positions is maintained if the number of visited positions is strictly limited . the authors acknowledge the support by the polish national science centre ( ncn ) under the grant number dec-2011/03/d / st6/00413 . jam would like to acknowledge interesting discussions with m. mc gettrick and c. rver . bernardes and e. dos santos moreira . implementation of an intrusion detection system based on mobile agents . in _ proceedings of the international symposium on software engineering for parallel and distributed systems , 2000 _ , pages 158164 , 2000 . t. e. chapuran , p. toliver , n. a. peters , j. jackel , m. s. goodman , r. j. runser , s. r. mcnown , n. dallmann , r. j. hughes , k. p. mccabe , j. e. nordholt , c. g. peterson , k. t. tyagi , l. mercer , and h. dardy . optical networking for quantum key distribution and quantum communications . , 11:105001 , 2009 .
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we develop a model which can be used to analyse the scenario of exploring quantum network with a distracted sense of direction . using this model we analyse the behaviour of quantum mobile agents operating with non - adaptive and adaptive strategies which can be employed in this scenario .
we introduce the notion of node visiting suitable for analysing quantum superpositions of states by distinguishing between visiting and attaining a position .
we show that without a proper model of adaptiveness , it is not possible for the party representing the distraction in the sense of direction , to obtain the results analogous to the classical case .
moreover , with additional control resources the total number of attained positions is maintained if the number of visited positions is strictly limited .
+ keywords : quantum mobile agents ; quantum networks ; two - person quantum games = 1
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schrdinger s cat sums up one of the most striking and counter - intuitive features of quantum systems that is the ability to exist in coherent superpositions of states , which , in the classical world , would mutually exclude each other . while the conceptual ambiguities arising due to this phenomenon have been highly debated in the early days of quantum mechanics , during the last decades , it has been pointed out that quantum coherence might serve as valuable resource , especially for information processing . among the first suggestions in this direction were the brassard - bennett protocol and the deutsch jozsa algorithm promising respectively intrinsically eavesdrop - secure communication and an exponential speedup of computation by exploiting the quantum superposition principle @xcite . although theses schemes are of little practical use so far , they reveal the enormous potential of quantum technologies , which nowadays becomes all the more significant due to recent experiments showing the accessibility of quantum effects even under ambient conditions @xcite . information thermodynamics @xcite provides another , yet much less explored , area of research , which might benefit from the utilization of quantum coherence . the development of this field was originally triggered by maxwell s famous thought experiment challenging the second law by invoking a small intelligent being , which is able to separate the molecules of a gas in thermal equilibrium according to their velocity , thus establishing a spontaneous temperature gradient @xcite . building on maxwell s idea , szilrd invented a microscopic engine consisting of a single molecule confined in a container , which is in contact with a thermal reservoir of constant temperature @xcite . an external agent might operate this setup by first dividing the container in two chambers , second , detecting the position of the molecule and , third , adiabatically expanding the chamber the molecule was found in , thus extracting work from a single heat bath . half a century after its discovery , this apparent contradiction with the second law was resolved by bennett @xcite , who showed that , due to landauer s principle @xcite , the reduction of entropy associated with the measurement in the second step must be eventually compensated when the external agent discards the gathered information from its memory , which can not be inexhaustible . hence , effectively , the information acquired during the measurement is converted into work . meanwhile a fairly complete and experimentally confirmed theoretical framework exists @xcite at least for classical systems , which , on a general level , provides precise extensions of the second law accounting for information as a physical quantity thus relating it to traditional thermodynamic variables such as entropy and work . in the quantum realm , additional intricacies arise , which are not yet fully explored @xcite . as a consequence of the superposition principle even the hilbert space of a simple two - level system ( tls ) contains infinitely many orthonormal pairs of realizable states , each of which is associated with a specific observable , which , in principle , might be measured . moreover , according to the projection postulate , a measurement will typically alter the state of the system and thereby its mean energy . therefore , in strong contrast to the classical case , a projective quantum measurement is not only accompanied by a decrease in entropy but also by an intrinsic change in internal energy , which must be taken into account for thermodynamic considerations . jacobs argued that this energetic cost should be attributed to the external observer and derived an inequality , which incorporates it in an upper bound on the work extractable from a quantum system in thermal equilibrium after a single measurement @xcite . here , we go one step further by relaxing the assumption on the initial state and allowing multiple measurements in finite intervals . using a simple argument based on the first and the second law , we show that jacobs bound holds whenever the probability to obtain a certain outcome does not change from one measurement to the next . this result , in particular , implies a bound on the average work delivered by information driven quantum engines operating periodically and in finite time . moreover , it provides a natural definition for the efficiency of such machines . one of the first specific , fully quantum mechanical models for a measurement controlled device was proposed by lloyd @xcite . in the spirit of szilrd s pioneering work , he considers a single spin@xmath0 system in contact with a thermal heat bath . an external controller can extract work in form of photons from this system by measuring the energy of the spin and applying a @xmath1-pulse at the larmor frequency if the excited state is detected . after the spin - flip , or , if initially the ground state was observed , the system is allowed to return to thermal equilibrium , before the procedure repeats . lloyd demonstrates that his engine can completely convert the information acquired by the measurement into work . furthermore , he argues that the efficiency of this process will inevitably decrease , due to decoherence effects , if any observable different from energy is used to determine the state of the system . however , his reasoning strongly relies on the assumption that the spin has relaxed to thermal equilibrium before any measurement , which , in fact , would require an infinite waiting time . in this work , by generalizing the setup described above , we show that triggering a laser pulse by measuring an observable that does not commute with the hamiltonian of the system can enhance the efficiency if the model is operated in finite time . specifically , we investigate a quantum - optical tls , whose relaxation dynamics is modeled using a quantum master equation . after a projective von - neumann measurement , the system is assumed to be detached from the heat bath such that its time evolution during the laser pulse is governed by a time - dependent schrdinger equation . we note that such a separation of system and environment has recently been argued to be realistic in the context of quantum heat engines @xcite . for our model , we analytically calculate the time - dependent density matrix characterizing the system in the cyclic operation mode and numerically determine the optimal observable to control the feedback protocol as a function of the relaxation time and the spacing of the energy levels . our findings show that exploiting quantum coherences can enhance the efficiency of information engines beyond classically achievable values . the paper is structured as follows . as our first main result , we derive a new bound on the average work output of quantum information engines in section 2 . in section 3 , we introduce a specific model for such a machine and solve its dynamics . section 4 is devoted to the optimization of its efficiency . we conclude in section 5 . we begin this section by introducing a general scheme for a cyclic quantum information engine . to this end , we consider a finite quantum system with hamiltonian @xmath2 , which is in contact with a heat bath of temperature @xmath3 and whose density matrix is initially given by @xmath4 . this setup is now operated by an external agent in two steps . first , an instantaneous projective measurement of the observable @xmath5 is carried out , which yields the outcome @xmath6 and leaves the system in the state @xmath7 with probability @xmath8 here , the @xmath6 are the eigenvalues of @xmath5 , which we assume to be non - degenerate , and @xmath9 denotes the normalized eigenvector of @xmath5 corresponding to @xmath6 . second , to convert the acquired information into useful work , a control operation is applied to the system , which is conditioned on the result of the preceding measurement and leads to the evolved density matrix @xmath10,\ ] ] where @xmath11 , in principle , can be any positive , trace preserving map @xcite . practically , such an operation can be realized by intermediately manipulating the hamiltonian of the system or its coupling to the heat bath over a certain time interval . the agent now iterates the sequence of steps one and two , where , in the @xmath12^th^ operation cycle , the measurement outcome @xmath13 is obtained with probability @xmath14 . it is readily seen that these quantities fulfill the recursion relation @xmath15p_{m}^{(i-1)}\ ] ] with the conditional probability @xmath16 = { \langle\psi_{m'}|}\tilde{{\rho}}_{m}{|\psi_{m'}\rangle } = { \langle\psi_{m'}|}{\mathcal{v}}_{m}\big[{|\psi_{m}\rangle } { \langle\psi_{m}|}\big]{|\psi_{m'}\rangle},\ ] ] since the initial density matrix of each cycle is the result of the control operation applied in the foregoing one , as shown in figure [ fig : flow_diag ] . moreover , since the transition probability does not depend on the cycle index @xmath12 but rather is fully determined by the control operation @xmath11 and the observable @xmath5 , after sufficiently many iterations a stationary distribution @xmath17 satisfying @xmath18q_{m}\ ] ] will be approached . once this steady state is reached , the system works as a periodic information engine . is measured and the control operation @xmath11 conditioned on the measurement outcome @xmath6 is applied to the system . symbols are explained in the main text . ] for a thermodynamic analysis of the scheme outlined above , we have to calculate the changes in internal energy @xmath19\equiv { { { \rm tr}}\left\{h{\rho}\right\}}$ ] and entropy @xmath20\equiv-{k_\mathrm{b}}{{{\rm tr}}\left\{{\rho}\ln{\rho}\right\}}$ ] of the system associated with the steps one and two , where @xmath21 is boltzmann s constant . considering an operation cycle with initial density matrix @xmath22 and measurement outcome @xmath13 , we find @xmath23- e[\tilde{{\rho}}_{m } ] = { \langle\psi_{m'}|}h{|\psi_{m'}\rangle } -{{{\rm tr}}\left\{h\tilde{{\rho}}_{m}\right\ } } , \label{demeas}\\ \delta{s_{{\rm sys}}}^{{\rm meas}}(m',m ) & = { s_{{\rm sys}}}[{\rho}_{m'}]- { s_{{\rm sys}}}[\tilde{{\rho}}_{m } ] = { k_\mathrm{b}}{{{\rm tr}}\left\{\tilde{{\rho}}_{m}\ln\tilde{{\rho}}_{m}\right\ } } \label{dsmeas}\end{aligned}\ ] ] for the measurement and @xmath24- e[{\rho}_{m ' } ] = { { { \rm tr}}\left\{h\tilde{{\rho}}_{m'}\right\}}-{\langle\psi_{m'}|}h{|\psi_{m'}\rangle},\\ \delta{s_{{\rm sys}}}^{{\rm con}}({m ' } ) & = { s_{{\rm sys}}}[\tilde{{\rho}}_{m'}]-{s_{{\rm sys}}}[{\rho}_{m ' } ] = -{k_\mathrm{b}}{{{\rm tr}}\left\{\tilde{{\rho}}_{m'}\ln\tilde{{\rho}}_{m'}\right\}}\end{aligned}\ ] ] for the control operation , where we used @xmath25=0 $ ] due to @xmath26 representing a pure state . since the total entropy production during the control step @xmath27 must be nonnegative by virtue of the second law , it follows that the change in entropy of the heat bath @xmath28 is bounded from below by @xmath29 and thus the heat taken up by the system @xmath30 during the control operation @xmath31 is bounded from above by @xmath32 . consequently , the first law @xmath33 implies the bound @xmath34 on the work @xmath35 the agent can extract from the system using the operation @xmath31 . since the measurement outcome @xmath6 occurs with probability @xmath36 in the steady state , yields the bound @xmath37 on the average work extracted per operation cycle . furthermore , the average energetic cost and entropy reduction per cycle associated with the measurement read @xmath38 and @xmath39 respectively . here , @xmath40q_{m}$ ] is the probability to measure @xmath13 and @xmath6 in two consecutive operation cycles . inserting and into and and using the steady state condition as well as the sum rule @xmath41 = 1\ ] ] expressing probability conservation yields @xmath42 by comparing and with , we obtain the bound @xmath43 this inequality , which constitutes our first main result , provides a universal upper bound on the average work extractable per operation cycle in terms of quantities that are related to the measurement process only . it generalizes similar results obtained in @xcite for single stroke operations . following the arguments of jacobs @xcite , we consider the energetic cost of the measurement @xmath44 as work input provided by the measurement apparatus and thus infer from the natural definition @xmath45 for the efficiency , at which information is converted to work in cyclic quantum engines . we note that , while @xmath46 is readily seen to be always nonnegative , in contrast to the setup considered in @xcite , @xmath47 can , in principle , become negative , since , for finite cycle times , the system will typically not be in thermal equilibrium before the measurement is performed . moreover , the quantity @xmath47 is of pure quantum origin and vanishes in the quasi - classical situation , where the observable @xmath5 commutes with the hamiltonian of the system @xmath2 . as an application of the general theory discussed so far , we propose a generalization of a paradigmatic model for a quantum information engine originally invented by lloyd @xcite and analyze its thermodynamic properties . specifically , we consider an optical tls with hamiltonian @xmath48 where @xmath49 is the energetic spacing between the ground state @xmath50 and the excited state @xmath51 . the external agent measures the observable @xmath52 @xmath53 with eigenvalues @xmath54 and corresponding eigenvectors @xmath55 which reduce to the eigenstates of @xmath2 for @xmath56 . in order to extract work in form of photons , after a measurement of the state @xmath57 , the system is detached from the heat bath and a coherent laser pulse on resonance is applied for an interval @xmath58 . after a measurement of @xmath59 , the system is kept in contact with the thermal environment for a time @xmath60 without any action of the agent to allow the absorption of additional heat before the next measurement is carried out . . if the outcome of this measurement is @xmath61 , the internal energy of the tls is used to coherently amplify an externally generated laser pulse ( b ) . if the outcome is @xmath62 , the susceptibility of the system for further energy uptake is exploited to extract heat from the environment ( c ) . in any case , the density matrix at the end of the operation cycle serves as initial state for the subsequent one . for further explanations of the symbols , see main text . ] for a quantitative description of this procedure , which is summarized in figure [ fig : tls_scheme ] , we need to specify the control operations @xmath63 . during the interaction ( @xmath64 ) with the laser pulse , the density matrix @xmath65 of the system evolves unitarily according to the liouville - von neumann equation @xmath66 \equiv \mathcal{l}_+(\tau){\rho}(\tau),\ ] ] where , on the semiclassical level and within the rotating wave approximation , the time - dependent hamiltonian is given by @xmath67 with real rabi frequency @xmath68 and @xmath69 being the phase of the dipole matrix element @xcite . to describe the interaction of the tls with the heat bath during @xmath70 , we use the quantum optical master equation @xcite @xmath71 + \gamma n\left(l{\rho}(\tau)l^\dagger -\frac{1}{2}l^\dagger l{\rho}(\tau ) -\frac{1}{2}{\rho}(\tau)l^\dagger l\right)\nonumber\\ & \hspace{2.23 cm } + \gamma ( n+1)\left(l^\dagger{\rho}(\tau)l -\frac{1}{2}ll^\dagger{\rho}(\tau ) -\frac{1}{2}{\rho}(\tau)ll^\dagger\right)\nonumber\\ & \equiv { \mathcal{l}}_-{\rho}(\tau ) , \label{meq}\end{aligned}\ ] ] where @xmath72 is a lindblad operator , @xmath73 - 1)$ ] denotes the planck distribution evaluated for the level spacing @xmath74 and @xmath75 is a damping rate quantifying the coupling strength between the tls and the thermal reservoir . this time evolution equation , which is of lindblad form and therefore preserves trace and positivity of the density matrix , can be derived from a microscopic model in the weak coupling limit , where the role of the heat bath is played by the thermal radiation field , for details , see @xcite . such master equations are a well established method for the description of open quantum systems , which has previously lead to substantial insights in the context of quantum heat engines , see for example @xcite . in terms of the super operators @xmath76 , the control operations admit the formal representations @xmath77 = \overrightarrow{\mathcal{t}}e^{\int_0^{t_f}\!\!\ ! d\tau\;{\mathcal{l}}_+(\tau)}{\rho}\quad\text{and}\quad { \mathcal{v}}_-[{\rho } ] = e^{{\mathcal{l}}_- t_r}{\rho},\ ] ] where @xmath78 indicates time ordering . solving the equations and for a general initial condition yields the explicit expressions @xcite @xmath79 & = u{\rho}u^\dagger \quad\text{with}\quad\nonumber\\ u & = \cos\frac{\omega t_f}{2}\left ( e^{i\omega_0t_f/2}{|g\rangle}{\langleg| } + e^{-i\omega_0t_f/2}{|e\rangle}{\langlee|}\right)\nonumber\\ & \qquad -i\sin\frac{\omega t_f}{2}\left ( e^{i\left(\omega_0t_f/2-\phi\right)}l^\dagger + e^{-i\left(\omega_0t_f/2-\phi\right)}l\right ) \label{copp}\end{aligned}\ ] ] and @xcite @xmath80 = e^{{\mathcal{l}}_- t_r}{\rho } & = \frac{1}{4}\left(1+e^{-{\gamma}t_r}+2e^{-{\gamma}t_r/2 } \cos\omega_0 t_r\right){\rho}\nonumber\\ & + \frac{1}{4}\left(1+e^{-{\gamma}t_r}-2e^{-{\gamma}t_r/2 } \cos\omega_0 t_r\right)l_0{\rho}l_0\nonumber\\ & -\frac{1}{4}\left(\frac{\gamma}{{\gamma}}(1-e^{-{\gamma}t_r } ) -2ie^{-{\gamma}t_r/2}\sin\omega_0 t_r\right){\rho}l_0\nonumber\\ & -\frac{1}{4}\left(\frac{\gamma}{{\gamma}}(1-e^{-{\gamma}t_r } ) + 2ie^{-{\gamma}t_r/2}\sin\omega_0 t_r\right)l_0{\rho}\nonumber\\ & + ( 1-e^{-{\gamma}t_r})\left(\frac{\gamma ( n+1)}{{\gamma}}l^\dagger{\rho}l + \frac{\gamma n}{{\gamma } } l{\rho}l^\dagger\right ) , \label{copm}\end{aligned}\ ] ] where we introduced the abbreviations @xmath81 and @xmath82 . the work extracted within an operation cycle with measurement outcome @xmath61 can be determined form the first law @xmath83 - e[\tilde{{\rho}}_+ ] = e[{\rho}_+ ] - e[{\mathcal{v}}_+[{\rho}_+]],\ ] ] since the tls is decoupled from the environment and thus no heat is exchanged during the control operation @xmath84 . inserting @xmath85 and into gives @xmath86 to keep the subsequent analysis as simple as possible , from here onwards , we fix the pulse duration @xmath58 and the dipole phase @xmath87 such that assumes the maximal value @xmath88 with respect to these parameters , i.e. , we put @xmath89 this choice ensures that the tls ends up in the ground state after the laser pulse , i.e. , @xmath90 . furthermore , it leads to the fairly simple expressions @xmath91 & = \sin^2\frac{{\theta}}{2}\end{aligned}\ ] ] and @xmath92 & = \frac{1}{2}\left(1-\frac{\cos { \theta}}{2n+1}-e^{-{\gamma}t_r/2}\sin^2{\theta}\cos\omega_0 t_r -e^{-{\gamma}t_r}\cos{\theta}\left[\cos { \theta}-\frac{1}{2n+1 } \right ] \right)\label{condprob}\end{aligned}\ ] ] for the conditional probabilities defined in . since @xmath93 $ ] and @xmath94 $ ] are determined by the sum rules , the steady state probabilities @xmath95 can now be obtained from the fixed point condition . specifically , we find @xmath96}{1-p[+|+]+p[+|- ] } = 1-q_-.\ ] ] we are now ready to calculate the quantities entering the efficiency . first , the average work per cycle reads @xmath97 since no contribution arises from operation cycles with measurement outcome @xmath62 . second , the average energy spent on the measurement becomes @xmath98- e\big[{\mathcal{v}}_m[{\rho}_m]\big]\big)\nonumber\\ & = \langle w\rangle + q_-\frac{\hbar\omega_0}{2 } \left(1- e^{-{\gamma}t_r}\right)\left(\frac{1}{2n+1}-\cos{\theta}\right ) \label{etademeas}\end{aligned}\ ] ] upon using @xmath99 and the expressions and for the control operations . third , since @xmath100 represents a pure state due to the control operation @xmath84 being unitary , the average entropy reduction in the system associated with the measurement arises only from cycles with initial state @xmath101 . after some algebra again using , we thus obtain @xmath102\ln{\mathcal{v}}[{\rho}_-]\right\ } } \nonumber\\ & = \frac{q_-}{2}{k_\mathrm{b}}\left(\ln d + \sqrt{1 - 4d}\ln\left(\frac{1+\sqrt{1 - 4d } } { 1-\sqrt{1 - 4d}}\right)\right ) \qquad\text{with}\label{etadsmeas}\\ d&\equiv \frac{1}{4}\left(1- \left(e^{-{\gamma}t_r}\cos{\theta}-\frac{e^{-{\gamma}t_r}-1}{2n+1}\right)^2 - e^{-{\gamma}t_r}\sin^2{\theta}\right ) . \label{etaddef}\end{aligned}\ ] ] using the expressions - , the efficiency @xmath103 of this quantum optical information engine can be evaluated for any complete set of parameters comprising the level spacing @xmath74 , the temperature of the heat bath @xmath3 , the angle @xmath104 , the damping rate @xmath105 and the relaxation time @xmath60 . in this section , we focus on the question whether coherences , i.e. , a choice @xmath106 for the basis , in which the measurement is performed , can enhance the efficiency @xmath103 . in order to reduce the number of free parameters , we choose from now on the temperature such that @xmath107 leading to @xmath108 . we then find by inspection that @xmath103 depends only on @xmath104 and the two dimensionless parameters @xmath109 and @xmath110 . a numerical optimization procedure yields the maximal efficiency @xmath111 and optimal angle @xmath112 , which are both shown in upper panels of figure [ fig : eta_theta ] . these plots exhibit two qualitatively different regimes separated by @xmath113 . and @xmath114 . the upper panel shows the maximum efficiency @xmath115 on the left and the corresponding optimal angle @xmath116 on the right . along the dashed lines , the condition is fulfilled . the solid lines , which have a spacing of @xmath117 and constant offset @xmath118 , were introduced for graphical purposes . in the lower panel , the work output @xmath119 ( left ) and the average energy input required per measurement @xmath120 ( right ) is plotted in units of @xmath74.,scaledwidth=99.0% ] first , for @xmath121 , we recover the quasi - classical regime originally considered by lloyd @xcite , within which the tls can relax to thermal equilibrium in each operation cycle with measurement outcome @xmath62 . as argued in @xcite , the largest efficiency can then be obtained for @xmath56 . consistently , we observe that @xmath116 is effectively zero in this regime and , independent of @xmath110 , the optimal efficiency settles at the constant value @xmath122 second , in the coherent regime @xmath123 , we find a characteristic oscillatory pattern , which can be traced back to the structure of the conditional probability . the crucial role of this quantity , which is , in fact , the only ingredient of the efficiency depending on @xmath110 , can be explained by the following argument . if the tls is found in the state @xmath57 after being in contact with the thermal environment for the time @xmath60 , the heat absorbed during this period together with the energy invested for the measurement can be converted into work by applying a laser pulse . a measurement indicating the state @xmath59 , however , leads to another relaxation cycle , within which no work can be extracted and the previously gained information is inevitably wasted . consequently , the efficiency , at which acquired information is converted into work , can be expected to increase as the frequency of such idle cycles decreases . for @xmath123 , the corresponding probability @xmath124 & = 1- p[+|-]\nonumber\\ & = \frac{1}{2}\left ( 1+\frac{\cos{\theta}}{3}+e^{-3{\gamma}t_r/2}\sin^2{\theta}\cos { \omega}_0 t_r+ e^{- 3{\gamma}t_r}\cos{\theta}\left[\cos{\theta}-\frac{1}{3 } \right]\right ) \label{condprobic}\end{aligned}\ ] ] can be substantially reduced by the contribution proportional to @xmath125 , which arises solely due to quantum coherences and vanishes for @xmath56 . this effect becomes most pronounced for @xmath126 and @xmath127 accordingly , the hyperbolas are in good agreement with the local maxima of the efficiency @xmath115 in the @xmath128,@xmath129-plane and @xmath130 comes close to @xmath131 in their vicinity . the deviations from this pattern for @xmath132 can be explained by the remaining terms in , which come with a prefactor @xmath133 and thus give a non - negligible contribution only in these regions . most importantly , in this regime , we find as our second main result that the efficiency is enhanced by exploiting coherences . specifically , it can overcome the quasi - classical value and even approach its upper bound @xmath134 in the limit @xmath135 . the average work output for the optimal angle @xmath116 , @xmath136 , is plotted in the lower panel of figure [ fig : eta_theta ] . clearly , this quantity features the same characteristic dependence on @xmath137 and @xmath114 as the maximized efficiency @xmath115 . like @xmath115 , the average work @xmath138 exceeds its quasi - classical limit @xmath139 in the coherence - enhanced regime and becomes maximal in the same range of parameters like @xmath115 , i.e. , in the vicinity of the hyperbolas . finally , we consider the average energetic cost per measurement @xmath140 . this additional input is inevitably necessary for the exploitation of quantum coherence and therefore becomes non - negligible whenever @xmath116 significantly deviates from @xmath141 , hence , in particular , in the regions of the parameter space , where our numerical procedure reveals @xmath115 to be large . consequently , in the range of high efficiencies , the input of the device is mainly delivered by the measurement apparatus rather than the heat bath . this result underlines the crucial role of the measurement process in the quantum realm , which , besides delivering information , can alter the state of the system and thus bears the character of an additional control operation . in this paper , we have derived a universal upper bound on the average work output delivered in finite time by cyclically operating quantum information engines , which takes into account the energetic cost intrinsically associated with quantum measurements . this bound provides a benchmark for the performance of quantum mechanical information - to - energy converters , which , in contrast to their classical counterparts , see for example @xcite , can exploit the superposition principle and thus might be able to overcome classical limitations . we have explicitly investigated the benefit of quantum coherences in the second part of the paper by considering a specific model consisting of a quantum optical tls , which , conditioned on the outcome of a projective measurement , is repeatedly either coupled to a heat bath or used to amplify a coherent laser pulse . in the regime of long relaxation times , this setup corresponds to a model originally proposed by lloyd , whose properties are reproduced qualitatively in our analysis within this limit . we emphasize , however , that the definition of efficiency used in @xcite is different from ours , since it explicitly refers to landauer s principle by invoking the minimal heat that must be dissipated in a second heat bath of different temperature to achieve the entropy production necessary to reset the memory of the external agent . viewed in this way , the model acts effectively as a heat engine , whose efficiency is bounded by the carnot value . in our approach , we consider the system as an information engine and define its efficiency in terms of quantities directly associated with the system and the measurement process , leaving aside how the agent eventually erases the gathered information . in the coherent regime , which is characterized by short cycle times , we find that utilizing a non - classical observable @xmath52 , whose eigenstates are coherent superpositions of the energy eigenstates , can enhance the performance significantly . remarkably , it turns out that both , efficiency and average work output per cycle , can be substantially increased if the relaxation time is properly adjusted to the level spacing . since , in the corresponding regions of the parameter space , the optimal angle @xmath116 strongly deviates from the quasi - classical value @xmath141 and even comes close to @xmath131 , the device is then mainly supplied by the measurement apparatus rather than the heat bath . in fact , jacobs argued that , for thermodynamical consistency , this type of energy input must be considered as work rather than heat @xcite . it should , however , also be clearly distinguished from the work output extracted by external control operations . further clarification of the role of the measurement process in this context , using e.g. a scheme proposed in @xcite , constitutes an important and challenging subject for future research . 10 f. dolde , i. jakobi , b. naydenov , n. zhao , s. pezzagna , c. trautmann , j. meijer , p. 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|
a genuine feature of projective quantum measurements is that they inevitably alter the mean energy of the observed system if the measured quantity does not commute with the hamiltonian .
compared to the classical case , jacobs proved that this additional energetic cost leads to a stronger bound on the work extractable after a single measurement from a system initially in thermal equilibrium [ phys .
rev . a 80 , 012322 ( 2009 ) ] . here
, we extend this bound to a large class of feedback - driven quantum engines operating periodically and in finite time .
the bound thus implies a natural definition for the efficiency of information to work conversion in such devices . for a simple model consisting of a laser - driven two - level system ,
we maximize the efficiency with respect to the observable whose measurement is used to control the feedback operations .
we find that the optimal observable typically does not commute with the hamiltonian and hence would not be available in a classical two level system .
this result reveals that periodic feedback engines operating in the quantum realm can exploit quantum coherences to enhance efficiency .
= 1
| 11,540 | 261 |
@xcite attributed the production of the isotopes heavier than the iron group to three processes of nucleosynthesis , the @xmath10- and @xmath11-processes of neutron addition , and the @xmath0-process of proton addition . the conditions they specified for the @xmath0-process , proton densities @xmath13 g @xmath14 and temperatures @xmath15 k , were difficult to realize in nature and so other processes and sites were sought . @xcite and @xcite attributed the production of the @xmath0-process nuclei to photodisintegration , a series of ( @xmath16n ) , ( @xmath16p ) and ( @xmath17 ) reactions flowing downward through radioactive proton - rich progenitors from lead to iron . their `` @xmath7-process '' operated upon previously existing @xmath11-process seed in the star to make the @xmath0-process , and was thus `` secondary '' in nature ( or even `` tertiary '' since the @xmath11-process itself is secondary ) . it could only happen in a star made from the ashes of previous stars that had made the @xmath11-process . arnould suggested hydrostatic oxygen burning in massive stars as the site where the necessary conditions were realized ; woosley and howard , who discovered the relevant nuclear flows independently , discussed explosive oxygen and neon burning in a type ii supernova as the likely site . over the years , increasingly refined calculations showed that a portion of the @xmath0-nuclei could actually be produced as woosley and howard described ( e.g. * ? ? ? * ) . a nagging deficiency of @xmath0-process production in the mass range a = 92 - 124 still persisted though . the production of @xmath1mo posed a particular problem since , unless the star had previously experienced a strong @xmath11-process , enhancing the abundance of seed above a = 95 , there simply was not enough seed . in massive stars the @xmath11-process does not go above mass 90 and so the necessary seed enhancement does not occur . @xcite found that large abundances of some @xmath0-nuclei , and @xmath18mo in particular , could be synthesized in the neutrino - powered wind blowing from a young neutron star ( see also * ? ? ? * ) . while this wind had chiefly been seen as a way of making the @xmath10-process @xcite , for electron mole numbers , @xmath19 , the @xmath0-nuclei @xmath20zn , @xmath21se , @xmath22kr , @xmath23sr , and @xmath18mo were produced in great abundance . it is important to note in this regard that , while @xmath24 is nominally neutron rich ( @xmath25 = 0.5 corresponds to neutron , proton equality ) , it is still a lot more proton - rich than the @xmath0-nuclei themselves ( z / n for @xmath18mo = 0.457 ) , so the nucleonic gas contained some free protons . the @xmath0-nuclei here were also primary , in the sense that a star with no initial metallicity would still make the same composition in its neutrino wind . there were potential problems , however , in that the ejection of only a small amount of mass with @xmath25 just a little lower than 0.485 resulted in disastrous overproduction of n = 50 nuclei like @xmath26sr , @xmath27y , and @xmath28zr . also the neutron - rich wind failed to produce adequate amounts of p - process nuclei above a = 92 . though this paper focuses on early proton - rich outflows , the sn model we study is calculated to eject a sizable amount of neutron - rich material . it remains to be seen if very recent simulations predict neutron - rich outflows that satisfy the conditions needed for efficient synthesis of @xmath18mo , or if neutrino interactions facilitate production of @xmath29 in the neutron rich ejecta predicted by these models @xcite . based upon calculations by jim wilson , @xcite pointed out that @xmath25 in the wind would naturally evolve though the points necessary to make these @xmath0-nuclei and would actually start with a value greater than 0.5 . as other detailed models for core - collapse supernovae became available , nucleosynthesis was explored in this `` hot , proton - rich bubble '' by @xcite , @xcite , and @xcite . the latter two studies found substantial production of nuclei up to a = 84 , including some nuclei traditionally attributed to the @xmath0-process . it seems probable that these winds have also contributed appreciably to the solar abundances of @xmath30sc , @xmath31ti and @xmath20zn , and , possibly in a measurable way , to other rare abundances in metal poor stars . however , since these same nuclei were already made by other processes @xcite , there seemed to be no clear diagnostic of the proton - rich wind . here , following the suggestion of @xcite , we have revisited our calculations of the proton - rich wind including , in addition to the proton captures , the effect of a neutron flux created by p(@xmath5n . these neutrons have the effect of bridging the long - lived isotopes along the path of the @xmath6-process by ( n , p ) reactions and accelerating the flow to heavier elements . for our standard assumptions regarding expansion rate and neutrino fluxes @xcite , we find substantial production of @xmath0-process nuclei up to pd , whereas previously the heaviest major production was zn . if the entropy of the expansion is artificially increased by a factor of 3 to account for extra energy deposition in the wind @xcite , magnetic confinement @xcite , or alfvn wave dissipation @xcite , the production of @xmath0-nuclei extends up to a = 170 . interestingly , the relevant conditions , @xmath32 g @xmath14 and t = @xmath33 k resemble those originally proposed for the @xmath0-process by b@xmath34fh . key differences , however , are that all the species produced here are primary and the process occurs on a shorter time scale - just a few seconds - owing to the `` effective '' acceleration of weak decays by ( n , p ) reactions . our fiducial model for exploring nucleosynthesis in the proton - rich wind is the explosion of a 15@xmath35m@xmath36 star @xcite . an earlier paper ( * ? ? ? * henceforth paper i ) studied nucleosynthesis in this same model but did not account for the influence of neutrino interactions . the present study includes charged - current capture on free nucleons @xcite and neutral - current spallation of nucleons from alpha particles @xcite . many other details of the supernova model and associated nuclear - network calculations relevant to this work can be found in paper i. the ejecta of the deepest layers of the supernova can be divided into two categories - hot bubble ejecta and winds . material in the hot bubble originates from a region outside the neutron star that is driven convectively unstable by neutrino heating . this material does not have to escape the deep gravitational well found at the neutron star surface . as a result , modest conditions characterize these outflows : @xmath37 and @xmath38 . winds originate from the surface of the neutron star and are pushed outwards along gentle pressure gradients caused by neutrino heating @xcite . these outflows have relatively high entropies ( @xmath39 ) , high electron fractions ( @xmath25 as large as 0.57 ) , and short initial expansion timescales . tables 2 & 3 in paper i provide a brief summary of initial conditions in different bubble and wind trajectories . in the absence of neutrinos , very little synthesis occurs in the early proton - rich outflows for nuclei heavier than a= 64 . compared to observed solar abundances , proton - rich winds that are not subject to a neutrino fluence are copious producers of @xmath30sc and @xmath40 . bubble trajectories , on the other hand , tend to favor the synthesis of @xmath41 , @xmath42 and some co or ni isotopes . the nuclear flow stops at @xmath20ge for two reasons : the long weak decay lifetime ( with respect to the expansion timescale of the neutrino winds ) of this and other even - z even - n waiting point nuclei and the small q values for proton capture on these nuclei . the simulations of @xcite followed the explosion of the supernova in 2 dimensions until about 450 ms after core bounce . at this time , typical temperatures in the bubble trajectories were around 4 billion degrees . at the end of the 2d simulation the sn was mapped onto a 1d grid . the subsequent evolution of the sn , including winds emitted by the neutron star , was followed until about 1300 ms after core bounce . at this last time typical asymptotic temperatures in the early wind were just over 2 billion degrees . since most of the interesting nucleosynthesis occurs for @xmath43 , it is necessary to extrapolate outflow conditions to later times . details of the extrapolation were described in paper i which considered two estimates . in the first , the expansion already calculated is assumed to continue homologously with no deceleration . this gives a useful lower bound on the expansion timescale . however , in reality the outflow will quickly catch up to the outgoing shock . it is more reasonable then to assume that escaping matter enters a phase where it moves with the shock speed . conditions for this can be estimated from 1d supernova simulations . for the present paper we focus on the more realistic extrapolation that mimics the late time slowing of the wind . the temperature and density evolution was extrapolated as in paper i. specifically , the trajectories found by @xcite were smoothly merged with those calculated for the inner zone of the same @xmath44 star by @xcite . to avoid discontinuities in the entropy at the time where the two calculations were matched the density in the previous 1d calculation was changed to match that in the trajectories found by janka et al . our procedure likely leads to an underestimate of the expansion timescale at late times because the explosion energy and shock velocity in the calculations of woosley & weaver were somewhat larger than in the more recent 2d simulations . in order to include the influence of the neutrinos from the cooling neutron star additional assumptions are needed . for the neutrino temperatures and luminosities we used the same values calculated at approximately 1 second in the simulation of @xcite . these values are shown in table [ nutable ] and are assumed to remain constant . this may be a questionable assumption - neutrino spectra actually harden and their luminosities fall as the neutron star shrinks . however , it is estimated that uncertainties in the extrapolation of trajectories to low temperatures are greater than uncertainties arising from our simple treatment of the neutrino spectral evolution . to obtain the integrated neutrino exposure seen by outflowing material , it is also necessary to make assumptions about the evolution of the radial velocity . in all cases , it was assumed that the radial velocity at times greater than those followed in the simulations was constant at @xmath45 . this is close to the velocity of the outgoing shock in the 2d calculation of this fairly low energy explosion . again , there may be some inconsistency between our treatment of the late time expansion and our adopted asymptotic radial velocity . a more sophisticated approach might scale the expansion time estimated from the calculations of woosley & weaver to reflect the relatively small shock velocity found in the 2d simulations . it is shown in section [ modest ] that our calculations for nucleosynthesis are relatively insensitive to changes in the late - time outflow velocity . figure [ allfig ] shows results of our calculations that include neutrino captures for production factors characterizing nucleosynthesis in proton - rich outflows leaving the early sn . the production factor for an isotope @xmath46 is defined here as @xmath47 in this equation , the sum is over all trajectories , @xmath48 is the mass fraction of nuclide @xmath46 in the @xmath49th trajectory , @xmath50 is the mass fraction of nuclide @xmath46 in the sun @xcite , @xmath51 is the mass in the @xmath49th trajectory , and @xmath52 is the total mass ejected in the sn explosion . the lower panel in figure [ allfig ] shows production factors characterizing nucleosynthesis in just the hot bubble . this is comprised of the 40 trajectories described in table 1 of paper i. by comparing with table 5 from paper i we see that nucleosynthesis in the bubble material is not greatly changed when neutrino captures are included . this is because the bubble material is far from the neutron star and experiences few neutrino captures relative to the number of seed nuclei in this low entropy environment . the story is much different for the early wind shown in the middle panel of figure [ allfig ] . this wind is comprised of the 6 trajectories defined in table 2 of paper i. in these outflows , neutrinos convert free protons into neutrons . capture of these neutrons by ( n , p ) reactions on long - lived nuclei along the path of the @xmath6-process accelerate the progress to nuclei as heavy as pd . final conditions for these wind trajectories are provided in table [ 5512570_final ] . we defer a discussion of the nuclear flows and the potential for production of heavier elements to section [ broaderstudy ] . for comparison , we also show , in figure [ allfignoneutrinos ] , the integrated results ( summed over the same six wind trajectories ) of calculations that do _ not _ include neutrino capture on protons . apart from the neglect of neutrinos the trajectories studied here are identical to those described in the middle panel of figure [ allfig ] . the differences are dramatic . when neutrino captures are ignored nucleosynthesis stops some 40 mass units lower than when they are included . the calculations in the previous section are quite uncertain owing to both an uncertainty in the supernova explosion model and to our procedure for extrapolating expansion after the explosion calculation has stopped . modest alterations are reasonable to the asymptotic wind velocity , the electron fraction of the wind , and neutrino capture rates in the outflow . more extreme changes that might reflect the influence of novel physical processes operating in the early wind are explored in section [ broaderstudy ] . * influence of a larger asymptotic wind velocity + the supernova studied by janka et al . has a relatively low kinetic energy at infinity , 0.6@xmath53 erg . more energetic explosions would give rise to shock velocities larger than the @xmath54 adopted here . to estimate nucleosynthesis in more energetic supernovae , we show in the first column of table [ modesttable ] production factors for nuclei synthesized in a wind characterized by an asymptotic velocity of @xmath55 . this is more typical of supernovae with late time kinetic energies near @xmath56 erg . apart from the increase in asymptotic velocity all other properties of the 6 trajectories comprising the early wind were kept unchanged from those found by simulation . + by comparing with the middle panel of figure [ allfig ] , one sees that changing the asymptotic outflow velocity has little effect on nucleosynthesis . this may be partly an artifact of our definition of `` asymptotic '' as applying only to times after the last point given by the supernova simulation . to test this we also ran simulations where the outflow velocity was set to @xmath55 once the temperature of the wind fell below 2.5 billion degrees . again the differences were small . * effect of 5@xmath57 changes to @xmath25 + uncertainty in the neutrino spectral evolution or the dynamics of the wind near the neutron star could affect @xmath25 in the wind . the second and third columns of table [ modesttable ] show the influence of changing the electron fraction up or down by @xmath58 . this change was applied to all 6 trajectories comprising the early wind . other characteristics of these trajectories were left unchanged . + it is evident that relatively small changes to the electron fraction can have a big impact on nucleosynthesis . if @xmath25 is decreased by 5@xmath57 production of @xmath0-isotopes near mass 100 is lost and replaced by modest synthesis of some proton - rich kr and sr isotopes . a 5@xmath57 increase in @xmath25 leads to large production of pd and cd isotopes . the reason for the large impact of changing @xmath25 can be understood in terms of the number of protons available for capturing neutrinos . the most proton - rich trajectory found by simulations had @xmath59 . this corresponds to a mass fraction of free protons @xmath60 . increasing @xmath25 by five percent corresponds to a 40 percent increase in @xmath61 . * changes to the neutrino capture rates + the last two columns of table [ modesttable ] show the influence of halving and doubling the luminosity of electron neutrinos and anti - neutrinos . such large changes to the luminosities are probably unlikely , but might reflect plausible uncertainties in the local neutrino capture rate experienced by the wind . for example , the wind material might first catch up with the outgoing shock at a larger radius than found by simulations . our neglect of the temporal evolution and finer spectral details of the neutrinos might also result in modest changes to the capture rates . + from table [ modesttable ] it is seen that halving or doubling the charged current neutrino captures rates is roughly equivalent to decreasing or increasing @xmath25 by 5@xmath57 . galactic chemical evolution studies indicate that production factors in the whole star for isotopes exclusively produced in core - collapse supernovae must be of order 10 @xcite . as noted before @xcite , this implies that the current simulations predict a hot bubble ejecta that could explain the origin of @xmath40 and @xmath41 . implications for the early wind are more interesting . without any tuning or rescaling of wind conditions , the simulations of @xcite predict a wind that efficiently synthesizes several interesting @xmath0-nuclei - including the elusive isotopes @xmath62 and @xmath63 - in near solar relative proportions . overall these predict about 5 - 10 times too much yield of the most proton - rich stable ru and pd isotopes . in the previous section it was shown that small and plausible changes to the electron fraction can alleviate this overproduction . in this section we consider nucleosynthesis in outflows for which neutrino , proton , and neutron - induced reactions on heavy seed can produce still heavier nuclei . this may occur because the proton to seed ratio is higher - as happens if the entropy is higher - or the production of neutrons by p(@xmath64n is greater . reasons why the entropy might be higher are discussed in the conclusions . qualitatively , the nucleosynthesis we are describing occurs in three , or possibly four stages . first , in the outgoing wind , all neutrons combine with protons leaving an excess of unbound protons - much like in the big bang . as this combination of alphas and protons cools below @xmath65 k , a small fraction of the alphas recombine to produce nuclei in and slightly above the iron group - @xmath66ni , @xmath67zn , and @xmath20ge . flow beyond @xmath20ge is inhibited however by strong reverse flows , especially ( p,@xmath4 ) reactions . the second stage occurs as the temperature declines below @xmath68 k. a combination of ( p,@xmath69 and ( n , p ) reactions carries the flow , still close to the z = n line , to heavier nuclei . for a@xmath7092 the flow in the present calculations passes through the even - even n = z nuclei . after @xmath71 ( @xmath26ru ) the character of the flow changes . effective synthesis of the next even - even nucleus ( @xmath18pd ) is prevented in part by the small proton separation energy of @xmath72rh - the proton capture parent of @xmath18pd . as in the analogous @xmath10-process , just how far the flow goes in a particular trajectory depends on the proton - to - seed ratio and especially on the number of neutrons per seed produced by p(@xmath64n . all interesting nuclei in this stage are made as proton - rich progenitors . the third stage occurs as the temperature drops below @xmath73 k and charged - particle reactions freeze out . neutrons are still being produced by p(@xmath64n , however , albeit at a reduced rate ( both because the neutrino luminosity declines and the distance to the neutron star is getting greater ) . ( n , p ) reactions now drive material towards the valley of beta stability . because the nuclei involved are unstable to positron decay anyway , this only accelerates the inevitable . the atomic mass , a , does not change . it should be noted , however , that just as the @xmath10-process can contribute to nuclei made by the @xmath11-process that are unshielded against @xmath74 decay of more neutron - rich isobars , so too can the @xmath75 process considered here contribute to @xmath11-nuclei that are unshielded on the proton - rich side . that is , in addition to nuclei that are designated as `` @xmath76 '' , there may also be nuclei one should consider as `` @xmath77 '' . the fourth stage only occurs in the most extreme situation where the number of neutrons produced by neutrinos is quite large compared with the number of seed nuclei . then ( n , p ) reactions not only carry the flow at low temperature back to the valley of beta stability , but ( n,@xmath7 ) reactions carry it beyond - _ to the neutron rich side of the periodic chart _ , even in the presence of a large abundance of free protons . this is a novel version of the @xmath10-process that actually works best when the _ proton _ abundance is large but the temperature too low for proton addition . the protons are just a source of neutrons . the most interesting part of the nucleosynthesis occurs during the later stages of the outflow as the material cools . in the absence of an important neutrino flux the final isotopic yields are determined by an interplay between @xmath79 , @xmath80 and @xmath81 processes as well as details of how these reactions fall out of equilibrium . when neutrino capture on free protons is important , nuclei are pushed to higher isospin and mass via @xmath82 and ( n,@xmath7 ) reactions . as a first approximation neutrinos will be important if they create an appreciable number of neutrons per heavy nucleus . the ratio of created free neutrons to heavy nuclei is @xmath83 where @xmath84 is the mass fraction of elements heavier than @xmath4 particles and @xmath85 is an effective average atomic number . in eq . [ yneq ] @xmath86 is the net number of neutrinos captured per free proton at temperatures smaller than about @xmath87k . here @xmath88 is the rate at which each free proton captures @xmath89 s @xcite . in eq . [ lambdanu ] @xmath90 is the luminosity of electron anti - neutrinos , @xmath91 is an effective temperature for these neutrinos and @xmath10 is the radius of the material from the neutrino sphere . to estimate the relation between @xmath78 and nucleosynthesis consider a mass element of unit volume co - moving with the wind . the number of free protons in this mass element is @xmath92 these free protons are destroyed by anti - neutrino capture at a rate @xmath93 neutrons created when anti - neutrinos capture onto protons are subsequently absorbed by nuclei . the evolution of the free neutron abundance is then set by a competition between neutrino and nuclear processes @xmath94 here the sum is over all isotopes @xmath46 , @xmath95 and @xmath96 represent the atomic number and rate at which species @xmath46 absorbs free neutrons . as a matter of convention @xmath96 here represents the rate at which a single atom of species @xmath46 absorbs neutrons when the free neutron density is one mol per unit volume . if we introduce an average neutron absorption rate @xmath97 the destruction rate appearing on the right hand side of eq . [ xndot ] can be written @xmath98 neutrons are principally absorbed in ( n , p ) and @xmath99 reactions , with @xmath100 reactions playing a smaller role ( due to the larger coulomb barrier in the exit channel ) . table [ reprates ] shows rates for a sample of nuclei found in proton - rich outflows . at 2 billion degrees typical values of @xmath96 for even - even proton - drip line nuclei ( those bordering the line separating the proton - bound nuclei from the proton - unbound nuclei ) with mass near a@xmath10172 are around @xmath102 . this implies a very rapid neutron destruction rate . for example , at a typical density of @xmath103 g @xmath14 and a mass fraction of heavy nuclei equal to @xmath104 , a neutron is absorbed in less than a microsecond . since this is much shorter than the material expansion rate it is fair to treat the neutron abundance as being in equilibrium @xmath105 an estimate for the abundance of free neutrons also gives an estimate for the destruction rate of an atom of species @xmath46 : @xmath106 here we have defined an effective destruction rate that reflects the competition between different nuclear species for scarce neutrons @xmath107 which is enormous . this equation says for plausible outflow conditions a given species can be entirely destroyed by neutrino - produced neutrons in times as short as @xmath108 ms . it also implies a very small equilibrium neutron abundance : @xmath109 for @xmath110 g @xmath14 . one of the most interesting questions relates to how high in mass the nucleosynthesis will proceed . for the purpose of making first estimates we can suppose the starting inventory of nuclei to be concentrated near mass 60 . we will also suppose that @xmath96 is independent of species for nuclei with @xmath111 . in this case the number of neutrons captured by a heavy nucleus is just @xmath78 . a more careful treatment of @xmath96 is not presented since results from detailed network - based calculations are given in the next section . with the above assumptions the mass fraction of species @xmath49 is @xmath112 here we have defined a species to include all nuclei of a given isotone . depending on just how fast the effective absorption rates @xmath113 are one might or might not suppose that decay of nuclei with odd - n is dominated by weak processes . this is because odd - n drip - line nuclei can have rather fast @xmath81 rates ( see table [ reprates ] ) . when the neutron absorption rates are slower than these weak rates one would take @xmath114 in the above equation . though eq . [ howhigh ] is crude it can be used as a rough guide to gauge the influence of free neutrons created from neutrino captures . as an example , suppose one neutron is created per heavy nucleus ( @xmath115 ) . in this case eq . [ howhigh ] suggests that the relative mass fraction of nuclei that have captured a single neutron is about @xmath116 , while the relative mass fraction of nuclei that have captured 4 neutrons is about 20 times smaller . if we use a factor of ten decrease in mass fraction as a rough cutoff , this implies that an appreciable abundance of nuclei with mass up to a@xmath117 will be synthesized . here we have supposed that a unit increase in neutron number is accompanied by a unit increase in proton number . neutron capture on odd - n proton - drip line nuclei has been neglected since the @xmath81 rates of these nuclei are much faster than the assumed neutron capture rate of @xmath118sec . as another example , suppose that @xmath119 . in this case the relative mass fraction of nuclei that have captured 5 neutrons is about 20/@xmath120 , while the relative mass fraction of nuclei that have captured 11 neutrons is about 20 times smaller . this suggests appreciable synthesis of nuclei up to mass @xmath121 . here we have again assumed that the change in atomic number is twice the change in neutron number and that weak processes alone destroy odd - n nuclei . these assumptions probably lead to an overestimate for the increase in @xmath122 in this case because a neutron capture rate of @xmath123sec is comparable to the weak rates of many proton drip - line nuclei . the above considerations suggest a first constraint on conditions synthesizing @xmath0-nuclei with mass near 130 @xmath124 a second constraint comes from considering the evolution of the outflow at low temperatures . as @xmath125 falls below about 1.5@xmath126k charged - particle capture rates begin to freeze out . neutrons , on the other hand , are still rapidly absorbed . these neutron captures push the flow toward stability and away from progenitors of @xmath0-nuclei . if low - temperature neutrino - induced neutron production is significant , even the very neutron - rich @xmath10-process nuclei are synthesized . minimal destruction of @xmath0-nuclei implies a second constraint @xmath127 conversely , efficient synthesis of @xmath10-process isotopes in these proton - rich outflows requires the production of several neutrons per heavy nucleus at low temperatures . outflows characterized by the production of many free neutrons per heavy nucleus ( large @xmath78 ) can be realized in different ways . for example , the timescale characterizing the expansion of the outflow around the time that @xmath4-particles are synthesized might be small , the flow might be held close to the neutron star for an extended period , or @xmath25 might be very large . for simplicity we consider implications of changing the entropy of the outflow . apart from the influence of neutrino captures occurring at low temperature , the precise mechanism by which @xmath78 is increased is not so important for nucleosynthesis . changes to the entropy of the hot bubble are not considered . neutrino capture is not very pronounced in this portion of the outflow . as well , the hot bubble material does not begin close to the neutron star , so it is hard to see how an appreciable increase in the entropy of this material could be achieved . figure [ 2sfig ] shows the influence of doubling the entropy in the early wind . like the middle panel of figure [ allfig ] , this figure gives integrated production factors for a wind comprised of six trajectories . each of these trajectories has twice the entropy , but is otherwise identical to , a counterpart trajectory from the simulation . for definiteness the increase in entropy was assumed to influence only the density evolution . the evolution of temperature with time was assumed to be the same as that found from simulation . to a fair approximation doubling or tripling of the entropy corresponds to dividing the density by a factor of two or three . doubling the entropy in the early wind results in values of @xmath78 ranging from about 0.6 to 10 . the increased number of neutron captures results in efficient synthesis of nuclei as heavy as mass 125 . by contrast , for all of the unmodified wind trajectories @xmath78 is less than about 3 and efficient synthesis stops around ru . figure [ 3sfig ] shows the influence of tripling the entropy in the early wind . again - this shows integrated production factors for six trajectories that each have larger entropy , but that are otherwise identical to , unaltered trajectories described by the middle panel of figure [ allfig ] . these modified high entropy wind trajectories have values of @xmath78 ranging from about 1 to 22 . this results in efficient synthesis of nuclei as heavy as mass 170 . in other words , increasing the number of neutrons captured per heavy nucleus by 10 pushes the flow some 40 units higher in mass . though outflows predicted by simulations have values of @xmath78 that are too small to allow synthesis of @xmath128 @xmath0-nuclei , they naturally satisfy the constraint ( eq . [ con2 ] ) on the relative number of neutrino captures occurring while the wind material is so cold that charged particle captures have frozen out . figure [ ev1 ] shows the evolution of @xmath129 as a function of temperature for the wind outflow characterized by @xmath130 and @xmath131 . it is seen that the fraction of neutrino captures occurring at low temperatures smaller than 1.5 billion degrees is quite small , less than about @xmath58 . to illustrate the influence of neutrino captures occurring while the outflow is cold enough that charged particle reactions have frozen out we modified the entropy doubled version of trajectory 6 ( table [ 5512570_final ] ) to be held close to the neutron star at temperatures less than 1.5 billion degrees . a relatively modest modification in the outflow results in the capture of about 5 neutrons per heavy nucleus at temperatures lower than @xmath132k . the first four columns of table [ slowcompare ] shows a comparison between nucleosynthesis in this slow outflow and nucleosynthesis in an outflow with the nominal radial velocity of @xmath133 . since the only difference between these two trajectories is the evolution of radius with time at low temperatures , all differences in nucleosynthesis arise from late - time neutron production . it is seen that a couple of neutrons produced at the wrong time can be detrimental to the synthesis of some @xmath0-process isotopes . we also show in table [ slowcompare ] the influence of a great number of neutrons produced at low temperatures . again , this was studied by modifying just the radial profile at temperatures less than 1.5 billion degrees of the entropy doubled version of trajectory 6 . the last two columns of the table show nucleosynthesis in this trajectory in which about 20 free neutrons are created per heavy nucleus at low temperatures . most of the isotopes shown are @xmath10-process isotopes . it is perhaps remarkable that the 2nd and possibly 3rd @xmath10-process peak elements can be synthesized in these proton - rich environments . it may be difficult , however , to have ejecta that are both cold enough and close enough to the neutron star to experience the necessary neutrino irradiation . in all trajectories studies , regardless of initial electron fraction or entropy , nucleosynthesis begins with @xmath134c produced early on by the reaction sequence @xmath4(@xmath4n,@xmath7)@xmath135be(@xmath4,n)@xmath134c . by the time t@xmath136 the iron group has already been assembled . strong @xmath137 and pairs of ( p,@xmath7 ) and ( @xmath4,p ) reactions continue to populate the even - z even - n @xmath4-nuclei up to @xmath66ni and @xmath67zn . the flow mostly travels along the z = n line and does not stray more than two neutrons from it for any element up to zinc . this continues until the charged - particle reactions freeze out ( t@xmath138 ) . characteristics of the nucleosynthesis at lower temperatures depends sensitively on the influence of neutrino captures . to illustrate the influence of p(@xmath139n reactions we begin with a discussion of nucleosynthesis in trajectory 6 , which is characterized by the weak production of a few neutrons per heavy nucleus . important nuclear flows occurring when material in this trajectory has a temperature @xmath140k are shown in figure [ j570_s1_psp_zn2sn ] . it is seen that the dominant flows ( red arrows ) are due to proton - capture ( p,@xmath7 ) reactions . these can proceed until a proton unbound ( denoted by a white square ) or small ( blue ) @xmath141 energy is encountered . unlike the @xmath6-process , here we have a neutron abundance and though small it allows ( n , p ) or @xmath81 reactions to populate the next lowest isobar . the ( p,@xmath7 ) flow is governed by the separation energies . the end result for this trajectory is the production of the light @xmath0-process nuclei from kr to pd . the ( n , p ) reactions can continue to carry the flow even at low temperatures because such reactions on targets a few neutrons to the proton side of stability typically have positive q - values ( i.e. no thresholds ) . the flow to heavier nuclei eventually stops when the charged particle reactions freeze out ( t@xmath142 ) and at late times ( once the waiting points are passed ) when ( n , p ) and ( n,@xmath7 ) reactions or weak decay direct the flows toward stability . an interesting although unfortunate occurrence is the low ( relative to ru and pd ) production of the most abundant @xmath0-nucleus in nature , @xmath18mo . as the proton capture flow moves up the n=46 isotones ( see figure [ j570_s1_psp_zn2sn ] ) it is stopped in part because @xmath72rh has a small proton separation energy . this prevents efficient population of @xmath143 ( z = n=46 ) and so breaks the pattern of synthesizing the even - even n = z nuclei . as a consequence the flow detours towards stability until reaching n=47 and n=48 . the result is that the radioactive progenitor for @xmath18mo is now the odd - odd nucleus @xmath18rh . the flow moves very quickly through this nucleus ( as well as through @xmath18ru ) , and little is left for decay at the end of this trajectory . it is notable that the heavier @xmath0-nuclei , @xmath2ru and @xmath8pd , are co - produced in amounts that might explain their solar abundances . their radioactive progenitors are associated with nuclei in the two nearby closed shells . heavier @xmath144 nuclei ( @xmath8cd etc . ) are not made here because the flow failed to populate isotopes in the z=50 proton shell . @xmath18mo is the only one of these intermediate @xmath0-nuclei that ( for now ) appears to have an odd - z progenitor . we note however that the flow goes through regions where the possible error on @xmath141 is potentially large ( indicated by the t , meaning the value of @xmath141 was from an extrapolation from measured values ) . more accurate measurements here would be most welcome , and would have significant impact on our understanding of @xmath0-process nuclei and their solar abundances since this material is ejected ( unlike the case in x - ray bursts ) . as an example , the uncertainty in the proton separation energy of @xmath145 is about 600 kev . a plausible 1 mev increase in this separation energy results in a 50@xmath57 increase in the yield of @xmath18mo in trajectory 6 . figure [ j570_s2_tc2xe ] shows nucleosynthesis in a trajectory characterized by the production of many free neutrons per heavy nucleus . this trajectory has an initial electron fraction @xmath59 and entropy @xmath146 . it was constructed by doubling the entropy in the 2-d simulation of trajectory 6 ( table [ 5512570_final ] ) . for this modified outflow @xmath147 . when material in this modified trajectory reaches a temperature of about t@xmath148 , ( p,@xmath7 ) reactions on @xmath149sn ( with @xmath150 mev ) pierce the z=50 closed proton shell . at the time shown in figure [ j570_s2_tc2xe ] ( 2.21 sec , t@xmath151 ) , the charged particle reactions have frozen out , but the flow has entered an area where weak decay has yet to dominate . instead , ( n , p ) and ( n,@xmath7 ) reactions carry the flow rapidly toward stability . the @xmath0-nuclei of ru , pd , and cd are all made as radioactive progenitors in the closed neutron ( ru ) and proton ( pd & cd ) shells . we are in a very novel regime , where one can synthesize @xmath0-nuclei ( like @xmath152sn and @xmath9te ) via neutron capture reactions . current supernova simulations , without modification , provide the necessary conditions required to explain the origin of a number of @xmath0-process isotopes between a = 92 and 126 whose origin in nature has always been unclear . the site is the proton - rich bubble that powers the explosion and the early neutrino - powered wind that develops right behind it . the synthesis is primary , so a neutron star derived from a metal poor progenitor star would produce the same yields ( so long as the neutron star itself had the same properties ) . very metal deficient stars formed from these ejecta would be characterized by a excess of both @xmath0-process nuclei and @xmath10-process nuclei compared to the @xmath11-process , but since there is no element that is dominantly @xmath0-process , observational diagnostics may be difficult . in particular , large quantities of @xmath62 and @xmath153pd are produced in our calculations ( fig . [ allfig ] ) . synthesis of @xmath0-process isotopes as heavy as @xmath154 can also be achieved by only factor - of - two modifications to the entropy of the baseline simulation . it is interesting in this regard to note that an even larger increase in entropy is needed later in the _ neutron_-rich wind for the efficient synthesis of the @xmath10-process isotopes ( e.g. , * ? ? ? this is quite possibly informing us of some additional heating mechanism that operates in the mass outflow during the first few seconds of a neutron star s life . possible mechanisms are magnetic field entrainment of the outflowing matter @xcite , magnetic energy dissipation @xcite , acoustic energy input @xcite , or alfvn wave dampening @xcite . none of these were included in the present supernova model , but we varied the entropy to determine qualitatively their effect . in the more extreme , but still physically reasonable case that the entropy is multiplied by three , the synthesis extends all the way to @xmath155yb , with the accompanying production of many isotopes normally attributed to the @xmath11-process and even the @xmath10-process . somewhat disappointingly , none of our calculations produce a large overabundance of @xmath18mo compared to surrounding isotopes ( though some do make 10% of the necessary value ) . this may reflect either the fact that @xmath18mo has another origin , e.g. , the same neutrino - powered wind a few seconds later when @xmath25 = 0.485 , or uncertainties in the nuclear physics . in the current study , the @xmath18mo that is made is produced as the odd - odd progenitor @xmath18rh . this does not take advantage of the extra stability that would be afforded by an even - even nucleus like @xmath18pd , let alone the magic neutron shell of @xmath18mo itself . indeed the binding energies and lifetimes of nuclei in the vicinity of @xmath18pd are quite uncertain . an important aspect of the synthesis calculated here is that none of the @xmath0-nuclei are made as themselves ; all have proton - rich progenitors . many of these progenitors are so unstable that even their masses and lifetimes are not measured , let alone their cross sections for interacting with neutrons and protons . a similar situation is encountered in the @xmath6-process in type i x - ray bursts ( e.g. * ? ? ? * ) , a critical difference being that the isotopes made here are actually ejected and contribute to the solar inventory of heavy elements . the study of these is a major goal for nuclear astrophysics experiments of the future , like the rare isotope accelerator ( http://www.anl.gov/ria/index.html ) . for now , we can only note that these nuclear uncertainties are almost certainly responsible for a large fraction of the spread in production factors in , e.g. figs . [ allfig ] and [ 2sfig ] . this study has explored only a relatively limited set of outflow parameter space based upon simple modifications to trajectories found in one particular simulation . further studies will surely be carried out by us and others , but we have identified a key physical parameter , @xmath78 ( eq . 2 ) , which characterizes the solution for various combinations of time scale , @xmath25 , and entropy . @xmath78 is essentially a dimensionless measure of the number of neutrons produced by neutrino capture on protons compared to the number of heavy seed nuclei . surveys on a finer grid of @xmath78 than were used here will be interesting . this work was performed under the auspices of the u.s . department of energy by the university of california lawrence livermore national laboratory under contract w-7405-eng-48 . it was also supported , in part , by the scidac program of the us department of energy ( dc - fc02 - 01er41176 ) , the national science foundation ( ast 02 - 06111 ) , and nasa ( nag5 - 12036 ) and , in germany , by the research center for astroparticle physics ( sfb 375 ) and the transregional collaborative research center for gravitational wave astronomy ( sfb - transregio 7 ) . and density @xmath156 . the net nuclear flow ( in units of sec@xmath157 ) is defined as the product of abundance , density , and reaction rate in the forward ( charge or mass increasing ) direction minus a similar quantity for the inverse reaction . strong and electromagnetic flows begin at the center of a target nucleus and end as an arrow in the product nucleus . any flow that starts off - center represents weak decay . net nuclear flows are plotted in three strengths : red ( strong ) , green ( intermediate ) and blue ( weak ) , with values that are between a factor of 1.0 to 0.1 , 0.1 to 0.02 , and 0.02 to 0.01 of the value of the largest flow in the figure , respectively . the largest flow here is @xmath158zn(p,@xmath7)@xmath159ga ( @xmath160 sec@xmath157 ) . stable species are represented by a filled black square in the upper left corner . each nucleus is color coded according to the legend by the value of its proton separation energy . proton unbound nuclei are colored white . nuclei with @xmath161 mev are colored gray . a `` t '' is plotted in the upper right - hand corner for nuclei whose binding energy was extrapolated from measured masses @xcite . production factors at the time shown are given in the inset ( the stable isotopes depicted include the abundances of all radioactive progenitors that will eventually decay to them ) . as discussed in the text the classical @xmath162process waiting points ( @xmath20ge , @xmath163se , @xmath164kr , and @xmath165sr ) are bypassed by ( n , p ) reactions . [ j570_s1_psp_zn2sn ] ] . at the time shown here t@xmath151 , @xmath166 , and @xmath167 . the reactant mass fractions are x(p)=0.122 , x(n)=@xmath168 , and x(@xmath4)=0.865 . the largest abundance is x(@xmath169sn)@xmath170 , the largest flow depicted is @xmath171sn(n,@xmath7)@xmath149sn ( @xmath172 sec@xmath157 ) . the charged particle reactions have frozen out , leaving ( n , p ) and ( n,@xmath7 ) reactions to carry the flow rapidly towards stability ( before the onset of weak decay ) . this allows @xmath144nuclei like @xmath152sn and @xmath9te to be made as themselves via neutron capture reactions . [ j570_s2_tc2xe ] ] ccccccccc 1 & 0.539 & 54.8 & 0.078 & 0.614 & 0.307 & 0.244 & 80 & 0.2 + 2 & 0.548 & 58.0 & 0.095 & 0.714 & 0.190 & 0.135 & 71 & 0.4 + 3 & 0.551 & 76.7 & 0.101 & 0.822 & 0.075 & 0.043 & 57 & 1.7 + 4 & 0.551 & 71.0 & 0.102 & 0.796 & 0.101 & 0.063 & 62 & 1.1 + 5 & 0.556 & 74.9 & 0.113 & 0.831 & 0.054 & 0.025 & 46 & 2.9 + 6 & 0.558 & 76.9 & 0.115 & 0.840 & 0.043 & 0.014 & 33 & 3.2 + cccccccccc @xmath174ru & 2.09 & @xmath23sr & 1.22 & @xmath8pd & 3.14 & @xmath23sr & 1.25 & @xmath8pd & 3.07 + @xmath8pd & 2.06 & @xmath175kr & 0.93 & @xmath153cd & 2.72 & @xmath175kr & 0.91 & @xmath176ru & 2.90 + @xmath176ru & 1.86 & @xmath22kr & 0.92 & @xmath176ru & 2.69 & @xmath176ru & 0.82 & @xmath153cd & 2.71 + @xmath23sr & 1.72 & @xmath31ti & 0.74 & @xmath174ru & 2.65 & @xmath174ru & 0.71 & @xmath174ru & 2.66 + @xmath177mo & 1.36 & @xmath165se & 0.69 & @xmath169cd & 2.10 & @xmath22kr & 0.65 & @xmath169cd & 2.14 + @xmath175kr & 1.20 & @xmath176ru & 0.56 & @xmath178pd & 1.98 & @xmath31ti & 0.59 & @xmath23sr & 1.94 + @xmath179mo & 0.99 & @xmath21se & 0.39 & @xmath23sr & 1.79 & @xmath165se & 0.56 & @xmath178pd & 1.92 + @xmath180nb & 0.89 & @xmath20zn & 0.34 & @xmath181ru & 1.76 & @xmath177mo & 0.55 & @xmath177mo & 1.86 + @xmath153cd & 0.87 & @xmath177mo & 0.30 & @xmath177mo & 1.70 & @xmath8pd & 0.39 & @xmath181ru & 1.82 + @xmath182ru & 0.83 & @xmath164ge & 0.23 & @xmath182ru & 1.61 & @xmath20zn & 0.26 & @xmath183ru & 1.77 + cccc @xmath184 & 6.4@xmath185 & 4.5@xmath186 & 0.01 + @xmath187 & 7.7@xmath185 & 9.6@xmath186 & 7.24 + @xmath188 & 7.6@xmath185 & 1.1@xmath189 & 0.02 + @xmath190 & 1.0@xmath191 & 2.0@xmath189 & 8.67 + @xmath192 & 1.6@xmath191 & 4.5@xmath189 & 0.18 + @xmath193 & 3.3@xmath194 & 1.7@xmath195 & 0 . + ccccccc @xmath169cd & 7.33 & @xmath196sn & 7.20 & @xmath197sn & 6.45 + @xmath153cd & 6.99 & @xmath198 in & 7.08 & @xmath199sn & 6.22 + @xmath9te & 6.87 & @xmath152sn & 7.00 & @xmath200cd & 6.13 + @xmath198 in & 6.82 & @xmath169cd & 6.87 & @xmath149pd & 6.01 + @xmath152sn & 6.75 & @xmath201sn & 6.85 & @xmath202sb & 6.01 + @xmath8pd & 6.50 & @xmath9te & 6.55 & @xmath203pd & 5.89 + @xmath201sn & 6.48 & @xmath8pd & 6.43 & @xmath204rh & 5.87 + @xmath199xe & 6.41 & @xmath153cd & 6.34 & @xmath178ru & 5.80 + @xmath196sn & 6.37 & @xmath174ru & 6.25 & @xmath205cd & 5.76 + @xmath149cd & 6.31 & @xmath206xe & 6.00 & @xmath207sb & 5.71
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one of the outstanding unsolved riddles of nuclear astrophysics is the origin of the so called `` p - process '' nuclei from a = 92 to 126 .
both the lighter and heavier @xmath0-process nuclei are adequately produced in the neon and oxygen shells of ordinary type ii supernovae , but the origin of these intermediate isotopes , especially @xmath1mo and @xmath2ru , has long been mysterious . here
we explore the production of these nuclei in the neutrino - driven wind from a young neutron star .
we consider such early times that the wind still contains a proton excess because the rates for @xmath3 and positron captures on neutrons are faster than those for the inverse captures on protons .
following a suggestion by @xcite , we also include the possibility that , in addition to the protons , @xmath4-particles , and heavy seed , a small flux of neutrons is maintained by the reaction p(@xmath5n .
this flux of neutrons is critical in bridging the long waiting points along the path of the @xmath6-process by ( n , p ) and ( n,@xmath7 ) reactions . using the unmodified ejecta histories from a recent two - dimensional supernova model by @xcite , we find synthesis of @xmath0-rich nuclei up to @xmath8pd . however , if the entropy of these ejecta is increased by a factor of two , the synthesis extends to @xmath9te
. still larger increases in entropy , that might reflect the role of magnetic fields or vibrational energy input neglected in the hydrodynamical model , result in the production of numerous @xmath10- , @xmath11- , and @xmath0-process nuclei up to a @xmath12 170 , even in winds that are proton - rich .
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the study of processor shared queues has received much attention over the past 45 or so years . the processor sharing ( ps ) discipline has the advantage over , say , first - in first - out ( fifo ) , in that shorter jobs tend to get through the system more rapidly . ps models were introduced during the 1960 s by kleinrock ( see @xcite , @xcite ) . in recent years there has been renewed attention paid to such models , due to their applicability to the flow - level performance of bandwidth - sharing protocols in packet - switched communication networks ( see @xcite-@xcite ) . perhaps the simplest example of such a model is the @xmath1-ps queue . here customers arrive according to a poisson process with rate parameter @xmath2 , the server works at rate @xmath3 , there is no queue , and if there are @xmath4 customers in the system each gets an equal fraction @xmath5 of the server . ps and fifo models differ significantly if we consider the `` sojourn time '' . this is defined as the time it takes for a given customer , called a `` tagged customer '' , to get through the system ( after having obtained the required amount of service ) . the sojourn time is a random variable that we denote by @xmath6 . for the simplest @xmath1 model , the distribution of @xmath6 depends on the total service time @xmath7 that the customer requests and also on the number of other customers present when the tagged customer enters the system . one natural variant of the @xmath1-ps model is the finite population model , which puts an upper bound on the number of customers that can be served by the processor . the model assumes that there are a total of @xmath8 customers , and each customer will enter service in the next @xmath9 time units with probability @xmath10 . at any time there are @xmath11 customers being served and the remaining @xmath12 customers are in the general population . hence the total arrival rate is @xmath13 $ ] and we may view the model as a ps queue with a state - dependent arrival rate that decreases linearly to zero . once a customer finishes service that customer re - enters the general population . the service times are exponentially distributed with mean @xmath14 and we define the traffic intensity @xmath15 by @xmath16 . this model may describe , for example , a network of @xmath8 terminals in series with a processor - shared cpu . this may be viewed as a closed two node queueing network . the finite population model does not seem amenable to an exact solution . however , various asymptotic studies have been done in the limit @xmath17 , so that the total population , or the number of terminals , is large . if @xmath8 is large it is reasonable to assume either that @xmath18 , the arrival rate of each individual customer , is small , of the order @xmath19 , or that the service rate @xmath3 is large , of the order @xmath20 . then @xmath16 will remain @xmath21 as @xmath17 . previous studies of the finite population model were carried out by morrison and mitra ( see @xcite-@xcite ) , in each case for @xmath17 . for example , the moments of the sojourn time @xmath6 conditioned on the service time @xmath7 are obtained in @xcite , where it was found that the asymptotics are very different according as @xmath22 ( called `` normal usage '' ) , @xmath23 ( called `` heavy usage '' ) , or @xmath24 ( called `` very heavy usage '' ) . in @xcite the unconditional sojourn time distribution is investigated for @xmath17 and the three cases of @xmath15 , in @xcite the author obtains asymptotic results for the conditional sojourn time distribution , conditioned on the service time @xmath7 , in the very heavy usage case @xmath24 , and in @xcite the results of @xcite are generalized to multiple customer classes ( here the population @xmath8 is divided into several classes , with each class having different arrival and service times ) . in @xcite the authors analyze the multiple class model and obtain the unconditional sojourn time moments for @xmath17 in the normal usage case , while in @xcite heavy usage results are obtained . in this paper we study the spectral structure of the finite population model as @xmath17 . we denote the sojourn time by @xmath25 and its conditional density we call @xmath26 with @xmath27.\ ] ] here @xmath28 denotes the number of other customers present in the system immediately before the tagged customer arrives , and thus @xmath29 . then we define the column vector @xmath30 . @xmath31 satisfies a system of odes in the form @xmath32 where @xmath33 is an @xmath34 tridiagonal matrix , whose entries depend on @xmath16 and @xmath8 . the eigenvalues of @xmath33 are all negative and we denote them by @xmath35 @xmath36 with the corresponding eigenvectors being @xmath37 . we shall study this eigenvalue problem for @xmath17 and three cases of @xmath15 : @xmath22 , @xmath24 and @xmath38 . in each case we obtain expansions of the @xmath39 and then the @xmath40 , for various ranges of @xmath41 . often the eigenvectors can be expressed in terms of hermite polynomials for @xmath17 . since @xmath33 is a finite matrix the spectrum is purely discrete , but as the size of the matrix becomes large we sometimes see the eigenvalues coalescing about a certain value . ordering the eigenvalues as @xmath42 , the tail behavior of @xmath26 and @xmath43 for @xmath44 is determined by the smallest eigenvalue @xmath45 , where @xmath43 is the unconditional sojourn time density with @xmath46.\ ] ] it is interesting to note that while previous studies ( see @xcite-@xcite ) of the finite population model lead to the scaling @xmath23 , the spectrum involves the transition scale @xmath38 . our basic approach is to use singular perturbation methods to analyze the system of odes when @xmath8 becomes large . the problem can then be reduced to solving simpler , single differential equations whose solutions are known , such as hermite equations . our analysis does make some assumptions about the forms of various asymptotic series , and about the asymptotic matching of expansions on different scales . we also comment that we assume that the eigenvalue index @xmath47 is @xmath21 ; thus we are not computing the large eigenvalues here . this paper is organized as follows . in section 2 we state the mathematical problem and obtain the basic equations . in section 3 we summarize our final asymptotic results for the eigenvalues and the ( unnormalized ) eigenvectors . the derivations are relegated to section 4 . some numerical studies appear in section 5 ; these assess the accuracy of the asymptotics . throughout the paper we assume that time has been scaled so to make the mean service time @xmath48 . the conditional sojourn time density @xmath26 satisfies the following linear system of ordinary differential equations , or , equivalently , differential - difference equation : @xmath49p_n(t),\;0\le n\le n-1.\ ] ] the above holds also at @xmath50 if we require @xmath51 to be finite , and when @xmath52 ( [ s2_fs_rec ] ) becomes @xmath53 the initial condition at @xmath54 is @xmath55 and we note that ( [ s2_fs_pn ] ) follows from integrating ( [ s2_fs_rec ] ) from @xmath54 to @xmath56 . here we focus on obtaining the eigenvalues ( and corresponding eigenvectors ) for the matrix @xmath57 , which corresponds to the difference operator in the right - hand side of ( [ s2_fs_rec ] ) , specifically @xmath58\ ] ] which has size @xmath34 . it follows that the solution of ( [ s2_fs_rec])-([s2_fs_pn ] ) has the spectral representation @xmath59 here @xmath35 are the eigenvalues of @xmath33 , indexed by @xmath60 and ordered as @xmath61 , @xmath40 is the eigenvector corresponding to eigenvalue @xmath39 , and the spectral coefficients @xmath62 in ( [ s2_fs_sum ] ) can be calculated from ( [ s2_fs_pn ] ) , hence @xmath63 from ( [ s2_fs_rec ] ) we can easily establish orthogonality relations between the @xmath64 , and these lead to an explicit expression for the @xmath62 : @xmath65 we note that the tail of the sojourn time , in view of ( [ s2_fs_sum ] ) , is given by @xmath66 this asymptotic relation holds for @xmath67 and large times . our analysis will assume that @xmath0 but for sufficiently large @xmath68 ( [ s2_fs_sim ] ) must still hold . we shall study the behavior of @xmath69 for @xmath17 and for various ranges of the parameter @xmath15 . we shall show that the behavior of the eigenvalues is very different for the cases @xmath22 , @xmath24 , and @xmath70 ( more precisely @xmath38 ) . furthermore , within each range of @xmath15 the form of the expansions of the eigenfunctions @xmath40 is different in several ranges of @xmath41 . our analysis will cover each of these ranges , but we restrict ourselves to the eigenvalue index @xmath47 being @xmath21 . note that the matrix @xmath33 has @xmath71 eigenvalues so that @xmath47 could be scaled as large as @xmath20 . we are thus calculating ( asymptotically for @xmath17 ) only the first few eigenvalues and their eigenfunctions . obtaining , say , the large eigenvalues , would likely need a very different asymptotic analysis . when @xmath72 we shall see that the eigenfunctions and their zeros are concentrated in a narrow range of @xmath73 , which represents the fraction of the customer population that is using the processor . since @xmath40 are functions of the discrete variable @xmath41 , by `` zeros '' we refer to sign changes of the eigenvectors . if the eigenvalue index @xmath47 were scaled to be also large with @xmath8 , we would expect that these sign changes would be more frequent , and would occur throughout the entire interval @xmath74 . we comment that to understand fully the asymptotic structure of @xmath75 for @xmath17 requires a much more complete analysis than simply knowing the eigenvalues / eigenfunctions , as the spectral expansion may not be useful in certain space / time ranges . however , ( [ s2_fs_sim ] ) will always apply for sufficiently large times , no matter how large @xmath8 is , as long as @xmath8 is finite . we also comment that in ( [ s1_pt ] ) , by results in @xcite , we have @xmath76=\frac{(\frac{\rho}{n})^n(n-1)!}{(n-1-n)!}\bigg/\sum_{l=0}^{n-1}\frac{(\frac{\rho}{n})^l(n-1)!}{(n-1-l)!},\ ] ] which says that the distribution of @xmath28 coincides with the steady state distribution of @xmath11 in a finite population queue with population @xmath71 . the tail of the unconditional sojourn time is then @xmath77e^{-\nu_0t},\;t\to\infty,\ ] ] where @xmath78^{-1}$ ] . we give results for the three cases : @xmath22 , @xmath24 and @xmath38 . within each case of @xmath15 the eigenvectors @xmath40 have different behaviors in different ranges of @xmath41 . for @xmath22 the eigenvalues are given by @xmath79 where @xmath80 and @xmath81 we observe that the leading term in ( [ s31_nu ] ) ( @xmath82 ) is independent of the eigenvalue index @xmath47 and corresponds to the relaxation rate in the standard @xmath1 queue . for the standard @xmath1-ps model ( with an customer population ) , it is well known ( see @xcite , @xcite ) that the tail of the sojourn time density is @xmath83t^{-5/6},\ ; t\to\infty\ ] ] where @xmath84 is a constant . this problem corresponds to solving an infinite system of odes , which may be obtained by letting @xmath17 in the matrix @xmath33 ( with a fixed @xmath22 ) . the spectrum of the resulting infinite matrix is purely continuous , which leads to the sub - exponential and algebraic factors in ( [ s31_ptsim ] ) . the result in ( [ s31_nu ] ) shows that the ( necessarily discrete ) spectrum of @xmath85 has , for @xmath22 and @xmath17 , all of the eigenvalues approaching @xmath86 , with the deviation from this limit appearing only in the third ( @xmath87 ) and fourth ( @xmath19 ) terms in the expansion in ( [ s31_nu ] ) . note that @xmath88 in ( [ s31_nu ] ) is independent of @xmath47 . comparing ( [ s31_ptsim ] ) to ( [ s2_fs_ptsim ] ) with ( [ s31_nu ] ) and @xmath89 , we see that the factors @xmath90t^{-5/6}$ ] in ( [ s31_ptsim ] ) are replaced by @xmath91 $ ] . note that ( [ s31_ptsim ] ) corresponds to letting @xmath17 and then @xmath44 in the finite population model , while ( [ s2_fs_ptsim ] ) has @xmath44 with a finite large @xmath8 . the expansion in ( [ s31_nu ] ) breaks down when @xmath47 becomes very large , and we note that when @xmath92 , the three terms @xmath93 , @xmath94 and @xmath95 become comparable in magnitude . the expansion in ( [ s31_nu ] ) suggests that @xmath47 can be allowed to be slightly large with @xmath8 , but certainly not as large as @xmath20 , which would be needed to calculate all of the eigenvalues of @xmath33 . as we stated before , here we do not attempt to get the eigenvalues of large index @xmath47 . now consider the eigenvectors @xmath37 , for @xmath22 and @xmath17 . these have the expansion @xmath96\ ] ] where @xmath41 and @xmath97 are related by @xmath98 and @xmath99 @xmath100\frac{d}{dz}\big[e^{-z^2/4}\mathrm{he}_j(z)\big]+\big(\frac{\alpha}{2}-\frac{2}{3}\beta\big)ze^{-z^2/4}\mathrm{he}_j(z),\ ] ] with @xmath101 here @xmath102 is the @xmath103 hermite polynomial , so that @xmath104 and @xmath105 . the constant @xmath106 is a normalization constant which depends upon @xmath47 , @xmath15 and @xmath8 , but not @xmath41 . apart from the factor @xmath107 in ( [ s31_phi ] ) we see that the eigenvectors are concentrated in the range @xmath108 and this corresponds to @xmath109 . the zeros of the hermite polynomials correspond to sign changes ( with @xmath41 ) of the eigenvectors , and these are thus spaced @xmath110 apart . we next give expansions of @xmath40 on other spatial scales , such as @xmath67 , @xmath111 and @xmath112 , where ( [ s31_phi ] ) ceases to be valid . these results will involve normalization constants that we denote by @xmath113 , but each of these will be related to @xmath114 , so that our results are unique up to a multiplicative constant . note that the eigenvalues of @xmath33 are all simple , which can be shown by a standard sturm - liouville type argument . for @xmath115 ( hence @xmath116 ) we find that @xmath117 where @xmath118 @xmath119^{2j+1}}\exp\big[(2j+1)(1-\sqrt{\rho})^{1/4}\sqrt{x}\big].\ ] ] the normalization constants @xmath114 and @xmath120 are related by @xmath121,\ ] ] by asymptotic matching between ( [ s31_phi ] ) and ( [ s31_phisim ] ) . next we consider @xmath112 and scale @xmath122 . the leading term becomes @xmath123\ ] ] with @xmath124\nonumber\\ & & + \frac{\rho-4\sqrt{\rho}+2}{2\rho}\log\big[\frac{\rho\xi+2 - 2\sqrt{\rho}+\sqrt{\rho^2\xi^2 + 4\rho\xi(1-\sqrt{\rho})}}{2(1-\sqrt{\rho})}\big]\nonumber\\ & & + \frac{1}{2}\log\bigg[\frac{(\rho-2\sqrt{\rho}+2)\xi+2(1-\sqrt{\rho})-(\sqrt{\rho}-2)\sqrt{\rho\xi^2 + 4\xi(1-\sqrt{\rho})}}{2(1-\sqrt{\rho})}\bigg],\nonumber\end{aligned}\ ] ] @xmath125,\ ] ] @xmath126\ ] ] and @xmath127 can be written as the integral @xmath128\ ] ] where @xmath129\bigg\}\ ] ] and @xmath130 and @xmath131 are , respectively , the derivatives of ( [ s31_f ] ) and ( [ s31_f1 ] ) . note that @xmath132 and @xmath133 are independent of the eigenvalue index @xmath47 , while @xmath134 and @xmath135 do depend on it . the constants @xmath136 and @xmath120 are related by @xmath137 , by asymptotic matching between ( [ s31_phisim ] ) and ( [ s31_phileading ] ) . finally we consider the scale @xmath67 . the expansion in ( [ s31_phisim ] ) with ( [ s31_fg ] ) develops a singularity as @xmath138 and ceases to be valid for small @xmath139 . for @xmath67 we obtain @xmath140 where the contour integral is a small loop about @xmath141 , and by asymptotic matching of ( [ s31_phisim ] ) as @xmath142 with ( [ s31_phisim3 ] ) as @xmath143 , @xmath144 we next consider @xmath17 with @xmath24 . the eigenvalues are now small , of the order @xmath19 , with @xmath145 note that now we do not see the coalescence of eigenvalues , as was the case when @xmath22 , and the eigenvalue index @xmath47 appears in the leading term in ( [ s31_nuj ] ) . the form in ( [ s31_nuj ] ) also suggests that the tail behavior in ( [ s2_fs_sim ] ) is achieved when @xmath146 . now the zeros of the eigenvectors will be concentrated in the range where @xmath147 , and introducing the new spatial variable @xmath148 , with @xmath149 we find that @xmath150 where @xmath102 is again the @xmath103 hermite polynomial , and @xmath114 is again a normalization constant , possibly different from ( [ s31_phi ] ) . on the @xmath151-scale with @xmath73 we obtain @xmath152 and @xmath114 and @xmath120 are related by @xmath153 by asymptotic matching between ( [ s31_phijx ] ) and ( [ s31_phijk1 ] ) . note that @xmath154 and for the first two eigenvectors ( @xmath155 ) , ( [ s31_phijx ] ) is a special case of ( [ s31_phijk1 ] ) . the expression in ( [ s31_phijk1 ] ) holds for @xmath156 and for @xmath157 , but not for @xmath158 or @xmath159 . for the latter we must use ( [ s31_phijx ] ) and for @xmath67 we shall show that @xmath160 where @xmath161 is a closed loop that encircles the branch cut , where @xmath162 and @xmath163 $ ] , in the @xmath164-plane , with the integrand being analytic exterior to this cut . by expanding ( [ s31_phiint ] ) for @xmath165 and matching to ( [ s31_phijk1 ] ) as @xmath166 we obtain @xmath167 in contrast to when @xmath22 , the eigenvectors now vary smoothly throughout the interval @xmath74 ( cf . ( [ s31_phijk1 ] ) ) but their sign changes are all concentrated where @xmath159 and the spacings of these changes are of the order @xmath168 , and approximately the same as the spacings of the zeros of the hermite polynomials ( cf . ( [ s31_x ] ) and ( [ s31_phijx ] ) ) . finally we consider the case @xmath70 and introduce the parameter @xmath170 by @xmath171 this case will asymptotically match , as @xmath172 , to the @xmath24 results and , as @xmath173 , to the @xmath22 results . now a two - term asymptotic approximation to the eigenvalues is @xmath174 where @xmath175 @xmath176 @xmath177 thus given @xmath170 we must solve ( [ s31_gammaa ] ) to get @xmath178 and then compute @xmath179 and @xmath180 . we can explicitly invert ( [ s31_gammaa ] ) to obtain @xmath181^{1/3}.\ ] ] we also note that if @xmath182 we have @xmath183 and then @xmath184 , @xmath185 , and @xmath186 . the expression in ( [ s31_nu2term ] ) shows that the eigenvalues are small , of the order @xmath187 , and to leading order coalesce at @xmath188 . the second term , however , depends linearly on the eigenvalue index @xmath47 . the expansions of the eigenvectors will now be different on the four scales @xmath67 , @xmath189 , @xmath190 and @xmath112 . it is on the third scale that the zeros of @xmath40 become apparent , and if we introduce @xmath191 by @xmath192 we obtain the following leading order approximation to the eigenvectors @xmath193 thus again the hermite polynomials arise , but now on the scale @xmath194 , which corresponds to @xmath195 . the spacing of the zeros ( or sign changes ) of @xmath40 for the case @xmath38 is thus @xmath196 which is comparable to the case @xmath24 , and unlike the case @xmath22 where ( cf . ( [ s31_ny ] ) ) the spacing was @xmath110 . on the spatial scale @xmath197 , ( [ s31_phieig ] ) ceases to be valid and then we obtain @xmath198 where this applies for all @xmath199 except @xmath200 , where ( [ s31_phieig ] ) holds , and @xmath201 is given by @xmath202 @xmath203,\ ] ] where @xmath204 is as in ( [ s31_fa ] ) . note that @xmath201 is independent of the eigenvalue index @xmath47 , and if @xmath205 then @xmath206 , which follows from ( [ s31_gammaa ] ) and ( [ s31_fa ] ) . thus the discriminant in ( [ s31_fpm ] ) vanishes when @xmath205 , and in fact it has a double zero at this point . thus the sign switch in @xmath207 as @xmath208 crosses @xmath209 is needed to smoothly continue this function from @xmath210 to @xmath211 . the function @xmath212 is given by @xmath213^{j}\exp\bigg\{\int_0^v\big[\frac{1}{4w}-\frac{j}{w - a}-h^{(1)}_+(w , j)\big]dw\bigg\},\ ; 0\le v < a\ ] ] and @xmath214^{j}\exp\bigg\{\int_0^a\big[\frac{1}{4w}-\frac{j}{w - a}-h^{(1)}_+(w , j)\big]dw\bigg\}\nonumber\\ & & \times\exp\bigg\{-\int_a^v\big[\frac{j}{w - a}+h^{(1)}_-(w , j)\big]dw\bigg\},\ ; v > a.\end{aligned}\ ] ] we can show that @xmath215 as @xmath216 and @xmath217 as @xmath218 , so that the integrals in ( [ s31_gv < a ] ) and ( [ s31_gv > a ] ) are convergent at @xmath219 and @xmath220 . by expanding ( [ s31_phiv ] ) for @xmath218 we can establish the asymptotic matching of the results on the @xmath208-scale ( cf . ( [ s31_phiv ] ) ) and the @xmath191-scale ( cf . ( [ s31_phieig ] ) ) and then relate the constants @xmath114 and @xmath120 , leading to @xmath221dw\bigg\}.\ ] ] next we consider the scale @xmath222 with @xmath223 , and now the eigenvectors have the expansion @xmath224 where @xmath225 and @xmath226\ ] ] where @xmath180 and @xmath179 are as in ( [ s31_fa ] ) and ( [ s31_ga ] ) . the constants @xmath136 and @xmath120 are related by @xmath227/3}a^{-g(j,\gamma)+\gamma f(\gamma)+j-1/4}\exp\bigg\{\int_0^a\big[\frac{1}{4w}-\frac{j}{w - a}-h^{(1)}_+(w , j)\big]dw\bigg\}\nonumber\\ & & \times\exp\bigg\{n^{1/3}\big[\frac{f(\gamma)}{3}\log n - f(\gamma)\log a-\frac{1-\gamma f(\gamma)}{a}+\int_0^a\mathcal{f}_+'(w)dw\big]\bigg\}.\end{aligned}\ ] ] for @xmath67 , ( [ s31_phisim3 ] ) holds with @xmath228 , but now @xmath229^jk_1 $ ] . to summarize , we have shown that the eigenvalues have very different behaviors for @xmath22 ( cf . ( [ s31_nu ] ) ) , @xmath24 ( cf . ( [ s31_nuj ] ) ) and @xmath38 ( cf . ( [ s31_nu2term ] ) ) . in the first case , as @xmath17 , the eigenvalues all coalesce about @xmath86 which is the relaxation rate of the standard @xmath1 model . however , higher order terms in the expansion show the splitting of the eigenvalues ( cf . @xmath230 and @xmath231 in ( [ s31_nu ] ) ) , which occurs at the @xmath87 term in the expansion of the @xmath39 . for @xmath24 the eigenvalues are small , of order @xmath19 , but even the leading term depends upon @xmath47 ( cf . ( [ s31_nuj ] ) ) . when @xmath232 there is again a coalescence of the eigenvalues , now about @xmath233 where @xmath178 is given by ( [ s31_ab ] ) . the splitting now occurs at the first correction term , which is of order @xmath19 . we also note that the leading order dependence of the @xmath69 on @xmath47 occurs always in a simple linear fashion . our analysis will also indicate how to compute higher order terms in the expansions of the @xmath39 and @xmath40 , for all three cases of @xmath15 and all ranges of @xmath41 . ultimately , obtaining the leading terms for the @xmath40 reduces in all 3 cases of @xmath15 to the classic eigenvalue problem for the quantum harmonic oscillator , which can be solved in terms of hermite polynomials . we proceed to compute the eigenvalues and eigenvectors of the matrix @xmath33 above ( [ s2_fs_sum ] ) , treating respectively the cases @xmath22 , @xmath24 and @xmath38 , in subsections 4.1 - 4.3 . we always begin by considering the scaling of @xmath41 , with @xmath8 , where the oscillations or sign changes of the eigenvectors occur , and this scale also determines the asymptotic eigenvalues . then , other spatial ranges of @xmath41 will be treated , which correspond to the `` tails '' of the eigenvectors . we recall that @xmath234 so that @xmath22 means that the service rate @xmath3 exceeds the maximum total arrival rate @xmath235 . when @xmath22 and @xmath17 the distribution of @xmath28 in ( [ s2_fs_probn ] ) behaves as @xmath236\sim ( 1-\rho)\rho^n$ ] for @xmath67 , which is the same as the result for the infinite population @xmath1-ps queue . we introduce @xmath237 as in ( [ s31_ny ] ) , and set @xmath238 and @xmath239 . then setting @xmath240 in ( [ s2_fs_rec ] ) and noting that changing @xmath41 to @xmath241 corresponds to changing @xmath97 to @xmath242 , we find that @xmath243\bigg(\frac{\sqrt{n}}{\sqrt{1-\sqrt{\rho}}}+yn^{3/8}+1\bigg)\phi_j(y).\nonumber\end{aligned}\ ] ] here we also multiplied ( [ s2_fs_rec ] ) by @xmath244 , which will simplify some of the expansions that follow . letting @xmath17 in ( [ s41_expany ] ) leads to @xmath245 so we conclude that @xmath246 as @xmath17 and @xmath22 . the form of ( [ s41_expany ] ) , which has the small parameter @xmath247 , then suggests that we expand @xmath248 as @xmath249 and we also expand @xmath250 as @xmath251 then we obtain from ( [ s41_expany ] ) the limiting ode @xmath252\phi_j^{(0)}(y)=0,\ ] ] where we set , for convenience , @xmath253 . letting @xmath254 be the differential operator @xmath255f(y)$ ] , at the next two orders ( @xmath256 and @xmath257 ) we then obtain @xmath258 and @xmath259 note that the coefficients @xmath260 in ( [ s41_expandcy ] ) will depend upon @xmath15 and also the eigenvalue index @xmath47 . we also require the eigenfunctions @xmath248 to decay as @xmath261 . changing variables from @xmath97 to @xmath164 with @xmath262 we see that @xmath263 , where @xmath264f(z)$ ] . thus solving ( [ s41_limitode ] ) corresponds to @xmath265 , which is a standard eigenvalue problem . the only acceptable solutions ( which decay as @xmath266 ) correspond to @xmath267 and the ( unnormalized ) eigenfunctions are @xmath268 here @xmath102 is the hermite polynomial , which satisfies @xmath269 for @xmath266 . we proceed to compute the correction term @xmath270 in ( [ s41_expandphiy ] ) , and also @xmath271 and @xmath231 in ( [ s41_expandcy ] ) . in terms of @xmath164 , ( [ s41_order1 ] ) becomes @xmath272=-\frac{c_3}{2\sqrt{\rho}(1-\sqrt{\rho})^{3/4}}\phi_j^{(0)}-\frac{(1-\sqrt{\rho})^{1/8}}{4\sqrt{2}}z^3\phi_j^{(0)}+\frac{\sqrt{\rho}}{\sqrt{2}(1-\sqrt{\rho})^{7/8}}\frac{d}{dz}\phi_j^{(0)}\ ] ] where @xmath273=f''(z)+\big(j+1/2-{z^2}/{4}\big)f(z)$ ] , in view of ( [ s41_c2 ] ) . we determine @xmath274 by a solvability condition for ( [ s41_lphi1 ] ) . we multiply ( [ s41_lphi1 ] ) by @xmath275 and integrate from @xmath276 to @xmath277 , and use the properties of hermite polynomials ( see @xcite ) . then we conclude that @xmath278 for all @xmath47 . thus there is no @xmath279 term in the expansion in ( [ s31_nu ] ) . to solve for @xmath270 we write the right - hand side of ( [ s41_lphi1 ] ) as @xmath280 where @xmath281 and @xmath282 are as in ( [ s31_ab ] ) . then we can construct a particular solution to ( [ s41_lphi1 ] ) ( with @xmath278 ) in the form @xmath283 where @xmath284 , @xmath285 , and @xmath286 are determined from @xmath287 solving ( [ s41_abc ] ) leads to @xmath270 as in ( [ s31_phi1 ] ) . to compute @xmath288 , the @xmath19 term in ( [ s31_nu ] ) , we use the solvability condition for the equation ( [ s41_order2 ] ) for @xmath289 . we omit the detailed derivation . we note that ( [ s31_c4 ] ) is singular in the limit @xmath290 , while @xmath230 in ( [ s31_c1c2 ] ) vanishes in this limit . also , @xmath288 is quadratic in @xmath47 while @xmath291 is linear in @xmath47 , so that ( [ s31_nu ] ) becomes invalid both as @xmath290 and as the eigenvalue index @xmath47 becomes large . we next consider the @xmath40 on the spatial scales @xmath111 , @xmath112 and @xmath67 . for @xmath115 ( @xmath116 ) , the expansion in ( [ s31_phi ] ) ceases to be valid . we expand the leading term @xmath292 in ( [ s31_phi ] ) as @xmath293 to obtain @xmath294,\ ] ] which suggests that we expand the eigenvector @xmath40 in the form ( [ s31_phisim ] ) on the @xmath139-scale , noticing also that @xmath295 . we set @xmath296 in ( [ s2_fs_rec ] ) with @xmath39 given by ( [ s31_nu ] ) and @xmath40 having the form in ( [ s31_phisim ] ) . for @xmath17 we obtain the following odes for @xmath297 and @xmath298 : @xmath299 ^ 2+(\sqrt{\rho}-1)x-\frac{1}{x}+\frac{c_1}{\sqrt{\rho}}=0\ ] ] and @xmath300g_j(x)=0.\ ] ] using @xmath88 and @xmath230 in ( [ s31_c1c2 ] ) , ( [ s41_xf(x)ode ] ) and ( [ s41_xg(x)ode ] ) can be easily solved and the results are in ( [ s31_fg ] ) . we note that ( [ s41_xf(x)ode ] ) can be rewritten as @xmath301 ^ 2=\big(\sqrt{1-\sqrt{\rho}}\sqrt{x}-1/\sqrt{x}\big)^2.$ ] after taking the square root , we choose the solution with the negative sign since @xmath297 should achieve a maximum at @xmath302 . now consider the scale @xmath303 . letting @xmath304 in the exponential terms @xmath305 in ( [ s31_phisim ] ) ( with ( [ s31_fg ] ) ) and noticing that @xmath306 , we conclude that the expansion on the @xmath151-scale should have the form in ( [ s31_phileading ] ) . using @xmath296 with ( [ s31_nu ] ) and ( [ s31_phileading ] ) in ( [ s2_fs_rec ] ) and expanding for @xmath17 , we obtain the following odes for @xmath132 , @xmath133 , @xmath134 and @xmath135 : @xmath307 @xmath308f_1'+\frac{c_1}{\sqrt{\rho}}=0,\ ] ] @xmath309f_2'+\frac{c_2(j)}{\sqrt{\rho}}=0,\ ] ] and @xmath310g'+\bigg\{\frac{c_4(j)+\rho}{\sqrt{\rho}}-e^{f'}-\frac{1}{\xi}e^{-f'}+\big(1-\frac{\sqrt{\rho}}{2}\xi\big)\big[f''+(f_1')^2\big]\bigg\}g=0.\ ] ] solving ( [ s41_odef ] ) for @xmath311 leads to @xmath312 integrating the logarithm of ( [ s41_ef ] ) leads to ( [ s31_f ] ) . from ( [ s41_ef ] ) we also have @xmath313 , and then solving ( [ s41_odef1])-([s41_odeg ] ) leads to ( [ s31_f1])-([s31_g ] ) . finally we consider the scale @xmath67 . we assume the leading order approximation of @xmath40 is @xmath314.\ ] ] then using @xmath296 in ( [ s2_fs_rec ] ) with @xmath40 in ( [ s41_phin ] ) and letting @xmath17 , we obtain the following limiting difference equation for @xmath315 : @xmath316 with @xmath317 finite ( thus ( [ s41_depsi2 ] ) holds for all @xmath318 ) . from ( [ s41_depsi2 ] ) we conclude that @xmath315 is independent of the eigenvalue index @xmath47 , except via a multiplicative constant . solving ( [ s41_depsi2 ] ) with the help of generation functions leads to ( [ s31_phisim3 ] ) . when @xmath24 the distribution of @xmath28 in ( [ s2_fs_probn ] ) is approximately , for @xmath17 , a gaussian which is centered about @xmath319 and this corresponds to the fraction of the population that is typically served by the processor . we thus begin by considering the scale @xmath147 , setting @xmath320 we thus scale the eigenvalue parameter @xmath321 to be small , of order @xmath19 , which is necessary to obtain a limiting differential equation . setting @xmath322 and omitting for now the dependence of @xmath321 and @xmath323 on the index @xmath47 , ( [ s2_fs_rec ] ) becomes @xmath324+\psi\big(x-\frac{1}{\sqrt{n}}\big)\nonumber\\ & & -\psi(x)-\frac{1}{n(1-\rho^{-1})+\sqrt{n}x}\psi\big(x-\frac{1}{\sqrt{n}}\big).\end{aligned}\ ] ] then expanding @xmath325 and @xmath326 as @xmath327 and multiplying ( [ s42_psixexpan ] ) by @xmath8 we obtain the limiting ode @xmath328 this can be easily transformed to the hermite equation by setting @xmath329 . in terms of @xmath330 , ( [ s42_psiode ] ) becomes @xmath331 which is the standard hermite equation ( in contrast to the parabolic cylinder equation in ( [ s41_limitode ] ) ) . equation ( [ s42_psixtildode ] ) admits polynomial solutions provided that @xmath332 which yields the leading order eigenvalue condition , and then the solution to ( [ s42_psixtildode ] ) is @xmath333 from ( [ s42_nu*poly ] ) and ( [ s42_nuxexpan ] ) we have thus derived ( [ s31_nuj ] ) , while ( [ s31_phijx ] ) follows from ( [ s42_psi0 ] ) . higher order terms can be obtained by refining the expansion of ( [ s42_psixexpan ] ) using ( [ s42_nuxexpan ] ) , which will lead to inhomogeneous forms of the hermite equation ; the correction terms @xmath334 for @xmath335 will follow from appropriate solvability conditions . next we consider ( [ s2_fs_rec ] ) on a broader spatial scale , introducing @xmath73 , which is essentially the fraction of the population in the system ( not counting the tagged customer ) . letting @xmath336 in ( [ s2_fs_rec ] ) and scaling again @xmath337 leads to @xmath338\big[\varphi\big(\xi+\frac{1}{n}\big)-\varphi(\xi)\big]+\big(1-\frac{1}{n\xi+1}\big)\varphi\big(\xi-\frac{1}{n}\big)-\varphi(\xi).\ ] ] for @xmath17 ( [ s42_phixiexpan ] ) leads to the limiting differential equation @xmath339\varphi'(\xi)+\big(\nu_*-\frac{1}{\xi}\big)\varphi(\xi)=0\ ] ] with solution @xmath340 we argue that if the right - hand side of ( [ s42_phixisolu ] ) is to be real and finite for all @xmath74 we must have @xmath341 a non - negative integer , and this regains the eigenvalue condition in ( [ s31_nuj ] ) . alternately , since we have already fixed @xmath342 by considering the scale @xmath343 , we can view ( [ s42_phixisolu ] ) as simply giving the approximation to the eigenfunctions on the @xmath151-scale , as given by ( [ s31_phijk1 ] ) . note that for @xmath89 and @xmath344 the zeros of ( [ s42_phixisolu ] ) and ( [ s42_psi0 ] ) coincide , but for @xmath345 the expression in ( [ s42_phixisolu ] ) has a zero of order @xmath47 at @xmath346 while ( [ s42_psi0 ] ) has @xmath47 simple zeros at points where @xmath347 is at a zero of @xmath348 . the expression in ( [ s42_phixisolu ] ) vanishes as @xmath166 for @xmath24 , and we thus need another expansion for the @xmath40 for small values of @xmath151 . we re - examine ( [ s2_fs_rec ] ) on the scale @xmath67 . setting @xmath349 and @xmath350 , ( [ s2_fs_rec ] ) becomes @xmath351 and if @xmath352 for @xmath17 with @xmath67 , the leading term must satisfy @xmath353 solving ( [ s42_qleading ] ) using generating functions or contour integrals leads to the formula in ( [ s31_phiint ] ) . to analyze the cases @xmath182 and @xmath169 we first note that when @xmath290 the eigenvalues for @xmath22 , which concentrate about @xmath86 , behave as @xmath354 ( here first @xmath17 , then @xmath290 ) . the result for @xmath24 ( cf . ( [ s31_nuj ] ) ) leads to @xmath355 as @xmath356 , which again shows that the eigenvalues begin to coalesce . also , @xmath357 balances @xmath358 when @xmath38 and this suggests the appropriate scaling for the transition region where @xmath169 . we thus define @xmath170 by @xmath359 the behavior of @xmath39 for @xmath360 indicates that the eigenvalues on the transition scale coalesce about some value that is @xmath187 as @xmath17 , so we set @xmath361 asymptotic matching as @xmath362 to the case @xmath24 and as @xmath173 to the case @xmath22 leads to the following behaviors of @xmath363 @xmath364 this analysis suggests that the eigenvalues may be expanded in the form ( [ s31_nu2term ] ) , where the correction term @xmath365 can also be argued by matching higher order terms in the expansions of @xmath39 for @xmath360 . to argue what the appropriate scaling of @xmath41 should be when @xmath38 , we examine ( [ s31_ny ] ) as @xmath290 , which becomes @xmath366 similarly , as @xmath356 the scaling in ( [ s31_x ] ) becomes @xmath367 in view of ( [ s43_ngamma ] ) and ( [ s43_nx ] ) we scale @xmath41 as in ( [ s31_u ] ) , where @xmath284 is to be determined . by asymptotic matching @xmath178 must behave as @xmath368 and @xmath369 . we then set @xmath370 where @xmath371 is another parameter , that will be determined to insure that @xmath372 satisfies a limiting differential equation for @xmath17 . for now we suppress the dependence of @xmath321 and @xmath372 on the eigenvalue index @xmath47 , and use ( [ s43_pntu ] ) in ( [ s2_fs_rec ] ) to obtain @xmath373\nonumber\\ & & -\frac{1}{n^{2/3}a+\sqrt{n}u+1}\exp\big(-\frac{\widetilde{a}}{n^{1/3}}\big)\phi\big(u-\frac{1}{\sqrt{n}}\big)\nonumber\\ & & + \frac{\gamma}{n^{1/3}}\big[\phi\big(u+\frac{1}{\sqrt{n}}\big)-\phi(u)\big]\nonumber\\ & & + \big[\exp\big(\frac{\widetilde{a}}{n^{1/3}}\big)-1\big]\big[1+\frac{\gamma}{n^{1/3}}\big]\phi\big(u+\frac{1}{\sqrt{n}}\big)\nonumber\\ & & -\big[\exp\big(\frac{\widetilde{a}}{n^{1/3}}\big)-1\big]\big(\frac{a}{n^{1/3}}+\frac{u}{\sqrt{n}}+\frac{1}{n}\big)\big(1+\frac{\gamma}{n^{1/3}}\big)\phi\big(u+\frac{1}{\sqrt{n}}\big)\nonumber\\ & & + \big[\exp\big(-\frac{\widetilde{a}}{n^{1/3}}\big)-1\big]\phi\big(u-\frac{1}{\sqrt{n}}\big).\end{aligned}\ ] ] the equation in ( [ s43_phiuexpan ] ) is an exact transformation of ( [ s2_fs_rec ] ) using the scaling in ( [ s31_u ] ) , ( [ s43_gamma ] ) and ( [ s43_pntu ] ) , and we rearranged the terms in the right - hand side of ( [ s43_phiuexpan ] ) in such a way that they are easier to expand for @xmath17 . next we assume that @xmath39 can be expanded in the form in ( [ s31_nu2term ] ) . with the help of taylor expansions , we see that the right side of ( [ s43_phiuexpan ] ) will have terms that are @xmath187 , @xmath374 , and @xmath19 , with the rest being @xmath375 , and this would balance the error term(s ) in the eigenvalue expansion in ( [ s31_nu2term ] ) . balancing the @xmath187 terms , which includes @xmath179 in ( [ s31_nu2term ] ) , leads to @xmath376\phi(u),\ ] ] the @xmath374 terms lead to @xmath377 and the @xmath19 terms give @xmath378\phi(u).\ ] ] in view of ( [ s43_fphiu ] ) we must have @xmath379 which is one equation relating @xmath179 , @xmath284 and @xmath371 to the `` detuning '' parameter @xmath170 . in order to obtain the second order equation in ( [ s43_n-1term ] ) as the leading term we must set @xmath380 and @xmath381 this yields two additional relations between @xmath371 and @xmath284 , and if we eliminate @xmath371 we obtain precisely the equation ( [ s31_gammaa ] ) that relates @xmath170 and @xmath284 , and then setting @xmath382 in ( [ s43_faa ] ) leads to @xmath383 , which is ( [ s31_fa ] ) . finally , using @xmath382 in ( [ s43_n-1term ] ) and setting @xmath384 leads to @xmath385\widetilde{\phi}(u)=0.\ ] ] if we further scale @xmath191 as @xmath386 ( [ s43_phitilde ] ) becomes @xmath387\widetilde{\phi}=0,\ ] ] which is the parabolic cylinder equation in standard form . solutions that have appropriate decay as @xmath191 ( or @xmath388 ) @xmath389 require that @xmath390 then the solution to ( [ s43_odephibar ] ) is proportional to a hermite polynomial , with @xmath391 , so we have derived ( [ s31_ga ] ) and ( [ s31_phieig ] ) . in view of the form of ( [ s43_phiuexpan ] ) , we can expand the eigenfunctions in powers of @xmath392 and calculate higher order correction terms , obtaining an expansion of the form @xmath393 . it is likely that the expansion of the eigenvalues @xmath39 involves only powers of @xmath394 . from ( [ s31_gammaa ] ) we can easily obtain @xmath395 as @xmath362 , and @xmath396 as @xmath173 , which can be used to verify the matching conditions in ( [ s43_fgammasim])-([s43_nx ] ) . we have thus shown that analysis of the case @xmath169 is quite intricate , but ultimately , with the appropriate scaling , we again reduced the problem to the standard eigenvalue problem for the hermite or parabolic cylinder equations . next we examine the eigenvectors @xmath40 on the scales @xmath189 , @xmath112 and @xmath67 , since ( [ s31_phieig ] ) no longer applies in these ranges . we first consider the scale @xmath197 . letting @xmath397 in ( [ s31_phieig ] ) suggests that on the @xmath208-scale the eigenvector @xmath40 is in the form ( [ s31_phiv ] ) . using ( [ s31_phiv ] ) in ( [ s2_fs_rec ] ) and noticing that changing @xmath41 to @xmath241 corresponds to changing @xmath208 to @xmath398 , we obtain the following odes for @xmath399 and @xmath400 : @xmath401 ^ 2+(\gamma - v)\mathcal{f}'(v)+f(\gamma)-\frac{1}{v}=0,\ ] ] @xmath402 ^ 2}{2\mathcal{f}'(v)-v+\gamma}\mathcal{g}(v , j)=0.\ ] ] solving ( [ s43_fode ] ) and ( [ s43_gode ] ) leads to the results in ( [ s31_mathcalf])-([s31_gv > a ] ) . next we consider the scale @xmath303 with @xmath403 . letting @xmath404 in ( [ s31_mathcalf ] ) and ( [ s31_gv > a ] ) , we find that the eigenvectors have an expansion in the form in ( [ s31_phixi ] ) . using ( [ s31_phixi ] ) in ( [ s2_fs_rec ] ) yields @xmath405 and @xmath406g(\xi , j)=0,\ ] ] which can be easily solved to give ( [ s31_fxi ] ) and ( [ s31_gxi ] ) . finally we consider the @xmath67 scale . this is necessary since @xmath400 in ( [ s31_gv < a ] ) is singular as @xmath216 , with @xmath407 . we assume that the leading order approximation to @xmath40 is @xmath408 and use this in ( [ s2_fs_rec ] ) . similarly to the analysis of @xmath67 in subsection 4.1 , we find that @xmath409 satisfies @xmath410 , which leads to the contour integral in ( [ s31_phisim3 ] ) with @xmath228 . we assess the accuracy of our asymptotic results , and their ability to predict qualitatively and quantitatively the true eigenvalues / eigenvectors . recall that @xmath40 is the @xmath411 eigenvector with @xmath412 and @xmath413 . in figure [ f1 ] we plot the exact ( numerical ) @xmath414 for @xmath415 $ ] , where @xmath416 and @xmath417 . our asymptotic analysis predicts that @xmath414 will undergo a single sign change , and on the scale @xmath343 , ( [ s31_phijx ] ) shows that @xmath414 should be approximately linear in @xmath148 , with a zero at @xmath418 which corresponds to @xmath419 ( @xmath420 ) , in view of ( [ s31_x ] ) . the exact eigenvector undergoes a sign change when @xmath41 changes from 74 to 75 , in excellent agreement with the asymptotics . while the eigenvector is approximately linear near @xmath419 , figure [ f1 ] shows that it is not globally linear , and achieves a minimum value at @xmath421 . but our analysis shows that on the @xmath422 scale , we must use ( [ s31_phijk1 ] ) , and when @xmath423 and @xmath344 , ( [ s31_phijk1 ] ) achieves a minimum at @xmath424 , which corresponds to @xmath425 . this demonstrates the necessity of treating both the @xmath148 and @xmath151 scales . in figure [ f2 ] we retain @xmath416 and @xmath423 , but now plot the second eigenvector @xmath426 . we see that typically @xmath427 , and its graph is approximately tangent to the @xmath41-axis near @xmath428 . in figure [ f3 ] we `` blow up '' the region near @xmath419 , plotting @xmath426 for @xmath429 $ ] . now we clearly see two sign changes , and these occur as @xmath41 changes from 69 to 70 , and 79 to 80 . our asymptotic analysis suggests that near @xmath430 the eigenvector is proportional to @xmath431 , which has zeros at @xmath432 , and in view of ( [ s31_x ] ) this corresponds to @xmath433 , which is again in excellent agreement with the exact results . in the range @xmath434 $ ] figure [ f3 ] shows roughly a parabolic profile , as predicted by ( [ s31_phijx ] ) , but the larger picture in figure [ f2 ] again demonstrates that the @xmath151-scale result in ( [ s31_phijk1 ] ) must be used when @xmath148 is further away from @xmath435 , as ( [ s31_phijk1 ] ) will , for example , predict the minimum value seen in figure [ f2 ] . next we consider @xmath182 , maintaining @xmath416 . now the asymptotic result in the main range is in ( [ s31_u ] ) and ( [ s31_phieig ] ) , and when @xmath182 we have @xmath436 . in figure [ f5 ] we plot @xmath414 ( @xmath344 ) in the range @xmath437 $ ] , and we see a single sign change when @xmath41 increases from 26 to 27 . for @xmath344 the asymptotic formula in ( [ s31_phieig ] ) predicts a zero at @xmath438 , which corresponds to @xmath439 . in figure [ f6 ] we have @xmath440 and @xmath441 $ ] ; there are two sign changes , between @xmath442 and 22 , and @xmath443 and 35 . now @xmath444 and the approximation in ( [ s31_phieig ] ) has zeros where @xmath445 , which corresponds to @xmath446 , leading to the numerical values of @xmath447 and @xmath448 . next we consider @xmath22 . now the main range is the @xmath97-scale result in ( [ s31_phi])-([s31_ab ] ) . for @xmath22 we now plot the `` symmetrized '' eigenvector(s ) @xmath449 . in figures [ f8 ] and [ f9 ] we always have @xmath450 and @xmath416 . figure [ f8 ] has @xmath451 for @xmath452 $ ] and shows a single sign change between @xmath453 and 16 . the leading term in ( [ s31_phi ] ) predicts a sign change at @xmath454 , so that @xmath455 . since @xmath456 in ( [ s31_phi1 ] ) ( with ( [ s31_ab ] ) ) , the correction term would improve on the accuracy of the sign change prediction . figure [ f9 ] plots @xmath457 and shows two sign changes , between @xmath458 and 13 , and @xmath459 and 23 . now the leading term in ( [ s31_phi ] ) has zeros at @xmath460 which correspond to the numerical values @xmath461 and @xmath462 from ( [ s31_ny ] ) . we note that figures [ f8 ] and [ f9 ] show that the eigenvector becomes very small as @xmath41 increases toward 99 , and this is indeed predicted by the expansion in ( [ s31_phileading ] ) , which applies on the @xmath73 scale . we have thus shown that our asymptotic results predict quite well the qualitative properties of the eigenvectors @xmath40 for @xmath8 large and moderate @xmath47 . we get very good quantitative agreement when @xmath24 , but less so when @xmath182 or @xmath22 , for the moderately large value @xmath416 . next we consider the accuracy of our expansions for the eigenvalues @xmath39 . in table [ table1 ] we take @xmath450 and increase @xmath8 from 10 to 100 , and give the two - term , three - term , and four - term asymptotic approximations in ( [ s31_nu ] ) for the zeroth eigenvalue @xmath45 , along with the exact numerical result . we also give the relative errors , defined as @xmath463 . we note that the leading term would simply give @xmath464 , independent of @xmath8 , which is not very accurate . table [ table1 ] shows that the correction terms in ( [ s31_nu ] ) do lead to accurate approximations , and indeed when @xmath416 the four - term approximation is accurate to three significant figures . in table [ table2 ] we take @xmath417 , in which case we only computed the leading term in ( [ s31_nuj ] ) , but now the eigenvalue index @xmath47 appears to leading order . table [ table2 ] gives the exact values of @xmath45 and @xmath465 , along with the leading order approximations in ( [ s31_nuj ] ) , for @xmath8 increasing from 10 to 100 . the agreement is again very good , with the relative errors decreasing to about @xmath466 when @xmath416 . we next do some numerical studies to see how rapidly the unconditional sojourn time density settles to its tail behavior , for problems with moderately large @xmath8 . here we compute @xmath43 exactly ( numerically ) , using ( [ s2_fs_sum ] ) and ( [ s2_fs_probn ] ) in ( [ s1_pt ] ) , and compare the result to the approximation in ( [ s2_fs_ptsim ] ) , which uses only the zeroth eigenvalue @xmath45 . tables [ table3]-[table5 ] compare the exact @xmath467 $ ] to the corresponding approximation from ( [ s2_fs_ptsim ] ) , and we note that both must approach @xmath45 as @xmath44 . our asymptotic analysis predicts that for @xmath17 and @xmath22 , the @xmath89 term in ( [ s2_fs_sum ] ) should dominate for times @xmath468 , while if @xmath38 or @xmath24 , the @xmath89 term dominates for times @xmath469 . in table [ table3 ] we take @xmath450 and let @xmath470 and @xmath471 . for @xmath470 , the largest eigenvalue is @xmath472 , which we list in the last row of the table , and the second largest eigenvalue is @xmath473 . for @xmath474 , we have @xmath475 and @xmath476 . in table [ table4 ] we take @xmath182 . now @xmath477 and @xmath478 when @xmath470 , and @xmath479 and @xmath480 when @xmath474 . in table [ table5 ] we take @xmath423 and @xmath481 , @xmath482 when @xmath470 , and @xmath483 , @xmath484 when @xmath474 . tables [ table3]-[table5 ] show that the approximation resulting from ( [ s2_fs_ptsim ] ) is quite accurate , though it may take fairly large times before the exact and approximate results ultimately reach the limit @xmath45 . the relative errors improve as we go from @xmath450 to @xmath182 to @xmath423 , which is again consistent with our asymptotic analysis , as when @xmath22 there is the most coalescence ( for @xmath17 ) of the eigenvalues , making it hard to distinguish @xmath45 from the others . + 5 & 0.5398 & 0.5415 & 0.32% & 0.3383 & 0.3387 & 0.14% + 10 & 0.3362 & 0.3363 & 0.03% & 0.2024 & 0.2024 & 0.01% + 15 & 0.2679 & 0.2680 & 2.4e-05 & 0.1570 & 0.1570 & 1.5e-05 + 20 & 0.2338 & 0.2338 & 2.2e-06 & 0.1343 & 0.1343 & 1.6e-06 + 30 & 0.1996 & 0.1996 & 1.9e-08 & 0.1207 & 0.1207 & 1.7e-07 + 50 & 0.1722 & 0.1722 & 1.6e-12 & 0.1002 & 0.1002 & 2.1e-10 + 100 & 0.1517 & 0.1517 & @xmath4861.0e-12 & 0.0934 & 0.0934 & 2.6e-12 + @xmath487 & 0.1312 & 0.1312 & & 0.0662 & 0.0662 & + 2 l. kleinrock , _ analysis of a time - shared processor _ , naval research logistics quarterly 11 ( 1964 ) , 59 - 73 . + l. kleinrock , _ time - shared systems : a theoretical treatment _ , j. acm 14 ( 1967 ) , 242 - 261 . + d. p. heyman , t. v. lakshman , and a. l. neidhardt , _ a new method for analysing feedback - based protocols with applications to engineering web traffic over the internet _ acm sigmetrics ( 1997 ) , 24 - 38 . + l. massouli and j. w. roberts , _ bandwidth sharing : objectives and algorithms _ , proc . ieee infocom , new york , ny , usa ( 1999 ) , 1395 - 1403 . + m. nabe , m. murata , and h. miyahara , _ analysis and modeling of world wide web traffic for capacity dimensioning of internet access lines _ , evaluation , 34 ( 1998 ) , 249 - 271 . + d. mitra and j. a. morrison , _ asymptotic expansions of moments of the waiting time in closed and open processor - sharing systems with multiple job classes _ , adv . in appl . probab . 15 ( 1983 ) , 813 - 839 . + j. a. morrison and d. mitra , _ heavy - usage asymptotic expansions for the waiting time in closed processor - sharing systems with multiple classes _ , adv . in appl . 17 ( 1985 ) , 163 - 185 . + j. a. morrison , _ asymptotic analysis of the waiting - time distribution for a large closed processor - sharing system _ , siam j. appl . 46 ( 1986 ) , 140 - 170 . + j. a. morrison , _ moments of the conditioned waiting time in a large closed processor - sharing system _ , stochastic models 2 ( 1986 ) , 293 - 321 . + j. a. morrison , _ conditioned response - time distribution for a large closed processor - sharing system in very heavy usage _ , siam j. appl . 47 ( 1987 ) , 1117 - 1129 . + j. a. morrison , _ conditioned response - time distribution for a large closed processor - sharing system with multiple classes in very heavy usage _ , siam j. appl . 48 ( 1988 ) , 1493 - 1509 . + k. c. sevcik and i. mitrani , _ the distribution of queueing network states at input and output instants _ , j. acm 28 ( 1981 ) , 358 - 371 . + f. pollaczek , _ la loi dattente des appels tlphoniques _ , c. r. acad . paris 222 ( 1946 ) , 353 - 355 . + j. w. cohen , _ on processor sharing and random service ( letter to the editor ) _ , j. appl . prob . 21 ( 1984 ) , 937 . + w. magnus , f. oberhettinger , and r. p. soni , formulas and theorems for the special functions of mathematical physics , springer - verlag , new york , 1966 .
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we consider sojourn or response times in processor - shared queues that have a finite population of potential users .
computing the response time of a tagged customer involves solving a finite system of linear odes . writing the system in matrix form , we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large .
this corresponds to finite population models where the total population is @xmath0 . using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation , such as the hermite equation .
the dominant eigenvalue leads to the tail of a customer s sojourn time distribution . +
* keywords : * finite population , processor sharing , eigenvalue , eigenvector , asymptotics .
| 19,894 | 200 |
a relativistic motion of grb sources was advocated by @xcite from that the energies released exceed by many orders of magnitude the eddington luminosity for a stellar - mass object , especially if grbs are at cosmological distances ( see also @xcite ) . the detection by cgro / egret of photons with energy above 1 mev during the prompt burst emission ( e.g. @xcite ) shows that grb sources are optically thin to such photons . together with the sub - mev burst isotropic - equivalent output of @xmath1 ergs ( e.g. @xcite ) and the millisecond burst variability timescale , the condition for optical thickness to high energy photons gives another reason why grbs must arise from ultra - relativistic sources , moving at lorentz factor @xmath2 ( e.g. @xcite ) . the same conclusion is enforced by the measurement of a relativistic expansion of the radio afterglow source . that expansion was either measured directly , as for grb 030329 ( @xmath3 ) , whose size increased at an apparent speed of 5c , indicating a source expanding at @xmath4 at 12 months @xcite , or was inferred from the rate at which interstellar scintillation @xcite quenches owing to the increasing source size , as for grb 970508 , whose expansion speed is inferred to be close to @xmath5 at 1 month @xcite . the adiabatic dynamical evolution of a blast - wave , @xmath6 , where @xmath7 is the mass of the ambient medium , leads to @xmath8 for a homogeneous medium ( @xmath9 being the observer - frame photon arrival time ) . then , @xmath10 extrapolated to the burst time implies @xmath11 . extrapolating to such early times is justified by that most optical afterglow light - curves display a power - law decay starting after the burst , which sets a lower limit on the source lorentz factor at that time . whether the grb ejecta are a cold baryonic outflow accelerated by the adiabatic losses of fireball s initial thermal / radiation energy ( e.g. @xcite ) , or relativistic pairs formed through magnetic dissipation in a poynting outflow , as in the electromagnetic model of @xcite , their interaction with ambient medium will drive two shocks : a reverse shock crossing the ejecta and a forward - shock sweeping the circumburst medium , as illustrated in figure [ bw ] . both shocks energize their respective media , accelerate relativistic particles and generate magnetic fields through some plasma - instability related process , such as the two - stream weibel instability driven by an anisotropic particle distribution function @xcite . the original magnetic field of the fireball at @xmath12 cm becomes too weak by the time the fireball reaches the @xmath13 cm radius ( where the burst and afterglow emissions are produced ) for the synchrotron emission to account for the sub - mev burst emission and for the longer - wavelength , ensuing afterglow emission , even if the fireball was initially magnetically dominated @xcite . the evolution of the synchrotron and inverse - compton fluxes produced by the blast - wave at a fixed frequency ( i.e. the light - curve ) is determined by how the characteristics of the spectrum ( break frequencies and peak flux ) change with time . figure [ spek ] shows the expected afterglow synchrotron spectrum , whose characteristics depend on the blast - wave radius , number of radiating electrons , their distribution with energy , magnetic field strength , and lorentz factor . if the typical electron energy and magnetic field energy correspond to some fixed fraction of the post - shock energy , or if they start from such a fixed fraction and then evolve adiabatically ( as for adiabatically colling ejecta ) , then the afterglow light - curve depends on ( 1 ) the evolution of the blast - wave lorentz factor , the blast - wave radius being @xmath14 ( with @xmath9 the photon - arrival time measured since burst trigger ) ( 2 ) the spectrum of the blast - wave emission ( i.e. the distribution of electrons with energy ) , and , in the case of the reverse - shock , ( 3 ) the evolution of the incoming mass . the power - law deceleration of the blast - wave ( @xmath15 for @xmath16 , where @xmath17 is the radial stratification of the ambient medium density ) and the power - law afterglow spectrum ( @xmath18 ) are two factors which lead to a power - law afterglow light - curve ( @xmath19 ) , with the decay index @xmath20 being a linear function of the spectral slope @xmath21 . these are the only two factors at work for the forward - shock emission and the ejecta emission during the adiabatic cooling phase ( which starts when the reverse shock has crossed the ejecta shell ) , the two models that yield power - law afterglow light - curves in the most simple and natural way . in contrast , for the reverse - shock emission ( i.e. the ejecta emission while the shock exists ) , the light - curve depends also on the radial distribution of ejecta mass and of their lorentz factor , thus the observed power - law light - curves require additional properties to be satisfied by the relativistic ejecta . such properties seem _ ad - hoc _ when it comes to explaining single power - law afterglows whose flux displays an unchanged decay over 24 decades in time ( such as the x - ray afterglows of grbs 050801 , 050820a , 06011b , 060210 , 060418 , 061007 ) , but they also provide the flexibility required to account for the prevalent x - ray afterglow light - curves that exhibit one or more breaks . i consider first the afterglow emission at early times , when the blast - wave is sufficiently relativistic that the observer receives boosted emission from a region of half - angle opening @xmath22 ( as seen from the center of the blast - wave ) that is smaller than the half - aperture @xmath23 of the collimated outflow . in that case , the observer does not `` see '' yet the angular boundary of the outflow and the received emission is as bright as for a spherical blast - wave . the evolution of the spectral characteristics of the emission from ejecta and the swept - up ambient medium are presented below , the resulting power - law decay indices of the synchrotron flux being listed in table 1 . for a short - duration ejecta release , the reverse shock crosses the ejecta shell over an observer - frame time that depends primarily on the ejecta lorentz factor @xmath24 : @xmath25 s , where @xmath26 is the isotropic - equivalent ejecta kinetic energy in @xmath27 erg , @xmath28 is the ambient medium density in protons per @xmath29 , and @xmath30 . in this case , the reverse shock is semi - relativistic and , more likely , radiates below the optical . after @xmath31 , the input of energy into the shocked structure ceases and the blast - wave begins to decelerate . if the ejecta release is an extended process , the deceleration timescale depends primarily on the duration @xmath32 over which the ejecta are expelled : @xmath33 . in this case , the reverse shock is relativistic and could produce a bright optical emission . the separation between these two cases is set by @xmath34 , `` short - duration ejecta release '' meaning @xmath35 s. for a wind medium , the deceleration timescale is @xmath36 s for a short - duration ejecta release , where @xmath37 is the wind density parameter , normalized to that resulting for @xmath38 being ejected at a terminal velocity of 1000 km / s , @xmath31 being the same as for a homogeneous medium in the case of a long - duration ejection ( which occurs for @xmath39 s ) . at the deceleration radius , @xmath40 of the ejecta energy has been transferred to the swept - up ambient medium , which moves at @xmath41 for @xmath42 and a lower @xmath43 for @xmath44 . taking into account that the energy per particle in the post forward - shock gas is @xmath43 , it follows that , at @xmath31 , the ejecta mass is larger than that of the forward shock by at most a factor @xmath45 . this implies that , at @xmath31 , the peak flux of the reverse - shock emission spectrum is a factor @xmath46 larger than the peak flux of the forward - shock spectrum , hence , the optical flash from the reverse shock could be up to 5 magnitudes brighter than the optical emission from the forward shock . as mentioned above , when there is a reverse shock crossing the ejecta , its emission flux should depend on the density and lorentz factor of the incoming ejecta . semi - analytical calculations of the ejecta synchrotron emission when there is a reverse shock have been done by @xcite and @xcite for a density and lorentz factor of the incoming ejecta tailored to produce x - ray light - curve plateaus , hydrodynamical calculations of the reverse - shock dynamics have been presented by @xcite , and calculations of the ejecta emission after the reverse shock has crossed the ejecta ( i.e. during adiabatic cooling ) have been published by @xcite . however , analytical calculations of the ejecta emission while there is a reverse shock @xcite have yet to be done . assuming a uniform ejecta density & lorentz factor and an extended ejecta release , for which the reverse shock is relativistic and the shocked ejecta are slightly decelerated even before @xmath31 , owing to the progressive dilution of the incoming ejecta , i find that the _ reverse - shock _ peak flux @xmath47 and break frequencies @xmath48 ( injection ) and @xmath49 ( cooling ) evolve as @xmath50 , @xmath51 , @xmath52 for a homogeneous medium and @xmath53 , @xmath54 , @xmath55 for a wind . as for the _ ejecta _ emission decay during the adiabatic cooling phase , the evolution of the spectral characteristics is approximately @xmath56 , @xmath57 for a homogeneous medium , and @xmath58 , @xmath59 for a wind . for @xmath60 , the cooling ejecta flux should decay with an index @xmath61 , for either type of medium . above @xmath48 but below @xmath49 , the decay index is @xmath62 for a homogeneous medium and @xmath63 for a wind . depending on the treatment of the ejecta dynamics and adiabatic cooling , other researchers reached slightly different results @xmath64 with @xmath65 and @xmath66 in @xcite , @xmath67 and @xmath68 in @xcite , @xmath69 and @xmath70 in @xcite . after @xmath49 falls below @xmath71 owing to adiabatic cooling , the observer receives no emission from the area of angular opening @xmath22 moving directly toward the observer ( because of the exponential cut - off of the synchrotron emissivity above the synchrotron peak ) but receives emission from the fluid moving at increasing angles larger than @xmath22 . that emission ( called _ large - angle _ emission , lacking a better name ) was released at the same time as the emission from angles less than @xmath22 , but arrives later at observer because of the spherical curvature of the emitting surface and finite speed of light , and is less beamed relativistically . as shown by @xcite , the large - angle emission is characterized by @xmath72 , @xmath73 , the power - law decay index being @xmath74 ( see also @xcite ) these are general results , arising only from relativistic effects , and independent of the emission process . the only assumption made in its derivation is that the surface emissivity properties are angle - independent . before deceleration of the blast - wave begins , the shocked ambient medium moves at a constant @xmath75 , if the reverse shock is semi - relativistic , or is slowly decelerating as @xmath76 , if the reverse shock is relativistic . for a _ semi - relativistic _ reverse shock , the spectral characteristics of the _ pre - deceleration _ forward - shock synchrotron emission evolve as @xmath77 , @xmath78 , @xmath79 for a homogeneous medium , and @xmath80 , @xmath81 , @xmath82 for a wind . for a _ relativistic _ reverse shock , the above scalings become @xmath83 , @xmath84 , @xmath52 for @xmath85 , and @xmath53 , @xmath86 , @xmath55 for @xmath87 . the forward - shock synchrotron emission _ after deceleration _ has received the most attention ( e.g. @xcite ) . under the usual assumptions of constant blast - wave energy and micro - physical electron and magnetic field parameters , the forward - shock peak flux and spectral break frequencies evolution is @xmath88 , @xmath89 , @xmath90 for a homogeneous medium , and @xmath91 , @xmath89 , @xmath92 for a wind . from here , it follows that the flux below @xmath48 should rise slowly as @xmath93 for a homogeneous medium or be constant for a wind medium . for @xmath94 , the forward - shock flux decay is a power - law of index @xmath95 , with @xmath96 if ambient medium has a wind - like stratification ( as expected for a massive stellar long - grb progenitor ) and if @xmath97 , @xmath98 for a homogeneous medium ( which , surprisingly , is more often found to be compatible with the observed afterglows than a wind ) if @xmath97 , and @xmath99 if @xmath100 , for any type of medium . all the above results hold for a spherical outflow or a collimated one before the jet boundary becomes visible to the observer . at the jet - break time @xmath101 , when deceleration lowers the jet lorentz factor to @xmath102 , the emission from the jet edge is no longer relativistically beamed away from the direction toward the observer . at @xmath103 , the lack of emitting fluid at angles larger than @xmath23 leads to a steepening of the afterglow decay by @xmath104 for a homogeneous medium and @xmath105 for a wind . simultaneously , the lateral spreading of the jet becomes important and leads to a faster deceleration of the jet , which switches from a power - law in the blast - wave radius to an exponential @xcite , yielding an extra steepening of the afterglow decay of magnitude smaller or comparable to @xmath106 above . together , these two _ jet effects _ lead to @xmath107 , @xmath108 , @xmath109 , rather independent of the ambient medium stratification , and a post jet - break forward - shock flux decay of index @xmath110 below @xmath48 , while for @xmath111 , one obtains @xmath112 below @xmath49 and @xmath113 above @xmath49 . & & & & + model & @xmath114&@xmath115&@xmath116&@xmath117&@xmath118 & @xmath114&@xmath115&@xmath116&@xmath117&@xmath118 + rs(1 ) & 1/5 & 1/15 & 1/5 & 3/5 & 3/5 & 1/9 & 5/9 & @xmath119 & 0 & @xmath120 + rs(2 ) & 0.3 & 0.3 & @xmath121 & & @xmath122 & 0.3 & 0.3 & @xmath123 & & @xmath122 + fs(1 ) & -3 & -11/3 & -3 & -2 & -2 & -1/3 & 1/3 & @xmath21 & -1/2 & @xmath124 + fs(2 ) & -1 & -19/15 & @xmath125 & -1/5 & @xmath126 & -1/9 & 5/9 & @xmath127 & 0 & @xmath128 + fs(3 ) & -1/2 & -1/6 & @xmath129 & 1/4 & @xmath130 & 0 & 2/3 & @xmath131 & 1/4 & @xmath132 + @xcite were the first to predict the existence of radio afterglows following the burst phase . @xcite have analyzed two models for the reverse shock emission and one for the forward shock , predicting long - lived optical afterglows with a flux decaying as a power of time . the first detection of an afterglow and measurement of a power - law flux decay followed soon ( grb 970228 @xcite ) , with many other optical @xcite and x - ray afterglows @xcite having been observed until today . in general , the broadband ( radio , optical , x - ray ) emission of grb afterglows display the expected power - law spectra and light - curves , as well as other features , which , in chronological order of their _ prediction _ are : radio scintillation ( @xcite & @xcite ) , optical counterpart flashes ( @xcite & @xcite ) , jet - breaks ( @xcite & @xcite ) , dimmer afterglows for short bursts ( @xcite & @xcite ) . grb afterglows display sufficient diversity ( e.g. wide luminosity distributions at all observing frequencies , non - universal shock micro - physical parameters ) and puzzling features ( slowly - decaying radio fluxes , x - ray light - curve plateaus , chromatic x - ray light - curve breaks ) to challenge the standard external - shock model and warrant various modifications . below , i discuss some of these issues . according to the temporal scalings identified in the previous section , the light - curves of grb afterglows should display rises in the early phase , if the forward - shock emission is dominant ( because the reverse - shock flux is most often expected to decay ) , followed by a decay , both being power - laws in time . furthermore , the broadband afterglow spectra are expected to be rising at ( radio ) frequencies below the spectrum peak and fall - off at higher ( x - ray ) photon energies . also expected is that the light - curve decay indices @xmath20 and spectral slopes @xmath21 satisfy one or more closure relationships and that , there is a positive correlation between @xmath20 and @xmath21 ( from that @xmath133 $ ] ) . figure [ aglows ] illustrates the flux power - law decays and power - law spectra typically observed for grb afterglows . the light - curves chosen there display long - lived power - law decays , but many afterglows exhibit more diversity , their light - curves showing two or three power - law decays , sometimes even rising at earlier times , rarely exhibiting brightening episodes ( in optical or x - ray ) or sudden drops ( in x - ray ) . the broadband spectrum of grb afterglow 030329 shown in figure [ aglows ] displays an optically thick part ( to self - absorption ) in the radio , at earlier times , a constant peak flux up to 10 days , during which the radio flux rises slowly , followed by a decreasing peak flux and decreasing radio flux . for @xmath60 , as required by the radio spectrum ( right panel ) , the rise of the radio flux , @xmath134 ( left panel ) , is consistent with that expected from a decelerating forward - shock interacting with a homogeneous medium ( @xmath135 ) , and marginally consistent with the forward - shock pre - deceleration emission for either a semi - relativistic reverse shock and wind medium ( @xmath136 ) or a relativistic reverse shock and homogeneous medium ( @xmath137 ) . the evolution of spectral breaks is generally hard to determine observationally and use for identifying the correct afterglow model : self - absorption affects only the early radio emission , when the large flux fluctuations are caused by interstellar scintillation , while the cooling break is too shallow and evolves too slowly to be well measured even if it fell in the optical or x - ray bands . the best prospects for this test is offered by the injection break , which should cross the radio domain at tens of days , when the scintillation amplitude is reduced by the larger source size . sufficient radio coverage to construct radio afterglow spectra at many epochs and determine the peak frequency @xmath48 and flux @xmath47 is rarely achieved . grb 030329 is one such case ( figure [ radiopk ] ) , the evolution of @xmath48 being slower than expected for a spherical blast - wave or a jet that does not expand ( yet or ever ) laterally ( for either , @xmath138 for any medium stratification ) , while that of @xmath47 is close to that expected for a jet spreading laterally ( for which @xmath107 ) . thus , the evolutions of @xmath48 and @xmath47 for grb afterglow 030329 seem mutually inconsistent . the slower - than - expected evolution of @xmath48 requires that shock micro - physical are not constant ( as assumed in the standard model ) , but the evolution of @xmath47 can be accounted for by a spherical blast - wave provided that the ambient medium density decreases as @xmath139 . rising optical light - curves have been seen for more than a dozen afterglows up to 1 ks after trigger ( figure [ rises ] ) . if interpreted as the due to the pre - deceleration emission from the forward shock ( e.g. @xcite ) , they require smaller than average initial ejecta lorentz factors ( if the reverse shock is semi - relativistic ) or longer - lasting ejections . the existence of energetic ejecta with a significantly smaller ejecta lorentz factor could also explain the late ( 12 d ) sharp rise displayed by the optical afterglow of grb 970508 ( figure [ aglows ] ) . however , late - rising afterglows could also be due to a structured outflow endowed with two `` hot spots '' , one moving directly toward the observer and giving the prompt grb emission , and another one moving slightly off the direction toward the observer @xcite , at an angle @xmath140 , its emission becoming visible when the outflow lorentz factor decreases to @xmath141 . the same result could be accomplished with an axially symmetric outflows having a bright core that yields the burst emission and a bright ring that produced the rising afterglow when it becomes visible . thus , a late - rising afterglow may be a relativistic effect rather than the signature of some ejecta with a lower initial lorentz factor , either shock ( reverse or forward ) being a possible origin of the rising afterglow . in fact , this model is found to account better for the peak luminosity peak time anti - correlation exhibited by a dozen optical afterglows with early , fast rises than the pre - deceleration external - shock model @xcite , although it should be noted that only half of that correlation is real ( i.e. optical peaks do not occur later and are not brighter than a certain linear limit in log - log space ) while the other half is just an observational bias , as there are many optical afterglows exhibiting decaying fluxes from first measurement that fall below the peak flux peak time relation found for the fast - rising afterglows ( figure [ rises ] ) . most afterglow observations were made during the decay phase , where @xmath20 and @xmath21 are expected to be correlated and satisfy one or more closure relationships . the left panel of figure [ ab ] shows the temporal and spectral indices of optical and x - ray afterglows measured before the jet break and the expectations for the ( post - deceleration ) forward - shock model ( as it seems more likely that the reverse shock dominates the afterglow emission only until at most 1 ks ) . surprisingly , no significant correlation can be seen between @xmath20 and @xmath21 , which may be taken either as indication that the standard forward - shock model does not account for the diversity of afterglows ( e.g. departures from its assumptions of constant shock parameters would be required to explain the decays below the `` s1 '' model , which are too slow ) or that more than one variant of it realized which , combined with a small baseline in @xmath20 and @xmath21 , requires a much larger sample to reveal the expected underlying correlation . a bright optical emission arising from the reverse shock was predicted by @xcite and may have been observed for the first time in the optical counterpart ( i.e. during the burst ) accompanying grb 990123 @xcite and in the early afterglow emission following grb 021211 , but without any further candidates until recently . lacking a continuous , long - lived injection of new ejecta , and because of the adiabatic cooling , the reverse - shock emission should be confined to the early afterglow . then , the early afterglow emissions of grb 990123 and 021211 being brighter than the extrapolation of the later flux and decaying faster are two reasons for attributing those two early optical emissions to the reverse shock . the larger brightness ( by 2.5 mag for 990123 and by 1 mag for 021211 ) could be explained by that the number of ejecta electrons is , at deceleration ( i.e. around burst end ) , larger by a factor @xmath142 than in the forward shock , and by a smaller factor at later times ( as for 021211 ) , with some relative dimming of the reverse shock optical flux attributed to the peak of the reverse - shock synchrotron spectrum being lower than that of the forward shock . however , both the above reasons disappear if the origin of time is not at trigger but sometime later , e.g. at 3040 s ( corresponding to the peak of grb 990123 optical flash ) . then , the early optical emission appears as a small deviation of the extrapolation of the late flux and the entire afterglow light - curve is consistent with a single power - law , indicating a unique dissipation mechanism . thus , absent spectral information , the evidence for a reverse shock origin of the early optical emissions of grb 990123 and 021211 is circumstantial . such spectral information has been acquired only recently , for the early optical emission of grb afterglows 061126 and 080319b . for the former , @xcite finds that the steeper - decaying early ( up to 200 s ) optical emission is harder than at later times ( after 1 ks ) . the indices @xmath143 and @xmath144 of the early optical emission of grb 061126 are consistent with the closure relation expected for adiabatic cooling ejecta . the spectral evolution observed simultaneously with the slowing of light - curve decay suggests the emergence of a different component after 1 ks . the optical afterglow of grb 080319b also displayed a spectral hardening simultaneous with the reduction in the flux decay rate @xcite , supporting a reverse - shock origin of the early fast - decay phase and forward - shock origin of the later slower - decaying emission . however , the decay at early times is too fast ( for the measured spectral slope ) to be attributed to the adiabatic cooling of ejecta . in fact , the decay index @xmath145 is consistent with the expectations for the large - angle emission released during the burst . however , that does not exclude a reverse - shock origin of the early optical flux , as the cooling frequency may have fallen below the optical , revealing the large - angle emission . the slow softening of the optical spectrum of grb afterglow 080319b after 1 ks was a surprise . if the rather flat ( @xmath146 ) spectral slope at 1 ks were due to the peak energy of the forward - shock synchrotron spectrum being in the optical , then a much faster softening is expected , given that the injection frequency evolution @xmath89 is also fast . energy injection in the blast - wave or an increasing electron / magnetic shock parameters could account for slow decrease of @xmath48 required by the slow spectral softening of grb 080319b afterglow optical emission . a tight collimation of grb ejecta , into a jet of half - aperture less than 10 degrees , is desirable to reduce the isotropic - equivalent grb output , reaching @xmath147 erg , to lower values , below @xmath148 erg , compatible with what the mechanisms for production of relativistic jets by solar - mass black - holes can yield . besides the @xmath149 closure relationship being satisfied self - consistently ( i.e. by a forward - shock model with same features before and after the jet - break ) , _ achromaticity _ of the break ( i.e. simultaneous occurrence at all frequencies ) is an essential test of this model . the steepening of the afterglow flux decay due to collimation of ejecta was predicted by @xcite and was observed for the first time in the optical emission of grb afterglow 990123 @xcite . about 3/4 of well - monitored pre - swift optical afterglows displayed jet - breaks at 0.53 day , as shown in the compilation of @xcite . the x - ray coverage of pre - swift afterglow extended over at most 1 decade in time and was insufficient to test for the existence of an x - ray light - curve break simultaneous with that seen in the optical . a smaller fraction , between 1/3 and 2/3 , of swift x - ray afterglows also exhibit jet - breaks @xcite , defined as a steepening occurring after 0.1 day from a power - law decay with @xmath150 to one with @xmath151 . comparing the post jet - break temporal and spectral indices with the expectations for the forward - shock model ( right panel of figure [ ab ] ) shows that that model accounts for observations of post jet - break decays if jets are both both spreading and conical . however , just as for pre jet - break decays , the expected @xmath149 correlation is not seen . while there are many examples of potential jet - breaks in the x - ray light - curve monitored by swift , few are sufficiently well - monitored in the optical to test for the achromaticity of the break . figure [ jets ] shows the only 3 afterglows sufficiently sampled and followed sufficiently late to search for achromatic light - curve breaks . besides the simultaneity of the optical and light - curve breaks , note the equality of the pre and post - break decay indices . the smaller fraction of swift x - ray afterglows that exhibit jet - breaks , relative to that of pre - swift such optical afterglows , could be due to swift detecting and localizing afterglows that are fainter than those followed in the optical prior to swift . the argument here is that , if all jets had the same energy ( e.g. @xcite ) , then the afterglow flux and jet - break time should be anti - correlated : @xmath152 and @xmath153 ( for a wind - like medium ) , leading to @xmath154 , where @xmath155 is the forward - shock s kinetic energy per solid angle , thus dimmer afterglows should display later jet - breaks . as shown in figure [ ox ] , the afterglows with jet - breaks at 0.310 days are brighter by a factor @xmath0 than those without jet - breaks detected until 10 days , thus the anti - correlation between afterglow flux and jet - break time expected for a universal jet energy is confirmed , even though jet energies inferred from the timing of afterglow light - curve breaks have a broad distribution ( e.g. @xcite ) . that ratio of 10 between the average brightness of afterglows with breaks and of those without breaks until 10 days and the above - derived @xmath154 imply that the latter type of afterglows should display a break at 3100 days , which could be missed if monitoring does not extend for sufficiently long times . for this reason , some of the dimmer x - ray afterglows detected by swift may have breaks that are too late to be observed , leading to an apparent paucity of swift x - ray afterglows with jet - breaks , as noted by @xcite . the long - time monitoring of radio afterglows showed that often there is an incompatibility between the radio and optical flux decays . after the peak of the forward - shock synchrotron spectrum falls below the radio domain , which should happen within @xmath0 days and is , indeed , observed in the radio spectra of grb afterglows 970508 @xcite , 021004 , and 030329 ( figure [ radiopk ] ) , the radio and optical flux decays are expected to be similar , up to a difference @xmath156 that could occur if the cooling break is in between radio and optical and if the jet is not laterally spreading . in a set of nine pre - swift afterglows with long temporal coverage at both frequencies , i find that the above expectation is met by only four : grb afterglows 980703 , 970508 , 000418 , and 021004 , but that the radio flux of grb afterglows 991208 , 991216 , 000301c , 000926 , and 010222 decay much slower than in the optical , with @xmath157 , respectively @xcite ( see also @xcite ) . for all the above three cases for which the optical and radio decays are well - coupled , the decays are slower than @xmath158 , indicating a wide jet , while for all the five cases of decoupled radio and optical decays , the optical displays a decay steeper than @xmath158 after a @xmath159 day break that could be interpreted as a jet - break . thus , whenever the optical flux decays fast , there seems to be a mechanism which produces radio emission in excess of that expected for the forward - shock model . an example of each type of radio afterglow is shown in figure [ or ] . because the slow radio flux decay is observed _ at the same time _ as the faster optical decay , the decoupling of radio and optical light - curves can not be attributed to energy injection in the blast - wave , to a structured outflow , or to evolving micro - physical parameters ( mechanisms which have also been used to explain the slow early decays seen in swift x - ray afterglow plateaus ) , nor to the transition to non - relativistic dynamics . instead , the decoupled radio and optical light - curve decays may indicate that these emissions arise from different parts of the relativistic outflow . that could happen if the outflow endowed with angular structure ( in the sense that its kinetic energy per solid angle is anisotropic ) , with a core that dominates the optical emission , yielding the jet - break , and an outer , wider envelope that produces the radio emission . the problem with this model @xcite is that , optical and radio afterglows being long - lived , the spectral break frequencies of the core and envelope emissions evolve substantially , making it impossible for their emissions to be dominant over such long timescales at only one frequency , i.e. without `` interfering '' with the emission of the other part of the outflow . shortly put , it is quite likely that the emission from the radio envelope would soon dominate the optical emission from the core and change the initially steep optical flux decay into a slower one . ( while that is a general issue for explaining decoupled afterglow light - curves with a structured outflow , @xcite shows that it can be avoided for grb 080319b , whose optical and x - ray light - curve decays are decoupled for until 1 day ) . another possibility is that the optical emission arises in the forward shock while the radio is from the reverse shock . for adiabatic cooling , the ejecta emission should decay slowly , even when observations are at a frequency below that of the spectral peak , thus a reverse shock energizing the ejecta is required to account for the flat or slowly rising part of radio light - curves ( up to about 10 days ) . in this case , the light - curve decay depends on the law governing the injection of fresh ejecta into the reverse shock , the observed radio light - curve indices being close to the closure relations derived in the previous section for a uniform radial distribution of the incoming ejecta mass . in the reverse - forward shock model for afterglows with different radio and optical light - curve decays , the cross - interference issue may also exist , as the forward - shock synchrotron peak flux , being larger than @xmath160 mjy at 1 day ( to account for the @xmath161 20 magnitude optical flux ) , could over - shine the reverse - shock radio emission at some later time , when the peak energy of the forward - shock emission spectrum reaches the radio domain . this issue is best addressed with numerical calculations of the blast - wave dynamics and radiation . in this way , i found @xcite that the most likely solution for the decoupled radio and optical light - curves is that the radio afterglow emission is dominated by the reverse shock during the first decade in time , with the forward - shock emission peaking in the radio at about 100 days , overtaking that from the reverse shock sometime during the second decade . by itself , each component would display a decay faster than observed , but their sum resembles a shallow power - law over two decades in time . the swift satellite has opened a new temporal window for observations of x - ray afterglows , which previously were monitored by bsax only after 8 hours after trigger . the major surprise ( i.e. a feature not predicted ) in swift observations was that , although it appeared that the x - ray afterglow emission at hours and days extrapolated back to the burst time would match the grb flux , implying a smooth transition from counterpart to afterglow emission , the x - ray flux from burst end to several hours is much less than that back - extrapolation , displaying at 0.310 ks a phase of slow decay , with @xmath162 . in fact , that should have been a partial surprise because bsax has observed a sharply decaying grb tail in at least three cases , indicating that a phase of slow x - ray decay must exist at the burst end . figure [ 0315 ] illustrates the `` plateau '' phase observed for grb afterglow 050315 . in the simplest form of the blast - wave model , the magnitude of the light - curve decay steepening at the end of the plateau requires the peak of the synchrotron spectrum to fall below the x - ray band at the end of the plateau . however , that explanation is ruled out by that , observationally , the plateau end is most often not accompanied by the a spectral evolution @xcite , although exceptions exist @xcite . because the plateau phase is followed by a `` normal '' decay , compatible with the expectations of the standard forward - shock model , it is natural to think that departures from the assumptions of that standard model are the cause of x - ray plateaus : ( 1 ) increase of the average energy per solid angle of the blast - wave area visible to the observer by means of ( 1a ) energy injection in the blast - wave owing to some late ejecta catching - up with the forward - shock @xcite , or by absorbing low - frequency electromagnetic radiation from a millisecond pulsar @xcite , ( 1b ) an anisotropic outflow @xcite , ( 2 ) evolving shock micro - physical parameters @xcite , and ( 3 ) blast - wave interacting with an `` altered '' ambient medium , shaped by a grb precursor @xcite . the effect on the afterglow flux decay of the above mechanisms for a variable `` apparent '' kinetic energy of the blast - wave were first investigated by @xcite , the x - ray plateaus discovered by swift several years later providing the first tentative confirmation that those mechanisms may be at work . however , the discovery of _ chromatic _ light - curve breaks at the end of the x - ray plateau @xcite , which are not seen in the optical as well , soon showed that neither of the above mechanisms for x - ray plateaus provide a complete picture of the afterglow phenomenon , as in all those models the break should be _ achromatic _ , manifested at all frequencies . evolution of shock parameters for electron and magnetic field energies could `` iron out '' the optical light - curve break produced by the other mechanisms listed above , provided that the cooling frequency is between optical and x - ray ( to allow a way of decoupling the optical and x - ray light - curves ) , however there is no reason for their evolution to conspire and hide the optical light - curve break so often ( universality of the required micro - physical parameter evolutions with blast - wave lorentz factor would provide some support to this contrived model ) . to date , i find that there are 11 good cases of chromatic x - ray breaks , 6 good cases of achromatic breaks , and 3 afterglows displaying well - coupled optical and x - ray light - curves : a single power - law decay , of same decay index at both frequencies , extending over over three decades in time . figure [ xbr ] shows two examples of achromatic breaks , one with discrepant post - break optical and x - ray decay indices , and one chromatic x - ray break . to explain the chromatic x - ray breaks , @xcite and @xcite have proposed that the _ entire _ afterglow emission is produced by the reverse shock and have shown that , by placing either the injection or the cooling frequency between optical and x - ray , decoupled light - curves can be obtained . just as energy injection in the blast - wave , this mechanism relies on the existence of a long - lived central engine that expels ejecta until the last afterglow measurement ( the existence of a reverse shock at days is also required by the slow decays seen in a couple of radio afterglows ) . on the other hand , the transition observed in two optical afterglows from a fast - decaying phase to one of slower decay at 1 ks , accompanied by spectral evolution , argue in favour of the reverse - shock emission being dominant only up to 1 ks , after which it seems more natural to attribute the afterglow emission to the forward - shock . furthermore , the results shown by @xcite show a softening of the reverse - shock optical spectrum at the transition from fast to slow decay , which is in contradiction with the hardening observed in grb afterglows 061126 and 080319b . thus , the reverse - shock model may not provide a correct description of the entire afterglow emission . on average , the x - ray to optical flux ratio is larger for afterglows with chromatic x - ray breaks than for afterglows with coupled optical and x - ray light - curves ( i.e. with achromatic breaks or single power - law decays ) , as shown in figure [ fxfo ] . this indicates that chromatic x - ray breaks are due to a mechanism whose emission over - shines an underlying one only in the x - rays , but not in the optical . thus , it seems that the diversity of optical vs. x - ray light - curve behaviours should be attributed to the existence of two mechanisms for afterglow emission and not to a unique origin . so far , three proposals along that line have been put forth : dust - scattering of the blast - wave emission , bulk and inverse - compton scattering of the same emission , and a central - engine mechanism that could be the same internal shocks in a variable wind that are believed to produce the prompt burst emission @xcite . @xcite have proposed that x - ray plateaus result from scattering by dust in the host galaxy , much like the expanding rings produced by dust - scattering in our galaxy in grb afterglows 031203 @xcite and 050713a @xcite . however , for dust - scattering , harder photons are those scattered at a smaller angle , thus they arrive earlier at observer , leading to a strong spectral softening of the x - ray light - curve plateau , of @xmath163 , and to a strong dependence of the plateau duration on the photon energy , @xmath164 , both of which are clearly refuted by afterglow observations @xcite . @xcite have proposed that , in some afterglows , the `` central engine '' makes a substantial contribution to the afterglow x - ray flux . this model requires a central engine that operates until the last afterglow detection ; the dissipation mechanism may be shocks in a variable outflow , which can also account for the bright and short - lived flares observed in many swift x - ray afterglows ( e.g. @xcite ) . given that the forward - shock model with cessation of energy injection at the plateau end can explain the achromatic breaks and that the standard forward - shock model accounts for the coupled single power - law light - curves , it would be more desirable to identify a mechanism for producing decoupled x - ray and optical light - curves that is related to the forward shock and which dominates its emission only occasionally . bulk and inverse - compton scattering of the forward - shock photons by an outflow interior to the blast - wave is such a mechanism . in this model @xcite , all the afterglow emission originates in the forward shock , which explains so naturally the long - lived , power - law decay of grb afterglows , coupled optical and x - ray light - curves resulting when the scattered emission is dimmer than the forward - shock s , while chromatic x - ray light - curve breaks occur when the scattered emission is dominant in the x - ray . for this model to work , the scattering outflow must be almost purely leptonic , to ensure a sufficiently high ( sub - unity ) optical depth to electron scattering to account for the observed x - ray flux , assuming that the kinetic energy of the scattering outflow is not much larger than that of the forward shock . the same outflow also injects energy into the blast - wave and , if that energy is larger than the forward shock s , then it mitigates the blast - wave deceleration , producing a light - curve plateau ending with an achromatic break when the injected energy falls below that of the forward shock and stops being dynamically important . therefore , a delayed outflow is the origin of both chromatic and achromatic light - curve breaks , the former occurring when the scattered emission is dominant , while the latter happening when the forward - shock emission is dominant . still , the achromatic break of grb 050730 ( figure [ xbr ] ) , followed by an x - ray flux decay much steeper than that of the optical can not be explained with only cessation of energy injection at the time of the break , and requires an extra feature , that the shock micro - physical parameters are not constant . the above scattering model also explains late x - ray flares , which arise from dense or hot ( i.e. with relativistic electrons ) sheets within the outflow . when dominant , the scattered x - ray emission received at time @xmath9 reflects the properties ( density , lorentz factor ) of the outflow at @xmath165 behind the forward shock , sharp drops of the x - ray flux as that observed for grb afterglow 070110 at 30 ks being due to a gap in the scattering outflow . for an instantaneous release of all the ejecta , the kinematics of outflow radial - spreading owing to different initial lorentz factors , followed by deceleration of the forward - shock , leads to that , when the ejecta of lorentz factor @xmath166 catch up with the forward shock , moving at @xmath167 , the lorentz factor contrast is @xmath168 ( i.e. 2 for a homogeneous medium and @xmath169 for a wind ) . that ratio is too small for bulk - scattering to boost enough the forward - shock emission to dominate that arriving directly from the forward shock . but , if the scattering outflow was energized by internal shocks , inverse - compton scatterings by relativistic electrons ( of comoving frame energy @xmath170 ) can achieve that goal . in fact the properties of the scattered emission depend only on the product @xmath171 . thus , for a sudden release of ejecta to lead to a sufficiently bright scattered emission , the scattering outflow should be hot . alternately , if the scattering outflow is cold , then the larger ratio @xmath172 necessary for the scattered flux to over - shine that from the forward shock requires a long - lived engine . as a general test of all models that explain decoupled optical and x - ray afterglows by attributing them to different mechanisms , there should not be any afterglows whose optical and x - ray light - curves evolve from decoupled ( i.e. with a chromatic break ) to coupled ( i.e. with an achromatic break ) , or vice - versa , unless one of the light - curves displays a sudden flux or spectral change that would indicate a second mechanism becoming dominant . so far , i find only two cases of afterglows whose optical and x - ray light - curves evolve from decoupled to coupled ( grb afterglows 070110 and 080319b ) , but their x - ray light - curves display , indeed , a sharp drop ( at 20 ks , in both cases ) . the temporal and spectral properties of grb afterglows are , in general , consistent with those predicted for the synchrotron emission from the blast - wave produced when highly relativistic ejecta ( initial lorentz factor above 100 ) interact with the ambient medium . as expected from shocks accelerating particles with a power - law distribution with energy , power - laws are observed in the optical and x - ray afterglow continua . a spectral softening ( i.e. decrease of peak frequency ) is expected owing to the blast - wave deceleration and is observed in radio afterglows spectra and in the behaviour of afterglow radio light - curves , which rise slowly until the synchrotron spectrum peak reaches the radio domain and fall - off afterward . rising afterglow light - curves are seen at early times in the optical , and are consistent with the pre - deceleration emission from the forward shock , although a structured outflow with a hot - spot that gradually becomes visible to the observer is also possible , in which case the rising afterglow emission could also be explained with the reverse shock . much more often , afterglow light - curves display power - law decays of indices that are not correlated with the spectral slopes , as would be expected for the external - shock model . that inconsistency could be due to a substantial intrinsic scatter in decay indices and spectral slopes , owing to more than one variant of the blast - wave model occurring in grb afterglows , combined with a small range of those indices being realized . bright optical flashes accompanying the burst emission were predicted to arise from the reverse shock , owing to the larger number of ejecta electrons than in the forward shock . fast - falling optical light - curves have been observed at 1001000s in two afterglows ( 991023 and 021211 ) , followed by a slower of the decay , which was taken as evidence for the reverse - shock emission dominating the early afterglow , although spectral information was not available to test that hypothesis . more recently , the early optical spectral slopes were measured for two afterglows ( 061126 and 080319b ) , the decay of the former being consistent with that from adiabatically - cooling ejecta , while the later is faster than expected and consistent with it being the large - angle emission released at an earlier time . for the above two optical afterglows with spectral information at early times , the slowing of the optical flux decay at 1 ks is accompanied by a spectral evolution , which indicates the transition from one mechanism to another . most naturally , that is the transition from ejecta emission to forward - shock emission , with the reverse - shock emission being relevant for the optical afterglow only during its early phase . evidence for a reverse - shock emission is also provided by the slow radio flux decays observed after 10 day in several afterglows . adiabatically cooling ejecta would yield a decay faster than observed , particularly if the synchrotron cooling frequency were to fall below the radio , thus a reverse - shock accelerating ejecta electrons is required to operate for days and produce a radio emission decaying much slower than the optical at the same time , the latter being attributed to the forward shock . together with the above conclusion regarding the contribution of the reverse shock to the early optical afterglow , this suggests that the reverse shock is the main afterglow source for a duration that decreases with observing frequency , perhaps never being dominant in the x - rays and having no connection with the chromatic x - ray light - curve breaks seen at @xmath0 ks in most afterglows . that the x - ray - to - optical flux - ratio is larger ( by a factor 5 ) for afterglows with chromatic x - ray light - curve breaks than for those with coupled optical and x - ray light - curves ( i.e. with achromatic breaks or similar power - law decays ) , indicates the existence of a different mechanism producing the x - ray emission of afterglows with chromatic x - ray breaks , coupled light - curves resulting when the emission from that novel mechanism is negligible . long - lived internal shocks or scattering of the blast - wave emission by an outflow located behind it are two possibilities that could explain chromatic x - ray breaks , as well as the flares seen in many x - ray light - curves . the latter mechanism is also related to energy injection in the blast - wave , whose cessation accounts naturally for achromatic light - curve breaks . thus , in the scattering model , the diversity of optical and x - ray light - curve relative behaviours is attributed to the interplay between the scattered and direct blast - wave emissions , combined with the changing dynamics of the blast - wave produced when the scattering outflow brings into the shock more energy than already existing . on energetic grounds , grb outflows should be collimated into jets narrower than 10 degrees . the steepening of the afterglow flux decay when the jet boundary becomes visible to the observer was another major prediction confirmed by observations . just as for the pre jet - break phase , more than one jet model is required to account for the measured decay indices ( given the observed spectral slopes ) . swift x - ray afterglows display light - curve jet - breaks less often than pre - swift optical afterglows , which could be due to that the former afterglows ( being dimmer ) arise from wider jets whose jet - breaks occur later and could , thus , be missed more often .
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+ i discuss some theoretical expectations for the synchrotron emission from a relativistic blast - wave interacting with the ambient medium , as a model for grb afterglows , and compare them with observations . an afterglow flux evolving as a power - law in time , a bright optical flash during and after the burst , and light - curve breaks owing to a tight ejecta collimation are the major predictions that were confirmed observationally , but it should be recognized that light - curve decay indices are not correlated with the spectral slopes ( as would be expected ) , optical flashes are quite rare , and jet - breaks harder to find in swift x - ray afterglows . the slowing of the early optical flux decay rate is accompanied by a spectral evolution , indicating that the emission from ejecta ( energized by the reverse shock ) is dominant in the optical over that from the forward shock ( which energizes the ambient medium ) only up to 1 ks .
however , a long - lived reverse shock is required to account for the slow radio flux decays observed in many afterglows after @xmath0 day .
x - ray light - curve plateaus could be due to variations in the average energy - per - solid - angle of the blast - wave , confirming to two other anticipated features of grb outflows : energy injection and angular structure .
the latter is also the more likely origin of the fast - rises seen in some optical light - curves . to account for the existence of both chromatic and achromatic afterglow light - curve breaks
, the overall picture must be even more complex and include a new mechanism that dominates occasionally the emission from the blast - wave : either late internal shocks or scattering ( bulk and/or inverse - compton ) of the blast - wave emission by an outflow interior to it .
address= isr-1 , los alamos national laboratory , los alamos , nm 87545 , usa
| 14,631 | 496 |
when monte carlo methods are used to compute the spectra of astronomical sources , it is advantageous to work with _ indivisible _ monochromatic packets of radiant energy and to impose the constraint that , when interacting with matter , their energy is conserved in the co - moving frame . the first of these constraints leads to simple code and the second facilitates convergence to an accurate temperature stratification . for a static atmosphere , the energy - conservation constraint automatically gives a divergence - free radiative flux even when the temperature stratification differs from the radiative equilibrium solution . a remarkable consequence is that the simple @xmath0-iteration device of adjusting the temperature to bring the matter into thermal equilibrium with the monte carlo radiation field results in rapid convergence to the close neighbourhood of the radiative equilibrium solution ( lucy 1999a ) . an especially notable aspect of this success is that this temperature - correction procedure is geometry - independent , and so these methods readily generalize to 2- and 3-d problems . for an atmosphere in differential motion , the energy - conservation constraint yields a radiative flux that is rigorously divergence - free in every local matter frame . determining the temperature stratification by bringing matter into thermal equilibrium with such a radiation field - i.e. , by imposing radiative equilibrium in the co - moving frame - is an excellent approximation if the local cooling time scale is short compared to the local expansion time scale . this condition is well satisfied for the spectrum - forming layers of supernovae ( sne ) and of hot star winds ( klein & castor 1978 ) . the constraint that the energy packets be indivisible is advantageous from the point of view of coding simplicity . the interaction histories of the packets are then followed one - by - one as they propagate through the computational domain , with there being no necessity to return to any of a packet s interactions in order to continue or complete that interaction . this is to be contrasted with a monte carlo code that directly simulates physical processes by taking its quanta to be a sampling of the individual photons . absorption of a monte carlo quantum is then often followed by the emission of several quanta as an atom cascades back to its ground state . multiple returns to this interaction are then necessary in order to follow the subsequent paths of each of these cascade quanta . the resulting coding complexity is of course compounded by some of these quanta creating further cascades . although coding simplicity argues strongly for indivisible packets , a counter argument is the apparent implied need to approximate the treatment of line formation . thus , in monte carlo codes for studying the dynamics of stellar winds ( abbott & lucy 1985 ; lucy & abbott 1993 ) or for synthesizing the spectra of sne ( lucy 1987 ; mazzali & lucy 1993 ) , the integrity of the packets could readily be maintained since lines were assumed to form by coherent scattering in the matter frame . but significantly , an improved sn code has recently been described ( lucy 1999b ) in which branching into the alternative downward transitions is properly taken into account without sacrificing indivisibility . accordingly , an obvious question now is whether monte carlo techniques can be developed that enforce energy - packet indivisibility and yet do not have to adopt _ any _ simplifications with regard to line formation . if this can be achieved , then monte carlo codes for general nlte transfer problems become feasible . as discussed in sect . 1 , it is common in monte carlo transfer codes to quantize radiation into monochromatic energy packets . but matter is not quantized , neither naturally into individual atoms nor artificially into parcels of matter . instead , the continuum description of matter is retained , with macroscopic absorption and scattering coefficients governing the interaction histories of the energy packets . nevertheless , it now proves useful to imagine that matter is quantized into _ macro - atoms _ whose properties are such that their interactions with energy packets asymptotically reproduce the emissivity of a gas in statistical equilibrium . but these macro - atoms , unlike energy packets , do not explicitly appear in the monte carlo code . as conceptual constructs , they facilitate the derivation and implementation of the monte carlo transition probabilities that allow in an accurate treatment of line formation . the general properties of macro - atoms are as follows : \1 ) each macro - atom has discrete internal states in one - to - one correspondence with the energy levels of the atomic species being represented . \2 ) an inactive macro - atom can be activated to one of its internal states @xmath1 by absorbing a packet of kinetic energy or a packet of radiant energy of an appropriate co - moving frequency . \3 ) an active macro - atom can undergo an internal transition from state @xmath1 to any other state @xmath2 without absorbing or emitting an energy packet . \4 ) an active macro - atom becomes inactive by emitting a packet of kinetic energy or a packet of radiant energy of an appropriate co - moving frequency . \5 ) the de - activating packet has the same energy in the macro - atom s frame as the original activating packet . figure 1 illustrates these general rules . an inactive macro - atom , with internal states shown schematically , encounters a packet of energy @xmath3 and is activated to one of these states . the active macro - atom then undergoes two internal transitions before de - activating itself by emitting a packet of energy @xmath3 . subsequently , energy packets will in general be referred to as @xmath4-packets but also as @xmath5- or @xmath6-packets when specifying their contents to be radiant or kinetic energy , respectively . 2 , the concept of a macro - atom was introduced by stating some general properties concerning its interaction with @xmath4-packets . the challenge now is to derive explicit rules governing a macro - atom s activation , its subsequent internal transitions , and its eventual de - activation . asymptotically , the result of obeying these rules must be the emissivity corresponding to statistical equilibrium . for the moment , we drop the notion of a macro - atom and consider a real atomic species interacting with its environment . let @xmath7 denote the excitation _ plus _ ionization energy of level @xmath1 and let @xmath8 denote the radiative rate for the transition @xmath9 . the rates per unit volume at which transitions into and out of @xmath1 absorb and emit radiant energy are then @xmath10 respectively , where @xmath11 . note the summation convention adopted for the suffix @xmath12 , which ranges over all levels @xmath13 , including those of lower ions . similarly , below , the suffix @xmath14 implies summation over all levels @xmath15 , including those of higher ions . the corresponding rates at which kinetic energy is absorbed from , or contributed to , the thermal pool by transitions to and from level @xmath1 are @xmath16 where @xmath17 is the collisional rate per unit volume for the transition @xmath9 . if we now define the total rate for the transition @xmath9 to be @xmath18 , then the net rate at which level @xmath1 absorbs energy is @xmath19 this is an identity that follows directly from the defining eqs.(1 ) and ( 2 ) ; it is therefore quite general and does not assume statistical equilibrium . we now assume that the level populations @xmath20 are in statistical equilibrium . for level @xmath1 , this implies that @xmath21 a useful alternative representation of statistical equilibrium is obtained by multiplying eq.(4 ) by @xmath7 and then eliminating the term @xmath22 using eq.(3 ) . the result can be written in the form @xmath23 eq.(4 ) , the conventional equation of statistical equilibrium , balances the rates at which basic atomic processes excite and de - excite level @xmath1 . as such , it directly relates to nature s quantization of radiation into photons and of matter into atoms . in contrast , eq.(5 ) , though mathematically equivalent , deals with macroscopic energy flow rates in a finite volume element . these flows can now be quantized into indivisible @xmath4-packets . moreover , we can think of the volume element as a macro - atom with discrete energy states . eq.(5 ) expresses the fact that in statistical equilibrium the contribution from level @xmath1 to the energy content of unit volume is stationary . in consequence , the net rate at which level @xmath1 gains energy - the right - hand side of eq.(5 ) - equals the net rate of loss - the left - hand side . but the importance here of eq.(5 ) lies in the various terms contributing to gains and losses by level @xmath1 and their relevance for constucting transition rules for macro - atoms . the net rate of gain comprises the expected absorption terms @xmath24 and @xmath25 plus the terms @xmath26 and @xmath27 that clearly represent energy flows into @xmath1 from upper and lower levels . similarly , the net rate of loss comprises the expected emission terms @xmath28 and @xmath29 plus the terms @xmath30 and @xmath31 representing energy flows out of @xmath1 to upper and lower levels . the above remarks imply definitive values for the energy flows between level @xmath1 and other levels . but this is not true . if eq.(4 ) is rewritten as @xmath32 then comparison with eq.(5 ) shows immediately that an arbitrary quantity of energy @xmath33 may be added to @xmath7 and @xmath34 without invalidating this equation . but this merely shifts the zero point of the energy scale for excitation and ionization , which we are always free to do . nevertheless , this freedom implies a corresponding indefiniteness in the energy flow rates between levels . notwithstanding this indefiniteness , we now interpret eq.(5 ) in terms of macro - atoms absorbing and emitting @xmath4-packets or undergoing transitions between internal states . in this interpretation , the terms @xmath24 and @xmath25 obviously represent the activation of macro - atoms to state @xmath1 due to the absorption of @xmath5-packets and of @xmath6-packets , respectively . now consider an ensemble of active macro - atoms in state @xmath1 . for this ensemble to reproduce the behaviour of the real system , the relative numbers of the macro - atoms that subsequently de - activate with the emission an @xmath5- or @xmath6-packet or which make a transition to another internal state must be proportional to the relative values of the corresponding terms on the left - hand side of eq.(5 ) . accordingly , for an individual macro - atom in state @xmath1 , the probabilities that it de - activates with the emission of an @xmath5-packet or a @xmath6-packet are @xmath35 where @xmath36 similarly , the probabilities that it makes an internal transition to _ particular _ upper or lower states are @xmath37 unlike transition probabilities for real atoms , these analogues for macro - atoms depend on ambient conditions . consequently , in the course of a nlte calculation , they are iterated on just as are eddington factors in various other radiative tranfer schemes ( auer & mihalas 1970 ; hummer & rybicki 1971 ) . moreover , as with eddington factors , the monte carlo transition probabilities are dimensionless ratios that are likely to converge faster than do their dimensional numerators and denominators . when eq.(5 ) is summed over all energy levels , the energy flows between different levels cancel , giving @xmath38 thus , in statistical equilibrium , the energy stored in the form of excitation and ionization is stationary . for the macro - atoms , this is obeyed rigorously by each activation - de - activation event since the emitted packet s energy equals that of the absorbed packet - see figure 1 . monte carlo transition probabilities have been defined in sect . 3 , but their non - negativity was not established . of concern in this regard is stimulated emission when level populations are inverted . however , in anticipation of this issue , radiative rates were introduced without specifying whether stimulated emission contributes positively to @xmath8 or negatively to @xmath39 . we now exploit this flexibility in order to avoid negative probabilities . in the general case , inverted level populations may occur - i.e. , @xmath40 for some @xmath41 . in order to prevent the probabilities becoming negative when levels invert , stimulated emissions must be added to spontaneous emissions and _ not _ treated as negative absorptions . accordingly , for bound - bound ( b - b ) transitions , the radiative rates per unit volume are defined to be @xmath42 where @xmath43 and @xmath44 are the mean intensities averaged over the line s emission and absorption profiles - see mihalas ( 1978 , p78 ) . similarly , for free - bound ( f - b ) and bound - free ( b - f ) transitions , we define @xmath45 here @xmath46 and @xmath47 are the rate coefficients for spontaneous and stimulated recombinations to level @xmath1 , and @xmath48 is the uncorrected rate coefficient for photoionizations from level @xmath1 . each of these three quantities can be expressed as an integral over frequency involving the b - f absorption coefficient for an atom excited to level @xmath1 - see mihalas ( 1978 , pp130 - 131 ) . for collisions , a population inversion gives a negative rate if de - excitations are treated as negative excitations . this is avoided by defining @xmath49 with these expressions for the radiative and collisional rates , the probabilities defined by eqs.(7 ) and ( 9 ) are non - negative provided only that the @xmath7 s are non - negative . this latter condition is satisfied with the standard convention that the ground state of the neutral atom has zero excitation energy . because @xmath50 and therefore @xmath24 are here defined without correcting for stimulated emission , the macroscopic line- and continuum - absorption coefficients that determine the flight paths of @xmath5-packets must also be defined without this correction . this ensures a positive absorption coefficient even for a transition with a population inversion . if the monte carlo transition probabilities result in a macro - atom de - activating radiatively from state @xmath1 , the next step is to determine the frequency of the photons comprising the emitted @xmath5-packet . first we suppose that @xmath1 corresponds to a bound level . because @xmath51 and therefore @xmath28 here include stimulated emission , the process that radiatively de - activates the macro - atom may be either a spontaneous or a stimulated emission . the ratio of the probabilities of these alternatives is @xmath52 , where @xmath53 are the contributions to @xmath54 from spontaneous and stimulated emissions . knowing @xmath55 , we can choose between the two alternatives with a standard monte carlo procedure . thus , if @xmath56 is a random number from the interval @xmath57 , we select spontaneous emission if @xmath58 and stimulated otherwise . having thus decided the emission process , we must next choose a downward transition . for spontaneous line emission , the transition @xmath9 is selected with probability @xmath59 . for stimulated emission , on the other hand , the selection probability is @xmath60 . with the transition thus determined , the frequency @xmath61 of the @xmath5-packet is selected by randomly sampling the line s emission profile @xmath62 . thus , if @xmath56 again denotes a random number from @xmath57 , then @xmath61 is determined by the equation @xmath63 this equation can of course always be solved numerically for @xmath61 . however , elegant and efficient procedures for sampling standard profiles are available ( lee 1974a , b ) . now we consider a macro - atom that de - activates from a continuum state @xmath64 . in this case , the probabilities of spontaneous and stimulated emission are in the ratio @xmath65 , where @xmath66 are the contributions to @xmath67 from spontaneous and stimulated emissions . thus @xmath61 is selected by first deciding between spontaneous and stimulated emission and then randomly sampling the energy distribution of the chosen process s recombination continua . if the above selection procedure rules that an @xmath5-packet is emitted spontaneously , then a new direction of propagation is chosen in accordance with this process s isotropic emission . on the other hand , for stimulated emission , the new direction of propagation is that of the stimulating photon . thus , the new direction will be in solid angle @xmath68 at @xmath69 with probability @xmath70 , where @xmath61 is the frequency of the emitted @xmath5-packet . accordingly , a monte carlo code that treats stimulated emission separately must store a complete description of the radiation field - i.e. , @xmath71 . for problems where population inversions are not anticipated , we can usefully make the traditional assumption that lines have identical emission and absorption profiles and treat stimulated emissions as negative absorptions - see mihalas ( 1978 , p78 ) . the radiative rates for b - b transitions are then @xmath72 similarly , for f - b and b - f transitions , we define @xmath73 where the photionization coefficient is now corrected for stimulated recombinations . for collisions , the absence of population inversions allows us to treat de - excitations as negative excitations without the risk that eqs.(7 ) and ( 9 ) will give negative probabilities . accordingly , we now define @xmath74 this then implies that @xmath75 and therefore also @xmath76 for all @xmath1 . energy transfer from the radiation field to the thermal pool then occurs _ explicitly _ only via f - f absorptions . because @xmath77 and therefore @xmath24 are here defined with the correction for stimulated emission included , the macroscopic line- and continuum - absorption coefficients must also include this correction . in the posited absence of population inversions , these absorption coefficients are positive . because @xmath51 and therefore @xmath28 now exclude stimulated emission , the process that radiatively de - activates a macro - atom is always a spontaneous emission . if @xmath1 is a bound state , the frequency @xmath61 of the emitted @xmath5-packet is then decided as follows : the transition @xmath9 is selected with probability @xmath78 , and then @xmath61 is selected by randomly sampling this transition s emission profile , as in sect . 4.1.3 . for de - activation from a continuum state , @xmath61 is selected by randomly sampling the energy distribution of the spontaneous recombination continua . because the de - activating process is in this case spontaneous emission , the new direction of propagation is selected according to isotropic emission . thus , we now do not need to store @xmath71 . in fact , from the monte carlo radiation field generated at one iteration , we only require the mean intensities @xmath79 . these allow us to compute transition probabilities from eqs.(7 ) and ( 9 ) for use during the next iteration . the procedures described in sects . 4.1 and 4.2 apply to both static and moving media . but for some important problems involving moving media , a substantial speeding up of the calculation with negligible loss of accuracy is possible by applying sobolev s theory of line formation . in doing so , we take advantage of a small dimensionless quantity - the ratio of a line s doppler width to the typical flow velocity , which implies an essentially constant velocity gradient over the zone in which a given pencil of radiation interacts with a particular line . the monte carlo codes for hot star winds and sne cited in sect . 1 treat line formation in the sobolev approximation . the simplest case of this kind is that of homologous spherical expansion , as is commonly assumed for sne . this case will be treated here since it will be used in the test calculations of sect . generalization to a spherically - symmetric stellar wind is readily carried out by referring to castor & klein ( 1978 ) . we also assume no population inversions and so treat stimulated emissions as negative absorptions , as in sect . 4.2 . the radiative rates for b - b transitions are then @xmath80 here @xmath81 is the mean intensity at the far blue wing of the transition @xmath82 , and @xmath83 is the sobolev escape probability for this transition , given by @xmath84 \;\;\ ; , \ ] ] where @xmath85 , the transition s sobolev optical depth , is @xmath86 with @xmath87 being the elapsed time since the sn exploded . for f - b and b - f transitions , the rates are as in eq.(18 ) . for collisions , the rates are as in eq.(19 ) . the absorption of @xmath5-packets by lines is determined by the sobolev optical depths given by eq.(22 ) . absorption of an @xmath5-packet to the continuum is determined by the conventional macroscopic absorption coefficient corrected for stimulated emission . the frequency of an emitted @xmath5-packet is decided as follows : for de - activation from a bound state @xmath1 , the transition @xmath9 is selected with probability @xmath88 , where @xmath28 is evaluated with eq.(1 ) using the decay rates from eq.(20 ) , and the emitted packet is assigned frequency @xmath89 - i.e. , it is in the far red wing of a line whose emission profile is approximated by a delta function . the packet s next possible b - b transition is therefore with the next line to the redward of @xmath90 ( abbott & lucy 1985 ) . for de - activation from a continuum state , the new frequency is , as in sect . 4.2.3 , selected by randomly sampling the energy distribution of the spontaneous recombination continua . if an @xmath5-packet is emitted from a continuum state , the new direction of propagation is selected according to isotropic emission since the emission in this case is spontaneous . for de - activation from a bound state , the emission is also isotropic since , for homologous expansion , there is no kinematically - preferred direction . this is not true for a stellar wind . the monte carlo transition probabilities derived in sect . 3 are designed to reproduce asymptotically the emissivity of an atomic species whose level populations are in statistical equilibrium . to test this , we now consider one - point problems with specified and fixed ambient conditions . such tests sensibly precede application to a general nlte problem , for then the local ambient conditions are everywhere being adjusted iteratively as the global solution is sought . in the initial tests , the monte carlo transition probabilities are applied to the model fe ii ion with @xmath91 levels used previously ( lucy 1999b ) to investigate the accuracy of approximate treatments of line formation in sne envelopes . the energy levels of the fe ii ion and the f - values for permitted transitions were extracted from the kurucz bell ( 1995 ) compilation by m.lennon ( munich ) . einstein a - values for forbidden transitions are from quinet et al.(1996 ) and nussbaumer & storey ( 1988 ) . collision strengths , needed for sect.5.1.5 , are from zhang & pradhan ( 1995 ) and van regemorter ( 1962 ) . in the first fe ii test , we neglect collisional excitations and , as previously ( lucy 1999b ) , take the ambient radiation field determining the quantities @xmath81 in eq.(20 ) to be @xmath92 with @xmath93 and dilution factor @xmath94 , corresponding to @xmath95 . the density parameter is @xmath96 , and the time since explosion is @xmath97 . with parameters specified , this one - point statistical equilibrium problem - eq.(4 ) for @xmath98 levels plus a normalization constraint - is non - linear in the unknowns @xmath20 because the rate coefficients in eq.(20 ) depend on the @xmath20 through the sobolev escape probabilities . fortunately , repeated back substitutions give a highly accurate solution @xmath99 in @xmath100 10 iterations . with @xmath99 determined , the fe ii level emissivites @xmath28 and absorption rates @xmath24 can be computed from eq.(1 ) . we now test the monte carlo transition probabilities by seeing how accurately they reproduce these values @xmath28 . note that it is sufficient to test _ level _ emissivities since if these are exact so also are the line emissivities computed as described in sect . 4.3.3 . in the following monte carlo experiment , @xmath101 packets of radiant energy are absorbed and subsequently emitted by a macro - atom representing a macroscopic volume element of fe ii ions in the ambient conditions specified above . the energies of these packets are taken to be equal and given by @xmath102 , where @xmath103 . the calculation proceeds step - by - step as follows : \1 ) @xmath104 of the packets activate the macro - atom to internal state @xmath1 . \2 ) the transition probabilities @xmath105 , @xmath106 and @xmath107 for a macro - atom in state @xmath1 are computed from eqs.(7 ) and ( 9 ) . \3 ) the transition probabilities sum to one , so each corresponds to a segment @xmath108 of the interval ( 0,1 ) . a particular transition is therefore selected by computing a random number @xmath56 in ( 0,1 ) and finding in which segment it falls . \4 ) if the selected transition is the de - activation of the macro - atom , we update @xmath109 to @xmath110 and then return to step 3 ) to process the next activation of state @xmath1 , or to step 2 ) to process the first of the packets that activate the macro - atom to state @xmath111 . \5 ) if the selected transition is an internal transition to state @xmath2 , then we return to step 2 ) with @xmath2 replacing @xmath1 . \6 ) when all @xmath101 packets have been processed , the final elements of the vector @xmath109 are the estimates of the level emissivities @xmath28 as a single measure of the accuracy of the estimated level emissivities , we compute the quantity @xmath112 this is the mean of the absolute fractional errors of the @xmath109 when weighted by @xmath28 . figure 2 shows the values of @xmath113 , expressed as percentage errors , found in a series of trials with @xmath101 increasing from @xmath114 to @xmath115 . the values of @xmath113 decrease monotonically with increasing @xmath101 , falling to @xmath116 percent for @xmath117 . more importantly , the errors accurately follow an @xmath118 line , as expected if the only source of error are the sampling error at step 3 ) of the monte carlo experiments . accordingly , to the accuracy of these experiments , macro - atoms obeying the transition probabilities derived in sect . 3 do indeed reproduce the emissivity of a gas in statistical equilibrium . also included in fig.2 are values of @xmath113 obtained when the transition probabilities are computed with excitation energies @xmath7 increased by 5ev . this is to investigate the consequences of the dependence of the energy flow terms in eq.(5 ) - and therefore also of the transition probabilities - on the zero point of the scale of excitation energy . these results also track an @xmath118 line and so indicate that the predicted emissivities are asymptotically independent of the zero point . but since the open circles are marginally higher , there is an indication that increasing the zero point gives slighty less accurate emissivities at a given @xmath101 . in the monte carlo codes for hot star winds and sne cited in sect . 1 , line formation is treated approximately , with either resonant scattering or downward branching being assumed . for both assumptions , @xmath119 , corresponding to a macro - atom for which de - activation always immediately follows activation - i.e. , @xmath120 for all @xmath1 . in this case , as indicated on fig.2 , @xmath121 percent . thus , when the points in fig.2 drop below this value , the success must be due to the internal , radiationless transitions governed by the probabilities @xmath106 and @xmath107 . the above experiments show that despite the formidable complexity of its level structure the fe ii ion s reprocessing of radiation is accurately simulated by the monte carlo transition probabilities . nevertheless , from a computational standpoint , a remaining concern is how many internal transitions - or jumps - does this require ? to answer this , the number of jumps before de - activation was recorded for each absorbed packet in the @xmath117 trial and used to derive @xmath122 , the number of packets requiring @xmath2 jumps . from @xmath122 , we find that the expected number of jumps is @xmath123 and that the probability of immediate de - activation - i.e. , zero jumps - is @xmath124 . evidently , fears of numerous , time- consuming internal transitions are ill - founded . figure 3 is a logarithmic plot of @xmath122 . this reveals a power - law decline with increasing @xmath2 but with alternating deviations indicating that an even number of jumps before de - activation is favoured . a simple model suggests the origin of this curious behaviour . consider a 3-level atom with @xmath125 and suppose that level 2 is metastable with @xmath126 . because @xmath127 , the macro - atom can only be activated to state 3 ; and because @xmath128 , the macro - atom can not de - activate from state 2 . moreover , since @xmath129 , eq.(9 ) gives @xmath130 , and so state 1 of the macro - atom can not be reached . accordingly , following activation at state 3 , the macro - atom de - activates with probability @xmath131 or jumps to state 2 with probability @xmath132 , from whence it returns to state 3 with probability @xmath133 . it is now simple to prove that the probabilty of @xmath2 jumps before de - activation is @xmath134 if @xmath2 is even , and @xmath135 if @xmath2 is odd . the fe ii ion s numerous low - lying metastable levels are presumably playing the role of level 2 and thereby favouring an even number of jumps . histograms @xmath122 have also been computed for two other cases . first , the above trial was repeated with the @xmath7 s increased by 5ev as in sect . this change increases @xmath136 - to 4.54 - as expected since the probabilities @xmath106 and @xmath107 are thereby increased and @xmath105 correspondingly decreased . evidently , the standard choice of energy - level zero point leads to the most computationally - efficient set of transition probabilities . in the second case , @xmath137 is decreased from 0.5 to 0.067 , corresponding to @xmath138 . this change decreases @xmath136 - from 2.19 to 1.29 - as expected given the weakening of the radiative excitation rates . in the above experiment , the emission derives entirely from radiative excitation since collisions were neglected . now we consider the opposite extreme by setting the ambient radiation field to zero but including collisions . the only parameters of this test are the electron temperature and density , and these are assigned the values @xmath139 and @xmath140 . the resulting statistical equilibrium problem is linear and so solved without iteration . for this solution , accurate values of the level emissivities @xmath28 are again computed from eq.(1 ) . the next step is to derive estimates of the level emissivities by repeating the monte carlo experiment of sect.5.1.2 . the only changes needed are the following : first , since the solution has population inversions the general formulation of sect . 4.1 must be adopted to avoid negative probabilities . secondly , since a macro - atom is now always activated by a @xmath6-packet , their energies are taken to be @xmath141 , where @xmath142 . correspondingly , at step 1 ) of the experiment , @xmath143 . thirdly , since a macro - atom can now de - activate by emitting either an @xmath5- or a @xmath6-packet , only the former results in an updating of @xmath109 . the emission of a @xmath6-packet represents the return of energy @xmath3 to the therrmal pool . apart from these changes , the convergence experiment proceeds as in sects . 5.1.2 and 5.1.3 . the result is a plot similar to fig.2 , but with @xmath144 percent for @xmath117 . evidently , the monte carlo transition probabilities are equally applicable to problems where collisional excitation is a source of emission . although the fe ii experiments demonstrate the validity of the monte carlo transition probabilities , a test including b - f and f - b transitions is of interest . accordingly , a convergence experiment at one point in a sn s envelope has also been carried out for a 15-level model of the h atom , with level 15 being the continuum @xmath64 . the 14 bound levels correspond to principal quantum numbers @xmath145 , with each level having consolidated statistical weight @xmath146 . as for fe ii , the ambient radiation field incident on the blue wings of the b - b transitions is @xmath147 , but now with @xmath148 and @xmath149 . however , beyond the lyman limit , we assume zero intensity , so that photoionizations occur only from excited states . correspondingly , recombinations to @xmath150 are excluded on the assumption of immediate photoionization . collisional excitations and ionizations are neglected . the density parameter is @xmath151 , the electron temperature @xmath152 , and the time since explosion @xmath153 . with parameters specified , this non - linear statistical equilibrium problem can also be solved with repeated back substitutions , giving a highly accurate solution @xmath99 in @xmath154 iterations . with @xmath99 determined , monte carlo experiments as described in sect . 5.1.2 were carried out to test if level emissivities are also recovered in this case . in fig.4 , two such trials , with @xmath155 and @xmath156 , are compared with the exact solution . the results show that excellent agreement is achieved for @xmath157 . note in particular the success with @xmath158 , which is the rate of ionization energy loss due to recombinations , and with @xmath159 , whose very low value is due to the strong trapping of @xmath160 photons . thus far , a monte carlo procedure has been used to validate the transition probabilities developed in sect.3 . this has the advantage of following closely and therefore illustrating their use in realistic nlte calculations . but for feasible values of @xmath101 , sampling errors limit the accuracy of such tests . in order to test to higher precision , approximate level emissivities @xmath161 can be computed recursively according to the following scheme : @xmath162 where @xmath105 is the radiative de - activation probability from eq.(7 ) and @xmath163 is the increment at cycle @xmath5 to the summation approximating the rate at which level @xmath1 gains energy - i.e. the right - hand side of eq.(5 ) . this increment is derived from the previous increment by applying the transition probabilities from eq.(9 ) . thus @xmath164 and the recursion cycles are initiated by setting @xmath165 this procedure is now applied to the fe ii test problem of sect . as with that experiment , the accuracy of the vectors @xmath161 are measured by computing @xmath113 defined by eq.(23 ) . for @xmath166 , @xmath113 drops below the value 0.36 percent found in sect.5.1.3 with @xmath117 - see fig.2 . as the recursion procedure continues further , @xmath113 decreases monotonically until at @xmath167 it drops to a value of @xmath168 , at which point machine precision or accumulated roundoff errors halt further progress . this test clearly confirms and strengthens the earlier tests of the monte carlo transition probabilities . the experiments of sect.5 demonstrate that , when computed with the exact level populations @xmath99 , the monte carlo transition probabilities applied to indivisible @xmath4-packets reproduce the exact level emissivities as @xmath169 . but this success , though necessary , does not of itself imply that the technique will be successful when applied to nlte problems . for example , if the monte carlo emissivities were to undergo large changes in response to small changes in @xmath20 , then we would reasonably suspect that the iterations inevitably required for a nlte problem would converge very slowly - or might even diverge . on the other hand , if the emissivities are insensitive to changes in @xmath20 , then the prospects for successful applications are excellent . this crucial question of sensitivity can be investigated by repeating the calculations of fe ii emissivities reported in sect.5.1 , but with @xmath20 perturbed away from @xmath99 . a convenient way of doing this is to replace @xmath99 by the boltzmann distribution at excitation temperature @xmath170 . then , for given @xmath170 , the corresponding level emissivities @xmath109 are obtained from a monte carlo trial with @xmath171 packets , and so are negligibly affected by sampling errors ( cf . fig.2 ) . now , for the given @xmath170 , we can also compute @xmath28 , the level emissivities predicted by the fundamental formulae - eqs.(1 ) and ( 20 ) in this case . this represents the standard approach to nlte transfer problems whereby the radiation field is computed from the radiative transfer eq . ( rte ) with emissivity coefficients evaluated using the current estimates of @xmath20 . thus by comparing these two emissivity estimates @xmath109 and @xmath28 , we can see whether this monte carlo technique is potentally capable of yielding a superior estimate of the radiation field . in fig.5 , the quantities @xmath109 and @xmath28 obtained for @xmath172 are plotted against @xmath173 , the exact statistical equilibrium level emissivities - i.e. , the values corresponding to @xmath99 . remarkably , figure 5 shows that the monte carlo emissivities are far less sensitive to the departure of @xmath20 from @xmath99 than are the emissivities computed directly from the fundamental formula . for the most part , the @xmath109 are in error by @xmath174 dex , with little evidence of bias , while the @xmath28 are systematically offset by @xmath175 dex . to investigate whether this insensitivity is characteristic of the monte carlo procedure , the above test is now repeated with @xmath170 ranging from @xmath176 to @xmath177 and the resulting mean errors defined by eq.(23 ) plotted in fig.6 . we see that @xmath28 gives reasonably accurate emissivities only in the immediate neighbourhood of the minimum at @xmath178 . on the other hand , the values @xmath109 are accurate to @xmath179 dex across the entire range . the causes of these astonishing differences in sensitivity are of considerable interest . for @xmath28 , the strong sensitivity to @xmath170 is readily understood . because the sum @xmath180 , an error in the population of the emitting level translates directly into an error in @xmath28 . now consider @xmath109 . this quantity is determined by the rate at which active macro - atoms reach state @xmath1 , and this happens by direct absorptions of packets into this state or by transitions from other states . either way , the accuracy of the source vectors @xmath24 and @xmath25 is clearly fundamental to the accuracy of the vector @xmath109 . but the dominant contributors to the elements of these source vectors - see eqs.(1 ) and ( 2 ) - are transitions from the ground state and from low - lying metastable levels , and the estimated populations of these levels are unlikely to be seriously in error . in particular , with an assumed boltzmann distribution over excited states , the @xmath20 of these low levels is insensitive to @xmath170 and do not differ much from @xmath99 . in contrast , the populations of high levels are quite likely to be badly estimated and are acutely sensitive to @xmath170 . another way of appreciating the differences in these approaches to calculating emissivities is as follows . the monte carlo procedure applies only to a state of statistical equilibrium and , as such , constrains every level s emissivity to be consistent with the rates of processes populating that level . in contrast , the fundamental emissivity formula applies also to states out of statistical equilibrium and so takes no account of whether the levels populations can be maintained . accordingly , with this monte carlo technique , the principle of statistical equilibrium is _ incorporated _ ( approximately ) as the radiation field is being calculated . on the other hand , when emissivities are computed from the fundamental formula , any consideration of statistical equilibrium is effectively being deferred until the updated radiation field has been determined . the likely beneficial impact of this insensitivity on the iterations needed to derive nlte solutions is worth stressing . with the conventional rte approach , an erroneously overpopulated upper level @xmath1 pollutes the radiation field with spurious line photons at frequencies @xmath181 , and these are sources of excitation for level @xmath1 when level populations are next solved for . similarly , an erroneously overpopulated upper ion pollutes the radiation field with recombination photons that are subsequent sources of photoionization for the lower ion . to some degree , therefore , such errors are _ self - perpetuating _ and so are not rapidly eliminated . this persistency contributes to the slow convergence typical of nlte codes . in contrast , with the monte carlo approach , this pollution does not happen and so - for sufficiently large @xmath182 - a high quality radiation field is obtained immediately provided that the initial populations of the low - lying levels are estimated sensibly . the monte carlo transition probabilities allow statistical equilibrium to be incorporated into the calculation of radiation fields for nlte problems . moreover , this is achieved without imposing the constraint of radiative equilibrium . accordingly , in principle at least , the technique applies equally to problems with non - radiative heating , such as stellar chromospheres . in the absence of non - radiative heating , a nlte transfer problem must be solved subject to the constraint of radiative equilibrium . the incorporation of this _ additional _ constraint into the macro - atom formalism is readily understood . first suppose that collisional processes are neglected . the absorbed and the emitted @xmath4-packets are then always @xmath5-packets and they have identical energies - see fig.1 . thus , the constraint of radiative equilibrium is obeyed rigorously since it holds exactly for every activation - de - activation event , all of which are of the form @xmath183 , where @xmath184 denotes an active macro - atom . note also that since active macro - atoms do not appear spontaneously within the computational domain ( d ) , every monte carlo quantum s interaction history starts and ends as an @xmath5-packet crossing a boundary of d. now suppose that collisions are included . in this case , a macro - atom activated by an @xmath5-packet can de - activate itself by emitting a @xmath6-packet , so that radiative equilibrium no longer holds exactly for each individual activation - de - activation event . however , the emitted @xmath6-packet is re - absorbed _ in situ _ by another macro - atom and thereby ( eventually ) converted into an @xmath5-packet . since this has the same energy as the original @xmath5-packet , radiative equilibrium holds for every sequence of in situ events that starts with the absorption of an @xmath5-packet and ends with the next emission of an @xmath5-packet . a typical in situ sequence is @xmath185 . if such sequences are abbreviated as @xmath186 \rightarrow r$ ] , we see that the inclusion of collisions has not fundamentally changed the procedure and that radiative equilibrium is still rigorously obeyed . in the presence of non - radiative heating , the nlte problem is not subject to the additional constraint of radiative equilibrium . statistical equilibrium is incorporated with the macro - atom formalism as before , and the challenge now is to incorporate the creation of radiant energy within d due to the additional heating . this is accomplished by allowing for the spontaneous and random appearance within d of active macro - atoms with their number , locations and internal states @xmath1 all controlled by the collision source vector @xmath25 - cf . note that because this sampling of @xmath25 takes full account of the collisional creation of excitation , the emission of a @xmath6-packet is not now followed by its in situ re - absorption ; instead , the interaction history of that monte carlo quantum then ends and its energy is added to the thermal pool ( cf . sect.5.1.5 . ) . the radiation field generated by this procedure is not divergence - free but reflects the collisional creation of radiant energy due to an elevated temperature profile maintained by the non - radiative heating . the limited aim of this paper has been to see if monte carlo transfer codes whose quanta are indestructable energy packets can be constructed without resorting to simplified treatments of line formation . to this end , the concept of a macro - atom has been introduced and rules established governing its activation and de - activation as well as its transitions between internal states . these rules - the monte carlo transition probabilities - have been derived by demanding that the macro - atom s emission of @xmath5-packets asymptotically reproduces the local emissivity of a gas in statistical equilibrium ; and these rules validity has been confirmed with one - point test problems . evidently , the next step is to implement these transition probabilities in a code to solve a realistic nlte problem for a stratified medium and thus to investigate the practicality of this technique for problems of current interest . in a companion paper , a monte carlo nlte code treating the formation of h lines in a type ii sn envelope will be described and used to illustrate the convergence behaviour of iterations to obtain both the level populations and the temperature stratification . abbott d.c . , lucy l.b . 1985 , apj 288 , 679 auer l.h . , mihalas d. 1970 , mnras 149 , 65 hummer d.g . , rybicki g.b . 1971 , mnras , 152 , 1 klein r.i . , castor j.i . 1978 , apj 220 , 902 kurucz r.l . , bell b. , 1995 , kurucz cd - rom no . 23 lee j .- s . 1974a , apj 187 , 159 lee j .- s . 1974b , apj 192 , 465 lucy l.b . , 1987 , in : danziger i.j . eso workshop on sn 1987a , p. 417 1999a , a&a 344 , 282 lucy l.b . 1999b , a&a 345 , 211 lucy l.b . , abbott d.c . 1993 , apj 405 , 738 mazzali p.a . , lucy l.b . 1993 , a&a 279 , 447 mihalas d. , 1978 , stellar atmospheres ( 2nd ed . ) . w.h.freeman & co. , san francisco
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transition probabilities governing the interaction of energy packets and matter are derived that allow monte carlo nlte transfer codes to be constructed without simplifying the treatment of line formation .
these probabilities are such that the monte carlo calculation asymptotically recovers the local emissivity of a gas in statistical equilibrium .
numerical experiments with one - point statistical equilibrium problems for fe ii and hydrogen confirm this asymptotic behaviour .
in addition , the resulting monte carlo emissivities are shown to be far less sensitive to errors in the populations of the emitting levels than are the values obtained with the basic emissivity formula .
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based on low temperature resistivity and heat capacity measurements in applied magnetic fields ybagge was recently classified as a new heavy fermion material with long range , possibly small moment , magnetic order below 1 k @xcite that shows magnetic field induced non - fermi - liquid ( nfl ) behavior @xcite . the critical field required to drive ybagge to the field - induced quantum critical point ( qcp ) is anisotropic ( @xmath3 45 koe , @xmath4 80 koe ) and conveniently accessible by many experimental groups @xcite . ybagge is one of the _ rarae aves _ of intermetallics ( apparently only second , after the extensively studied ybrh@xmath5si@xmath5 @xcite ) a _ stoichiometric _ , yb - based , heavy fermion ( hf ) that shows magnetic field induced nfl behavior and as such is suitable to serve as a testing ground for experimental and theoretical constructions relevant for qcp physics . among the surfeit of detailed descriptions developed for a material near the antiferromagnetic qcp we will refer to the outcomes @xcite of two more general , competing , pictures : in one viewpoint the qcp is a spin density wave ( sdw ) instability @xcite of the fermi surface ; within the second picture that originates in the description of heavy fermions as a kondo lattice of local moments @xcite , heavy electrons are composite bound states formed between local moments and conduction electrons and the qcp is associated with the breakdown of this composite nature . it was suggested @xcite that hall effect measurements can help distinguish which of these two mechanisms may be relevant for a particular material near a qcp . in the sdw scenario the hall coefficient is expected to vary continuously through the quantum phase transition , whereas in the composite hf scenario the hall coefficient is anticipated to change discontinuously at the qcp . perhaps more importantly , in both scenarios a clear and sharp change in the field dependent hall effect ( for the field - induced qcp ) is anticipated to occur at low temperatures , near the critical field value . although hall effect measurements appear to be a very attractive method of gaining insight into the nature of the qcp , one has to keep in mind that an understanding of the different contributions to the measured hall coefficient , in particular in magnetic or strongly correlated materials , is almost inevitably difficult and potentially evasive @xcite . therefore measurements on samples well characterized by other techniques @xcite as well as comparison with non - magnetic as well as non - hf members of the same series can be beneficial . in this work we present temperature- and field- dependent hall effect measurements on ybagge single crystals . the non - magnetic member of the same ragge ( r = rare earth ) series , luagge , and the magnetic , essentially non - hybridizing , tmagge were used for common sense checks , or calipers , of the ybagge measurements . ybagge , luagge and tmagge single crystals in the form of clean , hexagonal - cross - section rods of several mm length and up to 1 mm@xmath6 cross section were grown from high temperature ternary solutions rich in ag and ge ( see @xcite for details of the samples growth ) . their structure and the absence of impurity phases were confirmed by powder x - ray diffraction . temperature and field dependent hall resistivity , @xmath7 , and auxiliary high field magnetization measurements were performed down to 1.9 k , in an applied magnetic field of up to 140 koe in a quantum design ppms-14 instrument . for ybagge hall measurements were extended down to 0.4 k using the he-3 option of the ppms-14 . a four probe , ac technique ( @xmath8 = 16 hz , @xmath9 = 1 - 0.1 ma ) , was used for the hall measurements . samples were polished down to a plate - like shape with thicknesses of 0.3 - 0.4 mm . pt leads were attached to the sample with epotek h20e silver epoxy so that the current was flowing along the crystallographic @xmath10 axis . for the @xmath11 case hall resistivity ( @xmath12 ) was measured in the hexagonal crystallographic plane ( approximately along @xmath13 $ ] direction ) with the magnetic field applied perpendicular to both the current and the hall voltage directions ( approximately along @xmath14 $ ] direction ) ( see the lower inset to figure [ rhlutm ] ) . in the @xmath2 case , current was flowing in the hexagonal plane , approximately in @xmath13 $ ] direction , the hall voltage was measured along @xmath14 $ ] direction . due to rod - like morphology of the crystals , samples that were cut and polished for @xmath2 measurements were smaller and the error bars in the absolute values ( due to geometry and position of the contacts ) are larger than for the @xmath11 data sets . to eliminate the effect of inevitable ( small ) misalignment of the voltage contacts , the hall measurements were taken for two opposite directions of the applied field , @xmath15 and @xmath16 , and the odd component , @xmath17 was taken as the hall resistivity . to determine the hall resistivity in the limit of low field , linear fits of the initial ( linear ) parts of the @xmath18 data in both quadrants were used . he-4 hall measurements for ybagge and luagge were performed on two samples of each material , the results were the same within the error bars in sample dimensions and contact position measurements . during the measurements particular care was taken to avoid rotation and/or misplacement of the tmagge sample due to its magnetic anisotropy . the field - dependent hall resistivity for luagge for @xmath11 is shown in the upper inset to fig . [ rhlutm](a ) for several temperatures . @xmath12 is only slightly non - linear in field over the whole temperature range . this minor non - linearity causes some difference in the @xmath19 _ vs. _ @xmath20 data obtained in different applied fields ( fig . [ rhlutm](a ) ) . the hall coefficient , @xmath21 , is measured to be negative . the overall temperature dependence is monotonic , slow and featureless with approximately a factor of two increase in the absolute value of @xmath19 from room temperature to low temperatures . this temperature - dependency of the hall coefficient of the non - magnetic material possibly reflects some details of its electronic structure ( for example , comparable factor of 2 changes in @xmath22 were recently observed in latin@xmath23 , t = rh , ir , co , @xcite ) . overall the temperature- and field - dependence of the hall coefficient for tmagge ( fig . [ rhlutm](b ) ) is similar to that of luagge with two main differences : ( i)the long range order and metamagnetism of tmagge @xcite is reflected in hall measurements as a low temperature decrease in @xmath24 and as anomalies in @xmath25 for @xmath26 2 k that are consistent with the fields of the metamagnetic transitions ; ( ii)the absolute values of the @xmath27 data for tmagge are a factor of 3 - 4 smaller than for luagge . the temperature dependent hall coefficient and the dc susceptibility data for ybagge with the same orientation of the magnetic field with respect to the crystallographic axis are shown in fig . [ rhmtyb ] . the susceptibility , @xmath28 , is field - independent above 50 k ( _ i.e. _ @xmath29 is linear below 140 koe in this temperature range ) and is similar to the data reported in @xcite . the hall coefficient , @xmath22 , is field - independent above approximately 25 k. the temperature dependencies of the susceptibility and the hall coefficient at high temperatures closely resemble each other . at low temperatures a field - dependent maximum in @xmath22 ( see inset to fig . [ rhmtyb ] ) is observed . qualitatively the temperature dependence of the hall coefficient is consistent with the picture presented in @xcite ( see also @xcite for a comprehensive review ) . within this picture the temperature dependence of the hall coefficient in heavy fermion materials is a result of two contributions : a residual hall coefficient , @xmath30 , and a hall coefficient due to the intrinsic skew scattering , @xmath31 . the residual hall coefficient is ascribed to a combination of the ordinary hall effect and residual skew scattering by defects and impurities and , to the first approximation , is considered to be temperature - independent , although , realistically , both the ordinary hall effect and the residual skew scattering may have weak temperature dependence . the temperature - dependent , intrinsic skew scattering contribution ( @xmath32 ) at high temperatures ( @xmath33 , where @xmath34 is the kondo temperature ) increases as the temperature is lowered in a manner that is mainly due to the increasing magnetic susceptibility . at lower temperatures @xmath32 passes through a crossover regime , then has a peak at a temperature on the order of the coherence temperature , @xmath35 , and finally , on further cooling rapidly decreases ( in the coherent regime of skew scattering by fluctuations ) to zero ( _ i.e. _ @xmath22 ultimately levels off to the @xmath36 value at very low temperatures @xcite ) . in the high temperature ( @xmath33 ) limit we can ( very roughly , within an order of magnitude ) separate these two contributions to the observed temperature - dependent hall coefficient using a phenomenological expression @xmath37 @xcite with the temperature - dependent skew scattering contribution written as @xmath38 where @xmath39 , @xmath40 is the curie constant , and @xmath41 is the weiss temperature . using @xmath42 = -15.1 k from @xcite we can plot @xmath43 _ vs. _ @xmath44 ( fig . [ rht ] ) and from the linear part of the curve we can estimate @xmath45 0.02 n@xmath46 cm / oe and @xmath47 -0.17 n@xmath46 cm it seems peculiar that our estimate of @xmath48 for ybagge differs noticeably from the hall coefficient measurements for luagge and tmagge ( see fig . [ rhlutm ] ) . regarding this discrepancy it should be mentioned that besides possible experimental ( mainly geometrical ) errors these three materials may have different residual skew scattering and , additionally , as indicated by the preliminary results of band structure calculations @xcite , the density of states at the fermi level can be considerably different for all three compounds under consideration . although the magnetic susceptibility , @xmath49 of tmagge above the nel temperature has a clear curie - weiss behavior @xcite , in contrast to the case of ybagge , the temperature dependence of the hall coefficient for tmagge ( fig . [ rhlutm](b ) ) does not have a similar functional form . the reason for this difference is apparently the very small skew scattering contribution ( @xmath50 ) to the hall coefficient in tmagge . similarly small couplings of local moment magnetism with the hall effect has been seen in other rare earth intermetallics , _ e.g. _ rni@xmath5b@xmath5c ( r = rare earth ) borocarbides @xcite . in order to further explore the low temperature behavior of the hall coefficient , measurements down to 0.4 k were performed . the results ( on a semi -_log _ scale ) are shown in fig . [ rhltyb ] . the data taken in applied fields of 75 koe and higher show the expected levelling off of the @xmath24 as @xmath51 . it is noteworthy that the measured value of @xmath52 is close to the aforementioned estimate of the residual hall coefficient . this agreement suggests that at the lowest temperatures the hall coefficient is dominated by @xmath48 and , barring the residual skew scattering contribution , can probe the concentration of the electronic carriers . whereas the higher field values of the hall coefficient vary smoothly with temperature ( fig . [ rhltyb ] ) , the low field data , below @xmath53 3 k , show large variations . although the signal to noise ratio in the low field measurements is inherently lower , these variations appear to be above the noise level ( fig . [ rhltyb ] , inset ) and the peaks slightly above 0.6 k and 1.0 k are understood as the signatures of the magnetic transitions in ybagge @xcite that are suppressed ( in this orientation ) when a 75 koe , or higher , magnetic field is applied . to further study the field - induced qcp in ybagge , field dependent hall resistance measurements were performed at different temperatures ( fig . [ rhhyb ] ) . although the theoretical constructions are usually formulated in terms of the hall _ coefficient _ , not hall _ resistivity _ , in the case of ybagge the magnetic field itself is a control parameter for the qcp that makes the proper definition of the hall coefficient ambiguous . we will continue presenting our data as hall resistivity , since it is a quantity unambiguously extracted from the measurements , and leave the discussion on the suitable definition of the hall coefficient for the appendix . for temperatures at and above @xmath5410 k , the @xmath25 behavior is monotonic and , at higher temperatures , eventually linear ( fig . [ rhhyb](b ) ) . this type of behavior has been observed in a number of different materials in the paramagnetic state @xcite . the low temperature evolution of the @xmath25 behavior is more curious ( fig . [ rhhyb](a ) ) and ought to be compared with the phase diagram obtained for ybagge ( @xmath11 ) in @xcite ( an augmented version of which is shown in fig . [ pd ] below ) . the lines in fig . [ rhhyb](a ) roughly connect the points according to the phase lines in @xcite ( see also fig . [ pd ] below ) . it can be seen that the lower @xmath55 magnetically ordered phase line possibly has ( despite the scattering of the points ) correspondent features in @xmath25 , and the coherence line in @xcite ( and fig . [ pd ] ) roughly corresponds to the beginning of the high field linear behavior in @xmath25 . on the other hand , the higher @xmath55 magnetically ordered phase line can not unambiguously be associated with any feature in @xmath25 curves . the most interesting feature shown in fig . [ rhhyb](a ) though is the presence of the pronounced peak , or local maximum , in @xmath25 that occurs at @xmath56 koe for the @xmath26 0.4 k curve and can be followed up to temperatures above long range magnetic order transition temperatures . for @xmath26 2.5 k a broad , local maximum in @xmath12 , centered at @xmath57 koe can just barely be discerned . as temperature is reduced this feature sharpens and moves down in field . for @xmath26 1 k the local maximum in @xmath12 is clearly located at @xmath58 koe and by @xmath59 k @xmath12 has sharpened almost to the point of becoming discontinuous with @xmath60 koe . the temperature dependence of @xmath61 is shown in fig . [ pd ] clearly demonstrating that as @xmath62 , @xmath63 for the qcp . independent of any theory these data clearly show that ( i ) @xmath12 is an extremely sensitive method of determining @xmath64 of qcp , ( ii ) @xmath61 has a clear temperature dependence , and ( iii ) the qcp influences @xmath12 up to @xmath65 k , a temperature significantly higher than the @xmath66 antiferromagnetic ordering temperature . the new phase line ( shown as stars in fig . [ pd ] ) associated with @xmath12 maximum is distinct from the lines inferred from @xmath67 and @xmath68 data @xcite . as @xmath51 this line approaches @xmath64 , but for finite @xmath20 it is well separated from the coherence line that was determined by the onset of @xmath69 resistivity behavior . this new @xmath61 line rather clearly locates @xmath64 at @xmath70 koe , the field at which the long range antiferromagnetic order appears to be suppressed . since the response of ybagge to an applied magnetic field is anisotropic @xcite , it is apposite to repeat the hall measurements for the magnetic field applied parallel to the crystallographic @xmath10-axis . the temperature - dependent hall coefficient taken in different applied fields is presented in fig . [ rhltybc ] ( the low - field data were obtained as described above ) . the @xmath24 behavior for @xmath2 is qualitatively similar to that for @xmath11 with a broad maximum being shifted to @xmath71 k ( as compared to @xmath72 k for @xmath11 ) and being less sensitive to the applied field . the low temperature , field - dependent hall resistivity for @xmath2 is shown in fig . [ rhhybc ] . in many aspects the overall behavior is similar to that for @xmath11 : there are no apparent features associated with the phase lines derived from magnetoresistance and specific heat measurements @xcite ( shown as lines in fig . [ rhhybc ] ) , however there is the presence of a pronounced minimum in @xmath25 that occurs at @xmath73 koe for the @xmath26 0.4 k curve and can be followed up to the temperatures well above the zero - applied - field magnetic transition temperatures . for @xmath26 2 k a broad , local minimum in @xmath12 , centered at @xmath74 koe can still be recognized and at @xmath75 k a local minima occurs just at the edge of our field range . as temperature is reduced this feature sharpens and moves down in field . the temperature dependence of @xmath76 is shown in fig . [ pdc ] clearly demonstrating that , akin to the @xmath11 case , as @xmath51 , @xmath77 for the qcp . the @xmath25 behavior for this orientation is more complex , and there is an additional , broad maximum in lower fields ( @xmath58 koe at 0.4 k ) that fades out with increasing temperature . this highly non - monotonic in field behavior is the origin of the dissimilarities in the low temperature @xmath24 data ( fig . [ rhltybc ] ) taken in different applied fields . the high field minimum in @xmath25 ( fig . [ rhhybc ] ) defines a new phase line ( shown as stars in fig . [ pdc ] ) which is clearly different from the lines inferred from @xmath67 and @xmath68 data @xcite . as @xmath51 this line approaches @xmath64 , but for finite @xmath20 it is well separated from the coherence line that was determined by the onset of @xmath69 resistivity behavior . for this orientation of the applied field this new @xmath76 line rather clearly locates @xmath64 at @xmath78 koe , the field at which the long range antiferromagnetic order appears to be suppressed . it should be noted that the new lines in the @xmath55 phase diagrams were established from different types of extrema in @xmath25 , _ maximum _ for @xmath11 and _ minimum _ for @xmath2 . we neither consider this difference as a reason for particular discomfort nor do we necessarily view it as a potential clue to deeper understanding of the nature of the field - induced qcp in this material . the preliminary band structure calculations @xcite on luagge , the non - magnetic analogue of the title compound , suggest that the members of the ragge series have a complex fermi surface consisting of multiple sheets . in such a case a change in the fermi surface may possibly have different signatures in the hall measurements with different field orientation . in addition , existing qcp models appear not to be at the level of considering different shapes and topologies of the fermi surfaces . whereas these new , @xmath79 lines on the @xmath80 phase diagrams ( figs . [ pd ] and [ pdc ] ) appear to be closely related with the qcp their detailed nature and temperature dependencies will require further experimental and theoretical attention . the temperature- and field - dependent hall resistivity have been measured for ybagge single crystals with @xmath11 and @xmath2 orientation of the applied magnetic field . the temperature dependent hall coefficient of ybagge behaves similarly to other heavy fermion materials . low temperature , field - dependent measurements reveal a local maximum ( @xmath11 ) or minimum ( @xmath2 ) in @xmath25 for @xmath65 k that occurs at a value that approaches @xmath81 koe ( @xmath11 ) and @xmath82 koe ( @xmath2 ) as @xmath51 . these data indicate that ( i ) the hall resistivity is indeed a useful measurement for the study of qcp physics and ( ii ) the influence of the qcp extends to temperatures significantly higher than the @xmath66 antiferromagnetic ordering temperature . coleman _ et al . _ @xcite suggest that @xmath83 data ( where @xmath84 is a control parameter , _ i.e. _ @xmath15 in our case ) can be used to distinguish between two possible qcp scenarios : diffraction off of a critical spin density wave or a breakdown of the composite nature of the heavy electron , with the former manifesting a change of slope at @xmath85 and the latter manifesting a divergence in the slope of @xmath83 at @xmath85 . since in our case the magnetic field is itself the control parameter , it in not clear if @xmath21 , @xmath86 or just simply @xmath12 should be used for comparison with the theory . @xmath87 curves determined by two aforementioned ways are presented in fig . [ drhhyb ] ( @xmath11 ) and fig . [ drhhybc ] ( @xmath2 ) . for both definitions and both orientations the evolution of a clear feature in @xmath87 ( defined as a local extremum for @xmath19 and as a mid - point between two different field - dependent regimes for @xmath88 ) replicates ( albeit with slight @xmath15-shift ) the behavior of the hall resistivity ( figs . [ rhhyb](a ) , [ rhhybc ] ) . given that the new phase line in figs . [ pd ] and [ pdc ] is fairly insensitive to the data analysis we feel that the use of @xmath25 data is currently the least ambiguous data set to analyze . on the other hand , if the form of the anomaly near @xmath64 is to be analyzed in detail it will be vital to have a more detailed theoretical treatment of magneto - transport in field - induced qcp materials . it is tempting to say that for the case of applied field as a control parameter the quantity @xmath89 ( rather than @xmath90 ) serves the role of the low - field hall coefficient and should be compared with the prediction of the models . if this point of view is accepted , then for @xmath11 the shape and evolution of the @xmath89 curves ( fig . [ drhhyb](b ) ) suggest that possibly the composite fermion model of the qcp is more relevant to the case of ybagge , although for @xmath2 the shape and evolution of the @xmath89 curves ( fig . [ drhhybc](b ) ) are at variance with the simple theoretical views . the lack of the @xmath91 k data and an absence of more detailed , realistic - fermi - surface - tailored , model do not allow us to choose the physical picture of the field - induced qcp in ybagge unambiguously . ames laboratory is operated for the u.s . department of energy by iowa state university under contract no . this work was supported by the director for energy research , office of basic energy sciences . we thank h. b. rhee for assistance in growing some of the additional crystals needed for this work . , of luagge measured in different applied fields ( @xmath11 ) . upper inset : field - dependent hall resistivity of luagge measured at different temperatures . lower inset : the sample , current and applied field geometry used during the measurements . ( b ) similar data for tmagge.,title="fig:",width=377 ] , of luagge measured in different applied fields ( @xmath11 ) . upper inset : field - dependent hall resistivity of luagge measured at different temperatures . lower inset : the sample , current and applied field geometry used during the measurements . ( b ) similar data for tmagge.,title="fig:",width=377 ] , of ybagge measured in different applied fields ( @xmath11 ) . inset : enlarged , low temperature part of the data . lower panel : dc susceptibility of ybagge ( @xmath15 along @xmath14 $ ] direction ) . the low h label in the legend refers to the low field hall resistivity ( see experimental section ) and for susceptibility measured in @xmath15 = 1 koe.,width=453 ] ) measured in different applied fields down to 0.4 k. open symbols - he-4 measurements ( 2 - 300 k ) , filled symbols - measurements using he-3 option ( 0.4 - 10 k ) . inset : enlarged low temperature part of the low field data with the estimated error bars.,width=453 ] ) measured at different temperatures : ( a)low temperature data : the curves , except for @xmath20 = 0.4 k , are shifted by @xmath93 cm increments for clarity ; the lines represent the phase lines from the phase diagram in fig . 10(a ) of the ref . 2 ; the triangles mark the position of the local maximum in @xmath25 ; ( b)intermediate and high temperature data . note : @xmath94 k data is shown in both plots for reference and is un - shifted in ( b ) . , title="fig:",width=340 ] ) measured at different temperatures : ( a)low temperature data : the curves , except for @xmath20 = 0.4 k , are shifted by @xmath93 cm increments for clarity ; the lines represent the phase lines from the phase diagram in fig . 10(a ) of the ref . 2 ; the triangles mark the position of the local maximum in @xmath25 ; ( b)intermediate and high temperature data . note : @xmath94 k data is shown in both plots for reference and is un - shifted in ( b ) . , title="fig:",width=340 ] phase diagram for @xmath95 . long range magnetic order ( lrmo ) and the coherence temperature lines marked on the phase diagram are taken from ref . 2 . filled stars and corresponding dashed line as a guide to the eye are defined from the maximum in the @xmath25 curves.,width=453 ] ) ; the curves , except for @xmath20 = 0.4 k , are shifted by @xmath93 cm increments for clarity ; the lines represent the phase lines from the phase diagram in fig . 10(b ) of the ref . 2 ; the triangles mark the position of the peak in @xmath25 . , width=453 ] phase diagram for @xmath96 . long range magnetic order ( lrmo ) and the coherence temperature lines marked on the phase diagram are taken from ref . filled stars are defined from the minimum in the @xmath25 curves . dashed lines are guides to the eye.,width=453 ] ) , defined as ( a ) @xmath21 and ( b ) @xmath97 , measured at different temperatures . the curves , except for @xmath20 = 0.4 k , are shifted by ( a ) 0.02 n@xmath46 cm and ( b ) 0.2 n@xmath46 cm increments for clarity ; the triangles mark the position of the feature in @xmath87 : a local maximum in @xmath19 and a mid - point of the transition between two different field - dependent regimes ( see _ e.g. _ 0.8 k curve)in @xmath88 . curves in the ( b ) panel were obtained by differentiation of the 5-adjacent - points - smoothed @xmath25 data . small downturn at @xmath98 koe in some @xmath88 curves ( panel ( b ) ) is most likely an artifact of using digital smoothing and differentiation.,title="fig:",width=283 ] ) , defined as ( a ) @xmath21 and ( b ) @xmath97 , measured at different temperatures . the curves , except for @xmath20 = 0.4 k , are shifted by ( a ) 0.02 n@xmath46 cm and ( b ) 0.2 n@xmath46 cm increments for clarity ; the triangles mark the position of the feature in @xmath87 : a local maximum in @xmath19 and a mid - point of the transition between two different field - dependent regimes ( see _ e.g. _ 0.8 k curve)in @xmath88 . curves in the ( b ) panel were obtained by differentiation of the 5-adjacent - points - smoothed @xmath25 data . small downturn at @xmath98 koe in some @xmath88 curves ( panel ( b ) ) is most likely an artifact of using digital smoothing and differentiation.,title="fig:",width=283 ] ) , defined as ( a ) @xmath21 and ( b ) @xmath97 , measured at different temperatures . the curves , except for @xmath20 = 0.4 k , are shifted by ( a ) 0.02 n@xmath46 cm and ( b ) 0.1 n@xmath46 cm increments for clarity ; the triangles mark the position of the feature in @xmath87 : a local minimum in @xmath19 and a mid - point of the transition between two different field - dependent regimes ( see _ e.g. _ 0.8 k curve)in @xmath88 . curves in the ( b ) panel were obtained by differentiation of the 5-adjacent - points - smoothed @xmath25 data . small downturn at @xmath98 koe in some @xmath88 curves ( panel ( b ) ) is most likely an artifact of using digital smoothing and differentiation.,title="fig:",width=283 ] ) , defined as ( a ) @xmath21 and ( b ) @xmath97 , measured at different temperatures . the curves , except for @xmath20 = 0.4 k , are shifted by ( a ) 0.02 n@xmath46 cm and ( b ) 0.1 n@xmath46 cm increments for clarity ; the triangles mark the position of the feature in @xmath87 : a local minimum in @xmath19 and a mid - point of the transition between two different field - dependent regimes ( see _ e.g. _ 0.8 k curve)in @xmath88 . curves in the ( b ) panel were obtained by differentiation of the 5-adjacent - points - smoothed @xmath25 data . small downturn at @xmath98 koe in some @xmath88 curves ( panel ( b ) ) is most likely an artifact of using digital smoothing and differentiation.,title="fig:",width=283 ]
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temperature- and field - dependent hall effect measurements are reported for ybagge , a heavy fermion compound exhibiting a field - induced quantum phase transition , and for two other closely related members of the ragge series : a non - magnetic analogue , luagge and a representative , good local moment , magnetic material , tmagge . whereas the temperature dependent
hall coefficient of ybagge shows behavior similar to what has been observed in a number of heavy fermion compounds , the low temperature , field - dependent measurements reveal well defined , sudden changes with applied field ; in specific for @xmath0 a clear local maximum that sharpens as temperature is reduced below 2 k and that approaches a value of 45 koe - a value that has been proposed as the @xmath1 quantum critical point .
similar behavior was observed for @xmath2 where a clear minimum in the field - dependent hall resistivity was observed at low temperatures .
although at our base temperatures it is difficult to distinguish between the field - dependent behavior predicted for ( i ) diffraction off a critical spin density wave or ( ii ) breakdown in the composite nature of the heavy electron , for both field directions there is a distinct temperature dependence of a feature that can clearly be associated with a field - induced quantum critical point at @xmath1 persisting up to at least 2 k.
| 8,472 | 334 |
a subset @xmath4 of a metric space is a _ @xmath0-distance set _ if there are exactly @xmath0 non - zero distances occuring between points of @xmath4 . we also call a @xmath5-distance set an _ equilateral set . _ in this paper we find upper bounds for the cardinalities of @xmath0-distance sets in _ minkowski spaces _ , i.e. finite - dimensional banach spaces ( see theorems [ tha ] to [ up ] ) , and make a conjecture concerning tight upper bounds . in euclidean spaces @xmath0-distance sets have been studied extensively ; see e.g. @xcite , and the books @xcite and ( * ? ? ? * and f3 ) . for general @xmath1-dimensional minkowski spaces it is known that the maximum cardinality of an equilateral set is @xmath6 , with equality iff the unit ball of the space is a parallelotope , and that if @xmath7 , there always exists an equilateral set of at least @xmath8 points @xcite . it is unknown whether there always exists an equilateral set of @xmath9 points ; see @xcite and ( * ? ? ? * , p. 308 problem 4.1.1 ) . however , brass @xcite recently proved that for each @xmath10 there is a @xmath11 such that any @xmath1-dimensional minkowski space has an equilateral set of at least @xmath10 points . see @xcite for problems on equilateral sets in @xmath12 spaces . equilateral sets in minkowski spaces have been used in @xcite to construct energy - minimizing cones over wire - frames . see also @xcite . as far as we know , @xmath0-distance sets for @xmath13 have not been studied in spaces other than euclidean . our main results are the following . [ tha ] if the unit ball of a @xmath1-dimensional minkowski space is a parallelotope , then a @xmath0-distance set in @xmath14 has cardinality at most @xmath2 . this bound is tight . [ cor1 ] given any set @xmath4 of @xmath10 points in a @xmath1-dimensional minkowski space with a parallelotope as unit ball , there exists a point in @xmath4 from which there are at least @xmath15 distinct non - zero distances to points in @xmath4 . this bound is tight . [ thb ] the cardinality of a @xmath0-distance set in a @xmath3-dimensional minkowski space is at most @xmath16 , with equality iff the space has a parallelogram as unit ball . [ cor2 ] given any set of @xmath10 points in a @xmath3-dimensional minkowski space , there exists a point in @xmath4 from which there are at least @xmath17 distinct non - zero distances to points in @xmath4 . [ up ] the cardinality of a @xmath0-distance set in a @xmath1-dimensional minkowski space is at most @xmath18 . in the light of theorems [ tha ] and [ thb ] and the results of @xcite , we make the following the cardinality of a @xmath0-distance set in any @xmath1-dimensional minkowski space is at most @xmath19 , with equality iff the unit ball is a parallelotope . as mentioned above , @xcite shows that this conjecture is true for @xmath20 . by theorem [ thb ] the conjecture is true if @xmath21 , and by theorem [ tha ] if the unit ball is a parallelotope . in the sequel , @xmath22 is a @xmath1-dimensional minkowski space with norm @xmath23 , @xmath24 is the closed ball with centre @xmath25 and radius @xmath26 , and @xmath27 the _ unit ball _ of the space . recall that two @xmath1-dimensional minkowski spaces are isometric iff their unit balls are affinely equivalent ( by the mazur - ulam theorem ; see e.g. ( * ? ? ? * theorem 3.1.2 ) ) . in particular , a minkowski space has a parallelotope as unit ball iff it is isometric to @xmath28 , where @xmath29 . we define a _ cone _ ( or more precisely , an _ acute cone _ ) @xmath30 to be a convex set in @xmath31 that is positively homogeneous ( i.e. , for any @xmath32 and @xmath33 we have @xmath34 ) and satisfies @xmath35 . recall that such a cone defines a partial order on @xmath31 by @xmath36 . we denote the cardinality of a set @xmath4 by @xmath37 . for measurable @xmath38 , let @xmath39 denote the lebesgue measure of @xmath4 . for later reference we state lyusternik s version of the brunn - minkowski inequality ( see ( * ? ? ? * theorem 8.1.1 ) ) . if @xmath40 are compact , then @xmath41 if equality holds and @xmath42 , then @xmath43 and @xmath44 are convex bodies such that @xmath45 for some @xmath46 and @xmath47 . we may assume without loss of generality that the space is @xmath48 . we introduce partial orders on @xmath31 following blokhuis and wilbrink @xcite . for each @xmath49 , let @xmath50 be the partial order with cone @xmath51 for each @xmath25 in a @xmath0-distance set @xmath4 , let @xmath52 be the length of the longest descending @xmath50-chain starting with @xmath25 , i.e. @xmath52 is the largest @xmath53 such that there exist @xmath54 for which @xmath55 . since @xmath56 , for all distinct @xmath57 there exists @xmath58 such that @xmath59 or @xmath60 . exactly as in @xcite , it follows that the mapping @xmath61 is injective , and thus @xmath62 , where @xmath63 it remains to show that @xmath64 . suppose not . then for some @xmath65 and some @xmath58 there exist @xmath66 such that @xmath67 . since @xmath4 is a @xmath0-distance set , @xmath68 for some @xmath69 . also , @xmath70 . now note that if @xmath71 with @xmath72 , then @xmath73 and @xmath74 are @xmath50-incomparable ; in particular , @xmath75 . therefore , @xmath76 , a contradiction . the set @xmath77 is a @xmath0-distance set of cardinality @xmath2 . note that it is not difficult to see that in fact the only @xmath0-distance sets of cardinality @xmath2 are of the form @xmath78 for some @xmath79 and @xmath80 . consider the mapping @xmath81 in the proof of theorem [ tha ] . if @xmath53 is the length of the longest @xmath82-chain over all @xmath58 , then @xmath83 . thus there is a @xmath82-chain @xmath84 of length @xmath85 . by the last paragraph of the proof of theorem [ tha ] , the distances @xmath86 are all distinct . any @xmath87 such that @xmath88 has exactly @xmath15 distinct distances in the norm @xmath89 . the following corollary is easily gleaned from the proof of theorem [ tha ] . [ cor ] suppose that @xmath90 is a family of cones in a minkowski space @xmath91 satisfying @xmath92 and @xmath93 then a @xmath0-distance set in @xmath91 has cardinality at most @xmath94 . [ metriclemma ] let @xmath4 be a @xmath0-distance set in a metric space @xmath95 with distances @xmath96 . if @xmath97 , then for some @xmath98 , the relation @xmath99 is an equivalence relation . the relation @xmath100 is reflexive and symmetric . if it is not transitive , there exist @xmath101 such that @xmath102 and @xmath103 . thus @xmath104 . if this holds for all @xmath98 , we obtain @xmath105 . [ up1 ] the cardinality of a @xmath0-distance set in a @xmath1-dimensional minkowski space is at most @xmath106 . let @xmath107 be a @xmath0-distance set with distances @xmath96 . set @xmath108 . then we have @xmath109 also , @xmath110 , since if @xmath111 , there exist @xmath58 and @xmath112 such that @xmath113 , @xmath114 . thus @xmath115 therefore , @xmath116 substituting and into the brunn - minkowski inequality @xmath117 we obtain @xmath118 , and @xmath119 . if @xmath120 , there is nothing to prove . otherwise , @xmath121 , and by lemma [ metriclemma ] , @xmath122 is an equivalence relation for some @xmath123 . by induction on @xmath0 we obtain that each equivalence class , being an @xmath58-distance set , has at most @xmath124 points . by choosing a representative from each equivalence class , we obtain a @xmath125-distance set with at most @xmath126 points . therefore , @xmath127 . in the proof of theorem [ thb ] , we need the following geometric lemma , which is a modification of ( * ? ? ? * corollary 3.2.6 ) in @xmath3 dimensions . [ auerbach ] let @xmath128 be the convex hull of @xmath129 and @xmath130 the square @xmath131 ^ 2 $ ] . for any symmetric convex disc @xmath132 in @xmath133 there exists an invertible linear transformation taking @xmath132 to @xmath134 such that @xmath135 and such that any straight - line segment contained in the boundary of @xmath134 lies completely in one of the four coordinate quadrants . we consider all triangles with vertices @xmath136 , where @xmath25 and @xmath137 are on the boundary of @xmath132 . by compactness there exist @xmath138 and @xmath139 such that the area of the triangle is a maximum . then @xmath140 is a support line of @xmath132 at @xmath138 , since otherwise we can replace @xmath138 by a point on the side of the line opposite @xmath141 to enlarge the area of the triangle . similarly , @xmath142 is a support line of @xmath132 at @xmath139 . since @xmath132 is symmetric , it follows that @xmath132 is contained in the parallelogram @xmath143 . see figure [ auerfig ] . if @xmath138 is an interior point of a straight - line segment contained in the boundary of @xmath132 , we may shift @xmath138 to a boundary point of such a segment , without changing the area of the triangle . thus @xmath132 is still contained in a parallelogram as above . a similar remark holds for @xmath139 . we now apply the linear transformation sending @xmath138 and @xmath139 to the standard unit vectors @xmath144 and @xmath145 , respectively ( see figure [ auerfig2 ] ) . we have to find two cones @xmath146 and @xmath147 satisfying and of corollary [ cor ] . by lemma [ auerbach ] we may replace the space by an isometric space @xmath148 such that the unit ball @xmath44 of @xmath23 lies between @xmath128 and @xmath130 , and such that any straight - line segment contained in the boundary of the unit ball lies completely in a quadrant of the plane . we provisionally let @xmath146 be the closed first quadrant , and @xmath147 the closed second quadrant . see figure [ auerfig2 ] . then is satisfied . the only way that could fail is if there is a straight - line segment contained in the boundary of the unit ball parallel to either the x - axis or the y - axis , lying in @xmath146 or @xmath147 . if there is a segment in the boundary of the unit ball in @xmath146 parallel to the x - axis , say , we remove the positive x - axis @xmath149 from @xmath146 . if in this case there were another straight - line segment in the boundary parallel to the x - axis in @xmath147 , then there would be a straight - line segment in the boundary lying in the first and second quadrants , giving a contradiction . thus we do not have to remove the negative x - axis from @xmath147 , and is still satisfied . we do the same thing for segments parallel to the y - axis , and for @xmath147 . in the end , the modified @xmath146 and @xmath147 satisfy and , and we deduce @xmath150 from corollary [ cor ] . if equality holds , then the mapping @xmath151 in the proof of theorem [ tha ] is a bijection from @xmath4 to @xmath152 . we now denote a point @xmath65 by @xmath153 , where @xmath154 . suppose that two of the distances @xmath155 ( @xmath156 ) are equal , say @xmath157 with @xmath158 . then , since @xmath159 , we have @xmath160 , contradicting . it follows that the distances @xmath155 , @xmath156 are distinct , and thus are exactly the @xmath0 different distances in increasing order . similarly , the distances @xmath161 , @xmath162 are in increasing order . if @xmath163 , the three points @xmath164 again contradict . thus these distances are @xmath165 in increasing order , etc . in the end we find that @xmath166 for all @xmath58 . thus @xmath167 , by the triangle inequality . using the brunn - minkowski inequality as in the proof of lemma [ up1 ] , we find that equality holds in and , implying that for @xmath168 we have @xmath169 , and @xmath170 and @xmath171 are homothetic . thus @xmath171 is a ball that is perfectly packed by smaller balls . by a result of @xcite , this implies that the unit ball is a parallelogram . follows from the proof of theorem [ thb ] in the same way that theorem [ cor1 ] follows from theorem [ tha ] . lemma [ up1 ] already gives part of the theorem . for the remaining part we apply corollary [ cor ] . in order for a cone @xmath30 to satisfy , it is sufficient that @xmath172 to see this , suppose that @xmath30 does not satisfy the condition in , i.e. there exist distinct @xmath173 such that @xmath174 and @xmath175 . let @xmath176 , @xmath177 , @xmath178 , and @xmath179 . then @xmath180 , and @xmath181 implying @xmath182 . in order for to be satisfied too , we need a cover of the unit sphere by sets such that , if they are extended to positive cones , are convex . we do this with the following construction : let @xmath183 be a maximal set of unit vectors satisfying @xmath184 for all @xmath185 . then for any unit vector @xmath25 there exists @xmath58 such that @xmath186 or @xmath187 . for @xmath188 , let @xmath189 be the cone generated by @xmath190 i.e. @xmath191 . then the @xmath189 s satisfy by the maximality of @xmath132 . each @xmath189 satisfies : let @xmath192 , where @xmath193 and @xmath194 . then @xmath195 also , since @xmath196 we obtain @xmath197 , and @xmath198 . a volume argument gives the upper bound for @xmath199 : the balls @xmath200 have disjoint interiors and are contained in the ball @xmath201 . therefore , @xmath202 giving @xmath203 . this paper is part of the author s phd thesis written under supervision of prof . w. l. fouch at the university of pretoria . i thank the referees as well as graham brightwell for their suggestions on the layout of the paper .
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a subset of a metric space is a _ @xmath0-distance set _ if there are exactly @xmath0 non - zero distances occuring between points .
we conjecture that a @xmath0-distance set in a @xmath1-dimensional banach space ( or _ minkowski space _ ) , contains at most @xmath2 points , with equality iff the unit ball is a parallelotope .
we solve this conjecture in the affirmative for all @xmath3-dimensional spaces and for spaces where the unit ball is a parallelotope .
for general spaces we find various weaker upper bounds for @xmath0-distance sets .
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computer simulation is widely regarded as complementary to theory and experiment @xcite . at present there are only a few physical phenomena that can not be simulated on a computer . one such exception is the double - slit experiment with single electrons , as carried out by tonomura and his co - workers @xcite . this experiment is carried out in such a way that at any given time , only one electron travels from the source to the detector @xcite . only after a substantial ( approximately 50000 ) amount of electrons have been detected an interference pattern emerges @xcite . this interference pattern is described by quantum theory . we use the term `` quantum theory '' for the mathematical formalism that gives us a set of algorithms to compute the probability for observing a particular event @xcite . of course , the quantum - mechanics textbook example @xcite of a double - slit can be simulated on a computer by solving the time - dependent schrdinger equation for a wave packet impinging on the double slit @xcite . alternatively , in order to obtain the observed interference pattern we could simply use random numbers to generate events according to the probability distribution that is obtained by solving the time - independent schrdinger equation . however , that is not what we mean when we say that the physical phenomenon can not be simulated on a computer . the point is that it is not known how to simulate , event - by - event , the experimental observation that the interference pattern appears only after a considerable number of events have been recorded on the detector . quantum theory does not describe the individual events , e.g. the arrival of a single electron at a particular position on the detection screen @xcite . reconciling the mathematical formalism ( that does not describe single events ) with the experimental fact that each observation yields a definite outcome is often referred to as the quantum measurement paradox and is the central , most fundamental problem in the foundation of quantum theory @xcite . if computer simulation is indeed a third methodology to model physical phenomena it should be possible to simulate experiments such as the two - slit experiment on an event - by - event basis . in view of the fundamental problem alluded to above there is little hope that we can find a simulation algorithm within the framework of quantum theory . however , if we think of quantum theory as a set of algorithms to compute probability distributions there is nothing that prevents us from stepping outside the framework that quantum theory provides . therefore we may formulate the physical processes in terms of events , messages , and algorithms that process these events and messages , and try to invent algorithms that simulate the physical processes . obviously , to make progress along this line of thought , it makes sense not to tackle the double - slit experiment directly but to simplify the problem while retaining the fundamental problem that we aim to solve . the main objective of the research reported in this paper is to answer the question : `` can we simulate the single - photon beam splitter and mach - zehnder interferometer experiments of grangier et al . @xcite on an event - by - event basis ? '' . these experiments display the same fundamental problem as the single - electron double - slit experiments but are significantly easier to describe in terms of algorithms . the main results of our research are that we can give an affirmative answer to the above question by using algorithms that have a primitive form of learning capability and that the simulation approach that we propose can be used to simulate other quantum systems ( including the double - slit experiment ) as well . in section [ illu ] we introduce the basic concepts for constructing event - based , deterministic learning machines ( ) . an essential property of these machines is that they process input event after input event and do not store information about individual events . a can discover relations between input events ( if there are any ) and responds by sending its acquired knowledge in the form of another event ( carrying a message ) through one of its output channels . by connecting an output channel to the input channel of another we can build networks of . as the input of a network receives an event , the corresponding message is routed through the network while it is being processed and eventually a message appears at one of the outputs . at any given time during the processing , there is only one input - output connection in the network that is actually carrying a message . the process the messages in a sequential manner and communicate with each other by message passing . there is no other form of communication between different . although networks of can be viewed as networks that are capable of unsupervised learning , there have very little in common with neural networks @xcite . the first described in section [ illu ] is equivalent to a standard linear adaptive filter @xcite but the that we actually use for our applications do not fall into this class of algorithms . in section [ ndim ] we generalize the ideas of section [ illu ] and construct a which groups @xmath0-dimensional data in two classes on an event - by - event basis , i.e. , without using memory to store the whole data set . we demonstrate that this is capable of detecting time - dependent trends in the data and performs blind classification . this example shows that can be used to solve problems that have no relation to quantum physics . in section [ qi ] we show how to construct -networks that generate output patterns that are usually thought of as being of quantum mechanical origin . we first build a -network that simulates photons passing through a polarizer and show that quantum theory describes the output of this deterministic , event - based network . then we describe a -network that simulates a beam splitter and use this network to build a mach - zehnder interferometer and two chained mach - zehnder interferometers . we demonstrate that quantum theory also describes the behavior of these networks . quantum theory gives us a recipe to compute the frequency of events but does not predict the order in which the events will be observed @xcite . in genuine experiments the detection of events appears to be random @xcite , in a sense which , as far as we know , has not been studied systematically . in our simulation approach , this apparent randomness can be accounted for by a marginal modification of the , as explained in section [ slm ] . this modification does not change the deterministic character of the learning process . it merely randomizes the order in which the activate their output channels . a summary and outlook is given in section [ summ ] . we consider a that has one input and two output channels labeled by @xmath1 ( see fig . 1 ) . the internal state of the after processing the @xmath2-th input event ( @xmath3 ) is uniquely defined by the real variable @xmath4 . at the next event @xmath5 the receives as input a real number @xmath6 . for simplicity , but without loss of generality , we assume that @xmath7 $ ] . the responds by sending a message containing @xmath6 through one of the two output channels @xmath8 . the selects the output channel @xmath9 or @xmath10 by minimizing the cost function @xmath11 defined by @xmath12 updates its internal state according to the rule @xmath13 and sends a message with the input value @xmath6 on the selected output channel @xmath14 . the parameter @xmath15 that enters eqs . and controls the decision process . for simplicity we assume that @xmath16 is fixed during the operation of the machine . it is easy to see that @xmath9 if @xmath17 and @xmath10 if @xmath18 . thus , for this particular we have @xmath19 hence the update rule eq . can be written as the familiar recursion @xmath20 the solution of eq . reads @xmath21 where @xmath22 denotes the initial value of the internal variable . as an illustration of how this learns , we consider the most simple example where @xmath23 for all @xmath24 . then from eq . we find that @xmath25 as @xmath15 , we conclude that @xmath26 . thus the `` learns '' the value of the input variable @xmath27 . from eq . it follows that @xmath28 ( @xmath29 ) implies @xmath30 ( @xmath31 ) . hence @xmath4 approaches @xmath27 monotonically ( and @xmath32 is the same for all @xmath2 ) . therefore , if @xmath33 , the always sends the value of @xmath34 through the same output channel . ( 8,6 ) ( -5,1 ) by passing the input to one of the two output channels @xmath8 . the value of @xmath14 depends on the current state of the , encoded in the variable @xmath35 , the input @xmath6 , and the update rule eq . in which @xmath16 appears as a control parameter . right : evolution of the internal variable @xmath35 as a function of the number of events @xmath2 . solid line : @xmath36 for @xmath37 and @xmath38 for @xmath39 ; dashed line : random sequence of @xmath40 . , title="fig:",width=340 ] ( 4,0 ) by passing the input to one of the two output channels @xmath8 . the value of @xmath14 depends on the current state of the , encoded in the variable @xmath35 , the input @xmath6 , and the update rule eq . in which @xmath16 appears as a control parameter . right : evolution of the internal variable @xmath35 as a function of the number of events @xmath2 . solid line : @xmath36 for @xmath37 and @xmath38 for @xmath39 ; dashed line : random sequence of @xmath40 . , title="fig:",width=340 ] a distinct feature of this machine is its ability to adapt to changes in the input pattern . we illustrate this important property by two examples . let @xmath41 for @xmath42 and @xmath43 for @xmath44 . during the first 1000 events the machine will learn @xmath45 . after 1000 events only @xmath46 is being presented as input . then , the machine `` forgets '' @xmath45 and learns @xmath46 as shown in the right panel of fig . [ exam01 ] . in this simulation @xmath47 . alternatively , if @xmath34 is a random sequence of @xmath48 ( each with the same probability ) the machine has to learn @xmath45 and @xmath46 simultaneously . because of this it can not `` forget '' and it ends up oscillating around the mean of the input values ( zero in this example ) as illustrated in the right panel of fig . [ exam01 ] . let us now assume that our machine has reached this oscillating state . all input events @xmath49 give @xmath50 and hence the machine sends @xmath46 over the @xmath51 channel . a second machine attached to this channel only receives @xmath46 events and will learn @xmath46 . this suggests that a network of these machines can be used as an adaptive classifier . ( 8,12 ) ( -4.5,3 ) . right : evolution of the internal variables @xmath35 of the as a function of the number of events @xmath2 . the machine number is used to label the corresponding line . top right : first three ; bottom right : third - level . , title="fig:",width=302 ] ( 4,6 ) . right : evolution of the internal variables @xmath35 of the as a function of the number of events @xmath2 . the machine number is used to label the corresponding line . top right : first three ; bottom right : third - level . , title="fig:",width=340 ] ( 4,0 ) . right : evolution of the internal variables @xmath35 of the as a function of the number of events @xmath2 . the machine number is used to label the corresponding line . top right : first three ; bottom right : third - level . , title="fig:",width=340 ] consider the network of three layers of shown in the left panel of fig . [ exam03 ] . each machine in the network learns the average of the numbers it receives at its input channel and sends the numbers which are smaller ( larger or equal ) than the number it learned to the -1 ( + 1 ) output channel . in our numerical experiments we set @xmath47 . we start with 5000 events of random numbers @xmath52 , each occurring with equal probability . machine 1 learns the average ( zero in this example ) and sends the negative ( positive ) @xmath6 over the @xmath53 ( @xmath51 ) channel to the input of machine 2 ( 3 ) . machine 2 ( 3 ) learns -0.50 ( 0.50 ) , as shown in the top right panel of fig . [ exam03 ] , and sends -0.75 ( 0.25 ) over its -1 output channel and -0.25 ( 0.75 ) over its + 1 output channel . machines 4 to 7 learn -0.75,-0.25,0.25 and 0.75 , respectively , as shown in the bottom right panel of fig . [ exam03 ] . each of these machines forwards the received input on its + 1 ( -1 ) output channel if the initial value of its internal variable is smaller ( larger ) than the received input value . let us now assume that after 5000 events the input data set changes to @xmath54 . as can be seen from the right panel of fig . [ exam03 ] , machines 1 , 3 and 7 `` forget '' the number they learned and replace it by -0.0625 , 0.375 and 0.50 , respectively . all other machines are unaffected because they never get 0.50 as input . after another 5000 events we change the set of input values once more , this time to @xmath55 , i.e. , we add one element . now , machine 1 learns -0.17 , machine 2 learns -0.53 and the internal state of machine 3 remains unchanged . machine 4 can now receive two numbers on its input channel , namely -0.75 and -0.60 . as a consequence , machine 4 learns -0.675 , i.e. , the average of the two possible input numbers . machine 4 puts -0.60 on its + 1 output channel and -0.75 on its -1 output channel . in order for the network to learn all the numbers of the input set , we would have to attach one extra to each output channel of machine 4 . ( 8,6 ) ( -5,0 ) of the machine defined by eqs . and . the input events are @xmath56 , @xmath47 , and the initial value @xmath57 . for @xmath58 the internal variable @xmath4 oscillates about @xmath27 . for @xmath59 the sequence of increments ( @xmath9 ) and decrements ( @xmath10 ) of @xmath4 repeats itself after 8 events ( data not shown ) . lines are guides to the eyes . right : the number of increments of the internal variable ( @xmath9 ) divided by the total number of events as a function of the value of the input variable @xmath27 . bullets : each data point is obtained from a simulation of 1000 events with a fixed , randomly chosen value of @xmath60 , using the last 500 events to count the number of @xmath9 events . solid line : @xmath61 . , title="fig:",width=340 ] ( 4,0 ) of the machine defined by eqs . and . the input events are @xmath56 , @xmath47 , and the initial value @xmath57 . for @xmath58 the internal variable @xmath4 oscillates about @xmath27 . for @xmath59 the sequence of increments ( @xmath9 ) and decrements ( @xmath10 ) of @xmath4 repeats itself after 8 events ( data not shown ) . lines are guides to the eyes . right : the number of increments of the internal variable ( @xmath9 ) divided by the total number of events as a function of the value of the input variable @xmath27 . bullets : each data point is obtained from a simulation of 1000 events with a fixed , randomly chosen value of @xmath60 , using the last 500 events to count the number of @xmath9 events . solid line : @xmath61 . , title="fig:",width=340 ] for the defined by eqs . and , formulating the operation of the through the minimization of the difference between the input and internal variable may seem a little superfluous and indeed , for this particular machine it is . however , this formulation is a convenient starting point for defining machines that can perform more intricate tasks . for instance , let us make an innocent looking change to the update rule eq . by writing @xmath62 and replace the cost function eq . by the corresponding expression @xmath63 for @xmath9 we have @xmath64 and for @xmath10 we have @xmath65 . therefore , if @xmath66 and @xmath67 , the internal variable will always be in the range @xmath68 $ ] . at each event the internal variable either increases by @xmath69 ( if @xmath9 ) or decreases by @xmath70 ( if @xmath10 ) . in both cases this change is always nonzero , except if @xmath71 which can only occur if @xmath72 . the ratio of the step sizes is @xmath73 . the machine defined by eqs . and behaves differently from the machine defined by eqs . and . to see this , it is instructive to consider the case @xmath74 for all @xmath24 ( the case @xmath75 can be treated in the same manner ) . for concreteness we assume that @xmath76 . at the first event , minimization of eq . yields @xmath77 and @xmath78 . in other words , the internal variable @xmath79 moves towards @xmath27 . as long as @xmath80 , the selects @xmath9 , always increasing its internal variable @xmath4 . for some some @xmath81 we must have @xmath82 . then , making another move in the positive @xmath79-direction allows for two different decisions . if the error that results is larger than the error that is obtained by moving in the negative direction the decides to set @xmath10 . otherwise it makes another move in the positive @xmath79-direction ( @xmath9 ) . in any case , for some @xmath83 the machine will select @xmath10 . note that when this happens , we must have @xmath84 and @xmath85 . this implies that after this @xmath2-th event ( that we denote by @xmath86 ) the internal variable will oscillate ( forever ) around the input value @xmath27 . this process is illustrated in fig . [ figline ] ( left ) . for @xmath87 we have @xmath88 . thus , if @xmath89 , the amplitude of the oscillations is small . the `` learns '' the input value @xmath27 and the ratio of the increments to decrements is @xmath90 . in this stationary regime of oscillating behavior , the number of times the actives the + 1 ( -1 ) channel is given by @xmath61 ( @xmath91 ) . the simulation results shown in fig . [ figline ] ( right ) confirm the correctness of this analysis . for a fixed ( unknown ) value of the input variable , the rate at which the machine defined by the rules eqs . and activates one of its output channels is determined by the value of its internal variable . therefore , this rate reflects the value that the machine has learned by processing the input events . depending on the application , the message that is sent through the active output channel can contain @xmath92 or the input value @xmath6 ( there is nothing else that can be send ) . obviously we can make the learning process more precise by increasing @xmath93 . of course , a larger value of @xmath16 also results in slower learning : in general it will take more events for the internal variable to reach the value where it starts to oscillate . in going from the first to the second example of section [ illu ] we changed the update rule such that the variable @xmath4 is constrained to lie in the interval @xmath68 $ ] . we now consider the two - dimensional analogue of the described in section [ illub ] for which the internal vector @xmath94 and input vector @xmath95 represent points on a circle . this receives as input a sequence of angles @xmath96 defined by @xmath97 and responds by activating one of the two output channels . for all @xmath98 , the update rules are defined by @xmath99 where @xmath100 and @xmath15 . in order that the internal vector @xmath101 stays on the unit circle we must have @xmath102 \pm\sqrt{1-\alpha^2+\alpha^2[x_{1,n}^2\theta_{n+1 } + x_{2,n}^2(1-\theta_{n+1 } ) ] } . \label{circ2}\ ] ] substitution of eq . in eq . gives us four different rules : @xmath103 where @xmath104 takes care of the fact that for each choice of @xmath105 , the has to decide between two quadrants . the cost function is defined by @xmath106 obviously , the cost function eq . is nothing but the inner product of the vectors @xmath107 and @xmath108 . the new internal state itself is determined by calculating the cost eq . for each of the four candidate update rules listed in eq . and selecting the rule that yields the minimum cost . note that the minimum of the cost function eq . does not depend on the length of the vector of input variables @xmath95 . from eq . it follows that if @xmath109 . the value of @xmath110 is obtained by rescaling of @xmath111 and @xmath112 is adjusted such that @xmath113 . for @xmath114 we interchange the role of the first and second element of @xmath107 . ( 8,6 ) ( -5,0 ) ( 4,0 ) in general the behavior of the defined by rules eqs . and is difficult to analyze without the use of a computer . however , for a fixed input vector @xmath115 it is clear what the will try to do : it will minimize the cost eq . by rotating its internal vector @xmath107 to bring it as close as possible to @xmath116 . however , @xmath107 will not converge to a limiting value but instead it will keep oscillating about the input value @xmath116 . an example of a simulation is given in fig . [ c30 ] ( left ) . for a fixed input vector @xmath115 the reaches a stationary state in which its internal vector oscillates about @xmath116 . in this stationary state the output signal consists of a finite sequence of ones and zeros . the repeats this sequence over and over again . obviously , the whole process is deterministic . the details of the approach to the stationary state depend on the initial value of the internal vector @xmath117 , but the stationary state itself does not . these observations are of much more general nature than the example given in fig . [ c30 ] ( left ) suggests . in fact , as the applications discussed below amply illustrate , the stationary - state analysis is a very useful tool to predict the behavior of the . assuming that @xmath89 and that we have reached the stationary regime in which the internal vector performs small oscillations about @xmath118 , a simple calculation shows that @xmath119 in the stationary regime , we have @xmath120 where @xmath121 ( @xmath122 ) is the number of @xmath109 ( @xmath114 ) events . from eq . it then follows immediately that @xmath123 and @xmath124 . the results of this analysis are in excellent agreement with the simulation results shown in fig . [ c30 ] ( right ) . the conventional approach to regard the variables @xmath105 as input is fundamentally different from the approach adopted in this paper . this can be seen by reformulating the update rules in terms of difference equations and to assume that the @xmath100 are independent , uniform random variables with mean @xmath125 . the four rules eq . can be written as @xmath126 formally eq . has the same structure as eq . . averaging over many realizations of @xmath127 and taking the limit @xmath128 we obtain @xmath129 in other words , a machine that operates according to the rules eq . and receives as input the random sequence @xmath105 will ( on average ) approach a state in which the direction of its internal vector gives us an estimate of the @xmath130 . in contrast , a that minimizes the cost eq . and updates its internal state according to eq . responds on either output channel @xmath109 or output channel @xmath114 , with a frequency that is directly related to the difference between the current input angle and the angle defined by the internal vector . consider a sequence of events , characterized by vectors @xmath131 for @xmath98 . the vector @xmath108 is the input for the . the internal state of the is described by a @xmath0-dimensional unit vector @xmath132 . we define the @xmath133 candidate update rules @xmath134 by @xmath135 note that @xmath136 implies @xmath137 for each of the @xmath133 update rules . the responds to the input @xmath108 by selecting from the @xmath133 possible rules in eq . , the update rule that minimizes the cost @xmath138 and by sending a message containing @xmath108 ( or , depending on the application , @xmath107 ) on one of its output channels . note that the minimum of the cost function eq . does not depend on the length of the vectors @xmath107 or @xmath108 . disregarding the variables @xmath139 that merely serve to determine the sign of @xmath140 there are @xmath0 rules . hence there can be as many as @xmath0 output channels . however , depending on the application , it may be expedient to reduce the number of output channels by arranging them in groups . the analyzed in the previous subsections have one input channel that receives input and two output channels , only one of which sends out data ( a message ) at a particular event . an obvious generalization is to construct that accept , at a given instance , input from one out of two different sources . this is absolutely necessary if we want to build machines in which events can communicate or , in physical terms , interact with each other . we now demonstrate that the that we introduced above already have the capability to let events interact with each other . therefore we do not need to add a new feature or rule to the . consider a that has two input channels 0 and 1 and an internal vector @xmath141 with @xmath142 components . at the @xmath5-th event , either input channel 0 receives the two - component vector @xmath143 or input channel 1 receives the two - component vector @xmath144 . in the former case the transforms this input into the input vector @xmath145 of four elements by using the current internal vector as a source for the missing elements . similarly , in the latter case the input vector becomes @xmath146 . then the uses @xmath147 to determine the cost and selects the update rule according to the procedure described in section [ hyp ] ( with @xmath147 replacing @xmath108 ) . this learns the two - dimensional vectors @xmath143 and @xmath144 separately , as if it consists of two separate , independent two - dimensional , with the additional crucial feature that the internal vector represents a point on a 4-dimensional unit sphere . it is not difficult to imagine what this does in the case that it receives events on only one of the two input channels ( say 0 ) . irrespective of the initial value of the internal vector @xmath148 , the will always select the update rule with @xmath149 ( see eq . ) and the two components @xmath150 and @xmath151 will vanish exponentially fast with increasing @xmath2 ( recall that @xmath15 ) . thus , after a few events the internal state of the indicates that the receives events on only one channel . if the machine receives input on both channels ( but never simultaneously ) , eq . implies that the only scales the two components of the internal state that it uses to provide the missing elements for building the input @xmath147 . therefore , in the stationary regime , the length of the two - dimensional vector @xmath94 ( @xmath152 ) is proportional to the number of events on input channel 0 ( 1 ) . furthermore the number of @xmath149 ( @xmath153 ) events is approximately equal to the number of events on input channel 0 ( 1 ) . although this may seem a very elementary form of communication , it is sufficient to construct that perform very complicated tasks . the described above are simple deterministic machines that make decisions . the responds to the input event by choosing from all possible alternatives , the internal state that minimizes the error between the input and the internal state itself . then the sends a message through one of its output channels . the message contains information about the decision the took while updating its internal state and , depending on the application , also contains other data that the can provide . by updating its internal state , the learns " about the input it receives and by sending messages through one of its two output channels , it tells its environment about what it has learned . in the sequel we will call such a machine a * deterministic learning machine * ( dlm ) . for a particular choice of the update rule ( see section [ illua ] ) , the performs linear estimation but as the other examples of this section amply demonstrate , minor modifications to this rule and/or cost function yield that may behave in a substantially different manner . the of section [ illua ] learns about the input data by moving a point on a line . obviously , this point separates two parts of the line . the generalization to @xmath0-dimensional space is a @xmath154-dimensional hyperplane that divides the space into two parts . thus , to interpret two - dimensional data the should learn a line instead of a point . we represent the line by a segment @xmath155 defined by its mid - point @xmath156 and its direction @xmath157 . as the receives an event @xmath108 , i.e. a point in a two - dimensional plane , the updates its internal line segment @xmath155 and sends the information describing @xmath155 through the -1 ( + 1 ) channel , depending on whether it lies on the left ( right ) side of the line . the update procedure consists of two steps . first we define two support points @xmath158 and @xmath159 on either side of @xmath156 along the direction @xmath157 by @xmath160 and we update the two support points according to @xmath161 where @xmath15 controls the learning process . then we compute the new mid - point and direction of the line segment : @xmath162 from eq . it follows that the support point farthest away from @xmath108 makes the largest move . therefore , as new input data is received by the , both the mid - point and the direction of the line segment change . note that the update rule eq . is non - linear in the difference between internal and input vector . although a linear update rule also works , our numerical experiments ( results not shown ) indicate that the non - linear rule eq . performs much better . in general @xmath156 will converge to the mean of the input vectors and @xmath158 and @xmath159 will be pulled most strongly in the direction of largest variance . therefore @xmath155 will be ( approximately ) perpendicular to the largest principal component of the covariance matrix of the input data . in other words , the defined above can find the eigenvector that corresponds to the largest eigenvalue of the covariance matrix by processing data points in a sequential manner , i.e. , without actually having to compute the elements of the covariance matrix . ( 14,10 ) ( 0,5 ) and means @xmath163 . each panel shows the output of the -based classifier after it has processed , point - by - point , the 100 data points shown . the classifier smoothly follows the rotation of the means . in contrast to the event - by - event processing of the -based classifier , the principal - component - based classifier processes the whole set of 100 data points simultaneously . , title="fig:",width=264 ] ( 5,5 ) and means @xmath163 . each panel shows the output of the -based classifier after it has processed , point - by - point , the 100 data points shown . the classifier smoothly follows the rotation of the means . in contrast to the event - by - event processing of the -based classifier , the principal - component - based classifier processes the whole set of 100 data points simultaneously . , title="fig:",width=264 ] ( 10,5 ) and means @xmath163 . each panel shows the output of the -based classifier after it has processed , point - by - point , the 100 data points shown . the classifier smoothly follows the rotation of the means . in contrast to the event - by - event processing of the -based classifier , the principal - component - based classifier processes the whole set of 100 data points simultaneously . , title="fig:",width=264 ] ( 0,0 ) and means @xmath163 . each panel shows the output of the -based classifier after it has processed , point - by - point , the 100 data points shown . the classifier smoothly follows the rotation of the means . in contrast to the event - by - event processing of the -based classifier , the principal - component - based classifier processes the whole set of 100 data points simultaneously . , title="fig:",width=264 ] ( 5,0 ) and means @xmath163 . each panel shows the output of the -based classifier after it has processed , point - by - point , the 100 data points shown . the classifier smoothly follows the rotation of the means . in contrast to the event - by - event processing of the -based classifier , the principal - component - based classifier processes the whole set of 100 data points simultaneously . , title="fig:",width=264 ] ( 10,0 ) and means @xmath163 . each panel shows the output of the -based classifier after it has processed , point - by - point , the 100 data points shown . the classifier smoothly follows the rotation of the means . in contrast to the event - by - event processing of the -based classifier , the principal - component - based classifier processes the whole set of 100 data points simultaneously . , title="fig:",width=264 ] as an illustration of the capabilities of the introduced in this section , let us consider a classification task in which we want to blindly group events into two categories . the input data @xmath143 are generated through a gaussian random process described by : @xmath164 where @xmath165 is a uniform random bit . the random numbers @xmath166 and @xmath167 are drawn from the normal distribution @xmath168 . in our numerical example we take @xmath169 and @xmath47 . from eq . it is clear that the input events consist of points in a plane that are drawn from one of two ( @xmath170 ) gaussian distributions , the centers of which rotate with a period of 10000 events . the mean of all input data is @xmath171 and there is no preferred direction of largest variance . the reason of course is that the center of the gaussian distributions slowly moves on the unit circle . clearly , this kind of classification task can only be performed by permanently updating the estimate of the direction and that is exactly what the does . in fig . [ 2dp2 ] we present results of a blind classification experiment that illustrates the operation of the defined by the rules eqs . the processes event - by - event , each time updating its estimate for the separatrix . for comparison we also show the result obtained by the principal component analysis @xcite using as input the group of 100 most recent data points processed by the . the differences between both classifiers are rather small so that it is clear that the -based classifier performs very well . the two - dimensional described above can easily be extended to a that processes @xmath0-dimensional input data . instead of a line segment the has to learn a segment of a @xmath154-dimensional hyperplane . this can be done by extending the procedure used in the two - dimensional case . the hyperplane segment is described by a mid - point @xmath156 and @xmath172 orthonormal directions @xmath173 for @xmath174 . we choose @xmath0 points @xmath175 on the hyperplane defined by @xmath176 and @xmath156 such that the distance between each pair of points is one . as new input data @xmath108 is received by the these points are updated according to ( the generalization of ) eq . . as in the two - dimensional case , from the updated points we can calculate the new mid - point and the new directions . however , unlike in the two - dimensional case , these directions do not need to be orthonormal . the orthonormality is then restored by using the ( modified ) gramm - schmidt procedure @xcite . ( 14,7 ) ( -1.75,0 ) ( 7,0 ) [ p25 ] ( 14,7 ) ( -1.,0 ) ( 7,0 ) [ 3pol ] we demonstrate that the defined by eqs . and and a passive element that performs a plane rotation are sufficient to perform a deterministic simulation of the quantum theory @xcite of photon polarization . we start by recalling some elementary facts about photon polarization @xcite . some optically active materials like calcite split an incoming beam of light into two spatially separated beams @xcite . the light intensity of these beams is related to the angle of polarization @xmath177 of the electromagnetic wave , relative to the orientation @xmath178 of the material @xcite . we disregard all imperfections of real experiments and assume that the experimental data are in exact agreement with the wave mechanical theory . then the intensities @xmath179 of beam 0 and @xmath180 of beam 1 are given by @xcite @xmath181 respectively . if the incident beam has a random polarization , averaging of eq . over all @xmath177 shows that half of the light intensity will go to beam 0 and the other half to beam 1 . if the conventional light source is replaced by a source that emits one photon at a time , the photon leaves the material either in the direction of beam 0 or beam 1 , never in both @xcite . collecting photons over a sufficiently long period shows that eq . still gives the number of photons detected in the direction of beam 0 ( 1 ) , divided by the total amount of detected photons @xcite . quantum theory @xcite describes the polarization in terms of a two - dimensional ( complex - valued ) vector and the action of the material is to rotate this vector by an angle @xmath178 ( set by the experimentalist ) @xcite . the probability to observe photons in beam 0 ( 1 ) is given by the square of the 0-th ( 1-st ) element of the vector @xcite . in addition , as the photon leaves the material in beam 0 ( 1 ) , its polarization is @xmath178 ( @xmath182 ) @xcite . thus the piece of material can be used to prepare and also determine the polarization of the photons and is called a `` polarizer '' @xcite . according to quantum theory @xcite , the polarizer rotates the vector of polarization amplitudes in the following manner @xcite : @xmath183 still according to quantum theory @xcite , the intensity in beam 0 ( 1 ) is given by @xmath184 ( @xmath185 ) . an incident beam with an angle of polarization @xmath177 is described by the vector @xmath186 . from eq . we obtain @xmath187 and hence @xmath188 and @xmath189 , in agreement with eq . . we now construct a simple deterministic machine that generates events of which the distribution agrees with the probability distributions predicted by quantum theory @xcite . the layout of this `` polarizer '' is shown in fig . [ figpol ] . the incoming event ( photon ) carries an ( unknown ) angle @xmath190 . the purpose of the passive element @xmath191 is to perform a rotation @xmath192 of the input vector @xmath193 by the angle @xmath178 . the resulting vector @xmath194 is sent to the input of a that operates according to eqs . and . if @xmath109 , the responds by sending the vector @xmath195 through the output channel 0 . if @xmath114 , the responds by sending the vector @xmath196 through the output channel 1 . clearly this procedure is strictly deterministic . we emphasize that the processes information event by event and does not store the data contained in each event . in fig . [ p25 ] ( right ) we show simulation results for the machine depicted in fig . [ figpol ] ( left ) . each data point represents the intensity in beam 0 ( 1 ) , i.e. , the number of @xmath197 @xmath198 events divided by the total amount of events . the machine is initialized once by choosing a random direction of the vector @xmath148 . the angle of rotation @xmath178 is kept fixed for 1000 events , then a uniform random number is used to select another direction , and this procedure is repeated 100 times . in all these numerical experiments we set @xmath47 . fig . [ p25 ] shows the results for two different numerical experiments : in the first set of 100 runs , the direction of polarization @xmath177 of the incoming photons is also determined by means of uniform random numbers . in the second set of 100 runs , the direction of polarization of the incoming photons is fixed ( @xmath199 ) . from fig . [ p25 ] ( right ) it is clear that quantum theory @xcite provides a very good description of the input - output behavior of the shown in fig . [ figpol ] ( left ) . as a second illustration we use the same to simulate an experiment with three polarizers described by feynman @xcite . the diagram of this experiment is shown in fig . [ fig3pol ] . a randomly polarized beam of photons passes through the first polarizer ( without loss of generality we set its angle @xmath200 equal to zero ) . each output channel is used as input to another polarizer . both these polarizers are tilted by the same angle @xmath201 . according to quantum theory @xcite , the intensity at the output of these four channels is ( from top to bottom , see fig . [ fig3pol ] ) @xmath202 , @xmath203 , @xmath203 , and @xmath202 . the results of our numerical experiments are shown in fig . the simulation procedure is the same as the one used to generate the data of fig . also in these numerical experiments we set @xmath47 . we emphasize once more that the randomness in these discrete - event simulations only enters through the characterization of the photon source and through our procedure of selecting the direction of the polarizer for each set of 1000 events . actually , the latter only serves to counter the possible objection that the apparent quantum mechanical behavior would be caused by monotonically changing the direction of the polarizers . as in the previous example , it is clear that quantum theory @xcite describes the input - output behavior of the three- network very well . we now show that two @xmath142 and two passive devices that perform a plane rotation by @xmath204 are sufficient to build a network that behaves as if it where a single - photon beam splitter . first we describe the network and then we demonstrate that it acts as a beam splitter . ( 14,7 ) ( -1.75,0 ) ( 7,0 ) [ one - bs ] ( 14,7 ) ( -1.75,0 ) ( 7,0 ) [ figmz ] the network shown in fig.[figbs ] has two input channels ( 0 and 1 ) and two output channels ( 0 and 1 ) . the network receives events at one of the two input channels . each input event carries information in the form of a two - dimensional unit vector . either input channel 0 receives @xmath95 or input channel 1 receives @xmath205 . the input is fed into the device described in section [ twoone ] . the purpose of this front - end is to transform the information contained in two - dimensional input vectors ( of which only one is present for any given input event ) , into a four - dimensional unit vector . the four - dimensional internal vector of this device is split into two groups of two - dimensional vectors @xmath206 and @xmath207 and each of these two - dimensional vectors is rotated by @xmath204 . put differently , the four - dimensional vector is rotated once in the ( 1,4)-plane about @xmath204 and once in the ( 3,2 ) plane about @xmath204 . the order of the rotations is irrelevant . the resulting four - dimensional vector is then sent to the input of a second @xmath142 . this back - end sends @xmath208 through output channel 0 if it used rule @xmath149 ( see eq . ) to update its internal state . otherwise it sends @xmath209 through output channel 1 . the operation of the network depicted in fig.[figbs ] can be analyzed analytically if we disregard transient effects and assume that the information carried by events on channel 0 ( 1 ) is given by @xmath210 ( @xmath211 ) . we denote by @xmath212 the number of events on input channel 0 divided by the total number of events . then , the number of events on input channel 1 is given by @xmath213 . in the stationary regime , the internal state @xmath214 of the front - end ( see fig.[figbs ] ) learns @xmath215 . carrying out the two plane rotations of @xmath204 we see that the back - end receives as input the four - dimensional vector @xmath216 . in the stationary regime , the internal vector @xmath217 of the back - end oscillates about @xmath216 . therefore , in the stationary regime and for fixed two - dimensional vectors on input channels 0 and 1 , the input - output relation of the bs network of fig . [ one - bs ] can be written as @xmath218 using two complex numbers instead of four real numbers eq . can also be written as @xmath219 in quantum theory @xcite the presence of photons in the input modes 0 or 1 is represented by the probability amplitudes ( @xmath220 @xcite . according to quantum theory @xcite , the probability amplitudes ( @xmath221 of the photons in the output modes 0 and 1 of a beam splitter are given by @xcite @xmath222 identifying @xmath223 with @xmath224 and @xmath225 with @xmath226 it is clear that by construction , the network in fig . [ figbs ] will allow us to simulate a beam splitter , not by calculating the amplitudes eq . but by a deterministic event - by - event simulation . in fig . [ one - bs ] ( right ) we present results of discrete - event simulations using the network depicted in fig . [ figbs ] ( left ) . before the simulation starts , the internal vectors of the are given a random value ( on the unit sphere ) . each data point represents 10000 events . all these simulations were carried out with @xmath47 . for each set of 10000 events , a uniform random number in the range @xmath227 $ ] generates two angles @xmath228 and @xmath229 . input channel 0 receives @xmath230 with probability @xmath231 . input channel 1 receives @xmath232 with probability @xmath233 . random processes only enter in the procedure to generate the input data . the network processes the events sequentially and deterministically . from fig . [ one - bs ] it is clear that the output of the deterministic -based beam splitter reproduces the probability distributions as obtained from quantum theory @xcite . ( 14,14 ) ( -1,7 ) ( 7,7 ) ( -1,0 ) ( 7,0 ) ( 14,14 ) ( -1,7 ) ( 7,7 ) ( -1,0 ) ( 7,0 ) in quantum physics @xcite , single - photon experiments with one beam splitter provide direct evidence for the particle - like behavior of photons @xcite . the wave mechanical character appears when one performs single - particle interference experiments . in this subsection we construct a network that displays the same interference patterns as those observed in single - photon mach - zehnder interferometer experiments @xcite . the schematic layout of the network is shown in fig . [ figmz ] . not surprisingly , it is exactly the same as that of a real mach - zehnder interferometer . the bs network described in the previous subsection is used for the beam splitters . the phase shift is taken care of by a passive device that performs a plane rotation . clearly there is a one - to - one mapping from each relevant component in the interferometer to a processing unit in the network . recall that the processing units in the network only communicate with each other through the message ( photon ) that propagates through the network . according to quantum theory @xcite , the probability amplitudes ( @xmath221 of the photons in the output modes 0 ( @xmath234 ) and 1 ( @xmath235 ) of the mach - zehnder interferometer are given by @xcite @xmath236 note that in a quantum mechanical setting it is impossible to simultaneously measure ( @xmath237 , @xmath238 ) and ( @xmath239 , @xmath240 ) : photon detectors operate by absorbing photons . however , in our deterministic , event - based simulation there is no such problem . in fig . [ one - mz ] we present a small selection of simulation results for the mach - zehnder interferometer built from . we assume that input channel 0 receives @xmath230 with probability one and that input channel 1 receives no events . this corresponds to @xmath241 . we use uniform random numbers to determine @xmath228 . in all these simulations @xmath47 . the data points are the simulation results for the normalized intensity @xmath242 for i=0,2,3 as a function of @xmath243 . lines represent the corresponding results of quantum theory @xcite . from fig . [ one - mz ] it is clear that quantum theory provides an excellent description of the deterministic , event - based processing by the network . the examples presented in fig . [ one - mz ] do not rule out that there may be settings for the angles @xmath228 , @xmath244 and @xmath200 for which quantum theory fails to give a good description of the behavior of the network . however extensive series of simulations show that this is not the case . instead of presenting the results of these simulations we will demonstrate that quantum theory @xcite also describes the stationary - state input - output behavior of more extended networks . as an example we consider the network depicted in fig . [ fig2mz ] . obviously this network maps exactly onto two chained mach - zehnder interferometers @xcite . now there are seven parameters @xmath231 , @xmath228 , @xmath229 , @xmath244 , @xmath200 , @xmath245 , and @xmath246 that may be varied , so simply plotting selected cases is not the proper procedure to establish that quantum theory describes the stationary - state behavior of the network . therefore we adopt the following strategy . for each set of 10000 events , we use seven random numbers to fix the parameters @xmath231 , @xmath228 , @xmath229 , @xmath244 , @xmath200 , @xmath245 , and @xmath246 . then we collect the data for these 10000 events and compare the intensity in output channel 0 ( @xmath247 ) and 1 ( @xmath248 ) with the corresponding results of quantum theory @xcite . the latter is given by @xmath249 for each choice of @xmath250 we compute the differences @xmath251 and @xmath252 . @xmath247 ( @xmath248 ) is the number of events in the output channel 0 ( 1 ) of the third beam splitter . @xmath253 is the total number of events ( 10000 in this case ) . in fig . [ two - mz ] we show @xmath251 as a function of @xmath231 , @xmath254 , @xmath255 , and @xmath256 . in all these simulations @xmath47 . once again it is clear that quantum theory @xcite provides a very good description of a -based simulation of two chained mach - zehnder interferometers . all simulations that we presented in this section have been performed for @xmath47 . from the description of the learning process it is clear that @xmath16 controls the rate of learning or , equivalently , the rate at which learned information can be forgotten . furthermore it is evident that the difference between a constant input to a and the learned value of its internal variable can not be smaller than @xmath257 . in other words , @xmath16 also limits the precision with which the internal variable can represent a sequence of constant input values . on the other hand , the number of events has to balance the rate at which the can forget a learned input value . the smaller @xmath257 is , the larger the number of events has to be for the to adapt to changes in the input data . we use the last example of section [ mzi ] to illustrate the effect of changing @xmath16 and the total number of events @xmath258 . in fig . [ two - mz - b ] we show the results of repeating the procedure used to obtain the data shown in fig . [ two - mz ] but instead of @xmath47 and @xmath259 events per data point , we used @xmath260 and @xmath261 event per data point . as expected , the difference between the simulation data and the results of quantum theory decreases if @xmath257 decreases and @xmath258 increases accordingly . comparing fig . [ two - mz ] with fig . [ two - mz - b ] it is clear that the decrease of this difference is roughly proportional to the inverse of the square root of the number of events . note that each data point in fig . [ two - mz ] is generated without the use of random processes . in the stationary regime , the sequence of messages that a ( network ) generates is strictly deterministic . for some applications , e.g. for quantum physics @xcite , it may be desirable to randomize these sequences . a marginal modification turns a into a stochastic learning machine ( slm ) . here the term _ stochastic _ does not refer to the learning process but to the method that is used to select the output channel that will carry the outgoing message . in the stationary regime the components of the internal vector represent the probability amplitudes . comparing the ( sums of ) squares of these amplitudes with a uniform random number @xmath262 gives the probability for sending the message over the corresponding output channel . for instance , in the case of the beam splitter bs ( see fig . [ figbs ] ) we replace the back - end dlm by a slm . this slm will send a message over output channel 0 if @xmath263 . otherwise it will activate output channel 1 . although the learning process of this modified bs network is still deterministic , in the stationary regime the output messages are randomly distributed over the two output channels . of course , the distribution of output messages is the same as that of the original -network . replacing by slms in a -network changes the order in which messages are being processes by the network but leaves the content of the messages intact . therefore , in the stationary regime , the distribution of messages over the outputs of the slm - network is essentially the same as that of the original dlm network . as an illustration of the use of slms , we replace the two back - end in the mach - zehnder interferometer network ( see fig . [ figmz ] ( left ) ) by their `` randomized '' version and repeat the procedure that generates the data of fig . [ figmz ] ( right ) . the results of these simulations are shown in fig . [ one - mz - random ] . not unexpectedly , the randomness in the output channel selection is reflected by a ( small ) increase of the scatter on the data points . in this simulation , the output channels 0 and 1 of each beam splitter are activated in a random manner and the functional dependence of @xmath237 , @xmath238 , @xmath264 and @xmath265 on @xmath178 is still in full agreement with quantum theory @xcite . in other words , this slm - network performs a genuine , event - by - event simulation of the ideal ( perfect detectors , etc . ) version of both the single - photon beam splitter and mach - zehnder interferometer experiments by grangier et al @xcite . we have proposed a new procedure to construct deterministic algorithms that have primitive learning capabilities . we have used these algorithms to build deterministic learning machines ( dlms ) . a dlm learns by processing event after event but does not store the data contained in an individual event . connecting the input of a dlm to the output of another dlm yields a locally connected network of dlms . a dlm within the network locally processes the information contained in an event and responds by sending a message that may be used as input for another dlm . a distinct feature of a dlm network is that at any given time , only one event ( message ) is propagating through the network . the dlms process messages in a sequential manner and only communicate with each other by message passing . we have demonstrated that dlm networks can discover relationships between successive events ( see section [ ndim ] ) and that certain classes of dlm networks exhibit behavior that is usually only attributed to quantum systems . in sections [ qi ] and [ slm ] we have presented dlm networks that simulate quantum interference on an event - by - event basis . more specifically , we map each physical part of the real mach - zehnder interferometer onto a dlm and the messages ( phase shifts in this case ) are carried by photons . no ingredient other than simple geometry is used to specify the update rules of the dlms . as the network processes event after event , the network generates output events that build an interference pattern that is described by the quantum theory @xcite of the single - photon beam splitter and mach - zehnder interferometer . to illustrate that dlm networks are indeed capable of simulating quantum interference on an event - by - event basis we also simulate an experiment involving three beam splitters ( i.e. two chained mach - zehnder interferometers ) and demonstrate that quantum theory @xcite also describes the behavior of this network . the results presented in sections [ qi ] and [ slm ] suggest that we may have discovered a systematic procedure to construct algorithms that simulate quantum phenomena using deterministic , local , and event - by - event - based processes . we emphasize that our approach is not a proposal for another interpretation of quantum mechanics . our approach is not an extension of quantum theory in any sense : the probability distributions of quantum theory appear as the result of a deterministic , causal learning process , and not vice versa ( see section [ qi ] ) @xcite . our results suggest that quantum mechanical behavior may originate from an underlying deterministic process @xcite . indeed , it is somewhat ironic that in order to mimic the apparent randomness with which events are observed in experiments , we have to explicitly randomize the output of the dlms to mask the underlying deterministic processes ( see section [ slm ] ) . to the best of our knowledge , this paper contains the first demonstration that quantum interference can be simulated on an event - by - event basis using local , causal , and deterministic processes , and without using concepts such as wave fields or particle - wave duality . at this point it may be worthwhile to recall what a dlm actually does . in a simple physical picture , a dlm is a device ( e.g. beam splitter , polarizer ) that exchanges information with the particles that pass through it . the dlm tries to do this in an effective manner . it learns by comparing the message carried by an event with predictions based on the knowledge acquired by the dlm during the processing of previous events . effectively this comparison amounts to a minimization of the squared error ( see section [ illu ] ) . schrdinger used exactly the same principle to derive his famous equation @xcite but called this approach `` unverstndlich '' in a subsequent publication @xcite . in a future publication we will show that the approach introduced in this paper can be employed to perform event - based simulations of a universal quantum computer @xcite . it has been shown that the time evolution of the wave function of a quantum system can be simulated on a quantum computer @xcite . therefore it should be possible to compute the real - time dynamics of these systems ( including the double - slit experiment mentioned in the introduction ) through discrete - event simulation by constructing appropriate dlm networks . we thank s. miyashita for extensive discussions . we make a distinction between quantum theory and quantum physics . we use the term _ quantum theory _ when we refer to the mathematical formalism , i.e. , the postulates of quantum mechanics ( with or without the wave function collapse postulate ) @xcite and the rules ( algorithms ) to compute the wave function . the term _ quantum physics _ is used for microscopic , experimentally observable phenomena that do not find an explanation within the mathematical framework of classical mechanics . an interactive program that performs the event - based simulations of a beam splitter , one mach - zehnder interferometer , and two chained mach - zehnder interferometers can be found at http://www.compphys.net/dlm m. nielsen and i. chuang , _ quantum computation and quantum information _ , cambridge university press , cambridge ( 2000 ) g. t hooft , `` determinism beneath quantum mechanics '' , quant - ph/0105105 g. t hooft , `` quantum mechanics and determinism '' , quant - ph/0212095 e. schrdinger , ann . * 79 * , 361 ( 1926 )
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we propose and analyse simple deterministic algorithms that can be used to construct machines that have primitive learning capabilities .
we demonstrate that locally connected networks of these machines can be used to perform blind classification on an event - by - event basis , without storing the information of the individual events .
we also demonstrate that properly designed networks of these machines exhibit behavior that is usually only attributed to quantum systems .
we present networks that simulate quantum interference on an event - by - event basis .
in particular we show that by using simple geometry and the learning capabilities of the machines it becomes possible to simulate single - photon interference in a mach - zehnder interferometer . the interference pattern generated by the network of deterministic learning machines is in perfect agreement with the quantum theoretical result for the single - photon mach - zehnder interferometer . to illustrate that networks of these machines are indeed capable of simulating quantum interference we simulate , event - by - event , a setup involving two chained mach - zehnder interferometers .
we show that also in this case the simulation results agree with quantum theory .
# 1 # 1#1 # 1#1 # 1#1 # 1#2#1 # 2 # 1([#1 ] )
| 16,381 | 287 |
starting from the pioneering measurements of the @xmath2 mass difference @xmath3 and of the cp - violating parameter @xmath4 , continuing with the precision measurements of the @xmath5 mixing parameters @xmath6 and @xmath7 and with the recent determination of the @xmath8 oscillation frequency @xmath9 and the first bounds on the mixing phase @xmath10 , until the very recent evidence of @xmath11 mixing , @xmath1 processes have always provided some of the most stringent constraints on new physics ( np ) . for example , it has been known for more than a quarter of century that supersymmetric extensions of the standard model ( sm ) with generic flavour structures are strongly constrained by @xmath12 mixing and cp violation @xcite . the constraints from @xmath12 mixing are particularly stringent for models that generate transitions between quarks of different chiralities @xcite . more recently , it has been shown that another source of enhancement of chirality - breaking transitions lies in the qcd corrections @xcite , now known at the next - to - leading order ( nlo ) @xcite . previous phenomenological analyses of @xmath1 processes in supersymmetry @xcite were affected by a large uncertainty due to the sm contribution , since no determination of the cabibbo - kobayashi - maskawa @xcite ( ckm ) cp - violating phase was available in the presence of np . a breakthrough was possible with the advent of @xmath13 factories and the measurement of time - dependent cp asymmetries in @xmath13 decays , allowing for a simultaneous determination of the ckm parameters and of the np contributions to @xmath1 processes in the @xmath14 and @xmath15 sectors @xcite . furthermore , the tevatron experiments have provided the first measurement of @xmath16 and the first bounds on the phase of @xmath8 mixing . combining all these ingredients , we can now determine allowed ranges for all np @xmath1 amplitudes in the down - quark sector . to complete the picture , the recent evidence of @xmath17 mixing allows to constrain np contributions to the @xmath18 amplitude @xcite . our aim in this work is to consider the most general effective hamiltonian for @xmath1 processes ( @xmath19 ) and to translate the experimental constraints into allowed ranges for the wilson coefficients of @xmath19 . these coefficients in general have the form @xmath20 where @xmath21 is a function of the ( complex ) np flavour couplings , @xmath22 is a loop factor that is present in models with no tree - level flavour changing neutral currents ( fcnc ) , and @xmath23 is the scale of np , _ i.e. _ the typical mass of the new particles mediating @xmath1 transitions . for a generic strongly - interacting theory with arbitrary flavour structure , one expects @xmath24 so that the allowed range for each of the @xmath25 can be immediately translated into a lower bound on @xmath23 . specific assumptions on the flavour structure of np , for example minimal @xcite or next - to - minimal @xcite flavour violation ( mfv or nmfv ) , correspond to particular choices of the @xmath21 functions , as detailed below . our study is analogous to the operator analysis of electroweak precision observables @xcite , but it provides much more stringent bounds on models with non - minimal flavour violation . in particular , we find that the scale of heavy particles mediating tree - level fcnc in models of nmfv must lie above @xmath26 tev , making them undetectable at the lhc . this bound applies for instance to the kaluza - klein excitations of gauge bosons in a large class of models with ( warped ) extra dimensions @xcite . flavour physics remains the main avenue to probe such extensions of the sm . the paper is organised as follows . in sec . [ sec : exp ] we briefly discuss the experimental novelties considered in our analysis . in sec . [ sec : mi ] we present updated results for the analysis of the unitarity triangle ( ut ) in the presence of np , including the model - independent constraints on @xmath1 processes , following closely our previous analyses @xcite . in sec . [ sec : eh ] we discuss the structure of @xmath19 , the definition of the models we consider and the method used to constrain the wilson coefficients . in sec . [ sec : results ] we present our results for the wilson coefficients and for the scale of np . conclusions are drawn in sec . [ sec : concl ] . we use the same experimental input as ref . @xcite , updated after the winter @xmath27 conferences . we collect all the numbers used throughout this paper in tables [ tab : expinput ] and [ tab : hadinput ] . we include the following novelties : the most recent result for @xmath28 @xcite , the semileptonic asymmetry in @xmath29 decays @xmath30 @xcite and the dimuon charge asymmetry @xmath31 from d@xmath32 @xcite and cdf @xcite , the measurement of the @xmath29 lifetime from flavour - specific final states @xcite , the determination of @xmath33 from the time - integrated angular analysis of @xmath34 decays by cdf @xcite , the three - dimensional constraint on @xmath35 , @xmath36 , and the phase @xmath37 of the @xmath29@xmath38 mixing amplitude from the time - dependent angular analysis of @xmath39 decays by d@xmath32 @xcite . .values of the experimental input used in our analysis . the gaussian and the flat contributions to the uncertainty are given in the third and fourth columns respectively ( for details on the statistical treatment see @xcite ) . see text for details . [ cols="<,^,^,^ " , ] + we conclude that any model with strongly interacting np and/or tree - level contributions is beyond the reach of direct searches at the lhc . flavour and cp violation remain the main tool to constrain ( or detect ) such np models . weakly - interacting extensions of the sm can be accessible at the lhc provided that they enjoy a mfv - like suppression of @xmath1 processes , or at least a nmfv - like suppression with an additional depletion of the np contribution to @xmath40 . we thank guennadi borissov , brendan casey , stefano giagu , marco rescigno , andrzej zieminski , and jure zupan for useful discussions . we aknowledge partial support from rtn european contracts mrtn - ct-2004 - 503369 `` the quest for unification '' , mrtn - ct-2006 - 035482 `` flavianet '' and mrtn - ct-2006 - 035505 `` heptools '' . 99 j. r. ellis and d. v. nanopoulos , phys . b * 110 * ( 1982 ) 44 ; j. f. donoghue , h. p. nilles and d. wyler , phys . b * 128 * ( 1983 ) 55 ; j. m. frere and m. belen gavela , phys . b * 132 * ( 1983 ) 107 ; 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we update the constraints on new - physics contributions to @xmath0 processes from the generalized unitarity triangle analysis , including the most recent experimental developments . based on these constraints ,
we derive upper bounds on the coefficients of the most general @xmath1 effective hamiltonian .
these upper bounds can be translated into lower bounds on the scale of new physics that contributes to these low - energy effective interactions .
we point out that , due to the enhancement in the renormalization group evolution and in the matrix elements , the coefficients of non - standard operators are much more constrained than the coefficient of the operator present in the standard model .
therefore , the scale of new physics in models that generate new @xmath1 operators , such as next - to - minimal flavour violation , has to be much higher than the scale of minimal flavour violation , and it most probably lies beyond the reach of direct searches at the lhc .
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since its isolation on an adhesive tape @xcite , graphene has quickly become a material of intensive attention . many researches have revealed various interesting aspects of the material ; see e.g. refs . @xcite for reviews . one of the most interesting features emerges particularly when we apply a magnetic field to it @xcite . the landau levels due to the magnetic field form a structure different from the simple two - dimensional electron gas in that there are levels of zero energy and in that the non - zero energy levels are spaced not equally but proportionally to the square root of the level number . in the present paper , we apply to graphene yet another ingredient of recent interest , namely the @xmath0 symmetry @xcite . in order to attract attention of condensed - matter physicists , let us briefly describe the @xmath0 symmetry here . it refers to the parity - and - time symmetry of a hamiltonian . the simplest example of the @xmath0-symmetric hamiltonian may be the two - by - two matrix @xmath4 which we can interpret as a two - site tight - binding model : the two sites are coupled with a real coupling parameter @xmath5 ; the first site has a complex potential @xmath6 , which can represent injection of particles from the environment , because the amplitude of the wave vector would increase in time as @xmath7 if the site were isolated ; the potential @xmath8 of the second site can represent removal of the particles to the environment . the @xmath9 operator swaps the first and second sites , which is represented by the linear operator @xmath10 the @xmath11 operator is complex conjugation , which is an anti - linear operator . it is easy to confirm that the hamiltonian @xmath12 in eq . satisfies @xmath13 the parity operation @xmath9 swaps @xmath6 and @xmath8 but the time operation @xmath11 switches them back to the original . this is what we mean by the @xmath0 symmetry of the hamiltonian . the hamiltonian @xmath12 has the eigenvalues @xmath14 which are real for @xmath15 , although @xmath12 is non - hermitian ; @xmath16 . for @xmath17 , on the other hand , the eigenvalues become complex ( pure imaginary in this specific model ) . this transition between real and complex eigenvalues physically means the following . in the strong - coupling case @xmath15 , the particles injected to the first site can flow abundantly into the second site , where they are removed at the same rate as the injection . this constitutes a stationary state of a constant flow , which is indicated by the reality of the eigenvalues . in the weak - coupling case @xmath17 , on the other hand , the particles tend to build up in the first site , while they keep becoming scarcer in the second site . this instability is indicated by the non - reality of the eigenvalues . the first situation is often called the @xmath0-symmetric phase , whereas the second one is the @xmath0-broken phase . at the transition point @xmath18 , not only the two eigenvalues coalesce with each other , but the corresponding two eigenvectors become parallel . this is therefore not the standard degeneracy , but often called an exceptional point @xcite , which has a huge literature recently , including experimental studies @xcite . at the exceptional point , the eigenvectors are not complete and the hamiltonian @xmath12 is not diagonalizable . ( in fact , non - hermitian matrices are _ generally diagonalizable _ except at the exceptional points . ) the transition at an exceptional point between the two phases indeed happens in a very wide class of @xmath0-symmetric operators . suppose that a general @xmath0-symmetric hamiltonian has an eigenvector @xmath19 with an eigenvalue @xmath20 : @xmath21 inserting the symmetry relation , we have @xmath22 where we used the fact @xmath23 . when the eigenvalue @xmath20 is real , the operator @xmath0 passes it , yielding @xmath24 which means that @xmath25 is also an eigenvector with the same eigenvalue . if we assume no degeneracy of the eigenvalue @xmath20 for simplicity , we conclude that @xmath26 ; indeed , we can choose the phase of @xmath19 so that we can make @xmath27 ; namely , the eigenvector is @xmath0-symmetric . this is what happens in the @xmath0-symmetric phase . when the eigenvalue @xmath20 is complex , on the other hand , we have , instead of eq . , @xmath28 where @xmath29 denotes the complex conjugate of @xmath20 . this means that we always have a complex - conjugate pair of eigenvalues @xmath20 and @xmath29 with the eigenvectors @xmath19 and @xmath25 ; each eigenvector is not @xmath0-symmetric anymore in spite of the fact that the hamiltonian is still @xmath0-symmetric . this is what happens in the @xmath0-broken phase . in typical situations including the example , two neighboring real eigenvalues in the @xmath0-symmetric phase , as we tune system parameters , are attracted to each other , collide at the exceptional point , and then become a pair of complex - conjugate eigenvalues in the @xmath0-broken phase , which repel each other ; see fig . [ fig - exceptional point ] . -symmetric and @xmath0-broken phases . as we tune system parameters from the @xmath0-symmetric phase to the exceptional point and further on to the @xmath0-broken phase , two real eigenvalues neighboring on the real axis are attracted to each other ( indicated by the blue horizontal arrows on the real axis ) , collide at the exceptional point ( indicated by a green dot ) , and become a complex - conjugate pair , which repel each other ( indicated by the red vertical arrows).,scaledwidth=40.0% ] questions of interest include the following : is it possible to formulate a standardized quantum mechanics for non - hermitian but @xmath0-symmetric hamiltonians with real energy eigenvalues , namely in the @xmath0-symmetric phase ? what is the general theoretical structure of the @xmath0-broken phase , on the other hand ? a more specific subject of study is to find @xmath0-symmetric models that describe physically interesting situations . in the present paper , we introduce the potential @xmath6 to one sublattice of graphene under a magnetic field and the potential @xmath8 to its other sublattice , which constitutes a @xmath0-symmetric situation . it is quite common to introduce a staggered chemical potential to graphene , that is , @xmath30 to one sublattice and @xmath31 to the other sublattice , which may be indeed realized by hexagonal lattice of boron - nitride @xcite , in which boron atoms are on the a sublattice and nitride atoms are on the b sublattice . our @xmath0-symmetric situation may be also realized in the following way : suppose that we put a hexagonal lattice of two elements , such as boron - nitride , on a substrate ; assume that the substrate is an electron - doping material for one element but hole - doing for the other . this can materialize our @xmath0-symmetric situation . for completeness , we should observe that few other @xmath0 , or non - hermitian , versions of the graphene have been proposed in recent years , but in a different spirit with respect to ours @xcite . let us now describe the model in more detail . graphene forms a hexagonal lattice , which is a bipartite lattice ; see fig . [ fig2 ] . a unit cell , indicated by red broken lines in fig . [ fig2 ] , consists of two sites , one on the a sublattice and the other on the b sublattice . if we assume that the electrons , specifically @xmath32 electrons , hop only to nearest - neighbor lattice points , those on a site on the a sublattice hop only to sites on the b sublattice , and vice versa . the tight - binding hamiltonian in the real - space representation therefore is of the following form : on the diagonal , we have two - by - two blocks @xmath33 which is the local hamiltonian inside a unit cell with the chemical potentials @xmath34 and @xmath35 for the a and b sublattices , respectively , and with the non - zero off - diagonal intra - unit - cell hopping elements @xmath36 between the two sites . in addition , the total hamiltonian has the inter - unit - cell hopping elements @xmath36 between different unit cells . by fourier transforming the basis set with respect to the unit cells , we end up with the block - diagonalized hamiltonian @xmath37 with @xmath38 where @xmath39 and @xmath40 are indicated in fig . we thereby have two energy eigenvalues for each wave number @xmath41 , which form two energy bands in the two - dimensional wave - number space , as shown in fig . [ fig - bands ] for @xmath42 . with @xmath42 . the energy unit in the vertical axis is given by @xmath36 , while the unit of the wave numbers @xmath43 and @xmath44 are given by the inverse of the lattice constant , which we put to unity here.,scaledwidth=45.0% ] among the blocks of the block - diagonalized hamiltonian , the most important are the blocks of the two specific wave numbers , namely the dirac points @xmath1 and @xmath2 , respectively specified by @xmath45 at which the energy eigenvalues are degenerate to zero for @xmath42 . because the fermi energy for graphene is zero , these points control the elementary excitation of graphene . the upper and lower energy bands touch at these points , as can be seen in fig . [ fig - bands ] , forming dirac cones around the points , which are schematically shown in fig . [ fig4](a ) . and @xmath2 points for @xmath42 . the fermi energy of graphene is zero , which coincides with the dirac points @xmath1 and @xmath2 . ( b ) the landau levels are formed under a magnetic field . the levels are spaced proportionally to @xmath46 ; see sec . [ sectsam ] for the definition of the quantum number @xmath47 . ( c ) shifts of the landau levels for @xmath48 . the levels are spaced as @xmath49 . the central landau level @xmath50 has already become complex . ( d ) further shifts for @xmath51 . the levels with @xmath52 have collided with each other and become complex.,scaledwidth=60.0% ] in the standard graphene , therefore , the low - energy excitations follow relativistic quantum mechanics ; this is one big feature of graphene , namely , the desktop relativity . as we predicted above , we apply two ingredients to the graphene tight - binding model , namely a magnetic field and a @xmath0-symmetric chemical potential . first , the spectrum is quantized to the landau levels under a magnetic field . focusing on the dirac cones around the @xmath1 and @xmath2 points , we can write down the effective hamiltonian as in eq . below . as is well studied ( see e.g. ref . @xcite ) , which we will repeat in our way in sec . [ sectsam ] , the landau levels are not equally spaced as in the standard two - dimensional electron gas , but spaced proportionally to @xmath46 , as shown schematically in fig . [ fig4](b ) ; see sec . [ sectsam ] for the definition of the quantum number @xmath47 . each landau level has an infinite number of degeneracy because of another quantum number @xmath53 . we then further apply the @xmath0-symmetric potential to the model . we set the potentials to @xmath54 for the a sublattice and @xmath55 for the b sublattice , as is represented in eq . below . let us define the @xmath9 operation as the mirror reflection with respect to the horizontal axis of fig . [ fig2 ] ; it then swaps the a and b sublattices with each other , changes the sign of the potentials @xmath56 , which is represented by the transformation @xmath57 the @xmath11 operation , which is the complex conjugation , then changes the hamiltonian back to the original one . see the end of sec . [ sec1.1 ] for a possible materialization of the @xmath0-symmetric situation . we will show in sec . [ sec3 ] that the landau levels are then spaced proportionally to @xmath49 ( after proper parameter normalization ) under a set of biorthogonal eigenstates . therefore , as we increase the potential @xmath3 , two landau levels labeled by @xmath58 approach each other , collide with each other when @xmath59 , which is an exceptional point , and then split into a pair of two pure imaginary eigenvalues @xmath60 ; see fig . [ fig4](c d ) . we will deduce that at this exceptional point , the eigenvectors of the two landau levels become parallel , which makes the set of biorthogonal eigenstates incomplete . note that each landau level still has an infinite number of degeneracy because of the other quantum number @xmath61 . note also that the central level @xmath50 becomes a complex eigenvalue as soon as we introduce the @xmath0-symmetric potential @xmath3 ; see fig . [ fig4](b c ) . we show that the central level @xmath50 of the @xmath1 point coalesces with the central level @xmath50 of the @xmath2 point and becomes complex as @xmath56 . this particular coalescence , however , is not an exceptional point but a degeneracy because it occurs in the hermitian limit @xmath62 . these are probably the main results of the present paper . this article is organized as follows : in the next section [ sectsam ] we briefly review the hermitian version of the model under a magnetic field and some of its main mathematical characteristics . in sec . [ sec3 ] we introduce our @xmath0-symmetrically deformed version of the model with @xmath56 and consider the consequences of this deformation . our conclusions and future perspective are given in sec . [ sec5 ] . to make the paper self - contained , we have added [ appa ] with some useful facts for non - hermitian hamiltonians . let us first consider a layer of graphene in an external constant magnetic field along @xmath63 : @xmath64 , which can be deduced from @xmath65 with a vector potential in the symmetric gauge , @xmath66 . the hamiltonian for the two dirac points @xmath1 and @xmath2 can be written as @xcite @xmath67 where , in the units @xmath68 , we have @xmath69 while @xmath70 is just its transpose : @xmath71 . here @xmath72 and @xmath73 are the canonical , hermitian , two - dimensional position and momentum operators , which satisfy @xmath74=[y , p_y]=i{1 \!\ ! 1}$ ] with all the other commutators being zero , where @xmath75 is the identity operator in the hilbert space @xmath76 . the factor @xmath77 is the so - called fermi velocity . the scalar product in @xmath78 will be indicated as @xmath79 . let us now introduce the parameter called the magnetic length , @xmath80 , as well as the following canonical operators : @xmath81 these operators can be used to define two different pairs of bosonic operators : we first put @xmath82 and @xmath83 , and then @xmath84 the following commutation rules are satisfied : @xmath85=[a_y , a_y^\dagger]=[a_1,a_1^\dagger]=[a_2,a_2^\dagger]={1 \!\ ! 1 } , \label{22}\end{aligned}\ ] ] with the other commutators being zero . in terms of these operators , @xmath86 appears particularly simple . indeed , we find @xmath87 for @xmath88 and @xmath89 for @xmath90 . note that @xmath91 and @xmath92 are different expressions of the same hamiltonian . it is evident that @xmath93 , and a similar conclusion can also be deduced for @xmath70 . it is also clear that neither @xmath91 nor @xmath94 depends on @xmath95 and @xmath96 , so that their eigenstates possess a manifest degeneracy . the same is true for @xmath92 nor @xmath97 , which do not depend on @xmath98 and @xmath99 . however , from now on , we will essentially concentrate on @xmath100 and @xmath94 , except for what is discussed in appendix b. most of what we are going to discuss from now on can be restated easily for @xmath101 and @xmath97 . for instance , the eigenvectors of @xmath101 could be found from those of @xmath94 , replacing the operators @xmath95 and @xmath96 with @xmath98 and @xmath99 , and vice versa . now , let @xmath102 be the non - zero vacuum of @xmath95 and @xmath98 : @xmath103 . then we introduce , as usual , @xmath104 the set @xmath105 is an orthonormal basis for @xmath78 , being the same as the one for a two - dimensional harmonic oscillator . rather than working in @xmath78 , in order to deal with @xmath91 it is convenient to work in a different hilbert space , namely the direct sum of @xmath78 with itself , @xmath106 : @xmath107 in the new hilbert space @xmath108 , the scalar product @xmath109 is defined as @xmath110 and the square norm is @xmath111 , for all @xmath112 , @xmath113 in @xmath108 . introducing now the vectors @xmath114 we have an orthonormal basis set @xmath115 for @xmath108 . this means , among other things , that @xmath116 is complete in @xmath108 : the only vector @xmath117 which is orthogonal to all the vectors of @xmath116 is the zero vector . in view of application to graphene it is more convenient to use a different orthonormal basis of @xmath108 , the set @xmath118 , where @xmath119 quite often , in the rest of the paper , we call this vector simply @xmath120 . for @xmath121 , we have @xmath122 it is easy to check that these vectors are mutually orthogonal , normalized in @xmath108 , and complete . hence , @xmath123 is an orthonormal basis , as stated before . this is not surprising , since its vectors are indeed the eigenvectors of @xmath91 : @xmath124 where @xmath125 . more compactly we can simply write @xmath126 . we see explicitly that the eigenvalues have an infinite degeneracy with respect to the quantum number @xmath61 , which can be removed by using the angular momentum @xcite . we will not consider this aspect here , since it is not relevant for us . of course , both @xmath116 and @xmath123 can be used to produce two different resolutions of the identity . indeed we have @xmath127 for all @xmath117 . * remark : * what we have seen so far can be easily adapted to the analysis of the hamiltonian for the other dirac cone , @xmath94 , which is simply the transpose of @xmath91 , and , as we have already pointed out , also to @xmath92 and @xmath97 . we will say more on the other dirac cone in sec . [ sec3.2 ] , in the presence of the @xmath0-symmetric potential . we now introduce the @xmath0-symmetric chemical potential to eq . ( [ 23 ] ) as follows : @xmath128 where @xmath3 is assumed to be a strictly positive ( real ) quantity . as we show the details in [ appb ] , for @xmath62 , the _ set _ of dirac cones at @xmath1 and @xmath2 are time - reversal symmetric as well as parity symmetric . for @xmath129 , it observes neither symmetries but does the @xmath0 symmetry . an easy extension of the standard arguments allows us to deduce that the general expression of the eigenvectors are still , as in the case with @xmath62 , of the form ( [ 28 ] ) , but with some essential difference , which is also reflected in the form of the eigenvalues . in particular we first find that @xmath130 which reduces to the known value if @xmath131 , and which is still independent of @xmath61 . a major difference appears as follows : if @xmath132 , then the values of the energy are real ; we are in the @xmath0-symmetric region . as soon as @xmath133 , however , the energy turns out to be complex , and we are in the @xmath0-broken region . we will come back to this later on . going now to the eigenvectors , we first observe that @xmath134 is an eigenvector of @xmath135 with the eigenvalue @xmath136 . on the other hand , we can prove that there is no non - zero eigenstate corresponding to @xmath137 . in fact , if we assume that such a non - zero vector @xmath138 does exist , it must satisfy the equation @xmath139 , which implies in turn that @xmath140 and @xmath141 should satisfy the equations @xmath142 and @xmath143 . acting on this last with @xmath98 and using the first , we obtain @xmath144 , so that @xmath145 and therefore @xmath146 . then we have @xmath147 for this last equality to be satisfied , we must have @xmath148 . hence @xmath141 must be zero . the fact that @xmath149 also is now a consequence of the equality above @xmath143 , at least if @xmath129 . then the trivial vector @xmath150 is the only solution that satisfies the equation @xmath139 . in the limit @xmath62 , on the other hand , the equation @xmath143 does not imply that @xmath149 and in fact a nontrivial ground state in this case does exist , as discussed in sec . [ sectsam ] . the reason for this is that , if @xmath62 , there is no difference between @xmath151 and @xmath152 , which are both zero . as for the levels with @xmath121 , the normalized eigenstates are deformed versions of those in eq . ( [ 28 ] ) . more in detail , defining the following quantities , which are in general complex , @xmath153 we can write @xmath154 with these definitions we have @xmath155 it is easy to see what happens for @xmath156 , since this can be recovered from @xmath135 just replacing everywhere @xmath3 with @xmath157 . in particular , since the eigenvalues are quadratic in @xmath3 , @xmath135 and @xmath156 turn out to be isospectral . concerning the eigenstates , these are deduced from the eigenvectors @xmath158 just with the same substitution . more in details , calling @xmath159 for all @xmath121 , we can write @xmath160 and @xmath161 analogously to what happens for @xmath135 , only one ground state of @xmath156 does exist , which coincides with @xmath162 above . however , the corresponding eigenvalue is now @xmath137 , so that we conclude that @xmath163 . on the other hand , no non - zero eigenvector does exist which corresponds to @xmath136 . we thus deduce a similar situation with respect to the one observed for @xmath135 . we therefore conclude that the case @xmath50 is really exceptional ; indeed we have @xmath164 , while neither @xmath165 nor @xmath166 do exist . this should be remembered in the rest of the paper , since all the formulas considered from now on , and in particular those in section [ sec3.1 ] , are valid only when these particular vectors are not involved . * remarks : * ( i ) in the limit @xmath167 , all the result reduces to the ones discussed in sec . [ sectsam ] . in particular the fact that @xmath163 agrees with the fact that , in this limit , @xmath168 . it is also interesting to observe that the coefficients @xmath169 and @xmath170 simply returns @xmath171 or @xmath172 , as in formula ( [ 28 ] ) . ( ii ) the choice of normalization in ( [ 35 ] ) and ( [ 38 ] ) is such that @xmath173 . we prefer this choice , rather than the one which could also be used which makes the scalar product between @xmath158 and @xmath174 equal to unity , since this biorthogonality strongly refer to the value of @xmath3 . this will be evident in the next section . let us call @xmath175 and @xmath176 . because of their particular forms and because of the orthogonality of the vectors @xmath177 , it is clear that @xmath178 for all @xmath179 , and for all choices of @xmath180 and @xmath181 . it is also possible to check that , @xmath182 therefore , eigenstates of @xmath135 corresponding to different eigenvalues are not mutually orthogonal . this is not surprising , since @xmath135 is not hermitian in the present settings . however , we can check that the orthogonality is recovered when @xmath3 is sent to zero , i.e. , when @xmath135 becomes hermitian . what still remains , as quite often in situations like ours , is the possible biorthogonality of the sets @xmath183 and @xmath184 . in fact , this is not so automatic , and needs some care . the point is the following : if @xmath12 is not hermitian but two of its eigenvalues @xmath185 and @xmath186 are real , then the states @xmath187 and @xmath188 that satisfy @xmath189 and @xmath190 are guaranteed to be mutually orthogonal . if @xmath185 or @xmath186 , or both , are complex , on the other hand , this is no longer granted in general . we will show that in our particular situation of the @xmath0-broken region , the biorthogonality of the sets @xmath183 and @xmath184 is recovered only when properly pairing the eigenstates . first we observe that @xmath191 can only be different from zero if @xmath192 . otherwise these scalar products are all zero . now , if we compute @xmath193 for instance , we deduce that , neglecting an unnecessary multiplication factor , @xmath194 the result of this computation depends on the values of @xmath58 and @xmath3 . in fact , we can check that for @xmath132 , we have @xmath195 , but for @xmath196 this is not true . hence @xmath197 similarly we can check that @xmath198 is zero for @xmath132 , but is not zero otherwise . it is also interesting to notice that a completely opposite result is deduced in the @xmath0-broken region , i.e. for purely imaginary eigenvalues . in fact , for @xmath133 , we deduce that the different pair satisfies @xmath199 so that they are biorthogonal , while they are in general not for @xmath200 : @xmath201 for @xmath200 . these results are of course related to the reality of the eigenvalues of @xmath135 and @xmath156 . in fact , when @xmath132 , the eigenvalues @xmath202 are all real , and we know for general reasons that @xmath158 must be orthogonal to @xmath203 , but not , in general , to @xmath174 . on the other hand , when the eigenvalues are purely imaginary , @xmath158 are necessarily orthogonal to @xmath174 , but not to @xmath203 . this has consequences on the possibility of introducing a metric , at least in the way which is discussed in ref . @xcite for instance ; see [ appa ] . here , in fact , the intertwining operator between @xmath135 and @xmath156 has the formal expression @xmath204 ( we recall that @xmath205 ) . however this operator acts in different ways depending on whether we are in the @xmath0-symmetric or @xmath0-broken region . for instance , for @xmath200 ( @xmath0-symmetric region ) , @xmath206 is proportional to @xmath207 . however , for @xmath133 ( @xmath0-broken region ) , we can find that @xmath206 is proportional to @xmath208 . therefore , except for some multiplicative coefficients which can be fixed properly , @xmath209 can change eigenstates @xmath210 of @xmath156 either into the eigenstates @xmath207 or into the eigenstates @xmath208 of @xmath135 , depending on the parameter region . of course , a similar behavior is expected for an operator @xmath211 defined in analogy with @xmath209 . an interesting issue to consider now is the completeness of the sets @xmath183 and @xmath184 . in many applications in quantum mechanics with non - hermitian hamiltonians the eigenvectors of a given @xmath12 and @xmath212 are , in fact , non - orthogonal but complete in their hilbert space . what may or may not be true is that they are also bases for such a hilbert space @xcite . hence , it is surely worth to investigate this kind of properties for @xmath183 and @xmath184 . in the present case , we obtain the following interesting result : if @xmath3 is such that @xmath213 is not a natural number , then @xmath183 and @xmath184 are complete in @xmath108 . if , on the other hand , @xmath213 is a positive integer number @xmath214 , then @xmath183 and @xmath184 are not complete . let @xmath215 be a vector which is orthogonal to all the eigenvectors @xmath158 . we would like to show if and in which condition @xmath216 is zero . first of all , since @xmath217 in particular , it follows that @xmath218 for all @xmath219 . moreover , we also have , for @xmath121 and for all @xmath61 , @xmath220 then , by subtraction , we have @xmath221 but , since @xmath222 , it follows that , if @xmath213 is not equal to any natural numbers , then @xmath223 for all @xmath61 and for all @xmath121 . then , because of the completeness of the set @xmath224 , we conclude that @xmath225 . this result , together with ( [ 310 ] ) , now implies that @xmath226 for all @xmath61 and for @xmath121 . since we also have that @xmath218 , however , it follows that @xmath227 after using again the completeness of @xmath224 . hence @xmath228 . let us now check what happens if @xmath229 , for some particular natural number @xmath214 . in this case we find that @xmath230 for all @xmath61 , and therefore @xmath231 we see that we are _ losing one vector _ , so that it is not really surprising that the set @xmath183 ceases to be complete . in fact , a simple computation shows that , for instance , the non - zero vector @xmath232 is orthogonal to all the eigenvectors @xmath158 as well as to @xmath233 for all fixed @xmath219 . a similar proof can be repeated for the set @xmath184 . * remarks : * ( i ) the content of this proposition can be understood in terms of exceptional points : we have an exceptional point when @xmath229 , for a natural number @xmath214 , while no exceptional point exists if @xmath213 is not natural . it is exactly the presence of an exceptional point which makes two eigenvectors collapse into a single one , and this prevent @xmath183 to be a basis . \(ii ) the above result implies that in order for @xmath183 or @xmath184 to be bases for @xmath108 , @xmath3 must be such that its square is not a natural number , since any basis must be , first of all , complete and , in this situation , our sets are not . on the other hand , whenever @xmath213 is not an integer , @xmath183 or @xmath184 could be bases , but the question is , for the time being , still open . we believe that , even if this is often not so for non - hermitian hamiltonians @xcite , it is probably true in the present situation . it is now interesting to observe that the results that we have deduced so far can be easily adapted to the other dirac cone at @xmath2 . this is because the hamiltonian @xmath234 in this case is simply the transpose of @xmath135 in eq . ( [ 31 ] ) . hence we have @xmath235 if we now compare the generic eigenvalue equations for @xmath236 and @xmath234 , @xmath237 it is easy to see that the second equation is mapped into the first one if we put @xmath238 , @xmath239 and @xmath240 . hence the conclusion is that the eigenvectors of @xmath234 are just those which we have deduced previously after this changes , and that the eigenvalues are just those of @xmath236 but with signs exchanged . more in details we find that , for all @xmath219 and @xmath121 , @xmath241 where @xmath242 and @xmath243 when @xmath50 we have @xmath244 where @xmath245 , and @xmath246 combining @xmath247 for @xmath234 with @xmath136 for @xmath236 ( see below eq . ) , we see that these two eigenvalues become complex as in fig . [ fig - exceptional point ] but without the horizontal arrows . notice that , similarly to what happened for @xmath236 , the hamiltonian @xmath234 has no ( non - zero ) eigenstate corresponding to @xmath248 . similar features as those considered for @xmath236 arise also here , as for instance the completeness of the sets of eigenstates of @xmath234 and of its adjoint , and the conclusions do not differ from what we have found so far ; we will not repeat similar considerations here . in this paper we have considered an extended non - hermitian version of the graphene hamiltonian close to the dirac points @xmath1 and @xmath2 . on a mathematical side we have shown that , depending on the value of the parameter @xmath3 measuring this non - hermiticity , exceptional points may arise , which breaks down the existence of a basis for @xmath108 . in fact , the set of eigenstates of @xmath236 is not even complete at the exceptional points . we have also deduced an interesting behavior concerning the zeroth eigenvalues and eigenvectors of the model : while @xmath162 does exist , no @xmath166 can be found in @xmath108 , at least if @xmath249 . similarly , @xmath250 does exist , but @xmath165 does not . hence , introducing @xmath3 in the hamiltonian creates a sort of asymmetry between the plus and the minus eigenstates , at least for the ground state . this asymmetry disappears as soon as @xmath3 is sent to zero . if we compare the conclusion for the @xmath0-symmetric graphene with the physical view of the simplest case that we described in introduction , we may say the following . the electrons doped on one sublattice may not be carried to the other sublattice through the central channels @xmath50 as soon as we introduce the @xmath0-symmetric chemical potential . the other channels remain open until @xmath251 , when the corresponding @xmath58th channel is closed . it may be an interesting future work to drive the system around an exceptional point to see the state swapping @xcite . this work was supported by national group of mathematical physics ( gnfm - indam ) . f.b . also acknowledges partial support by the university of palermo . this paper does not contain any computational solution . this work did not involve any active collection of human data . this work does not have any experimental data . we have no competing interests . fb cured the mathematical part of the paper , with the help of nh . nh cured the physical interpretation of the results , with the help of fb . both authors gave final approval for publication . this work was partly supported by gnfm - indam and by the university of palermo . we here briefly describe the general notion of the intertwining operator that we have introduced in sec . [ sec3.1 ] . in order to avoid mathematical problems , we focus here on finite - dimensional hilbert spaces . in this way our operators are finite matrices . the main ingredient is an operator ( i.e. a matrix ) @xmath12 , acting on the vector space @xmath252 , with @xmath253 and with exactly @xmath254 distinct eigenvalues @xmath20 , @xmath255 , where the hermitian conjugate @xmath256 of @xmath12 is the usual one , i.e. the complex conjugate of the transpose of the matrix @xmath12 . because of what follows , and in order to fix the ideas , it is useful to remind here that the hermitian conjugate @xmath257 of an operator @xmath258 is defined in terms of the _ natural _ scalar product @xmath259 of the hilbert space @xmath260 : @xmath261 , for all @xmath262 , where @xmath263 , with obvious notation . in this appendix we will restrict to the case in which all the eigenvalues @xmath20 are real , and with multiplicity one . hence @xmath264 the set @xmath265 is a basis for @xmath252 , since the eigenvalues are all different . then an unique biorthogonal basis of @xmath78 , @xmath266 , surely exists @xcite : @xmath267 , for all @xmath268 . it is easy to check that @xmath269 is automatically an eigenstate of @xmath256 , with eigenvalue @xmath270 : @xmath271 using the bra - ket notation we can write @xmath272 , where , for all @xmath273 , we define @xmath274 . we now introduce the ` intertwining ' operators @xmath275 and @xmath276 , following ref . these are bounded positive , hermitian , invertible operators , one the inverse of the other : @xmath277 . moreover @xmath278 and we also get the following intertwining relations involving @xmath12 , @xmath256 , @xmath279 and @xmath211 : @xmath280 notice that the second equality follows from the first one , by left and right multiplying @xmath281 with @xmath279 . to prove the first equality , we first observe that @xmath282 for all @xmath283 . hence our claim follows because of the basis nature of @xmath284 . * remark : * it might be interesting to recall that the intertwining operators , such as @xmath279 and @xmath211 , are quite useful in quantum mechanics , @xmath0-symmetric or not @xcite , in order to deduce eigenvectors of certain hamiltonians connected by intertwining relations . for instance , let us assume that @xmath285 is an eigenstate of a certain operator @xmath286 with eigenvalue @xmath20 : @xmath287 , and let us also assume that two other operators @xmath288 and @xmath258 exist such that @xmath289 and that the intertwining relation @xmath290 is satisfied . this is exactly what happens in ( [ a4 ] ) , identifying @xmath258 with @xmath211 , @xmath286 with @xmath12 and @xmath288 with @xmath256 . then , it is a trivial exercise to check that the non - zero vector @xmath291 is an eigenstate of @xmath288 , with eigenvalue @xmath20 . indeed we have @xmath292 note that the fact that @xmath286 and @xmath288 are hermitian or not and the fact that @xmath20 is real or not play no role . note also that the fact that @xmath293 can be deduced out of @xmath285 simply by applying @xmath258 , is exactly what happens in our situation ; see eq . ( [ a3 ] ) . this explains why the intertwining operators are so important in concrete applications ; they can be used , for instance , to find eigenstates of new operators starting from eigenstates of old ones . * remark : * it is probably worth mentioning that not all we have discussed here can be easily extended if @xmath294 . for instance , considering the intertwining relations in ( [ a4 ] ) , if , for instance , @xmath12 and @xmath279 are unbounded , taken @xmath295 , the domain of @xmath279 , there is no reason _ a priori _ for @xmath296 to belong to @xmath297 , so that @xmath298 needs not to be defined . in this appendix we will briefly discuss the role of the @xmath11 and @xmath9 symmetries in our model . the @xmath11 operator works as follows : @xmath299 note that , as expected for physical reasons , the time - reversal operator flips the magnetic field too . we therefore have @xmath300 when we apply @xmath11 to @xmath301 we have @xmath302 we thus realize that the time reversal of the dirac cone at @xmath1 is the negative of the dirac cone at @xmath2 . for @xmath62 , we can flip the sign by the diagonal unitary transformation @xmath303 as in @xmath304 . similarly we have @xmath305 . note that @xmath91 and @xmath92 are different expressions of the same hamiltonian , expressions which depend on the direction of the magnetic field along @xmath63 . the model for @xmath62 is time - reversal symmetric in this sense . under @xmath11 , the dirac cone at @xmath1 is transformed to the one at @xmath2 , which in turn is transformed to the one at @xmath1 . therefore , the set of the two dirac cones for @xmath62 has the time - reversal symmetry . the time - reversal symmetry is broken when @xmath129 because @xmath306 . this is also true for the parity operation @xmath307 for which we have @xmath308 for @xmath62 , this is isomorphic to @xmath97 but for @xmath129 , @xmath309 is isomorphic to @xmath310 , which is not equal to @xmath311 . k. s. novoselov , a. k. geim , s. v. morozov , d. jiang , m. i. katsnelson , i. v. grigorieva , s. v. dubonos , a. a. firsov , two - dimensional gas of massless dirac fermions in graphene , nature * 438 * , 197200 ( 2005 ) . c. dembowski , h .- grf , h. l. harney , a. heine , w. d. heiss , h. rehfeld , a. richter , experimental observation of the topological structure of exceptional points , phys . lett . * 86 * , 787790 ( 2001 ) . m. brandstetter , m. liertzer , c. deutsch , p. klang , j. schberl , h. e. treci , g. strasser , k. unterrainer , s. rotter , reversing the pump dependence of a laser at an exceptional point , nature commun . * 5 * , 4034 ( 2014 ) . c. r. dean , a. f. young , i. meric , c. lee , l. wang , s. sorgenfrei , k. watanabe , t. taniguchi , p. kim , k. l. shepard , j. hone , boron nitride substrates for high - quality graphene electronics , nature nanotech . * 5 * , 722726 ( 2010 ) . l. ci , l. song , c. jin , d. jariwala , d. wu , y. li , a. srivastava , z. f. wang , k. storr , l. balicas , f. liu , p. m. ajayan , atomic layers of hybridized boron nitride and graphene domains , nature materials * 9 * , 430435 ( 2010 ) . m. yankowitz , j. xue , d. cormode , j. d. sanchez - yamagishi , k. watanabe , t. taniguchi , p. jarillo - herrero , p. jacquod , b. j. leroy , emergence of superlattice dirac points in graphene on hexagonal boron nitride , nature physics * 8 * , 382386 ( 2012 ) . f. bagarello , m. g. gianfreda , @xmath313deformed and susy - deformed graphene : first results , in _ non - hermitian hamiltonians in quantum physics _ - selected contributions from the 15th international conference on non - hermitian hamiltonians in quantum physics , palermo , italy , 18 - 23 may 2015 , springer ( 2016 ) f. bagarello , deformed canonical ( anti-)commutation relations and non - hermitian hamiltonians , in _ non - hermitian operators in quantum physics : mathematical aspects _ , f. bagarello , j. p. gazeau , f. h. szafraniek and m. znojil , eds . , john wiley and sons , new jersey ( 2015 )
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we propose a @xmath0-symmetrically deformed version of the graphene tight - binding model under a magnetic field .
we analyze the structure of the spectra and the eigenvectors of the hamiltonians around the @xmath1 and @xmath2 points , both in the @xmath0-symmetric and @xmath0-broken regions .
in particular we show that the presence of the deformation parameter @xmath3 produces several interesting consequences , including the asymmetry of the zero - energy states of the hamiltonians and the breakdown of the completeness of the eigenvector sets .
we also discuss the biorthogonality of the eigenvectors , which turns out to be different in the @xmath0-symmetric and @xmath0-broken regions . * @xmath0-symmetric graphene under a magnetic field * + fabio bagarello + dipartimento di energia , ingegneria dellinformazione e modelli matematici , + facolt di ingegneria , universit di palermo , + i-90128 palermo , italy + e - mail : fabio.bagarello@unipa.it + home page : www.unipa.it/fabio.bagarello naomichi hatano + institute of industrial science , university of tokyo , + komaba 4 - 6 - 1 , meguro , tokyo 153 - 8505 , japan + e - mail : hatano@iis.u-tokyo.ac.jp +
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proteins are essential building blocks of living organisms . they function as catalyst , structural elements , chemical signals , receptors , etc . the molecular mechanism of protein functions are closely related to their structures . the study of structure - function relationship is the holy grail of biophysics and has attracted enormous effort in the past few decades . the understanding of such a relationship enables us to predict protein functions from structure or amino acid sequence or both , which remains major challenge in molecular biology . intensive experimental investigation has been carried out to explore the interactions among proteins or proteins with other biomolecules , e.g. , dnas and/or rnas . in particular , the understanding of protein - drug interactions is of premier importance to human health . a wide variety of theoretical and computational approaches has been proposed to understand the protein structure - function relationship . one class of approaches is biophysical . from the point of view of biophysics , protein structure , function , dynamics and transport are , in general , dictated by protein interactions . quantum mechanics ( qm ) is based on the fundamental principle , and offers the most accurate description of interactions among electrons , photons , atoms and even molecules . although qm methods have unveiled many underlying mechanisms of reaction kinetics and enzymatic activities , they typically are computationally too expensive to do for large biomolecules . based on classic physical laws , molecular mechanics ( mm ) @xcite can , in combination with fitted parameters , simulate the physical movement of atoms or molecules for relatively large biomolecular systems like proteins quite precisely . however , it can be computationally intractable for macromoelcular systems involving realistic biological time scales . many time - independent methods like normal mode analysis ( nma ) @xcite , elastic network model ( enm ) @xcite , graph theory @xcite and flexibility - rigidity index ( fri ) @xcite are proposed to capture features of large biomolecules . variational multiscale methods @xcite are another class of approaches that combine atomistic description with continuum approximations . there are well developed servers for predicting protein functions based on three - dimensional ( 3d ) structures @xcite or models from the homology modeling ( here homology is in biological sense ) of amino acid sequence if 3d structure is not yet available @xcite . another class of important approaches , bioinformatical methods , plays a unique role for the understanding of the structure - function relationship . these data - driven predictions are based on similarity analysis . the essential idea is that proteins with similar sequences or structures may share similar functions . also , based on sequential or structural similarity , proteins can be classified into many different groups . once the sequence or structure of a novel protein is identified , its function can be predicted by assigning it to the group of proteins that share similarities to a good extent . however , the degree of similarity depends on the criteria used to measure similarity or difference . many measurements are used to describe similarity between two protein samples . typical approaches use either sequence or physical information , or both . among them , sequence alignment can describe how closely the two proteins are related . protein blast @xcite , clustalw2 @xcite , and other software packages can preform global or local sequence alignments . based on sequence alignments , various scoring methods can provide the description of protein similarity @xcite . additionally , sequence features such as sequence length and occurrence percentage of a specific amino acid can also be employed to compare proteins . many sequence based features can be derived from the position - specific scoring matrix ( pssm ) @xcite . moreover , structural information provides an efficient description of protein similarity as well . structure alignment methods include rigid , flexible and other methods . the combination of different structure alignment methods and different measurements such as root - mean - square deviation ( rmsd ) and z - score gives rise to various ways to quantify the similarity among proteins . as per structure information , different physical properties such as surface area , volume , free energy , flexible - rigidity index ( fri ) @xcite , curvature @xcite , electrostatics @xcite etc . can be calculated . a continuum model , poisson boltzmann ( pb ) equation delivers quite accurate estimation for electrostatics of biomolecules . there are many efficient and accurate pb solvers including pbeq @xcite , mibpb @xcite , etc . together with physical properties , one can also extract geometrical properties from structure information . these properties include coordinates of atoms , connections between atoms such as covalent bonds and hydrogen bonds , molecular surfaces @xcite and curvatures @xcite . these various approaches reveal information of different scales from local atom arrangement to global architecture . physical and geometrical properties described above add different perspective to analyze protein similarities . due to the advance in bioscience and biotechnology , biomolecular structure date sets are growing at an unprecedented rate . for example , the http://www.rcsb.org/pdb/home/home.do[protein data bank ( pdb ) ] has accumulated more than a hundred thousand biomolecular structures . the prediction of the protein structure - function relationship from such huge amount of data can be extremely challenging . additionally , an eve - growing number of physical or sequence features are evaluated for each data set or amino - acid residue , which adds to the complexity of the data - driven prediction . to automatically analyze excessively large data sets in molecular biology , many machine learning methods have been developed @xcite . these methods are mainly utilized for the classification , regression , comparison and clustering of biomolecular data . clustering is an unsupervised learning method which divides a set of inputs into groups without knowing the groups beforehand . this method can unveil hidden patterns in the data set . classification is a supervised learning method , in which , a classifier is trained on a given training set and used to do prediction for new observations . it assigns observation to one of several pre - determined categories based on knowledge from training data set in which the label of observations is known . popular methods for classification include support vector machine ( svm ) @xcite , artificial neural network ( ann ) @xcite , deep learning @xcite , etc . in classification , each observation in training the set has a feature vector that describes the observation from various perspectives and a label that indicates to which group the observation belongs . a model trained on the training set indicates to which group a new observation belongs with feature vector and unknown label . to improve the speed of classification and reduce effect from irrelevant features , many feature selection procedures have been proposed @xcite . machine learning approach are successfully used for protein hot spot prediction @xcite . the data - driven analysis of the protein structure - function relationship is compromised by the fact that same protein may have different conformations which possess different properties or delivers different functions . for instance , hemoglobins have taut form with low affinity to oxygen and relaxed form with high affinity to oxygen ; and ion channels often have open and close states . different conformations of a given protein may only have minor differences in their local geometric configurations . these conformations share the same sequence and may have very similar physical properties . however , their minor structural differences might lead to dramatically different functions . therefore , apart from the conventional physical and sequence information , geometric and topological information can also play an important role in understanding the protein structure - function relationship . indeed , geometric information has been extensively used in the protein exploration . in contrast , topological information has been hardly employed in studying the protein structure - function relationship . in general , geometric approaches are frequently inundated with too much geometric detail and are often prohibitively expensive for most realistic biomolecular systems , while traditional topological methods often incur in too much reduction of the original geometric and physical information . persistent homology , a new branch of applied topology , is able to bridge traditional geometry and topology . it creates a variety of topologies of a given object by varying a filtration parameter , such as a radius or a level set function . in the past decade , persistent homology has been developed as a new multiscale representation of topological features . the 0-th dimensional version was originally introduced for computer vision applications under the name size function " @xcite and the idea was also studied by robins @xcite . the persistent homology theory was formulated , together with an algorithm given , by edelsbrunner et al . @xcite , and a more general theory was developed by zomorodian and carlsson @xcite . there has since been significant theoretical development @xcite , as well as various computational algorithms @xcite . often , persistent homology can be visualized through barcodes @xcite , in which various horizontal line segments or bars are the homology generators which survive over filtration scales . persistence diagrams are another equivalent representation @xcite . computational homology and persistent homology have been applied to a variety of domains , including image analysis @xcite , chaotic dynamics verification @xcite , sensor network @xcite , complex network @xcite , data analysis @xcite , shape recognition @xcite and computational biology @xcite . compared with traditional computational topology @xcite and/or computational homology , persistent homology inherently has an additional dimension , the filtration parameter , which can be utilized to embed some crucial geometric or quantitative information into the topological invariants . the importance of retaining geometric information in topological analysis has been recognized @xcite , and topology has been advocated as a new approach for tackling big data sets @xcite . recently , we have introduced persistent homology for mathematical modeling and prediction of nano particles , proteins and other biomolecules @xcite . we have proposed molecular topological fingerprint ( mtf ) to reveal topology - function relationships in protein folding and protein flexibility @xcite . we have employed persistent homology to predict the curvature energies of fullerene isomers @xcite , and analyze the stability of protein folding @xcite . more recently , we have introduced resolution based persistent topology @xcite . most recently , we have developed new multidimensional persistence , a topic that has attracted much attention in the past few years @xcite , to better bridge geometry and traditional topology and achieve better characterization of biomolecular data @xcite . we have also introduced the use of topological fingerprint for resolving ill - posed inverse problems in cryo - em structure determination @xcite . the objective of the present work is to explore the utility of mtfs for protein classification and analysis . we construct feature vectors based on mtfs to describe unique topological properties of protein in different scales , states and/or conformations . these topological feature vectors are further used in conjugation with the svm algorithm for the classification of proteins . we validate the proposed mtf - svm strategy by distinguishing different protein conformations , proteins with different local secondary structures , and proteins from different superfamilies or families . the performance of proposed topological method is demonstrated by a number of realistic applications , including protein binding analysis , ion channel study , etc . the rest of the paper is organized as following . section [ sec : methods ] is devoted to the mathematical foundations for persistent homology and machine learning methods . we present a brief description of simplex and simplicial complex followed by basic concept of homology , filtration , and persistence in section [ persistenthomology ] . three different methods to get simplicial complex , vietoris - rips complex , alpha complex , and ech complex are discussed . we use a sequence of graphs of channel proteins to illustrate the growth of a vietoris - rips complex and corresponding barcode representation of topological persistence . in section [ svm+roc ] , fundamental concept of support vector machine is discussed . an introduction of transformation of the original optimization problem is given . a measurement for the performance of classification model known as receiver operating characteristic is described . section [ feature+preprocessing ] is devoted to the description of features used in the classification and pre - processing of topological feature vectors . in section [ sec : numerical ] , four test cases are shown . case 1 and case 2 examine the performance of the topological fingerprint based classification methods in distinguishing different conformations of same proteins . in case 1 , we use the structure of the m2 channel of influenza a virus with and without an inhibitor . in case 2 , we employ the structure of hemoglobin in taut form and relaxed form . case 3 validates the proposed topological methods in capturing the difference between local secondary structures . in this study , proteins are divided into three groups , all alpha protein , all beta protein , and alpha+beta protein . in case 4 , the ability of the present method for distinguishing different protein families is examined . this paper ends with some concluding remarks . this section presents a brief review of persistent homology theory and illustrates its use in proteins . a brief description of machine learning methods is also given . the topological feature selection and construction from biomolecular data are described in details . * simplex * a @xmath0-simplex denoted by @xmath1 is a convex hull of @xmath2 vertices which is represented by a set of points @xmath3 where @xmath4 is a set of affinely independent points . geometrically , a @xmath5-@xmath6 is a line segment , a @xmath7-simplex is a triangle , a @xmath8-simplex is a tetrahedron , and a @xmath9-simplex is a @xmath10-cell ( a four dimensional object bounded by five tetrahedrons ) . a @xmath11face of the @xmath0-simplex is defined as a convex hull formed from a subset consisting @xmath12 vertices . * simplicial complex * a simplicial complex @xmath13 is a finite collection of simplices satisfying two conditions . first , faces of a simplex in @xmath13 are also in @xmath13 ; second , intersection of any two simplices in @xmath13 is a face of both the simplices . the highest dimension of simplices in @xmath13 determines dimension of @xmath13 . * homology * for a simplicial complex @xmath13 , a @xmath0-chain is a formal sum of the form @xmath14 $ ] , where @xmath15 $ ] is oriented @xmath0-simplex from @xmath13 . for simplicity , we choose @xmath16 . all these @xmath0-chains on @xmath13 form an abelian group , called chain group and denoted as @xmath17 . a boundary operator @xmath18 over a @xmath0-simplex @xmath1 is defined as , @xmath19,\ ] ] where @xmath20 $ ] denotes the face obtained by deleting the @xmath21th vertex in the simplex . the boundary operator induces a boundary homomorphism @xmath22 . an very important property of the boundary operator is that the composition operator @xmath23 is a zero map , @xmath24+\sum_{j > i}(-1)^i(-1)^{j-1}[u_0, ... ,\widehat{u_j}, ... \widehat{u_i}, ... u_k ] \\ & = 0 \end{aligned}\ ] ] a sequence of chain groups connected by boundary operation form a chain complex , @xmath25{}}c_n(\mathcal{k})\xrightarrow{\makebox[.27in]{$\partial_n$}}c_{n-1}(\mathcal{k})\xrightarrow{\makebox[.27in]{$\partial_{n-1}$}}\cdots\xrightarrow{\makebox[.27in]{$\partial_1$}}c_0(\mathcal{k } ) \xrightarrow{\makebox[.27in]{$\partial_0$}}0.\ ] ] the equation @xmath26 is equivalent to the inclusion @xmath27 , when @xmath28 and @xmath29 denotes image and kernel . elements of @xmath30 are called @xmath0th cycle group , and denoted as @xmath31=ker@xmath18 . elements of @xmath32 are called @xmath0th boundary group , and denoted as @xmath33=im@xmath34 . a @xmath0th homology group is defined as the quotient group of @xmath31 and @xmath33 . @xmath35 the @xmath0th betti number of simplicial complex @xmath13 is the rank of @xmath36 , @xmath37 betti number @xmath38 is finite number , since @xmath39 . betti numbers computed from a homology group are used to describe the corresponding space . generally speaking , the betti numbers @xmath40 , @xmath41 and @xmath42 are numbers of connected components , tunnels , and cavities , respectively . * filtration and persistence * a filtration of a simplicial complex @xmath13 is a nested sequence of subcomplexes of @xmath13 . @xmath43 with a filtration of simplicial complex @xmath13 , topological attributes can be generated for each member in the sequence by deriving the homology group of each simplicial complex . the topological features that are long lasting through the filtration sequence are more likely to capture significant property of the object . intuitively , non - boundary cycles that are not mapped into boundaries too fast along the filtration are considered to be possibly involved in major features or persistence . equipped with a proper derivation of filtration and a wise choice of threshold to define persistence , it is practicable to filter out topological noises and acquire attributes of interest . the @xmath44-persistent @xmath0th homology group of @xmath45 is defined as @xmath46 where @xmath47 and @xmath48 . the consequent @xmath44-persistent @xmath0th betti number is @xmath49 . a well chosen @xmath44 promises reasonable elimination of topological noise . * vietoris - rips complex * based on a metric space @xmath50 and a given cutoff distance @xmath51 , an abstract simplicial complex can be built . if two points in @xmath50 have a distance shorter than the given distance @xmath51 , an edge is formed between these two points . consequently , simplices of different dimensions are formed and a simplicial complex is built . for a point cloud data , natural metric space based on euclidean distance or other metric spaces based on alternative definition of distance can be used to build a vietoris - rips complex . for example , any correlation matrix can be used directly to form a vietoris - rips complex . figure [ fig : ex1 ] illustrates growth of vietoris - rips complex along with increment of @xmath51 over the point set of @xmath52 atoms from m2 chimera channel . there are many ways of constructing complex other than vietoris - rips complex , including alpha complex , cech complex , cw complex , etc . in the present work , we used vietoris - rips complex in part because of its intuitive nature and in part because of the moderate size of the systems we studied . the computational topology package javaplex@xcite was used for computation of persistent homology . the results were represented in the form of barcodes @xcite . figure [ fig : ex2 ] illustrates barcodes computed from a point cloud data extracted from @xmath53 atoms of protein i d 2ljc . svm is a machine learning method that can be applied to classification and regression problems . it computes a hyperplane which maximizes margin between positive and negative training sets . in this work , classification svm type 1 , also known as c - support vector classification ( c - svc ) @xcite is used . for the problem of classification , with pre - determined classes , a classifier is trained on a data set with the description of samples and their classes and it predicts the class of a new observation . the input for svm is a set of samples . each sample has a feature vector that describes the properties of the sample and a label that implies to which class the sample belongs . given the input which is the training set , svm will generate a hyperplane in the feature space or higher dimensional spaces depending on which kernel it uses that separates the classes . for two - class svm , it looks for a hyperplane @xmath54 that separates the classes . the determination of the coefficients @xmath55 and @xmath56 breaks down to a constrained optimization problem as @xmath57 subject to @xmath58 where @xmath59 denotes the feature vector of the @xmath21th sample , @xmath60 is the label of the @xmath21th sample which takes value of either @xmath5 or @xmath61 , and @xmath62 is a penalty coefficient for misclassified points . to handle linearly inseparable data , one maps the data into a higher dimensional space as @xmath63 with @xmath64 . since in the optimization problem and in scoring function of the classifier , the operator used is dot product , @xmath65 does not need to be explicitly found . a decaying kernel @xmath66 function is used to represent @xmath67 . commonly used kernel functions include linear function : @xmath68 , polynomial : @xmath69 , radial basis functions ( rbfs ) such as gaussian @xmath70 . in fact , the admissible kernels of fleibility - rigidity index ( fri ) @xcite work too . in this work , the gaussian kernel is used and a 5-fold cross validation was applied to search for optimized training parameters for problems with large amount of samples . to solve the optimization problem , the original problem is transformed into the corresponding lagrange dual problem . for a contained optimization problem @xmath71 the lagrange function of this problem is defined as @xmath72 where @xmath73 and @xmath74 are lagrange multipliers . the lagrange dual problem is defined as @xmath75 where @xmath76 . the lagrange function of the original optimization problem ( [ eq : opt-1 ] ) is formulated as @xmath77 tthe corresponding dual problem with karush - kuhn - tucker conditions is defined as @xmath78 the dual problem can be solved with sequential minimal optimization ( smo ) method @xcite . roc is a plot that visualizes the performance of a binary classifier @xcite . a binary classifier uses a threshold value to decide the prediction label of an entry . in testing process , we define true positive rate ( tpr ) and false positive rate ( fpr ) for the testing set . @xmath79 an roc space is a two dimensional space defined by points with @xmath80 coordinate representing fpr and @xmath81 coordinate representing tpr . in the prediction process of a binary classifier , a score is assigned to a sample by the classifier . a test sample may be labeled as positive or negative with different threshold value used by the classifier . corresponding to a certain threshold value , there is a pair of fpr and tpr values which is a point in the roc space . all such points will fall in the box @xmath82\times[0,1]$ ] . points above the diagonal line @xmath83 are considered as good predictors and those below the line are considered as poor predictors . if a point is below the diagonal line , the predictor can be inverted to be a good predictor . for points that are close to the diagonal line , they are considered to act similarly to random guess which implies a relatively useless predictor . roc curve is obtained by plotting fpr and tpr as continuous functions of threshold value . the area between roc curve and @xmath80 axis represents probability that the classifier assigns higher score to a randomly chosen positive sample than to a randomly chosen negative sample if positive is set to have higher score than negative . the area under the curve ( auc ) of roc is a measure of classifier quality . intuitively , a higher auc implies a better classifier . in this work , algebraic topology is employed to discriminate proteins . specifically , we compute mtfs through the filtration process of protein structural data . mtfs bear the persistence of topological invariants during the filtration and are ideally suited for protein classification . to implement our topological approach in the svm algorithm , we construct protein feature vectors by using mtfs . we select distinguishing protein features from mtfs . these features can be both long lasting and short lasting betti 0 , betti 1 , and betti 2 intervals . table [ tab : features ] lists topological features used for classification . detailed explanation of these features is discussed . the length and location value of bars are in the unit of angstrom ( ) for protein data .
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protein function and dynamics are closely related to its sequence and structure
. however prediction of protein function and dynamics from its sequence and structure is still a fundamental challenge in molecular biology .
protein classification , which is typically done through measuring the similarity between proteins based on protein sequence or physical information , serves as a crucial step toward the understanding of protein function and dynamics .
persistent homology is a new branch of algebraic topology that has found its success in the topological data analysis in a variety of disciplines , including molecular biology .
the present work explores the potential of using persistent homology as an independent tool for protein classification . to this end
, we propose a molecular topological fingerprint based support vector machine ( mtf - svm ) classifier .
specifically , we construct machine learning feature vectors solely from protein topological fingerprints , which are topological invariants generated during the filtration process . to validate the present mtf - svm approach , we consider four types of problems .
first , we study protein - drug binding by using the m2 channel protein of influenza a virus .
we achieve 96% accuracy in discriminating drug bound and unbound m2 channels .
additionally , we examine the use of mtf - svm for the classification of hemoglobin molecules in their relaxed and taut forms and obtain about 80% accuracy .
the identification of all alpha , all beta , and alpha - beta protein domains is carried out in our next study using 900 proteins .
we have found a 85% success in this identification . finally , we apply the present technique to 55 classification tasks of protein superfamilies over 1357 samples . an average accuracy of 82% is attained .
the present study establishes computational topology as an independent and effective alternative for protein classification .
key words : persistent homology , machine learning , protein classification , topological fingerprint .
* running title : topological protein classification *
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the light source of our experiment is a continuous - wave ( cw ) ti : sapphire laser operating at the wavelength of 860 nm . in addition to the setup shown in fig . 1a , there are two optical cavities which are omitted from fig . 1a . one cavity is a second - harmonic generator , which is a bow - tie - shaped cavity and contains a periodically - poled ktiopo@xmath1 ( ppktp ) crystal as a nonlinear optical medium . the resulting continuous output beam at the wavelength of 430 nm is , after a frequency shift by an acousto - optic modulator , directed as a pump beam to each memory cavity , which contains a periodically - poled ktiopo@xmath1 ( ppktp ) crystal as a nonlinear optical medium and works as a non - degenerate optical parametric oscillator ( nopo ) . the pumping power at each memory cavity is about 3 mw . optionally , the pump beams can be individually blocked after a herald in order to prevent further production of photons , but this was not employed in this demonstration , because the probability of such unwanted events is small . the other cavity is a mode - cleaning cavity , by which the transverse mode of the local oscillators is purified to a tem@xmath2 mode in order to maximize the interference visibility at the homodyne detections . the power of each local oscillator is about 18 mw . the concatenated - cavity - based memory systems ( memory-1 , memory-2 ) have the same design , similar to our previous experiment @xcite but with parameters slightly different from before . the design of the concatenated - cavity - based memory system is as follows . the transmissivity of the coupling mirror that couples the memory cavity and the shutter cavity is 1.6 % and the transmissivity of the outcoupling mirror at the exit of the shutter cavity is 24.7% . the former transmissivity corresponds to a compromise between long lifetime and phase - locking stability , while the latter is set in accordance with the former in order to operate the system nearly at the critical damping condition @xcite . the memory cavity has a free spectrum range ( fsr ) of 214.1 mhz . spontaneous parametric down conversion inside the memory cavity produces signal and idler photons which are separated by this fsr . the shutter cavity contains an electro - optic modulator ( eom ) , which is a rbtiopo@xmath1 ( rtp ) crystal with an aperture size of 4 mm @xmath3 4 mm . it is driven by a high - voltage switch ( bergmann messgerte entwicklung kg ) , whose voltage is around 900 v to match the frequency shift to the fsr of the memory cavity . in order to stabilize the whole setup , all of the resonant frequencies of the cavities and the phases of the local oscillators are each electronically controlled by using a feedback loop . each controller is composed of an analog feedback controller , an error detector of locking , and a digital scanner for the error recovery . the error detection signals from all of the controllers are brought together by logical or gates , and then sent to the timing controller in order to pause the measurement when there is an error . the timing controller is a field - programmable gate array ( fpga ) ( virtex-4 , xilinx ) , which processes the heralding signals and controls the timing of the photon release . the same timing controller actually also controls the sequence of the switching between the feedback phase and the measurement phase , as explained below . for the analog feedback control , we monitor the optical systems by means of bright beams . however , such bright beams , except for the local oscillator beams and the pump beams , represent an extra complication in our single - photon - level experiment . therefore we cyclically switch the optical systems . one phase is the feedback phase where the bright beams are injected to the cavity systems . the other is the measurement phase where the bright beams are blocked and the two - photon interference is tested . the switching rate is 5 khz , and the duty cycle is 40% for the measurement phase . as mentioned above , the fpga controls the release timing of photons . the fpga clock frequency is 100 mhz , and thus , in our system , the release timings are synchronized with 10 ns intervals . note that the timing jitter of the driving signals in the timing controller was negligibly small ( less than 1 ns ) , compared to the width of the wavepackets of photons ( about 100 ns ) . the photon arrival times of the two inputs are matched by adjusting the electric cable lengths from the timing controller to the eoms , as well as the optical path lengths from the exits of the shutter cavities to the balanced beam splitter of the hom interferometer . the lengths of the two output arms to the data - storage oscilloscope are also matched . these were tested in the preliminary experiment . in the hom experiment , the photons are released when both heralding events of memory-1 and memory-2 happen within 2 @xmath4 . if the second herald does not occur by 2 @xmath4 , the timing controller becomes idle and waits for 5 @xmath0s , which is long enough compared to the memory lifetime of about 2 @xmath0s . during this dead time , the heralded photon is almost lost and the memory systems are almost initialized again . in the preliminary experiment , we tested our memory systems by using a highly reflective mirror as the replaceable mirror ( rm ) . for each memory-@xmath5 , we repeated single - photon generation and homodyne detection 43,404 times for each variable set storage time @xmath6 after the herald , changed from 0 ns to 500 ns at 50-ns intervals . from the results , we estimated the shapes of the wavepackets @xmath7 by utilizing principal component analysis @xcite , as shown in fig . 1b , and calculated the single - photon purities @xmath8 ( corresponding to the respective single - photon fractions ) , as shown in fig . the single - photon purities are shown to degrade for longer storage times indicating a finite memory lifetime , but the wigner - function - negativity condition @xcite with purities above 0.5 is still satisfied for up to about 0.4 @xmath0s . as shown in fig . 1c , the maximum storage time @xmath9 of the synchronization can be set much longer than this 0.4 @xmath0s , thus preserving the negativity . the shapes of the estimated wavepackets @xmath7 , shown in fig . 1b , are independent of @xmath6 , excluding pre - leakage . furthermore , they are almost the same between memory-1 and memory-2 . the overlap @xmath10 was 0.992 . we defined the signal wavepacket for the hom experiment as an average temporal mode : @xmath11/ \sqrt{\int dt | f_1(t ; 0 ) + f_2(t ; 0)|^2}$ ] . the overlap between the signal wavepacket and the estimated wavepacket released from the individual memory-@xmath5 after @xmath6 , defined as @xmath12 , was larger than 0.96 for up to @xmath6 = 500 ns , as listed in supplementary table [ table : modefunction ] . this high value contributes to the visibility of the hom interference and the purities of the input states . the estimated purities @xmath8 corresponding to the estimated wavepackets @xmath7 in fig . 1bare listed in table [ table : modefunction ] , corresponding to the exact values of the points in fig . s1a . in order to exclude the influence of the mode mismatches on the input purities , we also calculated purities @xmath13 by using the time - shifted signal wavepacket @xmath14 , as also shown in table [ table : modefunction ] . from the values below , we can see that the contribution of the pre - leakage is very small . the decay of the purity is thus due to the optical losses in each memory system . these results are fitted with an exponential decay curve @xmath15 , where @xmath16 means the initial purity and @xmath17 means the memory lifetime for the memory system . in the hom experiment , we synchronized the photon sources , created an initial dual - single - photon state , and then obtained an output hom state by replacing the highly - reflective rm in fig . 1a with a balanced beam splitter . the storage time of each event was recorded together with the homodyne outcomes . then , in order to observe the dependence on @xmath9 , we analyzed those events associated with a storage time between 0 and @xmath9 . figure s1b shows the obtained number of events with respect to each @xmath9 . it indicates a linear increase of the dual - heralding event rate for any extension of @xmath9 . here we discuss how the maximum storage time @xmath9 is dependent on the dual - heralding event rate . we assume that the individual heralding events of the memories occur at random and independently , i.e. they follow the poisson distributions with average event rates @xmath18 for each memory-@xmath5 ( @xmath19 ) . considering the ( dimensionless ) measurement duty @xmath20 and ignoring the 5-@xmath0s dead time for simplicity , the theoretical dual - heralding event rate is given by @xmath21 , when @xmath9 is much smaller than @xmath22 and @xmath23 . here the factor of 2 effectively stems from the fact that the first heralding event may occur in either of memory-1 and memory-2 . this linear relationship of the dual - heralding event rate for @xmath9 can be confirmed in fig . the experimental parameters were @xmath24 counts per second ( cps ) , and @xmath25 . from above , the predicted dual - heralding event rate for @xmath26 1.8 @xmath4 is @xmath27 cps , which agrees well with the experimental value of about 90 cps . in order to estimate the enhancement by the memories in the dual - heralding event rate , here we discuss the coherence time of the wavepacket without synchronization , from the aspect of the 0.5-bound of the wigner - function - negativity condition . however , we note that this is a problem dependent on how the signal wavepacket is defined . a reasonable definition of the signal wavepacket is to set the temporal origin of the signal wavepaket to the mean of the two heralding event timings : when individual heralding events happen at @xmath28 and @xmath29 , then a signal wavepacket @xmath30 has a balanced overlap with both of @xmath31 and @xmath32 . by using this definition , we calculate the average purity for various timing - mismatched events @xmath33 as a function of the maximum time difference @xmath34 . then we find that the average purity crosses the 0.5-bound of the wigner - function - negativity condition at @xmath34 = 72 ns . comparing this @xmath34 of 72 ns with the @xmath9 of 1.8 @xmath4 for the synchronized case , and taking into account the linear dependence on @xmath9 , the memory enhancement in the event rate can be estimated at a factor of 25 . let us now discuss the image of the hom state in the wave basis by considering wavefunctions ( probability amplitudes ) in fig . s2 . theoretically , vacuum _ math _ , single - photon _ math _ , and two - photon _ math _ states have zero , one , and two nodes in their schrdinger standing wave functions _ math _ for the field quadrature amplitude _ math _ , respectively ( fig . the two - mode wave functions of separable states _ math _ , _ math _ , and _ math _ are products of the single - mode wave functions ( figs . s2b d ) . when _ math _ and _ math _ are coherently superimposed to obtain the hom state , _ math _ , the probability amplitudes at the origin destructively or constructively interfere depending on _ math_. as a result , the superimposed wave function corresponds to a four - leaf clover for _ math _ ( fig . s2e ) and a concentric pattern for _ math _ ( fig . s2f ) . comparing fig . s2e with fig . s2d representing the input state _ math _ , we can see that the interference at the balanced beam splitter for _ math _ corresponds to a 45@xmath35 rotation of the pattern . the probability distributions _ math _ are the absolute squares of the wave functions ( figs . s2g h ) . [ [ experimental - quadrature - distributions - for - the-36-measurement - bases ] ] experimental quadrature distributions for the 36 measurement bases ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ in the hom experiment , the optical phases @xmath36 of the two measured quadratures @xmath37 , @xmath38 are independently set to @xmath39 , @xmath40 , @xmath41 , @xmath42 , @xmath43 , and @xmath44 . the quadrature distributions for the tested @xmath45 combinations are shown in fig . these are the results for a @xmath9 of 400 ns as a typical example . the distributions for equal relative phases , as can be seen in the plots along the diagonal of fig . s3 , are almost the same , thus reflecting the phase - insensitivity of the input single - photon states . the characteristic results of the output for the relative phase _ math _ of @xmath39 and _ math _ are collected in fig . s4 , together with those of phase - insensitive input state . [ [ quantitative - evaluation - of - the - entanglement - of - the - output - hom - state ] ] quantitative evaluation of the entanglement of the output hom state ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ here we quantitatively evaluate the output state with regards to its correlations , interference visibility , and entanglement . the intensity cross - correlation factor , which corresponds to the depth of the hom dip , @xmath46 , is 0.24 @xmath47 0.01 . the interference visibility , defined as @xmath48 is 0.902 @xmath47 0.005 . this is much higher than the phase - randomized coherent light limit of 0.5 , which confirms the non - classicality of the output state . in order to verify the quantum entanglement in the two output ports , we calculated the entanglement criterion @xcite . the logarithmic negativity @xcite of the output state was @xmath49 , where the superscript of @xmath50 means partial transpose with respect to mode 1 . in order to exclude the contribution of undesired photon - number components to the entanglement , we also calculated the logarithmic negativity of the output state after numerically applying local filters @xmath51 to individual modes . the local filters do not increase the entanglement but just remove irrelevant photon - number components and transform the output state to @xmath52 . here , @xmath53 is the filtered density operator of @xmath54 in the hilbert space spanned by @xmath55 , and @xmath56 , @xmath57 is the probability fraction of the filtered components which was experimentally 0.470 , and @xmath58 is a non - entangled state in the orthogonal hilbert space . for this filtered output state , the logarithmic negativity was @xmath59 . from comparison of these two values , we can see the main contribution of the entanglement came from the superposition of @xmath60 and @xmath61 , but there is also a small discrepancy of the log - negativities which is mainly due to the higher - order photon components @xmath62 and @xmath63 of the input states . the non - zero log - negativities of the filtered and unfiltered states confirm the entanglement of our output hom state . by using the density matrices of the input states ( fig . s5 a , b ) , we theoretically calculated a density matrix of the output state with the ideal balanced beam splitter ( fig . s5 e , f ) . compared to this simulation , the density matrix of the experimental output hom state has a similar structure ( fig . s5 c , d ) . additional important information about the phase - space characteristics of the entangled hom state can be obtained by analyzing its wigner function . we calculated the wigner function of the output state in order to verify the strong non - classicality of this state . the wigner function of a two - mode state @xmath64 is defined ( @xmath65 ) in four - dimensional space as @xmath66 where @xmath67 correspond to a pair of rotated quadratures . in fig . s6 , the wigner function of the ideal hom state and that of the experimental output state are shown , for different vertical scales . as the wigner function is very sensitive to the non - classical properties of a quantum state , its value quickly changes owing to experimental imperfections . however , the wigner function of the output state still shows the distinctive shape of the ideal hom state : it has a positive value at the origin and negative values at four points around the origin in fig . s6c , which come from the strongly non - classical single - photon inputs , thus confirming that the output state maintains the strong non - classicality after the photon interference .
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a fundamental element of quantum information processing with photonic qubits is the nonclassical quantum interference between two photons when they bunch together via the hong - ou - mandel ( hom ) effect .
ultimately , many such pure photons must be processed in complex interferometric networks , and for this it is essential to synchronize the arrival times of the flying photons preserving their purity . here
we demonstrate for the first time the hom interference of two heralded , pure optical photons synchronized through two independent quantum memories . controlled storage times up to 1.8 @xmath0s for about 90 events per second were achieved with purities sufficiently high for a negative wigner function confirmed with homodyne measurements .
optical photons are a fundamental resource to encode flying quantum bits for quantum communication and computation .
in particular , in linear - optics quantum information processing @xcite , universal two - qubit gates rely upon nonclassical quantum interferences , where photons tend to bunch due to their bosonic nature .
the elementary manifestation for this is the so - called hong - ou - mandel ( hom ) effect @xcite : when two indistinguishable single photons _
math _ enter a balanced beam splitter , they bunch in either of the two output ports , resulting in a hom state _ math _ with some relative phase _ math_. for large - scale quantum computation , many pure single photons must be available simultaneously at the input ports of large interferometric networks , in order to apply the corresponding gate sequences at the same time on all the initial qubits .
numerous tests of the hom effect have been performed over the past few decades mainly in order to characterize single - photon sources , such as parametric down converters @xcite , trapped single neutral atoms @xcite , ions @xcite , atomic ensembles @xcite , quantum dots @xcite , and nitrogen vacancy centers in diamond @xcite .
however , the simultaneous occurrence of two single photons at a beam splitter has depended on a random coincidence between two independent statistical sources .
one possible way toward scalability is to combine statistical photon sources with quantum memories , by which a photon is stored until the other photon is available too @xcite .
however , such quantum storage must not be at the expense of the single - photon purity . only in a very recent experiment ,
optical single - photon states sufficiently pure to show a negative dip in their wigner functions @xcite were released in a controlled fashion ( quasi on demand ) from a memory system based on optical cavities @xcite or atomic ensembles @xcite . here
we report the next significant step beyond this : the controlled hom interference of two nearly pure photons that emerge from two independent quantum memories employing the cavity - storage method @xcite .
the resulting states still have single - photon purities above 0.5 , which is sufficient for a negative wigner function .
thanks to the memories , utilizing controlled storage times of up to 1.8 @xmath0s , the output hom state can be synchronized .
compared to previous works with atomic memories , the memory times of our all - optical system are of similar order @xcite , however , the purities of the synchronized photons are raised to an unprecedented , qualitatively different level .
we believe that this controlled , almost on - demand demonstration of the hom effect represents a breakthrough toward scaling up photonic quantum interference experiments , with direct applications in linear - optics quantum computation @xcite , quantum communication @xcite , and boson sampling @xcite .
another important aspect of our approach , making it distinct from all postselective schemes based on particle - like click - by - click photon detections , is the characterization of the resulting hom state from a wave - like perspective using homodyne measurements of field quadrature amplitudes . here
we will show that a characteristic pattern in the wave basis survives even after the active synchronization with quantum memories .
a fundamental feature of quantum mechanics is the wave - particle duality , and our demonstration looks at the famous hom effect from a completely wave - like angle .
this is not only of fundamental interest but also practically important for optical hybrid quantum information processing , where both continuous wave and discrete particle properties are exploited for quantum state preparation , processing , and detection @xcite . since the first demonstration @xcite
, the hom effect has been always demonstrated as a dip in the coincidence probability of photon detections at both output ports of the beam splitter .
however , the hom dip only reflects a particle - like aspect of the hom state , in which photonic particle bunching becomes manifest .
the more general , actual quantum nature of that prominent optical quantum state , such as quantum entanglement , can not be revealed by correlation measurements in a fixed particle basis .
the wave - basis image of the hom state that we present here as a counterpart to the particle - basis hom dip is a correlation pattern similar to a four - leaf clover that vanishes and reappears depending on the relative phase _ math _ @xcite .
the phase - dependent pattern is actually sufficient for fully characterizing the hom state and obtaining its density matrix @xcite .
however , the pattern is very fragile against optical losses and detection noises mostly because the homodyne detection is sensitive to vacuum fluctuations , unlike the hom dip whose shape is in principle unchanged even for large optical losses .
therefore , in order to observe the phase - sensitive clover pattern , highly pure single photons must be prepared simultaneously and detected with very low - noise homodyne detectors , which became possible only very recently @xcite .
our experimental setup is schematically shown in fig . 1a , where every one of the two single - photon sources is enclosed by a memory system ( memory-1 or memory-2 ) which is composed of two concatenated cavities @xcite .
each individual single - photon creation corresponds to an ordinary quantum optical heralding scheme , where photons are probabilistically but simultaneously produced in pairs by nonlinear optical effects and one photon serves as the herald of the other @xcite .
single - photon sources based on such nonlinear optical effects have good controllability of the wavelength unlike other on - demand - type sources .
the special interference inside the concatenated cavities ensures that the photon to be measured is released to the outside while the photon to be prepared stays inside . after heralding ,
the stored single photon is released on demand by rapidly switching the cavity resonance via an electro - optic effect @xcite , which is also different from a simple storage - loop switching scheme @xcite . in a preliminary experiment
, we tested the performance of the individual memory systems by using a highly reflective mirror as the replaceable mirror ( rm ) in fig .
1a .
for both input ports , we estimated the wavepackets of the released single photons for various fixed storage times after the heralding @xcite .
figure 1b shows the longitudinal modes of the released wavepackets in the time domain , vertically shifted depending on the corresponding storage times .
it can be confirmed that the single - photon wavepackets are correctly shifted by the memories without deformation , and also that the wavepackets from the two independent memories are almost identical .
this sameness ( indistinguishability ) of the wavepackets is critical for the hom interference effect . in the hom experiment
, we synchronized the photon sources and then obtained an output hom state .
we acquired two - mode quadrature data for 390,636 events when both single photons were heralded within 2 @xmath0s .
this large number of events was acquired within only 3 hours .
the storage time of each event was recorded together with the homodyne outcomes .
then , in order to observe the dependence on the maximum storage time _ math _
, we analyzed those events associated with a storage time between 0 and _ math_. figure 1c shows the individual input single - photon purities , calculated with the maximum - likelihood method @xcite for each _ math_. although the single - photon purities degrade for longer storage times due to a finite memory lifetime @xcite , the purities were kept above the 0.5-bound of the wigner - function - negativity condition for up to about 1.8 @xmath0s , which corresponds to about 90 events per second .
the coherence time of the wavepackets for the 0.5 criteria without synchronization is estimated at 72 ns @xcite , and thus the memory enhancement in the event rate can be calculated as a factor of 25 .
let us now discuss the image of the hom state in the wave basis , as shown in fig . 2 .
theoretically , vacuum _ math _ , single - photon _ math _ , and two - photon _ math _ states have one , two , and three peaks in their quadrature distributions , respectively , and thus the two - dimensional distributions of the separable states _ math _ and _ math _ look like those in fig .
2a and fig . 2b , respectively .
when _ math _ and _ math _ are coherently superimposed to obtain the hom state , _ math _ , the probability amplitudes at the origin destructively or constructively interfere depending on _ math _ @xcite . as a result ,
the remaining distribution patterns are the four - leaf clover for _ math _ ( fig .
2c ) and a concentric pattern for _ math _ ( fig .
2d ) .
these characteristic distribution patterns are indeed observed in our experiment by homodyne detections . in figs .
3a , b , the distributions of the output for _ math _ and _ math _ are shown , together with the individual histograms of the single - mode quadratures .
these are the results for a _ math _ of 400 ns as a typical example .
the output barely exhibits side fringes reflecting a large two - photon component ( top and right panels in figs .
3a , b ) .
the two - mode distributions of the output completely change depending on _ math_. the four - leaf clover is most pronounced at _ math _ whereas the clover totally disappears at _ math _ as expected from figs .
2c , d @xcite .
these distributions are totally different from that of the case without synchronization ( fig .
3c ) .
the results here are only particular examples , and the full results are presented in the supplementary information @xcite , together with those for the phase - independent input dual - single - photon state .
so far , we have discussed the relative phase _ math _ of the hom state _
math _ , where _ math _ is the phase - shift operator acting on the second mode . from a different viewpoint , this can be reinterpreted as a phase shift of the measured quadratures _ math _ for the fixed state _ math _ , since _
math_. in the latter interpretation , the wave - basis two - mode distributions with various measurement phases contain the complete information for estimating the quantum state @xcite . in fig .
3d , the density matrix of the output state _ math _ is shown in the number basis .
the ideal density operator of the hom state is _ math _ , and this structure also becomes apparent in the experimental output state _ math _ : the diagonal elements of _ math _ and _ math _ indicate the photon bunching effect on the _ math_-component of the input @xcite , while the presence of the off - diagonal , negative elements of _ math _ and _ math _ proves that the bunched components are quantum - mechanically superimposed rather than classically , incoherently mixed .
these off - diagonal elements are ( necessary and ) sufficient for the entanglement between the two output modes .
quantitatively @xcite , a logarithmic negativity _ math _ is obtained , while _ math _ in the ideal case .
moreover , strong nonclassicality of the output state is evident from its negative wigner function _
math_. the cross section _
math _ has negative values in a structure reflecting the four - leaf - clover pattern .
a further evaluation can be found in the supplementary information @xcite . in conclusion , we have experimentally demonstrated the famous hom interference effect in a way that is potentially scalable for large - scale quantum information processing . to achieve this ,
two nearly pure single photons enter a beam splitter almost on demand after their synchronized release from two independent optical ( cavity - based ) quantum memories .
we have also shown the wave - like aspect of the resulting hom state via homodyne detections , characterized by a phase - dependent appearance of a four - leaf clover pattern .
the quality of the states is sufficient to exhibit a negative wigner function and entanglement even after the memory storage .
the hom state interpreted as a noon state @xcite is also potentially useful in quantum metrology and sensing .
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[ [ acknowledgments ] ] acknowledgments : + + + + + + + + + + + + + + + + this work was partly supported by gia , pdis , apsa of the mext of japan , refost of japan , the scope program of the mic of japan . p.v.l . was supported in germany by qcom ( bmbf ) and hipercom ( era - net chist - era ) .
k.m . acknowledges support from jsps .
* supplementary information for + synchronization of optical photons for quantum information processing * + + + + + + s addtoresettablesection makecaption#1#2 tempboxa*#1 . *
# 2 tempboxa > # 2 minipagefalse
| 4,634 | 4,677 |
massive stars play a fundamental role in driving the energy flow and material cycles that influence the physical and chemical evolution of galaxies . despite receiving much attention , their formation process remains enigmatic . observationally , the large distances to the nearest examples and the clustered mode of formation make it difficult to isolate individual protostars for study . it is still not certain , for instance , whether massive stars form via accretion ( similar to low mass stars ) or through mergers of intermediate mass stars . advances in instrumentation , have enabled ( sub ) arcsecond resolution imaging at wavelengths less affected by the large column densities of material that obscure the regions at shorter wavelengths . recent observations exploiting these capabilities have uncovered the environment surrounding _ individual _ massive protostellar systems . from analysis of @xmath42.3 @xmath0 m co bandhead emission , @xcite have inferred keplerian disks very closely surrounding ( within a few au ) four massive young stellar objects , while interferometric , mm - continuum observations , find the mass - function of protostellar dust clumps lies close to a salpeter value down to clump radii of 2000au @xcite . these high resolution observations point toward an accretion formation scenario for massive stars . further discrimination between the two competing models is possible by examining the properties , in particular the young stellar populations , of hot molecular cores . the mid - infrared ( mir ) window ( 7 - 25 @xmath0 m ) offers a powerful view of these regions . the large column densities of material process the stellar light to infrared wavelengths , and diffraction limited observations are readily obtained . recent observations indicate that class ii methanol masers exclusively trace regions of massive star formation @xcite and are generally either not associated or offset from uchii regions @xcite . @xcite ( hereafter m05 ) have carried out multi - wavelength ( mm to mir ) observations toward five star forming complexes traced by methanol maser emission to determine their large scale properties . they found that maser sites with weak ( @xmath510mjy ) radio continuum flux are associated with massive ( @xmath650m@xmath7 ) , luminous ( @xmath610@xmath8l@xmath7 ) and deeply embedded ( a@xmath940 mag ) cores characterising protoclusters of young massive ( proto)stars in an earlier evolutionary stage than uchii regions . the spatial resolution of the observations ( @xmath68@xmath2 ) was , however , too low to resolve the sources inside the clumps . details of the regions from observations in the literature are described in m05 . we have since observed three of the m05 regions at high spatial resolution to uncover the embedded sources inside the cores at mir wavelengths . the data were obtained with michelle . ] on the 8-m , gemini north telescope in queue mode , on the 18@xmath10 , 22@xmath11 and 30@xmath10 of march 2003 . each pointing centre was imaged with four n band silicate filters ( centred on 7.9 , 8.8 , 11.6 and 12.5 @xmath0 m ) and the qa filter ( centred on 18.5 @xmath0 m ) with 300 seconds on - source integration time . g173.49 and g188.95 were observed twice on separate nights and g192.60 observed once . the n and q band observations were scheduled separately due to the more stringent weather requirements at q band . the standard chop - nod technique was used with a chop throw of 15@xmath2 and chop direction selected from msx images of the region , to minimise off - field contamination . the spatial resolution calculated from standard star observations was @xmath4 0.36@xmath2 at 10 @xmath0 m and @xmath4 0.57@xmath2 at 18.5 @xmath0 m . the 32@xmath2x24@xmath2 field of view fully covered the dust emission observed by m05 in each region . particular care was taken to determine the telescope pointing position but absolute positions were determined by comparing the mir data to sensitive , high resolution , cm continuum , vla images of the 3 regions ( minier et al . in prep ) . similar spatial distribution and morphology of the multiple components allowed good registration between the images . the astrometric uncertainty in the vla images is @xmath41@xmath2 . flux calibration was performed using standard stars within 0.3 airmass of the science targets . there was no overall trend in the calibration factor as a result of changes in airmass throughout the observations . the standard deviation in the flux of standards throughout the observations was found to be 7.4 , 3.1 , 4.4 , 2.4 and 9% for the four n - band and 18.5 @xmath0 m filters respectively . the statistical error in the photometry was dominated by fluctuations in the sky background . upper flux limits were calculated from the standard deviation of the sky background for each filter and a 3@xmath12 upper detection limit is used in table 1 . similarly , a 3@xmath12 error value is quoted for the fluxes in table 1 ( typical values for the n and q band filters were 0.005 and 0.03 jy respectively ) . the flux densities for the standard stars were taken from values derived on the gemini south instrument , t - recs which shares a common filter set with michelle . regions confused with many bright sources were deconvolved using the lucy - richardson algorithm with 20 iterations . this was necessary to resolve source structure and extract individual source fluxes . the instrumental psf was obtained for each filter using a bright , non - saturated standard star . the results were reliable and repeatable near the brighter sources when using different stars for the psf and observations of the objects taken over different nights . as a further check , the standard stars were used to deconvolve other standards and reproduced point sources down to 1% of the peak value after 20 iterations , so only sources greater than 3% of the peak value were included in the final images . the resulting deconvolutions are shown in fig 1 . [ tab_sources ] [ cols="^,^,^,^,^,^,^,^,^,^,^,^ " , ] as the large scale clump dust and gas morphology appears simple and centrally peaked ( see m05 ) , we make the reasonable assumption that the protocluster centres coincide with the central peak of dust emission . the spatial distribution of the point sources within the protocluster is similar between the clumps with close point sources toward the cluster centre . the methanol masers are found closest to the brightest mir point source ( within the assumed 1@xmath2 pointing error from image registration ) . these sources have temperatures sufficient to evaporate methanol ice from the dust grains into the gas phase ( @xmath690k ) as well as sufficient luminosity of ir photons to pump the masing transition conditions models suggest are required for such emission @xcite . it is known that more massive stars favour cluster centres ( e.g. @xcite ) , but it is unclear whether they form there or migrate in from outside . we have used the simple - harmonic model of ballistic motion developed by @xcite to consider the motion of sources within the cores . using the measured column density and radius from m05 ( listed in table 2 ) , the time required for migration from the edge to the centre is @xmath4 @xmath13 years . this is comparable to the predicted hmc lifetime of 10@xmath14 years @xcite so we can not rule out the possibility of migration within the clumps . any sources having migrated to the centre in this way would have acquired a velocity of @xmath4 2 kms@xmath15 with respect to the clump . massive stars in clusters are observed to have a high companion star fraction @xcite . in the m16 cluster , @xcite observed massive stars ( earlier than b3 ) with visual companions separated by 1000 - 3000au . if multiple systems are bound from birth , it is likely some of the sources we have observed will belong to multiple systems , even though the companions may lie below the detection limit . however , all three regions show two or more point sources at close angular separation ( see insets of figure 1 ) corresponding to linear separations of 1700 to 6000au . we can not determine whether these stars are physically bound or simply close due to projection effects but we can calculate the instrumental sensitivity required to confirm or deny the association . assuming they are physically bound in a keplerian orbit , the maximum proper motions ( projection angle = 0@xmath16 ) of @xmath4 0.1 mas / year are too small to be detected on short temporal baselines . the maximum velocity difference ( projection angle = 90@xmath16 ) @xmath4 2 kms@xmath15 is achievable by high spectral resolution observations of any line features . the mass distribution of stars is generally well described as a power law through the initial mass function ( imf ) . given the mass of gas available to form stars , we may estimate the likelihood that a cluster will end up with the most massive stars that are observed in it . the fraction of gas that forms stars is given by the star formation efficiency ( sfe ) and is observationally found to be less than 50% in any cloud and to be @xmath17 33% for nearby embedded clusters @xcite . for a cluster whose total stellar mass is 120 ( 50 , 320 ) m@xmath7 ( equivalent to the gas mass determined for the three cores ) , @xcite estimate that the mean maximum mass that a star may have in it is 10 ( 5 , 20 ) m@xmath7 . this is comparable to the largest observed mass in two out of the three cases . however , we also observed several other stars in each cluster so can estimate the probability of generating stars of equal or greater mass than the remaining mass distribution . we did this by running monte - carlo simulations to populate 10@xmath14 clusters using @xcite , @xcite and @xcite imfs until the available gas mass was exhausted . we only considered clusters which contained a star of at least equal mass to the most massive observed . the simulations show that even using the salpeter form of the imf ( most biased toward forming high - mass stars ) and allowing 50% of the gas to form stars , it is difficult to generate the observed mass distributions ( probabilities @xmath18 10@xmath19 , 10@xmath20 , 10@xmath15 for the three cores respectively ) . by itself , this may not be significant for a single cluster . however , since the probability is low for all three sources studied , it is unlikely that the mass distribution of the most massive stars can be produced by sampling a standard form of the imf from the reservoir of gas available for star formation . this conclusion would not hold if there was a substantial stellar mass already in the cluster that remains unseen , or if much of the original gas mass had already been dispersed from the core due to star formation . the former requires a sfe close to unity and given the relatively quiescent state of the cores , the latter seems unlikely . a larger sample of young , massive protoclusters is required to draw general conclusions . however , in all three hot molecular cores traced by methanol maser emission we have found : * multiple , mir sources which can be separated into three morphological types : unresolved point sources ( p ) ; unresolved point source with weak surrounding extended emission ( pe ) and extended sources ( e ) . * the point sources lie at close angular separations . future high spatial and spectral resolution observations may be able to determine whether or not they are physically bound . * the methanol masers are found closest to the brightest mir point source ( within the assumed 1@xmath2 pointing accuracy ) . * cooler , extended sources dominate the luminosity . * the time scale for a source at the core edge to migrate to the centre is comparable to the hot molecular core lifetime , so it is not possible to rule out large protostellar motions within the core . * from the derived gas mass of the core and mass estimates for the sources , monte carlo simulations show that it is difficult to generate the observed distributions for the most massive cluster members from the gas in the core using a standard form of the imf . this conclusion would not hold , however , if most of the original gas has already formed stars , or has been dispersed such that the original core mass is much greater than now observed . s.l . would like to thank alistair glass , scott fisher , tony wong and melvin hoare for helpful discussion of the data and scientific input . we thank the anonymous referee for the thorough response and insightful comments . this work was made possible by funding from the australian research council and unsw . the gemini observatory is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : nsf ( usa ) , pparc ( uk ) , nrc ( canada ) , conicyt ( chile ) , arc ( australia ) , cnpq ( brazil ) and conicet ( argentina ) . b. t. , 1989 , interstellar extinction in the infrared . infrared spectroscopy in astronomy , proceedings of the 22nd eslab symposium held in salamanca , spain , 7 - 9 december , 1988 . edited by b.h . kaldeich . esa sp-290 . european space agency , 1989 . , p.93 , pp 93+ g. , bouvier j. , eislffel j. , simon t. , 2001 , in asp conf . ser . 243 : from darkness to light : origin and evolution of young stellar clusters statistical properties of visual binaries as tracers of the formation and early evolution of young stellar clusters .
|
we present high resolution , mid - infrared images toward three hot molecular cores signposted by methanol maser emission ; g173.49 + 2.42 ( s231 , s233ir ) , g188.95 + 0.89 ( s252 , afgl-5180 ) and g192.60 - 0.05 ( s255ir ) . each of the cores was targeted with michelle on gemini north using 5 filters from 7.9 to 18.5 @xmath0 m .
we find each contains both large regions of extended emission and multiple , luminous point sources which , from their extremely red colours ( @xmath1 ) , appear to be embedded young stellar objects .
the closest angular separations of the point sources in the three regions are 0.79 , 1.00 and 3.33@xmath2 corresponding to linear separations of 1,700 , 1,800 and 6,000au respectively .
the methanol maser emission is found closest to the brightest mir point source ( within the assumed 1@xmath2 pointing accuracy ) .
mass and luminosity estimates for the sources range from 3 - 22 m@xmath3 and 50 - 40,000 l@xmath3 . assuming the mir sources are embedded objects and
the observed gas mass provides the bulk of the reservoir from which the stars formed , it is difficult to generate the observed distributions for the most massive cluster members from the gas in the cores using a standard form of the imf .
masers stars : formation techniques : high angular resolution stars : early type stars : mass function infrared : stars .
| 3,528 | 396 |
even though the mn containing oxides with the perovskite - like structure have been studied for more than a half century,@xcite various phase transitions occurring on doping in these materials are not fully understood . in particular , lamno@xmath0 exhibit rich and interesting physical properties because of the strong interplay between lattice distortions , transport properties and magnetic ordering . this compound also have a very rich phase diagram depending on the doping concentration , temperature and pressure ; being either antiferromagnetic ( af ) insulator , ferromagnetic ( f ) metal or charge ordered ( co ) insulator.@xcite the magnetic behavior of the lamno@xmath0 perovskite is particularly interesting , because the jahn - teller ( jt ) distortion is accompanied by the so - called a - type antiferromagnetic ( a - af ) spin ( moment ) and c - type orbital ordering ( oo ) , i.e , alternative occupation of @xmath5 and @xmath6 in the @xmath1 plane and same type of orbital occupation perpendicular to the @xmath1 plane.@xcite recently manganites have also been subjected to strong interest due to their exhibition of negative colossal magnetoresistance ( cmr ) effects.@xcite in particular the perovskite - oxide system la@xmath7ae@xmath8mno@xmath0 , where @xmath9 is a divalent alkali element such as ca or sr , have attracted much attention primarily due to the discovery of a negative cmr effect around the ferromagnetic transition temperature @xmath10 , which is located near room temperature.@xcite the mutual coupling among the charge , spin , orbital and lattice degrees of freedom in perovskite - type manganites creates versatile intriguing phenomena such as cmr,@xcite field - melting of the co and/or oo state(s ) accompanying a huge change in resistivity,@xcite field - induced structural transitions even near room temperature,@xcite field control of inter - grain or inter - plane tunneling of highly spin - polarized carriers,@xcite etc . several mechanisms have been proposed for cmr , such as double exchange,@xcite dynamical jt effect,@xcite antiferromagnetic fluctuation,@xcite etc . , but no consensus has been attained so far about the importance of those mechanisms . since the spin , charge , orbital and structural ordering phenomena may affect cmr at least indirectly it is important to obtain a full understanding of the mechanism stabilizing the observed a - af order in the undoped insulating mother compound lamno@xmath0 . it has been suggested that an understanding of hole - doped lamno@xmath0 must include , in addition to the double - exchange mechanism,@xcite strong electron correlations,@xcite a strong electron - phonon interaction@xcite and cooperative jt distortions associated with mn@xmath11 . several theoretical studies have been made on this material using the mean - field approximation@xcite , numerical diagonalization,@xcite gutzwiller technique,@xcite slave - fermion theory,@xcite , dynamical mean - field theory@xcite , perturbation theory@xcite and quantum monte - carlo technique.@xcite nevertheless it is still controversial as to what is the driving mechanism of the experimentally established properties , particularly the strongly incoherent charge dynamics , and what the realistic parameters of theoretical models are . by calculating and comparing various experimentally observed quantities one can get an idea about the role of electron correlations and other influences on the cmr effect in these materials . hence , theoretical investigations of ground state and excited state properties are important to understand the exotic physical properties of these materials . the importance of spin and lattice degrees of freedom on the metal - insulator transition in lamno@xmath0 is studied extensively.@xcite popovic and satpathy@xcite showed how the cooperative jt coupling between the individual mno@xmath12 centers in the crystal leads to simultaneous ordering of the distorted octahedron and the electronic orbitals . it is now accepted that oo and magnetic ordering ( mo ) are closely correlated and that the anisotropy in the magnetic coupling originates from oo.@xcite so in order to understand the origin of oo , it is important to study the energetics of different spin - ordered states . ahn and millis@xcite calculated the optical conductivity of lamno@xmath0 using a tight - binding parameterization of the band structure . they noted a troubling discrepancy with lsda band - theory calculations@xcite of the optical conductivity and concluded with the need for further work to find the origin of the difference . hence , accurate calculations of optical properties is expected to give more insight into the origin of the discrepancy . an appreciable faraday rotation has been observed in hole - doped lamno@xmath0 thin films@xcite and hence it has been suggested that these ferromagnetic films may be used to image vortices in high - temperature superconductors . further , due to the half - metallic behavior in the f phase of lamno@xmath0 one can expect a large magneto - optical effect . for this reason , we have calculated the optical and magneto - optical properties of this material . the simultaneous presence of strong electron - electron interactions within the transition - metal 3@xmath2 manifold and a sizable hopping interaction between transition metal ( @xmath13 = ti@xmath14cu ) 3@xmath2 and o 2@xmath15 states are primarily responsible for the wide range of properties exhibited by transition - metal oxides . often the presence of a strong intraatomic coulomb interaction makes a single - particle description of such systems inadequate . due to this deficiency , the density - functional calculations often fail@xcite to predict the insulating behavior of lamno@xmath0 . to correct this deficiency of the local spin - density approximation ( lsda ) to give the right insulating properties of the perovskites , lsda+@xmath16 theory@xcite is applied , where @xmath16 is the intrasite coulomb repulsion . from experimental and theoretical studies it has been believed that the electron correlations in the la@xmath13o@xmath0 ( @xmath13 = ti@xmath14cu ) series are very important and should be considered more rigorously beyond lsda . hence , lda+@xmath16 is applied@xcite to obtain magnetic moments and fundamental gap in good agreement with experiment . the calculations for la@xmath13o@xmath0 by solovyev _ @xcite showed that the correlation correction was significant for ti , v , co but less important for mn . however , in this study the calculated intensity of the optical conductivity were found to be much smaller than the experimental results in the whole energy range . hu _ et al._@xcite reported that to get the correct experimental ground state for lamno@xmath0 , it is necessary to take jt distortion , electron - electron correlation and af ordering simultaneously into consideration . from the observation of large on - site coulomb @xmath16 and exchange @xmath17 , obtained from `` constrained lda calculations '' , satpathy _ et al._@xcite indicated the importance of correlation effects in lamno@xmath0 . held and vollhardt,@xcite using the dynamical mean - field theory , emphasize the importance of electronic correlations from the local coulomb repulsion for understanding the properties of manganites . et al._@xcite pointed out that the electron correlations remain strong even in the metallic state of doped manganites . photoemission studies on doped manganites gave an electron - electron correlation effect with @xmath16 = 7.5ev for the coulomb repulsion.@xcite in contrast to the above observations , it has been shown that many aspects of the ground state as well as single - electron excited state properties of lamno@xmath0 and related compounds can be described satisfactorily in terms of lsda energy bands.@xcite density functional calculations also show strong couplings between lattice distortions , magnetic order , and electronic properties of lamno@xmath0 . in particular , it is found that without lattice distortions lamno@xmath0 would have a f metallic ground state , and even if forced to be a - af , it would still be metallic.@xcite further , sarma _ et al._@xcite indicated that the electron - electron correlation is unimportant due to a relatively large hopping parameter @xmath18 and a large screening effect . the lsda studies@xcite show that substantially hybridized bands derived from majority - spin mn @xmath4 and o @xmath15 states dominate the electronic structure near the fermi energy ( @xmath19 ) . x - ray absorption spectroscopy ( xas ) data show that several apparent peaks exist up to 5ev above @xmath19 , but some disputes still remain about the origins of these conduction - band peaks.@xcite the reported positions of the @xmath20 bands in lamno@xmath0 differ.@xcite the above mentioned studies indicate that it is important to study the significance of correlation effects in lamno@xmath0 . some of the features lacking in most of the theoretical studies on lamno@xmath0 originate from the fact that they have often resulted from the use of the atomic - sphere approximation ( asa ) , i.e. the calculations have not included the nonspherical part of the potential and also used a minimal basis set . further the cubic perovskite structure is frequently assumed@xcite and the significant structural distortions are not taken into account . apart from these , owing to the presence of magnetic ordering , relativistic effects such as spin - orbit coupling may be of significance in this material , which was not been included in earlier studies . moreover , it is shown that instead of using the uniform electron gas limit for exchange and correlations ( corresponding to lsda ) one can improve the outcome by including the inhomogeneity effects through the generalized - gradient approximation ( gga).@xcite to overcome the above mentioned deficiencies we have used a generalized - gradient - corrected , relativistic full - potential method with the experimentally observed orthorhombic distorted perovskite structure as the input in the present calculation . the rest of the paper is organized as follows . the structural aspects and the computational details about the calculations of the electronic structure , energetics of different magnetic phases , optical properties , magneto - optical properties , xps and xanes features are given in sec.[sec : details ] . in sec.[sec : resdis ] we give the orbital , angular momentum and site - projected density of states ( dos ) for lamno@xmath0 in the ground state , the spin - projected dos for various magnetic state . the calculated band structure for the a - af and f phases of the distorted orthorhombic structure are given . the role of the structural distortion on the electronic structure , optical and magnetic properties is analyzed . calculated magnetic properties are compared with available experimental and theoretical studies . the origin of excited state properties such as xps , bis , xanes , optical properties and magneto - optical properties is analyzed through the electronic structure and compared with available experimental spectra . finally in sec.[sec : con ] we have summarized the important findings of the present study . lamno@xmath0 is stabilizes in the orthorhombic gdfeo@xmath0-type structure@xcite ( comprising four formula units ; space group @xmath21 ) as shown in fig.[fig : str ] . it can be viewed as a highly distorted cubic perovskite structure with a quadrupled unit cell ( @xmath22 , @xmath22 , 2@xmath23 where @xmath23 is the lattice parameter of the cubic perovskite structure . the structural parameters used in the calculations are taken from a 4.2k neutron diffraction study@xcite with @xmath24 = 5.742 , @xmath25 = 7.668 and @xmath26 = 5.532 and atom positions : la in 4c ( 0.549 0.25 0.01 ) , mn in 4a ( 0 0 0 ) , o(1 ) in 4c ( @xmath140.014 0.25 @xmath140.07 ) and o(2 ) in 8d ( 0.309 0.039 0.244 ) . the electronic configuration of mn@xmath11 in lamno@xmath0 is postulated as @xmath27 and hence , it is a typical jt system . basically , two different types of distortions are included in the structure shown in fig.[fig : str ] . one is a tilting of the mno@xmath12 octahedra around the cubic [ 110 ] axis as in gdfeo@xmath0 so that the mn - o - mn angle changes from 180@xmath28 to @xmath29 160@xmath30 which is not directly related with the jt effect , but is attributed to the relative sizes of the components , say , expressed in terms of the tolerance factor @xmath31 , where @xmath32 , @xmath33 and r@xmath34 are the ionic radii for la , mn and o respectively giving @xmath35 = 0.947 for lamno@xmath0 . the rotation of the mno@xmath12 octahedra facilitates a more efficient space filling . the second type of crystal distortion in lamno@xmath0 is the deformation of the mno@xmath12 octahedra caused by the jt effect , viz . originating from orbital degeneracy . this may be looked upon as a cooperative shifting of the oxygens within the @xmath1 plane away from one of its two nearest neighboring mn atoms towards the others , thus creating long and short mn - o bond lengths ( modified from 1.97 for cubic case to 1.91,1.96 and 2.18 for the orthorhombic variant ) perpendicularly arranged with respect to the mn atoms . the long bonds can be regarded as rotated 90@xmath28 within @xmath1 on going from one mn to the neighboring mn.@xcite the structural optimization@xcite wrongly predicts that the ground state of lamno@xmath0 is close to that of cubic perovskite with the f state lower in energy than the experimentally observed af state . hence it is important to study the relative stability between the orthorhombic and cubic phases for different magnetic arrangements . consequently we have made calculations both for the orthorhombic ( @xmath21 ) as well as the ideal cubic perovskite variants . for the calculations of the undistorted cubic variant we have used the experimental equilibrium volume for the orthorhombic structure . when lamno@xmath0 is in the af state there are three possible magnetic arrangements according to interplane and intraplane couplings within ( 001 ) plane . ( i ) with interplane af coupling and intraplane f coupling the a - af structure arises . ( ii ) the opposite structure of a - af , where the interplane coupling is f , but the intraplane coupling is af is called c - af structure . in the c - type cell all atoms have two f and four af nearest neighbors whereas the reverse is true for a - type af . ( iii ) if both the inter- and intraplane couplings are af the g - af structure should arise.@xcite in the g - type af lattice , each mn atom is surrounded by six mn neighbors whose spins are antiparallel to the chosen central atom . among the several possible magnetic orderings , the experimental studies show that for lamno@xmath0 the a - af ordering is the ground state with a nel temperature of 140k . in the cubic case we have made calculations for the af structures in the following way . the a - af structure contains two formula units , and is a tetragonal lattice with @xmath24 = @xmath25 = @xmath23 and @xmath26 = 2@xmath23 . the c - af structure is also tetragonal with @xmath24 = @xmath25 = @xmath36 and @xmath26 = a@xmath37 . finally , the g - af structure is a face centered cubic structure with @xmath24 = @xmath25 = @xmath26 = 2@xmath38 with eight formula units , where two nonequivalent 3@xmath2 metals with spins up and down replace na and cl in an nacl configuration . this magnetic structure can be viewed as consisting of two interpenetrating face - centered lattices with opposite spin orientation . these investigations are based on _ ab initio _ electronic structure calculations derived from spin - polarized density - functional theory ( dft ) . for the xanes , orbital - projected dos and the screened plasma frequency calculations we have applied the full - potential linearized - augmented plane wave ( fplapw ) method@xcite in a scalar - relativistic version , i.e. without spin - orbit ( so ) coupling . in the calculation we have used the atomic sphere radii ( @xmath39 ) of 2.2 , 2.0 and 1.5 a.u . for la , mn and o , respectively . since the spin densities are well confined within a radius of about 1.5 a.u . , the resulting magnetic moments do not depend strongly on the variation of the atomic sphere radii . the charge density and the potentials are expanded into lattice harmonics up to @xmath40 = 6 inside the spheres and into a fourier series in the interstitial region . the initial basis set included 4@xmath41 , 4@xmath15 , 3@xmath2 functions and 3@xmath41 , 3@xmath15 semicore functions at the mn site , 6@xmath41 , 6@xmath15 , 5@xmath2 valence and 5@xmath41 , 5@xmath15 semicore functions for the la site and 2@xmath41 , 2@xmath15 and 3@xmath2 functions for the o site . this set of basis functions was supplemented with local orbitals for additional flexibility in representing the semicore states and for relaxing the linearization errors generally . due to the linearization errors dos are reliable to about 1@xmath142ry above the fermi level . so , after selfconsistency was achieved for this basis set we included a few high - energy local orbitals ; 6@xmath2- , 4@xmath42-like function for la , 5@xmath41- , 5@xmath15-like functions for mn and 3@xmath15-like functions for o atoms . the effects of exchange and correlation are treated within the generalized - gradient - corrected local spin - density approximation using the parameterization scheme of perdew _ et al._@xcite to ensure convergence for the brillouin zone ( bz ) integration 243*k * points in the irreducible wedge of first bz were used . self - consistency was achieved by demanding the convergence of the total energy to be smaller than 10@xmath43ry / cell . this corresponds to a convergence of the charge below 10@xmath44 electrons / atom . the full - potential lmto calculations@xcite presented in this paper are all electron , and no shape approximation to the charge density or potential has been used . the base geometry in this computational method consists of a muffin - tin part and an interstitial part . the basis set is comprised of augmented linear muffin - tin orbitals.@xcite inside the muffin - tin spheres the basis functions , charge density and potential are expanded in symmetry - adapted spherical harmonic functions together with a radial function and a fourier series in the interstitial . in the present calculations the spherical - harmonic expansion of the charge density , potential and basis functions were carried out up to @xmath40 = 6 . the tails of the basis functions outside their parent spheres are linear combinations of hankel or neumann functions depending on the sign of the kinetic energy of the basis function in the interstitial region . for the core charge density , the dirac equation is solved self - consistently , i.e. no frozen core approximation is used . the calculations are based on the generalized - gradient - corrected - density - functional theory as proposed by perdew _ et al._@xcite the so term is included directly in the hamiltonian matrix elements for the part inside the muffin - tin spheres , thus doubling the size of the secular matrix for a spin - polarized calculation . even though the la @xmath45 states are well above @xmath19 their contribution to the magnetic and structural properties are very important.@xcite so we have included these orbitals in all our calculations . moreover , the present calculations make use of a so - called multibasis , to ensure a well converged wave function . this means that we use different hankel or neuman functions each attaching to its own radial function . we thus have two @xmath46 , two @xmath47 , two 6@xmath15 , two 5@xmath2 and two 4@xmath42 orbitals for la , two @xmath48 , two @xmath47 and three 3@xmath2 orbitals for mn , two @xmath49 , three @xmath50 and two @xmath51 orbitals for o in our expansion of the wave function . in the method used here , bases corresponding to multiple principal quantum numbers are contained within a single , fully hybridizing basis set . the direction of the moment is chosen to be ( 001 ) . the calculations were performed for the cubic perovskite structure as well as the orthorhombic gdfeo@xmath0-type structure in nonmagnetic ( p ) , f , a - af , c - af and g - af states . the * k*-space integration was performed using the special point method with 284 * k * points in the irreducible part of first bz for the orthorhombic structure and the same density of * k * points were used for the cubic structure in the actual cell as well as the supercell . all calculations were performed using the experimental structural parameters mentioned in sec.[ssec : str ] for both the nonmagnetic and spin - polarized cases . using the self - consistent potentials obtained from our calculations , the imaginary part of the optical dielectric tensor and the band structure of lamno@xmath0 were calculated for the a - af and f cases . the density of states were calculated for the p , f , a - af , c - af and g - af phases in the cubic as well as orthorhombic structure using the linear tetrahedron technique . once energies @xmath52 and functions @xmath53 for the @xmath54 bands are obtained self consistently , the interband contribution to the imaginary part of the dielectric functions @xmath55(@xmath56 ) can be calculated by summing transitions from occupied to unoccupied states ( with fixed * k * vector ) over bz , weighted with the appropriate matrix element for the probability of the transition . to be specific , the components of @xmath55(@xmath56 ) are given by @xmath57 here ( @xmath58,@xmath59,@xmath60 ) = * p * is the momentum operator and @xmath61 is the fermi distribution . the evaluation of the matrix elements in eq . ( [ e2 ] ) is done over the muffin - tin and interstitial regions separately . further details about the evaluation of matrix elements are given elsewhere.@xcite due to the orthorhombic structure of lamno@xmath0 the dielectric function is a tensor . by an appropriate choice of the principal axes we can diagonalize it and restrict our considerations to the diagonal matrix elements . we have calculated the three components @xmath62 , @xmath63 and @xmath64 of the dielectric constants corresponding to the electric field parallel to the crystallographic axes @xmath24 , @xmath25 and @xmath26 , respectively . the real part of the components of the dielectric tensor @xmath65(@xmath56 ) is then calculated using the kramer - kronig transformation . the knowledge of both the real and imaginary parts of the dielectric tensor allows the calculation of important optical constants . in this paper , we present the reflectivity @xmath66 ) , the absorption coefficient @xmath67 ) , the electron energy loss spectrum @xmath68 ) , as well as the refractive index @xmath54 and the extinction coefficient @xmath69 . the calculations yield unbroadened functions . to reproduce the experimental conditions more correctly , it is necessary to broaden the calculated spectra . the exact form of the broadening function is unknown , although comparison with measurements suggests that the broadening usually increases with increasing excitation energy . also the instrumental resolution smears out many fine features . to simulate these effects the lifetime broadening was simulated by convoluting the absorptive part of the dielectric function with a lorentzian , whose full width at half maximum ( fwhm ) is equal to @xmath70ev . the experimental resolution was simulated by broadening the final spectra with a gaussian , where fwhm is equal to 0.02 ev . the magneto - optic effect can be described by the off - diagonal elements of the dielectric tensor which originate from optical transitions with a different frequency dependence of right and left circularly polarized light because of so splitting of the states involved . for the polar geometry , the kerr - rotation ( @xmath71 ) and ellipticity ( @xmath72 ) are related to the optical conductivity through the following relation : @xmath73 where @xmath74 in terms of conductivities are @xmath75 for small kerr - angles , eq . ( [ eq : kerr1 ] ) can be simplified to@xcite @xmath76 the magnetic circular birefringence , also called the faraday rotation @xmath77 , describes the rotation of the polarization plane of linearly polarized light on transmission through matter magnetized in the direction of the light propagation . similarly , the faraday ellipticity @xmath78 , which is also known as the magnetic circular dichroism , is proportional to the difference of the absorption for right- and left - handed circularly polarized light.@xcite thus , these quantities are simply given by@xcite @xmath79 where @xmath26 is the velocity of light in vacuum , and @xmath2 is the thickness of the thin film . within the so - called single - scatterer final - state approximation@xcite ( free propagation of the photoelectrons through the crystal , loss of * k*-dependent information and neglect of surface effects ) the photocurrent is a sum of local ( atomic like ) , partial ( @xmath80-like ) dos weighted by cross sections ( transition probabilities ) . as the theoretical framework of the xps intensity calculations is given earlier,@xcite only a brief description of the main points is outlined here . for high incident energies of xps ( @xmath29 1.5kev ) the low - energy electron - diffraction function@xcite can be simplified and the fully relativistic angle - integrated intensity @xmath81 can be written as @xmath82 @xmath83 where @xmath56 denotes the energy of the incident photons . this expression has been cast into a fermi golden - rule form , where @xmath84 are the partial dos functions for the @xmath85 channel at the @xmath86 site , and are obtained from full - potential lmto calculation . the partial , angular momentum - dependent cross sections @xmath87 of the species @xmath88 can be obtained from @xmath89^{2}\ ] ] where @xmath90 in eq.[eqn : sigma ] the radial functions @xmath91 and @xmath92 differ conceptually only by different single - site scattering normalizations . the radial gradients @xmath93 refer to the @xmath88-th scattering potential . the wigner 3j - symbols @xmath94 automatically take care of the dipole selection rules . the relativistic cross sections @xmath95 are calculated using the muffin - tin part of the potential over the energy range @xmath96 of the dos functions for the fixed incident photon energy @xmath56 . this expression has been evaluated for lamno@xmath0 from the potentials and dos functions by a fully relativistic full - potential lmto self - consistent calculation . because the cross sections ( matrix elements ) are energy dependent , the theoretically predicted spectra will depend on the energy chosen for the calculation . to be consistent with the reported xps data , we have made all calculations with the fixed incident photon energy @xmath56 = 1253.6ev ( mg@xmath97 line ) used in the experimental study . the finite lifetime of the photoholes was taken into account approximately by convoluting the spectra using a lorentzian with a energy - dependent half - width . the fwhm of this lorentzian is zero at @xmath19 and increases linearly with the binding energy , as 0.2(@xmath98 ) . in addition to the lorentzian life - time broadening , the spectra were broadened with a gaussian of halfwidth 0.8ev to account for spectrometer resolution . the procedure we adapt to calculate xps and bis spectra is as follows . first , we take the partial dos functions ( @xmath99 for xps and @xmath100 for bis ) from a full - potential lmto calculations . we multiply these by the calculated @xmath80 dependent cross - sections for all band energies and sum them to get a total spectral - like function . next , to get a good fit to the data we broaden this with an energy - dependent lorentzian function to simulate what we call inherent life - time effects due to the coupling of the excited outgoing electron from the crystal . this broadening is zero at @xmath19 and goes as the square of the energy below @xmath19 . then we fit the leading edge to the experimental data with a gaussian broadening to simulate the instrument resolution . this broadening is same for all states . finally we add a background function to get the experimental profile . while the calculations were performed at the one - electron level , we believe this procedure should capture the essence of the photoemission process and lead to a meaningful comparison between calculated and observed spectra . the theoretical x - ray absorption spectra for lamno@xmath0 were computed within the dipole approximation from the flapw@xcite partial dos along the lines described by neckel _ et al._@xcite the intensity @xmath101 arising from transitions from initial vb states ( energy @xmath96 and and angular momentum @xmath80 ) to a final core state ( @xmath102,@xmath103 ) is given by @xmath104 where the matrix elements are given by @xmath105^{2 } } { \int_{0}^{r_{\tau}}p_{\ell^{\tau}}(r , e)^{2}dr}\ ] ] @xmath106 represent principal and angular momentum quantum numbers for the core states . @xmath107 is the partial dos of atom @xmath88 with angular momentum @xmath80 , @xmath108 and @xmath109 are the radial wave function and atomic sphere radii of atom @xmath88 . the transition coefficient @xmath110 can be calculated analytically according to the following equation , @xmath111 in the case of absorption spectra @xmath112 . to account for instrument resolution and life - time broadening of both core and valence states we have broadened the calculated spectra using the lorentzian function with fwhm of 1ev . let us first discuss qualitative distinctions among the electronic band structures of lamno@xmath0 with various magnetic configurations without considering the structural distortion . with full cubic symmetry , @xmath3 and e@xmath113 orbitals are three - fold and two - fold degenerate , respectively . the orbital - projected dos of mn 3@xmath2 electrons in the f phase of cubic lamno@xmath0 are shown in fig.[fig : cldos ] , where the @xmath3 states are away from @xmath19 and also rather narrow . however , the @xmath4 levels are broadly distributed in the dos profile . the electrons at @xmath19 have both @xmath114 and the @xmath115 electrons as shown in fig.[fig : cldos ] . against the pure ionic picture ( where the @xmath4 electrons only are expected to be closer to @xmath19 ) there is considerable amount of @xmath4 electrons present around @xmath146ev . these states originate from the covalent interaction between the mn @xmath4 electrons and o 2@xmath15 states which produce the @xmath4 bonding ( around @xmath146ev ) and antibonding ( above @xmath142ev ) hybrids . the @xmath3 states are energetically degenerate with the o 2@xmath15 states in vb indicating that there are finite covalent interactions between these states . owing to this covalent interaction along with the exchange interaction between the @xmath3 and @xmath4 states , a finite dos of @xmath3 electrons in the energy range @xmath147 to @xmath145ev is created . with @xmath19 positioned in the middle of the @xmath4 band , a jt instability is produced which causes the oxygen octahedra to distort and removes the orbital degeneracy . for the undistorted cubic perovskite structure , the total dos is shown in fig.[fig : ctdos ] for the nonmagnetic ( paramagnetic = p ) , f and the a- , c- and g- type af spin configurations . without the jt - caused lattice distortion , the lsda calculations show that lamno@xmath0 is metallic for all these magnetic states . in all the cooperative magnetic states , the hund splitting is large and induces an empty minority - spin band at the mn site . the dos for the p phase given in fig.[fig : ctdos ] shows a large value at @xmath19 . this is a favorable condition for providing a magnetic splitting and hence the cooperative magnetic phase is more stable than the p phase . from dos for the f state of the undistorted structure ( see fig.[fig : ctdos ] ) one can see that there is a finite number of states present in both spin channels . on the other hand the half - metallic character clearly appeared in the f state of the orthorhombic structure ( fig.[fig : tdos ] ) . the different hybridization nature of the cubic and the orthorhombic phases and the noninvolvement of jt effects in the cubic phase are the possible reasons for the absence of the half - metallic character in the cubic phase . the total dos of the af phases are much different from that of the f phase ( fig.[fig : ctdos ] ) . owing to large exchange splitting there is a smaller dos at the fermi level [ @xmath116 for the f and af phases compared with the p phase . the calculated value of @xmath117 for the p phase is 146.6states/(ry . f.u . ) and 16.7 , 20.5 , 25.1 and 17.8states/(ry f.u . ) for a- , c- and g - af as well as f phases , respectively . let us now focus on the role of the structural distortion on the electronic structure of lamno@xmath0 . it is believed that lamno@xmath0 is a charge - transfer - type ( ct ) insulator@xcite according to the zaanen - sawatzky - allen scheme,@xcite in which the lowest - lying gap transition corresponds to the ct excitation from the o 2@xmath15 to mn 3@xmath2 state and has four @xmath2 electrons per mn@xmath11 site with a configuration of @xmath118 . this electronic configuration with one electron for the two @xmath4 orbitals implies that lamno@xmath0 is a typical jt system . so , this electronic configuration is susceptible to a strong electron - phonon coupling of the jt type that splits the @xmath4 states into filled @xmath119 and empty @xmath120 , and thus , produces large asymmetric oxygen displacements . the important factors governing the formation of the electronic structure of orthorhombic lamno@xmath0 are the exchange splitting owing to spin polarization , the ligand field splitting of @xmath121 and @xmath3 states and the further splitting of the @xmath121 states owing to the jt distortion . early theoretical work focused on the undistorted perovskite aristotype structure and it was found that usual lsda can not produce the correct insulating ground state for lamno@xmath0.@xcite using the lapw method , pickett and singh@xcite obtained a gap of 0.12ev for the distorted lamno@xmath0 when they include the jt effect and a - af ordering . as noticed in earlier studies,@xcite two unexpected behaviors appear when one includes the structural distortion ; the stabilization of the a - af over the f phase and an opening of a gap and large rearrangement of bands . the valence band dos of lamno@xmath0 are derived primarily from mn 3@xmath2 and o 2@xmath15 admixture with dominant mn 3@xmath2 character ( fig.[fig : pdos ] ) . from the projected density of states ( pdos ) it can be seen that the mn 3@xmath2 @xmath14 o 2@xmath15 hybridization is spin dependent . for the case of the majority - spin channel both mn 3@xmath2 and o 2@xmath15 strongly mix with each other in the whole vb . on the contrary , owing to the presence of mn 3@xmath2 in the high spin state , the minority - spin vb is nearly empty for the mn 3@xmath2 states . hence , there is little overlap between mn 3@xmath2 and o 2@xmath15 states in the minority - spin channel . because of ct from la and mn to the o 2@xmath15 states , the latter are almost filled and their contribution to the unoccupied state is minimal . the o 2@xmath41 states are well localized and are present around @xmath1418ev . vb of lamno@xmath0 is generally composed of four regions . the lowest energy region contains mainly o 2@xmath41 bands , above this the la 5@xmath15 bands are distributed in the region between @xmath1417ev and @xmath1414ev . both the o 2@xmath41 and la 5@xmath15 bands are well separated from the bands in the vicinity of @xmath19 , and consequently they hardly contribute to the chemical bonding or to transport properties . the o 2@xmath15 bands are present in the energy range @xmath147ev to @xmath142ev and energetically degenerate with the mn 3@xmath2 states in the whole vb region indicating covalent interaction between mn and o in lamno@xmath0 . the present observation of strong covalency in the ground state of lamno@xmath0 is consistent with the conclusion drawn from photoemission and x - ray - absorption spectroscopy.@xcite the negligible contribution of la electrons in the vb region indicates that there is an ionic interaction between la and the mno@xmath12 octahedra . it has been pointed out by goodenough@xcite that the covalency between the a site and oxygen is important for the gdfeo@xmath0-type distortion . however our pdos profile shows that there is only a negligible amount of electrons present in vb from the a ( la ) site indicating that the covalent interaction between la and o is rather unimportant in lamno@xmath0 . from pdos along with the orbital - projected dos we see that both mn @xmath3 and @xmath4 electrons participate in the covalent interaction with the neighboring oxygens . the top of vb is dominated by the majority - spin mn 3@xmath2 states indicating the importance of mn 3@xmath2 states in transport properties such as cmr observed in hole - doped lamno@xmath0 . the bottom of cb is dominated by the minority - spin mn 3@xmath2 electrons and above the la 4@xmath42 electrons are present in a very narrow energy range between 2.6 and 3.5ev ( fig.[fig : pdos ] ) . o 3@xmath2 and mn 4@xmath15 states are found around 10ev above @xmath19 . as the mn @xmath2 states are playing an important role for the magnetism and other physical properties of lamno@xmath0 , it is worthwhile to investigate these in more detail . an approximately cubic crystal field stemming from the oxygen octahedron around mn would split the mn 3@xmath2 levels into @xmath3 and @xmath4 levels . for mn@xmath11 in lamno@xmath0 , three electrons would occupy localized @xmath3 levels , and one electron a linear combination of two @xmath4 levels . the general view of the electronic structure of lamno@xmath0 is that both @xmath3 and e@xmath113 orbitals hybridize with o 2@xmath15 orbitals , @xmath3 mainly with 2@xmath15 @xmath122 and @xmath4 mainly with 2@xmath15 @xmath123 . the @xmath3 electrons hybridize less with o 2@xmath15 states and hence may be viewed as local spins ( @xmath124 = 3/2 ) . in contrast to that , @xmath121 orbitals , which hybridize more strongly produce rather broad bands . the strong exchange interaction with the @xmath125 subbands along with the jt distortion lead to the splitting of @xmath4 into half occupied @xmath114 and unoccupied @xmath126 bands . our calculations show that @xmath3 bands forms intense peaks in both vb and cb , and majority - spin @xmath3 bands are almost completely filled owing to the high spin state of mn in lamno@xmath0 . contrary to the general opinion , our orbital projected dos ( fig.[fig : ldos ] ) show that both @xmath4 and @xmath3 electrons are present throughout vb and hence both types of electrons participate in the covalent bonding with oxygen . the orbital projected dos ( fig.[fig : ldos ] ) clearly shows that there is almost equal amounts of both @xmath4 and @xmath3 electrons at the top of vb and certainly not@xcite dominated by mn @xmath4 bands alone . the dos in fig.[fig : ldos ] @xmath119 states are shifted to a lower energy than @xmath120 and is more populated . a comparison of the orbital projected dos for the mn @xmath2 states of the f cubic phase with that of the af orthorhombic phase show significant differences . in the f cubic case the @xmath4 levels are well separated from the @xmath3 levels and there is no @xmath125 electrons present in the vicinity of @xmath19 . in the a - af orthorhombic phase both @xmath125 and @xmath114 electrons are present in the vicinity of @xmath19 . both @xmath4 and @xmath3 electrons are energetically degenerate throughout the vb range and in particular the electrons in the @xmath119 orbitals are well localized and degenerate with the electrons in @xmath127 and @xmath128 . importantly , the jt distortion split the @xmath4 states in the a - af orthorhombic phase and hence we find semiconducting behavior . photoemission studies@xcite show that the character of the bandgap of lamno@xmath0 is of the @xmath15-to-@xmath2 ct type . our site projected doss predict that there is considerable @xmath15 and @xmath2 character present at the top of vb as well as in the low energy region of cb . the vb photoemission study@xcite has been interpreted to suggest that dos closer to @xmath19 in vb contains contributions from mn @xmath114 . in contrast , our calculations ( fig.[fig : ldos ] ) show that dos closer to @xmath19 in vb contains contribution from both mn @xmath114 and @xmath125 electrons . owing to the jt distortion the @xmath4 electrons are subjected to an oo effect in lamno@xmath0 . as the jt distortion influences the stability of the orbital configuration , one can expect that it also should affect the electronic structure of lamno@xmath0 differently for different magnetic orderings . hence , we next focus our attention to the electronic structure of orthorhombic lamno@xmath0 in different magnetic configurations . the calculated total dos for orthorhombic lamno@xmath0 in the p , f and a- , c- and g - af arrangements are shown in fig.[fig : tdos ] . from dos at @xmath19 in the p phase , one can deduce pertinent information concerning the magnetism in lamno@xmath0 . this shows that lamno@xmath0 is a very favorable compound for cooperative magnetism since it has a large dos value at @xmath19 in the p state ( fig.[fig : tdos ] ) . owing to the splitting of @xmath4 states by the jt distortion , a gap opens up @xmath19 in doss for both the a - af and g - af phases ( fig.[fig : tdos ] ) . it is interesting to note that the cubic phase of lamno@xmath0 is always metallic irrespective of the magnetic ordering considered in the calculations . thus the structural distortion is the key ingredient to account for the stabilization of the insulating behavior of lamno@xmath0 , consistent with the earlier studies.@xcite from fig.[fig : tdos ] it is interesting to note that the p , f and c - af phases of lamno@xmath0 are found to exhibit metallic conduction even when we include the structural distortions in the calculation . this shows that apart from structural distortion , the af ordering also plays and important role for stabilizing the insulating behavior of lamno@xmath0 . usually , the @xmath4 splitting caused by the jt effect is somewhat underestimated in the asa calculations@xcite and a discrepancy could reflect an uncertainty introduced by the asa approach . the energy separation between the @xmath4 and @xmath3 levels , caused by the crystal - field splitting , is known to be larger than 1ev.@xcite owing to strong overlap between the @xmath4 and @xmath3 levels in the a - af orthorhombic phase , we are unable to estimate the crystal field splitting energy . however , owing to the well separated @xmath4 and @xmath3 levels in the cubic phase we estimated this energy to be @xmath290.96ev . the jt distortion lifts the degeneracy of the @xmath4 level . the @xmath4 level splitting in our calculation is found to be 0.278ev and this is nothing but the ( semiconducting ) bandgap in this material . the mn @xmath2 exchange splitting obtained from our calculation is 3.34ev and this is found to be in agreement with 3.5ev found by pickett and singh@xcite by a lapw calculation and 3.48ev reported by mahadevan _ et al._@xcite from lmto - asa calculations . the lmto - asa@xcite calculations reveals that the splittings between the spin - up and spin - down states on introduction of the orthorhombic distortion are slightly asymmetric for the two types of mn . however , our more accurate full potential calculations do not show any asymmetry of the splitting and the magnetic moments are completely canceled due to the af interaction between the mn ions . the total dos in fig.[fig : tdos ] show clearly that there is a gap opening near @xmath19 in the g - af phase with a value of 0.28ev , comparable with that of the a - af phase . as there is no bandgap in the c - af and f phases , the above results indicate that the af coupling between the layers plays an important role in opening up the bandgap in lamno@xmath0 . the dos for the f phase shows a half - metallic feature , viz . , the finite total dos around @xmath19 comes from one of the spin channels while there is a gap across @xmath19 for the other spin channel . both spin - up and spin - down e@xmath113 states extend over a wide energy range . owing to the f ordering within the @xmath1 plane in the a - af phase and within @xmath129 in the c - af phase , the dos for these phases bears resemblance to that of the f phase . on the other hand , dos for the g - af phase ( dominated by the af superexchange interactions ) is quite different from that of the a- , c - af and the f phases ( fig.[fig : tdos ] ) . moreover , in the g - af phase the width of the @xmath4 state is narrower than that in the a- , c - af and f phases and the partially filled @xmath4 states in these phases are well separated from the empty @xmath3 states . as the jt coupling between the @xmath4 electrons and the distortion modes for the mno@xmath12 octahedra plays an important role for the physical properties of lamno@xmath0 , a magnetic - property study of the undistorted cubic phase of lamno@xmath0 is important . we have calculated the total energies and magnetic moments for lamno@xmath0 in the undistorted cubic perovskite structure as well as for the orthorhombic structure with different magnetic configurations . the calculated total energies for the p , f , a- , c- and g - af states of lamno@xmath0 with the cubic perovskite structure relative to the a - af phase with the orthorhombic structure is given in table[table : de ] . according to these data the cubic phase of lamno@xmath0 should be stabilized in the f phase . stabilization of the f phase in the cubic structure concurs with earlier findings.@xcite the lapw calculations of hamada _ et al._@xcite for undistorted lamno@xmath0 show that the a - af phase is 1ev above the f phase , whereas pickett and singh@xcite found a difference of only 110mev . our calculations shows that the a - af phase is 60mev above the f phase in the cubic perovskite structure ( table[table : de ] ) . we used gga , so coupling and a large number of * k * points along with a well converged basis set . this may account for the difference between the present work and the earlier studies . our orbital projected dos for the @xmath2 electrons of mn ( fig.[fig : cldos ] ) show that the @xmath4 electrons are only distributed in the vicinity of @xmath19 in the majority - spin channel . the removal of the jt distortion enhances the exchange interaction originating from the @xmath4 states drastically and also reduces the negative exchange from the @xmath3 state . hence , the total interplane exchange interaction is positive and the system stabilizes in the f phase . the calculated total energies of various magnetic configurations for lamno@xmath0 in the orthorhombic structure ( table[table : de ] ) shows that the orthorhombic structure with the a - af ordering of the moments is the ground state for lamno@xmath0 . the stabilization of the a - af state is consistent with the neutron diffraction findings.@xcite further , our calculations predict that the f phase is only 24mev higher in energy than the ground state . on the other hand , the p phase is @xmath291.4ev higher in energy than the ground state . from the total energy of the various possible magnetic arrangements it is clear that intralayer exchange interactions in lamno@xmath0 are f and considerably stronger than the af interlayer couplings , the latter being very sensitive to lattice distortions . this observation is in agreement with the conclusion of terakura _ et al._@xcite from electronic structure studies and in qualitative agreement with inelastic neutron scattering results.@xcite the present observation of a small energy difference between the a - af and f phases is consistent with earlier theoretical studies.@xcite it should be noted that lmto - asa calculations@xcite predict the f state to be lower in energy than the af state also for the orthorhombic structure ( table[table : de ] ) . however , the lsda+@xmath16 calculations@xcite with lattice distortion yielded the correct ground state . hence , it has been concluded that the correlation effect plays an important role for obtaining the correct electronic structure of lamno@xmath0 . it should also be noted that the theoretically optimized crystal - structure theory predicts@xcite that f is more stable than the observed a - af state in lamno@xmath0 . on the other hand , our calculations , without the inclusion of the correlation effect , give the correct ground state for lamno@xmath0 indicating that the correlation effect is not so important in this case . if the interlayer f coupling is stronger than the intralayer af coupling the system will be stabilized in the c - af phase . however , our calculations show that c - af phase is higher in energy than the a - af and f phases . due to the jt instability of mn @xmath130 , a substantial energy gain is expected when we include the actual structural distortions in our calculations . hence , a gain of 0.323ev / f.u . is found for f and 0.407ev / f.u . for a - af ( table[table : de ] ) when the structural distortions were included . the magnetic ordering is such that the difference in total energy between f and a - af ( c - af ) give information about the exchange - coupling energy within the plane ( perpendicular to the plane for c - af ) . our calculations show that the af intraplane exchange interaction energy is smaller ( 24mev for the orthorhombic phase and 59mev for the cubic phase ) than the af interplane exchange interaction energy ( 41mev for the orthorhombic phase and 92mev for the cubic phase ) . the present findings are consistent with the experimental studies in the sense that neutron scattering measurements@xcite on a lamno@xmath0 single crystal showed a strong intraplane f coupling and a weak af interplane coupling . now we will try to understand the microscopic origin for the stabilization of a - af in lamno@xmath0 . in the ideal cubic lattice the hybridization between the mn @xmath3 and @xmath4 states is nearly vanishing . as expected for a half - filled @xmath3 band , the @xmath3-type interatomic exchange facilitates af ordering . our orbital - projected dos show that the @xmath120 electrons are distributed in the whole vb region and are mainly populated in the top of vb . the jt distortion induces orbital ordering in which the orbitals confined to the @xmath1 plane ( @xmath131 and @xmath6 ) are dominantly populated and the counter @xmath4 orbitals of @xmath119 symmetry are less populated in the vicinity of @xmath19 . in this case itinerant band ferromagnetism is operational in the @xmath1 plane and is responsible for the f ordering with in the plane . the electrons in the @xmath127 and @xmath119 orbitals are well localized and these make the interplane exchange interaction af , which in turn stabilizes the a - af phase . however , the electrons in the @xmath128 orbitals are well delocalized and have almost the same energy distribution as @xmath120 . this weakens the af coupling between the layers and in turn makes the energy difference between the f and a - af phases very small . the cmr effect in manganites may be understood as follows . when charges are localized by strong electron - electron interactions the system becomes an af insulator , but this state is energetically very close to the metallic f state in lamno@xmath0 . consequently , aligning of spins with an external magnetic field will activate the metallic f states and cause a large gain of kinetic energy , i.e. a cmr phenomenon . table[table : moment ] lists the calculated magnetic moment at the mn site in lamno@xmath0 for different spin configurations in the undistorted cubic perovskite structure as well as the distorted orthorhombic structure , including for comparison corresponding values from other theoretical studies and experimental neutron diffraction results . without hybridization , the mn spin moment should take an appropriate integer value ( 4@xmath132/mn atom in the case of high spin state ) and the oxygen moment should be negligible . owing to the covalent interaction between the mn @xmath2 and o @xmath15 states , experimental as well as theoretical studies will give smaller mn moments than predicted by the ionic model . for the f phase in the orthorhombic structure , mn polarizes the neighboring oxygens and the induced moment at the o(1 ) site is 0.07@xmath132/atom and at the o(2 ) 0.059@xmath132/atom and these moments are coupled ferromagnetically with the local moments of mn@xmath11 . there are small differences in the magnetic moments for the different magnetic arrangements indicating that these moments have a distinct atomic - like character . from table[table : moment ] it should be noted that the lmto - asa approach generally gives larger moments than the accurate full - potential calculation . the reason for this difference is that the full - potential calculations estimate the moments using the spin density within the muffin - tin spheres so that the spin - density in the interstitial region is neglected . our calculated magnetic moment at the mn site in the a - af orthorhombic structure is comparable with the experimental value ( table[table : moment ] ) . the electric field gradient for mn in the f phase obtained by the flapw calculation is 3.579 @xmath133 10@xmath134 v / m@xmath135 as compared with @xmath141.587 @xmath133 10@xmath134 v / m@xmath135 for the a - af phase . the asymmetry parameter @xmath136 of the electric field gradient tensor follows from the relation @xmath137 , which gives @xmath136=0.95 for mn in the a - af phase in good agreement with @xmath136 = 0.82 obtained from low temperature perturbed - angular - correlation spectroscopy.@xcite the magnetic hyperfine field at the mn site derived from our flapw calculation is 198 kg for the a - af phase and 176 kg for the f phase . a comment on the appearence of the f state by doping of divalent elements in lamno@xmath0 is appropriate . it has long been believed that the appearence of the f ground sate in metallic la@xmath7_ae_@xmath8mno@xmath0 can be explained by the double - exchange interaction of mn@xmath11 ( @xmath138 ) and mn@xmath139 ( @xmath140 ) . recent experiments and theoretical investigations have revealed many discrepancies in the simple double - exchange model . et al._@xcite pointed out that the simple double - exchange model is methodologically inappropriate for predicting the af insulating phase ( 0.1 @xmath141 0.15 , @xmath13 @xmath142 @xmath143 ) , for the canted af insulating phase with large canting angle or large magnetic moment estimated by kawano _ et al._@xcite in the f phase region . millis _ et al._@xcite proposed that in addition to the double - exchange mechanism , a strong electron - phonon interaction arising from the jt splitting of the outer mn @xmath2 level is important . as our calculations predict the cubic phase of lamno@xmath0 to be the f state , there must be another mechanism than the doping by the divalent elements which reduces the jt distortion and in turn increases the interlayer exchange coupling to stabilize the f state . moreover , our calculations show that within the @xmath1 plane the mn ions are f coupled without involvement of mn in different ionic states . if the double - exchange mechanism is operational in lamno@xmath0 one can expect a charge imbalance between the mn atoms in the @xmath1 plane ( with f alignment ) this suggests that the f coupling in lamno@xmath0 can be explained within the frame work of itinerant band ferromagnetism . the magneto - optical ( mo ) kerr - effect can be described by the off - diagonal elements of the dielectric tensor which in a given frequency region originates from optical transitions with different frequency dependence for right and left circularly polarized light caused by the so splitting of the states involved.@xcite considerable interest has recently been focused on studying mo properties of materials owing to their potential for application in rewritable high - density data storage . for magneto - optical information storage one requires materials with a large kerr - effect as well as perpendicular magnetic anisotropy . uniaxial magnetic anisotropy is observed for la@xmath144ca@xmath145mno@xmath0 films grown on srtio@xmath0 [ 001 ] substrates@xcite and confirmed theoretically.@xcite the energy difference between the a - af and f states of lamno@xmath0 is very small and hence a small magnetic field could drive a transition from a - af to f. also , it is interesting to note that lamno@xmath0 produced by annealing in a oxygen atomosphere shows@xcite a simple f structure below @xmath10 = 140k , with moments oriented along @xmath26 . manganites exhibiting the cmr effect are also in the f state and hence it is interesting to study the f phase of lamno@xmath0 in more detail . polar kerr - rotation measurements on hole doped lamno@xmath0 show that bi doping enhances the kerr - rotation to a value of 2.3@xmath30 at wavelength of 0.29 nm at 78k.@xcite it may be recalled that the large mo effect in mnptsb is usually linked with its half - metallic behavior.@xcite hence , as the f phase of lamno@xmath0 also possesses half - metallic behavior with a bandgap of 2.38ev ( between top of the minority - spin vb and the bottom of cb in fig.[fig : tdos ] at @xmath19 ) , it is interesting to examine its mo properties . in the case of f state lamno@xmath0 , our calculations predicted half - metallic behaviour and hence the intraband contribution will be of importance to predict the mo property reliably . therefore , we have taken into account the intraband contribution using the drude formula with the same relaxation time as earlier.@xcite we have calculated the unscreened plasma frequency by integrating over the fermi surface using the flapw method . for f state la@xmath146ca@xmath147mno@xmath0 pickett and singh predicted@xcite a plasma frequency of 1.9ev . several groups have reported an anomalously small drude weight in both la@xmath146ca@xmath147mno@xmath0 and la@xmath146sr@xmath147mno@xmath0.@xcite interpreting the latter findings in terms of enhanced optical mass , the effective mass values established optically are much greater than those derived from the specific - heat measurements.@xcite the theoretically established half - metallic feature of the f state in lamno@xmath0 suggests that the small drude weight is originating from large exchange splitting which contracts the drude contribution coming from the minority - spin channel . in accordance with the half - metallic nature of the f state , our calculated dos at @xmath19 is very small [ 3.36states/(ry f.u . ) ] as a result , the calculated unscreened plasma frequency along @xmath24 , @xmath25 and @xmath26 are 1.079 , 1.276 and 0.926ev , respectively . owing to the half - metallic nature of this material there is no intraband contribution arising from the majority - spin channel . hence , we conclude that the experimentally observed@xcite anomalously small drude weight in la@xmath146ca@xmath147mno@xmath0 and la@xmath146sr@xmath147mno@xmath0 is due to half - metallic behavior . our calculated mo spectra for f state lamno@xmath0 is given in fig.[fig : moke ] . our recent moke studies@xcite of fept shows that kerr - rotation spectra can be reliably predicted even up to as high energies as 10ev with the formalism adopted here . hence , we have depicted the calculated mo spectra up to 8ev in fig.[fig : moke ] . the kerr - rotation ( @xmath148 ) shows a positive or negative peak when the kerr - ellipticity ( @xmath72 ) passes through zero . the frequency dependent kerr - rotation and ellipticity spectra are experimentally established@xcite for hole - doped lamno@xmath0 . as there are no experimental mo spectra available for comparison with our theoretical spectra for ( pure ) f state lamno@xmath0 , we have made comparison with the experimetal spectrum for hole - doped lamno@xmath0 ( fig.[fig : moke ] ) . mo effects are proportional to the product of the so - coupling strength and the net electron - spin polarization . this makes mo effects sensitive to the magnetic electrons , i.e. the 3@xmath2 electrons of mn in lamno@xmath0 . in order to promote the understanding of the microscopic origin of the mo effect , we present the off - diagonal elements of the imaginary part of the dielectric tensor in fig.[fig : e2 ] . note that the @xmath149 spectra can have either positive or negative sign since it is proportional to the difference in the absorption of rcp and lcp light . the sign of @xmath149 is thus directly related to the spin polarization of the electronic states that contributes to the mo effect . owing to the half - metallic behavior of the f state of lamno@xmath0 the off - diagonal components of the dielectric tensor below 2.46ev originates only from the majority - spin electrons ( see fig.[fig : e2 ] ) . so , the peaks at 0.4 , 0.9 and 2ev in the kerr - rotation spectra stem from transition from mn @xmath150 to the hybridized mn@xmath14o mn(@xmath151o 2@xmath15)@xmath152 bands . the experimental @xmath149 spectra show@xcite two peaks , one at 1.2ev and the other at 3.5ev . these two peaks are assigned to be @xmath153 transitions involving @xmath154 minority - spin electrons . electrons , respectively . the absence of the 3.5ev peak in our theoretical spectra indicate that it is an effect of the hole - doping . it is experimentally observed that kerr - rotation spectra change significantly with the temperature@xcite as well as with the doping level.@xcite our calculated mo spectra are valid only for the stoichiometric f state lamno@xmath0 at low temperature . the polar mo kerr - rotation data for la@xmath146sr@xmath147mno@xmath0 at 78k was taken from the experimental spectra of popma and kamminga@xcite . even though the experimental mo spectra la@xmath155sr@xmath156mno@xmath0 were measured@xcite at 300k with a magnetic field of 2.2koe , our calculated kerr - rotation spectrum is comparable in the lower energy region ( fig.[fig : moke ] ) . the discrepancy between the experimental and theoretical kerr - spectra in the higher - energy region may be explained as a temperature and/or hole doping effect . we hope that our theoretical findings may motivate to measurements on fully magnetized lamno@xmath0 at low temperatures . et al._@xcite have performed theoretical calculation on mo properties such as kerr - rotation and ellipticity for lamno@xmath0 for different canted - spin configurations . as the f state is only 24mev above the ground state , it is quite possible that the f state can be stabilized experimentally . however , since we have studied the mo properties of f state lamno@xmath0 our result can not be compared with the findings of solovyev _ et al._@xcite lawler _ et al._@xcite found strong faraday rotation at 1.5ev ( @xmath157 1 @xmath133 10@xmath158 @xmath28/cm ) and 3ev ( @xmath159 4.10@xmath158 @xmath28/cm ) for la@xmath7ca@xmath8mno@xmath0 when ( 0.2 @xmath141 0.5 ) . our theoretical faraday rotation spectra also show two prominent peaks at 1.5 and 4ev , the latter having the highest value ( @xmath29 1.25@xmath13310@xmath160 @xmath28/cm ) in the spectrum . the lower - energy peak in the experimental spectrum@xcite of la@xmath7ca@xmath8mno@xmath0 is interpreted as associated with both ligand - to - metal charge transfer and @xmath2-@xmath2 transitions . however , there are as yet no experimental frequency - dependent faraday rotation and ellipticity measurements available . a deeper understanding of optical properties is important from a fundamental point of view , since the optical characteristics involves not only the occupied and unoccupied part of the electronic structure but also the character of the bands . so , in order to compare the band structure of lamno@xmath0 directly with experimental facts@xcite we have calculated the optical spectra for lamno@xmath0 ( for earlier theoretical studies see refs . ) . the experimental reflectivity spectra are rather confusing and controversial ( see the discussion in ref . ) . recent experiments@xcite indicate that the optical spectrum of the manganites is very sensitive to the condition at the surface . as the optical properties of materials originate from interband transitions from occupied to unoccupied bands it is more instructive to turn to the electronic energy - band structure . the calculated energy - band structure of lamno@xmath0 in the a - af state orthorhombic structure is shown in fig.[fig : bnd ] . from this illustration it is immediately clear that lamno@xmath0 is an indirect bandgap semiconductor where the bandgap is between the s - y direction on the top of vb and the @xmath161 point at the bottom of cb . there are two energy bands present in the top of vb , well separated from the rest of the vb . our detailed analysis shows that these two energy bands have mainly mn @xmath120 and @xmath128 character with a small contribution from o @xmath15 . the la 4@xmath42 electrons contribute with a cluster of bands between 2.5 to 3.5ev in cb ( fig.[fig : bnd ] ) . the mn 3@xmath2 and o 2@xmath15 electrons are distributed over the whole energy range of vb . saitoh _ et al._@xcite reported strong covalency and suggested that the energy gap in lamno@xmath0 should be considered as of the ct type . the bandgap estimated from our dos studies for a - af state lamno@xmath0 is 0.278ev and this is found to be in good agreement with the value of 0.24ev obtained from resistivity measurements by mahendiran _ et al._@xcite whereas jonker@xcite reported 0.15ev . however , our value for the direct gap ( 0.677ev ) between occupied and empty states at the same location in bz is too low in comparison with optical ( 1.1ev)@xcite and photoemission ( 1.7ev)@xcite measurements . ( it should be noted however that optical gaps are usually defined at the onset of an increase in spectral intensity in the measured optical variable . ) it is also useful to compare our calculated bandgap with other theoretical results . the lsda and lsda+@xmath16 calculations of yang _ et al._@xcite using the lmto - asa method gave a bandgap of 0.1 and 1.0ev , respectively . the lsda+@xmath162 approach ( where @xmath16 is applied only to the @xmath3 electrons)@xcite yielded a bandgap of 0.2ev . hence , our bandgap is somewhat larger than that of other lsda calculations . on the other hand , hartree fock calculations@xcite gave an unphysically large gap ( 3ev ) for lamno@xmath0 . as all the linear optical properties can be derived from @xmath162 we have illustrated this quantity in fig.[fig : e2 ] and we have compared our theoretical spectra with the experimental @xmath162 spectra derived from reflectivity measurements . the illustration shows that our calculated spectra are in good agreement with the experimental data at least up to 20ev . this indicates that unlike earlier reported @xmath55 spectra obtained from asa calculations@xcite , accurate full - potential calculations are able to predict the electronic structure of lamno@xmath0 reliably not only for the occupied states but also for the excited states . it is interesting to note that we are able to predict correctly the peaks around 1.5 , 4.7 , 8.8 and 20ev without the introduction of so - called scissor operations . this suggests that electron - correlation effects are less significant in lamno@xmath0 . the peak around 4.7ev is reasonably close to the experimental feature reported by of arima _ et al._@xcite ( fig.[fig : e2 ] ) . our theoretical peak at 8.8ev in the spectra is in quantitative agreement with both sets of experimental data included in the figure . our calculations are also able to predict correctly the experimentally observed peak at 20ev by jung _ et al._@xcite ( which arima _ et al._@xcite failed to record ) . now we will try to understand the microscopic origin of the optical interband transitions in lamno@xmath0 . the peak around 1.5ev has been assigned@xcite as intra - atomic @xmath163 transitions . note that such @xmath164 transitions are not allowed by the electric dipole selection rule , but it has been suggested@xcite that a strong hybridization of the @xmath4 bands with the o 2@xmath15 bands will make such @xmath164 transitions optically active . the interband transition between jt split bands gives low - energy transitions in the @xmath162 spectra and the sharp peak features present in @xmath165 and @xmath166 are attributed to such interband transition . the partial dos for lamno@xmath0 given in fig.[fig : pdos ] show that there is a considerable amount of o 2@xmath15 states present at the top of vb as well as at the bottom of cb arising from strong covalent interaction between mn @xmath2 and o @xmath15 states . hence , we propose that the lower energy peak in the @xmath167 spectra originates from [ ( mn @xmath120 @xmath128;o @xmath15 hybridized ) @xmath168 ( mn @xmath2;o @xmath15 ) ] optical interband transition . the peak around 4.7ev in the @xmath63 @xmath162 spectrum originates mainly from hybridized ( mn @xmath119,o @xmath15 ) states to unoccupied hybridized ( mn 3@xmath2;o @xmath15 ) states . all the majority - spin @xmath2 electrons participate in the interband transition from 3ev up to 10ev in in fig.[fig : e2 ] . in the cubic case , @xmath162 is a scalar , and in the orthorhombic case it is a tensor . so we have calculated the dielectric component with the light polarized along @xmath24 , @xmath25 and @xmath26 as shown in fig.[fig : e2 ] . the optical anisotropy in this material can be understood from our directional - dependent , optical dielectric tensor shown in fig.[fig : e2 ] . from the calculations we see that the @xmath162 spectra along @xmath24 and @xmath26 are almost the same . large anisotropy is present in the lower energy region of the @xmath162 spectra . in particular the sharp peaks present for @xmath62 and @xmath64 which is less pronounced for @xmath63 . the optical conductivity obtained by ahn and millis@xcite from tight - binding parameterization of the band structure also shows a sharp peak feature in the lowest energy part of @xmath169 whereas this feature is absent in their @xmath170 spectrum . our calculation predicts that the interband transition due to jt splitting is less pronounced in the @xmath55 spectrum corresponding to @xmath63 . in order to confirm these theoretical predictions polarized optical property measurements on lamno@xmath0 are needed . soloveyev@xcite suggested that the optical anisotropy in a - af state lamno@xmath0 is due to two factors : ( i ) owing to large exchange splitting the minority - spin @xmath119 states near @xmath19 will contribute less to @xmath171 . ( ii ) owing to a - af ordering and jt distortion , the contribution of @xmath119 character to the states with @xmath172 and @xmath119 symmetry is significantly reduced and hence the intensity of @xmath171 in the low - energy region should also be reduced . in order to understand the role of the jt distortion for the optical properties of lamno@xmath0 , we calculated also the optical dielectric tensor for lamno@xmath0 in the cubic perovskite structure with a - af ordering ( fig.[fig : e2 ] ) . just like for the orthorhombic a - af phase , there is large optical anisotropy present in the lower - energy ( @xmath142 2ev ) part of the spectrum . this indicates that the a - af spin ordering is responsible for the large anisotropy in the optical dielectric tensor which in turn is a result of different spin - selection rules applicable to in - plane and out - of - plane optical transitions . however , the optical anisotropy in the orthorhombic a - af phase is much larger than the corresponding undistorted cubic phase ( fig.[fig : e2 ] ) indicating that apart from a - af ordering , the jt distortion also contributes to large optical anisotropy in the lower energy region of the optical spectra of lamno@xmath0 . in order to compare our calculated spectra with those obtained by the lmto - asa method , we have plotted the optical dielectric tensors obtained by solovyev _ et al._@xcite from lsda calculation in fig.[fig : e2 ] . our calculated @xmath63 spectrum below 2ev is found to be in good agreement with the @xmath173 and @xmath174 spectra of these authors . however , their @xmath175 spectra is much smaller in the lower - energy region than our results as well as the experimental findings ( fig.[fig : e2 ] ) . from the optical spectrum calculated by the lda+@xmath16 method , solovyev _ et al._@xcite found that the low - energy part ( up to 3ev ) reflects excitations from o @xmath15 to an unoccupied band formed by alternating mn @xmath176 and @xmath177 orbitals . however , their study poorly described the higher - energy excitations . this discrepancy may be due to the limitations of asa or the minimal basis they have used in the calculation ( la 4@xmath42 states where not included ) . bouarab _ et al._@xcite calculated optical conductivity spectra for lamno@xmath0 . they used the lmto - asa method where nonspherical contributions to the potential are not included . further , these calculations are made with a minimal basis set and for the undistorted cubic perovskite structure rather than the actual distorted perovskite structure . hence , it does not make sense to compare our calculated optical spectra directly with the latter findings . as the linear optical spectra are directly measurable experimentally , we have reported our calculated optical spectra in fig.[fig : optic ] . also , to understand the anisotropy in the optical properties of lamno@xmath0 we show the linear optical spectra for lamno@xmath0 along the @xmath24 and @xmath25 axis in the same illustration . a large anisotropy in the optical properties for the low - energy region is clearly visible in the reflectivity as well as the extinction coefficient spectra . ( fig.[fig : optic ] shows the experimentally measured reflectivity by jung _ et al._@xcite , arima _ et al._@xcite and takenaka _ et al._@xcite in comparision with calculated spectra from our optical dielectric tensors . ) . the reflectivity measured by takenaka _ et al . _ above 8.4ev is found at a higher value than found in the spectra of other two experiments ( fig.[fig : optic ] ) . overall our calculated reflectivity spectra are found to be in good qualitative agreement with the experimental spectra up to 30ev in fig.[fig : optic ] . the peaks around 10 and 25ev in our reflectivity spectra concur with the findings of takenaka _ et al._.@xcite as the optical properties of lamno@xmath0 are anisotropic , it is particularly interesting to calculate the effective number of valence electrons , @xmath178 , participating in the optical transitions in each direction . hence , we have calculated ( see ref . ) the number of electrons participating in the optical interband transitions in different crystallographic directions , and the comparison between @xmath178 the @xmath62 and @xmath63 shows significant differences in the low - energy ( fig.[fig : optic ] ) . except for the reflectivity there is no optical properties measurements for lamno@xmath0 available . hopefully our findings will motivate such studies . in spite of intense experimental@xcite and theoretical@xcite efforts to understand the electronic structure of lamno@xmath0 , there are still a number of ambiguities concerning the vb features . both x - ray ( xps ) and ultraviolet photoemission spectroscopy ( ups ) experiments showed a double peak structure between @xmath1410ev and @xmath19.@xcite even though it is widely accepted that the double - peak structure arises from o @xmath15 and mn @xmath2 bands , an important controversy still remains in that some authors@xcite have argued that the o @xmath15 band lies below the mn @xmath3 bands , while others@xcite have suggested the opposite . in fig.[fig : xps ] we have compared our calculated value of vb xps intensity with the experimental spectra.@xcite in all the experimental studies , the main vb features have a marked increase in binding energy around 1.5@xmath142ev , remaining large in the region 2@xmath146ev , and falling off in the range 6@xmath148ev . the experimental xps data show three peak intensity features between @xmath147 and @xmath143ev . as can be seen from fig.[fig : xps ] , these three peaks are well reproduced in the calculated profile . the large experimental background intensity makes a direct comparison with the calculated peak feature around @xmath141.8ev difficult . the overall agreement between the theoretical and experimental positions of peaks and shoulders in the xps spectra is very satisfactory ( fig.[fig : xps ] ) . note that the experimental xps spectra does not exhibit any appreciable intensity in a correlation - induced satellite at higher - binding energies in contrast to intense satellite features usually found in transition metal monoxides , e.g. nio . thus , the good agreement between experimental and theoretical spectra indicates that the on - site coulomb correlation effect is not significant in lamno@xmath0 . knowledge of the theoretically calculated photoemission spectra ( pes ) has the advantage that one can identify contributions from different constituents to the pes intensity in different energy ranges . from fig.[fig : xps ] it is clear that the xps intensity in the energy range @xmath143.5ev to @xmath19 mainly originates from mn @xmath2 states . the o @xmath15 electrons contribute to the pes intensity in the energy range @xmath147 to 3.5ev . below @xmath144ev both la and mn atoms contribute equally to the pes intensity . pickett and singh@xcite compared their theoretical dos with pes and found significant differences . thus , the good agreement between the experimental and our theoretical xps indicates that the matrix - element effect is important in lamno@xmath0 . we now turn to a discussion of the bis spectra . the physical process in bis is as follows . an electron of energy @xmath179 is slowed down to a lower energy level @xmath180 corresponding to an unoccupied valence state of the crystal . the energy difference is emitted as bremsstrahlung radiation , which reflects the density of unoccupied valence states.@xcite our calculated bis spectrum is compared with the experimental spectrum@xcite in fig.[fig : xps ] which show that theory is able to reproduce the experimentally observed peak at 2ev . there is an intense peak above 3.5ev in the experimental spectrum which originates from the la 4@xmath42 electrons , but owing to its large intensity this is not shown in fig.[fig : xps ] . another experimental technique that provides information about cb is x - ray absorption spectroscopy . calculated o @xmath181 , mn @xmath181 and mn @xmath182 xanes spectra for lamno@xmath0 are shown in fig.[fig : xanes ] . because of the angular momentum selection rule ( dipole approximation ) , only o 2@xmath15 states and mn 4@xmath15 states contribute to the o @xmath183-edge and mn @xmath183-edge spectra , respectively . the mn 4@xmath41 and mn 3@xmath2 states contribute to the mn @xmath182 spectrum . the calculated x - ray absorption spectrum for the o @xmath183 edge ( vb @xmath168 1@xmath41 ) is given in the upper panel of fig.[fig : xanes ] along with available experimental data . the experimental spectrum has two prominent peaks in the low - energy range and both are well reproduced in our theoretical spectra even without including the core - hole effect in the calculation . as reflected in different mn - o distances there are two kinds ( crystallographically ) of oxygens present in lamno@xmath0 and results in considerable differences in peak positions in the xanes spectra . the calculated @xmath182 spectrum is shown in the middle panel of fig.[fig : xanes ] . due to the nonavailability of experimental mn @xmath182 spectra for lamno@xmath0 we have compared our calculated spectrum with that experimentally established@xcite for mn@xmath11 in mn@xmath184o@xmath0 . the calculated mn @xmath183-edge spectrum for lamno@xmath0 is given in the lower panel of fig.[fig : xanes ] along with the available experimental @xmath183-edge spectra . although dos are large for the empty @xmath114 and @xmath115 bands , they are not directly accessible in @xmath183-edge absorption via dipole transitions . thus the main @xmath183-edge absorption begins where the large part of the @xmath185 states are highly delocalized and extend over several mn atoms . et al._@xcite show that the intensity of the mn @xmath183-edge spectrum increases with an increasing degree of local lattice distortion and is insensitive to the magnetic order . the experimental mn @xmath183-edge spectra show two peaks in the lower energy region and these are well reproduced in our theoretical spectrum . in particular our mn @xmath183-edge spectrum is found to be in very good agreement with that of subias _ et al._@xcite in the whole energy range we have considered . the good agreement between experimental and calculated xanes spectra further emphasizes the relatively little significance of correlation effects in lamno@xmath0 and also shows that the gradient - corrected full - potential approach is able to predict even the unoccupied spectra quite correctly . transition metal oxides are generally regarded as a strongly correlated systems which are believed to be not properly treated by electronic structure theory . the present paper has demonstrated , however , that accurate full - potential calculations including so coupling and generalized - gradient corrections can account well for several features observed for lamno@xmath0 . we have presented a variety of results based on the _ ab initio _ local spin - density approach that are broadly speaking in agreement with experimental results for both ground and excited state properties . the present study strongly support experimental@xcite and theoretical models@xcite in which a strong coupling of the conduction electrons to a local jahn - teller distortion is considered to be important for understanding the basic physical properties of manganites . in particular we found the following : + 1 . the mn ions are in the high - spin state in lamno@xmath0 and hence the contribution to the minority - spin channel from the mn 3@xmath2 electrons are negligible small . + 2 . both mn @xmath3 and the @xmath4 electrons are present in the entire valence band region . the @xmath4 electrons with the @xmath186 symmetry are well localized compared with those of @xmath120 symmetry and hence the @xmath119 contribution in the vicinity of @xmath19 is negligible . on the other hand , the @xmath3 electrons with @xmath128 character have large electron distributions on the top of the valence band comparable with the distribution of @xmath120 electrons . the insulating behavior in lamno@xmath0 originates from the combined effect of jahn - teller distortion and a - type antiferromagnetic ordering . + 4 . without jahn - teller distortion lamno@xmath0 is predicted to be a ferromagnetic metal . + 5 . unlike the lmto - asa calculations,@xcite without inclusion of the correlation effects in the calculations we are able to predict the correct a - type antiferromagnetic insulating ground state for lamno@xmath0 indicating that the importance of correlation effect has been exaggerated for this material . the large changes in energy and the development of an energy gap resulting from the structural distortion along with exchange splitting indicate strong magnetostructural coupling . this may be the possible origin for the observation of cmr in hole - doped lamno@xmath0 . our calculation support the view point that the stabilization of the cubic phase is the prime cause for the occurrence of the ferromagnetic state by hole doping rather than the often believed double - exchange mechanism . the density 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[ cols="<,^,^,^,^,^,<",options="header " , ]
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the ground state and excited state properties of the perovskite lamno@xmath0 , the mother material of colossal magnetoresistance manganites , are calculated based on the generalized - gradient - corrected relativistic full - potential method .
the electronic structure , magnetism and energetics of various spin configurations for lamno@xmath0 in the ideal cubic perovskite structure and the experimentally observed distorted orthorhombic structure are obtained .
the excited state properties such as the optical , magneto - optical , x - ray photoemission ( xps ) , bremsstrahlung isochromat ( bis ) , x - ray absorption near edge structure ( xanes ) spectra are calculated and found to be in excellent agreement with available experimental results .
consistent with earlier observations the insulating behavior can be obtained only when we take into account the structural distortions and the correct antiferromagnetic ordering in the calculations .
the present results suggest that the correlation effect is not significant in lamno@xmath0 and the presence of ferromagnetic coupling within the @xmath1 plane as well as the antiferromagnetic coupling perpendicular to this plane can be explained through the itinerant band picture . as against earlier expectations ,
our calculations show that the mn 3@xmath2 @xmath3 as well as the @xmath4 electrons are present in the whole valence band region .
in particular significantly large amounts of @xmath3 electrons are present in combination with the @xmath4 electrons at the top of the valence band against the common expectation of presence of only pure @xmath4 electrons .
we have calculated the hyperfine field parameters for the a - type antiferromagnetic and the ferromagnetic phases of lamno@xmath0 and compared the findings with the available experimental results .
the role of the orthorhombic distortion on electronic structure , magnetism and optical anisotropy are analyzed .
| 32,575 | 491 |
let @xmath2 be the euclidean ball in @xmath3 ( @xmath4 ) centered at the origin with radius @xmath5 . let @xmath6 , @xmath7 and @xmath8 . consider local minimizers of the dirichlet functional @xmath9 over the closed convex set @xmath10 i.e. functions @xmath11 which satisfy @xmath12 this problem is known as the _ ( boundary ) thin obstacle problem _ or the _ ( elliptic ) signorini problem_. it was shown in @xcite that the local minimizers @xmath13 are of class @xmath14 . besides , @xmath13 will satisfy @xmath15 the boundary condition is known as the _ complementarity _ or _ signorini boundary condition_. one of the main features of the problem is that the following sets are apriori unknown : @xmath16 where by @xmath17 we understand the boundary in the relative topology of @xmath18 . the free boundary @xmath19 sometimes is said to be _ thin _ , to indicate that it is ( expected to be ) of codimension two . one of the most interesting questions in this problem is the study of the structure and the regularity of the free boundary @xmath19 . to put our results in a proper perspective , below we give a brief overview of some of the known results in the literature . the proofs can be found in @xcite and in chapter 9 of @xcite . we start by noting that we can extend solutions @xmath13 of the signorini problem to the entire ball @xmath2 in two different ways : either by even symmetry in @xmath20 variable or by odd symmetry . the even extension will be harmonic in @xmath21 , while the odd extension will be so in @xmath22 . in a sense , those two extensions can be viewed as two different branches of a two - valued harmonic function . this gives a heuristic explanation for the monotonicity of almgren s _ frequency function _ @xmath23 which goes back to almgren s study of multi - valued harmonic functions @xcite . in particular , the limiting value @xmath24 for @xmath25 turns out to be a very effective tool in classifying free boundary points . by using the monotonicity of the frequency @xmath26 , it can be shown that the rescalings @xmath27 converge , over subsequences @xmath28 , to solutions @xmath29 of the signorini problem in @xmath30 . such limits are known as _ blowups _ of @xmath13 at @xmath31 . moreover , it can be shown that such blowups will be homogeneous of degree @xmath32 , regardless of the sequence @xmath33 . it is readily seen from the the definition that the mapping @xmath34 is upper semicontinuous on @xmath19 . furthermore , it can be shown that @xmath35 for every @xmath25 and , more precisely , that the following alternative holds : @xmath36 this brings us to the notion of a regular point . a point @xmath37 is called _ regular _ if @xmath38 . by classifying all possible homogeneous solutions of homogeneity @xmath39 , the above definition is equivalent to saying that the blowups of @xmath13 at @xmath31 have the form @xmath40 after a possible rotation of coordinate axes in @xmath41 . in what follows , we will denote by @xmath42 the set of regular free boundary points , and call it the _ regular set _ of @xmath13 : @xmath43 the upper semicontinuity of @xmath44 , and the gap of values between @xmath39 and @xmath45 implies that @xmath42 is a relatively open subset of @xmath19 . besides , it is known that @xmath42 is locally a @xmath46 regular @xmath47-dimensional surface . in this paper , we are interested in the higher regularity of @xmath42 . since the codimension of the free boundary @xmath19 is two , this question is meaningful only when @xmath4 . in fact , in dimension @xmath48 the complete characterization of the coincidence set and the free boundary was already found by lewy @xcite : @xmath49 is a locally finite union of closed intervals . we will use fairly standard notations in this paper . by @xmath3 we denote the @xmath50-dimensional euclidean space of points @xmath51 , @xmath52 , @xmath53 . for any @xmath54 we denote @xmath55 and @xmath56 . we also identify @xmath57 with @xmath58 , thereby effectively embedding @xmath41 into @xmath3 . similarly , we identify @xmath59 with @xmath60 and @xmath61 . for @xmath62 , @xmath63 if @xmath31 is the origin , we will simply write @xmath64 , @xmath65 , @xmath66 and @xmath67 . let @xmath68 be the euclidean distance between two sets @xmath69 . in this paper we are interested in local properties of the solutions and their free boundaries only near regular points and therefore , without loss of generality , we make the following assumptions . we will assume that @xmath13 solves the signorini problem in @xmath70 and that all free boundary points in @xmath71 are regular , i.e. @xmath72 furthermore , we will assume that there exists @xmath73 with @xmath74 such that @xmath75 next we assume @xmath76 and that @xmath77 moreover , we will also assume the following nondegeneracy property for directional derivatives in a cone of tangential directions : for any @xmath78 , there exist @xmath79 and @xmath80 such that @xmath81 for any @xmath82 , where @xmath83 is the unit normal in @xmath41 to @xmath19 at @xmath31 outward to @xmath49 and @xmath84 for a unit vector @xmath85 . we explicitly remark that if @xmath13 is a solution to the signorini problem , then the assumptions - hold at any regular free boundary point after a possible translation , rotation and rescaling of @xmath13 ( see e.g. @xcite , @xcite ) . following the approach of kinderlehrer and nirenberg @xcite in the classical obstacle problem , we will use the partial hodograph - legendre transformation method to improve on the known regularity of the free boundary . the idea is to straighten the free boundary and then apply the boundary regularity of the solution to the transformed elliptic pde . this works relatively simply for the classical obstacle problem , and allows to prove @xmath86 regularity and even the real analyticity of the free boundary . in the signorini problem , the free boundary @xmath19 is of codimension two , and in order to straighten both @xmath19 and @xmath49 we need to make a partial hodograph transform in two variables . namely , for @xmath13 satisfying the assumptions in section [ sec : assumptions ] , consider the transformation @xmath87 consider also the associated partial legendre transform of @xmath13 given by @xmath88 formally , the inverse of @xmath89 is given by @xmath90 and we can recover the free boundary in the following way : @xmath91 however , we note that the mapping @xmath89 is only @xmath0 regular and even the local invertibility of such mapping is rather nontrivial . besides , even if one has a local invertibility of @xmath89 , the function @xmath92 will satisfy a degenerate elliptic equation , and apriori it is not clear if the equation will have enough structure to be useful . concerning the first complication , we make a careful asymptotic analysis based the precise knowledge of the blowups , and this does allow to establish the local invertibility of @xmath89 . [ thm : t - invert ] let @xmath13 be a solution of the signorini problem in @xmath70 , satisfying the assumptions in section [ sec : assumptions ] . then , there exists a small @xmath93 such that the partial hodograph transformation @xmath89 in is injective in @xmath94 . then via an asymptotic analysis of @xmath92 at the straightened free boundary points , we observe that the fully nonlinear degenerate elliptic equation for @xmath92 has a subelliptic structure , which can be viewed as a perturbation of the baouendi - grushin operator . then using the @xmath1 theory for the baouendi - grushin operator and a bootstrapping argument , we obtain the smoothness and even the real analyticity of @xmath92 . [ thm : fb - regul ] let @xmath13 be as in theorem [ thm : t - invert ] and @xmath92 be given by . then exists @xmath95 such that the mapping @xmath96 is real analytic on @xmath97 . in particular , @xmath98 is locally an analytic surface . the signorini problem is just one example of a problem with thin free boundaries . many problems with thin free boundaries arise when studying problems for the fractional laplacian and using the caffarelli - silvestre extension @xcite to localize the problem at the expense of adding an extra dimension ( which makes the free boundary `` thin '' ) . thus , the signorini problem can be viewed as an obstacle problem for half - laplacian , see @xcite . we hope that the methods in this paper can be used to study the higher regularity of the free boundary in many such problems . a different approach to the study of the higher regularity of thin free boundaries is being developed by de silva and savin @xcite . in particular , in @xcite they prove the @xmath86 regularity of @xmath99 free boundaries in the thin analogue of alt - caffarelli minimization problem @xcite . their method is based on schauder - type estimates rather than hodograph - legendre transform used in this paper . at about the same time as we completed this work , de silva and savin @xcite extended their approach to include the signorini problem , as well as lowered the initial regularity assumption on the free boundary to @xmath46 . the main ingredient in their proof is an interesting new higher order boundary harnack principle , applicable to regular , as well as slit domains ( see also @xcite ) . the paper is organized as follows : - in section [ sec : nond - prop ] we study the so - called @xmath39-homogeneous blowups of the solutions near regular points . this is achieved by a combination a boundary hopf - type principle in domains with @xmath46 slits , as well as weiss- and monneau - type monotonicity formulas . - in section [ sec : hodograph ] we introduce the partial hodograph transformation and show it is a homeomorphism in a neighborhood of the regular free boundary points . this is achieved by using the precise behavior of the solutions near regular free boundary points established in section [ sec : nond - prop ] . - in section [ sec : legendre - funct - nonl ] we consider the corresponding legendre transform @xmath92 and show some basic regularity of @xmath92 inherited from @xmath13 . - in section [ sec : smoothness ] we study the fully nonlinear pde satisfied by @xmath92 , which is the transformed pde of @xmath13 . we show the linearization of the pde is a perturbation of the baouendi - grushin operator . using the @xmath1 estimates available for this operator , and a bootstrapping argument , we obtain the smoothness of @xmath92 , which in turn implies the smoothness of the free boundary . - in section [ sec : analyticity ] , we give more careful estimates on the derivatives of @xmath92 which imply that @xmath92 , and consequently @xmath19 , is real analytic . we start our study by establishing a stronger nondegeneracy property for the tangential derivatives @xmath100 than the one given in . namely , we want to improve the lower bound in to a multiple of @xmath101 . to achieve this , we construct a barrier function by using the @xmath46-regularity of @xmath19 , to obtain a result that can be viewed as a version of the boundary hopf lemma for the domains of the type @xmath102 . to proceed , we introduce some notations . for @xmath103 , let @xmath104 [ lem : hopf - barrier ] there exists a continuous function @xmath105 on @xmath106 and a small @xmath107 such that @xmath108 and @xmath109 inspired by the construction in @xcite , we will show that the following function satisfies the conditions of the lemma : @xmath110 we next verify each of the properties in the lemma . first of all , is immediate . next , since @xmath111 , we have @xmath112 therefore , on @xmath113 one has @xmath114 hence there exists @xmath107 such that @xmath115 on @xmath116 . this implies . further , to show , notice that @xmath117 which yields @xmath118 for small @xmath119 , as claimed . next , we show . it is easy to check that @xmath120 and @xmath121 satisfy @xmath122 @xmath123 combining , we obtain that for @xmath124 @xmath125 \\\geq\frac{1}{2}\left(\hat f''_\alpha(u)+(2n-3)\frac{\hat f'_\alpha(u)}{u}\right)-\frac{1}{2}\left(\hat f''_\alpha(|x|^{1/2})+(2n-3)\frac{\hat f'_\alpha(|x|^{1/2})}{|x|^{1/2}}\right).\end{gathered}\ ] ] since @xmath126 and @xmath127 is decreasing on @xmath128 , then by we have @xmath129 in @xmath130 . this shows . finally , by , we have @xmath131 this completes the proof of the lemma . using the function constructed in lemma [ lem : hopf - barrier ] as a barrier , we have the improvement of the nondegeneracy for nonnegative harmonic functions in @xmath130 . [ cor : hopf1 ] let @xmath132 be a nonnegative superharmonic function in @xmath130 , @xmath133 and @xmath134 on @xmath135 . moreover , suppose that @xmath132 satisfies @xmath136 then there exists @xmath137 and @xmath138 such that @xmath139 let @xmath140 and @xmath119 be as in lemma [ lem : hopf - barrier ] . then by and the continuity of @xmath140 , there exists @xmath141 such that @xmath142 which combined with gives that @xmath143 then , by the maximum principle @xmath144 in particular , @xmath145 for @xmath146 with @xmath107 small . from lemma [ cor : hopf1 ] , we obtain the following nondegeneracy property for the tangential derivatives of the solution to the elliptic signorini problem . [ prop : hopf ] let @xmath13 be a solution to the elliptic signorini problem in @xmath70 satisfying the assumptions in section [ sec : assumptions ] . then for each @xmath82 and @xmath78 , there exist @xmath147 depending on @xmath148 , such that @xmath149 for each @xmath82 , we can rotate a coordinate system in @xmath41 so that @xmath150 . then we have @xmath151 where @xmath152 since @xmath13 is harmonic in @xmath153 and @xmath154 satisfies the nondegeneracy condition , then @xmath100 satisfies the assumptions of lemma [ cor : hopf1 ] , with a small difference that there is constant @xmath155 in the definition of the set @xmath156 above . however , by a simple scaling , we can make @xmath157 . thus , there exists @xmath158 and @xmath159 such that holds . for our further study , we need to consider the following rescalings @xmath160 with @xmath37 . note that these are different form rescalings in the sense that the @xmath161 norm of @xmath13 is not preserved under the rescaling , but it is better suited for the study of regular points . first , from the growth estimate ( see e.g. @xcite ) @xmath162 for @xmath82 , where @xmath163 and @xmath164 , we know that the family @xmath165 is locally uniformly bounded . moreover , by the interior @xmath166 estimate ( see e.g. @xcite ) @xmath167 we get that @xmath165 is uniformly bounded in @xmath168 . thus there exists @xmath169 such that @xmath170 in @xmath171 , for any @xmath172 over a certain subsequence @xmath173 . it is also immediate to see that @xmath174 is a global solution of the signorini problem , i.e. , a solution of in @xmath30 . furthermore , it is important to note that @xmath174 is nonzero , because of the nondegeneracy provided by proposition [ prop : hopf ] . sometimes we will refer to the function @xmath175 as the _ @xmath39-homogeneous blowup _ to indicate the way it was obtained . the following weiss - type monotonicity formula , whose proof can be found in @xcite , implies that @xmath174 is a homogeneous global solution of the signorini problem of degree @xmath39 . [ lem : weiss ] let @xmath13 be a nonzero solution of the signorini problem in @xmath176 . for any @xmath25 and @xmath177 define @xmath178 then @xmath179 is nondecreasing in @xmath180 . moreover , for a.e.@xmath181 we have @xmath182 furthermore , @xmath183 for @xmath184 if and only if @xmath13 is homogeneous of degree @xmath39 with respect to @xmath31 in @xmath185 , i.e. @xmath186 [ rem : weiss ] the weiss - type monotonicity formula above is specifically adjusted to work with rescalings . namely , by a simple change of variables , one can show that @xmath187 besides , by the definition of regular points , we also have that @xmath188 since @xmath189 . [ prop : unique ] let @xmath13 be a solution of the signorini problem in @xmath70 , satisfying the assumptions in section [ sec : assumptions ] . then there exist two positive constants @xmath190 and @xmath191 , depending only on @xmath13 such that if @xmath192 and that @xmath193 over a sequence @xmath28 , then @xmath194 with a constant @xmath195 satisfying @xmath196 we have already noticed at the beginning of section [ sec : blowups ] that @xmath175 is a nonzero global solution of the signorini problem . besides , by the weiss - type monotonicity formula , we will have @xmath197 for any @xmath198 . hence , by lemma [ lem : weiss ] , @xmath175 is a homogeneous of degree @xmath39 in @xmath199 . then , by proposition 9.9 in @xcite , we must have the form @xmath200 for some @xmath201 and a tangential unit vector @xmath202 . we claim that @xmath203 . indeed , we have @xmath204 provided @xmath205 so that @xmath206 . passing to the limit @xmath207 , we therefore have @xmath208 hence , for any @xmath78 @xmath209 this is possible only if @xmath203 . thus , we have the claimed representation @xmath210 the estimates on constant @xmath195 now follow from the growth estimate and the nondegeneracy proposition [ prop : hopf ] , which are preserved under the @xmath39-homogeneous blowup . [ rem : unique ] this proposition also holds for the rescaling family with varying centers : @xmath211 where @xmath212 , @xmath213 and @xmath214 . indeed , by lemma [ lem : weiss ] we have @xmath215 hence applying dini s theorem form the classical analysis to the family of monotone continuous functions @xmath216 on the compact set @xmath217 , we have that the above convergence is uniform on @xmath217 . hence passing to the limit @xmath218 , we obtain @xmath219 for any @xmath198 . arguing as in proposition [ prop : unique ] , we conclude that @xmath220 . in order to get the uniqueness of the blowup limit , we need to show that the constant @xmath195 in proposition [ prop : unique ] does not depend on the subsequence @xmath221 but only depends on @xmath31 . this is a consequence of the following monneau - type monotonicity formula @xcite . without apriori knowledge on the free boundary , this formula is known to hold only at so - called singular points , i.e. , @xmath37 with @xmath222 , @xmath223 . however , using the @xmath46 regularity of the free boundary , we will be able to establish this result also at regular points . [ lem : monneau ] let @xmath13 be a solution of the signorini problem in @xmath70 , satisfying the assumptions in section [ sec : assumptions ] . for any @xmath224 , @xmath164 , and a positive constant @xmath225 , we define @xmath226 where @xmath227 then there exists a constant @xmath228 which depends on the @xmath46 norm of @xmath229 , @xmath191 in , @xmath225 , and @xmath230 , such that @xmath231 is monotone nondecreasing for @xmath232 . for simplicity we assume @xmath233 and write @xmath234 , @xmath235 . letting @xmath236 and using the scaling properties of @xmath237 , we have @xmath238 next , we compute the weiss energy functional @xmath239 by remark [ rem : weiss ] , @xmath240 hence @xmath241 an integration by parts gives @xmath242 noticing that @xmath243 we have @xmath244 similarly , integrating by parts and using @xmath245 in @xmath65 , we obtain @xmath246 combining , we have @xmath247 since @xmath175 is homogeneous of degree @xmath39 , the second integral above is zero . moreover , using @xmath248 on @xmath66 , we have the third integral is equal to @xmath249 recalling , we have @xmath250 noticing that @xmath251 for @xmath164 by lemma [ lem : weiss ] and @xmath252 , @xmath253 on @xmath254 , we have @xmath255 using the growth estimate for @xmath13 and explicit expression for @xmath175 , we have @xmath256 since the free boundary @xmath257 is a @xmath46 graph and @xmath258 , we have @xmath259 where @xmath260 is a constant depending on @xmath261 . hence @xmath262 which implies the claim of the lemma with @xmath263 . we can now establish the main result of this section . [ prop : uni ] let @xmath13 be a solution of the signorini problem in @xmath70 , satisfying the assumptions in section [ sec : assumptions ] . then for @xmath224 , there exists a constant @xmath264 such that @xmath265 where @xmath266 moreover , the function @xmath267 is continuous on @xmath268 . furthermore , for any @xmath269 and @xmath5 , @xmath270 we first show @xmath271 does not depend on the converging sequences . given @xmath82 , let @xmath272 be a converging sequence such that @xmath273 in @xmath171 for some @xmath271 satisfying @xmath274 . by lemma [ lem : monneau ] , the mapping @xmath275 is nonnegative , monotone nondecreasing on @xmath276 , hence @xmath277 this implies @xmath271 does not depend on the converging sequences @xmath272 . next we show that @xmath271 depends continuously on @xmath278 . fix @xmath82 . for any @xmath279 , let @xmath280 be such that @xmath281 for @xmath282 fixed , by the continuity of @xmath13 we have @xmath283 is continuous on @xmath284 . moreover , from the explicit formulation of @xmath285 as well as the @xmath286 continuity of @xmath287 , the function @xmath288 is continuous . therefore , there exists a positive @xmath289 small enough , such that for all @xmath290 with @xmath291 , @xmath292 for @xmath293 , by lemma [ lem : monneau ] @xmath294 let @xmath295 in , then @xmath296 by the explicit expression of @xmath285 , there is a constant @xmath297 such that @xmath298 for any @xmath299 . this together with gives @xmath300 this shows the continuity of @xmath267 . finally , we show . in fact , @xmath301 is continuous on the compact set @xmath217 and it monotonically decreases to zero as @xmath302 decreases to zero for each @xmath31 . hence by dini s theorem , @xmath303 as @xmath295 uniformly on @xmath217 . thus one has @xmath304 it is not hard to see that @xmath305 are subharmonic in @xmath106 ( after the even reflection of @xmath13 about @xmath20 ) . hence by the @xmath306 estimates for the subharmonic functions we have @xmath307 by an interpolation of hlder spaces , i.e. for some absolute constant @xmath308 , @xmath309 for @xmath310 and @xmath311 , as well as a rescaling argument we obtain . the continuous dependence on @xmath31 of @xmath271 gives the uniqueness of the blowups with varying centers . [ cor : varicenter ] let @xmath13 be a solution of the signorini problem in @xmath70 , satisfying the assumptions in section [ sec : assumptions ] . let @xmath312 , such that @xmath313 as @xmath218 . let @xmath207 as @xmath218 . then for any @xmath310 and @xmath314 defined in theorem [ prop : uni ] , @xmath315 this follows from the uniform continuity of @xmath316 and theorem [ prop : uni ] . let @xmath13 be a solution of the signorini problem in @xmath70 satisfying the assumptions in section [ sec : assumptions ] . following the idea in the classical obstacle problem @xcite , we would like to use the method of partial hodograph - legendre transforms to study the higher regularity of the free boundary in the signorini problem . since @xmath19 has codimension two , the most natural hodograph transformation to consider is the one with respect to variable @xmath317 and @xmath20 : @xmath318 ( the reader can easily check that if we do the partial hodograph transform in @xmath317 variable only , it will still straighten @xmath19 , however , the image of @xmath319 will not have a flat boundary and this will render this transformation rather useless . ) by doing so , we hope that there exists a small neighborhood @xmath320 of the origin , such that @xmath89 is one - to - one on @xmath321 . however , due to the @xmath166 regularity of @xmath13 , the mapping @xmath89 is only @xmath0 near the origin . hence the simple inverse function theorem ( which typically requires the transformation @xmath89 to be from class @xmath322 ) can not be applied here . instead , we will make use of the blowup profiles , which contain enough information to catch the behavior of the solution near the free boundary points . before stating the main results , we make several observations . a. by the assumptions in section [ sec : assumptions ] , @xmath323 in @xmath324 and @xmath325 on @xmath254 . this together with the complementary boundary condition gives us @xmath326 + ( 300,163)(25,0 ) ( 0,0 ) . shown for even extension of @xmath13 in @xmath20 variable.,title="fig : " ] ( 55,90 ) ( 90,85)@xmath327 ( 188,0 ) . shown for even extension of @xmath13 in @xmath20 variable.,title="fig : " ] ( 226,120)@xmath328 ( 240,85)@xmath329 ( 130,120)@xmath330^{t}_{\substack{y''=x''\\y_{n-1}=u_{x_{n-1}}\\y_n = u_{x_n } } } & & } $ ] b. if we extend @xmath13 across @xmath331 to @xmath106 by the even symmetry or odd symmetry in @xmath20 , then the resulting function will satisfy @xmath332 hence @xmath13 is analytic in @xmath333 . from , @xmath89 is also analytic in @xmath333 . c. to better understand the nature of @xmath89 , consider the solution @xmath334 of the signorini problem and find an explicit formula for @xmath335 . a simple computation shows that using complex notations the mapping @xmath335 is given by @xmath336 where by the latter we understand the appropriate branch . loosely speaking , this tells that @xmath89 behaves like @xmath337 function in the last two variables . + the observation above also suggests us to compose @xmath89 with the mapping @xmath338 which can be expressed , by using complex notations ( denote @xmath339 by @xmath132 ) , as @xmath340 more explicitly , alongside @xmath89 , we consider the transformation @xmath341 @xmath342 now @xmath343 maps @xmath344 to the upper half space @xmath345 and @xmath254 to the hyperplane @xmath346 . moreover , @xmath343 straightens the free boundary @xmath19 as @xmath89 and satisfies @xmath347 d. the advantage of @xmath343 is that by arguing as in ( ii ) above and making odd and even extensions of @xmath13 in @xmath20 , we can extend it to a mapping on @xmath106 . moreover , this extension will be the same in both cases ( unlike for the mapping @xmath89 ) . in particular , this makes @xmath343 a single - valued mapping on @xmath106 , which is real analytic in @xmath348 . in what follows , we will prove the injectivity of @xmath343 in a neighborhood of the origin . for that we will need the make the following direct computations . ( 338,163)(25,0 ) ( 0,0),title="fig : " ] ( 55,90 ) ( 90,85)@xmath327 ( 188,0),title="fig : " ] ( 243,90 ) ( 278,85)@xmath349 ( 138,120)@xmath330^{t_1}_{\substack{w''=x''\\w_{n-1}=u_{x_{n-1}}^2-u_{x_n}^2\\w_n=-2u_{x_{n-1 } } u_{x_n } } } & & } $ ] [ prop : computeu0 ] let @xmath350 be given by @xmath351 for @xmath352 , @xmath353 $ ] . then we have the following identities @xmath354 direct computation . from now on , we will use a slightly different notation from theorem [ prop : uni ] to denote the blowup limit , i.e. for @xmath355 , we let @xmath356 the following proposition is a consequence of theorem [ prop : uni ] and the @xmath357 regularity of @xmath229 . [ prop : uniformity2 ] for any @xmath279 , there exists @xmath358 depending on @xmath359 and @xmath13 , such that for all @xmath360 and @xmath361 , we have : 1 . @xmath362 ; + 2 . @xmath363 has a harmonic extension at any @xmath364 with @xmath365 ( by making even or odd reflection about @xmath20 ) . hence if we let @xmath366 then for any multi - index @xmath367 with @xmath368 we have @xmath369 \(i ) given any @xmath279 , by there is a positive constant @xmath370 depending on @xmath13 such that for any @xmath371 , @xmath372 on the other hand , there exists @xmath373 depending on @xmath359 , modulus of continuity of @xmath271 and @xmath46 norm of @xmath229 such that @xmath374 taking @xmath375 we proved ( i ) . \(ii ) we observe that for small enough @xmath302 and @xmath376 , the rescaled free boundary @xmath377 satisfies @xmath378 when @xmath379 . this follows immediately from the assumption @xmath380 . the rest of ( ii ) then follows from the estimates for the higher order derivatives of harmonic functions . the next proposition is a consequence of proposition [ prop : computeu0 ] and proposition [ prop : uniformity2 ] , which is useful to understand the transformation @xmath89 . since @xmath89 fixes the first @xmath381 coordinates , it is more convenient to work on a `` tubular '' neighborhood of @xmath19 defined as follows : first we consider the projection map @xmath382 since @xmath383 , @xmath384 is continuous in @xmath106 . moreover , it is easy to verify that for some constant @xmath385 , @xmath386 next for @xmath387 , we let @xmath388 @xmath389 by the continuity of @xmath384 and , @xmath390 is a tubular neighborhood of the part of the free boundary @xmath19 lying in @xmath391 . [ prop : square ] given any @xmath279 , let @xmath392 be the same constant as in proposition [ prop : uniformity2 ] . then for any @xmath393 , we have @xmath394 \(i ) first we observe that latexmath:[\[\label{eq : ob1 } for @xmath393 , let @xmath396 applying proposition [ prop : uniformity2](i ) to @xmath397 we have , @xmath398 , @xmath399 together with implies @xmath400 rescaling back to @xmath13 and using , we obtain ( i ) . \(ii ) the proof of ( ii ) is similar to that of ( i ) . in @xmath401 , @xmath89 is a smooth mapping , and @xmath402 given @xmath403 , consider @xmath397 and rescaled point @xmath404 as above . by proposition [ prop : uniformity2](ii ) , @xmath405 this together with the explicit expression for @xmath406 in proposition [ prop : computeu0 ] gives @xmath407 it is easy to check the following rescaling property @xmath408 this combined with gives ( ii ) . now we are ready to prove the main theorem of the section . let @xmath409 [ thm : injective ] there exists a small constant @xmath95 , such that @xmath89 is a homeomorphism from @xmath410 to @xmath411 , which is relatively open in @xmath412 . moreover , if we extend @xmath89 to @xmath413 via the even ( odd ) reflection of @xmath13 about @xmath20 , then it is a diffeomorphism from @xmath414 ( @xmath415 ) onto an open subset in @xmath3 . by observation ( iii ) and ( iv ) , instead of @xmath89 , we consider the map @xmath416 which is first defined in @xmath417 as and then extended to @xmath106 via even or odd reflection of @xmath13 in @xmath20 variable . we will first show @xmath343 is a homeomorphism from @xmath390 to @xmath418 for sufficiently small @xmath419 . this is divided into three steps . _ there exists @xmath95 , such that for any @xmath360 and @xmath420 , @xmath421 is injective in @xmath422 . here @xmath423 in fact , by proposition [ prop : uniformity2 ] and the definition of @xmath343 , for any @xmath279 , there exists @xmath424 such that @xmath425 from proposition [ prop : computeu0 ] , @xmath426 is a nondegenerate linear map . hence if we take @xmath427 sufficiently small , @xmath428 will be injective on the compact set @xmath422 . _ @xmath343 is injective in @xmath390 for @xmath419 chosen as above . it is clear that @xmath343 is injective on @xmath19 , since it maps @xmath429 to @xmath61 . thus , it will be sufficient if we show the injectivity of @xmath343 in @xmath430 . indeed , suppose there exist @xmath431 such that @xmath432 . necessarily @xmath433 . let @xmath434 , @xmath435 , @xmath436 and w.l.o.g assume @xmath437 . a simple rescaling gives @xmath438 on the other hand , @xmath432 implies @xmath439 then recalling , by proposition [ prop : square](i ) for small enough @xmath359 ( by choosing @xmath419 small in step 1 ) one has @xmath440 note that @xmath441 and @xmath442 have first @xmath381 coordinates equal to zero , thus @xmath443 . but this contradicts the injectivity of @xmath444 on @xmath422 obtained in step 1 . _ @xmath343 is a homeomorphism from @xmath390 to @xmath418 . now @xmath445 is an injective map . moreover , it is continuous because of the continuity of @xmath446 . thus by the brouwer invariance of domain theorem @xmath418 is open and @xmath343 is a homeomorphism between @xmath390 and @xmath418 . now we proceed to show @xmath447 is a homeomorphism for @xmath419 chosen as above . indeed , recalling @xmath448 we obtain the injectivity of @xmath89 from the injectivity of @xmath343 . next we note that @xmath449 is injective in @xmath412 , which contains @xmath450 by observation ( i ) , thus @xmath451 for any subset @xmath452 . thus , @xmath89 is an open map from @xmath453 to @xmath412 , since @xmath343 is open and @xmath449 is continuous . combining with the continuity of @xmath89 , we obtain that @xmath89 is an homeomorphism from @xmath453 to @xmath454 . finally , we notice that @xmath89 is smooth on @xmath414 ( or @xmath455 ) after even ( or odd ) extension of @xmath13 about @xmath20 . moreover , by proposition [ prop : square](ii ) , @xmath456 is nonvanishing there for sufficiently small @xmath419 . hence @xmath89 is a diffeomorphism by the implicit function theorem . ( 200,200 ) ( 0,134 ) and extension of @xmath89.,title="fig : " ] ( 134,134 ) and extension of @xmath89.,title="fig : " ] ( 90,180)@xmath457^t & } $ ] ( 90,160)@xmath458^{t^{-1}}}$ ] ( -25,170)@xmath459 ( 0,67 ) and extension of @xmath89.,title="fig : " ] ( 120,67 ) and extension of @xmath89.,title="fig : " ] ( 90,110)@xmath457^t & } $ ] ( 90,90)@xmath458^{t^{-1}}}$ ] ( -25,100)@xmath460 ( 0,0 ) and extension of @xmath89.,title="fig : " ] ( 120,0 ) and extension of @xmath89.,title="fig : " ] ( 90,40)@xmath457^t & } $ ] ( 90,20)@xmath458^{t^{-1}}}$ ] ( -7,30)@xmath461 ( 200,30)@xmath462 next we construct a space @xmath461 by gluing two copies of @xmath453 , @xmath463 together properly and extend @xmath13 as well as @xmath89 to @xmath461 . the advantage of doing this is , the straightened free boundary @xmath464 will be contained in @xmath465 , which is an open neighborhood of the origin in @xmath3 . this transfers the boundary singularity into the interior singularity , which will be easier for us to deal with . more precisely , we fix the @xmath419 chosen in theorem [ thm : injective ] and consider @xmath466 , a family of subsets in @xmath3 , where @xmath467 define @xmath468 as follows : @xmath469 , where @xmath13 is the solution to the signorini problem satisfying the assumptions in the introduction ; @xmath470 is the even reflection of @xmath13 about @xmath20 ; @xmath471 and @xmath472 . let @xmath473 be the corresponding partial hodograph - legendre transform defined in . consider the disjoint union of @xmath474 : @xmath475 , and denote the elements of it by @xmath476 for @xmath477 , @xmath478 . define @xmath479 now we define an equivalence relation on @xmath480 as follows : @xmath481 it is easy to check from the definition of @xmath482 and theorem [ thm : injective ] that this equivalence relation identifies the points @xmath476 and @xmath483 if ( i ) @xmath484 , @xmath485 or @xmath486 ; ( ii ) @xmath487 , @xmath488 or @xmath489 . in particular , for @xmath490 , @xmath476 and @xmath483 are identified for all @xmath491 . let @xmath492 denote the quotient space . consider on @xmath461 : @xmath493 it is immediate that @xmath494 is continuous and injective . moreover , it is open from @xmath461 to @xmath3 by theorem [ thm : injective ] and the special way we glue @xmath474 . hence we obtain that @xmath494 is a homeomorphism from @xmath461 to @xmath495 . in particular , @xmath495 is an open neighborhood of the origin in @xmath3 , which contains the straightened free boundary @xmath464 . we still denote the set @xmath496 by @xmath19 . it is not hard to observe that @xmath497 is a double cover of @xmath498 with the covering map @xmath499 . hence @xmath497 can be given a smooth structure which makes @xmath500 into a local diffeomorphism . in the local coordinate charts @xmath501 , we have @xmath502 , where @xmath13 is the extended function via the even or odd reflection about @xmath49 or @xmath327 , hence @xmath503 is continuous on @xmath461 , smooth in @xmath497 and @xmath504 there . similarly , one can compute @xmath505 , which is a diffeomorphism on @xmath506 by theorem [ thm : injective ] ( apply theorem [ thm : injective ] for the extended @xmath13 ) . this shows that @xmath507 is a diffeomorphism . from now on , with slight abuse of the notation we will still denote @xmath503 by @xmath13 and @xmath494 by @xmath89 . in the following , we will simply write @xmath508 , @xmath509 , etc . while having in mind that we are taking the derivatives in the local coordinates . in this section we study the partial legendre transform of @xmath13 and the fully nonlinear pde it satisfies . we let @xmath510 which is an open neighborhood of the origin and @xmath511 be the straightened free boundary . for @xmath512 , we define the partial legendre transform of @xmath13 by the identity @xmath513 where @xmath514 . it is immediate to check the following properties of @xmath92 : a. @xmath92 is odd about @xmath515 and even about @xmath516 . b. @xmath92 is continuous in @xmath517 , smooth in @xmath518 and @xmath519 on @xmath520 . c. a direct computation shows that in @xmath521 @xmath522 hence @xmath523 can be written as @xmath524 the jacobian matrix of @xmath92 is then @xmath525 where @xmath526 since @xmath527 and its differential has an continuous extension to @xmath19 , this together with the continuity of @xmath523 and imply that @xmath528 . d. the restriction of @xmath523 to @xmath529 is given by @xmath530 which gives a local parametrization of the free boundary @xmath19 . thus , the regularity of the free boundary is directly related to the regularity of @xmath531 , restricted to @xmath529 . a direct computation using and shows that @xmath532 @xmath533 since @xmath534 in @xmath535 , the legendre function @xmath92 satisfies the following fully nonlinear equation in @xmath536 @xmath537 multiplying both sides of by @xmath538 we can write it in the form @xmath539 which can be further rewritten as @xmath540 where @xmath541 , @xmath542 , is the @xmath543 matrix @xmath544 in order to study the asymptotic of higher derivatives of @xmath92 at the straightened free boundary @xmath529 , we study the blowup of @xmath92 at points on @xmath520 . let @xmath545 be the legendre function of @xmath314 as in , where @xmath546 . it is not hard to compute that @xmath547 where @xmath548 is the unit outer normal of @xmath49 at @xmath31 . in particular , at the origin , @xmath549 for @xmath550 and @xmath62 , we consider the non - isotropic dilation @xmath551 and the rescaling at @xmath552 ( note @xmath553 ) @xmath554 from , and , one can easily check that @xmath555 here rescaling family @xmath556 and @xmath557 are defined on @xmath558 , where @xmath559 are the topological spaces obtained by gluing four rescaled copies @xmath560 together as in the construction of @xmath461 . to study the convergence of @xmath561 to @xmath545 , we first show a lemma which concerns about in the local coordinate charts the uniform convergence of @xmath562 to @xmath563 . the following two facts are easy to verify : a. @xmath557 is bijective and @xmath564 , where @xmath565 is the non - isotropic dilation in . in particular , we have @xmath566 . b. let @xmath567 and let @xmath568 then for any sufficiently small @xmath302 and @xmath569 , we have @xmath570 , i.e. @xmath557 always maps the copy @xmath571 into the corresponding `` quarter '' domain @xmath572 . let @xmath573 with @xmath574 denote the restriction of the covering map to @xmath575 . [ lem : dilation ] let @xmath576 and @xmath577 . then @xmath578 uniformly in any compact subset @xmath579 , @xmath580 . moreover , this convergence is also uniform for @xmath552 varying in a compact subset of @xmath529 . given @xmath576 and a compact set @xmath581 , by ( i ) and ( ii ) above , there is a positive @xmath582 , such that @xmath583 for any @xmath584 . by the rescaled version of proposition [ prop : uniformity2](i ) , there exists @xmath585 small such that @xmath586 , which is a bounded subset in @xmath3 , for any @xmath587 . we know from the @xmath588 convergence of @xmath556 to @xmath314 that @xmath589 moreover , the limit @xmath590 is continuous on @xmath572 by a direct computation . therefore , @xmath591 uniformly in @xmath592 . by theorem [ prop : uni ] and the continuous dependence of @xmath563 on @xmath31 , we have the above convergence is uniform for @xmath552 in any compact subset of @xmath529 . next we show the following compactness results . [ prop : compact - v ] let @xmath592 be a compact subset in @xmath593 . then for any multi - index @xmath230 , @xmath594 uniformly in @xmath592 . moreover , the above convergence is also uniform for @xmath552 varying in a compact subset of @xmath529 . given @xmath576 , let @xmath595 . for @xmath596 and @xmath597 , using and , we can easily conclude from lemma [ lem : dilation ] together with the uniform convergence of @xmath556 to @xmath314 that , @xmath598 converges to @xmath599 uniformly in @xmath592 . for @xmath600 , using and one can express @xmath598 in terms of @xmath601 with @xmath602 , i.e. for fixed @xmath230 , @xmath603 where @xmath604 is some polynomial . for @xmath605 compact , @xmath606 is also compact and @xmath607 . by the local uniform convergence of @xmath562 to @xmath563 ( lemma [ lem : dilation ] ) as well as the flatness of the free boundary @xmath608 ( i.e. the hausdorff distance between @xmath609 and @xmath608 goes to zero as @xmath302 goes to zero ) , there exists @xmath610 compact and @xmath611 small , such that for all @xmath612 , we have @xmath613 and @xmath614 note that implies that for any @xmath612 and @xmath580 , @xmath615 are harmonic in @xmath616 . thus for any multi - index @xmath230 , we have @xmath617 uniformly in @xmath616 . this combined with , and lemma [ lem : dilation ] gives the conclusion . due to theorem [ prop : uni ] and lemma [ lem : dilation ] , the above convergence is uniform in @xmath552 varying the compact subset of @xmath529 . from proposition [ prop : compact - v ] one can get continuous extension of higher order ( properly weighted ) derivatives of @xmath92 at @xmath529 . [ cor : extension ] for each @xmath576 , @xmath577 , we extend the following functions to @xmath552 by setting @xmath618 then after such extension the above functions are continuous on @xmath517 . the proof is based on the following two facts ( a ) ( b ) and a blowup argument . \(a ) for @xmath619 , @xmath620 , @xmath621 \(b ) from the @xmath46 regularity of @xmath19 and the explicit expression of @xmath545 , we have the map @xmath622 is continuous from @xmath529 to @xmath623 , where @xmath624 compact . this together with proposition [ prop : compact - v ] gives that for any multi - index @xmath230 @xmath625 we proceed to show the extended functions are continuous at @xmath576 . first they are continuous on @xmath529 from the @xmath286 dependence of @xmath83 on @xmath31 . next for @xmath626 , we use @xmath627 to denote @xmath628 and let @xmath629 then @xmath630 as @xmath631 , we have @xmath295 and @xmath632 . thus by ( b ) above , for a fixed @xmath624 , which is compact and contains the set @xmath633 , we have @xmath634 from the explicit expression of @xmath545 ( see ) we have @xmath635 this together with ( a ) and gives ( i)-(iv ) . in this section we show the fully nonlinear equation @xmath636 in has a subelliptic structure in @xmath462 . let @xmath637 be the space of @xmath638 symmetric matrices and we may consider @xmath639 as a smooth function on @xmath637 . let @xmath640 for @xmath641 be the linearization of @xmath639 at @xmath642 . a direct computation shows that for @xmath643 and @xmath644 , the linearization of @xmath639 at @xmath645 has the form @xmath646 observe that one can write @xmath647 , with @xmath648 and @xmath649 symmetric matrices of the following forms : @xmath650 @xmath651 where @xmath652 with @xmath653 for @xmath654 , and for @xmath655 , @xmath656 note that @xmath657 is smooth in @xmath658 due to the smoothness of @xmath92 there . moreover , by corollary [ cor : extension](iii)(iv ) and the intermediate value theorem , @xmath659 thus @xmath657 has a continuous extension on @xmath520 . in particular , @xmath660 hence , @xmath657 is positive definite in a small neighborhood of the origin , which implies that the linearized operator @xmath661 has a subelliptic structure near the origin . moreover , using we have @xmath662 which is the coefficient matrix for the _ baouendi - grushin type operator _ : @xmath663 this indicates us to view the linearization of @xmath639 in a neighborhood of the origin as a perturbation of baouendi - grushin type operator . we first recall the classical @xmath1 estimate for the baouendi - grushin operator . for @xmath664 , @xmath665 , @xmath666 , the baouendi - grushin operator is @xmath667 in order to study the weak solution associated with @xmath668 , it is natural to consider the following function space associated with the hrmander vector fields @xmath669 for @xmath670 , @xmath671 a bounded open subset , we define @xmath672 by theorem 1 in @xcite , @xmath673 is a separable banach space for @xmath674 with the norm @xmath675 denote by @xmath676 the closure of @xmath677 in @xmath673 . it is not hard to prove by using mollifier that if @xmath678 and has compact support in @xmath679 , then @xmath680 , @xmath674 . in this paper only the spaces with @xmath681 are involved . we list them below separately : @xmath682 to simplify the notation , we will denote @xmath683 we will need the sobolev embedding theorem for @xmath684 and the @xmath1 estimate for @xmath668 . similar results for more general subelliptic operators can be found in lots of literature like @xcite and @xcite . since our case is much simpler , we provide a relatively shorter and self - contained proof in the appendix . [ lem : embedding ] let @xmath685 be a bounded domain in @xmath686 . 1 . if @xmath687 , then @xmath688 for @xmath689 satisfying @xmath690 , i.e. there exists @xmath691 such that for all @xmath692 @xmath693 2 . if @xmath694 , then @xmath695 , i.e. there exists @xmath696 such that for all @xmath697 , @xmath698 see appendix . [ thm : grushin ] let @xmath685 be a domain in @xmath686 , @xmath699 and @xmath700 is even . let @xmath701 with @xmath702 . then there is a positive constant @xmath703 which only depends on @xmath384 such that @xmath704 see appendix . next we state the local @xmath1 estimates for the perturbed operator @xmath705 where @xmath706 can be decomposed into the form @xmath707 with @xmath708 and @xmath709 a positive definite matrix with continuous entries and for some small positive @xmath710 it satisfies @xmath711 where @xmath712 if @xmath713 and @xmath714 if @xmath715 . from now on , we will work on the following scale - invariant `` cylinder '' ( w.r.t @xmath668 ) centered at the origin : for @xmath62 @xmath716 [ prop : perturb - grushin ] let @xmath717 satisfy @xmath718 where @xmath719 is a perturbed baouendi - grushin operator given by . then if is satisfied for sufficiently small @xmath720 , there exists @xmath691 such that for any @xmath721 , @xmath722 we write @xmath723 if @xmath724 , then by @xmath725 hence if we choose @xmath726 in , for any @xmath727 we have @xmath728 now we remove the compact support condition . let @xmath721 be fixed and let @xmath729 . let @xmath730 be a smooth cut - off function in @xmath731 , where @xmath732 satisfy @xmath733 when @xmath734 , @xmath735 when @xmath736 ; @xmath737 when @xmath738 , @xmath739 when @xmath740 . moreover , @xmath741 , @xmath742 , @xmath743 , @xmath744 . let @xmath745 , then @xmath746 u.\end{aligned}\ ] ] by we have @xmath747u\|_{p,{\mathcal{c}}_{\sigma ' r}}\right).\ ] ] compute @xmath748u$ ] directly . using the estimates for the coefficient matrix @xmath749 with @xmath710 chosen less than 1 , as well as the cut - off functions , we obtain the following ( for simplicity we write @xmath750 ) @xmath751u\|_{p } \leq c\left(\frac{1}{(1-\sigma')r}\|\nabla_x u\|_p+\frac{1}{(1-\sigma')r^2}\||x|^2 \nabla_t u\|_p\right)+\\ + \frac{c}{(1-\sigma')^2r^2}\|u\|_p+\frac{c}{(1-\sigma')r}\||x|\nabla_t u\|_p,\end{gathered}\ ] ] where @xmath752 is some absolute constant . now using the interpolation between the classical sobolev spaces ( for @xmath753 ) and young s inequality we have for any @xmath279 , @xmath754 hence by rescaling and then taking the supreme in @xmath755 , we have @xmath756 @xmath757 @xmath758 combining - and choosing @xmath359 , @xmath759 small enough , depending on @xmath384 , we obtain the inequality . in this section we show that the legendre function @xmath92 which satisfies the fully nonlinear pde is smooth in a neighborhood of the origin . we will work on the non - isotropic cylinder at the origin : @xmath760 before proving the main theorem , we make the following two remarks : a. by corollary [ cor : extension ] and the discussion in section [ sec : subelliptic - coeff ] , there is @xmath761 small enough such that @xmath762 for any @xmath674 and the linearized operator @xmath661 can be viewed as a perturbation of the baouendi - grushin type operator in @xmath763 . b. we note the following rescaling property : if @xmath92 solves in @xmath763 , then @xmath764 with @xmath765 and @xmath766 the non - isotropic dilation in will solve @xmath767 hence by multiplying a nonzero constant , we may assume that the coefficient matrix @xmath768 is of the form @xmath707 in @xmath763 with @xmath649 continuous and satisfying for sufficiently small @xmath710 , where @xmath710 is chosen such that the @xmath1 estimate ( proposition [ prop : perturb - grushin ] ) applies . the idea to show the smoothness is then to apply iteratively to the first order difference quotient of @xmath769 , but each step we need to be careful that the non - homogeneous rhs coming from differentiation is bounded in @xmath1 . for notation simplicity , in what follows we will discuss the case when @xmath770 . then equation is simply @xmath771 the arguments for @xmath772 are the same . [ thm : smoothness - v ] let @xmath92 be the legendre function of @xmath13 defined in . let @xmath761 such that in @xmath763 , corollary [ cor : extension ] holds for @xmath92 and @xmath768 can be written in the form of with @xmath749 satisfying for sufficiently small @xmath710 . then @xmath92 is smooth at the origin . we let @xmath773 denote the first difference quotient of @xmath92 in @xmath774 direction . _ step 1 : _ show @xmath775 for any @xmath702 , @xmath587 . by corollary [ cor : extension ] , it is enough to show it for @xmath776 . in fact , @xmath762 implies that @xmath777 and @xmath778 on @xmath779 for @xmath780 . moreover , by taking @xmath781 on both sides of we get @xmath782 satisfies @xmath783 with @xmath784 since a translation in @xmath785 direction does not change the subelliptic structure of the operator ( by corollary [ cor : extension ] and the @xmath286 dependence of @xmath786 on @xmath31 ) , i.e. @xmath787 is still a perturbed baouendi - grushin operator in the form of , with @xmath749 satisfying in @xmath788 . then by proposition [ prop : perturb - grushin ] there exists @xmath691 such that for any @xmath789 @xmath790 note that the rhs of is uniformly bounded in @xmath140 . moreover , @xmath791 and @xmath792 ( here we slightly abuse of the notation to let @xmath793 denote the weighted second order derivatives and first order derivatives w.r.t . @xmath515 and @xmath516 ) . thus @xmath794 for any @xmath587 with @xmath795 depending on @xmath796 , @xmath797 and @xmath384 . _ step 2 : _ show @xmath798 , @xmath587 . take @xmath799 to both sides of . from step 1 , @xmath800 and it satisfies @xmath801 applying @xmath802 to with @xmath803 , then @xmath804 and it satisfies @xmath805 with @xmath806 to estimate the @xmath807 , we first notice that @xmath229 ( up to a translation @xmath808 ) is a summation of the following terms : @xmath809 next , since @xmath810 for any @xmath811 , then by hlder s inequality , for @xmath812 satisfying @xmath813 we have @xmath814 apply hlder to estimate @xmath815 . for some @xmath816 satisfying @xmath817 we have @xmath818 by corollary [ cor : extension](iv ) , the second term on the rhs of is bounded . from the boundedness of @xmath819 shown in step 1 , the third term is uniformly bounded in @xmath140 . hence combining we have @xmath820 is uniformly bounded in @xmath140 . similarly by using corollary [ cor : extension ] , and step 1 , we have @xmath821 , @xmath822 are uniformly bounded in @xmath140 . therefore , applying the @xmath1 estimate ( proposition [ prop : perturb - grushin ] ) to , one can find a constant @xmath823 independent of @xmath140 , such that for any @xmath721 , @xmath824 since the rhs is uniformly bounded in @xmath140 , this implies @xmath825 from we know @xmath826 , @xmath827 , @xmath828 . multiplying a cut - off function to extend the functions to @xmath3 and applying the sobolev embedding lemma [ lem : embedding](i ) we have , for @xmath829 ( with @xmath830 the homogeneous dimension associated with @xmath661 ) , @xmath831 repeat the above arguments starting from with @xmath832 replaced by @xmath833 ( note @xmath834 if @xmath835 ) . after finite steps @xmath700 ( which only depends on the dimension ) we will get @xmath826 , @xmath836 with @xmath816 larger than the homogeneous dimension @xmath830 , and hence by embedding lemma [ lem : embedding](ii ) are in @xmath837 . applying proposition [ prop : perturb - grushin ] again we obtain @xmath838 for @xmath702 . noting that @xmath839 is chosen arbitrary , we complete the proof for step 2 . _ show @xmath840 , @xmath841 with @xmath842 . first from step 1 , @xmath843 for @xmath844 with @xmath845 . this together with the boundedness of @xmath846 obtained in step 2 gives @xmath847 next we estimate @xmath848 with @xmath849 and @xmath828 . similar as , @xmath848 satisfies @xmath850 with @xmath851 by we immediately have @xmath852 is uniformly bounded in @xmath140 for any @xmath702 . applying the @xmath1 estimate ( proposition [ prop : perturb - grushin ] ) we have @xmath853 is uniformly bounded in @xmath140 , and therefore completes the proof for step 3 . _ show @xmath854 for @xmath855 , with @xmath856 depending on @xmath857 and dimension . this is done in the same way as for @xmath841 . more precisely , we first consider the equation of @xmath858 in @xmath859 @xmath860 with @xmath861 bounded uniformly in @xmath140 for some @xmath862 . then we apply the sobolev embedding lemma and the @xmath1 estimate iteratively to obtain the boundedness of @xmath863 with some @xmath864 , as well as the boundedness of @xmath865 for any @xmath702 . in particular , this combined with the fact that @xmath840 for all @xmath866 gives that @xmath867 is bounded with @xmath868 for any @xmath702 . next , we consider the equation for @xmath869 , @xmath870 , @xmath871 and @xmath842 . similar as in step 3 , one easily get the uniform boundedness of @xmath872 for any @xmath702 due to the boundedness of @xmath873 , @xmath874 . applying @xmath1 estimate again we obtain the conclusion . [ cor : freeboundary ] let @xmath13 be a solution to the signorini problem in @xmath70 , and let @xmath19 be the regular set of the free boundary . then @xmath19 is smooth . take @xmath37 a regular free boundary point , in a coordinate chart centered at @xmath31 , by @xcite @xmath19 can be locally expressed as the graph of a @xmath46 function @xmath875 with @xmath876 . consider in a neighborhood of @xmath31 the partial hodograph - legendre transform and the corresponding legendre function @xmath92 defined in section [ sec : legendre - transf - its ] . by theorem [ thm : smoothness - v ] , @xmath92 is smooth at the origin . since @xmath877 hence the smoothness of @xmath92 at the origin implies the smoothness of @xmath229 at @xmath878 . in this section , we show that the legendre transform @xmath92 is analytic in a neighborhood of the origin ( theorem [ thm : fb - regul ] ) . [ thm : main ] let @xmath92 be the legendre transform defined in section [ sec : legendre - funct - nonl ] . then @xmath92 is real analytic in a neighborhood of the origin . we first make some more assumptions and observations . + 1 . for simplicity we work on @xmath879 . by the scaling invariant property mentioned at the beginning of section [ sec : smoothness - free - bound ] we may assume @xmath92 is smooth in @xmath880 and solves the fully nonlinear equation there . denote @xmath881 from corollary [ cor : extension](iii)(iv ) and the intermediate value theorem , we have @xmath882 + 2 . the following multi - index notation will be used : + for a multi - index @xmath883 , @xmath884 let @xmath885 , @xmath886 be two multi - index in @xmath887 , we say @xmath888 iff @xmath889 , @xmath890 ; and @xmath891 iff @xmath892 , @xmath893 , where @xmath894 . the strategy to prove the analyticity is as follows : given @xmath895 , taking @xmath896 on both sides of and using the summation convention on @xmath897s , we obtain that @xmath769 satisfies @xmath898 where @xmath899 is the set of all permutations of @xmath900 . we will apply proposition [ prop : perturb - grushin ] for to get a fine estimate of the @xmath1 norm of @xmath769 . in order to do so , usually one needs a sequence of domains with properly shrinking radius as well as the corresponding sequence of cut - off functions . in this paper , we use the trick introduced in @xcite to avoid this technical trouble . in the following we take and fix a cut - off function @xmath901 which satisfies @xmath902 we will estimate @xmath903 with @xmath904 by @xmath903 with @xmath905 . + 4 . from now on we will simply write @xmath906 if the integral domain is @xmath907 . we will fix a @xmath384 larger than the homogeneous dimension @xmath830 . by a universal constant we mean an absolute constant which only depends on @xmath908 , @xmath237 , dimension ( which is @xmath909 in our setting ) or @xmath384 chosen , in particular independent of @xmath910 . the following observation will be useful in the proof : @xmath911 which implies @xmath912 hence for multi - index @xmath230 with @xmath913 , @xmath914 for some @xmath915 , @xmath916 with @xmath917 and @xmath918 . the following proposition is the main proposition of this section . [ prop : induction ] there exist universal constants @xmath919 , @xmath920 and @xmath921 such that for any @xmath922 @xmath923 theorem [ thm : main ] will follow from proposition [ prop : induction ] . indeed , by proposition [ prop : induction ] there exists a universal @xmath5 such that @xmath924 hence by the classical sobolev embedding @xmath925 for @xmath835 chosen above , one has @xmath926 hence @xmath92 is in gevrey class @xmath927 , which is the same as the class of real analytic functions . before proving proposition [ prop : induction ] , we first show a lemma on the @xmath928-norm of @xmath769 , which roughly speaking is a consequence of the sobolev embedding lemma ( lemma [ lem : embedding](ii ) ) . [ lem : embedding2 ] for @xmath929 , assume proposition [ prop : induction ] holds for @xmath930 and @xmath910 . then there is a universal constant @xmath931 such that @xmath932 \(i ) first by lemma [ lem : embedding2](ii ) , there is a universal constant @xmath752 such that @xmath933 applying to the rhs of above inequality , we have @xmath934 by proposition [ prop : induction](i ) for @xmath930 and @xmath910 , there exists a universal constant @xmath935 , which can be chosen larger than @xmath237 such that @xmath936 the estimate for @xmath937 follows from the classical sobolev embedding , , and the assumption . \(ii ) similar to ( i ) . \(iii ) use corollary [ cor : extension ] and ( ii ) above . this is done by induction . assume ( i ) and ( ii ) hold for @xmath938 . we want to show they hold for @xmath910 . let @xmath230 be a multi - index with @xmath904 . from , @xmath939 satisfies the following equation @xmath940 where @xmath941 by the @xmath1 estimate for the perturbed baouendi - grushin operator ( proposition [ prop : perturb - grushin ] ) , there is a universal @xmath942 such that @xmath943 the estimate of @xmath944 is standard . in fact , @xmath945 for some @xmath946 with @xmath947 . by the induction assumption ( i)(ii ) for @xmath930 , the above rhs is bounded by @xmath948 similarly there is some @xmath949 with @xmath950 such that @xmath951 hence @xmath952 next we estimate @xmath953 . _ proof of ( i ) . _ let @xmath954 , @xmath955 , @xmath956 . then @xmath957 we discuss the following two cases : @xmath958 . if @xmath959 , then @xmath960 which is by lemma [ lem : embedding2](iii ) and the induction assumption ( i ) for @xmath930 bounded by @xmath961 if @xmath962 , then @xmath963 which is by lemma [ lem : embedding2](iii)(i ) and the induction assumption ( i ) for @xmath930 bounded by @xmath964 hence @xmath965 by stirling s formula and the fact that @xmath966 , there is a universal constant @xmath967 such that @xmath968 hence @xmath969 @xmath970 or @xmath971 . we only discuss when @xmath972 . similarly we consider @xmath973 and @xmath974 . if @xmath973 , the estimate is indeed included in case 1 . if @xmath974 , then we simply have @xmath975 arguing similarly as in case 1 we have @xmath976 combining the above two cases we have , @xmath977 we combine , for @xmath978 and , and use the induction assumption ( i ) for @xmath979 to estimate @xmath980 . then @xmath981 where @xmath982 is a universal constant . by for @xmath978 , @xmath983 thus combining with induction assumption ( i ) for @xmath930 and @xmath984 , we have @xmath985 where @xmath986 . choosing @xmath987 we proved ( i ) . _ proof of ( ii ) . _ the key step is to estimate @xmath953 , which is done similarly as for ( i ) . more precisely , lemma [ lem : embedding2](i ) and ( with @xmath987 ) imply @xmath988 this together with the induction assumption ( ii ) up to @xmath930 yields , for any multi - index @xmath230 , @xmath989 , @xmath990 in the estimate of @xmath953 , we first consider when @xmath991 : by lemma [ lem : embedding2](ii ) and , @xmath992 hence @xmath993 since for @xmath230 with @xmath904 fixed , we have the following identity ( e.g. proposition 2.1 in @xcite ) : @xmath994 then if we let @xmath995 , @xmath996 , @xmath997 , the rhs of is bounded by @xmath998 by stirling s formula and if we still use @xmath999 to denote the universal constant from it , then the above quantity is bounded by @xmath1000 the arguments for the case @xmath1001 are exactly the same . hence @xmath1002 the rest of the proof for ( ii ) is the same as for ( i ) and we do not repeat here . in the appendix , we give a short proof for lemma [ lem : embedding ] ( the sobolev embeddings for @xmath1003 , see also @xcite ) and theorem [ thm : grushin ] ( the @xmath1 estimates for baouendi - grushin operator ) . both statements are well known , and available in much greater generality . checking that the general results apply , however , requires familiarity with the theory of subelliptic operators . for completeness and the convenience of the reader we provide complete proofs . we first prove ( i ) . suppose that @xmath1004 and extend @xmath13 to be zero outside @xmath679 . for every @xmath1005 , @xmath1006 with @xmath1007 , @xmath1008 where @xmath1009 is a @xmath322 function satisfying @xmath1010 , @xmath1011 . let @xmath1012 then by a direct computation @xmath1013 hence @xmath1014 which by a change of variable @xmath1015 gives @xmath1016 by young s inequality for convolution and fubini , for @xmath1017 , @xmath1018 observe that for @xmath1019 , @xmath1020 for any @xmath1021 , and for @xmath1022 , @xmath1023 . hence if we let @xmath1024 after integrating over @xmath1025 , by fubini and a change of variable we have @xmath1026 by the hardy - littlewood - sobolev inequality , for the chosen @xmath816 satisfying @xmath1027 we have @xmath1028 next we prove ( ii ) . since this property is local in nature , given @xmath1029 we may assume by multiplying a characteristic function that @xmath1030 in @xmath1031 . hence the integration in can be written from @xmath878 to @xmath155 for some @xmath1032 large enough . next we give a short proof for the @xmath1 estimate for baouendi - grushin operator @xmath668 . the idea is to treat @xmath668 as the projected operator of sub - laplacian on the heisenberg - reiter type group onto certain quotient space . the transference method in @xcite links the @xmath1 estimate on the group to the @xmath1 estimate for @xmath668 in the quotient space . consider @xmath1037 equipped with the group law @xmath1038 where @xmath1039 is the space of @xmath1040 real matrices , and @xmath1041 are understood as the matrix multiplication . let @xmath1042 . @xmath1043 is an example of heisenberg - reiter group , which is by @xcite a nilpotent lie group of step @xmath45 , with dilation @xmath1044 and homogeneous dimension @xmath1045 . it is also immediate that the lebesgue measure @xmath1046 is a left and right haar measure on @xmath1043 . now we recall several facts about the nilpotent lie groups with dilations . a direct computation shows that the horizontal vector fields in the lie algebra @xmath1047 that agree at the origin with @xmath1048 and @xmath1049 are @xmath1050 consider the sub - laplacian in @xmath1043 @xmath1051 it is easy to check that @xmath1052 , @xmath1053 are hrmander vector fields , then @xmath1054 is hypoelliptic . by theorem 2.1 in @xcite , there exists a unique fundamental solution of type @xmath45 ( i.e. smooth away from @xmath878 and homogeneous of degree @xmath1055 ) for @xmath1054 , which we denote by @xmath1056 . for @xmath1057 we have @xmath1058 . since @xmath1059 and @xmath1053 are left - invariant , then @xmath1060 where @xmath642 is a quadratic polynomial in @xmath1059 and @xmath1053 . by @xmath1061 estimates for the singular integral in homogeneous groups ( see for example xiii 5.3 in @xcite ) , we have @xmath1062 and in particular , using we have @xmath1063 now let @xmath1064 . one can easily verify that @xmath1065 is a closed subgroup of @xmath1043 with a bi - invariant measure the lebesgue measure @xmath1066 . let @xmath1067 be the quotient space and @xmath1068 the natural quotient mapping . given @xmath1069 , define @xmath1070 by theorem 15.21 in @xcite , the above correspondence @xmath1071 defines a linear mapping of @xmath1072 onto @xmath1073 . we identify @xmath1074 with @xmath1075 . a direct computation gives @xmath1076 since the lebesgue measure @xmath1077 and @xmath1078 are bi - invariant on @xmath1043 and @xmath1065 correspondingly , then by theorem 15.24 in @xcite , there exists a unique right - invariant measure ( up to a constant ) on @xmath1079 , which is necessarily the lebesgue measure @xmath1080 . consider the following vector fields , which are the ` push - down ' vector fields of @xmath1059 , @xmath1053 on @xmath1081 @xmath1082 @xmath1083 note that the baouendi - grushin operator can be written as @xmath1084 to see that they are indeed the push - down vector fields on @xmath1074 , we notice that for @xmath1057 , @xmath1085 similarly , using one can check that @xmath1086 hence @xmath1087 let @xmath919 be the representation of @xmath1043 acting on @xmath1088 given by the right translation , i.e. given @xmath1089 @xmath1090 it is easy to check that @xmath1091 are bounded from @xmath1088 to @xmath1088 by @xmath597 . given @xmath1092 , @xmath702 , we consider the following linear map @xmath89 on @xmath1088 : @xmath1093 notice that @xmath1043 is locally compact and by the convolution operator @xmath1094 is bounded in @xmath1095 . thus the transference method ( theorem 2.4 in @xcite ) applies and we have for @xmath1096 , @xmath1097 in particular if we take @xmath1098 , then @xmath1099 in the end we show that @xmath1100 . indeed , by and @xmath1101 by and fubini , then using and we have @xmath1102 observing that @xmath1103 , @xmath1104 , @xmath1105 , and @xmath1106 can be written as @xmath1107 for some quadratic polynomial @xmath642 , we complete the proof of the theorem .
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in this paper we study the higher regularity of the free boundary for the elliptic signorini problem . by using a partial hodograph - legendre transformation
we show that the regular part of the free boundary is real analytic .
the first complication in the study is the invertibility of the hodograph transform ( which is only @xmath0 ) which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points .
the second and main complication in the study is that the equation satisfied by the legendre transform is degenerate .
however , the equation has a subelliptic structure and can be viewed as a perturbation of the baouendi - grushin operator . by using the @xmath1 theory available for that operator , we can bootstrap the regularity of the legendre transform up to real analyticity , which implies the real analyticity of the free boundary .
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`` materials by design '' , the ability to design and create a material with specified correlated electron properties , is a long - standing goal of condensed matter physics . superlattices , in which one or more component is a transition metal oxide with a partially filled @xmath0-shell , are of great current interest in this regard because they offer the possibility of enhancing and controlling the correlated electron phenomena known @xcite to occur in bulk materials as well as the possibility of creating electronic phases not observed in bulk.@xcite following the pioneering work of ohtomo and hwang,@xcite heterostructures and heterointerfaces of transition metal oxides have been studied extensively . experimental findings include metal - insulator transitions,@xcite superconductivity , @xcite magnetism @xcite and coexistence of ferromagnetic and superconducting phases.@xcite solid solution in plane of carrier concentration ( changed by sr concentration ) and tilt angle in @xmath1 structure but with all three glazer s angles nearly equal . dashed line indicates relation between carrier concentration and rotation amplitude in physically occurring bulk solid solution . from ref . . ] in this paper we consider the possibility that appropriately designed superlattices might exhibit ferromagnetism . our work is partly motivated by a recent report@xcite of room - temperature ferromagnetism in superlattices composed of some number @xmath2 of layers of lavo@xmath3 ( lvo ) separated by one layer of srvo@xmath3 ( svo ) , even though ferromagnetism is not found at any @xmath4 in the bulk solid solution la@xmath5sr@xmath6vo@xmath3 . our study is based on a previous analysis@xcite of the possibility of obtaining ferromagnetism in variants of the crystal structure of bulk solid solutions of the form la@xmath5sr@xmath6vo@xmath3 . a key result of the previous work was that ferromagnetism is favored by a combination of large octahedral rotations and large doping away from the mott insulating lavo@xmath3 composition . a schematic phase diagram is shown in fig . [ fig : bulkpd ] . however , as indicated by the dashed line in the figure , in the physical bulk solid solution , doping away from the mott insulating concentration reduces the amplitude of the octahedral rotations so that the physical materials remain far from the magnetic phase boundary . the motivating idea of this paper is that in the superlattice geometry , octahedral rotation amplitude may be decoupled from carrier concentration . the rotations can be controlled by choice of substrate while the carrier concentration can be controlled by choice of chemical composition and may vary from layer to layer of a superlattice . in effect , an appropriately designed superlattice could enable the exploration of different paths in fig . [ fig : bulkpd ] . in this study , we combine single - site dynamical mean field approximation@xcite with realistic band structure calculations including the effects of the octahedral rotations to determine the ferromagnetic - paramagnetic phase diagram in superlattices with the crystal structures believed relevant@xcite to the experiments of ref . . unfortunately we find that the experimentally determined crystal structure is in fact less favorable to ferromagnetism than the one found in the bulk solid solution , but we indicate structures that may be more favorable . the paper has following structure . the model and methods are described in sec . [ sec : model ] . [ sec : cubicsuperlattice ] establishes the methods via a detailed analysis of the phase diagram of superlattices with no rotations or tilts . in sec . [ sec : tiltedsuperlattice ] we present the magnetic properties of superlattices with octahedral rotations similar to those observed experimentally . section [ sec : conclusions ] is a summary and conclusion . this paper builds on a previous study of the magnetic phase diagram of bulk vanadates.@xcite the new features relevant for the superlattices studied here are ( i ) the change in geometrical structure , including the differences from the bulk solid solution in the pattern of octahedral tilts and rotations and ( ii ) the variation of electronic density arising from superlattice structure . in the rest of this section we briefly summarize the basic theoretical methodology ( referring the reader to ref . for details ) , define the crystal structures more precisely , explain the consequences for the electronic structure and explain how the variation of density appears in the formalism . we study superlattices composed of layers of srvo@xmath3 ( svo ) alternating with layers of lavo@xmath3 ( lvo ) . if we idealize the structures as cubic perovskites , then the layers alternate along the @xmath7 $ ] direction . in bulk , svo crystallizes in the ideal cubic perovskite structure,@xcite while lvo crystallizes in a lower symmetry @xmath1 structure derived from the cubic perovskite via a four unit - cell pattern of octahedral tilts . @xcite the crystal structure of bulk solid solutions la@xmath5sr@xmath6vo@xmath3 interpolates between that of the two end - members with the rotation amplitude decreasing as @xmath4 increases . in the superlattice , the presence of a substrate and the breaking of translation symmetry can lead to different rotational distortions of the basic perovskite structure and also to a difference between lattice constants parallel and perpendicular to the growth direction . octahedral rotations in perovskites can be described using glazer s notation.@xcite in the coordinate system defined by the three v - o bond directions of the original cubic perovskite , there are 3 tilt angles @xmath8 and @xmath9 with corresponding rotation axes @xmath10,[010]$ ] and @xmath7 $ ] . the tilt is in - phase if successive octahedra rotate in the same direction , and anti - phase if they rotate in opposite directions . rotational distortions of the cubic perovskite @xmath11o@xmath3 structure may be denoted by @xmath12 where @xmath13 can be @xmath14 or @xmath15 denoting in - phase , anti - phase or no tilting , respectively and @xmath16.@xcite bulk lvo ( @xmath1 ) is of the type @xmath17 with @xmath18 and @xmath19.@xcite for superlattices , substrate - induced strain may change the situation in a way which depends on the growth direction . experiments@xcite confirm that the growth direction for the experimentally relevant superlattices is @xmath7 $ ] ( in the ideal cubic perovskite notation ) and we focus on this case here . recent experimental studies of superlattices@xcite and of lavo@xmath3 thin films , which apparently have the same growth direction,@xcite suggest that the rotations are of the type @xmath20@xcite and indicate that the dominant rotation is around the axis defined by the growth direction : @xmath21 and @xmath22 . this distortion pattern is different from that occurring in bulk . to explore its effects we set @xmath23 and consider the consequences of varying @xmath9 . in bulk la@xmath5sr@xmath6vo@xmath3 , while the 4-sublattice @xmath1 structure implies a difference in lattice constants , all v - o bond lengths are the same.@xcite the difference in lattice constants arises from a difference in tilting pattern . superlattices are typically grown on a substrate , and in epitaxial growth conditions the lattice constants perpendicular to the growth direction ( which we denote here by @xmath24 ) are fixed by the substrate , while the lattice parameter along the growth direction ( @xmath25 ) is free to relax . the result is a @xmath26 ratio typically @xmath27 contributed by both tilting and anisotropy in v - o bond lengths and possibly varying from layer to layer of the superlattice . for the experimentally studied superlattices , @xmath28 . @xcite the v - o bond lengths have not been determined but , as discussed in more detail in the appendix , our studies indicate that all v - o bonds have essentially the same length . further we show that a few percent differences have no significant effect on our study of ferromagnetism . in the rest of the paper we therefore ignore these distortions , setting all v - o bond lengths to be equal . we study superlattices designed to be similar to the system studied in ref . . in these superlattices , units of @xmath2 layers of lavo@xmath3 are separated by one layer of srvo@xmath3 . to define the superlattice , we begin from lvo in the appropriate bulk structure , then break translation invariance along the @xmath7 $ ] ( @xmath29-direction ) by replacing every ( m+1)@xmath30 lao plane with an sro plane . fig . [ fig : cartoon]a shows such a superlattice with @xmath31 . we assume that the superlattice is grown epitaxially so that in - plane bond lengths and other aspects of the local structure including rotations are the same for all layers . we therefore take the electron transfer integrals which define the band structure to the be same for all layers . in this case the electronic structure of a superlattice is defined by adding the electrostatic potentials of the sr and la ions to the basic translationally invariant hopping hamiltonian describing the bulk materials . in our calculations we follow the common practice in studies of early transition metal oxides by assuming that the energy splitting between transition metal @xmath0-bands and oxygen @xmath32-bands is large enough to justify the use of a `` frontier orbital '' model focusing on the @xmath32-@xmath0 antibonding bands which are mainly composed of vanadium @xmath33-symmetry @xmath0-states . the hamiltonian for the superlattice is thus @xmath34 where @xmath35 describes the electron - ion interaction and electron - electron interaction between different sites and @xmath36 describes the @xmath0-@xmath0 interactions , which we take to be on - site . @xmath37 is a tight binding model , derived by using maximally - localized wannier function ( mlwf ) techniques@xcite to fit the @xmath33-derived antibonding bands . the detailed procedure is described in our previous work.@xcite ( color online ) ( a ) schematic of superlattice lattice structure ( lavo@xmath3)@xmath38(srvo@xmath3)@xmath39 with @xmath31 . vanadium sites indicated as circles with charge density indicated by shading : heavy shading ( black online ) indicating higher charge density and light shading ( yellow online ) indicating lower charge density . lao and sro planes are shown as solid and dashed lines respectively . nearest neighbor ( @xmath40 ) and next - nearest neighbor ( @xmath41 ) hoppings between vanadium sites indicated by arrows . the numbers on the right are vo@xmath42 layer indices . ( b ) inset : @xmath43 hopping between @xmath33 orbital and @xmath32-orbital . main panel : two - dimensional nearest neighbor hopping @xmath40 made of two @xmath43 hoppings from @xmath44 orbital of one vanadium site to oxygen @xmath45 or @xmath46 orbital , then to @xmath44 orbital of another vanadium site . ] the kinetic hamiltonian has the quadratic form @xmath47 where @xmath48 and @xmath49 are electron creation and annihilation operators in reciprocal space with wavevector @xmath50 . @xmath51 and @xmath52 are orbital and layer indices , and @xmath53 is the spin index . we assume that the interaction takes the standard slater - kanamori form @xcite which following ref . we write as @xmath54 where the values of the on - site interaction @xmath55 and the hund s coupling @xmath56 are @xmath57 and @xmath58ev so that lvo is an insulator in bulk while svo is a metal . in the approximation employed here , the superlattice is defined by the coulomb interaction between the la / sr ions and electrons . this , and the off - site part of the electron - electron interaction is contained @xcite in @xmath59 to construct @xmath60 , we assume that the whole ion charge of svo or lvo unit cell comes into the sr or la site . consider srvo@xmath3 , the valence of v is @xmath61 ( @xmath62 . if this one @xmath0-electron is removed , the svo unit cell will have charge @xmath63 , hence , in our model , sr site has charge @xmath63 . similarly , lavo@xmath3 has v@xmath64 ( @xmath65 ) , thus la site has charge @xmath66 . as a result , @xmath60 has the form @xmath67 where @xmath68 is electron - occupation operator at v - site @xmath69 , @xmath70 is the relative dielectric constant . the part @xmath71 is the inter - site coulomb interaction of vanadium @xmath0-electrons @xmath72 @xmath71 is treated in the hartree approximation . note that in eq . , @xmath73 is the operator giving the total @xmath0-electron occupation of site @xmath69 , while @xmath74 is the expectation value of @xmath0-electron occupancy at site @xmath75 , which is determined self consistently . from @xmath35 , the coulomb potential @xmath76 for site @xmath69 is calculated using ewald summation.@xcite the dielectric constant @xmath70 is an important parameter in eqs . ( [ eqn : hel - ion ] , [ eqn : hel - el ] ) . it accounts for screening on the scale of a lattice constant so bulk measurements are not directly relevant and an appropriate value has not been determined . values ranging from @xmath77 to @xmath78 have been reported in the literature for similar systems.@xcite because the appropriate value of @xmath70 has not been determined , we have studied several cases and present results mainly for @xmath79 . we treat the on - site interaction terms using single - site dynamical mean field theory ( dmft)@xcite with the hybridization expansion continuous time quantum monte carlo ( ctqmc ) solver.@xcite the superlattice effect is taken into account by the coulomb potential @xmath80 . we use the superlattice dynamical mean field theory introduced by potthoff and nolting @xcite in the form given in ref . . here each v site @xmath69 has a self energy ( site local but dependent on site ) determined from the solution of a quantum impurity model which has parameters fixed by the dmft self - consistency equation linking the site local term of the lattice green function @xmath81 to the quantum impurity model green function@xcite @xmath82^{-1}\right\}_{ii},\ ] ] where @xmath83 is a site dependent quantity , diagonal in spin and orbital indices but linking different sites , derived from eqs . ( [ eqn : hel - ion ] , [ eqn : hel - el ] ) . the layers are coupled by a self - consistency condition which as discussed in refs . fixes both the hybridization function of the quantum impurity model and the layer - to - layer variation in the charge density . as described in ref . , it is advantageous to perform a site - local rotation to align the orbital basis to the local v - o bond directions of each octahedron before solving the impurity model . this reduces the sign problem in the ctqmc impurity solver and restores in - plane translation invariance in the sense of making the self - consistency equations the same for all sites in a given plane . in a superlattice composed of @xmath84 layers , it is in principle necessary to solve @xmath84 dynamical mean field problems , coupled by the self - consistency condition . however , we find ( see section [ sec : cubicsuperlattice ] ) that the susceptibility for a given layer of the superlattice may be determined from a bulk computation at the same local density and crystal structure . because the layer dependent density has no significant dependence on the temperature or the many - body physics , it may be determined once from a band structure calculation and then bulk results with the appropriate density for a wide range of temperature may be used to infer the curie temperature , substantially reducing the computational burden . the curie temperature for ferromagnetism is determined by extrapolating the inverse susceptibility @xmath85 to @xmath15 based on curie - weiss law @xmath86 . the test for the reliability of this method for @xmath87 has been done in ref . . a similar approach can be found in literature.@xcite in this section , we demonstrate that the magnetic phase diagrams of superlattice systems may be inferred , to reasonable accuracy , from the study of appropriately chosen bulk systems . this enables a considerable reduction in the computation resources required . ( color online ) panel ( a ) : non - interacting density of states for bulk system at carrier density @xmath88 . panels ( b ) : non - interacting density of states for different layers of ( lvo)@xmath3(svo)@xmath39 superlattice for two different values of dielectric constant @xmath89 ( solid ) and @xmath90 ( dashed ) with hopping parameters @xmath91ev and @xmath92ev . sro plane is between layers @xmath15 and @xmath93 ( the index is defined in fig . [ fig : cartoon ] ) . the fermi energy is at @xmath15 . ] we begin with a study of `` untilted '' or `` cubic '' superlattices : those in which all v - o - v bond angles are @xmath94 . we focus specifically on @xmath7 $ ] superlattices in which the unit cell contains @xmath2 layers lvo and one layer svo , where @xmath95 . for orientation , we present the density of states ( dos ) of the non - interacting system in fig . [ fig : dos ] . in obtaining these densities of states we used the simple tight binding parametrization . the dos for the bulk system is shown in panel ( a ) . one sees the typical three - fold degenerate dos for @xmath33 band , the van hove singularity is visible as a peak near the upper band edge . it is at high energy because the next - nearest neighbor hopping @xmath96 . the remaining panels show the layer - resolved densities of states for the @xmath31 superlattice . the upper two panels show layers sandwiched by la on both sides ; the lower two panels show the layers adjacent to the sro plane . the superlattice - induced changes in the density of states are seen to be relatively minor : the main effects are a weak splitting of the van hove peaks reflecting the breaking of translational invariance in the @xmath29-direction , and a relative shift in the positions of the van hove peaks arising from band bending associated with the different charges of the sr and la ions . [ fig : slchi ] shows the layer - resolved charge density and inverse susceptibility @xmath97 plotted against temperature for three different superlattice structures corresponding to @xmath98 . as expected from electrostatic considerations , the charge is lower for the vo@xmath42 planes nearer the sro layer and the charge variation between layers is controlled by the dielectric constant . the magnetization @xmath2 at the v sites on each layer was computed at field @xmath99 and the inverse susceptibility was obtained as @xmath100 . linearity was verified by repeating the computation using @xmath101 ( not shown ) . for the @xmath102 case ( fig . [ fig : slchi]a ) , we extended the computation to the lower temperature @xmath103ev ; for the other two cases @xmath104ev was the lowest temperature studied . the inverse susceptibilities are approximately linear in temperature at higher temperatures and in all cases , extrapolation to @xmath105 reveals @xmath106 , implying absence of ferromagnetism . ( color online ) temperature - dependent layer - resolved inverse magnetic susceptibilities for symmetry - inequivalent layers of untilted ( lvo)@xmath38(svo)@xmath39 superlattice structures with different numbers of lvo layers @xmath107 and @xmath108 . layer @xmath15 is adjacent to sro and layers @xmath109 and @xmath110 are between two lao layers . the relative dielectric constant is @xmath90 , magnetic field @xmath99 . the @xmath85 obtained from solution of bulk cubic systems with charge density set to the density on the given layer are also shown . `` bulk l0 '' ( `` bulkl2 '' ) denotes a calculation performed for a bulk system with density the same as for @xmath111 ( @xmath112 ) layer density . inset : the electron layer density distribution corresponding to the susceptibility plot , @xmath4-axis is the layer index , @xmath113-axis is the layer density . on - site interactions @xmath114ev , @xmath58ev . ] especially for the layer nearest the sro plane the @xmath115 curves exhibit weak upward curvature at the lowest temperatures studied . as shown in ref . , the curvature is a signature that the system is entering a fermi - liquid coherence regime . the fermi liquid coherence temperature is highest for the layers nearest the sro because the charge in these planes is farther from the @xmath116 mott insulating state . to verify this we followed ref . and computed the wilson ratio @xmath117 for each layer of the superlattice for the case @xmath118 , finding ( not shown ) that for each layer the @xmath117 extrapolates to @xmath109 at low temperature . the approach to the low temperature value is faster for layers with low density ( near sro planes ) than for layers with high density ( far from sro planes ) . @xmath119 is the value for a kondo lattice , while ferromagnetism is characterized by an @xmath120.@xcite we therefore believe that for `` untilted '' superlattices , the differences in @xmath115 among layers arise from differences in quasiparticle coherence scale , there is no evidence for ferromagnetism in this system , consistent with the solution of the corresponding bulk problem . to gain insight into the physics underlying the layer dependence of @xmath115 we have computed @xmath85 for the cubic bulk system ( @xmath121 , @xmath37 is constructed from the two - dimensional dispersion @xmath122 ) for carrier densities equal to those on the different vo@xmath42 layers . in fig . [ fig : slchi ] , we present bulk calculations for @xmath123 and @xmath124 corresponding to the densities calculated for layer @xmath15 and @xmath109 of the superlattice for all cases @xmath98 . for @xmath124 , bulk @xmath115 at @xmath125 and @xmath126ev are very close to those of @xmath112 layer of @xmath31 superlattice , which has the same density . for @xmath127 superlattices , bulk @xmath124 , @xmath85 ( not shown ) almost coincides with those of @xmath112 layer . for bulk @xmath123 , the difference between bulk and superlattice @xmath111 layer is small . these calculations demonstrate a general rule : within the single - site dmft approximation , the layer - resolved properties of a superlattice correspond closely to those of the corresponding bulk system at a density equal to that of the superlattice . ( color online ) comparison between bulk lvo partial dos ( positive curves ) and ( lavo@xmath3)@xmath3(srvo@xmath3)@xmath39 superlattice layer dos of layers near sro ( negative curves ) derived from band structure calculations ( dft+mlwf ) . both systems have the same lattice structure for each case : untilted structure for the top panel and @xmath128 structure ( glazer s notation @xmath20 ) with @xmath129 and @xmath130 and @xmath131 for other panels . the dos of bulk system is shifted towards higher energy so that bulk carrier density is the same as layer density of superlattices for the layers near sro ( @xmath132 ) . the vertical dashed line marks the fermi level . ] the superlattices of experimental relevance have crystal structures which are distortions of the `` untilted '' one , involving in particular a @xmath128 structure characterized by a rotational distortion of the @xmath20 type@xcite involving a large rotation about an axis approximately parallel to the growth direction and much smaller rotations about the two perpendicular axes . [ fig : dos_bulk_sl ] compares the non - interacting dos of bulk and ( lvo)@xmath3(svo)@xmath39 superlattice systems ( both with the same @xmath128 structure ) calculated using dft and a mlwf parametrization of the frontier bands . the dos of bulk system is shifted so that it has the same carrier density as layers of the superlattices near sro plane . for three different structures ( untilted structure and @xmath128 structure with @xmath133 and @xmath131 ) , the basic features of the partial dos are similar between bulk and superlattice . the translation symmetry breaking in @xmath29-direction leads to small extra peaks in the superlattice dos . these differences are smoothed out by the large imaginary part of the dmft self energy . because the dmft equations depend only on the density of states it is reasonable to expect that , as in the untilted case , they will therefore give the same results in the superlattice as in the bulk material with corresponding density of states . ( color online ) comparison in temperature dependent inverse susceptibility between bulk lvo ( solid lines ) and ( lavo@xmath3)@xmath3(srvo@xmath3)@xmath39 superlattice ( dashed lines ) . both have the same lattice structure @xmath128 with tilt angle @xmath134 and @xmath135 . bulk system has the same densities as those of layers of superlattice near and far from sro planes ( @xmath136 ) . left column : the plots in wide temperature range . right column : the expanded views near zero temperature . ] to verify that this is the case we have also compared bulk and superlattice susceptibilities for tilted structures . the four vo@xmath137 octahedra in a unit cell are related by rotation , so an appropriate choice of local basis means that only one calculation needs to be carried out for a given layer . [ fig : bulkvslayertilted ] compares the inverse susceptibilities for an @xmath31 superlattice to calculations performed on a bulk system with the same @xmath128 structure . in these calculations , we choose @xmath138 and dielectric constant @xmath139 . we see that in this case , as in the `` untilted '' case , the superlattice inverse susceptibilities @xmath85 are almost the same as those for bulk system calculated at the same density , with differences only resolvable in the expanded view for the largest tilt angles . in this section we present and explain our results for the magnetic phase diagram of ( lvo)@xmath38(svo)@xmath39 superlattices with the @xmath128 structure ( glazer s notation @xmath20 ) reported for the experimental systems.@xcite in these structures in - plane rotation along the growth direction @xmath141 $ ] is large @xmath130 ( presumably because of the strain imposed by the substrate ) , while the out - of - plane rotation is small ( @xmath129 ) perhaps because the system is free to relax along the growth direction . we concentrate on the effect of the large rotation by fixing the in - plane angles to @xmath142 while varying the out - of - plane angles over a wide range from @xmath143 . ( color online ) partial dos derived from dft+mlwf for `` bulk '' @xmath128 structure ( glazer s notation @xmath20 ) with @xmath129 and @xmath144 changing from @xmath145 to @xmath131 . only @xmath33 bands are plotted because @xmath146 bands are negligible in this range of energy . ] based on the results of section [ sec : cubicsuperlattice ] we generate a phase diagram for the superlattice from calculations for a bulk system which is a @xmath128 distortion of the ideal cubic perovskite structure of chemical composition lavo@xmath3 . the bulk system results are presented as a phase diagram in the plane of carrier concentration and @xmath9-rotation . specific layers of the superlattice will correspond to particular points on the phase diagram , with the layer dependent density fixed by number of lvo layers @xmath2 and the dielectric constant @xmath70 and the rotation fixed by the substrate lattice parameter . we use dft+mlwf methods to obtain the frontier orbital band structure for the @xmath33-derived antibonding bands [ fig : pdos_p21 m ] presents representative results for the orbitally resolved local density of states . in this figure the orbitals are defined with respect to the local basis defined by the 3 v - o bonds of a given vo@xmath137 octahedron . we define @xmath141 $ ] as the axis ( approximately parallel to the growth direction ) about which the large rotation occurs . [ fig : pdos_p21 m ] shows that @xmath147 and @xmath148 orbitals are almost degenerate , while @xmath44 orbital is strikingly different . the dos of @xmath44 orbital maintains the shape of a two - dimensional energy dispersion with a van hove peak well above the chemical potential , similar to the bulk cubic structure ( see e.g. fig . [ fig : dos]a ) . there are noticeable differences only at very high rotation angles . on the other hand , @xmath147 and @xmath148 orbitals are spread out with two small peaks , because hoppings along @xmath4 or @xmath113 directions ( more distorted ) are different from those along @xmath29-direction ( less distorted ) . when the distortion gets larger , the @xmath0-bandwidth becomes smaller , the @xmath44 peak gets larger and slightly closer to the fermi level , and @xmath147 and @xmath148 peaks near the fermi level also develop . based on @xmath149 generated by dft+mlwf , we carry out dmft calculations for in - plane rotation angle @xmath9 to get @xmath115 curves whose extrapolations define the curie temperatures @xmath87 . [ fig : tc_evolve ] shows how @xmath87 evolves when the rotation angle @xmath9 increases from @xmath150 to @xmath135 . in this figure , we consider two different carrier densities @xmath151 and @xmath152 , corresponding to the band structure prediction for the layer densities of layers near and far from sro planes in the superlattice . @xmath87 for @xmath153 is a slow function of rotation and is always negative for the range of @xmath9 under consideration , while @xmath87 for @xmath151 increases faster , so that the system becomes ferromagnetic when @xmath9 is between @xmath154 and @xmath131 . ferromagnetism is therefore expected only in superlattices with very large rotations , and then only in the layers with large hole doping ( i.e. the layers closest to the sro planes ) . ( color online ) inverse susceptibility @xmath115 vs. temperature @xmath155 for bulk @xmath128 structure of lavo@xmath3 at densities @xmath151 ( black circle solid lines ) and @xmath153 ( red diamond dashed lines ) for rotation angle @xmath9 increasing from @xmath156 . on - site interaction @xmath114ev and @xmath58ev . left column : the circles and diamonds are data points , the solid and dashed lines are fitted from these data points . right column : expanded view at small @xmath115 region . the vertical dashed line marks zero temperature . ] from a range of calculations such as those shown in fig . [ fig : tc_evolve ] we have constructed the superlattice magnetic phase diagram shown in fig . [ fig : phase_diagram ] . similar to ref . , there are uncertainties in our extrapolation for curie temperature , we consider @xmath157ev as the error bar for positions on the phase diagram . thus , @xmath158ev is considered as @xmath159 within the error bar . we see that ferromagnetism is favored only for very large rotations , much larger than the @xmath160 determined experimentally , and only for carrier concentrations far removed from @xmath116 . we may locate the experimentally studied superlattices on this phase diagram . for an @xmath31 superlattice , band structure calculations indicate layer densities @xmath161 for layers near sro plane and @xmath152 for the other layers . the experimentally determined rotation angle is @xmath162 . these two points are indicated by squares in fig . [ fig : phase_diagram ] . ( color online ) the magnetic phase diagram with @xmath4-axis carrier density @xmath163 and @xmath113-axis tilt and rotation angle along @xmath141 $ ] direction @xmath9 for bulk system lvo with the same type of distortion as for ( lvo)@xmath38(svo)@xmath39 superlattices ( @xmath128 structure ) , in - plane tilt angles @xmath164 . on - site interactions @xmath114ev , @xmath58ev . the white regime indicates absence of ferromagnetism ( @xmath165 ) , the colored regime indicates ferromagnetism with @xmath87 indicated by the color bar . also indicated are results for bulk la@xmath5sr@xmath6vo@xmath3 in the @xmath1 structure , from ref . . note that in the calculations for the @xmath1 structure all three tilt angles are almost the same . ] it is interesting to compare our results to those previously obtained @xcite for the bulk solid solution la@xmath5sr@xmath6vo@xmath3 ( @xmath1 structure ) . the dashed line in fig . [ fig : phase_diagram ] shows the theoretically estimated phase diagram for the bulk solid solution . we see that the bulk structure is more favorable for ferromagnetism than the superlattice structure . an important difference between the @xmath1 structure and the @xmath128 of the superlattice is that in the former case all three tilt angles are of comparable magnitude whereas in the @xmath128 structure only one rotation is large . we believe that this difference is responsible for the difference in phase boundary . in this paper , we have studied the possibility of ferromagnetism in superlattice structures of vanadium oxides derived from lavo@xmath3 and srvo@xmath3 . our investigation was based on the idea that ferromagnetism depends on an interplay between carrier density and octahedral rotation , and while these are coupled in bulk ( see the solid solution curve in fig . [ fig : phase_diagram ] ) they may be decoupled in the superlattice . in particular , the charge density varies across the superlattice , being lowest near the sro planes , while the rotation angle is controlled by the substrate . thus in an appropriately designed superlattice at least some portions of the system might be moved closer to ( or perhaps into ) the ferromagnetic region . in several important aspects this idea is consistent with calculations . we find that the local carrier density determines the local magnetic susceptibility ( see section [ sec : cubicsuperlattice ] ) and the density / tilt angle relationship may be significantly altered ( see solid line and square points in fig . [ fig : phase_diagram ] ) . however , we find that the @xmath128 octahedral rotation pattern characteristic of experimentally discovered superlattices is in fact less favorable to ferromagnetism than the @xmath1 pattern characteristic of bulk materials ( compare the phase boundaries in fig . [ fig : phase_diagram ] ) . thus while the general idea that an appropriately designed superlattice might provide conditions favorable for ferromagnetism thereby providing a potential explanation for the remarkable experimental report of room - temperature ferromagnetism in ( lavo@xmath3)@xmath38(srvo@xmath3)@xmath39 superlattices with @xmath166 by lders et . al.,@xcite ( even though there is no ferromagnetism in the bulk solid solution ) , our detailed findings are not consistent with the experimental result . our results indicate that designing ferromagnetism into a vanadate superlattice will require both large amplitude rotations about the growth axis and also substantial rotations about the other two axes . rotations about the growth axis arise from substrate - induced strain , so choosing substrates with smaller lattice parameter would be desirable . introduction of rotations about the orthogonal axes may be done by replacing the la with a smaller counterion such as y. our study has certain limitations . the calculations employ a frontier orbital model which includes only the @xmath33-derived antibonding bands . dft+dmft calculations based on correlated atomic - like @xmath0-states embedded in the manifold of non - correlated oxygen states provide a more fundamental description . our previous work@xcite indicates that the two models give very similar results if both calculations are tuned so that bulk lavo@xmath3 is a mott insulator , but the implications of the full ( but computationally very heavy ) dft+dmft procedure for the superlattice problem remain an open problem for future research . further , our calculations are based on the single - site dmft approximation , which includes all local effects but misses inter - site correlations . while it is generally accepted that these calculations give the correct trends and qualitative behavior , the quantitative accuracy of the methods is not known . unfortunately , as yet cluster extensions of dmft are prohibitively expensive for the multiband models considered here . the experimental results of lders et . al.@xcite therefore provide an interesting challenge to materials theory . they indicate that superlattices display ferromagnetism when the corresponding bulk solid solutions do not , whereas the present state of the art of real materials dynamical mean field calculations suggests that superlattices should be less likely to display magnetism than the corresponding bulk solid solutions . this discrepancy requires further investigation . we thank u. lders and j. okamoto for helpful conversations . we acknowledge support from doe - er046169 . htd acknowledges partial support from vietnam education foundation ( vef ) . we acknowledge travel support from the columbia - sorbonne - science - po ecole polytechnique alliance program and thank ecole polytechnique ( htd and ajm ) and jlich forschungszentrum ( htd ) for hospitality while portions of this work were conducted . a portion of this research was conducted at the center for nanophase materials sciences , which is sponsored at oak ridge national laboratory by the scientific user facilities division , office of basic energy sciences , u.s . department of energy . we use the code for ct - hyb solver@xcite written by p. werner and e. gull , based on the alps library.@xcite cccccccc atom & @xmath4 & @xmath113 & @xmath29 & atom & @xmath4 & @xmath113 & @xmath29 + la@xmath167 & 0 & 0.25 & 0 & la@xmath168 & 0.5 & 0.25 & 0.5 + v@xmath167 & 0.5 & 0 & 0 & v@xmath168 & 0 & 0 & 0.5 + o@xmath169 & 0.4662 & 0.25 & 0.0660 & o@xmath170 & 0.0392 & 0.25 & 0.4392 + o@xmath171 & 0.7638 & -0.0138 & 0.2362 & o@xmath172 & 0.2652 & -0.0493 & 0.2652 + + & @xmath173()&@xmath174( ) & @xmath175( ) & @xmath176 & @xmath24 ( ) & @xmath25 ( ) & @xmath26 ratio + exp . lvo thin film@xcite & 5.55 & 7.82 & 5.55 & @xmath177 & 3.91 & 3.945 & 1.008 + exp . superlattice@xcite & na & na & na & na & 3.88 & 3.95 & 1.018 + calculated with @xmath178 & 5.5988 & 7.8290 & 5.5821 & @xmath179 & 3.915 & 3.988 & 1.019 + calculated with @xmath180 & 5.5512 & 7.7623 & 5.5346 & @xmath179 & 3.881 & 3.954 & 1.019 + in this appendix , we present a more complete discussion of the strain - induced lattice distortions . the in - plane lattice constant of a superlattice epitaxially grown on a substrate matches that of the substrate and may therefore be different from the lattice constant preferred in a free - standing film or bulk material . the out - of - plane lattice constant is typically free to relax , and in the presence of an in - plane strain may also be different from that found in bulk materials . a difference in v - v distance may arise from a change in v - o bond length or from a difference in buckling of v - o bonds . we consider both possibilities here , but first remark that the main differences in structure between bulk and experimentally studied superlattices arise from differences in octahedral rotation . in the experimentally - studied superlattices , the in - plane v - v distance is in fact slightly less than the v - v distance in lvo . the v - o bond lengths have not been measured for the superlattice , but to a high degree of accuracy we are able to reconstruct the measured superlattice using the measured tilt angles given from experiments@xcite structure , assuming that all v - o bond lengths are equal . assuming the @xmath128 structure , we varied the in - plane and out - of - plane v - o bond lengths to fit the experimental data and found that @xmath181 only when the mean bond length @xmath0 is found in the range from @xmath182 to @xmath109 depending on which experimental result is fit but in all cases the v - o bond lengths are found to be equal to within an accuracy of @xmath183 . therefore , we believe that all the v - o bond lengths should , to a good approximation , be the same . the structure used in our calculations is presented in table . [ table : p21m_struct ] . although there are slight mismatches in in - plane angle and lattice constants , the @xmath181 ratio and bond angles are compatible with the experiment . ( color online ) partial dos for bulk lvo with @xmath184 with the @xmath26 ratio due to a change in v - o bonds ( panel ( a ) ) and to @xmath128 lattice structure with @xmath185 ( similar to superlattice structure)(panel ( b ) ) . the dashed blue curve is the @xmath44 orbital , the solid red curve is the degenerate @xmath147 ( or @xmath148 ) orbital . the dashed vertical line marks the fermi level . ] changing the amount of rotation has a different effect on the electronic structure than does changing the ratio of v - o bond lengths . [ fig : bondlength_pdos ] compares the partial dos for the two cases , using as example a hypothetical lavo@xmath3 crystal with @xmath184 . the upper panel presents the dos for the untilted structure with straight v - o - v bonds and the @xmath26 ratio induced by a difference in in - plane and out - of - plane v - o bond lengths . the lower panel presents the case of all equal v - o bonds , with the @xmath26 ratio produced by octahedral rotations about the @xmath29 axis . the densities of states are quite different , but can be understood from the simple energy dispersion @xmath186 where @xmath187 and @xmath188 are the in - plane and out - of - plane nearest neighbor hopping integrals and @xmath189 are the second neighbor hoppings . the lower band edge is assumed to be the same for all orbitals but we assume that the lattice distortions lead to different values for the in - plane and out of plane hoppings . ( color online ) inverse susceptibility vs. temperature for cubic structure of bulk hole - doped lvo . the in - plane and out - of - plane bondlengths are changed so that the octahedral volume is unchanged : tensile strain ( @xmath190 - black lines ) , no strain ( @xmath191 - red lines ) and compressive strain ( @xmath184 - blue lines ) . two levels of hole doping are considered : @xmath151 ( solid lines ) and @xmath153 ( dashed lines ) . these lines are linear fits for the data points . ] the lower band edge is defined to be zero and is independent of the distortion . the energy of the upper edge of the @xmath44 band is @xmath192 and of the @xmath193 bands is @xmath194 . the positions of the van hove singularities are at @xmath195 or @xmath196 . for @xmath44 band there is only one van hove peak , at @xmath197 ; while for @xmath198 band , there are two van hove peaks at @xmath199 and @xmath200 . when @xmath188 is different from @xmath187 , the difference in bandwidth of @xmath44 and @xmath148 orbitals is @xmath201 , which is also the distance between the two van hove peaks of @xmath148 band @xmath202 . with these definitions , we are in a position to understand the changes in the band structure . when the v - o bond lengths change ( fig . [ fig : bondlength_pdos]a ) so that the @xmath29-bond is longer and the in - plane bond is shorter but the octahedral volume is unchanged , the band structure calculation indicates that @xmath188 decreases but @xmath187 increases slightly . the difference between the bandwidth of the @xmath44 and @xmath198 bandwidths is @xmath201 which is the same as the splitting between the van hove peaks in the @xmath193 bands . on the other hand , if the @xmath26 ratio is produced by rotation , ( fig . [ fig : bondlength_pdos]b ) , the change is opposite . the in - plane hopping @xmath187 decreases because of the buckled in - plane v - o - v bonds , while the out - of - plane hopping @xmath188 is unchanged . the @xmath44 band therefore narrows substantially relative to the @xmath193 bands . in addition the splitting of the van hove peaks is greater . from the bandwidth of @xmath44 and @xmath148 bands ( fig . [ fig : bondlength_pdos]b ) , @xmath203ev , @xmath204ev , the van hove peak distance is @xmath205ev , which is compatible with the peak positions shown in fig . [ fig : bondlength_pdos]b . we tested with dmft calculations for the curie temperatures with the v - o bondlength changed . [ fig : chi_bondlength ] is the temperature - dependent inverse susceptibility derived from dmft for the bulk cubic structure with the @xmath26 ratio changing from 0.98 ( tensile strain ) to 1.02 ( compressive strain ) . for all the levels of hole doping under consideration , the results are nearly the same for every case of @xmath26 ratio . we conclude that even when the v - o bondlength changes within the physical range , the ferromagnetism is not affected . however , we also found that when the v - o bondlength is such that @xmath206 or @xmath207 , there is large orbital polarization and the ferromagnetism can be largely affected . but that range is unphysical and can be neglected in the context of this work .
|
motivated by recent reports ( phys .
rev . b**80 * * , 241102 ) of room - temperature ferromagnetism in vanadium - oxide based superlattices , a single - site dynamical mean field study of the dependence of the paramagnetic - ferromagnetic phase boundary on superlattice geometry was performed .
an examination of variants of the experimentally determined crystal structure indicate that ferromagnetism is found only in a small and probably inaccessible region of the phase diagram .
design criteria for increasing the range over which ferromagnetism might exist are proposed .
| 13,239 | 154 |
of the three most fundamental parameters of a star mass , age and composition age is arguably the most difficult to obtain an accurate measure . direct measurements of mass ( e.g. , orbital motion , microlensing , asteroseismology ) and atmospheric composition ( e.g. , spectral analysis ) are possible for individual stars , but age determinations are generally limited to the coeval stellar systems for which stellar evolutionary effects can be exploited ( e.g. , pre - main sequence contraction , isochronal ages , post - main sequence turnoff ) . individual stars can be approximately age - dated using empirical trends in magnetic activity , element depletion , rotation or kinematics that are calibrated against cluster populations and/or numerical simulations ( e.g. , @xcite ) . however , such trends are fundamentally statistical in nature , and source - to - source scatter can be comparable in magnitude to mean values . age uncertainties are even more problematic for the lowest - mass stars ( m @xmath5 0.5 m@xmath2 ) , as post - main sequence evolution for these objects occurs at ages much greater than a hubble time , and activity and rotation trends present in solar - type stars begin to break down ( e.g. , @xcite ) . for the vast majority of intermediate - aged ( 110 gyr ) , very low - mass stars in the galactic disk , barring a few special cases ( e.g. , low - mass companions to cooling white dwarfs ; @xcite ) age determinations are difficult to obtain and highly uncertain . ages are of particular importance for even lower - mass brown dwarfs ( m @xmath5 0.075 m@xmath2 ) , objects which fail to sustain core hydrogen fusion and therefore cool and dim over time @xcite . the cooling rate of a brown dwarf is set by its age - dependent luminosity , while its initial reservoir of thermal energy is set by gravitational contraction and hence total mass . as such , there is an inherent degeneracy between the mass , age and observable properties of a given brown dwarf in the galactic field population ; one can not distinguish between a young , low - mass brown dwarf and an old , massive one from spectral type , luminosity or effective temperature alone . this degeneracy can be resolved for individual sources through measurement of a secondary parameter such as surface gravity , which may then be compared to predictions from brown dwarf evolutionary models ( e.g. , @xcite ) . however , surface gravity determinations are highly dependent on the accuracy of atmospheric models , which are known to have systematic problems at low temperatures due to incompleteness in molecular opacities ( e.g. , @xcite ) and dynamic atmospheric processes ( e.g. , @xcite ) . discrete metrics such as the presence of absence of absorption ( depleted in brown dwarfs more massive than 0.065 m@xmath2 at ages @xmath6200 myr @xcite ) , are generally more robust but do not provide a continuous measure of age for brown dwarfs in the galactic field population . binary systems containing brown dwarf components can be used to break this mass / age degeneracy without resorting to atmospheric models . specifically , systems for which masses can be determined via astrometric and/or spectroscopic orbit measurements , and component spectral types , effective temperatures and/or luminosities assessed , can be compared directly with evolutionary models to uniquely constrain the system age ( e.g. , @xcite ) . furthermore , by comparing the inferred ages and masses for each presumably coeval component , such systems can provide empirical tests of the evolutionary models themselves . a benchmark example is the young ( @xmath7300 myr ) binary and perhaps triple brown dwarf system gliese 569b @xcite . with both astrometric and spectroscopic orbit determinations , and resolved component spectroscopy , this system has been used to explicitly test evolutionary model tracks and lithium burning timescales @xcite as well as derive component ages , which are found to agree qualitatively with kinematic arguments ( e.g. , @xcite ) . other close binaries with astrometric or spectroscopic orbits have also been used for direct mass determinations ( e.g. , @xcite ) , but these systems generally lack resolved spectroscopy and therefore precise component characterization . they have also tended to be young , preventing stringent tests of the long - term evolution of cooling brown dwarfs . older , nearby very low - mass binaries with resolved spectra ( e.g. , @xcite ) generally have prohibitively long orbital periods for mass determinations . recently , we identified a very low - mass binary system for which a spectroscopic orbit and component spectral types could be determined : the late - type source 2mass j03202839@xmath00446358 ( hereafter 2mass j0320@xmath00446 ; @xcite ) . our independent discoveries of this system were made via two complementary techniques . * hereafter bl08 ) identified this source as a single - lined radial velocity variable , with a period of 0.67 yr and separation @xmath70.4 au , following roughly 3 years of high - resolution , near - infrared spectroscopic monitoring ( see @xcite ) . * hereafter bu08 ) demonstrated that the near - infrared spectrum of this source could be reproduced as an m8.5 plus t5@xmath11 unresolved pair , based on the spectral template matching technique outlined in @xcite . the methods used by these studies have yielded both mass and spectral type constraints for the components of 2mass j0320@xmath00446 , and thus a rare opportunity to robustly constrain the age of a relatively old low - mass star and brown dwarf system in the galactic disk . in this article , we determine a lower limit for the age of 2mass j0320@xmath00446 by combining the radial velocity measurements of bl08 and component spectral type determinations of bu08 with current evolutionary models . our method is described in @xmath8 2 , which includes discussion of sources of empirical uncertainty and systematic variations from four sets of evolutionary models . we obtain lower limits on the age , component masses , and orbital inclination of the system , and compare our age constraint to expectations based on kinematics , magnetic activity and rotation of the primary component . in @xmath8 3 we discuss our results , focusing in particular on how future observations could provide bounded limits on the age and component masses of this system , and thereby facilitate tests of the evolutionary models themselves at late ages . evolutionary models predict the luminosities and effective temperatures of cooling brown dwarfs over time , parameters that have been shown to correlate well with spectral type ( e.g. , @xcite ) . luminosity is the more reliable parameter , being based on the measured distance and broad - band spectral flux of a source , as opposed to model - dependent determinations of photospheric gas temperature and/or radius . however , in the case of 2mass j0320@xmath00446 , neither distance nor component fluxes have been measured , the latter due to the fact that this system is as yet unresolved ( and for the near future , unresolvable ; see @xmath8 3 ) . we therefore used the component spectral types of this system and luminosity measurements for similarly - classified , single ( unresolved ) sources from @xcite and @xcite to estimate the component luminosities . for the m8.5 primary , there are 13 m8m9 field dwarfs with bolometric luminosities ( parallax distance and broad - band spectral flux measurements ) reported in the studies listed above . two of the sources the m8 lhs 2397a , a known binary @xcite ; and the m9 lp 944 - 20 , believed to be a younger system ( @xmath7500 myr , @xcite)are unusual sources and therefore excluded from this analysis . the mean bolometric magnitude of the remaining stars is @xmath9 = 13.36@xmath10.29 mag , corresponding to @xmath10 = -3.45@xmath10.12 . for the t5@xmath11 secondary , there are fewer field brown dwarfs with reliable luminosity measurements ( 1 t4.5 dwarf and 5 t6 dwarfs ) and these show considerably greater scatter in their bolometric magnitudes : @xmath9 = 17.2@xmath10.6 . this scatter may be due in part to unresolved multiplicity , which appears to be enhanced amongst the earliest - type t dwarfs @xcite . hence , we estimated the luminosity of 2mass j0320@xmath00446b using the @xmath11/spectral type relation of @xcite } = [ 1.37376e1 , 1.90250e-1 , 1.73083e-2 , 7.40013e-3 , -1.75144e-3 , 1.14234e-4 , -2.32248e-06 ] , where @xmath11 = @xmath12spt@xmath13 and spt(t0 ) = 10 , spt(t5 ) = 15 , etc . ] . a mean @xmath9 = 17.09@xmath10.29 ( @xmath10 = -4.94@xmath10.17 ) was adopted , where we have taken into account the uncertainty in the secondary spectral type and the @xmath11/spectral type relation ( 0.22 mag ) . this value agrees well with estimates from ( * ? ? ? * @xmath9 = 16.9@xmath10.4 ) and ( * ? ? ? * @xmath9 = 17.3@xmath10.6 ) . in order to assess systematic uncertainties in the derived age and component properties , we considered four different sets of evolutionary models in our analysis : the cloudless models of ( * ? ? ? * ; * ? ? ? * hereafter tucson models ) , the `` cond '' cloudless models of ( * ? ? ? * hereafter cond models ) , and the cloudless and cloudy models from ( * ? ? ? * hereafter sm08 models ) . all four sets of models assume solar metallicity , which is appropriate given that composite red optical and near - infrared spectra of 2mass j0320@xmath00446 show no indications of subsolar metallicity ( * ? ? ? the choice of `` cloudless '' evolutionary models ( referring to the absence of condensate clouds in atmospheric opacities ) is driven partly by their availability . in addition , the spectral energy distributions of the m8.5 and t5 components of 2mass j0320@xmath00446 are minimally affected by condensate cloud opacity ( e.g. , @xcite ) . however , cloud opacity in the intermediate l dwarf stage may slow radiative cooling during this phase and bias the inferred age of the t - type secondary ( sm08 ) , although @xcite have claimed that clouds have only a `` small effect '' on evolution . to test this possibility , we chose to examine both the cloudless and cloudy sm08 models , the latter of which takes into account photospheric cloud opacity in thermal evolution through the use of atmospheric models generated according to the prescriptions outlined in @xcite and sm08 . figure [ fig_models ] compares the luminosity estimates for the two components of 2mass j0320@xmath00446 to the evolutionary tracks of each model set . the luminosities ( and their uncertainties ) constrain the mass / age parameter space of each component , as illustrated in figure [ fig_mvst ] . component masses generally increase with system age , as more massive low mass stars and brown dwarfs take longer to radiate their greater reservoir of heat energy from initial contraction . the mass of the primary of this system reaches an asymptotic value of @xmath70.080.09 m@xmath2 for ages @xmath61 gyr , consistent with a hydrogen - fusing very low - mass star . if the system is younger than @xmath7400 myr , the primary could be substellar . note that ages @xmath5300 myr ( primary masses @xmath140.065 m@xmath2 ) can be ruled out based on the absence of absorption at 6708 in the unresolved red optical spectrum of this source @xcite . the mass of the secondary increases across the full age range shown in figure [ fig_mvst ] as this component is substellar up to 10 gyr . there is some divergence in the evolutionary tracks at late ages for this component , however ; the tucson models predict a mass near the hydrogen - burning limit , while the sm08 cloudless and cloudy models predict masses above and below the li - burning minimum mass , respectively . the kink in the mass / age relation of 2mass j0320@xmath00446b at ages of 200 - 300 myr , particularly in the cond and sm08 models , reflects the prolonged burning of deuterium in brown dwarfs with masses just above 0.013 m@xmath2 , producing higher luminosities at this temporary stage of evolution . the mass ratio of the system , @xmath15 m@xmath16/m@xmath17 , also increases as a function of age , ranging from @xmath70.2 at 100 myr to a maximum of @xmath70.8 at 10 gyr . with constraints in the mass / age phase space provided by the component luminosities and evolutionary models , we can now use the radial velocity orbit to break the mass / age degeneracy . the radial velocity variations measured by bl08 only probe the recoil velocity of the primary of the 2mass j0320@xmath00446 system . these observations provide a coupled constraint between the masses and inclination of the system : @xmath18 ( bl08 ) where @xmath19 is the inclination angle of the orbit , and m@xmath17 and m@xmath16 are the masses of the primary and secondary components in solar mass units , respectively . we can make a geometric constraint that @xmath20 , which yields a transcendental equation for the lower limit of the secondary component mass of the system as a function of the primary component mass . using our age - dependent lower bound for the latter based on the evolutionary models ( including luminosity uncertainties ) the constraint on @xmath21 from eqn . 1 translates into a minimum secondary mass as a function of age , as shown in figure [ fig_mvst ] . finally , the age at which the upper bound of the secondary component mass ( based on the evolutionary models ) crosses the radial velocity minimum mass line corresponds to the minimum age of the system . all four models predict a minimum age for 2mass j0320@xmath00446 in the range 1.72.2 gyr ( table [ tab_modelfit ] ) . this age is in qualitative and quantitative agreement with those inferred by bl08 from the kinematics of the 2mass j0320@xmath00446 system space motions of this system are @xmath22 = -38@xmath15 km s@xmath23 , @xmath24 = -20@xmath13 km s@xmath23 and @xmath25 = -32@xmath14 km s@xmath23 , where we assume an lsr solar motion of @xmath26 = 10 km s@xmath23 , @xmath27 = 5.25 km s@xmath23 and @xmath28 = 7.17 km s@xmath23 @xcite . @xcite , equation 8 , predicts an age @xmath291.6 gyr at the 95% confidence level for these kinematics . ] and stellar age / activity trends . in the case of the latter , the optical spectrum of 2mass j0320@xmath00446 shows no detectable h@xmath30 emission @xcite , even though @xmath2990% of nearby m8m9 dwarfs exhibit such emission @xcite . for comparison , @xcite estimate an `` activity lifetime '' ( i.e. , timescale for h@xmath30 emission to drop below detectable levels ) of 8@xmath31 gyr for m7 dwarfs . this age may be too high of an estimate for 2mass j0320@xmath00446 , as the increase in activity lifetimes for spectral types m2m7 observed by @xcite may not continue for later spectral types . magnetic field lines are increasingly decoupled from lower - temperature photospheres , and the frequency and strength of h@xmath30 emission decrease rapidly beyond type m7/m8 ( e.g. , @xcite ) . hence , the absence of magnetic emission from 2mass j0320@xmath00446a is merely indicative of an older age , as is its kinematics . rotation is a third commonly - used empirical age diagnostic for stars , based on the secular angular momentum loss observed amongst solar - type stars through the emission magnetized stellar winds ( e.g. , @xcite ) . however , so - called gyrochronological relations calibrated for these stars are not necessarily applicable to lower - mass objects , due both to the fully convective interiors of the latter and the decoupling of field lines from low - temperature atmospheres ( e.g. , @xcite ) . indeed , 2mass j0320@xmath00446a proves to be a relatively rapid rotator , with an equatorial spin velocity of @xmath32 = 16.5@xmath10.5 km s@xmath23 and a rotation period @xmath14 7.4 hr ( bl08 ) . for solar - type stars , this rapid rotation generally indicates a young age ; an extrapolation ( in color ) of gyrochronology relations by ( * ? ? ? * equation 3 , assuming @xmath33 @xmath34 2.1 for 2mass j0320@xmath00446a ; e.g. , @xcite ) yields an age of only @xmath70.1 myr . this is inconsistent with the absence of absorption , space kinematics and lack of magnetic emission from this source . clearly , rotation does not provide a useful age metric for the 2mass j0320@xmath00446 system , further emphasizing the breakdown of age / angular momentum trends in the lowest - mass stars . the derived minimum age and radial velocity constraints of the 2mass j0320@xmath00446 system allow us to constrain model - dependent minimum masses for its components as well ; m@xmath17 @xmath35 0.0800.082 m@xmath2 and m@xmath16 @xmath35 0.0530.054 m@xmath2 . the ranges in these values reflect variations between the four evolutionary models ( table [ tab_modelfit ] ) . note that there is essentially no difference in the minimum masses inferred from the cloudy and cloudless sm08 models . the minimum mass ratio of the system , @xmath36 0.600.62 , is also consistent across the evolutionary models , and in accord with the general preference of large mass ratio systems observed amongst very low - mass binaries ( @xmath2990% of known binaries with @xmath37 m@xmath2 have @xmath38 ; see @xcite ) . finally , while the minimum ages and masses of 2mass j0320@xmath00446 are inferred assuming @xmath20 , it is possible to constrain the maximum masses and minimum orbital inclination of this system assuming an upper age limit . adopting @xmath39 10 gyr , we infer @xmath40 0.800.86 , corresponding to @xmath41 53@xmath4259@xmath42 . this constraint is only slightly more restrictive than the @xmath43 lower limit determined by bl08 assuming that the fainter secondary must have a lower mass than the primary . it does not significantly improve the chances that this system is an eclipsing edge - on system ( @xmath70.3% by geometry ) . maximum primary ( 0.088 m@xmath2 ) and secondary masses ( 0.0660.075 m@xmath2 ) are effectively set by the luminosity constraints and evolutionary models . here we see the most significant difference between the models , a 13% discrepancy in the maximum mass of the secondary component , likely due to different treatments of light element fusion near the li- and h - burning minimum masses . this variation confirms the importance of older binary systems as tests of long - term brown dwarf evolution , particularly near fusion mass limits . we also note that there is little difference ( @xmath74% ) in the maximum masses inferred from the sm08 cloudy and cloudless models , illustrating again the negligible role of cloud opacity in the long - term evolution of brown dwarfs like 2mass j0320@xmath00446b . the combination of component luminosities , radial velocity orbit of the primary and evolutionary models have allowed us to estimate the minimum age of the 2mass j0320@xmath00446 system and its component masses . the ages are consistent between four evolutionary models of brown dwarfs , and are more precise ( although not necessarily more accurate ) than estimates based on as - yet poorly - constrained statistical trends in kinematics , magnetic activity and angular momentum evolution . however , a bounded age estimate remains elusive due to the unknown inclination of the system and determination of component masses . as discussed in bl08 , the inclination of the 2mass j0320@xmath00446 orbit is irrelevant if the radial velocity orbit of the secondary can be determined . in that case , one need only compare the derived system mass ratio and component luminosities to evolutionary models to obtain a bounded age estimate . measurement of the secondary motion in the @xmath44-band data of bl08 was not possible due to the very large flux contrast between the components ; bl08 rule out a contrast ratio of @xmath510:1 at these wavelengths ; spectral template fits from bu08 predict a contrast ratio of @xmath6350:1 . a more effective approach would be the acquisition of radial velocity measurements in the 1.2 - 1.3 @xmath4 band where the t dwarf secondary is considerably brighter and the contrast ratio is closer to 20:1 ( depending on the absolute magnitude scale ; see discussion in bu08 ) . at these contrasts , the radial velocity of the secondary can be measured using existing techniques for high - contrast spectroscopic binary systems ( e.g. , @xcite ) . alternately , if this system is observed to eclipse , then @xmath45 and the age of the system is uniquely determined . table [ tab_modelfit ] lists the ages corresponding to this scenario , ranging from 2.5@xmath46 gyr to 3.2@xmath47 gyr for the four models examined ( uncertainties include the full range of possible primary and secondary masses for which eqn . 1 and the luminosity constraints are satisfied ) . the relatively small age uncertainties estimated in this scenario ( 2560% ) are dominated by uncertainties in the component luminosities , which could be measured from primary and secondary eclipse depths over a broad range of optical and infrared wavelengths ( e.g. , @xcite ) . such measurements are currently more feasible than resolved imaging measurements , as the tight separation inferred from the radial velocity orbit , @xmath48 = @xmath49 0.4 au ( bl08 , assuming @xmath50 ) , implies a projected separation of @xmath517 mas , below the diffraction limit of the keck 10 m telescope at near - infrared wavelengths . yet the scientifically valuable measurements possible in an eclipsing scenario must be tempered by this scenario s low probability . for a maximum age of 10 gyr , we can only constrain the inclination of the 2mass j0320@xmath00446 system to @xmath3 , and hence an eclipse probability of @xmath70.3% . regardless of whether 2mass j0320@xmath00446 is an eclipsing pair , determination of its orbit inclination and/or component masses is a necessary step for testing brown dwarf evolutionary models at late ages , specifically through agreement of system parameters with model isochrones ( c.f .. , @xcite ) . the power of such a test is the long lever arm of time provided by older field binaries such as 2mass j0320@xmath00446 , resulting in large differences in luminosities and effective temperatures for the more common near equal - mass systems ( e.g. , @xcite ) . there may in fact be many such systems to exploit in this manner . simulations by bu08 of brown dwarf pairs in the vicinity of the sun predict that 12 - 25% of all m8l5 dwarf binaries have component spectral types that can be inferred from unresolved , near - infrared spectroscopy using the method outlined in @xcite . perhaps as many as 50% of these systems may be short - period radial velocity variables @xcite . identification and follow - up of these systems would complement the evolutionary model tests currently provided by younger systems @xcite and would more robustly address uncertainties associated with low - temperature light - element fusion , interior thermal transport , and substellar interior structure as their effects are compounded over time . the authors thank i. baraffe , a. burrows , m. marley and d. saumon for making electronic versions of their evolutionary models available ; and m. liu for identifying the roundoff errors in the @xmath11/spectral type relation in @xcite . we also thank our referee , k. luhman , for his helpful critique of the original manuscript . cb acknowledges support from the harvard origins of life initiative . this publication has made use of the vlm binaries archive maintained by nick siegler at http://www.vlmbinaries.org . burgasser , a. j. , reid , i. n. , siegler , n. , close , l. m. , allen , p. , lowrance , p. j. , & gizis , j. e. 2007 , in planets and protostars v , eds . b. reipurth , d. jewitt and k. keil ( univ . arizona press : tucson ) , p. 427 wilson , j. c. , miller , n. a. , gizis , j. e. , skrutskie , m. f. , houck , j. r. , kirkpatrick , j. d. , burgasser , a. j. , & monet , d. g. 2003 , in brown dwarfs ( iau symp . 211 ) , ed . e. martn ( san francisco : asp ) , p. 197 lcccc minimum age ( gyr ) & 1.7 & 2.2 & 2.2 & 2.0 + minimum @xmath51 & 0.80 ( 53@xmath42 ) & 0.82 ( 55@xmath42 ) & 0.83 ( 56@xmath42 ) & 0.86 ( 59@xmath42 ) + m@xmath17 ( m@xmath2 ) & 0.0820.088 & 0.0800.088 & 0.0820.088 & 0.0820.088 + m@xmath16 ( m@xmath2 ) & 0.0540.075 & 0.0530.070 & 0.0540.069 & 0.0540.066 + m@xmath16/m@xmath17 & 0.620.89 & 0.600.87 & 0.610.84 & 0.610.80 + age for @xmath52 ( gyr ) & 2.5@xmath46 & 3.2@xmath47 & 3.0@xmath53 & 2.8@xmath54 +
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2mass j03202839@xmath00446358ab is a recently identified , late - type m dwarf / t dwarf spectroscopic binary system for which both the radial velocity orbit for the primary and spectral types for both components have been determined . by combining these measurements with predictions from four different sets of evolutionary models ,
we determine a minimum age of 2.0@xmath10.3 gyr for this system , corresponding to minimum primary and secondary masses of 0.080 m@xmath2 and 0.053 m@xmath2 , respectively .
we find broad agreement in the inferred age and mass constraints between the evolutionary models , including those that incorporate atmospheric condensate grain opacity ; however , we are not able to independently assess their accuracy
. the inferred minimum age agrees with the kinematics and absence of magnetic activity in this system , but not the rapid rotation of its primary , further evidence of a breakdown in angular momentum evolution trends amongst the lowest luminosity stars . assuming a maximum age of 10 gyr
, we constrain the orbital inclination of this system to @xmath3 . more precise constraints on the orbital inclination and/or component masses of 2mass 0320@xmath00446ab , through either measurement of the secondary radial velocity orbit ( optimally in the 1.21.3 @xmath4 band ) or detection of an eclipse ( only 0.3% probability based on geometric constraints ) , would yield a bounded age estimate for this system , and the opportunity to use it as an empirical test for brown dwarf evolutionary models at late ages .
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modern astronomy requires state - of - the - art technology for the efficient detection of the faintest light from the farthest reaches of the universe . it is not uncommon for the technologies developed by astronomers to find uses in everyday life . rosenberg et al,@xcite have compiled numerous examples of such technology transfer including , but not limited to : ccds popularized by the _ hubble space telescope _ and now used in practically every digital camera ; wireless local area networking utilizing algorithms from image processing in radio astronomy ; computerized tomography in modern medical scanners based on aperture synthesis techniques from radio interferometry ; and gamma ray spectrometers for lunar / planetary surface composition analysis now used to probe historical buildings and artefacts . ongoing successes in sub - millimeter astronomy ( e.g. the _ _ herschel__@xcite and _ _ planck__@xcite space telescopes ) and the ever present demand for instruments with improved sensitivities and mapping speeds at terahertz ( thz ) frequencies have spurred the development of highly sensitive detectors , sophisticated optical components , cutting edge electronics , and advanced data processing techniques . kinetic inductance detectors ( kids ) are contemporary superconducting pair - breaking detectors that operate across the spectrum from x - ray to sub - thz frequencies@xcite . compared to alternative thz technologies such as semiconductor or transition edge sensor ( tes ) bolometers , kids are relatively simple to fabricate and read out . as such , they provide a practical and cost - effective solution to the manufacture and operation of the large format arrays required for advances in many fields of thz astronomy . a variant known as the lumped element kid , or lekid@xcite , has been demonstrated to provide state - of - the - art performance at millimeter - wavelengths@xcite and has seen first light as part of the nika@xcite instrument at the iram 30-m telescope . projects such as the next generation blast experiment@xcite , nika-2@xcite and a - mkid are currently under way to incorporate multi - kilopixel kid arrays into astronomical cameras with the potential for thz megapixel imaging within the next decade . beyond astronomy , the thz region of the electromagnetic spectrum ( 0.1 - 10thz ) has applications in a range of fields academic and industrial@xcite . in addition to the presence of a multitude of interesting spectral features , many typically opaque materials become transparent when viewed in this frequency range . various disciplines including biomedical sensing , non - destructive testing , and security screening now have the opportunity to benefit from the highly sensitive and highly multiplexable detector technology being developed by astronomers . for example , thz radiation is being used to study protein dynamics@xcite , to investigate thz induced dna damage@xcite , and as a potential imaging modality for the improved delineation of certain types of skin cancers@xcite . however , there are currently no off - the - shelf thz imaging spectrometers or cameras available to help proceed more rapidly with these investigations . the analysis and restoration of cultural artefacts benefits from the unique differential penetration of thz radiation , making it ideal for the non - destructive investigation of the internal paint layers in pieces of art@xcite . time domain techniques have been used to show that unique information can be gleaned at thz frequencies to verify the age , chemical composition and structure of works of art . however , such time - domain techniques are not necessarily time efficient . far larger potential demand is associated with the detection of hidden objects ( such as land mines@xcite ) , process control in manufacturing@xcite , and security screening@xcite . active mm - wave scanners are now widely deployed in airports across the globe and large format kid arrays could be used to produce systems with improved sensitivity at a comparable cost . the capacity for truly passive imaging , and the fast time - response of lekid detectors ( typically @xmath2sec ) enables , for the first time , the possibility of capturing images at video rate for so called `` walk through '' systems . this is regarded as desirable by ecac ( the european civil aviation conference ) and opens up the possibility of use in situations where requiring people to stand one - by - one in a booth is not practical@xcite . furthermore , multi - spectral observations would improve image contrast and reduce the number of false positives a common occurrence with current active systems . to demonstrate these capabilities , we have built and characterized a simple field scanning camera based on a 152 element linear array of lekids operating at 350ghz . the detectors have been optimized to perform under the optical loads present at ambient room temperatures ( @xmath1300k ) , which are substantially higher than the backgrounds present during astronomical observations . the instrument , in its present configuration , is comparable in performance to other recent passive imaging systems including those based on room temperature microbolometer fpas@xcite , cooled bolometer arrays@xcite , and superconducting tes arrays@xcite . in this article we describe the camera and its achieved performance as a quasi - video - rate system . we conclude with discussions of the improvements which will be implemented for the next generation camera in order to achieve a full video - rate , photon - noise limited imaging system . our goal was to demonstrate a video rate scanner capable of imaging variations in the thermal thz radiation received from a moving target ( a person ) with sufficient sensitivity to detect and identify concealed objects - akin to airport - style security scanners or other stand - off scanning instruments . the basic requirements were for a simple - to - use system with the necessary spatial resolution , scanning speed and sensitivity to identify objects of a few cm in size concealed behind clothing or inside bags or other luggage . the camera is designed to provide a @xmath3 m useful field of view ( typical of body scanners ) with operation at a distance of 3 - 5 m from the target and a linear resolution of roughly 1 cm . the camera observes in the 350ghz transparent atmospheric window with a @xmath4% wide band to minimize the loading and thermal fluctuations present at less transparent frequencies . as a demonstration system , quasi - video rate imaging with frames updating at least every second was deemed acceptable although the goal would be to reach full 25hz video rate . a 0.1k noise equivalent temperature difference , ne@xmath5 t , in each frame is required as this is necessary to identify the shape of concealed objects@xcite . finally , the superconducting detectors need to be operated at sub - kelvin temperatures , so the camera requires a completely dry cryogenic system which , unlike wet systems , can be easily deployed in the field . ideally , the noise performance of the system would be limited by variance in the arrival of photons from the source rather than from any components of the camera itself . the noise equivalent power due to photons measured at the detector focal plane in a diffraction limited optics system is given by@xcite @xmath6 where @xmath7 is planck s constant , @xmath8 is the frequency , @xmath9 is the optical bandwidth , and @xmath10 for a detector absorbing light in both polarizations . @xmath11 is the power in a band of width @xmath9 : @xmath12 where @xmath13 is the camera tendue , @xmath14 is the emissivity and @xmath15 is the blackbody radiance of a source at temperature @xmath16 , while @xmath17 and @xmath18 are respectively the optics and detector efficiencies . note that both shot and wave noise are accounted for in equation [ nep ] . the camera employs a linear array of detectors housed in a research cryostat retrofitted with a large ( 250 mm diameter ) window in the base . incoming radiation is coupled to the detectors through a refractive optics system and a flat beam - folding mirror . a thin horizontal section of the object plane is observed in any one instant and this section is scanned continuously in the vertical direction by oscillation of the beam - folding mirror . this is illustrated in figure [ fig : kidcam ] . the orientation of the mirror is recorded with an absolute encoder and images are reconstructed in real time by the acquisition electronics in a scheme no dissimilar to that of a common office desktop scanner . a photo of the current system ( left ) and a schematic ( right ) of the system overlaid with the zemax model and ray traces . the camera sees a small horizontal strip of the object plane in one instance with the oscillating fold mirror providing full sampling in the vertical direction . ] a fast ( @xmath19 ) triplet of high - density polyethylene ( hdpe ) lenses was designed to keep the optics simple and compact given limitations on where the focal plane array could be situated within the cryostat . to achieve the desired resolution at these frequencies , a large diameter ( 440 mm ) primary lens , l1 , was chosen . the focal distance of the camera is designed to be adjustable between 3 and 5 meters depending on the position of the secondary lens , l2 . at a distance of 3.5 m , the scannable field of view is 0.8@xmath201.6 m and the working depth of field is approximately @xmath21150 mm inside and outside of the focus . the third lens , l3 , visible in the cad model in figure [ fig : cryo - composite]a , is housed within the cryostat behind the hdpe window and a number of thermal blocking filters@xcite ( not shown in the figure ) . the lens and window absorptivities were measured in band and are non - negligible , with combined losses of up to 45% expected through the optics chain . furthermore , the hdpe components are not anti - reflection coated and are uncooled ( except l3 ) . consequently , stray light from these sources contributes significantly to detector loading . the oscillating beam - folding mirror is constructed from a thin , polished sheet of aluminum ( 800 mm long by 550 mm wide ) braced with strut profile and mounted to the camera s main frame via a set of bearings on the central horizontal axis . the oscillation is brought about by a crank wheel driven by a servo motor located behind the mirror . a small steel rod with bearings at each end connects the mirror to the wheel . the oscillation rate is controlled by a motor driver that is configured via usb from the control station . this mechanism can modulate the field of view at a maximum frequency of 2 - 3 frames per second , this ultimately limits the video rate output . cross sections of the cryostat and focal plane assembly . ( a ) the model shows the focal plane array ( green ) mounted via various thermally isolating support structures to the 4k base plate and connected to the helium-10 fridge ( yellow ) . optical paths from the focal plane , through the quasi - optical filters , baffles ( not shown ) , cold lens and the cryostat window are indicated by the transparent cones . ( b ) the schematic shows the coupling mechanism for a single array element . the aluminum lekid device is back - illuminated through the silicon substrate via a back - to - back conical feedhorn and the final band - defining filter . any radiation that is not absorbed by the detector is caught by a layer of silicon carbide infused epoxy at the back of the array packaging . ] a series of quasi - optical metal - mesh filters@xcite define the optical bandwidth of the system . currently , three low - pass edges with cut - offs at 630ghz , 540ghz , and 450ghz and two 10% wide band - pass filters define a combined 6% wide band centered at 347ghz . the additional bandpass filter was added as a precaution against detector saturation with the effect of reducing the overall bandwidth and the camera optical efficiency . the filter profiles were measured by a fourier transform spectrometer ( fts ) from 200ghz to 1thz with 1ghz resolution and are displayed in figure [ fig : filters ] . inset to the figure is a plot of the total transmittance of the filter stack . the peak in - band transmission is 45% and the out - of - band rejection at high frequency is better than 100db . the spectral transmittance of the band - defining quasi - optical filters . the product of these profiles is indicated by the black line and is re - plotted on a log scale in the inset to show the high out of band rejection . thermal filters are excluded for clarity as these have @xmath1100% transmission in band . ] the large cryostat window and the requirement for fast optics make the focal plane susceptible to off axis radiation . to lessen the effects , sic blackened metal baffles are arranged at the entrances to the three radiation shields and a copper horn - plate is mounted to the detector array at the focal plane , see figure [ fig : cryo - composite]b . the feedhorns are each approximately @xmath22 and although this helps prevent stray light reaching the detectors , there is a slight mismatch with the @xmath19 refracting optics . the cylindrical waveguides connecting the back - to - back horns admit at most two transverse electromagnetic modes , the te@xmath23 and tm@xmath24 modes . the number of detectors in the array needed to achieve the required performance is estimated . each lens in the system is characterized by its emissivity and transmission properties . l1 and l2 and the window operate at 300k , while l3 is estimated to be at 150k . having measured the hdpe transmission , we estimate the overall lens transmission using zemax . the overall instrument efficiency , including the filters , is 23% , corresponding to an expected optical load of 131pw per detector at the focal plane assembly . the photon noise ( including both the shot noise and wave noise components ) at the focal plane is then calculated from equation [ nep ] to be 3.0mk@xmath25 . this allows us to estimate that in order to achieve an image sensitivity of @xmath10.1k per frame at a 25hz frame rate and a 1 cm resolution , 150 detectors are required . note that to first order this is independent from the detector and optical efficiencies , as it is wave noise that currently dominates the noise budget . the detector array in use for this demonstration system is composed of 152 lekids arranged in 8 rows of 19 columns . the columns are skewed such that the instantaneous field of view is nyquist sampled in the horizontal direction ( see figure [ fig : array ] ) . the array and packaging . the detectors are arranged to nyquist sample a horizontal section of the object plane . for readout , each detector modulates a small range of the total bandwidth of the probe signal that propagates along the feedline . the feedline can be seen winding between the rows of detectors and is terminated to sma type connectors at each end . ] in general , a kinetic inductance detector ( kid ) is fabricated by patterning a thin film of superconducting material in such a way as to create an @xmath26 resonant circuit with frequency @xmath27 . the inductance of the superconductor , @xmath28 , has two key components , @xmath29 . these depend , respectively , on the shape of the deposited film and the density of cooper pairs in the film . photons that couple into the resonator with sufficient energy to overcome the superconducting gap will break cooper pairs into unbound pairs of quasiparticle excitations , leading to a decrease in @xmath30 . then , any variations in incident optical power are monitored by measuring the variations in @xmath30 . this is achieved by monitoring the complex transmission of a probe signal that is fed through a microwave transmission line adjacent to the resonator . multiple resonators , each with a different @xmath30 , may be coupled to the same transmission line and read out simultaneously with a superposition of probe signals . this inherent multiplexing capability considerably reduces the requirement for complex cryogenic circuitry . lumped - element kids as opposed to distributed kids are designed such that the absorbing element of the detector is part of the resonator structure itself . in this configuration it is possible to achieve very high filling factors in focal plane arrays without the need for additional coupling optics such as microlens or feedhorn arrays . note that the feedhorns used in this system are for stray light reduction and would not be necessary in a well baffled optical system . each lumped resonator in the current focal plane array has three sections : an inductor , an interdigital capacitor and a coupling capacitor . these are highlighted by the different colored sections in the design and the equivalent circuit in figure [ fig : detector]a . the inductor section is a 4th order hilbert curve which efficiently couples to both orthogonal polarizations of incoming radiation@xcite . variations in the length of the interdigital capacitor sections have been designed to set a range of resonant frequencies centered at 1.5ghz and each separated by 3mhz . the detectors are capacitively coupled to a coplanar waveguide ( cpw ) feedline , with the length of the coupling capacitor section and its distance from the feedline limiting the q - factor of the resonators to be of the order of 10,000 . the array is fabricated from a 40 nm aluminum film thermally deposited onto a 500@xmath0 m high resistivity float - zone silicon wafer . the array design was patterned into the aluminum in a single photolithographic cycle with a wet etch of orthophosphoric acid , nitric acid , and water in a 25:2:6 ratio . the cpw line is cross - bonded with wire bridges at regular intervals to ensure a constant potential across the ground plane , thus inhibiting problematic slotline modes in the cpw line . figure [ fig : cryo - composite]b shows a cross section of a single detecting element in the focal plane assembly . optical coupling is optimised by back - illumination of the detectors through the silicon substrate . thin film aluminum has a superconducting transition temperature of @xmath31k and kid arrays require cooling to at least @xmath32 in order to sufficiently reduce the density of quasiparticles in the superconducting film . the current system utilizes a cryomech pt400 series pulse - tube - cooler ( ptc ) and air - cooled compressor unit that operate off mains electricity only so that no liquid cryogens are required . a closed - cycle helium-10 adsorption fridge from chase cryogenics@xcite cools the focal plane assembly to the required sub - kelvin temperatures . thermometry and fridge - cycling are fully automated and may be monitored / controlled remotely . cool - down of the current system from room temperature takes around 36 hours with the ptc cold plate settling at 3.2k . the optical baffles on the radiation shields settle at 4.2k and 60k respectively , and the cold lens settles with a radial temperature gradient ranging between 100 - 150k . in the present configuration , fridge cycles continue for approximately 16 - 18 hours at 250mk and require up to 4 hours for recycling . the electronic readout system consists of cryogenic , warm and digital components ( see figure [ fig : readout ] ) , as well as a suite of software to control the camera components , to monitor the housekeeping system and to generate and display images in real - time . the nature of multi - channel kid read out is such that the complexity of the cryogenic electronics is reduced to an absolute minimum . aside from the detector array itself , a single attenuator , a single low noise amplifier , and a single set of coaxial cables are the only components required within the cold stages . an overview of the readout system . tones are output by the nikel digital electronics system , mixed up to the required kid resonant frequency range , and passed into the cryostat through a single coaxial cable . the probe signal is attenuated before reaching the detectors and then boosted by a si - ge cryogenic low noise amplifier at the 4kelvin stage . outside of the cryostat the signal is mixed back down to the daq band and read back in to the nikel electronics for spectral decomposition . the i and q components of up to 400 tones are sent over network to the control computer for processing and image generation . ] the cpw transmission is wire - bonded to sma connectors mounted to the copper array packaging . stainless steel semi - rigid coaxial cable then feeds out to the 4k stage . a cold attenuator on the input channel reduces the power ( and therefore the thermal noise ) of the multiplexed probe signal prior to the detector array . a high gain , low noise , caltech citlf4 sige amplifier on the output channel boosts the probe signal prior to readout . copper semi - rigid coaxial cable then feeds out to hermetic sma connectors on the cryostat exterior . a schematic of the cryogenic readout system is presented in figure [ fig : detector](b ) . a room temperature analog mixing circuit converts the probe signal to and from the 1.25 - 1.75ghz detector readout band and the 0 - 500mhz digital electronics band . an r&s smf100a signal generator is used as the lo input for a pair of marki iq mixers and a combination of amplifiers and variable attenuators are in place to balance the incoming and outgoing power levels . the digital system is a nikel@xcite ( new iram kid electronics ) frequency domain multiplexing system developed for the nika astronomical camera . it has the ability to output the in - phase and quadrature ( i and q ) components of the superposition of up to 400 cordic - generated tones across 500mhz of dac bandwidth . a single adc feeds into a polyphase filter bank and the resultant 400 independent decomposed i and q timestreams ( as well as the mirror encoder values and other housekeeping data ) are decimated and sent via the on - board computer over ethernet to the control station . the sample rate is limited to 477hz which provides a data rate of 24mbps . the control station is a desktop computer equipped with a custom software suite for control of the readout electronics , data acquisition , image generation and graphical display . the readout electronics system is initialized with commands sent over udp to the nikel on - board computer . the detector responses ( variations in @xmath30 , aka @xmath33 ) are computed from linear transformations of the raw i and q timestreams using coefficients from frequency sweep data taken across the resonators during initialization . a flat field calibration is performed at the start of each run where the detector responses are measured between a 30@xmath34c glow bar and a room temperature section of the field of view . the raw i and q , the transformed amplitude , phase , and @xmath33 , and the calibrated response time - streams can be accessed and displayed alongside their power spectral densities using the real - time plotting software kst . otherwise , image generation is performed on a scan - by - scan basis by reading the latest data , applying the transformations and calibrations , binning these products into a map , and updating the graphical interface with a new frame . broken or poorly performing detectors can cause blank or noisy columns in the image frames however these can be digitally filtered or interpolated over in real - time to improve the overall image quality . the optics system was tested with a measurement scheme based on raster scans of a chopped 50@xmath35c blackbody across the object plane . maps of the beam profiles for each working detector were made down to a 25db signal to noise level . a typical beam ( figure [ fig : beammap]a ) is approximately gaussian and the full width at half maximum ( fwhm ) is 11 mm at 3.5 m after deconvolution of the 10 mm diameter source aperture . this provides a resolution close to that expected for a diffraction limited system in this configuration , although , some channels show mild broadening and aberrations ( figure [ fig : beammap]b ) , particularly at one edge of the focal plane . there is also some indication of localized leakage from adjacent feedhorns at a level typically less than 5 - 10% . beam profile maps and central slices from two channels measured with a 50@xmath34c chopped blackbody source of 10 mm circular aperture . ( a ) is from a typical detector with a measured fwhm of 13 mm at a distance of 3.5 m . ( b ) is from a detector on the right edge of the array with a broader 15 mm fwhm beam and some strong aberration . both beams show some off - center low - level response attributable to light leaks from neighboring feedhorns . ] the operational yield of the current detector array is 85% with the majority of unusable pixels suffering from the effects of resonator overlap due to non - uniformity of the thickness / resistivity of the aluminum film . aside from this resonator clash there is no indication of any other electromagnetic cross coupling between resonators down to the measured 25db level . a noise power spectrum for a typical detector channel sampled at the maximum rate of 477hz is presented in the inset to figure [ fig : net ] . the spectrum shows white noise down to @xmath11hz . the excess below this knee frequency is attributed to the warm electronics system , as are the spurious components at 95.5hz and 191hz . these unwanted narrowband features are digitally filtered from the detector timelines prior to image generation . the filters are implemented as fifth - order , butterworth bandstop filters that operate on the timelines in the time domain on a frame - by - frame basis . the distribution of nets sampled at the white noise frequencies is indicated in the histogram in figure [ fig : net ] . the distribution is approximately log - normal with a peak net value of 6.1mk@xmath25 , a factor of 2 higher than the expected limit from photon noise in this system . the excess is thought to be due to stray infrared radiation leaking from the 4k stage . a normalized histogram of noise equivalent temperature measured at the white noise levels over each of the @xmath36 detectors . the distribution is well approximated by a log - normal function ( black curve ) , the modal value of which is 6.1mk@xmath25 . a noise spectrum , calculated from a typical detector timeline , is included in the inset . ] the constraint set by the scanning mechanism and the higher than expected noise currently limit the update rate to 2 frames per second for an ne@xmath37 t of 0.1k per frame with the camera in its present configuration . figure [ fig : composite ] shows a single frame taken from a combined `` three - color '' video . the sensitivity is clearly sufficient to identify objects that are invisible to thermal nir cameras and standard digital video cameras . a snap shot from a 2 fps video in which the 350ghz frames ( left ) were displayed simultaneously with frames from a standard web - cam ( center ) and a thermal nir camera ( right ) . objects such as ( a ) a wallet , ( b ) an air pistol , and ( c ) some loose change , are hidden by the high opacity of the coat at higher frequencies but become apparent at 350ghz . ] in most respects , the camera presented here has achieved the required specifications . the presence of parasitic optical loading on the detector array limits the noise performance so that full video rate could not be achieved even if a faster field modulation system was employed . however , a second generation system could overcome this in a number of ways . for example , by utilizing a reflective optics approach , especially one with a cold aperture stop within the cryostat . this would help to inhibit stray light loads on the detectors and also eliminate the requirement for feedhorn coupling . additionally , the field scanning mechanism of the present system is purely linear and thus does not employ any cross linking between detector channels . as such , the video frames suffer from vertical striping due to broken / noisy detectors and low frequency gain fluctuations between individual detectors . transitioning to a dual - axis circular or lissajous style scanning strategy would remedy this and is an advisable approach for any future system . a general purpose instrument similar to that presented here would benefit from a modular ( rather than fixed ) optics system . providing an additional image plane located externally to the cryostat would enable fast turnaround between a variety of application specific imaging formats without the need for any modification to the cryogenic platform . kinetic inductance detectors originally developed for far - infrared astronomy are now suitable for use in a range of applications requiring high sensitivity and/or fast mapping of objects at terahertz frequencies . the instrument presented here mimics stand - off imaging systems for the detection of concealed items but could easily be transformed for other applications by modification of the optics platform . this lekid based system operates close to the ideal photon noise limited sensitivity and is comparable in performance to the latest passive thz imaging systems . the development of larger kid arrays is ongoing and next generation instruments will benefit from order of magnitude increases in detecting elements with no considerable penalty in array fabrication or readout complexity . the authors acknowledge the uk science and technology facilities council for supporting the development of kid technology for astronomy applications . 40ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) link:\doibase 10.1051/0004 - 6361/201014759 [ * * , ( ) ] , link:\doibase 10.1051/0004 - 6361/201014519 [ * * , ( ) ] , link:\doibase 10.1051/0004 - 6361/201014535 [ * * , ( ) ] , link:\doibase 10.1051/0004 - 6361/201014698 [ * * , ( ) ] link:\doibase 10.1051/0004 - 6361/201116464 [ * * , ( ) ] , link:\doibase 10.1051/0004 - 6361/200912975 [ * * , ( ) ] link:\doibase 10.1051/0004 - 6361/200912853 [ * * , ( ) ] , link:\doibase 10.1038/nature02037 [ * * , ( ) ] link:\doibase 10.1063/1.2390664 [ * * , ( ) ] , link:\doibase 10.1086/674013 [ * * , ( ) ] , link:\doibase 10.1051/0004 - 6361/201014727 [ * * , ( ) ] , link:\doibase 10.1007/s10909 - 007 - 9685 - 2 [ * * , ( ) ] link:\doibase 10.1007/s10909 - 013 - 1069 - 1 [ * * , ( ) ] link:\doibase 10.1007/s10909 - 013 - 0985 - 4 [ * * , ( ) ] , link:\doibase 10.1142/s2251171714400017 [ * * , ( ) ] , link:\doibase 10.1109/22.989974 [ * * , ( ) ] link:\doibase 10.1109/jproc.2007.898906 [ * * , ( ) ] link:\doibase 10.1364/boe.4.000559 [ * * , ( ) ] \doibase 10.1023/a:1024409329416 [ * * , ( ) ] link:\doibase 10.1364/oe.21.017800 [ * * , ( ) ] link:\doibase 10.1117/12.503414 [ * * , ( ) ] link:\doibase 10.1002/jps.20225 [ * * , ( ) ] link:\doibase 10.1364/ao.49.00e106 [ * * , ( ) ] @noop `` , '' ( ) in link:\doibase 10.1109/comcas.2013.6685254 [ _ _ ] ( , ) in link:\doibase 10.1109/irmmw - thz.2014.6956015 [ _ _ ] ( , ) in link:\doibase 10.1117/12.852932 [ _ _ ] , ( , ) in link:\doibase 10.1117/12.976849 [ _ _ ] , ( , ) in @noop _ _ ( ) pp . link:\doibase 10.1364/ao.25.000870 [ * * , ( ) ] in link:\doibase 10.1117/12.673159 [ _ _ ] , , vol . ( ) p. in link:\doibase 10.1117/12.673162 [ _ _ ] , , vol . ( ) @noop ( ) , @noop link:\doibase 10.1088/1748 - 0221/7/07/p07014 [ * * , ( ) ] ,
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we have developed a passive 350ghz ( 850@xmath0 m ) video - camera to demonstrate lumped element kinetic inductance detectors ( lekids ) designed originally for far - infrared astronomy as an option for general purpose terrestrial terahertz imaging applications .
the camera currently operates at a quasi - video frame rate of 2hz with a noise equivalent temperature difference per frame of @xmath10.1k , which is close to the background limit .
the 152 element superconducting lekid array is fabricated from a simple 40 nm aluminum film on a silicon dielectric substrate and is read out through a single microwave feedline with a cryogenic low noise amplifier and room temperature frequency domain multiplexing electronics .
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modified theories of gravity have attained much attention after the discovery of expanding accelerated universe . the basic ingredient responsible for this tremendous change in cosmic history is some mysterious type force having repulsive nature dubbed as dark energy . the enigmatic nature of this energy has motivated many researchers to unveil its hidden characteristics which are still not known . modified gravity approach is considered as the promising and optimistic scenario among several other proposals that have been presented to explore the salient features of dark energy . these modified theories are established by adding or replacing curvature invariants and their corresponding generic functions in the einstein - hilbert action . lovelock theory of gravity is the direct generalization of general relativity ( gr ) in @xmath3-dimensions which coincides with gr in @xmath4-dimensions @xcite . the ricci scalar @xmath5 is known as first lovelock scalar while gauss - bonnet ( gb ) invariant is the second lovelock scalar yielding einstein - gauss - bonnet gravity in @xmath6-dimensions @xcite . the gb invariant is a linear combination with an interesting feature that it is free from spin-2 ghost instabilities defined as @xcite @xmath7 where @xmath8 and @xmath9 are the ricci and riemann tensors , respectively . this quadratic curvature invariant is a topological term in 4-dimensions which possesses trivial contribution in the field equations . to discuss the dynamics of gb invariant in 4-dimensions , there are two interesting scenarios either to couple @xmath1 with scalar field or to add generic function @xmath10 in the einstein - hilbert action . the first scheme naturally appears in the effective action in string theory which investigates singularity - free cosmological solutions @xcite . the second approach known as @xmath10 gravity is introduced as an alternative for dark energy which successfully discusses the late - time cosmological evolution @xcite . this modified theory of gravity is endowed with a quite rich cosmological structure as well as consistent with solar system constraints @xcite . the current cosmic accelerated expansion has also been discussed in modified theories of gravity involving the curvature - matter coupling . harko et al . @xcite established @xmath11 gravity to study the curvature - matter coupling . recently , we introduced the curvature - matter coupling in @xmath10 gravity named as @xmath0 theory of gravity @xcite . this coupling yields non - zero covariant divergence of the energy - momentum tensor and an extra force appears due to which massive test particles follow non - geodesic trajectories while geodesic lines of geometry are followed by the dust particles . shamir and ahmad @xcite constructed some cosmologically viable models in @xmath0 gravity using noether symmetry approach . it is mentioned here that cosmic expansion can be obtained from geometric as well as matter components in such coupling . the reconstruction as well as stability of cosmic evolutionary models in modified theories of gravity are the captivating issues in cosmology . in reconstruction technique , any known cosmic solution is used in the modified field equations to find the corresponding function which reproduces the given evolutionary cosmic history . in stability analysis , the isotropic and homogeneous perturbations are usually considered in which hubble parameter as well as energy density are perturbed to examine the background stability as time evolves @xcite . nojiri et al . @xcite formulated the reconstruction scheme to reproduce some cosmological models in @xmath12 gravity . elizalde et al . @xcite applied the same scenario for @xmath13cdm cosmology ( @xmath13 denotes cosmological constant while cdm stands for cold dark matter ) in @xmath14 gravity as well as in modified gb theories of gravity . the stability of power - law solutions are also discussed in modified gravity theories @xcite . sez - gmez @xcite explored the cosmological solutions in @xmath12 horava - lifshitz gravity and analyzed their stability against first order perturbations around frw universe . myrzakulov and his collaborators @xcite discussed the cosmological models and found that @xmath10 gravity could successfully explain the cosmic evolutionary history . jamil et al . @xcite reconstructed the cosmological models in @xmath11 gravity and found that numerical analysis for hubble parameter is in good agreement with observational data for redshift parameter @xmath15 . the stability of de sitter , power - law solutions as well as @xmath13cdm are analyzed in the context of @xmath14 gravity @xcite . salako et al . @xcite studied the cosmological reconstruction , stability as well as thermodynamics including first and second laws for @xmath13cdm model in generalized teleparallel theory of gravity . sharif and zubair @xcite demonstrated that @xmath11 gravity can reproduce @xmath13cdm model , phantom or non - phantom eras , de sitter universe and power - law cosmic history . they also analyzed the stability of reconstructed de sitter as well as power - law solutions . in this paper , we reconstruct various cosmological models including de sitter universe , power - law solutions , phantom or non - phantom eras and @xmath13cdm model in @xmath0 theory . we also analyze the stability against linear homogeneous perturbations for de sitter as well as power - law solutions . the paper has the following format . in section * 2 * , we formulate the modified field equations while section * 3 * is devoted to reconstruct some known cosmological solutions in this gravity . section * 4 * analyzes the stability of specific solutions against linear perturbations around frw universe model . the results are summarized in the last section . the action for @xmath0 gravity is defined as @xcite @xmath16 where @xmath17 and @xmath18 represent coupling constant , determinant of the metric tensor ( @xmath19 ) and lagrangian associated with matter distribution , respectively . varying eq.([1 ] ) with respect to @xmath19 , we obtain the field equations @xmath20f_{\mathcal{g}}(\mathcal{g},t ) \\\nonumber&-&[2rg_{\alpha\beta } \box-4r_{\alpha\beta}\box-2r\nabla_{\alpha}\nabla_{\beta } + 4r^{\mu}_{\beta}\nabla_{\alpha}\nabla_{\mu } + 4r^{\mu}_{\alpha}\nabla_{\beta}\nabla_{\mu}\\\label{2}&- & 4g_{\alpha\beta}r^{\mu\nu } \nabla_{\mu}\nabla_{\nu}+4r_{\alpha\mu\beta\nu } \nabla^{\mu}\nabla^{\nu}]f_{\mathcal{g}}(\mathcal{g},t)=0,\end{aligned}\ ] ] where @xmath21 ( @xmath22 denotes a covariant derivative ) and @xmath23 is the energy - momentum tensor . the expressions for @xmath23 and @xmath24 are @xcite @xmath25 where we have assumed that @xmath18 depends only on @xmath19 rather than its derivatives . the non - zero divergence of @xmath23 is given by @xmath26.\end{aligned}\ ] ] the above equations indicate that the complete dynamics of @xmath0 gravity is based on the suitable choice of @xmath18 . the energy - momentum tensor for perfect fluid is @xmath27 where @xmath28 and @xmath29 represent the four velocity , energy density and pressure of matter distribution , respectively . in this case , the expression for @xmath24 becomes @xmath30 where @xmath31 . the line element for frw universe model is given by @xmath32 where @xmath33 is the scale factor . using eqs.([5])-([7 ] ) in ( [ 2 ] ) , we obtain the corresponding field equation as follows @xmath34 where @xmath35 and dot represents derivative with respect to time . the non - zero continuity equation ( [ 4 ] ) takes the form @xmath36.\ ] ] the standard conservation law holds if right hand side of this equation vanishes . for equation of state @xmath37 ( @xmath38 is the equation of state parameter ) , eq.([9 ] ) yields @xmath39 with additional constraint @xmath40 we rewrite the above equations in terms of new variable @xmath41 known as e - folding instead of @xmath42 which is also related with redshift parameter @xmath43 as @xcite @xmath44 using the above definition of @xmath41 , eqs.([8 ] ) and ( [ 9 ] ) become @xmath45,\end{aligned}\ ] ] where @xmath46 and prime denotes derivative with respect to @xmath41 . the simplest choice of @xmath0 model is @xmath47 which possesses no direct non - minimally coupling between curvature and matter . for this particular model , the field equation ( [ 13 ] ) splits into a set of two ordinary differential equations as @xmath48 @xmath49 where @xmath50 and @xmath51 . the field equations for perfect fluid matter distribution in @xmath10 gravity is recovered if @xmath52 vanishes while gr is achieved for @xmath53 . in this section , we reproduce different cosmological scenarios including de sitter universe , power - law solutions , phantom / non - phantom eras and @xmath13cdm model in @xmath0 gravity . the de sitter cosmic evolution is interesting and well - known as it elegantly describes current expansion of the universe . the scale factor of this evolutionary model grows exponentially with constant hubble parameter @xmath54 , defined as @xcite @xmath55 where @xmath56 is an integration constant . equation ( [ 10 ] ) gives energy density of the form @xmath57 where @xmath58 and @xmath59 is a constant . using eqs.([1d ] ) and ( [ 2d ] ) in ( [ 8 ] ) , we obtain @xmath60 where @xmath61 is the gb invariant at @xmath54 . the solution of the above differential equation is @xmath62 where @xmath63 s @xmath64 are integration constants . since we have used the continuity equation ( [ 10 ] ) in eq.([3d ] ) , so we must constrain its solution . using the above equation with eq.([11 ] ) , we obtain the following functions @xmath65 where @xmath66 s @xmath67 are constants in terms of @xmath38 and @xmath68 given in appendix * a*. for the model ( [ 15 ] ) , we have @xmath69 where the first equation corresponds to de sitter universe in the absence of matter contents in @xmath10 gravity @xcite . using the constraint ( [ 11 ] ) , the second equation becomes @xmath70 the solution of eqs.([7d ] ) and ( [ 8d ] ) leads to @xmath71 where @xmath72 s are constants of integration . equations ( [ 4d ] ) and ( [ 9d ] ) indicate that de sitter expansion can also be described in @xmath0 gravity . power - law solutions have significant importance to discuss different evolutionary phases of the universe in modified theory . these solutions describe the decelerated as well as accelerated cosmic eras which are characterized by the scale factor as @xcite @xmath73 the cosmic decelerated phase is observed for @xmath74 including the radiation @xmath75 as well as dust @xmath76 dominated eras while @xmath77 covers the accelerated phase of the universe . for this scale factor , the gb invariant takes the form @xmath78 using eqs.([10 ] ) , ( [ 1p ] ) and ( [ 2p ] ) , the field equation becomes @xmath79 whose solution is given by @xmath80 where @xmath81 s are integration constants and @xmath82,\\\nonumber\gamma_{2}&=&\left[\frac{3}{4}\lambda \tilde{c}_{2}(1+\omega)\{2\tilde{c}_{2}\lambda(1+\omega)+2(\lambda-1 ) -8\}+\frac{1}{4}(\lambda-1)(\lambda-7)+4\right.\\\nonumber&+&\left . 8\tilde{c}_{2}(\lambda-1)\left(\frac{1+\omega}{1 - 3\omega}\right ) \right]^{\frac{1}{2}},\quad\gamma_{3}=-\frac{1}{2}\left(\frac{1 - 3\omega } { 1+\omega}\right),\quad\gamma_{4}=\frac{2\kappa^2}{\omega-3},\\\nonumber \gamma_{5}&=&\left(\frac{18\lambda^{3}(1 - 3\omega)^{\frac{3\lambda ( 1+\omega)-2}{3\lambda(1+\omega)}}}{3\lambda(1 - 3\omega)+4}\right ) \rho_{0}^{\frac{-2}{3\lambda(1+\omega)}},\quad\gamma_{6}=\frac{2 } { 3\lambda(1+\omega)}.\end{aligned}\ ] ] inserting eq.([4p ] ) in ( [ 11 ] ) , we obtain @xmath83 where @xmath84 s and @xmath85 s @xmath86 are given in appendix * a*. now we find the expression of @xmath0 for the choice of model ( [ 15 ] ) . the differential equation ( [ 3p ] ) yields two ordinary differential equations in variables @xmath1 and @xmath2 . the first is the second order cauchy - euler s equation related to curvature given by @xmath87 whose solution is given by @xmath88 where @xmath89 s are constants of integration which is consistent with power - law solutions in @xmath10 gravity @xcite . the second equation is obtained using the additional constraint ( [ 11 ] ) as @xmath90 the solution of this equation corresponds to matter distribution given by @xmath91}\left(\frac{t}{\rho_{0}(1 - 3\omega)}\right ) ^{\frac{2}{3\lambda(1+\omega)}},\end{aligned}\ ] ] where @xmath92 s @xmath93 are integration constants . consequently , @xmath0 model becomes @xmath94}\\\label{7p}&\times & \left(\frac{t}{\rho_{0}(1 - 3\omega)}\right)^{\frac{2}{3\lambda(1+\omega)}}.\end{aligned}\ ] ] thus , the power - law solutions are reconstructed which may be helpful to explore the expansion history of the universe in this modified theory of gravity . here , we reconstruct @xmath0 model which can explain the system including both phantom and non - phantom eras . in the einstein gravity , the hubble parameter describing the phantom as well as non - phantom matter distribution is given by @xcite @xmath95 where @xmath96 and @xmath97 represent the model parameter , energy densities of phantom and non - phantom matter fluids , respectively . when the scale factor is large , the first term on right hand side dominates which corresponds to the phantom era of the universe with @xmath98 . the non - phantom era in the early universe is observed for @xmath99 when the scale factor is small and the second term dominates . we rewrite @xmath100 in terms of a new function @xmath101 as @xmath102 so that eq.([1q ] ) becomes @xmath103 where @xmath104 and @xmath105 . the gb invariant takes the form @xmath106 inserting eq.([2q ] ) in ( [ 3q ] ) , we obtain a quadratic equation in @xmath107 whose solution is given by @xmath108 for the sake of simplicity , we consider @xmath109 so that it reduces to @xmath110 using eqs.([2q ] ) and ( [ 4q ] ) in ( [ 8 ] ) , we have @xmath111 which is a complicated partial differential equation whose analytical solution can not be found . to find the reconstructed @xmath0 model , we consider its particular form ( [ 15 ] ) which provides the following set of differential equations @xmath112 where we have used the additional constraint in the second equation . solving these equations , it follows that @xmath113-\frac{2\kappa^{2}t}{1 - 3\omega}.\end{aligned}\ ] ] where @xmath114 s are constants of integration . thus , phantom and non - phantom cosmic history can be discussed in @xmath0 gravity . now we apply the usual reconstruction technique to reproduce the @xmath13cdm cosmology in this gravity . in gr , the @xmath13cdm cosmological evolution is discussed by adding @xmath13 in the einstein - hilbert action whereas we reconstruct such evolution in the absence of @xmath13 in the action ( [ 1 ] ) . the hubble parameter for @xmath13cdm model is given by @xcite @xmath115 the first term indicates the contribution of cdm while the second term corresponds to @xmath13 . the hubble parameter in terms of gb invariant takes the form @xmath116 equation ( [ 1h ] ) gives @xmath117 using eqs.([2h ] ) and ( [ 3h ] ) in ( [ 13 ] ) , we obtain @xmath118 this equation can not be solved analytically hence we solve it for a specific @xmath0 model ( [ 15 ] ) with dust fluid . the corresponding equations become @xmath119 using constraint ( [ 11 ] ) , the solution of eq.([4h ] ) is given by @xmath120 where @xmath121 s are constants of integration while solution of eq.([5h ] ) can not be found . let us consider the case when @xmath122 with the assumption @xmath123 which reduces eq.([5h ] ) to @xmath124 whose solution yields @xmath125 where @xmath126 s are integration constants . consequently , @xmath0 takes the form @xmath127 provided that @xmath128 . here @xmath0 gravity can not explain @xmath13cdm cosmological evolution rather it corresponds to cdm evolution . in this section , we analyze stability of some cosmological evolutionary solutions about linear homogeneous perturbations in this modified gravity . we construct the perturbed field as well as continuity equations using isotropic and homogeneous universe model for both general and particular cases including de sitter and power - law solutions . we assume a general solution @xmath129 which satisfies the basic field equations for frw universe model in @xmath0 gravity . in terms of the above solution , the expressions for @xmath130 and @xmath131 are @xmath132 for any particular @xmath0 model that can regenerate the above solution ( [ 1 t ] ) , the following equation of motion as well as non - zero divergence of the energy - momentum tensor must be satisfied @xmath133,\end{aligned}\ ] ] where superscript @xmath134 denotes that the function and its corresponding derivatives are calculated at @xmath135 and @xmath136 . if the conservation law holds , we get energy density in terms of @xmath137 as @xmath138 the first order perturbations in hubble parameter and energy density are defined as @xmath139 where @xmath140 and @xmath141 are the perturbation parameters . in order to analyze first order perturbations about the solution ( [ 1 t ] ) , we apply the series expansion on the function @xmath0 as @xmath142 where @xmath143 involves the terms proportional to quadratic or higher powers of @xmath1 and @xmath2 while only the linear terms are considered . using eqs.([6 t ] ) and ( [ 7 t ] ) in ( [ 8 ] ) , we obtain the following perturbed field equation @xmath144 where @xmath145 s @xmath146 are given in appendix * a*. inserting these perturbations in eq.([9 ] ) , the perturbed continuity equation is @xmath147 where @xmath148 s are provided in appendix * a*. if the conversation law holds in this modified gravity , eq.([9 t ] ) reduces to @xmath149 the perturbed equations ( [ 8 t ] ) and ( [ 9 t ] ) are helpful to analyze the stability of any specific frw cosmological evolutionary model in @xmath0 gravity . for the particular model ( [ 15 ] ) , these perturbed equations reduce to @xmath150 where the coefficients of @xmath151 and their derivatives are expressed in appendix * a*. in the following subsections , we investigate the stability of de sitter and power - law solutions . consider the de sitter solution @xmath152 , the perturbed equation ( [ 8 t ] ) takes the form @xmath153 f_{\mathcal{g}t}^{0}-3456\rho_{*}h_{0}^{8}(1 - 3\omega)(1+\omega ) f_{\mathcal{gg}t}^{0}\right)\delta+12\rho_{*}h_{0}^{3}\\\nonumber & \times&(1 - 3\omega)f_{\mathcal{g}t}^{0}\dot{\delta}_{m}+\left(\kappa^{2 } \rho_{*}+\frac{1}{2}\rho_{*}(3-\omega)f_{t}^{0}+\rho_{*}^{2}(1 - 3\omega ) ( 1+\omega)f_{tt}^{0}\right.\\\nonumber&-&\left.12\rho _ { * } h_{0}^{2}(1 - 3\omega)[h_{0}^{2}+3(1+\omega)h_{0}^{2}]f_{\mathcal{g}t}^{0 } -36\rho_{*}^{2}h_{0}^{4}(1 - 3\omega)^{2}(1+\omega ) \right.\\\label{13t}&\times&\left.f_{\mathcal{g}tt}^{0}\right)\delta_{m}=0,\end{aligned}\ ] ] where the superscript @xmath154 represents that the function and its corresponding derivatives are evaluated at @xmath155 and @xmath156 . we consider the conserved perturbed equation for stability analysis since the de sitter solutions are constructed using the constraint ( [ 11 ] ) in the previous section . the numerical technique is used to solve eqs.([10 t ] ) and ( [ 13 t ] ) for the model ( [ 5d ] ) . the evolution of @xmath140 and @xmath141 are shown in figure * 1*. we consider @xmath157 and @xmath158 throughout the stability analysis of de sitter universe models whereas integration constants are @xmath159 and @xmath160 . figure * 1 * shows smooth behavior of @xmath140 ( left ) and @xmath141 ( right ) which do not decay in late times indicating that de sitter model ( [ 5d ] ) is unstable . the stability analysis of model ( [ 6d ] ) with same integration constants is shown in figure * 2*. in the left panel , it is observed that small oscillations are produced about @xmath161 while it decays in late times , thus the model ( [ 6d ] ) shows stable behavior against perturbations . for model ( [ 9d ] ) , eq.([13 t ] ) becomes @xmath162 figure * 3 * represents the behavior of @xmath140 and @xmath141 for model ( [ 9d ] ) with integration constants @xmath163 and @xmath164 . it is shown that oscillations in perturbation parameters are produced initially as shown in figure * 3*. this oscillating behavior is clearly observed in figure * 4 * which decays in future for both @xmath140 as well as @xmath141 and hence the solution becomes stable . here we investigate the stability of power - law solutions . these solutions describe the accelerated as well as decelerated cosmological evolutionary phases in the background of frw universe . we first consider the reconstructed power - law solution ( [ 5p ] ) and numerically solve eqs.([8 t ] ) and ( [ 10 t ] ) . for this model , we choose integration constants @xmath165 and @xmath166 figure * 5 * shows the oscillating behavior of perturbed parameters @xmath167 for the cosmic accelerated era with @xmath168 and @xmath169 . the perturbations around the power - law solutions decay in future leading to stable results . the radiation ( @xmath170 and @xmath171 ) as well as matter ( @xmath172 and @xmath173 ) dominated eras can not be discussed for the model ( [ 5p ] ) because singular as well as complex terms appear which lead to non - physical case . secondly , we consider the model ( [ 6p ] ) and analyze its behavior against linear perturbations . figure * 6 * shows the fluctuating behavior of considered perturbations in the cosmic accelerated phase with @xmath168 and @xmath169 . here , we choose @xmath174 and @xmath166 it is observed that the oscillating behavior disappears in future while both perturbation parameters will not decay in late times leading to unstable cosmological solutions . the considered model can not explain the cosmological evolution corresponding to matter and radiation dominated eras like previous model ( [ 5p ] ) . lastly , we explore the stability of model ( [ 7p ] ) with integration constants @xmath175 and @xmath176 . figure * 7 * represents the evolution of ( @xmath177 ) versus time for @xmath178 with @xmath168 . the left panel shows that the oscillations of @xmath140 decay in late times while fluctuations of @xmath141 remain present in future . since a complete perturbation against any cosmological solution includes the matter perturbations therefore , the solutions are unstable . in this paper , we have employed the reconstruction scheme to @xmath0 gravity in the background of isotropic and homogeneous universe model to reproduce some important cosmological models . the basic aspect of this modified gravity is the coupling between curvature and matter components which yields non - zero divergence of the energy - momentum tensor . we have imposed additional constraint to obtain the standard conservation equation which has been used to explain the cosmic evolution in this gravity . the de sitter and power - law solutions have been reconstructed for general as well as particular cases which are of great interest and have significant importance in cosmology . we have also reconstructed the @xmath0 model which can explain cosmic history of the phantom as well as non - phantom phases of the universe . similar reconstruction technique is carried out for @xmath13cdm model . in this case , we have found that the considered gravity fails to reproduce @xmath13cdm cosmology for both cases . for the specific form of function , this result is consistent with @xmath10 gravity in the absence of matter @xcite . on physical grounds , the stability analysis of different forms of generic function leads to classify the modified theories of gravity . we have applied the first order perturbations to hubble parameter and energy density to analyze the stability of models which reproduce de sitter and power - law cosmic history . we have perturbed the field equation as well as conservation law whose numerical solutions provide the stable / unstable results . * for the de sitter universe , the evolution of perturbation has been plotted against time as shown in figures * 1 * -*4*. these indicate that models ( [ 6d ] ) and ( [ 9d ] ) are stable against linear perturbations . * for the power - law universe , the stability analysis is given in figures * 5 * -*7*. it is found that @xmath0 gravity fails to reproduce matter and radiation dominated eras while stable results are obtained for accelerated phase of the universe for model ( [ 5p ] ) . we conclude that the cosmological reconstruction and stability analysis might restrict @xmath0 gravity in the background of frw universe . the expressions for @xmath179 s in eqs.([5d ] ) and ( [ 6d ] ) are @xmath180\\\nonumber & \times&[1+\omega-36c_{1}h_{0}^{4}(1 - 3\omega)]^{-2},\\\nonumber \xi_{2}&=&18c_{1}h_{0}^{4}[(1 - 11\omega)(1-\omega^{2})-8c_{1 } h_{0}^{4}\{2(2 + 59\omega^{2})-11\omega(5 - 3\omega^{2})\}]\\\nonumber & \times&[(1+\omega)(1 - 24c_{1}h_{0}^{4})\{1+\omega-6c_{1}h_{0}^{4 } ( 5 - 4\omega-33\omega^{2})\}]^{-1},\\\nonumber\xi_{3}&=&-[18c_{1 } h_{0}^{4}(1 - 32c_{1}h_{0}^{4})-3\omega\{1 - 6c_{1}h_{0}^{4}(3 - 352c_{1 } h_{0}^{4})\}\\\nonumber&-&2\omega^{2}\{1 - 9c_{1}h_{0}^{4 } ( 7 - 1248c_{1}h_{0}^{4})\}+\omega^{3}\{1 - 54c_{1}h_{0}^{4}(7 - 480c_{1 } h_{0}^{4})\}]\\\nonumber&\times&[(1 - 3\omega)(1 - 24c_{1}h_{0}^{4 } ) \{1+\omega-6c_{1}h_{0}^{4}(5 - 4\omega-33\omega^{2})\}]^{-1}.\end{aligned}\ ] ] the expressions for @xmath84 s and @xmath85 s in eqs.([5p ] ) and ( [ 6p ] ) are @xmath181 where @xmath182,\\\nonumber\gamma_{8}&=&\frac{\tilde{c}_{2 } } { 6\lambda}\left[6\tilde{c}_{2}\lambda(1+\omega)^{2}-3\lambda ( 1 + 5\omega+2\omega^2)+2(\gamma_{1}-\gamma_{2})\right].\end{aligned}\ ] ] the values of @xmath145 s in eq.([8 t ] ) are given as follows @xmath183f_{\mathcal{g}t}^{*}-864(1+\omega)(1 - 3\omega)\rho _ { * } h_{*}^{6}(4h_{*}^{2}\\\nonumber&+&\dot{h}_{*})f_{\mathcal{gg}t}^ { * } , \\\nonumber\chi_{4}&=&12(1 - 3\omega)\rho_{*}h_{*}^{3}f_{\mathcal{g}t}^ { * } , \\\nonumber\chi_{5}&=&\kappa^{2}\rho_{*}-\frac{1}{2}(\omega-3 ) \rho_{*}f_{t}^{*}+(1 - 3\omega)(1+\omega)\rho_{*}^{2}f_{tt}^{*}-12 ( 1 - 3\omega)\rho_{*}h_{*}^{2}\\\nonumber&\times&[(4 + 3\omega)h_{*}^{2 } + \dot{h}_{*}]f_{\mathcal{g}t}^{*}+288(1 - 3\omega)\rho_{*}h_{*}^{4 } ( 4h_{*}^{2}\dot{h}_{*}+2\dot{h}_{*}^{2}+h_{*}\ddot{h } _ { * } ) \\\nonumber&\times&f_{\mathcal{gg}t}^{*}-36(1+\omega ) ( 1 - 3\omega)^{2}\rho_{*}^{2}h_{*}^{4}f_{\mathcal{g}tt}^{*}.\end{aligned}\ ] ] the expressions for @xmath148 s are @xmath184f_{\mathcal{g}t}^ { * } -72\rho_{*}^{2}h_{*}^{2}(1 - 3\omega)(1+\omega)^{2}(4h_{*}^{2 } \\\nonumber&+&\dot{h}_{*})f_{\mathcal{g}tt}^{*}+576\rho_{*}h_{*}^{3 } ( 1+\omega)(4h_{*}^{2}+\dot{h}_{*})(4h_{*}^{2}\dot{h}_{*}+2 \dot{h}_{*}^{2}\\\nonumber&+&4h_{*}\ddot{h}_{*})f_{\mathcal{gg}t}^ { * } , \\\nonumber\upsilon_{2}&=&-12\rho_{*}h_{*}^{2}(1+\omega)\left[3h_{*}^{2 } ( 1-\omega)-4(2h_{*}^{2}+3\dot{h}_{*})\right]f_{\mathcal{g}t}^ { * } + 576\rho_{*}h_{*}^{2}\\\nonumber&\times&(1+\omega)(4h_{*}^{2 } \dot{h}_{*}+2\dot{h}_{*}^{2}+h_{*}\dot{h}_{*})f_{\mathcal{gg}t}^ { * } -72\rho_{*}^{2}h_{*}^{4}(1 - 3\omega)\\\nonumber&\times&(1+\omega^{2 } ) f_{\mathcal{g}tt}^{*},\\\nonumber\upsilon_{3}&=&24\rho_{*}h_{*}^{3 } ( 1+\omega)f_{\mathcal{g}t}^{*},\\\nonumber\upsilon_{4}&=&-\frac{3}{2 } \rho_{*}h_{*}(1+\omega)\left[(1-\omega)f_{t}^{*}+2\rho_{*}^{2 } ( 1+\omega)(1 - 3\omega)^{2}f_{ttt}^{*}\right]-\frac{15}{2 } \rho_{*}^{2}h_{*}\\\nonumber&\times&(1 - 3\omega)(1+\omega)^{2}f_{tt}^ { * } + 24\rho_{*}h_{*}(1+\omega)(4h_{*}^{2}\dot{h}_{*}+2\dot{h}_{*}^{2}+h _ { * } \ddot{h}_{*})\\\nonumber&\times&\left[f_{\mathcal{g}t}^{*}+\rho _ { * } ( 1 - 3\omega)f_{tt\mathcal{g}}^{*}\right],\\\nonumber\upsilon_{5 } & = & \rho_{*}\left(\kappa^2+\frac{1}{2}(3-\omega)f_{t}^{*}\right ) + ( 1+\omega)(1 - 3\omega)\rho_{*}^{2}f_{tt}^{*}.\end{aligned}\ ] ] for model ( [ 15 ] ) , the coefficients of @xmath151 have the following expressions @xmath185-\frac{15}{2 } \rho_{*}^{2}h_{*}\\\nonumber&\times&(1 - 3\omega)(1+\omega)^{2 } \mathcal{f}_{tt}^{*},\\\nonumber\hat{\upsilon}_{5}&=&\rho _ { * } \left(\kappa^2+\frac{1}{2}(3-\omega)\mathcal{f}_{t}^{*}\right ) + ( 1+\omega)(1 - 3\omega)\rho_{*}^{2}\mathcal{f}_{tt}^{*}.\end{aligned}\ ] ]
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the aim of this paper is to reconstruct and analyze the stability of some cosmological models against linear perturbations in @xmath0 gravity ( @xmath1 and @xmath2 represent the gauss - bonnet invariant and trace of the energy - momentum tensor , respectively ) .
we formulate the field equations for both general as well as particular cases in the context of isotropic and homogeneous universe model .
we reproduce the cosmic evolution corresponding to de sitter universe , power - law solutions and phantom / non - phantom eras in this theory using reconstruction technique . finally , we study stability analysis of de sitter as well as power - law solutions through linear perturbations .
* keywords : * reconstruction ; stability analysis ; modified gravity . + * pacs : * 04.50.kd ; 98.80.-k .
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when considering regression with a large number of predictors , variable selection becomes important . numerous methods have been proposed in the literature for the purpose of variable selection , ranging from the classical information criteria such as aic and bic to regularization based modern techniques such as the nonnegative garrote [ breiman ( @xcite ) ] , the lasso [ tibshirani ( @xcite ) ] and the scad [ fan and li ( @xcite ) ] , among many others . although these methods enjoy excellent performance in many applications , they do not take the hierarchical or structural relationship among predictors into account and therefore can lead to models that are hard to interpret . consider , for example , multiple linear regression with both main effects and two - way interactions where a dependent variable @xmath0 and @xmath1 explanatory variables @xmath2 are related through @xmath3 where @xmath4 . commonly used general purpose variable selection techniques , including those mentioned above , do not distinguish interactions @xmath5 from main effects @xmath6 and can select a model with an interaction but neither of its main effects , that is , @xmath7 and @xmath8 . it is therefore useful to invoke the so - called effect heredity principle [ hamada and wu ( @xcite ) ] in this situation . there are two popular versions of the heredity principle [ chipman ( @xcite ) ] . under _ strong heredity _ , for a two - factor interaction effect @xmath5 to be active both its parent effects , @xmath6 and @xmath9 , should be active ; whereas under _ weak heredity _ only one of its parent effects needs to be active . likewise , one may also require that @xmath10 can be active only if @xmath6 is also active . the strong heredity principle is closely related to the notion of marginality [ nelder ( @xcite ) , mccullagh and nelder ( @xcite ) , nelder ( @xcite ) ] which ensures that the response surface is invariant under scaling and translation of the explanatory variables in the model . interested readers are also referred to mccullagh ( @xcite ) for a rigorous discussion about what criteria a sensible statistical model should obey . li , sudarsanam and frey ( @xcite ) recently conducted a meta - analysis of 113 data sets from published factorial experiments and concluded that an overwhelming majority of these real studies conform with the heredity principles . this clearly shows the importance of using these principles in practice . these two heredity concepts can be extended to describe more general hierarchical structure among predictors . with slight abuse of notation , write a general multiple linear regression as @xmath11 where @xmath12 and @xmath13 . throughout this paper , we center each variable so that the observed mean is zero and , therefore , the regression equation has no intercept . in its most general form , the hierarchical relationship among predictors can be represented by sets @xmath14 , where @xmath15 contains the parent effects of the @xmath16th predictor . for example , the dependence set of @xmath5 is @xmath17 in the quadratic model ( [ 2way ] ) . in order that the @xmath16th variable can be considered for inclusion , all elements of @xmath15 must be included under the strong heredity principle , and at least one element of @xmath15 should be included under the weak heredity principle . other types of heredity principles , such as the partial heredity principle [ nelder ( @xcite ) ] , can also be incorporated in this framework . the readers are referred to yuan , joseph and lin ( @xcite ) for further details . as pointed out by turlach ( @xcite ) , it could be very challenging to conform with the hierarchical structure in the popular variable selection methods . in this paper we specifically address this issue and consider how to effectively impose such hierarchical structures among the predictors in variable selection and coefficient estimation , which we refer to as _ structured variable selection and estimation_. despite its great practical importance , structured variable selection and estimation has received only scant attention in the literature . earlier interests in structured variable selection come from the analysis of designed experiments where heredity principles have proven to be powerful tools in resolving complex aliasing patterns . hamada and wu ( @xcite ) introduced a modified stepwise variable selection procedure that can enforce effect heredity principles . later , chipman ( @xcite ) and chipman , hamada and wu ( @xcite ) discussed how the effect heredity can be accommodated in the stochastic search variable selection method developed by george and mcculloch ( @xcite ) . see also joseph and delaney ( @xcite ) for another bayesian approach . despite its elegance , the bayesian approach can be computationally demanding for large scale problems . recently , yuan , joseph and lin ( @xcite ) proposed generalized lars algorithms [ osborne , presnell and turlach ( @xcite ) , efron et al . ( @xcite ) ] to incorporate heredity principles into model selection . efron et al . ( @xcite ) and turlach ( @xcite ) also considered alternative strategies to enforce the strong heredity principle in the lars algorithm . compared with earlier proposals , the generalized lars procedures enjoy tremendous computational advantages , which make them particularly suitable for problems of moderate or large dimensions . however , yuan and lin ( @xcite ) recently showed that lars may not be consistent in variable selection . moreover , the generalized lars approach is not flexible enough to incorporate many of the hierarchical structures among predictors . more recently , zhao , rocha and yu ( @xcite ) and choi , li and zhu ( @xcite ) proposed penalization methods to enforce the strong heredity principle in fitting a linear regression model . however , it is not clear how to generalize them to handle more general heredity structures and their theoretical properties remain unknown . in this paper we propose a new framework for structured variable selection and estimation that complements and improves over the existing approaches . we introduce a family of shrinkage estimator that is similar in spirit to the nonnegative garrote , which yuan and lin ( @xcite ) recently showed to enjoy great computational advantages , nice asymptotic properties and excellent finite sample performance . we propose to incorporate structural relationships among predictors as linear inequality constraints on the corresponding shrinkage factors . the resulting estimates can be obtained as the solution of a quadratic program and very efficiently solved using the standard quadratic programming techniques . we show that , unlike lars , it is consistent in both variable selection and estimation provided that the true model has such structures . moreover , the linear inequality constraints can be easily modified to adapt to any situation arising in practical problems and therefore is much more flexible than the existing approaches . we also extend the original nonnegative garrote as well as the proposed structured variable selection and estimation methods to deal with the generalized linear models . the proposed approach is much more flexible than the generalized lars approach in yuan , joseph and lin ( @xcite ) . for example , suppose a group of variables is expected to follow strong heredity and another group weak heredity , then in the proposed approach we only need to use the corresponding constraints for strong and weak heredity in solving the quadratic program , whereas the approach of yuan , joseph and lin ( @xcite ) is algorithmic and therefore requires a considerable amount of expertise with the generalized lars code to implement these special needs . however , there is a price to be paid for this added flexibility : it is not as fast as the generalized lars . the rest of the paper is organized as follows . we introduce the methodology and study its asymptotic properties in the next section . in section [ sec3 ] we extend the methodology to generalized linear models . section [ sec4 ] discusses the computational issues involved in the estimation . simulations and real data examples are presented in sections [ sec5 ] and [ sec6 ] to illustrate the proposed methods . we conclude with some discussions in section [ sec7 ] . the original nonnegative garrote estimator introduced by breiman ( @xcite ) is a scaled version of the least square estimate . given @xmath18 independent copies @xmath19 of @xmath20 where @xmath21 is a @xmath22-dimensional covariate and @xmath0 is a response variable , the shrinkage factor @xmath23 is given as the minimizer to @xmath24 where , with slight abuse of notation , @xmath25 , @xmath26 , and @xmath27 is a @xmath22 dimensional vector whose @xmath28th element is @xmath29 and @xmath30 is the least square estimate based on ( [ 1.1 ] ) . here @xmath31 is a tuning parameter . the nonnegative garrote estimate of the regression coefficient is subsequently defined as @xmath32 , @xmath33 . with an appropriately chosen tuning parameter @xmath34 , some of the scaling factors could be estimated by exact zero and , therefore , the corresponding predictors are eliminated from the selected model . following yuan , joseph and lin ( @xcite ) , let @xmath15 contain the parent effects of the @xmath16th predictor . under the strong heredity principle , we need to impose the constraint that @xmath35 for any @xmath36 if @xmath37 . a naive approach to incorporating effect heredity is therefore to minimize ( [ shrink ] ) under this additional constraint . however , in doing so , we lose the convexity of the optimization problem and generally will end up with problems such as multiple local optima and potentially np hardness . recall that the nonnegative garrote estimate of @xmath38 is @xmath39 . since @xmath40 with probability one , @xmath6 will be selected if and only if scaling factor @xmath41 , in which case @xmath42 behaves more or less like an indicator of the inclusion of @xmath6 in the selected model . therefore , the strong heredity principles can be enforced by requiring @xmath43 note that if @xmath41 , ( [ strong ] ) will force the scaling factor for all its parents to be positive and consequently active . since these constraints are linear in terms of the scaling factor , minimizing ( [ shrink ] ) under ( [ strong ] ) remains a quadratic program . figure [ fig : strong ] illustrates the feasible region of the nonnegative garrote with such constraints in contrast with the original nonnegative garrote where no heredity rules are enforced . we consider two effects and their interaction with the corresponding shrinking factors denoted by @xmath44 , @xmath45 and @xmath46 , respectively . in both situations the feasible region is a convex polyhedron in the three dimensional space . ( left ) and the relaxed constraint @xmath47 ( right ) . ] similarly , when considering weak heredity principles , we can require that @xmath48 however , the feasible region under such constraints is no longer convex as demonstrated in the left panel of figure [ fig : weak ] . subsequently , minimizing ( [ shrink ] ) subject to ( [ weaknon ] ) is not feasible . to overcome this problem , we suggest using the convex envelop of these constraints for the weak heredity principles : @xmath49 again , these constraints are linear in terms of the scaling factor and minimizing ( [ shrink ] ) under ( [ weak ] ) remains a quadratic program . note that @xmath41 implies that @xmath50 and , therefore , ( [ weak ] ) will require at least one of its parents to be included in the model . in other words , constraint ( [ weak ] ) can be employed in place of ( [ weaknon ] ) to enforce the weak heredity principle . the small difference between the feasible regions of ( [ weaknon ] ) and ( [ weak ] ) also suggests that the selected model may only differ slightly between the two constraints . we opt for ( [ weak ] ) because of the great computational advantage it brings about . to gain further insight to the proposed structured variable selection and estimation methods , we study their asymptotic properties . we show here that the proposed methods estimate the zero coefficients by zero with probability tending to one , and at the same time give root-@xmath18 consistent estimate to the nonzero coefficients provided that the true data generating mechanism satisfies such heredity principles . denote by @xmath51 the indices of the predictors in the true model , that is , @xmath52 . write @xmath53 as the estimate obtained from the proposed structured variable selection procedure . under strong heredity , the shrinkage factors @xmath54 can be equivalently written in the lagrange form [ boyd and vandenberghe ( @xcite ) ] @xmath55 subject to @xmath56 for some lagrange parameter @xmath57 . for the weak heredity principle , we replace the constraints @xmath58 with @xmath59 . [ thm1 ] assume that @xmath60 and @xmath61 is positive definite . if the true model satisfies the strong / weak heredity principles , and @xmath62 in a fashion such that @xmath63 as @xmath18 goes to @xmath64 , then the structured estimate with the corresponding heredity principle satisfies @xmath65 for any @xmath66 , and @xmath67 if @xmath68 . all the proofs can be accessed as the supplement materials . note that when @xmath69 , there is no penalty and the proposed estimates reduce to the least squares estimate which is consistent in estimation . the theorems suggest that if instead the tuning parameter @xmath70 escapes to infinity at a rate slower than @xmath71 , the resulting estimates not only achieve root-@xmath18 consistency in terms of estimation but also are consistent in variable selection , whereas the ordinary least squares estimator does not possess such model selection ability . the nonnegative garrote was originally introduced for variable selection in multiple linear regression . but the idea can be extended to more general regression settings where @xmath0 depends on @xmath21 through a scalar parameter @xmath72 where @xmath73 is a @xmath74-dimensional unknown coefficient vector . it is worth pointing out that such extensions have not been proposed in literature so far . a common approach to estimating @xmath75 is by means of the maximum likelihood . let @xmath76 be a negative log conditional likelihood function of @xmath77 . the maximum likelihood estimate is given as the minimizer of @xmath78 for example , in logistic regression , @xmath79 more generally , @xmath80 can be replaced with any loss functions such that its expectation @xmath81 with respect to the joint distribution of @xmath20 is minimized at @xmath82 . to perform variable selection , we propose the following extension of the original nonnegative garrote . we use the maximum likelihood estimate @xmath83 as a preliminary estimate of @xmath84 . similar to the original nonnegative garrote , define @xmath85 . next we estimate the shrinkage factors by @xmath86 subject to @xmath87 and @xmath88 for any @xmath33 . in the case of normal linear regression , @xmath89 becomes the least squares and it is not hard to see that the solution of ( [ gnng ] ) always satisfies @xmath90 because all variables are centered . therefore , without loss of generality , we could assume that there is no intercept in the normal linear regression . the same , however , is not true for more general @xmath89 and , therefore , @xmath91 is included in ( [ gnng ] ) . our final estimate of @xmath92 is then given as @xmath93 for @xmath33 . to impose the strong or weak heredity principle , we add additional constraints @xmath94 or @xmath95 , respectively . theorem [ thm1 ] can also be extended to more general regression settings . similar to before , under strong heredity , @xmath96 subject to @xmath56 for some @xmath57 . under weak heredity principles , we use the constraints @xmath97 instead of @xmath98 . we shall assume that the following regularity conditions hold : @xmath99 is a strictly convex function of the second argument ; the maximum likelihood estimate @xmath83 is root-@xmath18 consistent ; the observed information matrix converges to a positive definite matrix , that is , @xmath100 where @xmath101 is a positive definite matrix . [ thm2 ] under regularity conditions , if @xmath62 in a fashion such that @xmath102 as @xmath18 goes to @xmath103 , then @xmath65 for any @xmath104 , and @xmath105 if @xmath68 provided that the true model satisfies the same heredity principles . similar to the original nonnegative garrote , the proposed structured variable selection and estimation procedure proceeds in two steps . first the solution path indexed by the tuning parameter @xmath34 is constructed . the second step , oftentimes referred to as tuning , selects the final estimate on the solution path . we begin with linear regression . for both types of heredity principles , the shrinkage factors for a given @xmath34 can be obtained from solving a quadratic program of the following form : @xmath106 where @xmath107 is a @xmath108 matrix determined by the type of heredity principles , @xmath109 is a vector of zeros , and @xmath110 means `` greater than or equal to '' in an element - wise manner . equation ( [ qpsvs ] ) can be solved efficiently using standard quadratic programming techniques , and the solution path of the proposed structured variable selection and estimation procedure can be approximated by solving ( [ qpsvs ] ) for a fine grid of @xmath34 s . recently , yuan and lin ( @xcite ) showed that the solution path of the original nonnegative garrote is piecewise linear , and used this to construct an efficient algorithm for building its whole solution path . the original nonnegative garrote corresponds to the situation where the matrix @xmath107 of ( [ qpsvs ] ) is a @xmath111 identity matrix . similar results can be expected for more general scenarios including the proposed procedures , but the algorithm will become considerably more complicated and running quadratic programming for a grid of tuning parameter tends to be a computationally more efficient alternative . . the objective function of ( [ qpsvs ] ) can be expressed as @xmath113 because @xmath114 does not depend on @xmath115s , ( [ qpsvs ] ) is equivalent to @xmath116\\[-8pt ] & & \qquad \mbox{subject to } \sum _ { j=1}^p \theta_j\le m \mbox { and } h\theta\succeq{\mathbf0},\nonumber\end{aligned}\ ] ] which depends on the sample size @xmath18 only through @xmath117 and the gram matrix @xmath118 . both quantities are already computed in evaluating the least squares . therefore , when compared with the ordinary least squares estimator , the additional computational cost of the proposed estimating procedures is free of sample size @xmath18 . once the solution path is constructed , our final estimate is chosen on the path according to certain criterion . such criterion often reflects the prediction accuracy , which depends on the unknown parameters and needs to be estimated . a commonly used criterion is the multifold cross validation ( cv ) . multifold cv can be used to estimate the prediction error of an estimator . the data @xmath119 are first equally split into @xmath120 subsets @xmath121 . using the proposed method , and data @xmath122 , construct estimate @xmath123 . the cv estimate of the prediction error is @xmath124 we select the tuning parameter @xmath34 by minimizing @xmath125 . it is often suggested to use @xmath126 in practice [ breiman ( @xcite ) ] . it is not hard to see that @xmath127 estimates @xmath128 since the first term is the inherent prediction error due to the noise , one often measures the goodness of an estimator using only the second term , referred to as the model error : @xmath129 clearly , we can estimate the model error as @xmath130 , where @xmath131 is the noise variance estimate obtained from the ordinary least squares estimate using all predictors . similarly for more general regression settings , we solve @xmath132 for some matrix @xmath107 . this can be done in an iterative fashion provided that the loss function @xmath89 is strictly convex in its second argument . at each iteration , denote @xmath133}_0,\theta^{[0]})$ ] the estimate from the previous iteration . we now approximate the objective function using a quadratic function around @xmath133}_0,\theta ^{[0]})$ ] and update the estimate by minimizing @xmath134}+\theta ^{[0]}_0\bigr ) \bigl[\mathbf{z}_i\bigl(\theta-\theta^{[0]}\bigr)+\bigl(\theta_0- \theta^{[0]}_0\bigr ) \bigr ] \\ & & \qquad{}+{\frac{1}{2 } } \ell''\bigl(y_i,\mathbf{z}_i\theta^{[0]}+\theta ^{[0]}_0\bigr ) \bigl[\mathbf{z}_i\bigl(\theta-\theta^{[0]}\bigr)+\bigl(\theta_0- \theta^{[0]}_0\bigr ) \bigr]^2 \biggr ) , \end{aligned}\ ] ] subject to @xmath87 and @xmath135 , where the derivatives are taken with respect to the second argument of @xmath136 . now it becomes a quadratic program . we repeat this until a certain convergence criterion is met . in choosing the optimal tuning parameter @xmath34 for general regression , we again use the multifold cross - validation . it proceeds in the same fashion as before except that we use a loss - dependent cross - validation score : @xmath137 in this section we investigate the finite sample properties of the proposed estimators . to fix ideas , we focus our attention on the usual normal linear regression . we first consider a couple of models that represent different scenarios that may affect the performance of the proposed methods . in each of the following models , we consider three explanatory variables @xmath138 that follow a multivariate normal distribution with @xmath139 with three different values for @xmath140 : @xmath141 and @xmath142 . a quadratic model with nine terms @xmath143 is considered . therefore , we have a total of nine predictors , including three main effects , three quadratic terms and three two - way interactions . to demonstrate the importance of accounting for potential hierarchical structure among the predictor variables , we apply the nonnegative garrote estimator that recognizes strong heredity , weak heredity and without heredity constraints . in particular , we enforce the strong heredity principle by imposing the following constraints : @xmath144 to enforce the weak heredity , we require that @xmath145 we consider two data - generating models , one follows the strong heredity principles and the other follows the weak heredity principles : the first model follows the strong heredity principle : @xmath146 the second model is similar to model i except that the true data generating mechanism now follows the weak heredity principle : @xmath147 for both models , the regression noise @xmath148 . for each model , 50 independent observations of @xmath149 are collected , and a quadratic model with nine terms is analyzed . we choose the tuning parameter by ten - fold cross - validation as described in the last section . following breiman ( @xcite ) , we use the model error ( [ me ] ) as the gold standard in comparing different methods . we repeat the experiment for 1000 times for each model and the results are summarized in table [ tab : ex1 ] . the numbers in the parentheses are the standard errors . we can see that the model errors are smaller for both weak and strong heredity models compared to the model that does not incorporate any of the heredity principles . paired @xmath150-tests confirmed that most of the observed reductions in model error are significant at the 5% level . = 9.2 cm 9.2cm@lccc@ & * no heredity * & * weak heredity * & * strong heredity * + + @xmath151 & 1.79 & 1.70 & 1.59 + & ( 0.05 ) & ( 0.05 ) & ( 0.04 ) + @xmath152 & 1.57 & 1.56 & 1.43 + & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) + @xmath153 & 1.78 & 1.69 & 1.54 + & ( 0.05 ) & ( 0.04 ) & ( 0.04 ) + + @xmath151 & 1.77 & 1.61 & 1.72 + & ( 0.05 ) & ( 0.05 ) & ( 0.04 ) + @xmath152 & 1.79 & 1.53 & 1.70 + & ( 0.05 ) & ( 0.04 ) & ( 0.04 ) + @xmath153 & 1.79 & 1.68 & 1.76 + & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) + for model i , the nonnegative garrote that respects the strong heredity principles enjoys the best performance , followed by the one with weak heredity principles . this example demonstrates the benefit of recognizing the effect heredity . note that the model considered here also follows the weak heredity principle , which explains why the nonnegative garrote estimator with weak heredity outperforms the one that does not enforce any heredity constraints . for model ii , the nonnegative garrote with weak heredity performs the best . interestingly , the nonnegative garrote with strong heredity performs better than the original nonnegative garrote . one possible explanation is that the reduced feasible region with strong heredity , although introducing bias , at the same time makes tuning easier . to gain further insight , we look into the model selection ability of the structured variable selection . to separate the strength of a method and effect of tuning , for each of the simulated data , we check whether or not there is any tuning parameter such that the corresponding estimate conforms with the true model . the frequency for each method to select the right model is given in table [ tab : ex1f ] , which clearly shows that the proposed structured variable selection methods pick the right models more often than the original nonnegative garrote . note that the strong heredity version of the method can never pick model ii correctly as it violates the strong heredity principle . we also want to point out that such comparison , although useful , needs to be understood with caution . in practice , no model is perfect and selecting an additional main effect @xmath154 so that model ii can satisfy strong heredity may be a much more preferable alternative to many . @lccc@ & * no heredity * & * weak heredity * & * strong heredity * + + @xmath151 & 65.5% & 71.5% & 82.0% + @xmath152 & 85.0% & 86.5% & 90.5% + @xmath153 & 66.5% & 73.5% & 81.5% + + @xmath151 & 65.5% & 75.5% & 0.00% + @xmath152 & 83.0% & 90.0% & 0.00% + @xmath153 & 56.5% & 72.5% & 0.00% + we also checked how effective the ten - fold cross - validation is in picking the right model when it does not follow any of the heredity principles . we generated the data from the model @xmath155 where the set up for simulation remains the same as before . note that this model does not follow any of the heredity principles . for each run , we ran the nonnegative garrote with weak heredity , strong heredity and no heredity . we chose the best among these three estimators using ten - fold cross - validation . note that the three estimators may take different values of the tuning parameter . among 1000 runs , 64.1% of the time , nonnegative garrote with no heredity principle was elected . in contrast , for either model i or model ii with a similar setup , less than 10% of the time nonnegative garrote with no heredity principle was elected . this is quite a satisfactory performance . the next example is designed to illustrate the effect of the magnitude of the interaction on the proposed methods . we use a similar setup as before but now with four main effects @xmath156 , four quadratic terms and six two - way interactions . the true data generating mechanism is given by @xmath157 where @xmath158 and @xmath159 with @xmath160 chosen so that the signal - to - noise ratio is always @xmath161 . similar to before , the sample size @xmath162 . figure [ fig : revsim1 ] shows the mean model error estimated over 1000 runs . we can see that the strong and weak heredity models perform better than the no heredity model and the improvement becomes more significant as the strength of the interaction effect increases . to fix the idea , we have focused on using the least squares estimator as our initial estimator . the least squares estimators are known to perform poorly when the number of predictors is large when compared with the sample size . in particular , it is not applicable when the number of predictors exceeds the sample size . however , as shown in yuan and lin ( @xcite ) , other initial estimators can also be used . in particular , they suggested ridge regression as one of the alternatives to the least squares estimator . to demonstrate such an extension , we consider again the regression model ( [ eq : effectsize ] ) but with ten main effects @xmath163 and ten quadratic terms , as well as 45 interactions . the total number of effects ( @xmath164 ) exceeds the number of observations ( @xmath162 ) and , therefore , the ridge regression tuned with gcv was used as the initial estimator . figure [ fig : aoasrevsim ] shows the solution path of the nonnegative garrote with strong heredity , weak heredity and without any heredity for a typical simulated data with @xmath165 . : solution for different versions of the nonnegative garrote . ] it is interesting to notice from figure [ fig : aoasrevsim ] that the appropriate heredity principle , in this case strong heredity , is extremely valuable in distinguishing the true effect @xmath166 from other spurious effects . this further confirms the importance of heredity principles . in this section we apply the methods from section [ sec2 ] to several real data examples . the first is the prostate data , previously used in tibshirani ( @xcite ) . the data consist of the medical records of @xmath167 male patients who were about to receive a radical prostatectomy . the response is the level of prostate specific antigen , and there are 8 explanatory variables . the explanatory variables are eight clinical measures : log(cancer volume ) ( lcavol ) , log(prostate weight ) ( lweight ) , age , log(benign prostatic hyperplasia amount ) ( lbph ) , seminal vesicle invasion ( svi ) , log(capsular penetration ) ( lcp ) , gleason score ( gleason ) and percentage gleason scores 4 or 5 ( pgg45 ) . we consider model ( [ 2way ] ) with main effects , quadratic terms and two way interactions , which gives us a total of 44 predictors . figure [ fig : prostate ] gives the solution path of the nonnegative garrote with strong heredity , weak heredity and without any heredity constraints . the vertical grey lines represent the models that are selected by the ten - fold cross - validation . to determine which type of heredity principle to use for the analysis , we calculated the ten - fold cross - validation scores for each method . the cross - validation scores as functions of the tuning parameter @xmath34 are given in the right panel of figure [ fig : prostate - cv ] . cross - validation suggests the validity of heredity principles . the strong heredity is particularly favored with the smallest score . using ten - fold cross - validation , the original nonnegative garrote that neglects the effect heredity chooses a six variable model : _ lcavol , lweight , lbph , gleason@xmath168 , lbph : svi _ and _ svi : pgg45_. note that this model does not satisfy heredity principle , because _ gleason@xmath168 _ and _ svi : pgg45 _ are included without any of its parent factors . in contrast , the nonnegative garrote with strong heredity selects a model with seven variables : _ lcavol , lweight , lbph , svi , gleason , gleason@xmath168 _ and _ lbph : svi_. the model selected by the weak heredity , although comparable in terms of cross validation score , is considerably bigger with 16 variables . the estimated model errors for the strong heredity , weak heredity and no heredity nonnegative garrote are @xmath169 , @xmath170 and @xmath171 , respectively , which clearly favors the methods that account for the effect heredity . to further assess the variability of the ten - fold cross - validation , we also ran the leave - one - out cross - validation on the data . the leave - one - out scores are given in the right panel of figure [ fig : prostate - cv ] . it shows a similar pattern as the ten - fold cross - validation . in what follows , we shall continue to use the ten - fold cross - validation because of the tremendous computational advantage it brings about . to illustrate the strategy in more general regression settings , we consider a logistic regression for the south african heart disease data previously used in hastie , tibshirani and friedman ( @xcite ) . the data consist of 9 different measures of 462 subjects and the responses indicating the presence of heart disease . we again consider a quadratic model . there is one binary predictor which leaves a total of 53 terms . nonnegative garrote with strong heredity , weak heredity and without heredity were applied to the data set . the solution paths are given in figure [ fig : heart - path ] . the cross - validation scores for the three different methods are given in figure [ fig : heart - cv ] . as we can see from the figure , nonnegative garrote with strong heredity principles achieves the lowest cross - validation score , followed by the one without heredity principles . to gain further insight on the merits of the proposed structured variable selection and estimation techniques , we apply them to seven benchmark data sets , including the previous two examples . the ozone data , originally used in breiman and friedman ( @xcite ) , consist of the daily maximum one - hour - average ozone reading and eight meteorological variables in the los angeles basin for 330 days in 1976 . the goal is to predict the daily maximum one - hour - average ozone reading using the other eight variables . the boston housing data include statistics for 506 census tracts of boston from the 1970 census [ harrison and rubinfeld ( @xcite ) ] . the problem is to predict the median value of owner - occupied homes based on 13 demographic and geological measures . the diabetes data , previously analyzed by efron et al . ( @xcite ) , consist of eleven clinical measurements for a total of 442 diabetes patients . the goal is to predict a quantitative measure of disease progression one year after the baseline using the other ten measures that were collected at the baseline . along with the prostate data , these data sets are used to demonstrate our methods in the usual normal linear regression setting . to illustrate the performance of the structured variable selection and estimation in more general regression settings , we include two other logistic regression examples along with the south african heart data . the pima indians diabetes data have 392 observations on nine variables . the purpose is to predict whether or not a particular subject has diabetes using eight remaining variables . the bupa liver disorder data include eight variables and the goal is to relate a binary response with seven clinical measurements . both data sets are available from the uci repository of machine learning databases [ newman et al . ( @xcite ) ] . we consider methods that do not incorporate heredity principles or respect weak or strong heredity principles . for each method , we estimate the prediction error using ten - fold cross - validation , that is , the mean squared error in the case of the four linear regression examples , and the misclassification rate in the case of the three classification examples . table [ tab : real ] documents the findings . similar to the heart data we discussed earlier , the total number of effects ( @xmath22 ) can be different for the same number of main effects ( @xmath1 ) due to the existence of binary variables . as the results from table [ tab : real ] suggest , incorporating the heredity principles leads to improved prediction for all seven data sets . note that for the four regression data sets , the prediction error depends on the scale of the response and therefore should not be compared across data sets . for example , the response variable of the diabetes data ranges from 25 to 346 with a variance of 5943.331 . in contrast , the response variable of the prostate data ranges from @xmath172 to 5.48 with a variance of 1.46 . @ld3.0d2.0d3.0d4.3cc@ * data * & & & & & & + boston & 506 & 13 & 103 & 12.609 & & 12.661 + diabetes & 442 & 10 & 64 & 3077.471 & & 3116.989 + ozone & 203 & 9 & 54 & 16.558 & & 15.397 + prostate & 97 & 8 & 44 & 0.624 & 0.632 & * 0.584 * + bupa & 345 & 6 & 27 & 0.287 & 0.279 & * 0.267 * + heart & 462 & 9 & 53 & 0.286 & 0.275 & * 0.262 * + pima & 392 & 8 & 44 & 0.199 & 0.214 & * 0.196 * + when a large number of variables are entertained , variable selection becomes important . with a number of competing models that are virtually indistinguishable in fitting the data , it is often advocated to select a model with the smaller number of variables . but this principle alone may lead to models that are not interpretable . in this paper we proposed structured variable selection and estimation methods that can effectively incorporate the hierarchical structure among the predictors in variable selection and regression coefficient estimation . the proposed methods select models that satisfy the heredity principle and are much more interpretable in practice . the proposed methods adopt the idea of the nonnegative garrote and inherit its advantages . they are easy to compute and enjoy good theoretical properties . similar to the original nonnegative garrote , the proposed method involves the choice of a tuning parameter which also amounts to the selection of a final model . throughout the paper , we have focused on using the cross - validation for such a purpose . other tuning methods could also be used . in particular , it is known that prediction - based tuning may result in unnecessarily large models . several heuristic methods are often adopted in practice to alleviate such problems . one of the most popular choices is the so - called one standard error rule [ breiman et al . ( @xcite ) ] , where instead of choosing the model that minimizes the cross - validation score , one chooses the simplest model with a cross - validation score within one standard error from the smallest . our experience also suggests that a visual examination of the solution path and the cross - validation scores often leads to further insights . the proposed method can also be used in other statistical problems whenever the structures among predictors should be respected in model building . in some applications , certain predictor variables may be known apriori to be more important than the others . this may happen , for example , in time series prediction where more recent observations generally should be more predictive of future observations . boyd , s. and vandenberghe , l. ( 2004 ) . _ convex optimization_. cambridge univ . press , cambridge . breiman , l. 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( 2004 ) . least angle regression ( with discussion ) . statist . _ * 32 * 407499 . fan , j. and li , r. ( 2001 ) . variable selection via nonconcave penalized likelihood and its oracle properties . _ j. amer . assoc . _ * 96 * 13481360 . george , e. i. and mcculloch , r. e. ( 1993 ) . variable selection via gibbs sampling . _ j. amer . statist . assoc . _ * 88 * 881889 . hamada , m. and wu , c. f. j. ( 1992 ) . analysis of designed experiments with complex aliasing . _ journal of quality technology _ * 24 * 130137 . harrison , d. and rubinfeld , d. ( 1978 ) . hedonic prices and the demand for clean air . _ journal of environmental economics and management _ * 5 * 81102 . hastie , t. , tibshirani , r. and friedman , j. ( 2003 ) . _ the elements of statistical learning : data mining , inference , and prediction_. springer , new york . joseph , v. r. and delaney , j. d. 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in linear regression problems with related predictors , it is desirable to do variable selection and estimation by maintaining the hierarchical or structural relationships among predictors . in this paper
we propose non - negative garrote methods that can naturally incorporate such relationships defined through effect heredity principles or marginality principles .
we show that the methods are very easy to compute and enjoy nice theoretical properties .
we also show that the methods can be easily extended to deal with more general regression problems such as generalized linear models .
simulations and real examples are used to illustrate the merits of the proposed methods . , .
| 12,135 | 131 |
long gamma - ray bursts ( lgrbs , see the reviews by * ? ? ? * ; * ? ? ? * ) are energetic radiation events , lasting between 2 and @xmath21000 seconds , and with photon energies in the range of kev mev . our current understanding of these sources indicates that the emission is produced during the collapse of massive stars , when the recently formed black hole accretes the debris of the stellar core . during the accretion , highly collimated ultrarelativistic jets consisting mainly of an expanding plasma of leptons and photons ( fireball ) are launched , which drill the stellar envelope . internal shocks in the fireball accelerate leptons and produce the @xmath3-ray radiation through synchrotron and inverse compton processes . external shocks from the interaction of the jets with the interstellar medium produce later emission at lower energies , from x - rays to radio ( afterglow ) . optical afterglow spectra allowed the measurement of lgrb redshifts @xcite , locating these sources at cosmological distances ( @xmath4 ) , and revealing that their energetics is similar to that of supernovae ( sne ) . some lgrbs have indeed been observed to be associated to hydrogen - deficient , type ib / c supernovae ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? afterglows allowed also the identification of lgrb host galaxies ( hgs ) , which turned out to be mostly low - mass , blue and subluminous galaxies with active star formation @xcite . although the general picture is clear enough , its details are still a matter of discussion . among other unanswered questions , the exact nature of the lgrb stellar progenitors is still being debated . stellar evolution models provide a rough picture of the production of a lgrb in a massive star . according to the _ collapsar _ model @xcite , lgrbs are produced during the collapse of single wolf - rayet ( wr ) stars . wr stars have massive cores that may collapse into black holes , and are fast rotators , a condition needed to support an accretion disc and launch the collimated jets . wrs have also large mass - loss rates , needed to lose their hydrogen envelope before collapsing , that would otherwise brake the lgrb jet . this model agrees with the observed association between lgrbs and hydrogen deficient sne . however , wrs large mass - loss rates imply large angular momentum losses that would brake their cores , which would inhibit the production of the lgrb . to overcome this problem , @xcite proposed low - metallicity wrs ( wos ) as progenitors . wos have lower mass - loss rates , diminishing the braking effect , but also preventing the loss of the envelope . another possibility was proposed by @xcite . according to these authors , low - metallicity , rapidly rotating massive stars evolve in a chemically homogeneous way , hence burning the hydrogen envelope , instead of losing it . low - metallicity progenitor models are consistent with different pieces of evidence . first , the works of @xcite and @xcite show that the collapse of high - metallicity stars produces mainly neutron stars , while those of low - metallicity stars form black holes . second , lgrb hgs have been found to be low - metallicity systems @xcite . finally , the analysis of the statistical properties of the population of lgrbs suggests that their cosmic production rate should increase with respect to the cosmic star formation rate at high redshift , which could be explained as an effect of the low metallicity of the progenitors , combined with the cosmic metallicity evolution @xcite . another possibility for wr to lose the envelope without losing too much angular momentum is to be part of binary systems as proposed by @xcite . understanding the nature of lgrb progenitors is beyond the interest of only stellar evolution , black hole formation , and high energy astrophysics . the visibility of lgrbs up to very high redshifts ( @xmath5 ) , allows their use as tools to explore star formation and galaxy evolution in the early universe . on the other hand , observations of the environment and hgs of lgrbs could reveal important clues about the progenitors of these phenomena . given that star formation shifts outward within a galaxy due to the depletion of gas in the central regions as the galaxy evolves , that the interstellar medium of galaxies is not chemically homogeneous , and that the chemical enrichment is affected by variations of the star formation rate and the production of different types of sne , it is expected that both the lgrb positions within a galaxy and the chemical properties of the environment in which lgrbs occur depend on redshift and on the metallicity of the lgrb progenitors . using high - precision astrometry , @xcite and @xcite have measured the positions of @xmath235 lgrbs with respect to the centres of their hosts , supporting the collapsar model against the ( now disproved ) neutron star merger model . the question of the metallicity dependence of lgrb progenitors could also be investigated comparing these data with model predictions . the chemical abundances of lgrb circumburst and hg environments were investigated by several authors @xcite . however , only in a few cases of low - redshift bursts a direct measure of the metallicity of the star - forming region that produced the lgrb is available . at intermediate redshift observers usually measure the mean hg metallicity , while at high redshift they must resort to grb - dla techniques , which give the metallicity of galactic clouds intercepting the line of sight to the lgrb , but not necessarily associated with the burst itself @xcite . in this paper , we use cosmological hydrodynamical simulations which include star formation and sn feedback to investigate the predictions of different progenitor scenarios regarding the positions of lgrbs and the chemical abundances of their environment . since galaxy formation is a highly non - linear process , cosmological numerical simulations @xcite are the best tools to investigate these lgrb properties . in the past , this method has been used by several authors to investigate different aspects of the lgrb environment and hgs . @xcite have shown that requiring hgs to have high star formation efficiency , the observed hg luminosity function can be reproduced . @xcite developed a monte carlo simulation to synthesize lgrb and hg populations in hydrodynamical simulations of galaxy formation , in the framework of the collapsar model . they have found that a bias to low - metallicity progenitors ( @xmath6 ) is needed to explain the observed properties of hgs . @xcite and @xcite used semi - analytical models of galaxy formation to study the properties of hg populations . particularly @xcite developed a new approach to model the detectability of the lgrbs . both teams explored models with mass and metallicity cut - offs for lgrb progenitors , finding that models with a metallicity cut - off could explain the hg properties , and hence supporting previous claims that lgrbs are biased tracers of star formation . however in these semi - analytical models , the spatial distribution of individual stellar populations within hgs can not be investigated . the chemical abundances of lgrb - dlas were investigated using numerical simulations by @xcite , finding that the clouds producing the absorption lie at galactocentric distances of the order of 1 kpc . in this work , we use cosmological numerical simulations of galaxy formation to construct synthetic lgrb populations , which allow us to investigate the properties of individual stellar populations within hgs . our simulations are similar to those of @xcite , but with a higher resolution , and include the effects of the energy feedback of sne into the interstellar medium . the metallicities of each stellar populations can be estimated , and used to construct different metallicity - dependent scenarios for lgrb production within the collapsar model . as stated by @xcite , the detectability of lgrbs and their hgs is an important aspect that should not be disregarded for a proper comparison with the observed samples , hence we included it in our population synthesis in the same way as these authors . this work is organized as follows . in sections [ sim ] and [ mod ] we describe the cosmological simulations of galaxy formation used , and our lgrb population synthesis models , respectively . we present our results and compare the to available observational data in section [ res ] . finally , in section [ con ] we present our conclusions . we analyse hydrodynamical cosmological simulations performed with a version of gadget-3 which includes star formation , metal - dependent cooling , chemical enrichment , multiphase gas and supernova feedback ( for further details see * ? ? ? * ; * ? ? ? * ) the simulated regions represent periodic volumes of 10 mpc @xmath7 side and are consistent with a @xmath8-cdm universe with the following cosmological parameters : @xmath9=0.7 , @xmath10=0.3 , @xmath11=0.04 , @xmath12=0.9 and @xmath13=100 @xmath14 where @xmath15=0.7 . the feedback model considers type ii and type ia supernovae ( snii and snia , respectively ) . the energy per sn event released into the interstellar medium is @xmath16 . the model assumes that stars with masses greater than @xmath17 end their life as snii with lifetime @xmath18yr . lifetimes for the progenitors of snia are randomly selected in the range @xmath19 gyr . the chemical yields for snii are given by @xcite while those of snia correspond to the w7 model of @xcite . initially gas particles are assumed to have primordial abundances of x@xmath20=0.76 and x@xmath21=0.24 . the chemical algorithm follows the enrichment by 12 isotopes : @xmath22h , @xmath23he , @xmath24c , @xmath25o , @xmath26 mg , @xmath27si , @xmath28fe , @xmath29n , @xmath30ne , @xmath31s , @xmath32ca and @xmath33zn @xcite . we would like to stress that this model has proven to be successfull at regulating the star formation activity and at driving powerful mass - loaded galactic winds without the need to introduce mass - depend parameters @xcite . we analyse two simulations : s230 and s320 , which have been also used by @xcite to study the tully - fisher relation obtaining very good agreement with observations . s230 has initially @xmath34 with dark matter masses of @xmath35 and initial gas mass of @xmath36 . s320 initially has @xmath37 with dark matter of @xmath38 and initial gas mass of @xmath39 . s320 was only run to @xmath40 because of lack of computational power . we use this simulation to assess possible numerical resolution problems . from the general mass distribution , we select virialized structures by using the friends - of - friends technique and then identify all substructures within the virial radius by applying the subfind algorithm @xcite . we select as simulated galaxies those substructures sampled with more than 3000 particles . @xcite found that the mass - metallicity relation ( mzr ) of galaxies in these simulations differs at low redshifts from that reported by @xcite so that galaxies have lower mean metallicity than observed although the shape of the observed mzr is very well reproduced . because of this , we renormalized the simulated abundances to make them consistent with observations . for that purpose , we adopted the results of @xcite who proposed a model to describe the evolution of observed mzr which matched available observations at z=0.07 , 0.7 and 2.2 . with this adjustement , our simulated mzr reproduce observations at different redshifts . for illustration purposes , we show our analysis at four redshifts : @xmath41 and @xmath42 . to construct synthetic lgrb populations from the stellar populations described by the simulations we adopt the collapsar model , in which lgrb progenitors are massive stars possibly with low metallicity . we investigate four scenarios in which progenitors have a mass greater than a certain minimum @xmath43 . for scenario 1 this is the only condition , while for the others a maximum metallicity @xmath44 is assumed for the progenitors . the values of @xmath43 and @xmath44 were taken from @xcite , who derived them by fitting the lgrb rate observed by batse , and are listed in table [ scenarios2 ] . we estimate the number of massive stars in each simulated galaxy at each analysed redshift by assuming a initial mass function given by @xcite . we included all stars born within @xmath45 . this time interval is larger than the mean lifetime of snii progenitors but it allow us to minimize numerical fluctuations and it is small compared to sfr variations . the selected progenitors defined the scenario 1 . for scenarios 2 , 3 , and 4 we impose a requirement on the mean metallicity , considering only new born stars with @xmath46 , respectively . following @xcite , for each stellar population represented by a particle @xmath47 with mass @xmath48 at redshift @xmath49 satisfying the above requirements , we calculated the number of lgrbs produced as the number of stars with @xmath50 , @xmath51 where @xmath52 is the initial mass function with @xmath53 and @xmath54 its lower and upper mass cut - offs , respectively . .properties of the four scenarios proposed for lgrbs . the values of minimal masses where taken from @xcite with the initial mass function of @xcite . [ cols="<,<,>,>,>,>,<,>,<,>",options="header " , ] the intrinsic lgrb rate for a stellar population in any scenario is then @xmath55 we are interested in computing observable properties of the stellar populations selected by lgrb observations , such as metallicites , @xmath1-elements abundances , and distances to their hg centre . as discussed by @xcite , selection effects introduced by observations can be modeled by weighting the properties of simulated stellar populations by their contribution to the total observed lgrb rate at the earth . we applied the method developed by @xcite to estimate the probability that a certain lgrb produced at a given @xmath49 could be observed at earth . however , there might be other biases introduced by observations which are difficult to model because of their dependence on sensitivity and spectral bands of the detectors and telescopes . particularly , the afterglow observations , on which the precise positioning of lgrbs is based , are usually made in the optical range and could be affected by dust absoption , biasing the samples towards low metallicity systems . as claimed by @xcite , about 40 per cent of lgrbs might be dust obscured . dust effects have not been included in our scenarios hence , caution should be taken when comparing our results with observations . we will point out posible dust effects when appropriate . at fixed redshift , the weights depend only on the intrinsic lgrb rate of the corresponding stellar population , becoming @xmath56 where the sum extends over all the stellar populations @xmath57 producing lgrbs at a given redshift . for any observable property @xmath58 of these stellar populations , its mean observed value at @xmath49 must then be @xmath59 we first investigate the spatial distribution of lgrbs in our scenarios . for this purpose , we calculate the distance between the lgrb and the centre of mass of its galaxy @xmath60 . to eliminate the effects produced by the growth of galaxies as the structure in the universe assembles , we normalize @xmath60 by taking the ratio @xmath61 , where @xmath62 is the optical radius of the galaxy , defined as the radius encompassing 83 per cent of its baryonic mass @xcite . in the fig . [ bnorm ] we present the distribution of @xmath61 for the lgrbs at different redshifts ( @xmath63 ) , weighted by their detectability as explained in section [ mod ] . we observe that lgrb progenitors tend to reside in the inner regions galaxies at high redshift , and to be progressively located at larger distances from the centre as redshift decreases . this is consistent with the fact that the main sites of star formation shift outwards as time evolves and the galactic structure gets assembled in a hierarchical fashion . this effect is stronger in our scenarios with higher @xmath64 , because low metallicity populations tend to be formed in the outer regions of galaxies which are less enriched since all simulated systems exhibit metallicity gradients . ( solid lines ) , 2 ( dashed lines ) , 1 ( dotted line ) , and 0 ( dot - dashed lines).,scaledwidth=45.0% ] in fig . [ bmedian ] , we plot the median values of @xmath61 in our scenarios as a function of redshift , together with the available observations of the lgrbs positions within their hosts @xcite . these authors measure the distance of lgrb to the centre of their hosts , projected onto the plane of the sky , and normalised by the galaxy half - light radius @xmath65 . as these authors point out , this normalisation is a crude way of deprojecting the values of @xmath60 . to transform them into @xmath61 , we assume that lgrb hosts can be modeled by an exponential disc , for which @xmath66 . [ bmedian ] shows that our scenarios are consistent with observations , except at very low redshifts in which the observed median value of @xmath61 drops abruptly , while our scenarios remain almost constant . by analysing the lgrbs contributing to the lowest-@xmath49 point in fig . [ bmedian ] , we find that almost half of them ( 3 out of 7 ) have @xmath61 values consistent with zero . interestingly , the hosts of these lgrbs show evidence of interaction or close companions . hence , the presence of nuclear star formation activity could be explained as triggered by galaxy interactions as suggested by observations @xcite and numerical simulations @xcite . then , the discrepancy can be attributed to the fact that our simulated galaxy sample does not reflect the effects of this mode of star formation at low redshift since our analysed galaxies are dominated by systems with low gas reservoir @xcite . a further piece of evidence for this explanation is provided by a recalculation of the lowest-@xmath49 point , excluding the three quoted lgrbs ( filled circle in fig . [ bmedian ] ) . the new point lies within @xmath67 of our scenarios , showing a better agreement than the original one . the large error bars of the observations , which originate in the low number of lgrbs with precise positions , prevent us from using a goodness - of - fit estimator to determine the scenario that better fits the observations . however the fact that the observed values are always higher than the predictions of scenarios sc1 and sc2 implies that it is very improbable that these scenarios could explain the observations . hence our results suggest that lgrb progenitors would have low metallicities ( @xmath68 ) . in the case that dust effects introduce important biases in the impact parameter distribution , the preference for low metallicity progenitors obtained from fig . [ bmedian ] would have to be re - considered . observations providing new insights on the location of dark grbs may help to resolve this issue . to analyse the chemical abundances of the lgrb progenitors in our scenarios , we use the ratio [ fe / h ] as a measure of the iron abundance , and [ si / fe ] as a measure of the relative abundance of @xmath1 elements to iron . in fig . [ feh ] we present the distribution of [ fe / h ] for the lgrb progenitors in our scenarios at different redshifts , wheighed by the detectability in the same way as in the previous section . we observe that the abundance of iron increases as redshift decreases in all scenarios . this can be understood in terms of the chemical evolution of the interstellar medium . as time evolves sne contribute to the enrichment of the medium with iron , hence stellar populations born at low redshifts exhibit higher iron abundances . this enrichment is stronger in scenarios sc1 and sc2 , where the metallicity cut - offs are not so restrictive . ( solid line ) , 2 ( dashed line ) , 1 ( dotted line ) , and 0 ( dot - dashed line).,scaledwidth=45.0% ] in fig . [ cum_sife ] we present the distribution of [ si / fe ] for lgrb progenitors . we observe that , for all our scenarios , they exibit a higher [ si / fe ] as redshift increases , indicating an enhacement in @xmath1 elements at high redshifts . this is consistent with the fact that snia and snii enrich the interstellar medium . due to the fact that snia progenitors have lifetimes @xmath69 while those of snii live only @xmath70 , at high redshift only the contribution of the latter to the interstelar medium enrichment is significant , rendering stellar populations rich in @xmath1 elements . as redshift decreases , the contribution of snia becomes important , decreasing the abundance of @xmath1 elements relative to iron . the trend of [ si / fe ] to decrease towards lower redshift is also observed in fig.[medians_sife_feh ] ( right panel ) , where we plot the median value of the ratio [ si / fe ] as a function of redshift . in this figure we also observe that the values of [ si / fe ] of the lgrb progenitors is lower and evolve more strongly with redshift in the scenarios where the metallicity cut - off is more restrictive . in the left panel of fig . [ medians_sife_feh ] we show the median value of the ratio [ fe / h ] as a function of redshift . we find that the median value of [ fe / h ] decreases with @xmath44 as expected for cut - offs progressively more restrictive in metallicity . these results indicate that the metallicity cut - off tends to eliminate old stellar populations highly enriched by snii , located mainly in the central regions of galaxies and originated in first outbreaks of star formation . this interpretation agrees with the shift of the normalized impact parameters of lgrb progenitors observed in fig . [ bnorm ] . for @xmath71 ( solid line ) , 2 ( dashed line ) , 1 ( dotted line ) , and 0 ( dot - dashed line ) in all scenarios proposed.,scaledwidth=45.0% ] aiming at understanding the relation between lgrbs and star formation , we analysed the spatial distribution and chemical abundances of stellar populations producing these phenomena . we investigated four different scenarios for the progenitors of lgrbs based on the collapsar model with different metallicity cut - offs . we compared the spatial distribution of lgrbs within their hgs in our scenarios and with the available observations . we found that in all our scenarios lgrb progenitors reside on average in the outer regions of their galaxies at low redshifts , shifting toward the centre as redshift increases . scenarios favouring low metallicity progenitors tend to produce lgrbs further out from the central regions than those allowing high metallicity progenitors . the confrontation of our models with available observations supports scenarios with low metallicity cut - offs , in agreement with previous results @xcite . particulary we best reproduce current available observations for a model where lgrb progenitors are massive stars with @xmath72 . further precise lgrb position measurements would help to confirm these trends . regarding [ fe / h ] abundances of the stellar populations producing lgrbs , we found that in all our scenarios [ fe / h ] increases as redshift decreases . this effect is less conspicuous in the scenarios with low metallicity progenitors , as in these cases the metallicity cut - off restricts the chemical abundances of the stellar populations producing lgrbs . the @xmath1-enhancement decreases with redshift in all our scenarios , as a result of the different contributions of snii and snia . contrary to the detected trend in [ fe / h ] , the @xmath1-enhancement shows a stronger evolution with redshift as @xmath44 decreases . as previously discussed , these chemical trends can be understood within the context of chemical evolution in hierarchical clustering scenarios . considering that the results on the spatial distribution of lgrb progenitors favours low - metallicity progenitor models , one would expect that the iron abundance of the stellar populations producing lgrbs remains low at all redshifts with little variations ( @xmath73}\sim-1 $ ] ) . on the other hand , one would expect that the @xmath1-enhancement strongly decreases with redshift ( by 0.2 dex between @xmath71 and @xmath74 ) . this means that , if lgrbs are produced by low metallicity massive stars , their location will be shifted on average from the central regions to the outskirts of galaxies . if lgrbs can trace the chemical properties of the interestelar medium , they may map different regions of galaxies at different redshifts . a test of these prediction could be set up as further dust - corrected measurements of the chemical abundances of the stellar populations producing lgrbs become available . ljp acknowledges funding by argentine anpcyt , through grant pict 2006 - 02015 and 2007 - 00848 . this work was partially supported by pict 2005 - 32342 , pict 2006 - 245 max planck and pip 2009 - 0305 .
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we analyse the spatial distribution within host galaxies and chemical properties of the progenitors of long gamma ray bursts as a function of redshift . by using hydrodynamical cosmological simulations which include star formation , supernova feedback and chemical enrichment and based on the hypothesis of the collapsar model with low metallicity
, we investigate the progenitors in the range @xmath0 .
our results suggest that the sites of these phenomena tend to be located in the central regions of the hosts at high redshifts but move outwards for lower ones .
we find that scenarios with low metallicity cut - offs best fit current observations .
for these scenarios long gamma ray bursts tend to be [ fe / h ] poor and show a strong @xmath1-enhancement evolution towards lower values as redshift decreases .
the variation of typical burst sites with redshift would imply that they might be tracing different part of galaxies at different redshifts .
[ firstpage ] gamma - rays : bursts galaxies : abundances , evolution
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since the beginning of accelerator physics , mass spectroscopy has been playing a leading role in the discovery of particle and resonance states , and understanding of the fundamental interactions in the standard model ( sm ) . for example , the first signals of charm and bottom quarks were in fact detected through the formation of @xmath0 and @xmath1 bound states . on the other hand , current colliders like the lhc , or the ilc in a farther future , will likely continue this discovery program beyond the sm . it is conceivable that new ( super ) heavy bound states can be formed and , contrary to e.g. the toponium system , their basic constituents are prevented from decaying before the binding is effective . the goal of this paper is to perfom a prospective study of the spectroscopy of such exotic massive states , by making several reasonable assumptions about the interacting potential among the new - physics constituents which may differ from standard qcd . furthermore , we will estimate leptonic decay widths of very heavy bound states by making specific assumptions on the quantum numbers of constituents , although not in a comprehensive way . during the last years , minimal extensions of the sm containing additional heavy particles charged under a new unbroken non - abelian gauge group @xmath2 with fermions _ q,@xmath3 _ have been proposed under the general name of @xmath4hidden valley models @xcite , which is a very general scenario containing such heavy particles but as well new sectors of lightweight particles to be observed . in these models all sm particles would be neutral under such the new @xmath2 group , while new particles charged under @xmath2 but neutral under the sm group would show up if large energy scales are probed . higher dimension operators , induced e.g. by a @xmath5 or a loop involving heavy particles carrying both @xmath6 and @xmath2 charges , should connect both sm and new physics sectors through rather weak interactions . in particular , if the @xmath2 group corresponds to @xmath7 , the fermions in the fundamental representation have been recently named as @xmath4quirks " @xcite or iquarks @xcite . @xcite actually this theory can be viewed as a certain limit of qcd where light quarks are removed and the typical scale @xmath8 where the new interaction becomes strong is much smaller than the heavy flavor masses . more generally , such kind of scenario can be put in correspondence with a class of @xmath4hidden valley models as pointed out in @xcite . in these models , quirks are defined to be new massive fermions transforming under both @xmath6 and a general ( not only @xmath9 ) non - abelian gauge group @xmath2 . it has been considered so far in the literature that @xmath8 is smaller than @xmath10 , but as well the particular case of hidden valleys with quirks in which @xmath8 is greater than @xmath10 has been studied in the literature @xcite,@xcite,@xcite . the name of _ infracolour _ is used to design of the new gluonic degrees of freedom when @xmath8 is much smaller than the quirk mass ; in this work we will refer to it as i - colour ( i - qcd ) in correspondence with the name quirk . in this section we will focus on the bound states of quirks . the phenomenology of such bound states has first studied in @xcite , and later an analysis on the spectroscopy of these systems was done by the authors of @xcite . + it is well - known that the large mass of the top quark in the sm prevents toponium to be formed since the constituents quarks would decay away too fast . a criterion for existence of such bound states is that the binding energy should be larger than the total decay width . for heavy onia states beyond the sm , however , the situation could be different . in particular , in the case of quirks @xcite,@xcite,@xcite , its decay is prevented from the conservation of a quantum number . regarding the dynamics of these quirks , according to @xcite , one can distinguish among three possible energy scales : the first one is the @xmath11 range , where the quirk strings are macroscopic ; the mesoscopic strings can be find at the range @xmath12 ( which is large compared to the atomic scales ) ; and finally we find the microscopic scale at @xmath13 . + on the other hand , assuming that the scale of i - colour is below the weak scale , bound states of the new sector are kinematically accessible to present and future collider experiments . however , since the sm particles are uncharged under i - colour , ( quirk ) loops would be required to couple both sectors leading to highly supressed production rates . moreover , from reference @xcite , quirks are defined to be charged under ( some ) gauge groups of the sm ( for example , charged under the electroweak interaction ) quirks could be pair produced through electroweak processes . + likewise , in ref.@xcite quirks are considered as vectors with respect to the electroweak gauge group without carrying qcd colour , but carrying i - colour charge . therefore notice that there is no yukawa coupling between the higgs boson and quirks . thus we discard the possible binding force which has been postulated for ultraheavy quarkonium taking over gluon - exchange . at this point , for the sake of clarity , it must be stressed that , taking simple assumptions , through this work we will focus on the case of uncoloured quirks . + in this scenario , quirks can be copiously pair - produced at the lhc not through qcd interactions but via electromagnetic and electroweak interactions . as quirks would be long - lived particles as compared to the collider / detection time scale , different detection strategies can be undertaken according to the possible aforementioned micro , meso or macroscopic regimes . + finally , notice that possible quirky signals of folded supersymmetry in colliders have been studied in @xcite , focusing on the scalar quirks ( _ squirks _ ) . contrary to usual supersymmetric partners of quarks , squirks ( choosing a simple scenario ) are expected to be uncolored , but instead charged under a new confining group , equivalent to i - qcd as introduced above . the study of spectroscopy performed in this paper is actually not sensitive to possible scalar nature of new fermions , and the main lines could be applicable to squirkonium as well . according to @xcite , when the quirk pair is produced an excited bound state can be formed with invariant mass given approximately by the total center of momentum energy of the hard partonic scattering giving raise to the pair . in the microscopic regime this bound state would loose energy by emitting i - glueballs and bremsstrahlung towards low - lying states . once they loose most of their kinetic energy these bound states ( to be dubbed quirkonium ) could decay via electroweak interaction . however , it is also possible other scenario in which the neutral and colourless quirk pair might have a prompt annihilation before they can loose energy enough to form the low lying quirkonium state . however , due to the non perturvative nature of the mechanisms it is difficult to estimate in advance the proportion of the events falling in each scenario , and therefore the possibility to detect these low lying states can not be discarded . through this work , in particular we will focus on neutral bound states which can decay , e.g. to final - state dileptons , providing a clean signature even admits a huge hadronic background as at the lhc experiments . + new particles with a mass of up to several hundred gev can be pair copiously produced at the lhc . one expects that quirks will be in general produced with kinetic energy quite larger than @xmath8 . a significant fraction of this energy should be lost by emission of photons and i - glueballs prior to pair - annihilation . the two quirks will fly off back - to - back , developing a i - qcd string or flux tube . in usual qcd with light matter the string is broken up promptly by creating light quark - antiquark pair ; in i - qcd this mechanism is practically absent . the two heavy ends of the string would continue to move apart , eventually stopping once all the kinetic energy was stored in the string . the quirks would be then pulled together by the string beginning an oscillatory motion . most examples of late - decaying particles that have been addressed in the literature yield missing energy , while quirks would annihilate into visible energy in most modes . besides , as explained in @xcite and @xcite , only i - colour singlet states could be observed . as commented before , in a optimistic scenario , the excited bound state will emit i - glueballs and bremsstrahlung , towards low - lying states ; then they would annihilate into a hard final state : di - lepton , di - jet , or di - photon . + if we want to investigate whether or not it is possible to disentangle different state levels under the assumption of a given ( large ) quirk mass and a specific form of new i - colour interaction , then we could focus on a dimuon signal from the annihilation of a _ narrow _ resonance , since it is the most promising channel @xcite ; it then becomes crucial if the level spacing of different @xmath14-wave states is enough to the foreseen mass resolution based on invariant mass reconstruction from a dimuon system . according to the reference @xcite the detection of bound states at hadron colliders is reliable because the signal production is strong and peaked in invariant mass , and the dominant background is electroweak and diffuse . on the other hand , dimuon backgrounds from sources other than drell - yan can be suppressed by requiring no extra hard jets or missing energy . besides , at the lhc , trigger and detector efficiencies are expected to be very high for high - mass dimuon events . + concerning other channels , it is expected that quirkonium annihilation into a electron pair could be also a useful signal since the invariant mass peak is expected to yield a similar peak to dimuon but smaller and wider due to detector effects . on the other hand , in the squirkonium case , according to the reference @xcite the radiative decay ( by soft radiation ) from the highly excited states to the ground state can be ultimately detected by means of unclustered soft fotons in the uncolored case . also , in ref . @xcite it is pointed out the possibility to use the invariant mass peak of @xmath15photon , since this channel dominates the squirkonium annhilation at or near the ground state . if necessary , all these signals could aid to distinguish among different states . + focusing again on the dimuon signal , as it can be seen from atlas and cms reports @xcite,@xcite one can consider a @xmath16 accuracy for the transverse momentum of muons even at high momentum . since the dimuon invariant mass should coincide to a good approximation , for small ( pseudo)rapidities , with the transverse momentum error , @xmath17 ( since @xmath18 ) ; letting @xmath19 vary along the range @xmath20 $ ] gev , the mass resolution should roughly take the values along the interval @xmath21 $ ] gev . hereafter we restrict our analysis to the range of @xmath8 given by @xmath22 but in the microscopic regime , namely @xmath23 as is well - known long ago , a non - relativistic treatment of the potential for conventional heavy quarkonium has proved to be suitable on account of the asymptotic freedom of qcd . moreover , one distinguishes between short- and long - distance dynamics of constituents in the bound state leading to an effective ( static ) potential of the type : @xmath24 in particular we will write @xmath25 where the first term with @xmath26 would correspond to a coulombic interaction , and the second one with @xmath27 to a linear confining interaction . in this work we will consider firstly a coulomb plus linear potential ( cpl ) with @xmath26 and @xmath27 ; later we will use a more general coulomb plus power law potential ( cpp ) with @xmath26 and @xmath28 as tentative possibilities . the motivation for the insertion of these power law potentials comes up from the clasical studies of quarkonia ( see refs . @xcite , @xcite , @xcite . ) in which are considered coulomb like , linear and cornell potentials , but as well the power law potentials are taken into account in order to cover possible deviations from them . in this way and focusing on the case of quirkonium , in reference @xcite a pure linear potential is taken into account , in reference @xcite a coulomb - like potential were considered . therefore tracking the same philosophy than in the quarkonia case possible deviations are also considered within this study . moreover , the interaction accounting for the above static potential can be parameterized by the fermion ( quirk ) mass @xmath29 , where @xmath30 and an additional @xmath31 gauge coupling ( @xmath32 stands for the i - colour number ) which can be related with a i - colour scale @xmath8 . in this work we will specify @xmath33 in eq.([eq : potential2 ] ) as @xmath34 to be interpreted as cpl ( @xmath27 ) and cpp ( @xmath28 ) potentials . here @xmath35 corresponds to the i - color string tension and we have introduced a i - color coupling @xmath36 , alongside a group theory factor @xmath37 , in close analogy to qcd potential models ; hence , making a simple assumption , such group factor is taken as a mirror from qcd potentials ( @xmath38 ) and included into the infracolour coupling : i.e. , in calculations we set @xmath39 . of course , other numerical choices for @xmath37 can be done , but @xmath36 depends on the @xmath8 scale which is actually uncertain as we shall discuss later . + on the other hand , in analogy to conventional qcd - inspired potential models , the i - colour string tension can be interpreted as a linear energy density ( @xmath40 ) , where @xmath41 and @xmath42 . hence the relation @xmath43 is expected to remain ( approximately ) valid , likewise the equivalent qcd expression @xmath44 ( also derived from lattice calculations @xcite ) , and finally @xmath45 ^ 2\ \sigma_{s}\ ] ] in other words , a proportionality depending on the respective @xmath8 and @xmath10 between both string tensions could be expected from the above arguments . basically , eq.([sigmalambda ] ) implies that @xmath35 and @xmath8 parameters are not independent of each other . for the sake of simplicity , the unknown proportionality factor will be set equal to unity , so that by fixing @xmath8 one gets @xmath35 ( for given @xmath10 and @xmath46 values ) . focusing on the @xmath8 scale , in this work in principle it corresponds to the microscopic @xmath47 scale depicted in ref.@xcite . numerically speaking , as it will be seen , the values of @xmath8 were taken below and above of the qcd scale in a bandwidth ; i.e. @xmath48 with @xmath49 to take into account the uncertainity about this quantity . in this way , the equation([sigmalambda ] ) can be regarded as a comparison between the strength of the linear potential in both sectors @xmath50 and the new gauge group @xmath51 , and it is intended to be an ansatz or a hint to determine numerically a proportionality between @xmath52 and @xmath35 , in which subsequently the numerical uncertainity about the proportionality factor is diluted , taking into account the lack of knowledge about @xmath8 . besides , this comparison between diferent theories can be viewed to some extent in a similar way than in classical physics , in which the strength of gravitational and the electrostatic forces are compared . + concerning the i - colour coupling constant , @xmath36 , it would be related with @xmath8 ; as we are dealing with a non - abelian i - colour binding force it implies @xmath36 is scale dependent . we will compute @xmath36 value at the running scale @xmath53 according to @xcite : @xmath54 where @xmath32 is the i - colour number , and @xmath55 the number of quirk generation at the running scale . from eq.([sigmalambda ] ) and eq.([eq : alpha_s ] ) , one can see that both parameters @xmath35 and @xmath36 are depending on @xmath8 , so that they are not independent quantities . nonetheless , all those constraints have to be taken with a grain of salt and one should consider as well values deviating from those given in eqs . ( [ sigmalambda ] - [ eq : alpha_s ] ) , as we will see later . in case of more quirk generations , additional active quirk should be taken into account at different energy scale thresholds . nevertheless , for the sake of simplicity , and in view of still many unknowns in the different models , we will assume @xmath56 and @xmath57 throughout this work . + as previously mentioned , we consider that the quirk mass lies in the range @xmath58 . therefore one can reasonably expect that the bound system indeed meets a truly non - relativistic regime , i.e. the relative quirk velocity @xmath59 in the center of mass frame being substantially smaller than the value for bottomonium ( @xmath60 ) . focusing on quirkonium , a formal derivation of such non - relativistic limit from the relativistic degrees of freedom can be found in @xcite ; besides according to reference @xcite and @xcite , the bound state is formed in a highly excited state then it decays to the lower states loosing the main part of its kinetic energy . therefore it is expected that in the lower levels near to the ground state this kinetic energy could be low enough to assume a non - relativistic aproximation . also , as we shall see later , the numerical results obtained for the expected quirk velocity @xmath61 in the com frame justify this approximation . since we are interested to perform spectroscopy for very heavy non - relativistic bound states , the schrdinger radial equation must be solved : in a analytical way it could be done by means of a expansion of the quirkonium wave function in a complete basis ; nevertheless here we will choose to solve it numerically , and therefore one should expect that the method to get the resulting mass spectroscopy followed in the _ qq - onia _ package @xcite based on the resolution of the schrdinger radial equation using the numerov @xmath62 technique , should work appropriately for our analysis of quirkonium . however , we are confronted here to the lack of experimental data to set the ground state of quirkonium , in sharp contrast with , e.g. , the bottomonium or charmonium systems . nevertheless , let us stress that in this work we are here mainly interested in estimating the mass spacing between different state levels rather than their absolute values . + as commented in the introduction , new interactions and particles can form very massive bound states . in this section we show the results for quirk @xmath63 bound state system by sweeping through the scale range , @xmath64 characterizing the i - colour force . first we will look at the results using a coulomb plus linear potential ( cpl ) ( with @xmath26 and @xmath27 ) ; later using a coulomb plus power law potential ( cpp ) ( with @xmath26 and @xmath28 ) . concerning the quirk mass , first we use @xmath65 gev , and later @xmath66 gev as representative values ( nevertheless in some calculations we will take additional values ) . the energy levels , @xmath67 shown in tables correspond to @xmath68 where @xmath69 ( or @xmath70 ) is the quirkonium mass level . in all cases , we set the ground state to be @xmath71 . besides as relevant calculations we will display also the squared radial wave function at the origin ( wfo ) ( or their derivatives ) , the size of each quirkonium level , and the mean value of the relative quirk velocity @xmath59 in their center of mass frame , since it is used in some calculations @xcite ; besides , the obtained velocities will check the non - relativistic approximation . let us start by considering the cpl potential . all tables cited in this and successive sections can be found in appendix . results for the quirkonium spectrum with @xmath73 gev and @xmath74 mev are shown in table [ table:1 ] . the corresponding parameters are @xmath75 and @xmath76 . concerning the mean radius , for the ground state we find a size similar to the bohr radius @xmath77 , and increases for higher states as expected up to @xmath78 ; moreover , with these parameters we can found ( @xmath79 ) states with sizes beyond @xmath80 , which is in accordance with @xmath81 . the quirk velocity in the com frame @xmath82 slowly increasing with the @xmath83 and @xmath84 quantum numbers ; these low @xmath59 values plainly justify the non relativistic regime resulting from the qq - onia package . + we provide the squared wfo and derivatives divided by powers of the quirk mass obtained in our calculations , following the same behaviour with @xmath83 and @xmath84 as found in standard quarkonium ( see for instance @xcite and references therein ) . from the ground state wfo value we realize that @xmath85 state follows mainly a coulombic behaviour @xcite . this is not the case for higher resonances , for the p states case we find that a coulombic ( derivative ) wfo underestimates the numerical value obtained from this potential . let us stress that the energy level spacing ( notably between s - wave resonances , of order of tens of mev ) would not permit the experimental discrimination by using the dimuon annihilation channel ( and likely any other else ) . table [ table:2 ] shows the results for @xmath86 ( @xmath87 and @xmath88 ) . the @xmath89 level spacings turn out to be somewhat larger than in the previous case but still not enough to permit experimental discrimination . something similar can be expected for @xmath90 mev as can be seen from table [ table:3 ] , with ( @xmath91 and @xmath92 ) . here we find lower values for sizes of resonances with respect to previous case ( as expected since @xmath8 increases ) . besides , we observe a wfo value for the ground state somewhat greater than the expected for a coulombic behaviour . the results shown in table [ table:4 ] ( appendix ) corresponds to the microscopic scale @xmath93 . here @xmath94 , and @xmath95 . the string tension turns to be much stronger than in the qcd case . the energy levels reach the gev scale and the wfo grow to the @xmath96 gev@xmath97 values ; according with previous trend the corresponding derivatives are growing also . the wfo value for the lowest state is @xmath98 times greater than the expected for a coulombic behaviour . concerning the level spacing this case is interesting since values among @xmath14-wave states turns out to be of order of @xmath99 gev , likely enough to be disentangled . we now set the quirk mass equal to 500 gev , so quirkonium mass is of order of the tev scale . in table [ table:5 ] the @xmath89 spectrum is shown for @xmath101 ( @xmath102 gev @xmath103 and @xmath104 ) . in table [ table:6 ] we show the results for @xmath93 . here @xmath105 , and @xmath106 gev @xmath103 . again , as in the @xmath107 gev case at this scale , the level spacing among @xmath14-wave states could be enough to distinguish experimentally these levels . + let us now give @xmath108 values in the long - range term of eq.([eq : potential3 ] ) different from unity . as in qcd , a larger ( smaller ) @xmath108 leads to stronger ( weaker ) long - distance interaction . the general trends are similar to the cpl case seen in the previous section . tables [ table:7 ] and [ table:8 ] show the @xmath89 spectrum for @xmath110 and @xmath111 respectively for @xmath112 . as we can see the sizes of bound states are similar to the bottomonium case @xcite,@xcite . tables [ table:9 ] and [ table:10 ] ( appendix ) provides again the corresponding @xmath89 spectrum for @xmath113 and @xmath100 , but this time having set @xmath114 . here we observe values of wfo for the ground state similar to the ones expected for a coulombic behaviour ; however higher resonances do not behave in this manner . to cover possible deviations from linear behaviour of the long distance part of the potential we analyze the cpp potential setting @xmath115 . tables [ table:11 ] and [ table:12 ] ( appendix ) show the @xmath89 spectrum for with @xmath101 for @xmath107 gev and @xmath116 gev respectively . tables [ table:13 ] and [ table:14 ] display the corresponding results for the @xmath89 spectrum for @xmath107 gev and @xmath116 gev with @xmath93 . + in order to compare the effect of the above mentioned potentials , in figure 1 we plot the @xmath117 level spacings @xmath118 of quirkonium found with the cpl and cpp potentials ( @xmath119 respectively ) for different @xmath29 and @xmath8 values . as far as we are interested in disentangling peaks of @xmath14-wave resonances , it becomes apparent that this would be only possible in some cases ( i.e. @xmath93 ) where the level spacing is @xmath120 gev or larger . finally , to take into account other contributions which could be entangled in the short distance part of the potential , we consider higher ( non perturbative ) @xmath121 effective values . in order to consider this scenario , we do not use the eq.([eq : alpha_s ] ) for @xmath36 but we take it as a free parameter . on the other hand we keep the explicit dependence of @xmath8 ( eq.([sigmalambda ] ) ) in @xmath35 . in this case we also increase the @xmath8 values from @xmath114 up to @xmath122 . by using _ qq - onia _ code we find the results with @xmath116 gev for the mass level spacing @xmath123 which are shown in table [ table:15 ] ( appendix ) . as we can see from these situations , we find separation between levels tens of gev , thus , in principle , we should be able to discriminate at least between these resonances . next let us analyze the wfo dependence w.r.t . the quirk mass using the above explained cpp potentials . here we focus on the @xmath85 ground level , by taking quirk mass values from @xmath124 up to @xmath125 . again we will take @xmath126 for each potential . for intermediate @xmath29 values @xmath35 does not change w.r.t . the mass values , however here @xmath127 changes for each case according to eq.([eq : alpha_s ] ) . figure 2 displays the obtained results . once computed the squared wfo , we can evaluate numerically the partial decay widths of neutral ( @xmath128 ) quirkonium ( @xmath129 ) to different final states . all of them are proportional to the ratio @xmath130 , where @xmath70 is the quirkonium mass . subsequently we make estimates of the respective branching ratios ( @xmath131 ) . + we will follow a similar treatment as the authors of @xcite,@xcite who considered the following @xmath129 decay modes : * decay to standard model fermion pairs ( @xmath132leptons and quarks ) @xmath133 where @xmath134 stands for functions containing the i - colour number , squared mass ratios @xmath135 , the quirk electric charge @xmath136 . the @xmath137 label means that those sm parameters involved in this calculation parameters are included . * decay to a @xmath138 pair @xmath139 where @xmath140 stands for a function entangling the i - colour number , squared mass ratios @xmath141 , and @xmath137 parameters . * decay to i - gluons ( @xmath142 ) . quirks couple to the i - gluon field of the @xmath31 with coupling strength @xmath143 , where @xmath144 is given by eq.([eq : alpha_s ] ) . @xmath145 + here , @xmath146 is a function of the i - colour number and the i - colour @xmath36 coupling . @xmath147 @xmath148 to make the reading easy , the explicit form of @xmath149 coefficients can be found in references @xcite,@xcite . once set the numerical values of parameters in the above expressions , the @xmath150 values from tables [ table:1 ] to [ table:14 ] ( appendix ) allow one to compute the decay widths of @xmath151 to @xmath137 quarks ( @xmath152 , [ @xmath153 if above the threshold ] ) , leptons ( @xmath154 ) , and other boson decays ( @xmath155 ) . the results are shown in table [ table:16 ] for cpl and cpp ( with @xmath156 ) potential using @xmath157 and @xmath158 at the above considered scales . in all cases the decay mode to @xmath137 quarks is the dominant channel . decay to leptons shares roughly with a @xmath159 for electron , muon and @xmath160 pair respectively . as we can see , for @xmath161 case if we take into account only the @xmath85 decay , the total width is quite narrow @xmath162kev , but we find similar values than in the heavy quarkonia case @xcite . for @xmath163 case the total width increases roughly one order of magnitude . nevertheless , if necessary , this analysis could be improved by adding upper @xmath117 levels contributions decays , which are proportional to @xmath130 . ] : for instance if we compute the whole @xmath117 contribution using the @xmath164 case , with a cpl potential we find a total width @xmath165 times the @xmath85 total width ; using cpp @xmath115 and @xmath109 potentials we find a factor @xmath166 and @xmath167 respectively . + concerning @xmath168-wave resonances ( @xmath169 ) , the corresponding widths satisfy @xmath170 so that , those contributions are suppressed with respect to the @xmath117 decays by a ( @xmath171 ) factor @xmath172 taking @xmath173 values in the cpl case from tables , we find @xmath174 $ ] in the @xmath175 case and @xmath176 $ ] for @xmath177 . regarding the dependence of the brs on the quirk mass ( @xmath131 are independent of the ratio @xmath130 , but @xmath70 enters also thorugh the functions @xmath178 ) : in the range of interest @xmath179 we find variations on the different @xmath131 less than a @xmath180 . we can also check the @xmath131 variations with the i - colour number @xmath32 : by replacing for instance in the above expressions @xmath181 , the @xmath131 to bosons varies mainly @xmath182 and the corresponding @xmath131 to @xmath137 quarks @xmath183 ( @xmath131 to leptons @xmath184 respectively ) . spectroscopy of exotic states might play a fundamental role in the discovery strategy of new physics at the lhc and ilc . in this paper we have focused on a simple extension of the sm , when a new @xmath185 gauge group is added to the sm . the new interaction and new associated fermions have been dubbed i - color , quirks respectively . we assume that quirks are colorless , but otherwise carry sm quantum numbers , thereby coupling to gauge @xmath186 and @xmath187 bosons . quirks can bind forming very peculiar structures reminding . in this work we have focused on neutral @xmath89 states called quirkonium , when the states are microscopic . we have performed a prospective study of quirkonium spectroscopy by employing a coulomb plus linear and coulomb plus power law potentials as representative possibilities with parameters according to i - qcd requirements , as well as other effective contributions to analyze their impact . taking into account the wide range where the @xmath89 bound state might be found , we have chosen the scale range , mev @xmath188 gev with different i - colour @xmath8 scales and quirk masses , finding sizes of several @xmath189 resonances and their squared wfo values of states ( or derivatives for @xmath169 ) . we also extracted the level spacing among resonances using different @xmath8 scenarios to determine whether or not it would be possible to discriminate different state levels . we also have computed total and partial decay widths . i am grateful to miguel angel sanchis - lozano for calling my attention to quirkonium systems , to point out all the details concerning detection by means of the dimuon channel , and many discussions . 1 m. j. strassler and k. m. zurek , phys . b * 651 * , 374 ( 2007 ) . [ arxiv : hep - ph/0604261 ] . j. kang , m.a . jhep * 0911*,065(2009).[arxiv:0805.4642[hep - ph ] ] . k. cheung , w - y . keung , t .- ch . yuan nucl.phys.b * 811*,274 ( 2009).[arxiv:0810.1524v2[hep - ph ] ] . time ago okun @xcite dubbed such new particles as @xmath4thetons in his pioneering study triggered by theoretical curiosity . okun , nucl . phys.b * 173*,1 ( 1980 ) . juknevich , d. melnikov , m . strassler , jhep 0907 ( 2009 ) 055 . [ arxiv:0903.0883[hep - ph ] ] . martin phys.rev . d83 ( 2011 ) 035019 . [ arxiv:1012.2072 [ hep - ph ] ] . kribs , t.s . roy , j. terning , k.m . zurek phys.rev . d81 ( 2010 ) 095001 [ arxiv:0909.2034 [ hep - ph ] ] . g. burdman , z. chacko , h. s. goh , r. harnik and c. a. krenke , phys . d * 78 * ( 2008 ) 075028 [ arxiv:0805.4667 [ hep - ph ] ] . r. harnik , t. wizansky . d80 ( 2009 ) 075015 , [ arxiv:0810.3948 hep - ph ] . atlas collaboration , atlas : detector and physics performance technical design report . vol.1,cern - lhcc-99 - 14 , atlas - tdr-14 . cms collaboration , `` cms physics technical design report volume i : detector performance and software '' , cern - lhcc-2006 - 001 ; cms - tdr-008 - 1 . d. flamm , f. schoberl , introduction to the quark model of elementary particles vol.1 , gordon and breach science publishers ( 1982 ) . c. quigg , j.l . rosner phys.rept . 56 ( 1979 ) 167 . eichten , c. quigg , phys.rev . d49 ( 1994 ) 5845 [ hep - ph/9402210 ] . donoghue , e. golowich , b.r . holstein , dynamics of the standard model ( cambridge monographs on particle physics ) cambridge university press ( 1996 ) . domenech - garret and m.a . sanchis - lozano , comput . . commun . * 180*,768 ( 2009 ) . [ arxiv:0805.2704 [ hep - ph ] ] . according to @xcite and assuming @xmath190 ( as a qcd mirror ) , for the lowest state @xmath191 and @xmath192 v. barger , e.w.n . glover , k. hikasa , w .- y . keung , m.g . olsson c.j . suchyta iii and x.r . tata , phys . rev.d * 35 * , 3366 ( 1987 ) [ erratum - ibid . d * 38*,1632(1988 ) ] . eitchten , k. gotfried , t. kinoshita , k . lane , t. yan , phys . d * 21 * , 203 ( 1980 ) . c. amsler et al . ( particle data group ) , physics letters b**667 * * , 1 ( 2008 ) . comparative plot of the @xmath117 energy levels w.r.t . the @xmath85 state , taken as the ground level , found with the cpl and cpp potentials ( @xmath119 respectively ) for different @xmath29 and @xmath8 values : m100 and m500 stands for the quirk mass value . l , 10l denotes @xmath193 , respectively.,title="fig:",width=302 ] comparative plot of the @xmath117 energy levels w.r.t . the @xmath85 state , taken as the ground level , found with the cpl and cpp potentials ( @xmath119 respectively ) for different @xmath29 and @xmath8 values : m100 and m500 stands for the quirk mass value . l , 10l denotes @xmath193 , respectively.,title="fig:",width=302 ] values of @xmath194 ( in @xmath195 ) corresponding to the @xmath85 level vs. @xmath196 ( in @xmath195 ) using cpp potentials with @xmath197 ( labeled as cpp05 , cpp15 respectively ) . l , 10l denote @xmath193 , respectively . the curve corresponding to @xmath27 lies in between.,width=377 ] .mass level spacings with respect to the ground state : @xmath198 ( mev ) , using a coulomb plus linear potential with @xmath73 gev ; @xmath199 mev ; @xmath200 ( in gev ) , and mean square radius ( in fm ) . [ cols="^,^,^,^",options="header " , ]
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in addition to well - motivated scenarios like supersymmetric particles , the so - called exotic matter ( quirky matter , hidden valley models , etc . ) can show up at the lhc and ilc , by exploring the spectroscopy of high mass levels and decay rates . in this paper
we use qcd - inspired potential models , though without resorting to any particular one , to calculate level spacings of bound states and decay rates of the aforementioned exotic matter in order to design discovery strategies .
we mainly focus on quirky matter , but our conclusions can be extended to other similar scenarios .
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synoptic surveys for gamma - ray bursts ( grbs ) , and subsequent ground - based observational follow - up at radio through optical wavelengths , has highlighted the importance of transient celestial phenomena ( masetti 2001 ) . the new parameter space of the transient cosmos has been emphasized in the design of future telescopes , such as the optical large synoptic survey telescope ( tyson & angel 2001 ) , and the radio square kilometer array ( van haarlem 1999 ) . while it is well documented that flat spectrum radio sources can be variable ( aller et al . 1985 ) , the areal density of such sources has not been well quantified through multi - epoch , wide field blind surveys . at high flux density levels ( @xmath8 mjy at 1.4 ghz ) , one can make a rough estimate of the areal density of variable radio sources by simply assuming that all flat spectrum sources are variable . for instance , the areal density of all sources @xmath9mjy is @xmath10 arcmin@xmath2 , and the fraction of flat spectrum sources is about 10@xmath11 , implying an areal density of variable radio sources of @xmath12 arcmin@xmath2 ( gruppioni et al . 1999 ; white et al . 1997 ; windhorst et al . 1985 ; hopkins et al . 2000 , 2002 ) . this number is consistent with the ( null ) results of frail et al . ( 1994 ) in their search for highly variable mjy - level sources associated with grbs . source populations at these high flux densities are dominated by agn . below about 1 mjy the slope of the source counts flattens , and star forming galaxies are thought to dominate the faint source population ( windhorst et al . 1985 ; georgakakis et al . 1999 ; hopkins et al . 2000 , 2002 ) . hence , when considering the areal density of variable sub - mjy radio sources , one can not simply extrapolate the results from high flux density source samples to low flux densities . knowledge of the areal density of variable sub - mjy radio sources is critical for setting the back - ground , or ` confusion ' , level for studies of faint variable source populations , such as grbs ( frail et al . . a recent comparison of the nvss and first surveys by levinson et al . ( 2002 ) sets a conservative upper limit of @xmath13 arcmin@xmath2 to the areal density of ` orphan ' grb radio afterglows ( i.e. grbs for which the @xmath14-ray emission is not beamed toward us ) with s@xmath15 mjy . they also argue that the areal density of radio supernovae will be considerably smaller . while the area of the sky covered by levinson et al . ( 2002 ) was much larger that the study presented herein , their flux density limit was higher than any grb radio afterglow yet recorded . in this paper we present a smaller area study , but we consider variable sources at flux density levels ( @xmath16mjy ) applicable to typical grb radio afterglows . in general , variability of radio sources at the sub - mjy level is an essentially unexplored part of parameter space a part of parameter space which may fundamentally drive the design of future radio telescopes , such as the ska ( carilli et al . herein we present the first study to delve into this part of parameter space , by exploring systematically the variability of the sub - mjy radio source population at 1.4 ghz . we examine variability on timescales of 17 months and 19 days . note that the lsst will probe similar variability timescales in the optical , with sampling on weekly to yearly timescales . observations were made using the using the vla at 1.4 ghz in the b configuration ( maximum baseline = 10 km ) . the region observed is in the lockman hole centered at : ( j2000 ) 10@xmath17 52@xmath18 56.00@xmath19 , 57@xmath20 29@xmath21 06.0@xmath22 . table 1 summarizes the observations . column 1 gives the observing date , column 2 gives the observed hour angle range , and column 3 gives the rms in the final image . these observations are part of a larger multiwavelength program to study the evolution of dusty star forming galaxies ( bertoldi et al . in prep ) . standard wide field imaging techniques were employed in order to generate an unaberrated image of the full primary beam of the vla ( fwhm = 32@xmath21 ) . the absolute flux density scale was set using 3c286 . we then generated a clean component model of the field using self - calibrated data taken on sept . the data from all days were then self - calibrated in amplitude and phase without gain renormalization using this model . this process should ensure that all the data are on the same flux scale . images made before and after this process showed that the absolute flux scale changed by at most 1@xmath11 . we also checked to see if variable sources could be removed ( or added ) due to using a single model to self - calibrate data from all the different days . components at the 0.1 to 0.2 mjy level were added to the self - calibration model at random positions in the field , and the self - calibration process was repeated . in no case was a new source generated . this gives us confidence that the self - calibration process is robust to small perturbations in the model , i.e. that the problem is over - constrained and that the input self - calibration model is dominated by non - varying sources . images for each day were generated using the wide field imaging capabilities in the aips task imagr ( perley 1999 ) . to remove problems with ` beam squint ' ( slightly different pointing centers for right and left circular polarizations ) the right and left polarizations were imaged separately . the images were then summed , weighted by the rms on each image . the final image using data from all the observing days is shown in figure 1 . the rms noise on this image is 7@xmath4jy beam@xmath23 and the restoring clean beam is circular with fwhm = 4.5@xmath22 . we searched for source variations over 17 months by comparing images made in april 2001 with those made in august / september 2002 , and also on timescales of 19 days by comparing images made on august 17 and september 5 , 2002 . we searched for variable sources out to the 10@xmath11 point of the primary beam , and we limited the analysis to sources with variations @xmath24 . images from all epochs were convolved to 6@xmath22 resolution to mitigate differences that might occur due to the non - linear process of deconvolution . the rms on the images used for variability analysis was 12.5 @xmath4jy for the 17 month comparison , and 17@xmath4jy for the 19 day comparison . the images from different epochs were both subtracted and averaged . a fractional variability image was then generated by dividing these two images , blanking at the 5@xmath25 level ( in absolute value ) . for the 17 months variability analysis @xmath25 was 17@xmath4jy on the differenced image , and 9@xmath4jy on the averaged image . the corresponding numbers for the 19 day analysis were 26@xmath4jy and 15@xmath4jy , respectively . the difference image for the 17 month analysis is shown in figure 2 . some artifacts are seen around the brighter extended sources arising from differences in deconvolution and residual calibration errors . beyond these artifacts , the difference images are remarkably free of sources . this result gives us confidence that the imaging process ( self - calibration and deconvolution ) does not generate spurious sources at the @xmath26 level , and tells us right away that the radio sky is not highly variable at the 100@xmath4jy level . variable sources were identified in the divided image . five variable sources were found at the @xmath27 level in the 17 month comparison , while four sources were found in the 19 day analysis . we then returned to the original images to find the flux densities of the sources at each epoch . table 2 lists source positions ( columns 1 and 2 ) , flux densities at the two epochs ( columns 3 and 4 ) , and the distance from the phase center ( column 5 ) for the 17 month analysis , and table 3 lists the corresponding values for the 19 day analysis . note that the source j1051 + 5734 was seen to vary on both timescales . in fact , this source varied between september 5 and september 9 from 1.53 mjy to 0.71 mjy . the september 2002 value listed in table 2 is the weighted average of these two measurements . we next consider the sensitivity of our observations to variable sources at some absolute level , @xmath28s . the analysis is complicated by the roughly gaussian roll - off of the primary beam of the vla , with fwhm @xmath29 . identification of variable sources was done using the non - primary beam corrected maps in order to have uniform noise across the field . of course , the final flux densities and noise levels for the variable sources were corrected for the primary beam attenuation . given a @xmath28s , one can calculate the maximum primary beam correction , f , for which a change in flux density @xmath28s could be detected at the 5@xmath25 level , where @xmath25 is the noise at the field center on the difference /@xmath28s . the value of f then sets the distance from the pointing center , r , to which such variation could have been detected , given the primary beam shape of the vla . values for @xmath28s , f , and r are listed in columns 1 , 2 , and 3 , respectively in table 4 . column 4 lists the number of variable sources over 17 months that meet these criteria . column 5 lists the number of sources within the specified radius with flux density , s @xmath30 @xmath28s/2 . this flux density sets the limit for 100@xmath11 variability , e.g. a source could be 100@xmath4jy on day 1 , and 0 @xmath4jy on day 2 , leading to a value on the difference image of 100 @xmath4jy and a value on the average image of 50@xmath4jy . column 6 lists the areal density of sources with s@xmath30@xmath28s/2 on our lockman hole image , while column 7 lists the corresponding values from the recent study of fomalont et al . ( 2002 ) comprised of a number of different fields . we have investigated the source sizes using the b array observations at 4.5@xmath22 resolution . two of the sources are partially resolved . for j1051 + 5708 gaussian fitting to the profile yields a peak surface brightness , i@xmath31 mjy beam@xmath23 , a total flux density , s@xmath32 mjy , and a ( deconvolved ) size of @xmath33 , with major axis position angle ( pa ) = @xmath34 . the corresponding numbers for j1055 + 5718 are : i@xmath35 mjy beam@xmath23 , s@xmath36 mjy , and @xmath37 at pa @xmath38 . the rest of the sources are unresolved , with upper limits between 1@xmath22 and 2@xmath22 depending on signal - to - noise . the nature of this analysis is such that we are not sensitive to very rapid variations , e.g. timescales of minutes or less . for instance , a 50 mjy flare of 1 min duration would average down to about 100@xmath4jy over 7 hours . while our nominal sensitivity would be adequate to detect such an event , the existence of such a transient source in the visibility data would lead to errors in the self - calibration and imaging process which would manifest themselves clearly on the images . of course , it is possible that such a bright , short timescale event was mis - identified as interference in the data editing process , and removed . for the analysis on 17 month timescales we could have detected sources with @xmath28s @xmath0jy out to a radius of 7.8@xmath21 , but none were detected . this sets an upper limit to the areal density of such sources of @xmath39 arcmin@xmath2 . another interesting point is that within this radius there are 46 sources between 50 and 100 @xmath4jy on the averaged image . hence , we set an upper limit of about 2@xmath11 to the fraction of sources in this flux density regime that are highly ( @xmath40 ) variable . these results are grossly consistent with the idea that below 1 mjy at 1.4 ghz the radio source population is dominated by star forming galaxies , as compared to agn which dominate at high flux densities ( hopkins 2000 , 2002 ) . the exact distribution of agn vs. starbursts vs. other source types as a function of radio flux density is not fully determined at this time , and is an area of active current research ( hopkins et al . 2002 ; georgakakis et al . 1999 ; richards 2001 ; fomalont et al . 2002 ) . as a rough guide we consider the models of hopkins et al . they suggests that at @xmath30 10 mjy the source population is 90@xmath11 steep spectrum radio sources , 10@xmath11 flat spectrum radio sources , and @xmath41 star forming galaxies . again , at high flux density levels the flat spectrum sources correspond to the variable radio source population . at 100 @xmath4jy the models suggests that the proportions change to roughly 80@xmath11 star forming galaxies , 15@xmath11 steep spectrum agn , and 5@xmath11 flat spectrum agn . these models have not considered truly transient source populations , such as grb radio afterglows , which have timescales of 10 s of days . the statistics of radio afterglows of grbs are such that in an area of 7.8@xmath21 radius one would expect to see @xmath42 sources above 0.1 mjy at 1.4 ghz at any given time , assuming grbs are highly beamed ( see below ) . hence , the fact that we did not see such a source is not surprising . more importantly , the results presented herein allow us to set a limit on the variable source confusion level for grb radio afterglow searches , again at a flux density level relevant to the observed population . for example , radio searches are needed for localizing an important subclass of afterglows known as ` dark grbs ' ( e.g. , djorgovski et al . 2001 ) , for which optical emission from the grb is not detected . it has been suggested that the absence of optical emission from these sources is the result of either dust obscuration , or the gunn - peterson effect , i.e. ly@xmath43 absorption by the neutral intergalactic medium . this latter effect would place the sources at @xmath44 ( fan et al . an upper limit to the areal density of @xmath39 arcmin@xmath2 for variable sources at the 100@xmath4jy level implies a lower limit of 200 arcmin@xmath45 to the area that can be safely searched at 1.4 ghz before one such source is expected to be detected by chance . for comparison , the typical grb error circle is @xmath4630 arcmin@xmath45 , but larger error circles are not uncommon . this result can also be used to derive a rough limit on the mean beaming angle for grbs ( perna & loeb 1998 ) . from a sample of 25 radio afterglows ( frail et al . in prep ) we estimate that 10 - 25% will be visible above the 100 @xmath4jy level at 1.4 ghz , with an average lifetime of one month . the grb event rate is approximately 600 per year ( fishman & meegan 1995 ) so we expect to find only @xmath47 arcmin@xmath2 . however , if grbs are highly beamed , as recent studies seem to suggest ( frail et al . 2001 ) , then our upper limit to areal density of @xmath39 arcmin@xmath2 implies a beaming factor @xmath48 , or a mean jet opening angle @xmath49 . this value is not very constraining compared to existing limits , but to our knowledge this is the first time a survey for variability has been done at the appropriate flux level for radio afterglows . more stringent limits will require sensitive , larger area surveys with existing or planned instruments ( totani & panaitescu 2002 ) . a final point we consider is cosmic variance . it is possible that the lockman hole region we have sampled was just statistically - poor in transient sources . a rough indication of the effect of cosmic variance comes from the overall source counts ( columns 6,7 in table 4 ) . the source counts we derive agree to within 20@xmath11 with those found by fomalont et al . ( 2002 ) in other areas of the sky . in general , it has been found that for 30@xmath21 fields - of - view the maximum field - to - field scatter in the sub - mjy source counts is about a factor two ( fomalont et al . we consider this an upper limit to the effect cosmic variance has on the areal density of variable source presented herein . the national radio astronomy observatory ( nrao ) is operated by associated universities , inc . under a cooperative agreement with the national science foundation . we thank f. owen , e. fomalont , r. becker , and j. condon for useful comments concerning this work , and the referee for careful and insightful criticism . hopkins , a.m. , afonso , j.m . , chan , b. , cram , l.e . , georgakakis , a. , mobasher , b. 2002 , in `` galaxy evolution : theory and observations '' , eds . v. avila - reese , c. firmani , c. frenk , & c. allen , revmexaa sc . tyson , a. & angel , r. 2001 , in ` the new era of wide field astronomy , asp conference series , vol . 232 , ' eds . roger clowes , andrew adamson , and gordon bromage , ( san francisco : astronomical society of the pacific ) , p.347
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we present 1.4 ghz vla observations of the variability of radio sources in the lockman hole region at the level of @xmath0jy on timescales of 17 months and 19 days .
these data indicate that the areal density of highly variable sources at this level is @xmath1 arcmin@xmath2 .
we set an upper limit of @xmath3 to the fraction of 50 to 100@xmath4jy sources that are highly variable ( @xmath5 ) .
these results imply a lower limit to the beaming angle for grbs of 1@xmath6 , and give a lower limit of 200 arcmin@xmath7 to the area that can be safely searched for grb radio afterglows before confusion might become an issue .
| 4,907 | 185 |
in this paper , we consider numerical methods for the computation of the @xmath0-gradient flow of a planar curve : @xmath1 where @xmath2 is a time - dependent planar curve . the gradient flow is energy dissipative , since @xmath3 = - \int |\operatorname{grad}e({\mathbf{u}})|^2 ds \le 0.\ ] ] here , @xmath4 is an energy functional , and @xmath5 is the frchet derivative with respect to the @xmath0-structure with line integral @xmath6 . thus , the curvature flow ( the curve shortening flow ) @xmath7 and the elastic flow ( the willmore flow ) @xmath8 have energy functionals @xmath9 = \int ds , \quad \text{and } \quad e[{\mathbf{u } } ] = \varepsilon^2 \int |{\bm{\upkappa}}|^2 ds + \int ds,\ ] ] where @xmath10 is the curvature vector , and @xmath11 is the tangential derivative ( example [ ex : gf ] ) . note that the elastic flow is a fourth - order nonlinear evolution equation . we consider a dissipative numerical scheme for , that is , a scheme which has the discrete energy dissipative property @xmath12 \le e[{\mathbf{u}}_h^n]$ ] at each time step . in general , a numerical method that retains a certain property for a target equation is called structure - preserving . it is known that the numerical solutions obtained by these methods are not only physically realistic but also have the advantage of numerical stability ( cf . in particular , structure - preserving methods are suitable for computations over long time intervals . here , we consider in more detail the energy dissipation for the gradient flow . as is well known , the ( classical ) solution of the curvature flow blows up in finite time . in particular , if the initial curve is self - crossing , a cusp appears . on the other hand , the elastic flow , which is a regularized version of , has a unique global solution @xcite . moreover , as @xmath13 , the solution of converges to that of under conditions when the classical solution of ( see @xcite ) exists . note that the elastic flow formally degenerates to the curvature flow as @xmath13 . let @xmath14 be the time at which a cusp appears . then , we are interested in determining the limit of the elastic flow as @xmath13 after time @xmath14 . therefore , the structure - preserving method is effective for investigating numerically the long - time behavior of the elastic flow with self - crossing initial curves . there are many works that consider the numerical computation of equations and ( for example , @xcite ) . however , none of them explicitly consider the discrete energy dissipation property . although some numerical examples in these works seem to be dissipative , no mathematical proofs are given . moreover , all of them use the p1-finite element method , which approximates solutions using polygonal curves . therefore , since they can not handle higher order derivatives directly , it is necessary to consider the mixed formulation for fourth - order equations . it is remarkable that these studies present numerical schemes for individual problems , and do not give a general framework for approximating gradient flows . there is a further issue regarding the arrangement of nodal points , when a curve evolution equation is discretized with polygonal curves . however , we do not consider this issue in detail in this paper . for gradient flows of graphs , there are some general frameworks to construct dissipative numerical schemes . in @xcite , a structure - preserving finite difference scheme , the so - called discrete variational derivative method , is proposed to approximate the solution of the equation @xmath15 over an interval in @xmath16 . here , @xmath17 is the frchet derivative of the energy functional @xmath4 with respect to the usual @xmath0-structure , i.e. , the @xmath0 space with respect to the lebesgue measure . finite element schemes , called discrete partial derivative methods ( dpdm ) , for the same problems are presented in @xcite . the main idea of these studies is to discretize the chain rule . in @xcite , discrete partial derivatives are introduced and a discretization of the chain rule is obtained . a similar approach is used in our scheme . see also @xcite for local discontinuous galerkin schemes for the above problems . in the present paper , we apply the idea of dpdm to the discretization of the time variable . we obtain the discrete chain rule formula with respect to the @xmath18-structure . in contrast to problems such as , our problem is accompanied by the line element @xmath6 , which increases the complexity of the problem . due to the inclusion of the line element , we can not use the dpdm for . therefore , we will present a generalization of the dpdm and derive a new scheme for weak forms of general gradient flows ( scheme [ scheme : semi - disc ] ) . for the approximation of curves , we use b - spline curves ( cf . @xcite ) . the b - spline approach ( also called nurbs in general ) is widely used to compute the solution to large - scale deformation problems , such as fluid - structure interaction ( see , e.g. , @xcite ) . this method is called isogeometric analysis @xcite . it is worth emphasizing that a b - spline curve of degree @xmath19 ( definition [ def : b - spline ] ) is a @xmath20 curve . hence we can directly address higher order derivatives , and we can derive the fully discretized scheme by the galerkin method ( scheme [ scheme : full - disc ] ) . this procedure is independent of the properties of the energy functional , and thus , our scheme gives a general framework for the approximation of gradient flows . we can also obtain the discrete energy dissipation with our scheme ( lemma [ lem : dissipation ] ) . in this study , we do not consider solvability or error estimates . this paper is structured as follows . in section [ section : preliminary ] , we present some necessary definitions and notation for gradient flows on planar curves ( subsection [ subsec : gf ] ) , and then illustrate the use of the dpdm ( subsection [ subsec : dpdm ] ) and formally define b - spline curves ( subsection [ subsec : bspline ] ) . in section [ section : scheme ] , we derive our energy dissipative scheme ( scheme [ scheme : full - disc ] ) under the framework of the dpdm , and introduce the discrete energy dissipation property ( lemma [ lem : dissipation ] ) . finally , we present some numerical examples of our scheme in section [ section : num_exp ] , and compute the elastic flow . a topology - changing solution and more complicated evolution are reported , which have not previously been shown in the literature . videos illustrating our method are available on youtube . in this subsection , we summarize the basic properties of the geometric gradient flows of planar curves . let @xmath21 and @xmath22 for @xmath23 , @xmath24 . here , the space @xmath25 is the @xmath26-th order sobolev space of @xmath0-type . note that the space @xmath27 is embedded into the space of the planar closed @xmath28-curves . we define an energy functional @xmath29 as @xmath9 = \int f({\mathbf{u}},{\mathbf{u } } ' , \dots , { \mathbf{u}}^{(m ) } ) ds , \quad { \mathbf{u}}\in { \mathbf{h}}^m_\pi , \label{eq : eng}\ ] ] where @xmath30 is the energy density function , and @xmath31 is the line element of the curve @xmath2 . let @xmath5 be the frchet derivative of the functional @xmath4 in the topology of @xmath18 . that is , @xmath32 then , the gradient flow for @xmath4 is represented by the following evolution equation : @xmath33 examples of the gradient flow are now considered . [ ex : gf ] 1 . ( curvature flow ) if @xmath34 = \int ds$ ] , then equation is the curvature flow @xmath35 where @xmath36 is the curvature vector , and @xmath37 is the arc - length parameter . 2 . ( elastic flow ) if @xmath34 = \varepsilon^2 \int |{\bm{\upkappa}}|^2 ds + \int ds$ ] , then equation is the elastic flow ( or willmore flow ) @xmath38 where @xmath39 , and @xmath40 . in this paper , we focus on the energy dissipation property , which is given as follows : @xmath3 = \int \operatorname{grad}e({\mathbf{u } } ) \cdot { \mathbf{u}}_t ds = - \int |\operatorname{grad}e({\mathbf{u}})|^2 ds \le 0.\ ] ] in this subsection , we introduce the dpdm , which was first presented in @xcite . let @xmath41 be an energy functional that is defined as @xmath42 = \int_0 ^ 1 g(u_\zeta , u_{\zeta\zeta } ) d\zeta , \quad u = u(\zeta ) \in h^2_\pi(0,1 ) , \label{eq : eng - graph}\ ] ] where @xmath43 is the energy density function . although we can consider more general energy functionals and density functions , we consider energy functionals @xmath4 such as for simplicity . let us denote the first variation of @xmath4 by @xmath44 , i.e. , @xmath45 the @xmath0-gradient flow for the energy @xmath4 is the evolution equation @xmath46 this equation also has the energy dissipation property as in . indeed , @xmath47 = \int_0 ^ 1 \frac{\delta e}{\delta u } u_t d\zeta = - \int_0 ^ 1 \left| \frac{\delta e}{\delta u } \right|^2 d\zeta \le 0.\ ] ] dpdm is an energy dissipative numerical scheme for the case of equation . in dpdm , the discrete partial derivatives @xmath48 and the others are defined as the functions that satisfy the following relation : @xmath49 for all @xmath50 . note that the partial derivatives which solve this relation may not be unique . when @xmath51 is written in a certain form , a method for deriving the partial derivatives is given in @xcite . next , we define the discrete analogue of the first variation as @xmath52 note that this function satisfies the relation @xmath42 - e[v ] = \int_0 ^ 1 \frac{\delta e_{\mathrm{d}}}{\delta(u , v ) } ( u - v ) d\zeta . \label{eq : disc - chain - graph}\ ] ] therefore , we can derive the discretized equation @xmath53 following @xcite , we can derive an energy dissipative time - discretization for as follows . the following scheme is the weak form of equation . let @xmath54 and @xmath55 be given . find @xmath56 that satisfies @xmath57 for all @xmath58 . we can now check the discrete energy dissipation property with the following lemma . here we give the proof for comparison with our scheme ( see lemma [ lem : dissipation ] ) . let @xmath59 and @xmath60 satisfy the relation . then , we have @xmath61 - e[u^n]}{\delta t } = - \left\| \frac{u^{n+1 } - u^n}{\delta t } \right\|_{l^2(0,1)}^2 \le 0\ ] ] for all @xmath54 and @xmath62 . let us write @xmath63 then , substituting @xmath64 into the weak form , , we can derive @xmath65 - e[u^n]}{\delta t } & = \int_0 ^ 1 \frac{g(u^{n+1}_\zeta , u^{n+1}_{\zeta\zeta } ) - g(u^{n}_\zeta , u^{n}_{\zeta\zeta})}{\delta t } d\zeta \\ & = \left ( \frac{\partial g_{\mathrm{d}}}{\partial(u^{n+1}_\zeta , u^n_\zeta ) } , \partial_{\mathrm{d}}u^n_\zeta \right ) + \left ( \frac{\partial g_{\mathrm{d}}}{\partial(u^{n+1}_{\zeta\zeta},u^n_{\zeta\zeta } ) } , \partial_{\mathrm{d}}u^n_{\zeta\zeta } \right ) \\ & = - \|\partial_{\mathrm{d}}u^n\|_{l^2(0,1)}^2.\end{aligned}\ ] ] note that the key point is substituting @xmath64 into . therefore , this proof can be derived also in the case of the galerkin method . in our scheme , we use b - spline curves to discretize the solution curves . we say that a set of points @xmath66 is a _ knot vector _ if @xmath67 for all @xmath68 . let @xmath69 , @xmath54 , and @xmath70 be a knot vector . 1 . the _ @xmath68-th b - spline basis function of degree @xmath19 _ with respect to @xmath71 is a piecewise polynomial function @xmath72 that is generated by the following formula : @xmath73 for @xmath74 and @xmath75 , where @xmath76 is the characteristic function of @xmath77 . here , if @xmath78 ( resp . @xmath79 ) , then the term @xmath80 ( resp . @xmath81 ) is null . 2 . a curve @xmath82 \to { \mathbb{r}}^2 $ ] is a _ b - spline curve of degree @xmath19 _ if @xmath2 is represented by @xmath83,\ ] ] for some knot vector @xmath71 and @xmath54 . the coefficient @xmath84 is called a _ control point_. in fact , if the knot vector is disjoint ( i.e. , @xmath85 ) , then it is known that @xmath72 is a @xmath20-function . for more details on the properties of b - spline functions , we refer the reader to @xcite . in the present paper , we only consider the periodic b - spline functions and curves . let @xmath86 \subset { \mathbb{r}}$ ] be an interval , @xmath87 , @xmath88 , and @xmath89 . we define a knot vector @xmath71 as @xmath90 and let @xmath72 be the corresponding b - spline basis function . note that @xmath91 $ ] . then , if @xmath92 , we can see that @xmath93 for @xmath94 and @xmath95 . therefore , the function @xmath96 , \\ n^\xi_{p , i+n}(\zeta ) , & \zeta \in [ \xi_{i+n},b ] , \\ 0 , & \text{otherwise } \end{cases } \quad i=1,2,\dots , p \label{eq : periodic - b - spline}\ ] ] is a periodic @xmath20-function in @xmath86 $ ] . the restriction @xmath97}$ ] for @xmath98 is also @xmath20-periodic on @xmath86 $ ] . then , we define a closed b - spline curve as follows . [ def : b - spline ] let @xmath86 \subset { \mathbb{r}}$ ] be an interval , @xmath87 , @xmath88 with @xmath99 , and @xmath89 . then , we define a _ periodic b - spline basis function of degree p _ @xmath100 by for @xmath101 and by @xmath102}$ ] for @xmath103 , where @xmath71 is a knot vector defined by . we also define a _ closed b - spline curve _ as a curve @xmath82 \to { \mathbb{r}}^2 $ ] expressed by @xmath104,\ ] ] for some @xmath105 . it is clear that a closed b - spline curve is a @xmath20-curve . in this section , we derive a numerical scheme for geometric gradient flows for the energy functional @xmath4 given by . we first consider the time discretization , and recall the idea of the dpdm . using a similar approach as for the dpdm , we derive a discretization of the chain rule . the definition of the partial derivatives is a discrete analogue of the chain rule formula @xmath106 for a smooth function @xmath107 . in our case , the corresponding chain rule can be expressed as @xmath3 = \int \operatorname{grad}e({\mathbf{u } } ) \cdot { \mathbf{u}}_t ds({\mathbf{u } } ) . \label{eq : chain}\ ] ] here , we denote the line element of @xmath2 by @xmath108 to emphasize the dependence on @xmath2 . now , we discretize the chain rule . we first change the time derivatives to time differences by expressing @xmath109 $ ] and @xmath110 as @xmath34 - e[{\mathbf{v}}]$ ] and @xmath111 , respectively . moreover , the line element @xmath108 should be changed appropriately . in the original formula , there is one function @xmath2 only . however , in the discretization , there are two functions @xmath2 and @xmath112 as in . therefore , we have some choices to discretize the term @xmath108 , for example , @xmath108 , @xmath113 , and @xmath114 . here , we use @xmath114 . then , we define a discrete gradient , @xmath115 , with a function that satisfies the following formula : @xmath9 - e[{\mathbf{v } } ] = \int \operatorname{grad}_{\mathrm{d}}e({\mathbf{u } } , { \mathbf{v } } ) ds\left ( \frac{{\mathbf{u}}+{\mathbf{v}}}{2 } \right ) , \quad \forall { \mathbf{u}},{\mathbf{v}}\in { \mathbf{h}}^m_\pi . \label{eq : disc - chain}\ ] ] thus , according to , the strong form of the time - discrete problem is written as follows : @xmath116 the discrete chain rule can then be expressed as @xmath9 - e[{\mathbf{v } } ] = \int_0 ^ 1 \left| \frac{{\mathbf{u}}_\zeta+{\mathbf{v}}_\zeta}{2 } \right| \operatorname{grad}_{\mathrm{d}}e({\mathbf{u } } , { \mathbf{v } } ) \cdot ( { \mathbf{u}}- { \mathbf{v } } ) d\zeta , \quad \forall { \mathbf{u}},{\mathbf{v}}\in { \mathbf{h}}^m_\pi , \label{eq : disc - chain2}\ ] ] and comparing with , we can derive the relationship between @xmath117 and the discrete first derivative @xmath118 as follows . @xmath119 here @xmath118 is a vector - valued function . letting @xmath120 , the energy @xmath4 is expressed by @xmath9 = \int_0 ^ 1 g({\mathbf{u } } , { \mathbf{u}}_\zeta , \dots , \partial_\zeta^m { \mathbf{u } } ) d\zeta,\ ] ] and thus the discrete first derivative is given by @xmath121 here , we define the ( vector - valued ) partial derivatives , @xmath122 as functions that satisfy the relation @xmath123 for all @xmath124 . note that , as in the previous case ( subsection [ subsec : dpdm ] ) , the partial derivative may not be unique . now , instead of solving , we may solve the equation @xmath125 and the weak form of gives our semi - discrete scheme for the gradient flow . note that the time increment @xmath126 can differ at each step . [ scheme : semi - disc ] find @xmath127 that satisfies @xmath128 for given @xmath129 . we now consider the full discretization of the gradient flow . let @xmath130 be the space of closed b - spline curves of degree @xmath19 as defined in definition [ def : b - spline ] . then , by the sobolev embedding theorem , @xmath131 if @xmath132 . thus , we can derive a fully discretized problem by the galerkin method . [ scheme : full - disc ] let @xmath88 , @xmath89 , and @xmath132 . assume @xmath133 is given . find @xmath134 that satisfies @xmath135 for all @xmath136 . then , we can establish the discrete energy dissipation property . [ lem : dissipation ] let @xmath137 and @xmath138 satisfy the relation . then , we have @xmath139 - e[{\mathbf{u}}_h^n]}{\delta t_n } = - \int_0 ^ 1 \left| \frac{{\mathbf{u}}^{n+1}_{h,\zeta } + { \mathbf{u}}^n_{h,\zeta}}{2 } \right| \left| \frac{{\mathbf{u}}_h^{n+1 } - { \mathbf{u}}_h^n}{\delta t_n } \right|^2 d\zeta \le 0.\ ] ] substituting @xmath140 into the scheme , we have @xmath141 - e[{\mathbf{u}}_h^n]}{\delta t_n},\end{aligned}\ ] ] by the definition of the partial derivatives . hence we can establish the desired assertion . in this section , we show some numerical examples of the elastic flow computed by our scheme . here , the functional is the elastic energy @xmath9 = \varepsilon^2 \int |{\bm{\upkappa}}|^2 ds + \int ds = \int_0 ^ 1 \left ( \varepsilon^2 \frac{\det({\mathbf{u}}_\zeta,{\mathbf{u}}_{\zeta\zeta})^2}{|{\mathbf{u}}_\zeta|^5 } + |{\mathbf{u}}_\zeta| \right ) d\zeta , \label{eq : elastic - energy}\ ] ] where @xmath142 it is known that equation has a unique time - global solution ( see , e.g. , ( * ? ? ? * theorem 3.2 ) ) . therefore , the turning number @xmath143 is invariant . to calculate the discrete partial derivatives for @xmath4 , let @xmath144 for @xmath145 . then , the energy density function for @xmath4 is @xmath146 . we can compute the partial derivatives of @xmath147 since @xmath148 which implies @xmath149 we can derive the partial derivatives of @xmath150 in several ways . in the following examples , these derivatives are computed by dividing @xmath151 as follows : @xmath152 the first and the second terms of the right - hand side are calculated as @xmath153 \\ & \times \big[v_{2,\zeta\zeta}(u_{1,\zeta}-v_{1,\zeta } ) - v_{1,\zeta\zeta}(u_{2,\zeta}-v_{2,\zeta } ) \\ & - u_{2,\zeta}(u_{1,\zeta\zeta}-v_{1,\zeta\zeta } ) + u_{1,\zeta}(u_{2,\zeta\zeta}-v_{2,\zeta\zeta } ) \big],\end{aligned}\ ] ] and @xmath154 respectively . although we have omitted them , we can derive partial derivatives of @xmath150 with these equations . before showing numerical examples , we recall the steady - state solutions for the elastic flow . it is known that steady closed curves of the elastic energy are circles of radius @xmath155 , the figure - eight - shaped curve with scale @xmath155 , and their multiple versions @xcite ( see figure [ fig : steady ] ) . their energies are @xmath156 = 4 \pi \varepsilon , \quad e[\text{eight - shaped } ] \approx \varepsilon \cdot 21.2075,\ ] ] respectively . the exact value of the latter energy is expressed by the elliptic integrals ( cf . @xcite ) . in our numerical examples , we set the time increment as @xmath157 and @xmath158 - e[{\mathbf{u}}_h^{n-1}]}{\delta t_{n-1 } } \right)^{-2 } \right\ } , \quad n \ge 1.\ ] ] we found these values empirically . we solved equation at each step with the newton method . moreover , in each step , when two adjacent control points are too close ( more precisely , when the distance is less than @xmath159 ) , one of them was removed . note that the energy may increase when control points are eliminated ; however , the shape of the curve will be less affected ( see figure [ fig : elim ] ) . with respect to the control points as shown in the upper figure . the right two figures show the b - spline curves of the same degree with respect to the control points except for the point @xmath160 . ] we show six examples here . in all examples , we use the b - spline curves of degree @xmath161 . therefore , every curve below is of class @xmath162 . videos of the following examples are available on youtube . [ ex : circle_0 ] the first example is shown in figure [ fig : circle_0 ] . the initial curve is a circle.the parameters are @xmath163 in this example , the elimination of control points was not necessary . figure [ fig : circle_0-curve ] shows the evolution of the curve at @xmath164 . the energy at @xmath165 is @xmath166 . note that the exact value of the energy at the steady state is @xmath167 . figure [ fig : circle_0-energy ] shows the evolution of the energy . the discrete energy dissipation property is clearly visible . in figure [ fig : circle_0-curve ] , one can observe that the curve shrinks as the curvature flow until @xmath168 , and it stops shrinking when the radius approaches @xmath155 . .45 .,title="fig : " ] .45 .,title="fig : " ] [ ex : eight_0 ] the second example is shown in figure [ fig : eight_0 ] . the initial curve is figure - eight - shaped . the parameters are @xmath169 as in the previous case , the elimination of control points is not necessary . figure [ fig : eight_0-curve ] shows the evolution of the curve at @xmath170 and figure [ fig : eight_0-energy ] shows the evolution of the energy . the energy at @xmath171 is @xmath172 . note that the exact value of the energy at the steady state is approximately @xmath173 . in figure [ fig : eight_0-curve ] , first the small loop ( the right loop ) shrinks faster than the larger one . when the scale of the right loop becomes @xmath155 , shrinking stops , and the left one begins to shrink . finally , the left one also stops shrinking , and the curve approaches the steady state . .45 .,title="fig : " ] .45 .,title="fig : " ] [ ex : double_0 ] the third example is shown in figure [ fig : double_0-curve ] . the initial shape of the curve is a cardioid - like curve as shown in figure [ fig : double_0 - 1 ] . the initial parameters of the curve are @xmath174 as in the previous cases , the elimination of control points is not necessary . figure [ fig : double_0-curve ] shows the evolution of the curve at @xmath175 and figure [ fig : double_0-energy ] shows the evolution of the energy . in this case , the steady - state is a double - looped circle with radius @xmath176 . therefore , the energy of the solution at @xmath165 ( @xmath177 ) is approximately twice the value of that of example [ ex : circle_0 ] . the behavior of the curve is similar to example [ ex : eight_0 ] . that is , first the smaller loop shrinks until the scale is approximately @xmath155 . then , the larger one shrinks and the curve approaches the steady state . .49 .,title="fig : " ] .49 .,title="fig : " ] + .49 .,title="fig : " ] .49 .,title="fig : " ] . ] [ ex : circle_1 ] this example shows a topology - changing solution . the initial curve is the one shown in figure [ fig : circle_1 - 1 ] , and figure [ fig : circle_1-curve ] shows its evolution . figures [ fig : circle_1-energy ] and show the evolution of the energy and the number of control points , respectively . the parameters are @xmath178 one can observe that the topology of the curve changes at around @xmath179 ( figures [ fig : circle_1 - 5 ] and ) . at the same time , the energy decreases drastically ( figure [ fig : circle_1-energy ] ) , and some control points become concentrated ( figure [ fig : circle_1-elim ] ) . as mentioned earlier , we implement an algorithm that deletes a control point if it is too close to the adjacent point . therefore , the elimination of control points occurs when the topology changes , and the number of control points finally converges to @xmath180 . .32 .,title="fig : " ] .32 .,title="fig : " ] .32 .,title="fig : " ] + .32 .,title="fig : " ] .32 .,title="fig : " ] .32 .,title="fig : " ] .45 .,title="fig : " ] .45 .,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] + .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] + .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] .32 at the times described as in the figures . the small circles represent control points , @xmath181 is the number of control points , and the gray colored curve is the solution.,title="fig : " ] [ ex : eight_1 ] the following two examples investigate problems with more complicated solutions . the initial shape of the curve is shown in figure [ fig : eight_1 - 1 ] , and figure [ fig : eight_1-curve ] shows its evolution . figures [ fig : eight_1-energy ] and show the evolution of the energy and the number of control points , respectively . the parameters are @xmath182 in this example , the topology of the curve changes frequently . for example , the loop in the upper left of the curve disappears at around @xmath183 . when the topology changes , the energy decreases rapidly as in example [ ex : circle_1 ] , and the number of control points also decreases at the same time . the final value of @xmath181 is @xmath184 . .32 .,title="fig : " ] .32 .,title="fig : " ] .32 .,title="fig : " ] + .32 .,title="fig : " ] .32 .,title="fig : " ] .32 .,title="fig : " ] .45 .,title="fig : " ] .45 .,title="fig : " ] [ ex : circle_3 ] the initial curve for the final example is shown in figure [ fig : circle_3 - 1 ] , and figure [ fig : circle_3-curve ] shows its evolution . figures [ fig : circle_3-energy ] and show the evolution of the energy and the number of control points , respectively . the parameters are @xmath185 the solution displays complicated behavior as in example [ ex : eight_1 ] , and the topology changes frequently . however , since the turning number of the initial curve is one , the steady state is a circle with radius @xmath155 . the energy and the number of control points decrease drastically when the topology changes , and the final number of control points is @xmath186 . .32 .,title="fig : " ] .32 .,title="fig : " ] .32 .,title="fig : " ] + .32 .,title="fig : " ] .32 .,title="fig : " ] .32 .,title="fig : " ] .45 .,title="fig : " ] .45 .,title="fig : " ] i would like to thank prof . yoshihiro tonegawa and dr . takahito kashiwabara for bringing this topic to my attention and encouraging me through valuable discussions . this work was supported by the program for leading graduate schools , mext , japan , and by jsps kakenhi ( no . 15j07471 ) .
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in this paper , we develop an energy dissipative numerical scheme for gradient flows of planar curves , such as the curvature flow and the elastic flow .
our study presents a general framework for solving such equations . to discretize time
, we use a similar approach to the discrete partial derivative method , which is a structure - preserving method for the gradient flows of graphs . for the approximation of curves , we use b - spline curves . owing to the smoothness of b - spline functions ,
we can directly address higher order derivatives . in the last part of the paper , we consider some numerical examples of the elastic flow , which exhibit topology - changing solutions and more complicated evolution .
videos illustrating our method are available on youtube .
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the origin and nature of quark and lepton masses and mixings remains one of the most intriguing questions left unanswered by the standard model ( sm ) of particle physics . within the sm , quark and lepton masses and mixings arise from yukawa couplings which are essentially free and undetermined . in extensions such as grand unified theories ( guts ) , the yukawa couplings within a particular family may be related , but the mass hierarchy between different families is not explained and supersymmetry ( susy ) does not shed any light on this question either . indeed , in the sm or guts , with or without susy , a specific structure of the yukawa matrices has no intrinsic meaning due to basis transformations in flavour space . for example , one can always work in a basis in which , say , the up quark mass matrix is taken to be diagonal with the quark sector mixing arising entirely from the down quark mass matrix , or _ vice versa _ , and analogously in the lepton sector ( see e.g. @xcite ) . this is symptomatic of the fact that neither the sm or guts are candidates for a theory of flavour . the situation changes somewhat once these theories are extended to include a family symmetry spontaneously broken by extra higgs fields called flavons . this approach has recently received a massive impetus due to the discovery of neutrino mass and approximately tri - bimaximal lepton mixing @xcite whose simple pattern strongly suggests some kind of a non - abelian discrete family symmetry might be at work , at least in the lepton sector , and , assuming a gut - type of structure relating quarks and leptons at a certain high energy scale , within the quark sector too . the observed neutrino flavour symmetry may arise either directly or indirectly from a range of discrete symmetry groups @xcite . examples of the direct approach , in which one or more generators of the discrete family symmetry appears in the neutrino flavour group , are typically based on @xmath3 @xcite or a related group such as @xmath4 @xcite or @xmath5 @xcite . models of the indirect kind , in which the neutrino flavour symmetry arises accidentally , include also @xmath6 @xcite as well as @xmath7 @xcite and the continuous flavour symmetries like , e.g. , @xmath8 @xcite or @xmath9 @xcite which accommodate the discrete groups above as subgroups @xcite . theories of flavour based on a spontaneously broken family symmetry are constructed in a particular basis in which the vacuum alignment of the flavons is particularly simple . this then defines a preferred basis for that particular model , which we shall refer to as the `` flavour basis . '' in such frameworks , the resulting low energy effective yukawa matrices are expected to have a correspondingly simple form in the flavour basis associated with the high energy simple flavon vacuum alignment . this suggests that it may be useful to look for simple yukawa matrix structures in a particular basis , since such patterns may provide a bottom - up route towards a theory of flavour based on a spontaneously broken family symmetry . unfortunately , experiment does not tell us directly the structure of the yukawa matrices , and the complexity of the problem , in particular , the basis ambiguity from the bottom - up perspective , generally hinders the prospects of deducing even the basic features of the underlying flavour theory from the experimental data . we are left with little alternative but to follow an _ ad hoc _ approach pioneered some time ago by fritzsch @xcite and currently represented by the myriads of proposed effective yukawa textures ( see e.g. @xcite ) whose starting assumption is that ( in some basis ) the yukawa matrices exhibit certain nice features such as symmetries or zeros in specific elements which have become known as `` texture zeros . '' for example , in his classic paper , fritzsch pioneered the idea of having six texture zeros in the 1 - 1 , 2 - 2 , 1 - 3 entries of the hermitian up and down quark yukawa ( or mass ) matrices @xcite . unfortunately , these six - zero textures are no longer consistent with experiment , since they imply the bad prediction @xmath10 , so texture zerologists have been forced to retreat to the ( at most ) four - zero schemes discussed , for example , in @xcite which give up on the 2 - 2 texture zeros allowing the good prediction @xmath11 . however , four - zero textures featuring zeros in the 1 - 1 and 1 - 3 entries of both up and down hermitian mass matrices may also lead to the bad prediction @xmath12 unless @xmath13 results from the cancellation of quite sizeable up- and down - type quark 2 - 3 mixing angles , leading to non - negligible induced 1 - 3 up- and down - type quark mixing @xcite . another possibility is to give up on the 1 - 3 texture zeros , as well as the 2 - 2 texture zeros , retaining only two texture zeros in the 1 - 1 entries of the up and down quark matrices @xcite . here we reject both of these options , and instead choose to maintain up to four texture zeros , without invoking cancellations , for example by making the 1 - 1 element of the up ( but not down ) quark mass matrix nonzero , while retaining 1 - 3 texture zeros in both the up and down quark hermitian matrices , as suggested in @xcite . in this paper we discuss phenomenologically viable textures for hierarchical quark mass matrices which have both 1 - 3 texture zeros and negligible 1 - 3 mixing in both the up and down quark mass matrices . such textures clearly differ from the textures discussed in @xcite and @xcite , but include some cases discussed in @xcite , as remarked above . our main contribution in this paper is to derive quark mixing sum rules applicable to textures of this type , in which @xmath0 is generated from @xmath1 as a result of 1 - 2 up - type mixing , in direct analogy to the lepton sum rules derived in @xcite . another important result of our study is to use the sum rules to show how the right - angled unitarity triangle , i.e. , @xmath2 , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element of either the up or down quark mass matrix being purely imaginary . fritzsch and xing have previously emphasized how their four - zero scheme with 1 - 1 and 1 - 3 texture zeros in the hermitian up and down mass matrices can be used to accommodate right unitarity triangles @xcite , but since their scheme involves large 2 - 3 and non - negligible 1 - 3 up and down quark mixing , our sum rules are not applicable to their case . therefore , the textures in refs . @xcite and @xcite do not allow us to explain @xmath2 by simple structures with a combination of purely real and purely imaginary matrix elements . recently , it has become increasingly clear that current data is indeed consistent with the hypothesis of a right unitarity triangle , with the best fits giving @xmath14 @xcite , and this provides additional impetus for our scheme . the phenomenological observation that @xmath15 has also motivated other approaches ( see e.g. @xcite ) which are complementary to the approach developed in this paper . the layout of the rest of the paper is as follows . in section 2 , we derive the quark mixing sum rules , assuming zero up and down quark 1 - 3 mixing angles . in section 3 , using the sum rules , we discuss the phenomenological viability of quark mass matrix textures with 1 - 3 texture zeros , show how modifications in the up sector can achieve viability , and show how @xmath2 allows each matrix element to be either real or purely imaginary . in section 4 , in the framework of guts , we discuss the implications of zero 1 - 3 mixing for the charged lepton and neutrino sectors , and show that the quark mixing sum rules may be used to yield an accurate prediction for the reactor angle . finally , section 5 concludes the paper . appendix a shows that textures with nonzero 1 - 3 elements in the up sector are disfavoured . the mixing matrix in the quark sector , the cabibbo - kobayashi - maskawa ( ckm ) matrix @xmath16 , is defined as the unitary matrix occurring in the charged current part of the sm interaction lagrangian expressed in terms of the quark mass eigenstates . these mass eigenstates can be determined from the mass matrices in the yukawa sector , namely @xmath17 where @xmath18 and @xmath19 are the mass matrices of the up - type and down - type quarks , respectively . the change from the flavour into the mass basis is achieved via bi - unitary transformations @xmath20 where @xmath21 , @xmath22 , @xmath23 and @xmath24 are unitary @xmath25 matrices . the ckm matrix @xmath26 ( in the `` raw '' form , i.e. before the `` unphysical '' phases were absorbed into redefinitions of the quark mass eigenstate field operators ) is then given by @xmath27 in this paper we shall use the standard ( or so - called particle data group ( pdg ) @xcite ) parameterisation for the ckm matrix ( after eliminating the `` unphysical '' phases ) with the structure [ updg ] u_ckm = r_23u_13r_12 , where @xmath28 denote real ( i.e. orthogonal ) matrices , and the unitary matrix @xmath29 contains the observable phase @xmath30 . for more details , see e.g. @xcite . other alternative parametrisations , motivated by the observation of @xmath2 ( see e.g. @xcite ) have been suggested , but we prefer to stick to the standard one here . let us now suppose that @xmath55 . from the sm point of view , this corresponds to just a convenient choice of basis but , as discussed in the introduction , it becomes a nontrivial assumption at the level of a specific underlying model of flavour . for models where zero 1 - 3 mixing is realised with flavour symmetries , see e.g. @xcite . for @xmath55 , eq . ( [ eq : param3 ] ) simplifies to [ eq : param5 ] u_ckm = u^u_l_12^^u^d_l_23 u^d_l_12 . then , by equating the right - hand sides of eqs . ( [ eq : param4 ] ) and ( [ eq : param5 ] ) and expanding to leading order in the small mixing angles , we obtain the following relations ( up to cubic terms in the physical quark mixing angles ) : [ f1 ] _ 23e^-i_23&= & _ 23^de^-i_23^d -_23^ue^-i_23^u , + [ f2 ] _ 13e^-i_13&= & -_12^ue^-i_12^u ( _ 23^de^-i_23^d - _ 23^ue^-i_23^u ) , + [ f3 ] _ 12e^-i_12&= & _ 12^de^-i_12^d -_12^ue^-i_12^u . let us first consider eq . ( [ f2 ] ) , which can be transformed into @xmath56 where @xmath57 and @xmath58 stand for the measurable 1 - 3 and 2 - 3 mixing angles in the quark sector , respectively . taking the modulus of eq . ( [ eq : theta13withphases ] ) , the 1 - 2 angle entering the up - sector rotation ( @xmath31 ) in the flavour basis obeys @xmath59 where the 1@xmath60 errors are displayed @xcite . similarly , combining eq . ( [ f3 ] ) with eq . ( [ eq : theta13withphases ] ) one receives @xmath61 this , together with the identification eq . ( [ eq : deltafromparam1 ] ) gives rise to the quark sector sum rule , the sum rule may be further simplified to @xmath62 . for similar considerations in the lepton sector , see e.g. @xcite . ] @xmath63 which is valid up to higher order corrections . the present best - fit value and the 1@xmath60 errors are also displayed . needless to say , the relations ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) apply at the scale at which the flavour structure emerges , often close to the scale of grand unification . thus , in principle , the renormalisation group ( rg ) effects should be taken into account . however , due to the smallness of the mixing angles in the quark sector and the hierarchy of the quark masses , the rg corrections to the above relations are very small and can be neglected to a very good approximation . it is interesting that , with the 1 - 2 mixing angles in the up and down sector derived from the physical parameters , the 1 - 2 phase difference in the up and down sectors can also be determined . indeed , combining all three equations ( [ f1])-([f3 ] ) , one obtains @xmath64 using eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) we can solve eq . ( [ eq : deltadu ] ) for @xmath65 and obtain ( at 1@xmath60 level ) @xmath66 which is remarkably close to @xmath67 . we emphasise that this is a consequence of zero 1 - 3 mixing in the up and down sectors , @xmath68 . we now show that , assuming quark textures with negligible 1 - 3 up and down quark mixing , corresponding to 1 - 3 texture zeros for hierarchical quark mass matrices , @xmath65 is approximately equal to @xmath69 . this comes from the definition of the unitarity triangle angle @xmath69 : @xmath70 for the second term in the argument , we can use eqs . ( [ f1 ] ) , ( [ f2 ] ) and ( [ f3 ] ) @xmath71 thus , one can see that the angle @xmath69 is nothing but the phase difference @xmath72 , corresponding to a very simple phase sum rule [ phase ] _ 12^d - _ 12^u . according to the phase sum rule in eq . ( [ phase ] ) , the experimental observation that @xmath2 , or the equivalent determination in eq . ( [ phasediff ] ) , suggests looking at quark mass matrices with 1 - 3 texture zeros and with @xmath74 or @xmath75 at the special values @xmath76 . this would correspond to a set of rather specific textures of the quark mass matrices with , for example , purely imaginary 1 - 2 elements in either @xmath18 or @xmath19 while the 2 - 2 elements remain real . for example , in @xcite the relation between the phases of the mixing angles and the phases of the matrix elements is discussed . for instance , the following patterns naturally emerge : @xmath77 and/or @xmath78 where @xmath79 and @xmath80 are real parameters , and the elements marked by `` * '' are irrelevant as long as the hierarchy of the mass matrix is large enough , or , equivalently , as long as the mixing angles in @xmath81 and @xmath82 are small . these textures are all phenomenologically viable , and consistent with @xmath83 , and their simple phase structure provides a post justification of our assumption of 1 - 3 texture zeros and negligible 1 - 3 up- and down - type quark mixing . however , the above textures are clearly not the most predictive ones and , for example , do not relate the up and down quark 1 - 2 mixing angles to masses . this requires additional assumptions , such as additional texture zeros and hermitian or symmetric matrices , as we now discuss . under the additional assumptions of symmetric or hermitian mass matrices in the 1 - 2 block and zero textures in the 1 - 1 positions of the quark mass matrices , i.e. , @xmath84 we obtain as additional predictions the gatto - sartori - tonin ( gst ) relations @xcite with 1@xmath60 errors displayed , @xmath85 here we already see a conflict in the up sector . the prediction for @xmath86 from the sum rule in eq . ( [ eq : quarkrelation ] ) is quite different ( several @xmath60 away ) from the gst relation above . that suggests that the texture in the up sector should be modified to be in good agreement with experiment . by contrast the prediction from the sum rule in eq . ( [ eq : quarksumrule ] ) for @xmath87 is in very good agreement ( within the errors ) with the gst result in eq . ( [ eq : m_d_over_m_s ] ) , and therefore it is quite plausible to keep the simple texture ansatz for the down sector . combining eqs . ( [ eq : m_u_over_m_c ] ) and ( [ eq : m_d_over_m_s ] ) with the sum rules in eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) , the two relations [ eq : qsplusgst1 ] |_12 - e^- i _ | = and [ eq : qsplusgst2 ] = emerge . we emphasize that these results do not hold for the textures in @xcite where the 2 - 3 up and down quark mixings are large and the 1 - 3 up and down quark mixings are non - negligible . ) . the blue lines indicate the predicted values of @xmath88 for given @xmath89 under the assumptions of section [ sec : sumruleplusgst ] , and the dashed horizontal and vertical black lines ( and solid black lines ) show the @xmath90 errors ( and best - fit values ) for @xmath88 and @xmath89 , respectively . ] the compatibility of eq . ( [ eq : qsplusgst1 ] ) with the experimental results for the down - type quark masses and mixing parameters @xcite is illustrated in fig . 1 . we note that rg running for the quark masses , as well as their potential susy threshold corrections , are very similar for the first two generations and thus cancel out in their ratio . for our estimates , we have considered the running quark masses at the top mass scale @xmath91 @xcite . @xmath88 is extracted for given @xmath89 . the solid blue line shows the relation for best - fit values of the parameters while the dashed blue lines indicate the range with @xmath90 errors included . the dashed horizontal and vertical black lines ( and solid black lines ) show the @xmath90 errors ( and best - fit values ) for @xmath88 and @xmath89 , respectively . the relation of eq . ( [ eq : qsplusgst1 ] ) is well compatible with the present data . future more precise experimental measurements ( for instance at lhcb or @xmath92 factories ) and , in particular , an improved knowledge on @xmath93 ( e.g. from lattice qcd ) are required to test it more accurately . in the following , we consider some examples of possible modifications to the textures in the up sector which are phenomenologically acceptable , while leaving the successful down sector texture unchanged , and retaining the successful real and imaginary scheme which leads to the right unitarity triangle . as discussed in appendix a , the idea of relaxing the up quark 1 - 3 texture zero is disfavoured , so we restrict ourselves to either relaxing the up quark 1 - 1 texture zero , or relaxing symmetry in the 1 - 2 up quark sector , as discussed below . one possible modification is to introduce a nonzero element in the 1 - 1 position of the up quark mass matrix , i.e.@xmath94 as a result , we obtain the up sector relation @xmath95 which allows to adjust @xmath96 , which is of the order of the up quark mass , while @xmath97 has to be equal to the value obtained in eq . ( [ eq : quarkrelation ] ) using the sum rule . for the down sector , there is still the successful prediction from eq . ( [ eq : m_d_over_m_s ] ) leading to the successful sum rule relation of eq . ( [ eq : qsplusgst1 ] ) . furthermore , as discussed in section [ sec : deltackm ] , the dirac phase of the ckm matrix is correct . we note that there exist several variants of the texture . for example , we can choose the 1 - 2 element of @xmath19 real , the 1 - 2 element of @xmath18 purely imaginary and all the other elements also real . these variants are valid as long as @xmath98 is real and @xmath99 is purely imaginary , or _ vice versa_. we emphasize that the elements marked by `` * '' are irrelevant as long as the hierarchy of the mass matrix is large enough , so they may be replaced by zeros or , if the matrices are hermitian , the 3 - 1 elements may be zero while the 3 - 2 elements are determined by hermiticity . since the sum rule in eq . ( [ f1 ] ) shows that @xmath1 is determined only by the difference in 2 - 3 mixing angles in the up and down sectors , it is also possible to set either @xmath100 or @xmath101 equal to zero without changing the physical predictions . in this way it is possible to arrive at some of the four - zero textures discussed , for example , in @xcite . however , we emphasize that here we are additionally assuming the real and imaginary structures consistent with the right unitarity triangle and this was not discussed in @xcite . a second option for a texture consistent with experimental data consists in relaxing the symmetry of the 1 - 2 block in the up sector , while keeping the texture zero in the 1 - 1 position : @xmath102 the two up - sector relations @xmath103 can be simultaneously fulfilled by choosing @xmath104 and @xmath105 appropriately . the prediction from eq . ( [ eq : m_d_over_m_s ] ) and the prediction for @xmath30 do not change and remain compatible with data . we note that there exist several variants of the texture . as before , it is sufficient to have @xmath98 real and @xmath99 purely imaginary , or _ vice versa_. extending the notion of zero 1 - 3 mixing to the lepton sector ( i.e. under the assumption of @xmath106 ) , the presently unknown mixing angle @xmath107 of the leptonic ( mns ) mixing matrix satisfies the relation ( analogous to eq . ( [ eq : quarkrelation ] ) ) ^mns_13 = ^mns_23 ^e_12 , where @xmath108 is the 1 - 2 mixing in the charged lepton mass matrix @xmath109 . this relation has emerged before , for example , in the context of lepton sum rules in @xcite . in many classes of gut models of flavour , the 1 - 2 mixing angles corresponding to @xmath109 and @xmath19 are related by a group theoretical clebsch factor , for example @xmath110 @xcite . in general , it is usually assumed that @xmath111 is of the order of the cabibbo angle , leading to a prediction @xmath112 @xcite . however , in the context of fritzsch - type textures , which are based on hermitian matrices with 1 - 1 , 2 - 2 and 1 - 3 texture zeros , this prediction can be made more precise by using the sum rule which relates @xmath111 to down - type quark masses . thus , applying eq . ( [ eq : quarksumrule ] ) at low energies and taking the present experimental data for the quark mixing angles , and for @xmath113 ( taken from @xcite ) , one can make the rather precise prediction [ eq : mns13mixing ] ^mns_13 = ( 2.84^+0.22_-0.18 ) ^which gives @xmath114 and holds under the assumption of texture zeros in the 1 - 3 elements of the mass matrices ( or more precisely @xmath115 ) and @xmath110 . of course , eq . ( [ eq : mns13mixing ] ) is only a single example out of a larger variety of predictions which may arise in unified flavour models ( see e.g. @xcite ) . we emphasise that the main use of eq . ( [ eq : quarksumrule ] ) in this context is that it allows to `` determine '' the down quark mixing @xmath111 , which is generically involved in relations between quark and lepton mixing angles , from measurable quantities . we note that in the lepton sector , the rg corrections ( see e.g. @xcite ) can be significant , depending on the absolute neutrino mass scale ( and on @xmath116 in a susy framework ) and other effects such as canonical normalisation on the mixing angles can also be sizeable @xcite . furthermore , relaxing the 1 - 1 texture zero in the up quark sector may switch on a nonzero 1 - 3 mixing angle in the neutrino sector via partially constrained sequential dominance @xcite . in this paper we have discussed phenomenologically viable textures for hierarchical quark mass matrices which have both 1 - 3 texture zeros and negligible 1 - 3 mixing in both the up and down quark mass matrices . such textures differ from the textures discussed in @xcite and @xcite . our main contribution in this paper has been to derive quark mixing sum rules applicable to textures of this type , in which @xmath0 is generated from @xmath1 as a result of 1 - 2 up - type mixing , in direct analogy to the lepton sum rules derived in @xcite . an important result of our study is to show how the right - angled unitarity triangle , i.e. , @xmath2 , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element of either the up or down quark mass matrix being purely imaginary . the experimental result that @xmath2 therefore provides an impetus for having hierarchical textures compatible with negligible 1 - 3 mixing in both the up and down quark mass matrices . this is probably the most important take - home message of this paper . the quark mixing sum rules in eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) relate the up and down quark 1 - 2 mixing angles to observable parameters in the ckm matrix . using these sum rules the four - zero texture with 1 - 1 and 1 - 3 texture zeros and a 2 - 1 symmetric or hermitian structure , is shown to be viable for the down quark sector but not for the up quark sector . however , it is possible to have four - zero textures compatible with our sum rules by , for example , filling in the up quark 1 - 1 texture zero , then having hermitian matrices with either of the 2 - 3 elements in the up or down sector set equal to zero as in @xcite . however , we emphasize that here we are additionally assuming the real and imaginary structures consistent with the right unitarity triangle and this was not discussed in @xcite . in the framework of guts , it is natural to have 1 - 3 texture zeros for both the quark and charged lepton sectors , and in such a case we have shown that the quark mixing sum rules may be used to yield an accurate prediction for the reactor angle ; see eq . ( [ eq : mns13mixing ] ) . however , we caution that this prediction is subject to considerable theoretical uncertainty due to the model dependence of the quark - lepton mixing angle relations , rg and canonical normalisation effects , as well as the assumption that the underlying neutrino 1 - 3 mixing angle is zero . indeed , relaxing the 1 - 1 texture zero in the up quark sector will switch on a nonzero 1 - 3 mixing angle in the neutrino sector via partially constrained sequential dominance @xcite . finally , we emphasise that the strategy of exploring particular textures for yukawa matrices , though necessarily rather _ ad hoc _ , is meaningful from the perspective of theories of flavour based on spontaneously broken family symmetry , where , in the flavour basis defined by the high energy theory , simple yukawa matrix structures are expected . indeed , the study of simple yukawa textures may provide the only bottom - up way of deducing a high energy theory of flavour from experimental data . we have shown that @xmath2 may provide a clue towards such a high energy theory of flavour via rather simple yukawa matrices involving 1 - 3 texture zeros whose nonzero elements are either real or purely imaginary . such patterns could be achieved , in principle , by appropriate alignment of the vacuum expectation values of flavour symmetry breaking flavon fields , and in particular their phases . it would be interesting to build a theory of flavour along these lines . we would like to thank jonathan flynn for discussions and christoph luhn for carefully reading the manuscript and providing helpful comments . we are indebted to nordita for the hospitality and support during the programme `` astroparticle physics a pathfinder to new physics '' held in stockholm in march 30 - april 30 , 2009 during which part of this study was performed . sfk is very grateful for the support and hospitality provided by the max - planck - institute of physics during his stay in munich while this paper was being finalized . sa and ms acknowledge partial support by the dfg cluster of excellence `` origin and structure of the universe . '' the work of mm is supported by the royal institute of technology ( kth ) , contract no . sii-56510 . sfk acknowledges partial support from the stfc rolling grant no . st / g000557/1 and a royal society leverhulme trust senior research fellowship . with nonzero 1 - 3 elements , @xmath30 depends not only on @xmath117 but also on other parameters ( in particular @xmath118 and @xmath119 ) and the simple quark mixing sum rules in eqs . ( [ eq : quarkrelation ] ) and ( [ eq : quarksumrule ] ) are no longer valid . examples of this type of texture include ( with real parameter @xmath120 ) @xmath121 and @xmath122 but also variations with different elements chosen either purely imaginary or real . we will demonstrate our approach for this case by means of the texture in eq . ( [ eq : futexture1 ] ) . the starting point is here ( similar to eqs . ( [ f1])-([f3 ] ) ) [ g1 ] _ 23e^-i_23&= & _ 23^de^-i_23^d -_23^ue^-i_23^u , + [ g2 ] _ 13e^-i_13&= & -_13^ue^-i_13^u -_12^u _ 23e^-i(_12^u + _ 23 ) , + [ g3 ] _ 12e^-i_12&= & _ 12^de^-i_12^d -_12^ue^-i_12^u , where we have also neglected terms of order @xmath123 . from our texture ansatz , we know the phases @xmath124 , @xmath125 and @xmath126 . for the values of @xmath127 , we take the values from the gst relations , i.e. eq . ( [ eq : m_u_over_m_c ] ) , which hold here because of the zeros in the 1 - 1 position and the symmetric structure for the first two generations . then we can calculate @xmath128 and @xmath30 in terms of the known quantities and obtain @xmath129 . this result is several standard deviations away from the measurements . beyond the particular example discussed above , we found that the inconsistency of the prediction for @xmath30 also appears in all other cases with @xmath130 , @xmath124 and @xmath131 @xmath132 . furthermore , the same happens for textures with @xmath133 and @xmath134 , where @xmath135 denotes the 1 - 3 element of @xmath19 . we conclude that under these conditions textures with nonzero 1 - 3 elements are disfavoured . s. f. king and c. luhn , arxiv:0908.1897 [ hep - ph ] . c. s. lam , arxiv:0907.2206 [ hep - ph ] . e. ma and g. rajasekaran , phys . d * 64 * ( 2001 ) 113012 [ arxiv : hep - ph/0106291 ] . g. altarelli , arxiv : hep - ph/0611117 ; g. altarelli , f. feruglio and y. lin , nucl . b * 775 * ( 2007 ) 31 [ arxiv : hep - ph/0610165 ] ; 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in analogy with the recently proposed lepton mixing sum rules , we derive quark mixing sum rules for the case of hierarchical quark mass matrices with 1 - 3 texture zeros , in which the separate up and down type 1 - 3 mixing angles are approximately zero , and @xmath0 is generated from @xmath1 as a result of 1 - 2 up type quark mixing . using the sum rules
, we discuss the phenomenological viability of such textures , including up to four texture zeros , and show how the right - angled unitarity triangle , i.e. , @xmath2 , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element being purely imaginary . in the framework of grand unified theories ,
we show how the quark and lepton mixing sum rules may combine to yield an accurate prediction for the reactor angle .
| 13,301 | 222 |
the spatial correlation function of galaxy clusters provides an important cosmological test , as both the amplitude of the correlation function and its dependence upon mean intercluster separation are determined by the underlying cosmological model . in hierarchical models of structure formation , the spatial correlation length , @xmath3 , is predicted to be an increasing function of cluster mass , with the precise value of @xmath3 and its mass dependence determined by @xmath4 ( or equivalently @xmath5 , using the constraint on @xmath6 from the local cluster mass function ) and the shape parameter @xmath7 . low density and low @xmath7 models generally predict stronger clustering for a given mass and a greater dependence of the correlation length upon cluster mass . in this paper we utilize the las campanas distant cluster survey ( lcdcs ) to provide a new , independent measurement of the dependence of the cluster correlation length upon the mean intercluster separation ( @xmath8 ) at mean separations comparable to existing abell and apm studies . we first measure the angular correlation function for a series of subsamples at @xmath9 and then derive the corresponding @xmath3 values via the cosmological limber inversion @xcite . the resulting values constitute the first measurements of the spatial correlation length for clusters at @xmath10 . popular structure formation models predict only a small amount of evolution from @xmath11 to the present - a prediction that we test by comparison of our results with local observations . the recently completed las campanas distant cluster survey is the largest published catalog of galaxy clusters at @xmath12 , containing 1073 candidates @xcite . clusters are detected in the lcdcs as regions of excess surface brightness relative to the mean sky level , a technique that permits wide - area coverage with a minimal investment of telescope time . the final statistical catalog covers an effective area of 69 square degrees within a @xmath13 strip of the southern sky ( @xmath14 @xmath15 mpc at @xmath1=0.5 for @xmath5=0.3 @xmath16cdm ) . gonzalez et al . ( 2001@xmath17 ) also provide estimated redshifts ( @xmath18 ) , based upon the brightest cluster galaxy ( bcg ) magnitude - redshift relation , that are accurate to @xmath215% at @xmath19 , and demonstrate the existence of a correlation between the peak surface brightness , @xmath20 , and velocity dispersion , @xmath21 . together these two properties enable construction of well - defined subsamples that can be compared directly with simulations and observations of the local universe . to compute the two - point angular correlation function , we use the estimator of landy & szalay ( 1993 ) . we measure the angular correlation function both for the full lcdcs catalog and for three approximately velocity dispersion - limited subsamples at @xmath1@xmath20.5 ( figure [ fig : fig1]@xmath17 ) . we restrict all subsamples to @xmath18@xmath220.35 to avoid incompleteness , while the maximum redshift is determined by the surface brightness threshold of the subsample . the angular correlation function for the entire lcdcs catalog is shown in the upper panel of figure [ fig : fig1]@xmath23 , with logarithmic angular bins of width @xmath24=0.2 . modeling this correlation function as a power law , @xmath25 a least - squares fit for all lcdcs clusters over the range 2@xmath26 - 5@xmath27 yields @xmath28=1.83@xmath290.12 and @xmath30=56@xmath31 . the angular correlation function for the lowest redshift subsample is shown in the lower panel of figure [ fig : fig1]@xmath23 , overlaid with a best - fit power law . we derive best - fit values both allowing @xmath28 to vary as a free parameter and also fixing @xmath28=2.1 equivalent to the best - fit value for the lowest redshift subsample and the best fit value for the rosat all - sky survey 1 bright sample @xcite . we then apply a correction to these best - fit values to account for the impact of false detections in the lcdcs catalog , which for this data set act to suppress the amplitude of the observed correlation function . if we assume that the contamination is spatially correlated and can be described by a power law with the same slope as the cluster angular correlation function ( a reasonable approximation because for galaxies which are likely the primary contaminant @xmath28@xmath21.8 - 1.9 [ e.g. 2 , 19 ] ) , then the observed value of @xmath32 is @xmath33 where @xmath34 is the fractional contamination . for detections induced by isolated galaxies of the same magnitude as bcg s at @xmath35 ( and identified as galaxies by the automated identification criteria described in gonzalez et al . ( 2001@xmath17 ) , we measure that @xmath36 is comparable to @xmath37 , the net clustering amplitude for all lcdcs candidates at 0.3@xmath0@xmath18@xmath00.8 . for detections identified as low surface brightness galaxies ( including some nearby dwarf galaxies ) we measure @xmath38 . while these systems are strongly clustered , we expect that they comprise less than half of the contamination in the lcdcs . for multiple sources of contamination the effective clustering amplitude @xmath39 , so the effective clustering strength of the contamination is @xmath40 even including the lsb s . the observed angular correlation function can be used to determine the three - space correlation length if the redshift distribution of the sample is known . this is accomplished via the cosmological limber inversion @xcite . for a power - law correlation function with redshift dependence @xmath41 , @xmath42 the corresponding comoving spatial correlation length is @xmath43 , and the limber equation is @xmath44 } \left [ \frac{\int_{z1}^{z2 } ( dn / dz)^2 e(z ) d_{a}(z)^{1-\gamma } f(z ) ( 1+z ) dz } { \left(\int_{z1}^{z2 } ( dn / dz ) dz\right)^2}\right]^{-1},\ ] ] where @xmath45 is the redshift distribution of the sample , @xmath46 is the angular diameter distance , and @xmath47 is defined as in peebles ( 1993 ) . because little evolution in the clustering is expected over the redshift intervals spanned by our subsamples ( see figure [ fig : cfvslocal ] ) , @xmath41 can be pulled out of the integral . for the lcdcs we estimate the true redshift distributions of our subsamples based upon the observed distribution of estimated redshifts . if we approximate the redshift error distribution as gaussian with @xmath48 at @xmath1=0.5 @xcite , then the actual redshift distribution @xmath45 is approximately equal to the observed redshift distribution @xmath49 for a given subsample convolved with this gaussian scatter . to test the validity of this approach , we also try modeling @xmath45 using the theoretical mass function of sheth & tormen ( 1999 ) convolved with redshift uncertainty . comparing these two methods we find that the derived spatial correlation lengths agree to better than 3% for all subsamples . ccccccc & & & + range & @xmath50 & @xmath3 & @xmath50 & @xmath3 & @xmath50 & @xmath3 + + 0.35 - 0.475 & 38.4 & 15.1@xmath51 & 33.8 & 13.2@xmath52 & 30.9 & 12.1@xmath53 + 0.35 - 0.525 & 46.3 & 18.5@xmath54 & 40.6 & 16.1@xmath55 & 36.9 & 14.7@xmath56 + 0.35 - 0.575 & 58.1 & 22.1@xmath57 & 50.8 & 19.1@xmath58 & 46.0 & 17.3@xmath59 + + + [ tab : ro ] table [ tab : ro ] lists the correlation lengths ( @xmath3 ) and mean intercluster separations ( @xmath50 ) that we derive for the three lcdcs subsamples . the values of @xmath3 and @xmath50 are cosmology - dependent , so we list these quantities for three different cosmologies @xmath16cdm ( @xmath5=0.3,@xmath7=0.2 ) , ocdm ( @xmath5=0.3,@xmath7=0.2 ) , and @xmath60cdm ( @xmath7=0.2 ) . we opt to fix @xmath28=2.1 when deriving the correlation lengths in table [ tab : ro ] because @xmath28 is strongly covariant with @xmath32 and hence poorly constrained by the lcdcs data set . in no instance does this choice alter the derived @xmath3 value by more than 10% from the value obtained when @xmath28 is treated as a free parameter , but we caution that it can systematically bias the observed dependence of @xmath3 upon @xmath50 . for subsamples with best - fit values of @xmath28@xmath222.1 , fixing gamma slightly increases the derived value of @xmath3 , which results in a mild steepening of the dependence of @xmath3 upon @xmath50 for the lcdcs subsamples . the cluster correlation function is predicted to not evolve significantly between @xmath1=0.5 and the present ; we thus compare the lcdcs results directly with local observations , as is shown in figure [ fig : cfvslocal ] for @xmath16cdm and @xmath60cdm . for both cosmologies the lcdcs values of @xmath3 are comparable to those from the edinburgh - durham galaxy catalogue @xcite , apm survey @xcite , and the mx survey northern sample @xcite , but smaller than those found by peacock & west ( 1992 ) for the abell catalog . also shown are the lowest @xmath50 data points for the xbac catalog @xcite , which probe higher masses than our study . we also plot theoretical predictions for comparison with the observational data . results from the virgo consortium hubble volume simulations are shown as a dot - dash line in figure [ fig : cfvslocal]@xmath17 @xcite . the other lines are analytic predictions based upon the work of sheth & tormen ( 1999 ) . like the apm data , the lcdcs results are consistent with the low - density models ( independent of @xmath61 ) . in contrast , the plotted @xmath60cdm model systematically underestimates both the local and @xmath1=0.5 data . @xmath60cdm can be made to match the data only by decreasing @xmath7 to values that are inconsistent with constraints from the galaxy power spectrum @xcite . to assess the robustness of these results , we also quantify potential systematic biases ( see gonzalez et al . 2001@xmath23 for more detail ) . we find that our results can only be significantly altered if the uncertainty in the estimated redshifts is underestimated . if so , then the correlation lengths we derive would be systematically too small ( by @xmath22 @xmath15 mpc if @xmath62=0.25 , for example ) . the second most significant potential systematic arises from fixing @xmath28 , which as mentioned above may lead us to overestimate the dependence of @xmath3 upon @xmath50 . treating @xmath28 as a free parameter would reduce the derived @xmath3 values for the three subsamples ( in order of increasing @xmath50 ) by 0% , 7% , and 10% , but would not qualitatively change our results . the las campanas distant cluster survey is the largest existing catalog of clusters at @xmath1@xmath220.3 , providing a unique sample with which to study the properties of the cluster population . we use the lcdcs to constrain the cluster - cluster angular correlation function , providing the first measurements for a sample with a mean redshift @xmath1@xmath630.2 . from the observed angular correlation function , we derive the spatial correlation length , @xmath3 , as a function of mean separation , @xmath50 . only modest evolution in the clustering amplitude is predicted between @xmath11 and present , and we thus compare our results directly with local data . we find that the lcdcs correlation lengths agree with results from local samples , and observe a dependence of @xmath3 upon @xmath50 that is comparable to the results of croft et al . ( 1997 ) for the apm catalog . this clustering strength , its dependence on number density , and its minimal redshift evolution are consistent with analytic expectations for low density models , and with results from the @xmath16cdm hubble volume simulations . consequently , while statistical uncertainty limits our ability to discriminate between cosmological models , our results are in concordance with the flat @xmath16cdm model favored by recent supernovae and cosmic microwave background observations @xcite . a final result of this analysis is that it demonstrates the utility of large catalogs like the lcdcs that are statistical in nature . while the properties of any particular cluster in the lcdcs catalog are rather uncertain , the properties of the sample as a whole are well - defined , which is sufficient for constraining properties such as the clustering strength and evolution in the comoving number density . this statistical approach requires a relatively small investment in telescope time , and so can be extended in the future to much larger samples than the lcdcs . abadi , m. g. , lambas , d. g. , & muriel , h. 1998 , , 507 , 526 cabanac , r. a. , de lapparent , v. , & hickson , p. 2000 , , 364 , 349 colberg et al . 2000 , , 319 , 209 croft , r. a. c. , dalton , g. b. , efstathiou , g. , sutherland , w. j. , & maddox , s. j. 1997 , , 291 , 305 de bernardis , p. et al . 2001 , astro - ph/0105296 efstathiou , g. , bernstein , g. , katz , n. , tyson , a. j. , & guhathakurta , p. 1991 , , 380 , 47 eisenstein , d. j. & zaldarriaga , m. 2000 , , 546 , 2001 gonzalez , a. h. , zaritsky , d. , dalcanton , j. j. , & nelson , a. e. 2001@xmath17 , accepted by gonzalez , a. h. , zaritsky , d. , & wechsler , r. 2001@xmath23 , submitted to hudon , j. d. , & lilly , s. j. 1996 , , 469 , 519 landy , s. d. , & szalay , a. s. 1993 , , 412 , 64 miller , c. j. , batuski , d. j. , slinglend , k. a. , hill , j. m. 1999 , , 523 , 492 moscardini , l. , matarrese , s. , de grandi , s. , & lucchin , f. 2000 , , 314 , 647 nichol , r. c. , collins , c. a. , guzzo , l. , lumsden , s. l. 1992 , , 255 , 21 peacock , j. a. & west , m. j. 1992 , , 259 , 494 peebles , p. j. e. 1980 , the large - scale structure of the universe ( princeton : princeton university press ) peebles , p. j. e. 1993 , physical cosmology ( princeton : princeton university press ) pryke , c. et al . 2001 , astro - ph/0104490 roche , n. , & eales , s. a. 1999 , , 307 , 703 riess , a. g. et al . 2001 , astro - ph/0104455 sheth , r. k. & tormen , g. 1999 , , 308 , 119
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we utilize a sample of galaxy clusters at 0.35@xmath0@xmath1@xmath00.6 drawn from the las campanas distant cluster survey ( lcdcs ) to provide the first non - local constraint on the cluster - cluster spatial correlation function .
the lcdcs catalog , which covers an effective area of 69 square degrees , contains over 1000 cluster candidates . estimates of the redshift and velocity dispersion exist for all candidates , which enables construction of statistically completed , volume - limited subsamples . in this analysis we measure the angular correlation function for four such subsamples at @xmath1@xmath20.5 . after correcting for contamination ,
we then derive spatial correlation lengths via limber inversion .
we find that the resulting correlation lengths depend upon mass , as parameterized by the mean cluster separation , in a manner that is consistent with both local observations and cdm predictions for the clustering strength at @xmath1=0.5 .
| 4,631 | 249 |
coined quantum walks ( qws ) on graphs were firstly defined in ref . @xcite and have been extensively analyzed in the literature @xcite . many experimental proposals for the qws were given previously @xcite , with some actual experimental implementations performed in refs . the key feature of the coined qw model is to use an internal state that determines possible directions that the particle can take under the action of the shift operator ( actual displacement through the graph ) . another important feature is the alternated action of two unitary operators , namely , the coin and shift operators . although all discrete - time qw models have the `` alternation between unitaries '' feature , the coin is not always necessary because the evolution operator can be defined in terms of the graph vertices only , without using an internal space as , for instance , in szegedy s model @xcite or in the ones described in refs . @xcite . more recently , the staggered quantum walk ( sqw ) model was defined in refs . @xcite , where a recipe to generate unitary and hermitian local operators based on the graph structure was given . the evolution operator in the sqw model is a product of local operators . moves the particle to the neighborhood of @xmath0 , but not further away . ] the sqw model contains a subset of the coined qw class of models @xcite , as shown in ref . @xcite , and the entire szegedy model @xcite class . although covering a more general class of quantum walks , there is a restriction on the local evolution operations in the sqw demanding hermiticity besides unitarity . this severely compromises the possibilities for actual implementations of sqws on physical systems because the unitary evolution operators , given in terms of time - independent hamiltonians having the form @xmath1 , are non - hermitian in general . to have a model , that besides being powerful as the sqw , to be also fitted for practical physical implementations , it would be necessary to relax on the hermiticity requirement for the local unitary operators . in this work , we propose an extension of the sqw model employing non - hermitian local operators . the concatenated evolution operator has the form @xmath2 where @xmath3 and @xmath4 are unitary and hermitian , @xmath5 and @xmath6 are general angles representing specific systems energies and time intervals ( divided by the planck constant @xmath7 ) . the standard sqw model is recovered when @xmath8 and @xmath9 . with this modification , sqw with hamiltonians encompasses the standard sqw model and includes new coined models . besides , with the new model , it is easier to devise new experimental proposals such as the one described in ref . @xcite . [ sqwwh_graph1 ] depicts the relation among the discrete - time qw models . szegedy s model is included in the standard sqw model class , which itself is a subclass of the sqw model with hamiltonians . flip - flop coined qws that are in szegedy s model are also in the sqw model . flip - flop coined qws using hadamard @xmath10 and grover @xmath11 coins , as represented in fig . [ sqwwh_graph1 ] , are examples . there are coined qws , which are in the sqw model with hamiltonians general class , but not in the standard sqw model , as for example , the one - dimensional qws with coin @xmath12 , where @xmath13 is the pauli matrix @xmath14 , with angle @xmath15 not a multiple of @xmath16 . those do not encompass all the possible coined qw models , as there are flip - flop coined qws , which although being built with non - hermitian unitary evolution , can not be put in the sqw model with hamiltonians . for instance , when the fourier coin @xmath17 is employed , where @xmath18 and @xmath19 , being @xmath20 the hilbert space dimension . the structure of this paper is as follows . in sec . [ sec2 ] , we describe how to obtain the evolution operator of the sqw with hamiltonians on a generic simple undirected graph . in sec . [ sec3 ] , we calculate the wave function using the fourier analysis for the one - dimensional lattice and the standard deviation of the probability distribution . in sec . [ sec4 ] , we characterize which coined qws are included in the class of sqws with hamiltonians . finally , in sec . [ sec5 ] we draw our conclusions . let @xmath21 be a simple undirected graph with vertex set @xmath22 and edge set @xmath23 . a tessellation of @xmath24 is a partition of @xmath22 so that each element of the partition is a clique . a clique is a subgraph of @xmath24 that is complete . an element of the partition is called a polygon . the tessellation covers all vertices but not necessarily all edges . let @xmath25 be the hilbert space spanned by the computational basis @xmath26 , that is , each vertex @xmath0 is associated with a vector @xmath27 of the canonical basis . each polygon spans a subspace of the @xmath25 , whose basis comprises the vectors of the computational basis associated with the vertices in the polygon . let @xmath28 be the number of polygons and let @xmath29 be a polygon for some @xmath30 . a unit vector _ induces _ polygon @xmath29 if the following two conditions are fulfilled : first , the vertices of @xmath29 is a clique in @xmath24 . second , the vector has the form @xmath31 so that @xmath32 for @xmath33 and @xmath34 otherwise . the simplest choice is the uniform superposition given by @xmath35 for @xmath33 . there is a recipe to build a unitary and hermitian operator associated with the tessellation , when we use the following structure : @xmath36 @xmath3 is unitary because the polygons are non - overlapping , that is , @xmath37 for @xmath38 . @xmath3 is hermitian because it is a sum of hermitian operators . then , @xmath39 . an operator of this kind is called an _ orthogonal reflection _ of graph @xmath24 . each @xmath29 induces a polygon and we say that @xmath3 induces the tessellation . the idea of the staggered model is to define a second operator that must be independent of @xmath3 . define a second tessellation by making another partition of @xmath24 with polygons @xmath40 for @xmath41 , where @xmath42 is the number of polygons . for each polygon @xmath40 , define unit vectors @xmath43 so that @xmath44 for @xmath45 and @xmath46 otherwise . the simplest choice is the uniform superposition given by @xmath47 for @xmath45 . likewise , define @xmath48 @xmath4 is an orthogonal reflection . to obtain the evolution operator we demand that the union of tessellations @xmath49 and @xmath50 should cover the edges of @xmath24 , where tessellation @xmath49 is the union of polygons @xmath29 for @xmath51 and tessellation @xmath50 is the union of polygons @xmath40 for @xmath41 . this demand is necessary because edges that do not belong to the tessellation union can be removed from the graph without changing the dynamics . the standard sqw dynamics is given by the evolution operator @xmath52 where the unitary and hermitian operators @xmath3 and @xmath4 are constructed as described in eqs . ( [ h_0 ] ) and ( [ h_1 ] ) . however , such graph - based construction of the operators does not correspond , in general , to the evolution of the real physical systems which are unitary but non - hermitian instead . actually , the unitary and non - hermitian operators do not have a nice representation as in eqs . ( [ h_0 ] ) and ( [ h_1 ] ) . in the following , we introduce and analyze a method for constructing `` physical evolutions '' using the graph - based unitary and hermitian operators . we define the staggered qw model with hamiltonians by the evolution operator @xmath53 where @xmath5 and @xmath6 are angles . @xmath54 can be written as @xmath55 the standard sqw model is obtained when @xmath56 and @xmath57 . the staggered qw model with hamiltonians is characterized by two tessellations and the angles @xmath5 and @xmath6 . the evolution operator is the product of two _ local _ unitary operators . local in the sense discussed before , that is , if a particle is on vertex @xmath0 , it will move to the neighborhood of @xmath0 only . some graphs are not 2-tessellable as discussed in ref . @xcite . in this case , we have to use more than two tessellations until covering all edges and eq . ( [ u ] ) must be extended accordingly . one of the simplest example of a sqw model with hamiltonians is the one - dimensional lattice ( or chain ) as in fig . [ 1dlattice ] . if we wish to use the minimum number of tessellations that cover all vertices and edges , the only choice are the two tesselations represented in the figure and correspond to two alternate interactions between first neighbors . therefore the evolution operator in the one - dimensional case with @xmath58 is given by @xmath59 where @xmath60 and @xmath61 for the sake of simplicity , we choose @xmath49 and @xmath50 to be independent from @xmath62 . @xmath54 is defined on hilbert space @xmath25 , whose computational basis is @xmath63 . while the diagonal forms of the hamiltonians ( [ h0 ] ) and ( [ h1 ] ) with @xmath64-eigenvectors ( [ eq : genual_0 ] ) and ( [ eq : genual_1 ] ) , respectively , are more appropriate to the qw related computations , one can not immediately see the connections to interactions energies that they usually represent . for actual implementations , it is more convenient to write down it in terms of bosonic operators as @xmath65 in that form the first term represents the occupations of each site and the second one represents hopping hamiltonians . note that since the qw models considered here are single particles quantum walks , the corresponding picture in terms of hamiltonians ( [ h00 ] ) and ( [ h11 ] ) implementation is to consider a single excitation in the encoding physical system . the joint hamiltonian @xmath66 describes a large number of physical systems , from cold atoms trapped in optical lattices @xcite to a linear array of electromechanical resonators @xcite . however the alternated action of the two local unitary operators in ( [ unit ] ) requires that the hamiltonians @xmath3 and @xmath4 be applied independently . this requires a more involved process of alternating interactions in the system , which demands an external control particular to each physical system . a proposal on how to implement it in a one dimensional array of coupled superconducting transmission line resonators is discussed elsewhere @xcite . to start our analysis , in fig . [ fig : sqwwh_graph2 ] we show the probability distribution for the 1d sqw with hamiltonians ( [ h0 ] ) and ( [ h1 ] ) after 60 steps with parameters @xmath67 , @xmath68 , and @xmath69 . the initial condition assumed was @xmath70 . a quantum walk with those parameters was analyzed by ref . note the typical profile , which is similar to the coined qw , but certainly not to the continuous - time qw @xcite . , @xmath68 , @xmath69 , and initial condition @xmath70 . ] in order to find the spectral decomposition of the evolution operator , we perform a basis change that takes advantage of the system symmetries . let us define the fourier basis by the vectors @xmath71 where @xmath72 $ ] . for a fixed @xmath73 , those vectors define a plane that is invariant under the action of the evolution operator , which is confirmed by the following results : @xmath74 where @xmath75 the analysis of the dynamics can be reduced to a two - dimensional subspace of @xmath25 by defining a reduced evolution operator @xmath76.\ ] ] @xmath77 is unitary since @xmath78 a vector in this subspace is mapped to hilbert space @xmath25 after multiplying its first entry by @xmath79 and its second entry by @xmath80 . the eigenvalues of @xmath77 ( the same of @xmath54 ) are @xmath81 , where @xmath82 note that @xmath83 in ( [ eq : a ] ) depends on @xmath73 , as well as others parameters . the non - trivial eigenvectors of @xmath77 are @xmath84 where @xmath85 the eigenvectors of the evolution operator @xmath54 associated with eigenvalues @xmath81 are @xmath86 and we can write @xmath87 , @xmath68 , @xmath69 , and initial condition @xmath88 . ] if we take @xmath89 as the initial condition , the quantum walk state at time @xmath90 is given by @xmath91 where @xmath92 and @xmath93 the probability distribution is obtained after calculating @xmath94 and @xmath95 . the probability distribution would not be symmetric in this case ( localized initial condition ) , as can be seen in fig . [ fig : sqwwh_graph2a ] . those results extend the corresponding ones obtained in ref . @xcite . the results of ref . @xcite can be extended in order to include parameter @xmath15 of the sqw with hamiltonians . the asymptotic expression for the odd moments with initial condition @xmath89 is @xmath96^{2n}dk+o(t^{2n-2}),\ ] ] and for the even moments is @xmath97 the square of the standard deviation is @xmath98 for @xmath99 and @xmath69 , it simplifies asymptotically to @xmath100 as function of @xmath15 and @xmath49 . the value of @xmath101 at the center of the plot is zero . ] [ fig : sqwwh_graph5 ] shows the plot of @xmath101 as function of @xmath15 and @xmath49 . the maximum value of @xmath101 is @xmath102 , which is achieved for the points on a circle with center at @xmath103 and radius @xmath104 , for instance , @xmath105 when @xmath106 and @xmath68 . when @xmath68 and @xmath69 , @xmath3 is the direct sum of pauli @xmath13 matrices @xmath107 , \ ] ] likewise @xmath4 , with a diagonal shift of one entry . any flip - flop coined qw on a graph @xmath21 with a coin operator of the form @xmath108 , where @xmath3 is an orthogonal reflection of @xmath24 , is equivalent to a sqw with hamiltonians on a larger graph @xmath109 . the procedure to obtain @xmath109 is described in ref . we briefly review it in the next paragraph . and a degree-3 vertex @xmath110 . edge @xmath111 has label @xmath112 . the other edges have labels @xmath113 to @xmath114 . ] , one has to replace a degree-@xmath115 vertex by a @xmath115-clique . the vertex labels of the enlarged graph have the form `` @xmath116 '' , where @xmath0 is the label of the vertex in the original graph and @xmath117 is the edge incident on @xmath0 . ] let @xmath118 be the vertex labels and let @xmath119 be the edge labels of graph @xmath24 . the action of the flip - flop shift operator on vectors of the computational basis associated with @xmath24 is @xmath120 where @xmath0 and @xmath110 are adjacent and @xmath112 is the label of the edge @xmath111 as shown in fig . [ fig : sqwwh_graph6a ] . @xmath121 as @xmath122 for all edges @xmath111 . the @xmath64-eigenvectors are @xmath123 and there is a @xmath64-eigenvector for each edge @xmath111 . we are assuming that @xmath124 . then @xmath125 @xmath126 induces the red polygons of fig . [ fig : sqwwh_graph6b ] . after replacing each degree-@xmath115 vertex of @xmath24 by a @xmath115-clique , we obtain graph @xmath127 of fig . [ fig : sqwwh_graph6b ] on which the equivalent sqw is defined . the degree-5 vertex is converted into a 5-clique and the degree-3 vertex is converted into a 3-clique . the vertex labels of @xmath127 have the form `` @xmath116 '' , where @xmath0 is the label of the vertex in the original graph and @xmath117 is the edge incident on @xmath0 . with this notation , it is straightforward to check that the unitary and hermitian operator that induces the red tessellation is @xmath126 given by eq . ( [ s ] ) , when we use vectors in uniform superposition . now , we can cast the evolution operator in the form demanded by the staggered model with hamiltonians . since @xmath128 , the shift operator can be put in the form e@xmath129 with @xmath130 and @xmath131 modulo a global phase . if the coin is @xmath108 and @xmath3 is an orthogonal reflection then any flip - flop coined qw on @xmath24 is equivalent to a sqw on @xmath127 with evolution operator @xmath132 operator @xmath3 induces the blue tessellation depicted in fig . [ fig : sqwwh_graph6b ] . it is known that grover s algorithm @xcite can be described as a coined qw on the complete graph using a flip - flop shift operator and the grover coin @xcite . therefore , grover s algorithm can also be reproduced by the sqw model @xcite . extensions of grover s algorithm analyzed by long _ et al . _ @xcite and hyer @xcite use operator @xmath133 where @xmath134 is the unit uniform superposition of the computational basis and @xmath135 is an angle , in place of the usual grover operator @xmath136 . this kind of extension can be reproduced by sqw model with hamiltonians because @xmath108 when @xmath3 is given by eq . ( [ h_0 ] ) can be written as @xmath137 modulo a global phase . we can choose values for @xmath5 and @xmath28 that reproduce eq . ( [ new_form ] ) . we have introduced an extension of the standard staggered qw model by using orthogonal reflections as hamiltonians . orthogonal reflections are local unitary operators in the sense that they respect the connections represented by the edges of a graph . besides , orthogonal reflections are hermitian by definition . this means that if @xmath3 is an orthogonal reflection of a graph @xmath24 , then @xmath1 is a local unitary operator associated with @xmath24 . in order to define a nontrivial evolution operator , we need to employ a second orthogonal reflection @xmath4 of @xmath24 . the generic form of the evolution operator of the sqw with hamiltonians for 2-tessellable graphs is @xmath138 , where @xmath5 and @xmath6 are angles . this form is fitted for physical implementations in many physical systems , such as , cold atoms trapped in optical lattices @xcite and arrays of electromechanical resonators @xcite . we have obtained the wave function of sqws with hamiltonians on the line and analyzed the standard deviation of the probability distribution . for a localized initial condition at the origin , the maximum spread of the probability distribution for an evolution operator of the form @xmath139 is obtained when @xmath106 . we have also characterized the class of coined qws that are included in the sqw model with hamiltonians and we have described how to convert those coined qws on a graph @xmath24 into their equivalent formulation in terms of sqws on an extended graph obtained from @xmath24 by replacing degree-@xmath115 vertices into @xmath115-cliques . as a last remark , we call attention that recently it was shown numerically that searching one marked vertex using the original sqw on the two - dimensional square lattice has no speedup compared to classical search using random walks @xcite . on the other hand , the sqw with hamiltonians with @xmath67 is able to find the marked vertex after @xmath140 steps at least as fast as the equivalent algorithm using coined quantum walks @xcite . rp acknowledges financial support from faperj ( grant n. e-26/102.350/2013 ) and cnpq ( grants n. 303406/2015 - 1 , 474143/2013 - 9 ) and also acknowledges useful discussions with pascal philipp and stefan boettcher . jkm acknowledges financial support from cnpq grant pdj 165941/2014 - 6 . mco acknowledges support by fapesp through the research center in optics and photonics ( cepof ) and by cnpq . dorit aharonov , andris ambainis , julia kempe , and umesh vazirani . quantum walks on graphs . in _ proceedings of the thirty - third annual acm symposium on theory of computing _ , stoc 01 , pages 5059 , new york , ny , usa , 2001 . acm . j. lozada - vera , a. carrillo , o. p. de s neto , j. khatibi moqadam , m. d. lahaye , and m. c. de oliveira . quantum simulation of the anderson hamiltonian with an array of coupled nanoresonators : delocalization and thermalization effects . , 3(9):116 , 2016 .
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quantum walks are recognizably useful for the development of new quantum algorithms , as well as for the investigation of several physical phenomena in quantum systems
. actual implementations of quantum walks face technological difficulties similar to the ones for quantum computers , though .
therefore , there is a strong motivation to develop new quantum - walk models which might be easier to implement . in this work
, we present an extension of the staggered quantum walk model that is fitted for physical implementations in terms of time - independent hamiltonians .
we demonstrate that this class of quantum walk includes the entire class of staggered quantum walk model , szegedy s model , and an important subset of the coined model .
| 6,171 | 161 |
there is a strong observational evidence that active galactic nuclei ( agn ) , x - ray transients and gamma - ray bursts ( grbs ) are associated with accretion onto black holes , and that these sources are able to form collimated , ultrarelativistic flows ( relativistic jets ) . the exact mechanisms to explain the production of jets are still uncertain , but they probably involve the interaction between a spinning black hole , the accretion disk and electromagnetic fields in strong gravitational fields ( see , for example , @xcite-@xcite and references therein ) . thus , a reasonably accurate general relativistic model of an agn would require an exact solution of einstein - maxwell field equations that describes a superposition of a kerr black hole with a stationary disk and electromagnetic fields . not even an exact solution of a stationary black hole - disk system has been found yet . solutions for static thin disks without radial pressure were first studied by bonnor and sackfield @xcite , and morgan and morgan @xcite , and with radial pressure by morgan and morgan @xcite . several classes of exact solutions of the einstein field equations corresponding to static thin disks with or without radial pressure have been obtained by different authors @xcite-@xcite . thin rotating disks were considered in @xcite . perfect fluid disks with halos @xcite and charged perfect fluid disks @xcite were also studied , whereas thick relativistic disks were reported in @xcite . several solutions of the einstein equations coupled to matter that represent disks has also been studied by the jena group @xcite-@xcite . the static superposition of a disk and a black hole was first considered by lemos and letelier @xcite . zellerin and semerk @xcite found a stationary metric that reduces to the superposition of a disk and a black hole in the static limit and thus may represent a stationary disk - black hole system . the analysis of their solution is complicated by the fact that the metric functions can not be analytically computed . for a survey on self gravitating relativistic thin disks , see for instance @xcite . the aim of this paper is to consider the gravitational field of an agn through a simple model : the _ static _ superposition of a black hole with a chazy - curzon disk and two rods placed on the symmetry axis , which will represent jets . our principal interest here is to see how the presence of the rods affect the matter properties and stability of the disk . the article is divided as follows . in sec . [ sec_disk ] we review the `` displace , cut and reflect '' method used to construct thin disks from a known solution of einstein field equations in weyl coordinates . [ sec_sup ] summarizes the formalism to superpose thin disks and other weyl solutions . [ sec_sol ] discusses schwarzschild solution and the metric of a finite rod in weyl coordinates . in sec . [ sec_model ] the results of sec . [ sec_sup ] and [ sec_sol ] are then applied to construct the superposition of disk , black hole and rods and the resulting energy - momentum tensor . in sec . [ sec_osc ] the disk stability is studied through small horizontal and vertical oscillations about equatorial circular geodesics . in sec . [ sec_orbits ] some geodesic orbits for the superposed metric are numerically calculated . finally , sec . [ sec_discuss ] is devoted to discussion of the results . we take units such that @xmath0 . in absense of matter , the general metric for a static axially symmetric spacetime in weyl s canonical coordinates @xmath1 is given by @xmath2 where @xmath3 and @xmath4 are functions of @xmath5 and @xmath6 only . einstein vacuum field equations for the metric eq . ( [ eq_weyl_metric ] ) yield @xcite @xmath7=\frac{1}{2}\int r \left [ ( \phi_{,r}^2-\phi_{,z}^2)\mathrm{d}r+ 2\phi_{,r}\phi_{,z}\mathrm{d}z \right ] \label{eq_eins2 } \mbox{.}\end{aligned}\ ] ] given a solution of eq . ( [ eq_eins1])-([eq_eins2 ] ) , one can construct a thin disk by using the well known `` displace , cut and reflect '' method , due to kuzmin @xcite . first , a surface @xmath8 is chosen so that it divides the usual space in two parts : one with no singularities or sources , and the other with them . then the part of the space with singularities or sources is disregarded . at last , the surface is used to make an inversion of the nonsingular part of the space . the result will be a space with a singularity that is a delta function with support on @xmath9 . the method is mathematically equivalent to make a transformation @xmath10 , where @xmath11 is a constant . the application of the formalism of distributions in curved spacetimes to the weyl metric eq . ( [ eq_weyl_metric ] ) is exposed in @xcite . one finds that the components of the distributional energy - momentum tensor @xmath12 $ ] on the disk are @xmath13 where @xmath14 is the dirac distribution with support on the disk and is understood that @xmath15 . the `` true '' energy density and azimuthal pressure are , respectively , @xmath16 to explain the disk stability in absence of radial pressure , one may assume the counterrotating hypothesis , where the particles on the disk move in such a way that there are as many particles moving in clockwise as in counterckockwise direction . the velocity @xmath17 of counterrotation of the particles in the disk is given by @xcite @xmath18 if @xmath19 , the particles travel at sublumial velocities . the specific angular momentum @xmath20 of particles on the disk moving in circular orbits along geodesics reads @xmath21 the stability of circular orbits on the disk plane can be determined with an extension of rayleigh criteria of stability of a fluid at rest in a gravitational field : @xmath22 @xcite . using eq . ( [ eq_ang_mom ] ) this is equivalent to @xmath23 an important property of the weyl metric eq . ( [ eq_weyl_metric ] ) is that the field equation ( [ eq_eins1 ] ) for the potential @xmath3 is the laplace equation in cylindrical coordinates . since laplace s equation is linear , if @xmath24 and @xmath25 are solutions , then the superposition @xmath26 is also a solution . the other metric function eq . ( [ eq_eins2 ] ) is nonlinear , and so can not be superposed . but one can show that the relation @xmath27=\nu[\phi_1]+\nu[\phi_2]+2\nu[\phi_1,\phi_2 ] \mbox{,}\ ] ] where @xmath28=\frac{1}{2}\int r[(\phi_{1,r}\phi_{2,r}-\phi_{1,z } \phi_{2,z})\mathrm{d}r+(\phi_{1,r}\phi_{2,z}+\phi_{1,z}\phi_{2,r})\mathrm{d}z ] \mbox{,}\ ] ] holds . other useful relations are given in @xcite . the energy - momentum tensor of the combined system disk and black hole has been computed by lemos and letelier @xcite . let @xmath29 and @xmath30 be the metric potentials of the disk and of the black hole , respectively . then the components @xmath12 $ ] of the superposition are @xmath31 \phi_{d , z } \delta(z ) \mbox { , } \label{eq_ttt_sup } \\ t^{\varphi}{}_{\varphi } & = e^{\phi_{d}+\phi_{bh}-\nu}r(\phi_{d}+\phi_{bh})_{,r } \phi_{d , z } \delta(z ) \mbox { , } \label{eq_tphi_sup } \\ t^r{}_r & = t^z{}_z=0 \mbox{,}\end{aligned}\ ] ] where @xmath32 $ ] , and again @xmath33 . the `` true '' energy density and azimuthal pressure read @xmath34 eq . ( [ eq_ttt_sup])-([eq_tphi_sup ] ) show that the potential of the black hole interacts with the disk and changes its matter properties . although eq . ( [ eq_ttt_sup])-([eq_p_sup ] ) have been derived for superposition of disk and black hole , they are also valid when the potential function @xmath30 is a sum of other weyl solutions , like the superposition of a black hole and rods . the schwarzschild black hole metric function @xmath30 in weyl coordinates is given by @xmath35 where @xmath36 and @xmath37 . the function @xmath38 can be related to the newtonian potential @xmath39 by @xmath40 thus , the metric potential @xmath41 of a finite rod of linear mass density @xmath42 lying on the @xmath6 axis and located along @xmath43 $ ] is @xmath44 \mbox{.}\ ] ] the calculation of the other metric function @xmath4 for eq . ( [ eq_sch1 ] ) , ( [ eq_bar1 ] ) and later for the superposed metric , is considerably simplified when one defines the following @xmath45 function @xmath46 where @xmath47 is an arbitrary constant . this function is a natural consequence of the formalism of the inverse scattering method @xcite . ( [ eq_sch1 ] ) and ( [ eq_bar1 ] ) can be rewritten as @xmath48 where we defined @xmath49 on using eq . ( [ eq_nu_rel1 ] ) @xmath50=\nu[\ln \mu_i]+\nu[\ln \mu_j]-2\nu[\ln \mu_i,\ln \mu_j ] \mbox{;}\ ] ] the result @xmath51=\ln ( \mu_i-\mu_j ) \mbox{,}\ ] ] which also follows from the inverse scattering method ; and the identity @xmath52 one obtains following expressions for the metric function @xmath4 @xmath53 \mbox { , } \label{eq_nu_bh}\\ \nu_{r } & = 4\lambda^2 \ln \left [ \frac{(r^2+\mu_3 \mu_4)^2}{(r^2+\mu_3 ^ 2)(r^2+\mu_4 ^ 2 ) } \right ] \mbox{. } \label{eq_nu_rod}\end{aligned}\ ] ] we now consider the superposition illustrated in fig . [ fig_1 ] : a black hole with mass @xmath54 whose center is on @xmath9 , two rods with equal mass density @xmath42 , each one with mass @xmath55 located along @xmath56 $ ] and @xmath43 $ ] on the @xmath6 axis , and a disk on the plane @xmath9 constructed with the `` displace , cut and reflect method '' from the chazy - curzon solution with mass @xmath57 , whose singularity lies on @xmath58 : @xmath59 it should be remembered that in weyl coordinates a black hole with mass @xmath54 is represented by a rod with length @xmath60 , thus in fig . [ fig_1 ] we put a dotted circle around the rod in the middle . , two rods and a chazy - curzon disk on the plane @xmath9 . ] such a configuration is not gravitationally stable : a consequence of the nonlinearity of eq . ( [ eq_eins2 ] ) is the appearence of gravitationally inert singular structures like struts between the rods and the black hole that keeps them appart ; also in the superposition of the disk with the black hole , superlumial regions @xmath61 exist because there is matter up to the event horizon . the metric function @xmath3 of the superposition can be expressed as @xmath62 with @xmath63 and @xmath64 . now we consider the case when both rods just touch the horizon of the black hole , that is , when @xmath65 . then @xmath66 and @xmath67 . from eq . ( [ eq_ttt_sup])-([eq_tphi_sup ] ) and eq . ( [ eq_vel ] ) , we get following conditions : @xmath68 \notag \\ & + 2\lambda ( \tilde{r}^2+\tilde{a}^2)^{3/2 } \left ( \sqrt{\tilde{r}^2+\tilde{c}_2 ^ 2 } -\tilde{c}_2\sqrt{1+\tilde{r}^2 } \right ) > 0 \mbox { , } \label{eq_cond_sigma } \\ p & > 0 \rightarrow \sqrt{\tilde{r}^2+\tilde{c}_2 ^ 2 } \left[\alpha \tilde{r}^2\sqrt{1+\tilde{r}^2 } + ( \tilde{r}^2+\tilde{a}^2)^{3/2 } \right ] \notag \\ & -2\lambda ( \tilde{r}^2+\tilde{a}^2)^{3/2 } \left ( \sqrt{\tilde{r}^2+\tilde{c}_2 ^ 2 } -\tilde{c}_2\sqrt{1+\tilde{r}^2 } \right ) > 0 \mbox { , } \label{eq_cond_p } \\ v^2 & < 1 \rightarrow \sqrt{\tilde{r}^2+\tilde{c}_2 ^ 2 } \left [ 2\alpha \tilde{r}^2\sqrt{1+\tilde{r}^2 } + ( \tilde{r}^2+\tilde{a}^2)^{3/2}(2-\sqrt{1+\tilde{r}^2 } ) \right ] \notag \\ & -4\lambda ( \tilde{r}^2+\tilde{a}^2)^{3/2 } \left(\sqrt{\tilde{r}^2+\tilde{c}_2 ^ 2}-\tilde{c}_2\sqrt{1+\tilde{r}^2 } \right ) < 0 \mbox { , } \label{eq_cond_v}\end{aligned}\ ] ] where @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 and @xmath74 . the conditions imposed are that of weak energy @xmath75 , azimuthal pressure @xmath76 and sublumial velocity @xmath77 of counterrotation of particles on the disk . for @xmath78 , all three conditions are satisfied . in the regions where @xmath19 , the weak energy condition is always satisfied , as can be seen by inequalities ( [ eq_cond_sigma ] ) and ( [ eq_cond_v ] ) . ( dotted curves ) and of @xmath79 ( solid curves ) for the chazy - curzon disk in presence of a black hole and two rods . we keep the mass of each rod constant @xmath80 and vary its mass density @xmath42 . in ( a)-(b ) we take , respectively , @xmath81 and @xmath82.,title="fig : " ] + ( dotted curves ) and of @xmath79 ( solid curves ) for the chazy - curzon disk in presence of a black hole and two rods . we keep the mass of each rod constant @xmath80 and vary its mass density @xmath42 . in ( a)-(b ) we take , respectively , @xmath81 and @xmath82.,title="fig : " ] fig . [ fig_2](a)-(c ) shows curves of @xmath83 ( dotted curves ) and of @xmath79 ( solid curves ) as function of @xmath42 and @xmath84 for three different values of @xmath85 . the mass of each rod is kept constant @xmath80 and the cut parameter takes values @xmath81 and @xmath82 in ( a ) and ( b ) , respectively . at the right of each dotted curve we have @xmath19 and the unstable regions of the disk appear between the curves of @xmath79 . we note that in general with increasing mass of the disk and smaller length of the rods , the disk becomes more unstable and the regions of superlumial velocity also increase . there is , however , an interval of values for the rod s mass density where the zone of stability is increased , as can be seen in the lower part of the curve @xmath79 for @xmath86 in fig . [ fig_2](a ) . this is probably due to the prolate quadrupole moment of the rods , which scale as @xmath87 , where @xmath88 is their length . thus , for larger rods , the effect of prolate deformations may overwhelm the effect of the oblate quadrupole moment of the disk , and increase stability ( see @xcite for a detailed discussion of the effect of quadrupolar fields on the stability of circular orbits ) . ( dotted curves ) and of @xmath79 ( solid curves ) for the chazy - curzon disk in presence of a black hole and two rods . we keep the length of each rod constant @xmath89 and vary its mass density @xmath42 ( or equivalent , its mass ) . in ( a)-(b ) we take , respectively , @xmath81 and @xmath82.,title="fig : " ] + ( dotted curves ) and of @xmath79 ( solid curves ) for the chazy - curzon disk in presence of a black hole and two rods . we keep the length of each rod constant @xmath89 and vary its mass density @xmath42 ( or equivalent , its mass ) . in ( a)-(b ) we take , respectively , @xmath81 and @xmath82.,title="fig : " ] fig . [ fig_3](a)-(b ) shows again curves of @xmath83 ( dotted curves ) and of @xmath79 ( solid curves ) as function of @xmath42 and @xmath84 for three different values of @xmath85 , but now the length of each rod is kept constant @xmath90 and the cut parameter takes values @xmath81 and @xmath82 in ( a ) and ( b ) , respectively . with increasing masses of the disk and of the rods , the zones of instability and superlumial velocity of the disk are enhanced . in fig . [ fig_4 ] ( a)-(d ) we plot the energy density @xmath91 , azimuthal pressure @xmath92 , square of the counterrotating velocity @xmath17 and specific angular momentum @xmath93 as functions of @xmath84 for @xmath82 , @xmath94 , @xmath89 ( constant length ) and different values of the rod s linear mass density @xmath42 . the curves were computed using eq . ( [ eq_ttt_sup])-([eq_p_sup ] ) , eq . ( [ eq_vel])-([eq_ang_mom ] ) and eq . ( [ eq_phi_sup ] ) . the expression for the corresponding metric function @xmath4 is given in the appendix . energy density is lowered for a fixed radius as the rods become more massive , while pressure is slightly increased . velocity of counterrotation and specific angular momentum are enhanced by increasing mass of the rods , as can also be deduced from fig . [ fig_3](b ) . , ( b ) azimuthal pressure @xmath95 , ( c ) counterrotating velocity @xmath96 and ( d ) specific angular momentum @xmath97 as functions of @xmath84 for @xmath94 , @xmath82 , @xmath89 and three different values for the rod s mass density @xmath42.,title="fig : " ] , ( b ) azimuthal pressure @xmath95 , ( c ) counterrotating velocity @xmath96 and ( d ) specific angular momentum @xmath97 as functions of @xmath84 for @xmath94 , @xmath82 , @xmath89 and three different values for the rod s mass density @xmath42.,title="fig : " ] + , ( b ) azimuthal pressure @xmath95 , ( c ) counterrotating velocity @xmath96 and ( d ) specific angular momentum @xmath97 as functions of @xmath84 for @xmath94 , @xmath82 , @xmath89 and three different values for the rod s mass density @xmath42.,title="fig : " ] , ( b ) azimuthal pressure @xmath95 , ( c ) counterrotating velocity @xmath96 and ( d ) specific angular momentum @xmath97 as functions of @xmath84 for @xmath94 , @xmath82 , @xmath89 and three different values for the rod s mass density @xmath42.,title="fig : " ] it is interesting to study the disk stability through the computation of horizontal ( epicyclic ) and vertical oscillation frequencies from perturbations of equatorial circular geodesics . semerk and ek @xcite have done such calculations for the superposition of a schwarzschild black hole with the lemos - letelier disk . they found that heavier disks are more stable with respect to horizontal perturbations near their inner rims , whereas they are less stable with respect to vertical perturbations . for astrophysical relevance , it is important to determine not only the stability of circular motion on the disk plane , but also stability in the vertical direction . using the perturbed equations for equatorial circular geodesics , the epicyclic frequency with respect to infinity @xmath98 and the vertical oscillation frequency with respect to infinity @xmath99 for the metric ( [ eq_weyl_metric ] ) are given by ( see @xcite for a detailed deduction ) @xmath100 \mbox{. } \label{eq_osc_v}\end{aligned}\ ] ] in eq . ( [ eq_osc_v ] ) the function @xmath101 is obtained from the limit @xmath102 and @xmath103 follows from eq . ( [ eq_eins1 ] ) . stable horizontal and vertical orbits are only possible where @xmath104 and @xmath105 , respectively . note that condition @xmath104 is equivalent to condition ( [ eq_stab ] ) which follows from rayleigh stability criteria . we compute first the frequencies for an isolated chazy - curzon disk , since is seems that such a calculation has not been done before for this class of disks . in fig . [ fig_5](a ) we plot the epicyclic frequency as functions of radius @xmath106 and cut parameter @xmath107 . for @xmath108 the disks always are stable and the epicyclic frequency is lowered for less relativistic disks . highly relativistic disks ( curve with @xmath109 for example ) develop annular regions of instability . the curves of vertical oscillation frequencies are depicted in fig . [ fig_5](b ) . we note that in this case highly relativistic disks are more stable in the vertical direction . in eq . ( [ eq_osc_v ] ) the term with @xmath101 is small compared to @xmath103 , thus if we consider only @xmath110 we note that vertical oscillations are zero at @xmath111 , so the regions of vertical stability are enlarged as the cut parameter @xmath11 is decreased . and ( b ) the vertical oscillation frequency @xmath112 for an isolated chazy - curzon disk.,title="fig : " ] + and ( b ) the vertical oscillation frequency @xmath112 for an isolated chazy - curzon disk.,title="fig : " ] now we consider the superposition of a curzon disk and a black hole _ without _ rods . [ fig_6 ] shows curves of ( a ) horizontal @xmath113 and ( b ) vertical @xmath114 oscillation frequencies of the disk with @xmath94 and four different values of the `` cut '' parameter @xmath115 . now we always have regions of horizontal instability that begin at the innermost stable circular orbit and decrease for less relativistic disks . with respect to vertical oscillations , it is seen from fig . [ fig_6](b ) that there are no regions of vertical instabilities . thus one can conclude that the black hole desestabilizes the curzon disk in the horizontal direction , whereas the opposite is true for the vertical direction . , @xmath81 , @xmath116 , @xmath117 and @xmath118.,title="fig : " ] + , @xmath81 , @xmath116 , @xmath117 and @xmath118.,title="fig : " ] , @xmath94 , @xmath82 , @xmath119 , @xmath120 and @xmath121.,title="fig : " ] + , @xmath94 , @xmath82 , @xmath119 , @xmath120 and @xmath121.,title="fig : " ] in fig . [ fig_7](a)-(b ) we graph again @xmath113 and @xmath114 , respectively , for the chazy - curzon disk with black hole and rods , with their length fixed and vary the linear mass density @xmath42 . as expected from the curves of fig . [ fig_3](b ) , the more massive the rods , the larger are the disk s unstable regions in the horizontal direction . the rods also tend to lower the vertical oscillation frequencies near the disk s center , but unstable regions do not appear . in the previous section perturbations of equatorial circular geodesics were used to discuss disk stability for the system disk + black hole + rods . now we solve numerically the geodesic equations of motion @xmath122 for metric eq . ( [ eq_weyl_metric ] ) , where @xmath123 are the christoffel symbols and the dot denote differentiation with respect to the proper time . defining the orthonormal tetrad @xmath124 where @xmath125 the tetrad components of the four - velocity @xmath126 read @xmath127 with @xmath128 . the specific energy and angular momentum of the test particle are @xmath129 as initial conditions we take a position at radius @xmath130 on the disk s plane and components of the four - velocity @xmath131 , where @xmath132 is equal to the tangential velocity of circular orbits at radius @xmath130 . we choose initial radii such that the energy is slightly higher than the escape energy . [ fig_8](a)-(b ) shows the orbits of particles in the presence of the black hole and curzon disk without rods . the parameters are @xmath94 , @xmath82 , @xmath133 , @xmath134 and different initial angles @xmath135 . fig . [ fig_8](a ) is a projection of the orbits on the @xmath136 plane . the coordinates have been transformed from weyl to schwarzschild coordinates @xmath137 via the relations @xmath138 and then to @xmath139 , @xmath140 and @xmath141 . in fig . [ fig_9](a)-(d ) we have computed some orbits now with the rods . the parameters are @xmath94 , @xmath82 , @xmath142 , @xmath143 , @xmath89 , @xmath134 and different initial angles @xmath135 . the orbit with @xmath144 has been placed in a separate graph for better visualization . for low initial angles , the rods have little effect on the trajectories , but this is not true as the particles approach the @xmath6 axis . the orbit in fig . [ fig_9](c)-(d ) even suggests that we can expect chaotic behaviour for orbits that pass very near the rods . in fact , it has been shown @xcite that prolate quadrupole deformations can introduce chaotic motion of geodesic test particles . in the oblate case , only regular motion was found . , @xmath82 , @xmath134 , @xmath133 . ( a ) projection on the @xmath136 plane of the curves in ( b).,title="fig : " ] + , @xmath82 , @xmath134 , @xmath133 . ( a ) projection on the @xmath136 plane of the curves in ( b).,title="fig : " ] , @xmath82 , @xmath143 , @xmath89 , @xmath134 @xmath142 . the curve for @xmath144 is displayed in ( c ) and ( d ) for better visualization.,title="fig : " ] , @xmath82 , @xmath143 , @xmath89 , @xmath134 @xmath142 . the curve for @xmath144 is displayed in ( c ) and ( d ) for better visualization.,title="fig : " ] + , @xmath82 , @xmath143 , @xmath89 , @xmath134 @xmath142 . the curve for @xmath144 is displayed in ( c ) and ( d ) for better visualization.,title="fig : " ] , @xmath82 , @xmath143 , @xmath89 , @xmath134 @xmath142 . the curve for @xmath144 is displayed in ( c ) and ( d ) for better visualization.,title="fig : " ] we presented a very simplified , although exact , general relativistic model of an active galactic nuclei based on a superposition of a schwarzschild black hole , a chazy - curzon disk and two rods placed on the symmetry axis , representing jets . we found that the presence of the rods enhances the disk regions with superlumial velocities . using an extension of rayleigh criteria of stability , it was found that in general the rods also increase the regions of instability , but when the rods are large and the disk s mass is low they can contribute to stabilize the disk . also disk stability in the vertical direction was studied through perturbation of circular geodesics . the rods contribute to lower the vertical oscillation frequencies near the disk s center . some geodesic orbits calculated numerically for the system black hole + disk + rods show the possibility of chaotic trajectories near the rods . the model here presented should be viewed as a first approach . as was stated in the introduction , more realistic models of active galactic nuclei should incorporate rotation and electromagnetic fields . however , the analysis of such a model would not be trivial , because of the large number of free parameters involved . * acknowledgments * d. v. thanks capes for financial support . p. s. l. thanks cnpq and fapesp for financial support . the metric function eq . ( [ eq_eins2 ] ) for the superposition of a black hole , two rods and a disk generated from chazy - curzon solution can be calculated as follows . we rewrite potential eq . ( [ eq_phi_sup ] ) as @xmath145 where @xmath146 with @xmath147 and @xmath148 . in the limit @xmath149 expression for @xmath150 reduces to the chazy - curzon disk eq . ( [ eq_curzon ] ) on @xmath151 . thus all terms can be expressed as @xmath45 potentials . using repeatedly properties ( [ eq_nu_rel2])-([eq_ident ] ) we get @xmath152=\nu [ \phi_{r1}]+\nu [ \phi_{bh}]+\nu [ \phi_{r2}]+\nu [ \phi_{d}^+ ] \notag \\ & + 2\nu [ \phi_{r1},\phi_{bh}]+2\nu [ \phi_{r1},\phi_{r2}]+ 2\nu [ \phi_{r1},\phi_{d}^+]+2\nu [ \phi_{bh},\phi_{r2 } ] \notag \\ & + 2\nu [ \phi_{bh},\phi_{d}^+]+2\nu [ \phi_{r2},\phi_{d}^+ ] \mbox { , } \label{eq_nu_sup}\end{aligned}\ ] ] with @xmath153= 4\lambda^2 \ln \left [ \frac{(r^2+\mu_3 \mu_4)^2}{(r^2+\mu_3 ^ 2)(r^2+\mu_4 ^ 2 ) } \right ] \mbox { , } \\ \nu & [ \phi_{bh}]= \ln \left [ \frac{(r^2+\mu_1 \mu_2)^2}{(r^2+\mu_1 ^ 2)(r^2+\mu_2 ^ 2 ) } \right ] \mbox { , } \\ \nu & [ \phi_{r2}]= 4\lambda^2 \ln \left [ \frac{(r^2+\mu_5 \mu_6)^2}{(r^2+\mu_5 ^ 2)(r^2+\mu_6 ^ 2 ) } \right ] \mbox { , } \\ \nu & [ \phi_{d}^{+}]=-\frac{m^2r^2}{[r^2+(z+a)^2]^2 } \mbox { , } \\ \nu & [ \phi_{r1},\phi_{bh}]=2\lambda \ln \left [ \frac{(r^2+\mu_1\mu_3)(r^2+\mu_2\mu_4)}{(r^2+\mu_1\mu_4)(r^2+\mu_2\mu_3 ) } \right ] \mbox { , } \\ \nu & [ \phi_{r1},\phi_{r2}]=4\lambda^2 \ln \left [ \frac{(r^2+\mu_3\mu_6)(r^2+\mu_4\mu_5)}{(r^2+\mu_3\mu_5)(r^2+\mu_4\mu_6 ) } \right ] \mbox { , } \\ \nu & [ \phi_{r1},\phi_{d}^+]=\frac{2\lambda m}{(a+c_1)(a+c_2)\sqrt{r^2+(a+z)^2 } } \left [ ( a+c_2)\sqrt{r^2+(c_1-z)^2 } \right . \notag \\ & \left . -(a+c_1)\sqrt{r^2+(c_2-z)^2}+(c_1-c_2)\sqrt{r^2+(a+z)^2 } \right ] \mbox{,}\\ \nu & [ \phi_{bh},\phi_{r2}]=2\lambda \ln \left [ \frac{(r^2+\mu_1\mu_5)(r^2+\mu_2\mu_6)}{(r^2+\mu_1\mu_6)(r^2+\mu_2\mu_5 ) } \right ] \mbox { , } \\ \nu & [ \phi_{bh},\phi_{d}^+]=\frac{m}{(a^2-m^2)\sqrt{r^2+(a+z)^2 } } \left [ ( a+m)\sqrt{r^2+(m+z)^2 } \right . \notag \\ & \left . -(a - m)\sqrt{r^2+(m - z)^2}-2m\sqrt{r^2+(a+z)^2}\right ] \mbox{,}\\ \nu & [ \phi_{r2},\phi_{d}^+ ] = \frac{2\lambda m}{(a - c_1)(a - c_2)\sqrt{r^2+(a+z)^2 } } \left [ ( c_2-a)\sqrt{r^2+(c_1+z)^2 } \right . \notag \\ & \left . -(c_1-a)\sqrt{r^2+(c_2+z)^2}+(c_1-c_2)\sqrt{r^2+(a+z)^2 } \right ] \mbox{.}\end{aligned}\ ] ] in the particular case @xmath65 @xmath154 and @xmath155 , and on @xmath9 , eq . ( [ eq_nu_sup ] ) simplifies to @xmath156= \notag \\ & \ln \left [ \frac{r^{16 \lambda^2 - 8\lambda+2}(r^2+\mu_1\mu_6)^{8\lambda^2 - 4\lambda}(r^2+\mu_2\mu_3)^{8\lambda^2 - 4\lambda } } { ( r^2+c_2 ^ 2)^{4\lambda^2}(r^2+m^2)^{4\lambda^2 - 4\lambda+1}(r^2+\mu_1\mu_3)^{8\lambda^2 - 4\lambda}(r^2+\mu_2\mu_6)^{8\lambda^2 - 4\lambda } } \right ] \notag \\ & -\frac{m^2r^2}{(r^2+a^2)^2}+\frac{8\lambda m}{(a^2-c_2 ^ 2)(a^2-m^2)\sqrt{r^2+a^2 } } \left [ c_2(a^2-m^2)\sqrt{r^2+c_2 ^ 2 } \right . \notag \\ & \left . -m(a^2-c_2 ^ 2)\sqrt{r^2+m^2}+(m - c_2)(a^2+mc_2)\sqrt{r^2+a^2 } \right ] \notag \\ & + 4mm\frac{(\sqrt{r^2+m^2}-\sqrt{r^2+a^2})}{(a^2-m^2)\sqrt{r^2+a^2 } } \mbox{.}\end{aligned}\ ] ] 99 w. kundt ( ed . ) , _ jets from stars and galactic nuclei _ , proceedings , lecture notes in physics , 471 , springer , 1996 . j. h. krolik , _ active galactic nuclei : from the central black hole to the galactic environment _ , princeton university press , princeton , new jersey , 1999 . r. d. blandford , _ prog . * 143 * , 182 ( 2001 ) . w. a. bonnor and a. sackfield , _ comm . * 8 * , 338 ( 1968 ) . t. morgan and l. morgan , _ phys . rev . _ * 183 * , 1097 ( 1969 ) . l. morgan and t. morgan , _ _ phys . rev . d__**2 * * , 2756 ( 1970 ) . d. lynden - bell and s. pineault , _ mon . not . r. astron . soc . _ * 185 * , 679 ( 1978 ) . p. s. letelier and s. r. oliveira , _ j. math _ * 28 * , 165 ( 1987 ) . j. p. s. lemos , _ class . quantum grav . _ _ 6 _ , 1219 ( 1989 ) . s. lemos and p. s. letelier , _ class . quantum grav . _ * 10 * , l75 ( 1993 ) . j. bik , d. lynden - bell and j. katz , _ _ phys . rev . d__**47 * * , 4334 ( 1993 ) . j. bik , d. lynden - bell and c. pichon , _ mon . not . r. astron . soc . _ * 265 * , 126 ( 1993 ) . s. lemos and p. s. letelier , _ _ phys d__**49 * * , 5135 ( 1994 ) . j. p. s. lemos and p. s. letelier , _ _ int . phys . d__**5 * * , 53 ( 1996 ) . g. gonzlez and o. a. espitia , _ phys . d _ * 68 * , 104028 ( 2003 ) . g. garca and g. gonzlez , _ phys . d _ * 69 * , 124002 ( 2004 ) . j. bik and t. ledvinka , _ phys . lett . _ * 71 * , 1669 ( 1993 ) . g. gonzlez and p. s. letelier , _ _ phys . rev . d__**62 * * , 064025 ( 2000 ) . d. vogt and p. s. letelier , _ _ phys . d__**68 * * , 08410 ( 2003 ) . d. vogt and p. s. letelier , _ exact relativistic static charged perfect fluid disks_. to appear in phys . d. g. gonzlez and p. s. letelier , _ _ phys . rev . d__**69 * * , 044013 ( 2004 ) . c. klein , _ class . quantum grav . _ * 14 * , 2267 ( 1997 ) . g. neugebauer and r. meinel , _ phys . lett . _ * 75 * , 3046 ( 1995 ) . c. klein and o. richter , _ * 83 * , 2884 ( 1999 ) . c. klein , rev . d__**63 * * , 064033 ( 2001 ) . j. frauendiener and c. klein , _ _ phys . rev . d__**63 * * , 084025 ( 2001 ) . c. klein , _ _ phys . rev . d__**65 * * , 084029 ( 2002 ) . c. klein , _ _ phys . d__**68 * * , 027501 ( 2003 ) . c. klein , _ ann . * 12 * ( 10 ) , 599 ( 2003 ) . t. zellerin and o. semerk , _ class . quantum grav . _ * 17 * , 5103 ( 2000 ) . v. karas , j. m. hur and o. semerk , _ class . quantum grav . _ * 21 * , r1 ( 2004 ) . h. weyl , _ ann . _ * 54 * , 117 ( 1917 ) . h. weyl , _ ann . phys . _ * 59 * , 185 ( 1919 ) . g. g. kuzmin , _ astron . * 33 * , 27 ( 1956 ) . j. p. s. lemos , _ mon . not . r. astron . _ * 230 * , 451 ( 1988 ) . lord rayleigh , _ proc . london a _ * 93 * , 148 ( 1917 ) ; see also l. d. landau and e. m. lifshitz , _ fluid mechanics _ ( pergamon press , oxford , 1987 ) , 27 . v. a. belinsky and v. e. zakharov , _ zh . eksp . _ * 75 * , 1955 ( 1978 ) [ sov jetp * 48 * , 985 ( 1978 ) ] . v. a. belinsky and v. e. zakharov , _ zh . eksp . fiz . _ * 77 * , 3 ( 1979 ) [ sov . phys . jetp * 50 * , 1 ( 1979 ) ] . p. s. letelier , _ phys . . d _ * 68 * , 104002 ( 2003 ) . o. semerk and m. ek , _ publ . soc . japan _ * 52 * , 1067 ( 2000 ) . e. gueron and p. s. letelier , _ phys . e _ * 63 * , 035201(r ) ( 2001 ) , * 66 * , 046611 ( 2002 ) .
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an exact but simple general relativistic model for the gravitational field of active galactic nuclei is constructed , based on the superposition in weyl coordinates of a black hole , a chazy - curzon disk and two rods , which represent matter jets .
the influence of the rods on the matter properties of the disk and on its stability is examined .
we find that in general they contribute to destabilize the disk .
also the oscillation frequencies for perturbed circular geodesics on the disk are computed , and some geodesic orbits for the superposed metric are numerically calculated .
pacs numbers : 04.20.jb , 04.40.-b , 98.58 fd , 98.62 mw
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the standard model is the theory of strong and electroweak interactions : they are described in terms of quarks and leptons , the basic constituents of matter , and gauge bosons , which are the carriers of the fundamental forces . this model is kept endlessly under inspection by comparing measured observables , such as couplings , masses , integral and differential cross sections or branching ratios , with the theory expectations . the interplay between theory , which is continuosly improving its predictions , and experiments , whose measurements are carried out with increasing accuracy , consolidates the standard model itself . at the same time , however , the standard model presents some drawbacks which open the road to new physics ; the forthcoming experiments at the large hadron collider will provide us with a unique chance to address these open problems . we present in the following our perspective , from both theoretical and experimental viewpoints , on the current situation of the standard model and discuss some of its open issues , which call for new physics extensions . quantum chromodynamics ( qcd ) , an unbroken renormalizable theory based on the colour @xmath0 gauge symmetry , is a main part of the standard model of particle physics , as it is the theory of strong interactions . at present and future colliders , having full control of qcd is fundamental , since events mediated by the strong interaction may mimic new physics signals . because of asymptotic freedom , at high energies qcd observables can be computed as fixed - order expansions in the coupling constant @xmath1 ( perturbative qcd ) . most cross sections have been computed at next - to - leading order ( nlo ) , whereas a few have been provided with next - to - next - to - leading order ( nnlo ) corrections as well . in fact , as nlo corrections can be as large as a factor of two , leading - order ( lo ) results can just estimate the order of magnitude of an observable , with the nlo giving the first reliable prediction . nnlo computations are nonetheless still necessary to reduce the theoretical uncertainty and weaken the dependence , _ e.g. _ , on the choice of parton distributions or fragmentation functions . among recent nlo calculations , we can mention the corrections to the production of three vector bosons at the lhc , namely @xmath2 , @xmath3 , @xmath4 and @xmath5 : the results on the transverse momentum @xmath6 spectrum show that the nlo has a relevant impact , indeed up to a factor of two , at small @xmath7 @xcite . while fixed - order calculations are reliable enough to predict inclusive observables , such as total cross sections and widths , differential distributions can exhibit contributions where @xmath1 is multiplied by logarithmic coefficients , which become large for soft or collinear parton radiation . resumming such terms is mandatory to describe exclusive quantities . a remarkable example , where the interplay between fixed - order and resummed computation is crucial , is higgs - boson production in the @xmath8 channel , the dominant one at the lhc . the higgs rapidity and transverse momentum spectra have been computed to nnlo accuracy ; moreover , the logarithms @xmath9 , large at small @xmath7 , where one is mostly sensitive to soft and collinear emissions , have been resummed in the next - to - next - to - leading logarithmic ( nnll ) approximation and included in the monte carlo code hqt @xcite . at low energy , however , one is faced with non - perturbative power corrections which can not be calculated in perturbative qcd . in fact , one may address non - perturbative phenomena , such as hadronisation or underlying event , following several models and approaches , which are typically based on fits to experimental data . as far as hadronisation is concerned , an alternative possibility consists in defining an effective strong coupling constant , free from the landau pole , which at large energy coincides with the standard coupling and at small energy includes non - perturbative power corrections . studies have shown that the effective - coupling model , used along with a resummed calculation in the framework of perturbative fragmentation functions , yields a reasonable description of data on @xmath10- and @xmath11-flavoured hadrons at lep and sld , without tuning any parameter to such data @xcite . another approach to non - perturbative qcd parametrises low - energy physics by means of the frozen coupling constant , defined as the integral of @xmath12 , where @xmath7 is some transverse momentum characteristic of the process , from zero to an upper limit , fitted to experimental data . in this way , one can , _ e.g. _ , study jet observables at the tevatron or the lhc , obtaining results in reasonable agreement with monte carlo generators , such as herwig or pythia , which implement their own models for hadronisation and underlying event , tuned to lep or tevatron data @xcite . along with the intensive applications of qcd to collider phenomenology , work has been lately undertaken to construct a dual model of qcd , in the fashion of the ads / cft correspondence . in the framework of such a model , called holographic qcd , one can perform a linear fit of the experimental spectrum of @xmath13 mesons and give predictions for masses and decay constants of the @xmath14 glueballs @xcite . the electroweak sector of the standard model is based on a @xmath15 symmetry , spontaneously broken into @xmath16 . although the electroweak theory has been successfully tested in several experimental environments , the higgs boson , a crucial ingredient of the model , as it is responsible of the mass generation mechanism , has not been discovered yet . there are , however , indications that a mass mechanism in the fashion of the higgs one should exist : for instance , the observed longitudinal polarisations of @xmath17 and @xmath18 bosons are indeed manifestations of a broken symmetry . furthermore , the relation between their masses , namely @xmath19 , @xmath20 being the weinberg angle , valid up to very small radiative corrections , confirms the so - called custodial @xmath21 symmetry of the standard model and that the higgs boson has to be a weak - isospin doublet . the lhc has been designed to solve the higgs problem , as it will be able to search for a standard model higgs boson , with a mass up to @xmath22 tev / c@xmath23 . on the theory side , the lack of the higgs discovery has opened the road to other scenarios beyond the standard model , including the possibility that the higgs may not exist ( higgsless models ) or that it is an approximate goldstone boson of a broken global symmetry ( composite higgs models ) . also , conceptual drawbacks , and especially the well - known hierarchy problem , namely the quadratical divergence of the higgs mass after radiative corrections , requiring fine tuning of several orders of magnitude , call for extensions of the standard model . among these , one of the most appealing and widely studied is surely supersymmetry , with its minimal formulation , the minimal supersymmetric standard model ( mssm ) . calculations and computing codes implementing supersymmetric processes have been available for a few years . work has been lately carried out to provide such computations with radiative corrections , which are likely to have a remarkable impact at the lhc , and must be taken into account in any reliable analysis . an example is given by the inclusion of nlo electroweak corrections to squark - antisquark pair production at hadron colliders , included in the monte carlo program prospino @xcite . a common feature of several new physics models is that they predict the existence of new heavy vector bosons @xmath24 . indeed , @xmath24 production in drell yan type processes , followed by leptonic decays , are among the first new physics signals possibly visible at the lhc . @xcite discusses @xmath24 production in the so - called left - right symmetric free - fermionic model , which enlarges the gauge structure of the standard model by an extra heterotic - string inspired @xmath25 ( see @xcite for more details ) . the free - fermionic model is constructed in such a way to suppress proton decay ; moreover , it is anomaly - free and consistent with the see - saw mechanism for neutrino masses , with family universality and yukawa - like couplings . within this model , @xmath24-production processes have been provided with qcd corrections to nnlo accuracy and should be visible at the lhc in the channels @xmath26 and @xmath27 . although the electroweak precision tests seem to favour a possibly light higgs , as will be discussed in detail in the next section , it is nonetheless still possible to accommodate a heavy higgs beyond the standard model , since new physics effects can contribute to electroweak observables and make a heavy higgs consistent with present data . an example is the so - called @xmath28susy model , based on an extension of the mssm , which predicts the existence of a heavy higgs @xmath29 , with mass , _ e.g. _ , 500 - 600 gev / c@xmath23 , mainly decaying , via @xmath30 , into a pair of lighter higgses @xmath31 , having mass @xmath32 200 - 300 gev / c@xmath23 . the properties of the lighter higgs @xmath31 are supposed to be pretty similar to the standard model one . in fact , new physics contributions , _ e.g. _ to the oblique parameters @xmath33 and @xmath34 , make @xmath28susy still compatible with the constraints of the electroweak precision tests , which do seem to favour a rather light higgs , but only within the standard model and without assuming any new physics effect @xcite . the @xmath28susy model is expected to be possibly observable at the lhc with 100 fb@xmath35 of integrated luminosity . nevertheless , a possible alternative is to try to describe the electroweak interactions and solve the hierarchy problem in the framework of the so - called higgsless models . a process which has been thoroughly investigated is @xmath36 scattering : in the standard model , the exchange of an intermediate @xmath29 moderates the growth of the cross section for longitudinal boson scattering , and the requirement of unitarity sets the limit @xmath37 tev / c@xmath23 on the higgs mass . in higgsless models , the starting point is typically a @xmath38 chiral symmetry , spontaneously broken to @xmath39 : in this way , one or more new heavy vector bosons replace the higgs and delay the unitarity issue to a few tev / c@xmath23 . ref . @xcite discusses the four - site higgsless model , which presents four new vector bosons , _ i.e. _ two charged @xmath40 and two neutral @xmath41 . at the lhc , @xmath42 and @xmath24 bosons can be produced in drell yan like processes , thus giving rise to visible resonances . thanks to these new gauge bosons , the problem of unitarity violation is delayed to energy scales which are much higher than those actually probed at the lhc . unlike other higgsless models , such vector bosons are not ` fermiophobic ' , as the electroweak precision data do allow sizeable couplings with fermions . before closing this section , we point out that , for the sake of experimental analyses , any new physics model needs to be implemented in monte carlo generators , in order to account for multiple initial- and final - state radiation , hadronisation and underlying event . in fact , work towards this direction has seen tremendous improvements in the latest few years . as for @xmath36 scattering , it is worthwhile to mention the phantom code , which implements final states with six fermions , and includes the option of no intermediate higgs boson as well . by using phantom , it will be possible to study @xmath36 scattering at the lhc and , in the phase of 100 fb@xmath35 luminosity , discriminate between , _ e.g. _ , a standard model scenario with a higgs of mass @xmath43 gev / c@xmath23 and the no - higgs case @xcite . the standard model continues to be explored in different experiments based on accelerators or not . the tevatron , colliding beams of protons and antiprotons with the currently highest energy of @xmath44 tev , represents a very favourable environment where to perform an extensive study of the major ingredients of the standard model , _ i.e. _ quarks and gauge bosons . figure [ tev_proc ] shows , as an example , the production cross section at the tevatron for different categories of processes : the production of @xmath10-hadrons , @xmath17 and @xmath18 bosons , events containing the top quark , and finally the possible production of the higgs boson . these and other processes are thouroughly studied by the two multi - purpose experiments operating at the tevatron , namely cdf and d . given the cross sections and the current integrated luminosity ( about 3 fb@xmath35 ) , the production of events containing @xmath17 or @xmath18 bosons is large , of the order of a few hundred thousand events . also , even if the tevatron can not be really considered a top factory , the production of top quarks is quite abundant , with yields , so far , of about @xmath45 @xmath46 pairs and about @xmath47 single - top events per year . on the other hand , the production of the higgs boson is expected to be very rare at the tevatron and only the total integrated luminosity before the final shutdown , which is foreseen to be about 6 - 8 fb@xmath35 , along with a full exploration and combination of all possible channels , might enable to obtain some results . the situation taking place at the tevatron will change drastically once the lhc , colliding beams of protons with @xmath48 tev , reaches its full power . the production cross sections at lhc will increase , with respect to the tevatron ones , by one order of magnitude for @xmath49 production and by two orders for top production . this , along with the higher - design instantaneous luminosity , will make lhc a real top - factory , and a promising environment for higgs discovery . given the different yields of the processes described in figure [ tev_proc ] , it is suitable to divide the experimental studies of the major standard model processes into three subfields : @xmath49 physics , production of top quarks and searches for the standard model higgs boson . in addition , we shall briefly discuss the exploration for new physics beyond the standard model . the study of the @xmath49 boson production , within the standard model , is crucial because it can provide accurate tests of both electroweak and strong interactions ; qcd corrections to the production cross section are in fact available in the nnlo approximation . cdf and d typically measure @xcite the production cross sections for leptonic final states , _ i.e. _ @xmath50 and @xmath51 , along with their ratio and the @xmath17-width @xmath52 . the size of the samples also enables the study of differential distributions . the measurement of the @xmath17 mass is very important , due to its link through radiative electroweak corrections to the higgs boson mass . the mass is measured at the tevatron using the large sample of inclusive @xmath17 events , where the @xmath17 decays into @xmath53 or @xmath54 . the @xmath17 transverse mass , @xmath55 , is reconstructed using the missing transverse energy , as a measure of @xmath56 , measuring the azimuthal angle @xmath57 between the lepton and the direction of the missing transverse energy , and is finally fitted to the expected distributions for different values of the @xmath17 mass . this measurement is a very delicate one , because it relies on a very precise calibration of the lepton momenta , and even the underlying event and additional @xmath58 interactions must be taken into account . with this technique , and using only @xmath59 of the luminosity currently available , cdf already reaches an accuracy better than what was measured by the various experiments at lep , thus improving the world average . the study of diboson production @xmath36 , @xmath60 and @xmath61 is accessible in spite of the relatively low production cross sections ( respectively about 10 , 4 and 1 pb ) . the investigation of these events is quite interesting not only in itself , but also because they represent relatively rare processes , with topologies similar to what can be expected for higgs boson production . in fact , @xmath60 production has been clearly observed and there are first hints of @xmath61 evidence . at the lhc , given the high energy and luminosity , a large number of @xmath17 and @xmath18 events will be collected shortly after the machine turn - on , with a very clean leptonic signature . even before providing accurate measurements of @xmath62 , @xmath63 and @xmath64 , and performing other tests of the electroweak theory , such events will be quite useful for calibration and alignment of the detector @xcite . once the detector is well understood , the study at the lhc of diboson production will become interesting to search for the higgs boson and for deviations from the standard model . the top quark , discovered thirteen years ago at the tevatron , is the heaviest quark and , for this reason , it plays an important role in loop corrections to several electroweak observables . indeed , together with the accurate measurement of the @xmath17 mass , the top - mass measurement constrains the mass of the higgs boson . the measurement of the top quark mass with the highest precision is one of the major goals of cdf and d . these two experiments already managed to measure the mass with a ( combined ) 1.4 gev / c@xmath23 precision , exceeding past expectations @xcite . the goal now is to reach a precision smaller than 1 gev / c@xmath23 , before the tevatron definitive turn - off . as for the production cross section , the uncertainty reached by cdf and d is of the order of 10% ( per experiment ) and comparable to the theoretical uncertainties , thus enabling a test of nlo / nll predictions . electroweak production of single top has a cross section about 1/3 of @xmath46 pair production . therefore , it has been more difficult to find evidence for it and to measure the single - top cross section , which gives also a direct measurement of the cabibbo kobayashi maskawa matrix element @xmath65 . in addition to mass and cross sections , other top properties are measured , such as its charge , lifetime , decay branching ratios and the @xmath17 helicity in top decays @xmath66 . also , due to its large mass , top quarks might couple to new physics at high energy scales and act as probes for these processes . for this reason , exotic decays or production mechanisms have been explored , but there is so far no evidence of heavy resonances or fourth - generation quarks . the measurement at the lhc of the top quark properties will reach high statistical accuracy , given the large yield expected for top events . precise measurements will require , however , a detailed modelling of the detector response , along with a good understanding of the sources of systematic uncertainty . top events themselves can be used , through _ in - situ _ calibration of the jet energy scale , to help reduce such uncertainties @xcite . curve derived from high-@xmath67 precision electroweak measurements , performed at lep and by sld , cdf , and d , as a function of the higgs boson mass , assuming the standard model . the yellow band on the left - hand side refers to the lep2 direct search limit . from @xcite.,width=226 ] curve derived from high-@xmath67 precision electroweak measurements , performed at lep and by sld , cdf , and d , as a function of the higgs boson mass , assuming the standard model . the yellow band on the left - hand side refers to the lep2 direct search limit . from @xcite.,width=226 ] the measurements of @xmath17 and top masses , together with other precision electroweak measurements , lead to the limit for the standard model higgs - boson mass of @xmath68 gev / c@xmath23 ( see figure [ blue ] ) , with a 95% c.l . upper limit of 160 gev / c@xmath23 . if one also includes the results from direct searches at lep , yielding @xmath69 gev / c@xmath23 , the 95% c.l . limit becomes 190 gev / c@xmath23 . such a ( relatively low ) expected mass sets higgs - boson production possibly within reach of the tevatron experiments , but , at the same time , the smallness of the production cross section makes it very hard to search for its evidence . this challenging task requires the experiments to aim at high sensitivity by combining all possible channels , improving efficiencies and reducing systematic uncertainties . specific strategies need to be implemented for a low mass higgs ( @xmath70 gev / c@xmath23 ) , where the associated production ( @xmath71 or @xmath72 ) has the best chances , and for a ( relatively ) high mass ( @xmath73 gev / c@xmath23 ) , where the channels @xmath74 are the most favourable ones . the cdf+d sensitivity reaches its optimum around 160 gev / c@xmath23 where they could be able to reach some results before the tevatron turn - off either in terms of exclusion or of evidence @xcite . as for cms and atlas , once the detector is well calibrated and its response known , they are expected to be sensitive to the discovery already with @xmath75 fb@xmath35 of data , if the higgs mass is above 140 gev / c@xmath23 , in the decay channels @xmath76 and @xmath74 . for lower higgs masses , more luminosity , about 5 - 10 fb@xmath35 , will be needed and several channels will have to be included , in order to be sensitive to its observation @xcite . as discussed in section 2 , many extensions of the standard model are introduced in order to address open issues , such as the hierarchy problem , or to provide different mechanisms of symmetry breaking or unify the fundamental forces . the searches currently performed at the tevatron are either inspired by a specific model or instead based on a specific signature . searches for supersymmetric particles , new gauge bosons , extra dimensions , leptoquarks or higgs compositeness belong to the first case . as for the second case , anomalous production of leptons , bosons and photons , as well as missing @xmath77 , have been searched . no signal of supersymmetry , exotic particles or anomalous production has yet been found at the tevatron and limits ( more or less stringent , depending on the cases ) have been set @xcite . new physics is searched also at hera where , for instance , limits on contact interactions have been set @xcite . given the large luminosity that will be integrated , the lhc experiments will have the potential for important discoveries , either within the susy framework @xcite or in other scenarios beyond the standard model @xcite . statistical uncertainties will not be a problem in most cases , but it will be crucial to understand the systematics in order to be able to find evidence of new signals . in this summary we have briefly reviewed the current status of the standard model of strong and electroweak interactions and underlined some of its drawbacks . we discussed improvements in the theory and their applications to collider phenomenology , as well as the most recent measurements carried out at present colliders , mostly at the tevatron accelerator , and commented on the features of future measurements which will be carried out at the lhc . on the theory side , qcd stands as a robust theory , which continuously yields predictions which are confirmed very well by the experiments . nevertheless , more accurate perturbative calculations and a better understanding and modelling of its non - perturbative aspects are still of great interests , in order to fully control the backgrounds for many new physics searches . as for the electroweak interactions , the ultimate confirmation of the standard model awaits the discovery of the higgs boson : as discussed , the lhc should be able to give a final answer to this point . moreover , the absence of any higgs signal so far , along with the open problems of the standard model , has pushed the development of a few new ideas on the electroweak symmetry breaking , which are theoretically well formulated and not in contradiction with the available electroweak precision tests . as only experimental measurements , _ e.g. _ the @xmath36 scattering cross section , can help to verify or disprove such models , we can just eagerly await for the next start of the lhc . 0 these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . http://lepewwg.web.cern.ch / lepewwg/. these proceedings . these proceedings . these proceedings . these proceedings . these proceedings . these proceedings .
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in this summary we present the current status of the standard model of strong and electroweak interactions from the theoretical and experimental point of view .
some discussion is also devoted to the exploration of possible new physics signals beyond the standard model .
[ 1999/12/01 v1.4c il nuovo cimento ]
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non - linear sigma models [ 1 - 3 ] are prototypes of a remarkable class of integrable two dimensional models which contain an infinite number of conserved local and non - local charges [ 4 - 7 ] . the algebraic relations obeyed by such charges are supposed to be an important ingredient in the complete solution of those models [ 8 - 11 ] . the local charges form an abelian algebra . opposing to that simplicity , the algebra of non - local charges is non - abelian and actually non - linear [ 12 - 28 ] . in ref.[29 ] the @xmath0 sigma model was investigated and a particular set of non - local charges called _ improved _ charges was found to satisfy a cubic algebra related to a yangian structure . in this work we intend to extend that result to the corresponding supersymmetric case [ 30 - 32 ] . the introduction of supersymmetry might have rendered a much more involved algebra [ 33 ] . however , it has been conjectured [ 29,32 ] that , in the sigma model , the algebra of supersymmetric non - local charges would remain the same as in the bosonic theory and we shall present results that confirm such conjecture . this paper is organized as follows . in sect.2 we briefly review the results from the purely bosonic theory . a graphic technique to compute charges and their algebra is introduced in sect.3 . in sect.4 we discuss the supersymmetric model and the main results of the paper . another application of graphic rules is shown in sect . 5 concerning the @xmath0 gross - neveu model . sect.6 is left for conclusions while an appendix contains examples of the graphic technique . the two - dimensional non - linear @xmath0 sigma model can be described by the constrained lagrangean = 12__i^_i , _ i=1^n ^2_i = 1 . associated to the @xmath0 symmetry we have a matrix - valued conserved curvature - free current @xmath1 = 0 \qquad , \end{aligned}\ ] ] whose components satisfy the algebra [ 29 ] @xmath2 where @xmath3 is the @xmath4 identity matrix . above we have introduced the intertwiner field ( j)_ij = _ i_j and the @xmath0 @xmath5-product defined in ref . [ 29 ] as ( ab)_ij , kla_ikb_jl - a_ilb_jk + a_jlb_ik - a_jkb_il . this model is known to have infinite non - local conserved charges . the standard set of charges can be iteratively built up by means of the potential method of brzin _ et . al . _ [ 5 ] . however , in ref . [ 29 ] an alternative set of _ improved _ charges @xmath6 has been defined and it was shown that they obey the non - linear algebra \ { q^(m)_ij , q^(n)_kl } = ( iq^(n+m ) ) _ ij , kl - _ p=0^m-1_q=0^n-1 ( q^(p ) q^(q ) q^(m+n - p - q-2))_ij , kl . these charges were named _ improved _ because they brought up an algebraic improvement : the non - linear part of the algebra is simply cubic , as opposed to the algebra of the standard charges previously used in the literature [ 14 ] . the jacobi identity and other properties of the improved cubic algebra were thoroughly discussed in ref . but there is a way to abbreviate that algebra , which is the first among the new results of this paper and which will be presented now . .5truecm we shall define a hermitean generator of improved charges q ( ) i + i_n=0^^n+1 q^(n ) , where @xmath7 will be called the spectral parameter . therefore one can summarize the algebra ( 6 ) as follows : i\ { q_ij(),q_kl ( ) } = ( f ( , ) q()- q ( ) ) _ ij , kl , where f(,)e ( q()q()^-1 -^-1 ) = i - _ m , n=0^^m+1^n+1q^(m)q^(n)^-1 -^-1 . the quadratic non - linearity encoded in @xmath8 can be related to the known yangian structure that underlies this model [ 17 - 26,29 ] . the advantage in writing the algebra as in ( 8) is not only aesthetic . recalling the monodromy matrix of standard charges , and its algebra expressed in terms of the classical @xmath9-matrix , @xmath10 \quad , \\ & & r(\lambda , \mu ) = { i_a\otimes i_a\over \lambda ^{-1}-\mu ^{-1}}\quad , \quad [ i_a , i_b ] = f_{abc}i_c \quad , \nonumber\end{aligned}\ ] ] we remark that the generator @xmath11 and the @xmath12-matrix play similar rles to those of the monodromy matrix and classical @xmath9-matrix in the standard approach [ 17 - 26 ] . we do not fully understand the relationship between ( 8) and ( 10 ) but we expect to be able to use this analogy to establish a precise translation between the different sets of charges [ 35 ] . we also hope that a complete knowledge about the conserved charges and their algebra will become an decisive ingredient in off - shell scattering calculations . .5truecm now let us consider the graphic methods announced in the introduction . we recall that in ref . [ 29 ] the improved charges were constructed by means of an iterative algebraic algorithm that uses @xmath13 as a step - generator , as indicated by the relation ( iq^(n+1 ) ) = linear part of \{q^(1),q^(n ) } . after a tedious calculation " the authors in ref . [ 29 ] managed to construct the charges @xmath14 and their algebra up to @xmath15 . in the next section we will present a _ graphic _ method that makes the calculation simpler , less tedious and convenient for a further supersymmetric extension . let us associate white and black semicircles to the @xmath0 current components , j -.1 cm 2truecm j -.1 cm a continuous line and an oriented line to the identity and the anti - derivative operator respectively , i .07 cm 2truecm 2 .07 cm the operator @xmath16 above follows the same convention adopted in ref . [ 29 ] , a(x ) = 12 y ( x - y)a(y ) , ( x)= . below one finds some diagrams and the corresponding expressions : @xmath17 we have noticed [ 29 ] that every improved charge can be written as an integral over symmetrized chains of @xmath18 s and @xmath19 s connected by the operator @xmath20 . therefore we can associate a diagram to each improved charge , as exemplified by the second non - local charge @xmath21 : q^(2)=dx @xmath22 if one is interested in constructing charges , there is an iterative graphic procedure , inspired by the lessons taken from ref . [ 29 ] and which will be described now . consider the following properties : .3truecm \(a ) the improved non - local charges have the general form @xmath23 where @xmath24 is a combination of terms which one can always write as j _ 2(j_s+s^^tj _ ) , where @xmath25 is some chain and @xmath26 its transposed . .5truecm ( b ) the algebraic definition of improved charges is ( iq^(n+1))_ij , kl= linear part of \{q_ij^(1),q_kl^(n ) } and we note that the linear part of @xmath27 comes exclusively from terms like dx ( \{q_ij^(1),(j_)_ka}2s_al-(k l ) ) . .5truecm ( c ) using the definition of @xmath13 and the elementary current algebra and dropping non - linear terms , we verify that dx\{q_ij^(1),(j__0)_ka}2s_al - ( k l ) =( i dx[j__12s+j__02(j__02s ) + 2js ] ) _ ij , kl . @xmath28 dx \{q_ij^(1),(j__1)_ka}2s_al - ( k l ) =( i dx[2j__0j2s+j__0 2(j__1s ) ] ) _ ij , kl . @xmath29 where some new symbols were introduced , @xmath30 -.1 cm 1truecm 1truecm -.1 cm 2= i the previous expressions justify the following prescription : .3truecm \i ) we start from the diagram of @xmath14 . .5truecm ii ) then we replace the left tip " of each chain according to the rules : .5truecm iii ) the resulting diagram corresponds to @xmath31 . .5truecm we remark that the substitution rules above can be directly read from the following basic brackets : @xmath32 in addition , one should not forget the constraints satisfied by the @xmath0 current @xmath33 , given below , _ + = j _ 2truecm -.1 cm 0.5 cm 2j__1js= ( j__1s+js ) 2truecm -.1 cm the half - white / half - black semicircle means @xmath34 or @xmath19 generically . we have tested the efficiency of this method : comparing to the explicit algebraic algorithm in ref . [ 29 ] we have taken much less time and space to construct the improved charges . for the sake of clarity we have gathered a few examples in appendix a. .3truecm we have also developed a diagrammatic technique to calculate the algebra itself . it can be seen as a set of contraction rules between the chains that constitute the charges . indeed , in computing the algebra of non - local charges we have to consider all possible contractions " ( i.e. dirac brackets ) between symmetrized chains . after some partial integrations we end up with elementary contractions of the following general kind : s_ia(x)t_bj(x ) \{(j_)_ab(x),(j_)_cd(y ) } u_kc(y)v_dl(y ) -(i j ) - ( k l ) . the current algebra ( 3 ) tells us that a contraction @xmath35 may produce a current - like term @xmath36 or a schwinger term . let us discuss the first kind , in which case ( 28 ) produces 4 terms : @xmath37_{ij , kl } \quad .\end{aligned}\ ] ] we can associate each of the 4 terms above to one of the 4 possible contractions between the 2 pairs of symmetrized chains . in the presence of a schwinger term we must take into account extra contributions involving the intertwiner and partial integrations . in any event , the contractions between chains can be resumed by the following rules : .5truecm * step 1 : choice * .2truecm in calculating @xmath38 we take one chain from @xmath39 and other from @xmath14 . then we pick up the internal " current components we intend to contract . this is symbolized by the generic diagram below : .5truecm * step 2 : isolation * .2truecm in each chain we must localize " the current components chosen in step 1 . this was explicitly made in ( 28 ) by means of partial integrations . within the diagrams this is achieved by inverting some arrows until all of them are pointing towards the chosen semicircle ( i.e. the current component we are isolating ) . eventually a minus sign will be picked up , depending on the number of inversions . finally have have this sort of diagrams : .2truecm the next step is just a graphic bending of chains , as a preparation to the final contraction . the chains from ( 31 ) should be bended in the following way : notice that the sub - chains @xmath40 and @xmath41 were transposed as eq . ( 31 ) demands . actually the graphic bending implies the transposition , as exemplified below where the transposed current components are naturally represented as @xmath42 .5truecm * step 4 : contraction * .2truecm finally we perform the contraction in ( 32 ) according to the rules below : where we introduced a symbol corresponding to the @xmath5-product , -.6 cm = ( ab)=(ba ) . for instance , a typical contraction between @xmath43 components would be @xmath44 = dx(su^^tt^^tj__0v ) . of course one must repeat all steps for every possible contraction . .2truecm the current @xmath33 obeys another constraint [ 27 ] involving the @xmath5-product , namely ( j_j)= 0 = -.6 cm which must also be taken into account . .3truecm we mention that the elementary contractions in ( 35 ) are nothing but the graphic representation of the current algebra ( 3 ) , where the diagrams containing the intertwiner field come from schwinger terms followed by partial integrations . these rules were applied to compute various brackets and in all cases the algebra ( 6 ) was confirmed . one can also find an example in appendix a. the most remarkable outcome of this graphic procedure is that it poses an easy and straightforward way to the supersymmetric extension and possibly other generalizations . the supersymmetric non - linear @xmath0 sigma model is defined [ 30 - 32 ] by the lagrangean _ s=__i^_i+ _ i /_i+ ( _ i_i)^2 , where @xmath45 are scalars and @xmath46 are majorana fermions satisfying the constraints _ i=1^n_i^2 - 1=0 , _ i=1^n_i_i=0 . we also have a conserved @xmath0 current @xmath47 which can be split into bosonic and fermionic parts @xmath48 whose curvature obeys the equation f _ = _ j_-_j_+2[j_,j_]= -(_b_- _ b _ ) . even though its curvature is not null , one can construct non - local conserved charges out of @xmath49 [ 30,31 ] . here we shall deal with an algebraic procedure to derive these charges . therefore it is necessary to start from the elementary @xmath0 current algebra , listed below , @xmath50_{ij , kl}(x)\delta(x - y ) \quad , \nonumber \\ & & \{(j_{{}_0})_{ij}(x),(j_{{}_1})_{kl}(y)\}= ( i\circ j_{{}_1})_{ij , kl}(x)\delta(x - y)+(i\circ j)_{ij , kl}(y)\delta'(x - y ) \quad , \nonumber \\ & & \{(j_{{}_1})_{ij}(x),(j_{{}_1})_{kl}(y)\}=0 \quad , \nonumber \\ & & \nonumber \\ & & \{(b_{{}_0})_{ij}(x),(b_{{}_0})_{kl}(y)\}= [ ( i\circ b_{{}_0})-(j\circ b_{{}_0})]_{ij , kl}(x)\delta(x - y ) \quad , \nonumber \\ & & \{(b_{{}_0})_{ij}(x),(b_{{}_1})_{kl}(y)\}= [ ( i\circ b_{{}_1})-(j\circ b_{{}_1})]_{ij , kl}(x)\delta(x - y ) \quad , \\ & & \{(b_{{}_1})_{ij}(x),(b_{{}_1})_{kl}(y)\}= [ ( i\circ b_{{}_0})-(j\circ b_{{}_0})]_{ij , kl}(x)\delta(x - y ) \quad , \nonumber \\ & & \nonumber \\ & & \{(j_{{}_0})_{ij}(x),(b_{{}_0})_{kl}(y)\}= ( j\circ b_{{}_0})_{ij , kl}(x)\delta(x - y ) \quad , \nonumber \\ & & \{(j_{{}_0})_{ij}(x),(b_{{}_1})_{kl}(y)\}= ( j\circ b_{{}_1})_{ij , kl}(x)\delta(x - y ) \quad , \nonumber \\ & & \{(j_{{}_1})_{ij}(x),(b_{{}_0})_{kl}(y)\}=0 \quad , \nonumber \\ & & \{(j_{{}_1})_{ij}(x),(b_{{}_1})_{kl}(y)\}=0 \quad , \nonumber\end{aligned}\ ] ] where the intertwiner and the @xmath5-product were already defined in ( 4 ) and ( 5 ) . the @xmath0 local charge and the first non - local charge are given [ 30,31 ] by the integrals @xmath51 some other supersymmetric _ standard _ non - local charges can be found in the literature [ 30 - 32 ] . however , as in the bosonic case , we are searching for _ improved _ charges satisfying the simplest algebra . using the algebraic method proposed in ref . [ 29 ] we have computed the improved charges and their brackets up to @xmath52 , finding the _ same cubic algebra _ given by ( 6 ) . calculation is hopelessly longer than in the bosonic theory , but we have been able to develop some graphic rules which rather simplified our work . this diagrammatic method is a direct extension of the one proposed for the bosonic theory . for instance , one can show that the supersymmetric step - generator @xmath13 satisfies the following algebraic relations : @xmath53 as in the bosonic model ( recall eqs.(23 - 25 ) ) these relations lead us to the proper transformation rules for the construction of charges . one can use the following iterative procedure : .3truecm \i ) we propose the symbolic notation @xmath54 b__0 -.1 cm , b__1 -.1 cm @xmath55 @xmath56 .5truecm ii ) we take the diagram associated to @xmath14 and replace the left tip " of each chain as follows : these transformations are a direct translation of eqs . ( 45 - 48 ) . .5truecm iii ) after using the constraints on @xmath33 and @xmath57 you will have the diagram of @xmath31 . .5truecm in order to calculate the algebra between the non - local charges , one should follow the same algorithm ( choice , isolation , bending and contraction ) , using the contraction rules below @xmath58 @xmath59 which is the graphic version of the algebra ( 43 ) . we also have a new constraint , b_j = jb_= 0 = -.1 cm to be added to the list in ( 26,27,38 ) . as before , the half - white / half - black triangle means @xmath60 or @xmath61 in general . .5truecm we have used this procedure to construct several charges and confirmed the algebra ( 6 ) . this is actually the main result of this paper , confirming previous conjectures . .3truecm to complete the algebraic analysis , we have also considered the conserved supersymmetry current and charge , given by @xmath62 using the equations of motion , we have checked that \ { q , q^(n ) } = 0 , n0 which means that every non - local charge is invariant under supersymmetry on - shell as already pointed in ref . therefore the non - local charges in the supersymmetric sigma model are all bosonic . however we must stress that this is not a general property : for instance , in ref . [ 34 ] one finds an integrable supersymmetric theory the supersymmetric two boson hierarchy containing fermionic non - local charges whose graded algebra exhibits cubic terms similar to those of eq . it would be very interesting to develop graphic rules for this kind of model [ 35 ] . this model consists of an @xmath63-plet of majorana fermions transforming as a fundamental representation of the @xmath0 group , with a quartic interaction . its lagrangean reads [ 7 ] _ _ gn = _ ( _ i_i)^2 , and it can be regarded as the limit of null bosonic field ( @xmath64 ) in the supersymmetric model ( 39 ) notice that the constraints ( 40 ) disappear . the noether current associated to the @xmath0 rotations is ( b_)_ij = -i_i__i , _ b^=0 and it satisfies the curvature - free condition _ b_-_b_+ [ b_,b _ ] = 0 and the algebraic relations @xmath65 as before , we may construct an infinite number of conserved non - local currents using the potential algorithm : we consider a conserved current @xmath66 and the corresponding potential @xmath67 , b_^(n)= _ then we define the current @xmath68 as b_^(n+1 ) = 2 ( _ + b _ ) ^(n ) . the properties ( 56 ) and ( 57 ) imply that @xmath68 is also conserved . starting out with @xmath69 we find an infinite number of conserved charges @xmath70 . after applying this algorithm to build up some of them , it is straightforward to check that this method is equivalent to the following graphic procedure : one chooses some symbols to represent @xmath60 and @xmath61 , for instance , b__0 -.1 cm 2truecm b__1 -.1 cm then one takes the sequence of chains associated to @xmath14 and uses the replacement rules for left - tips on the other hand , this is precisely the limit @xmath71 of the transformation rules ( 50 ) in the supersymmetric theory . .3truecm this provides an alternative derivation of the graphic rules to construct charges in the gross - neveu model . moreover it implies that those charges defined by the algorithm ( 60 ) are actually the improved charges and thus they must obey the cubic algebra ( 6 ) . diagrammatic methods are frequently used in physics to simplify long calculations and so we have proposed a graphic procedure to construct and compute the algebra of non - local charges in non - linear sigma models . applying such procedure we have been able to verify that the ( improved ) non - local charges in the supersymmetric @xmath0 sigma model obey a cubic yangian - like algebra which can be expressed as in eq . .2truecm one could easily recover the bosonic model from the supersymmetric theory by taking the no - fermion limit @xmath72 . moreover , the @xmath0 invariant gross - neveu model can be obtained after erasing the bosonic fields ( @xmath64 ) . the improved charges in these models may be different but their algebra is exactly the same . in the gross - neveu model , the improved charges could be computed by means of two different methods and therefore the corresponding graphic rules could be confirmed and further understood . we expect to find a similar confirmation in the sigma models but the presence of the intertwiner field within the diagrams has impeded us so far . .2truecm it is also interesting to consider the inclusion of wess - zumino terms , which modify the current algebra ( see for instance ref . [ 28,29 ] ) and derive the corresponding graphic rules . this problem and the general application of diagrammatic methods to integrable theories is presently under investigation [ 35 ] . 2truecm * acknowledgements * .5truecm we would like to thank e. abdalla and m.c.b . abdalla for helpful comments . we also thank j.c . brunelli for suggestions and for his participation in the early stages of this work . [ [ graphic - derivation - of - q1q1-in - the - non - linear - sigma - model ] ] graphic derivation of @xmath79 in the non - linear sigma model ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ notice that we have computed two minus signs : one from the inversion of an arrow ( along the isolation step ) and other from the transposition of @xmath80 ( during the bending step ) . the first contribution is this commented example may seem rather long , not revealing the actual power of the graphic method . but we can assure the reader that , after practicing the contraction rules for a while and skipping the intermediate comments we have been able to compute many dirac brackets very efficiently . maillet , j .- : hamiltonian structures for integrable classical theories from graded kac - moody algebras . 167b * , 401 - 405 ( 1986 ) ; new integrable canonical structures in two - dimensional models . b269 * , 54 - 76 ( 1986 ) . haldane , f.d . , ha , z.n.c . , talstra , j.c . , bernard , d. , pasquier , v. : yangian symmetry of integrable quantum chains with long - range interactions and a new description of states in conformal field theory . . lett . * 69 * , 2021 - 2025 ( 1992 ) . mackay , n.j . : on the bootstrap structure of yangian - invariant factorized @xmath25-matrices . print-92 - 0535 ( durham ) . bulletin board : hep - th - 9211091 . in tianjin 1992 , proceedings , differential geometric methods in theoretical physics 360 - 363 .
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we propose a graphic method to derive the classical algebra ( dirac brackets ) of non - local conserved charges in the two - dimensional supersymmetric non - linear @xmath0 sigma model . as in the purely bosonic theory
we find a cubic yangian algebra .
we also consider the extension of graphic methods to other integrable theories .
epsf.tex = 6.5 in = -.3 in = -.5 in j j__0 j j__1
| 7,227 | 126 |
majorana fermions ( mfs ) are real solutions of the dirac equation and which are their own antiparticles @xmath0 @xcite . although proposed originally as a model for neutrinos , mfs have recently been predicted to occur as quasi - particle bound states in engineered condensed matter systems @xcite . this exotic particle obeys non - abelian statistics , which is one of important factors to realize subsequent potential applications in decoherence - free quantum computation @xcite and quantum information processing @xcite . over the recent few years , the possibility for hosting mfs in exotic solid state systems focused on topological superconductors @xcite . currently , various realistic platforms including topological insulators @xcite , semiconductor nanowires ( snws ) @xcite , and atomic chains @xcite have been proposed to support majorana states based on the superconducting proximity effect . although various schemes have been presented , observing the unique majorana signatures experimentally is still a challenging task to conquer . mfs are their own antiparticles , and they are predicted to appear in tunneling spectroscopy experiments , in which majoranas manifest themselves as characteristic zero - bias peaks ( zbps ) @xcite . the theoretical predictions of zbps have been observed experimentally in snws which are interpreted as the signatures of mfs mourikv , dasa , dengmt , churchillhoh , finckadk . remarkably , nadj - perge et al . @xcite recently designed a chain of magnetic fe atoms deposited on the surface of an s - wave superconducting pb with strong spin - orbit interactions , and reported the striking observation of a zbp at the end of the atomic chains with stm , which provides evidence for majorana zero modes . however , these above experimental results can not serve as definitive evidences to prove the existence of mfs in condensed matter systems , and it is also a major challenge in these experiments to uniquely distinguish majoranas from conventional fermionic subgap states . the first reason is that the zero - bias conductance peaks can also appear in terms of the other mechanisms @xcite , such as the zero - bias anomaly due to kondo resonance @xcite and the disorder or band bending in the snw @xcite . the second one is that andreev bound states in a magnetic field can also exhibit a zero - energy crossing as a function of exchange interaction or zeeman energy @xcite , and therefore give rise to similar conductance features . as far as we know , most of the experimental evidences for majorana bound states largely relies on measurements of the tunneling conductance at present , and the observation of majorana signature based on electrical methods still remains a subject of debate . identifying mfs only through tunnel spectroscopy is somewhat problematic . therefore , to obtain definitive signatures of mfs , alternative setups or proposals for detecting mfs are necessary . here , we will propose an alternative all - optical scheme to detect mfs . benefitting from recent advances in nanotechnology and nanofabrication , nanostructures such as quantum dots ( qds ) and nanomechanical resonators ( nrs ) have been obtained significant progress in modern nanoscience and nanotechnology . qd , as a simple stationary atom with well optical property @xcite , lays the foundation for numerous possible applications urbaszekb . due to high natural frequencies and large quality factors of nrs @xcite , if qds coupled to nrs @xcite to form hybrid systems , the coherent optical properties will be enhanced remarkably , which will be an alternative ultrasensitive detection means . although probing mfs with qds @xcite have been proposed , we notice that all the schemes are still based on electrical means . in the present work , we propose an optical measurement scheme to detect the existence of mfs in iron chains on the superconducting pb surface @xcite via a coupled hybrid qd - nr system with optical pump - probe scheme @xcite . compared with electrical detection means where the qds are coupled to mfs via the tunneling @xcite , in our optical scheme , there is no direct contact between mfs and the hybrid qd - nr system , which can effectively avoid introducing other signals disturbing the detecting of mfs . the interaction between mfs in iron chains and qd in hybrid qd - nr system is mainly due to the dipole - dipole interaction , and the distance between the two systems can be adjusted by several tens of nanometers , therefore the tunneling between the qd and mfs can be neglected safely . in addition , the qd is considered as a two - level system rather than a single resonant level with spin - singlet state , and once mfs appear in the end of iron chains and couple to the qd , the majorana signature will be carried out via the coherent optical spectrum of the qd . the change in the coherent optical spectrum as a possible signature for mfs is another potential evidence in the iron chains . this optical scheme will provide another method for the detection of mfs , which is very different from the zero - bias peak in the tunneling experiments mourikv , dasa , dengmt , churchillhoh , finckadk , nadj - perges2 . furthermore , in order to investigate the role of the nr in the hybrid system , we further introduce the exciton resonance spectrum to detect mfs . the results shows that the vibration of the nr acting as a phonon cavity will enhance the exciton resonance spectrum significantly and make mfs more sensitive to be detectable . the technique proposed here provide a new platform for applications ranging from robust manipulation of mfs and mfs based quantum information processing . figure 1(b ) shows the schematic setup that will be studied in this work , where iron ( fe ) chains on the superconducting pb(110 ) surface nadj - perges2 , and we employ a two - level qd with optical pump - probe technology to detect mfs . the fe chain is ferromagnetically ordered nadj - perges2 with a large magnetic moment , which takes the role of the magnetic field in the nanowire experiments @xcite . different from the proposal of mourik et al . @xcite , this `` magnetic field '' is mostly localized on the fe chain , with small leakage outside , and superconductivity is not destroyed along the chain . in this setup , the energy scale of the exchange coupling of the fe atoms is typically much larger than that of the rashba spin - orbit coupling and the superconducting pairing . figure 1(c ) displays that a qd is implanted in the nr to form a coupled hybrid qd - nr system . the whole system includes two kinds of couplings which are qd - mf coupling and qd - nr coupling as shown in fig . 1(a ) . in the following , we will discuss the two kinds of coupling in detail , respectively . in the hybrid qd - nr system , the qd is modeled as a two - level system consisting of the ground state @xmath1 and the single exciton state @xmath2 at low temperatures zrennera , stuflers , and the hamiltonian of the qd can be described as @xmath3 with the exciton frequency @xmath4 , where @xmath5 and @xmath6 are the pseudospin operator describing the two - level exciton with the commutation relation @xmath7 = \pm s^{\pm } $ ] and @xmath8 = 2s^{z}$ ] . for the nr , the thickness of the beam is much smaller than its width , the lowest - energy resonance corresponds to the fundamental flexural mode that will constitute the resonator mode @xcite which can be characterized by a quantum harmonic oscillator with hamiltonian @xmath9 , where @xmath10 is the resonator frequency and @xmath11 is the annihilation operator of the resonator mode . since the flexion induces extensions and compressions in the structure @xcite , this longitudinal strain will modify the energy of the electronic states of qd through deformation potential coupling . then the coupling between the resonator mode and the qd is described by @xmath12 , where @xmath13 is the coupling strength between the resonator mode and qd wilson - raei . thus we obtain the hamiltonian of the coupled hybrid qd - nr system@xmath14 for the qd - mf coupling , as each mf is its own antiparticle , we introduce an operator @xmath15 with @xmath16 and @xmath17 to describe mfs . supposed that the qd couples to the nearby mf @xmath18 in the end of iron chains , then the hamiltonian is written by liude , flensbergk , leijnsem , caoys , lij @xmath19to detect mfs , it is helpful to switch the majorana representation to the regular fermion one via the exact transformation @xmath20 and @xmath21 @xmath22 , where @xmath23 and @xmath24 are the fermion annihilation and creation operators obeying the anti - commutative relation @xmath25 . accordingly , in the rotating wave approximation @xcite , the above hamiltonian can be rewritten as@xmath26where the first term gives the energy of mf with frequency @xmath27 and @xmath28 with the iron chains length ( @xmath29 ) and the pb superconducting coherent length ( @xmath30 ) . if the iron chains length ( @xmath29 ) is large enough , @xmath31 will approach zero . in the following , we will discuss the two conditions of @xmath32 and @xmath33 , and define the two conditions as coupled mfs ( @xmath34 ) and uncoupled mfs ( @xmath33 ) , respectively . the second term describes the coupling between the nearby mf and the qd with the coupling strength @xmath35 , where the coupling strength is related to the distance between the hybrid qd - nr system and the iron chains . it should be also noted that the term of non - conservation for energy , i.e. @xmath36 , is generally neglected . we have made the numerical calculations ( not shown in the following figures ) and shown that the effect of this term is too small to be considered in our theoretical treatment . currently , the optical pump - probe technique has become a popular topic , which affords an effective way to investigate the light - matter interaction . the optical pump - probe technology includes a strong pump laser and a weak probe laser @xcite . in the optical pump - probe technology , the strong pump laser is used to stimulate the system to generate coherent optical effect , while the weak laser plays the role of probe laser . therefore , the linear and nonlinear optical effects can be observed via the probe absorption spectrum based on the optical pump - probe scheme . xu et al . have obtained coherent optical spectroscopy of semiconductor qd when driven simultaneously by two optical fields @xcite . their results open the way for the demonstration of numerous quantum level - based applications , such as qd lasers , optical modulators , and quantum logic devices . in terms of this scheme , we apply the pump - probe scheme to the qd of the hybrid qd - nr system simultaneously . when the optical pump - probe technology is applied on the qd , the majorana signature will be carried out via the coherent optical spectrum . the hamiltonian of the exciton of the qd coupled to the two fields is given by @xcite @xmath37 , where @xmath38 is the dipole moment of the exciton , and @xmath39 is the slowly varying envelope of the field . therefore , we obtain the whole hamiltonian of the hybrid system as @xmath40 . in a rotating frame at the pump field frequency @xmath41 , we obtain the total hamiltonian of the system as @xmath42where @xmath43 is the detuning of the exciton frequency and the pump frequency , @xmath44 is the rabi frequency of the pump field , and @xmath45 is the detuning of the probe field and the pump field . @xmath46 is the detuning of the mf frequency and the pump frequency . actually , we have neglected the regular fermion like normal electrons in the nanowire that interact with the qd in the above discussion . to describe the interaction between the normal electrons and the exciton in qd , a tight binding hamiltonian of the whole iron chains is introduced chenhj . according to the heisenberg equation of motion and introducing the corresponding damping and noise terms , the quantum langevin equations of the whole system are derived as@xmath47@xmath48s^{-}+2(\beta f - i\omega _ { pu})s^{z}-\frac{2i\mu e_{pr}}{\hbar } e^{-i\delta t}s^{z}+\hat{% \tau}(t)\text{,}\]]@xmath49@xmath50where @xmath51 ( @xmath52 ) is the exciton spontaneous emission rate ( dephasing rate ) , @xmath53 is the position operator , @xmath54 is the decay rate of the nr , and @xmath55 is the decay rate of the mf . @xmath56 is the @xmath57-correlated langevin noise operator , which has zero mean @xmath58 and obeys the correlation function @xmath59 . the resonator mode is affected by a brownian stochastic force with zero mean value , and @xmath60 has the correlation function@xmath61,\]]where @xmath62 and @xmath63 are the boltzmann constant and the temperature of the reservoir of the coupled system . mfs have the same correlation relation as the resonator mode as @xmath64.\]]in eq.(9 ) and eq.(10 ) , both the nr and majorana mode will be affected by a thermal bath of brownian and non - markovian processes @xcite . in the low temperature , the quantum effects of both the majorana and nr mode are only observed in the case of @xmath65 and @xmath66 . due to the weak coupling to the thermal bath , the brownian noise operator can be modeled as markovian processes . in addition , both the qd - mfs coupling and qd - nr mode coupling in the hybrid system are stronger than the coupling to the reservoir that influences the two kinds coupling . in this case , owing to the second order approximation gardinercw , we can obtain the form of the reservoir that affects both the nr mode and majorana mode as eq.(9 ) and eq.(10 ) . to go beyond weak coupling , the heisenberg operator can be rewritten as the sum of its steady - state mean value and a small fluctuation with zero mean value@xmath67since the driving fields are weak , but classical coherent fields , we will identify all operators with their expectation values , and drop the quantum and thermal noise terms . simultaneously , inserting these operators into the langevin equations eqs.(5)-(8 ) and neglecting the nonlinear term , we can obtain two equation sets about the steady - state mean value and the small fluctuation . the steady - state equation set consisting of @xmath68 , @xmath69 and @xmath70 is related to the population inversion ( @xmath71 ) of the exciton which is determined by@xmath72 + 4w_{0}\gamma _ { 2}\omega _ { pu}^{2}(\delta _ { m}^{2}+\kappa _ { m}^{2}/4)=0.\end{gathered}\]]for the equation set of small fluctuation , we make the ansatz @xcite @xmath73 ( @xmath74 ) . solving the equation set and working to the lowest order in @xmath75 but to all orders in @xmath76 , we can obtain the linear susceptibility as @xmath77 , where @xmath78 is given by@xmath79\gamma _ { 2}}{\pi { % _ { 2}\pi _ { 4}^{\ast } -\lambda _ { 1}\lambda _ { 2}\pi _ { 1}\pi _ { 3}^{\ast } } } , \]]@xmath68 , @xmath70 and @xmath69 can be derived from the steady - state equations , and @xmath80 , @xmath81 , @xmath82 , @xmath83/(\gamma _ { 1}-i\delta ) $ ] , @xmath84/(\gamma _ { 1}-i\delta ) $ ] , @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 ( @xmath90 indicates the conjugate of @xmath91 ) . the imaginary and real parts of @xmath92 indicate absorption and dissipation , respectively . in addition , the average population of the exciton states can be obtained as@xmath93}{\pi { _ { 2}\pi _ { 4}^{\ast } -\lambda _ { 1}\lambda _ { 2}\pi _ { 1}\pi _ { 3}^{\ast } } } , \]]which is benefited for readout the exciton states of qd . for illustration of the numerical results , we choose the realistic hybrid systems of the coupled qd - nr system @xcite and the iron chains on the superconducting pb surface @xcite . for an inas qd in the coupled qd - nr system , we use parameters @xcite : the exciton relaxation rate @xmath94 ghz , the exciton dephasing rate @xmath95 ghz . the physical parameters of gaas nr are @xmath96 , @xmath97 , @xmath98 ghz , @xmath99 kg , @xmath100 , where @xmath101 , @xmath97 , and @xmath102 are the resonator frequency , the effective mass , and quality factor of the nr , respectively . the decay rate of the nr is @xmath103 @xmath104 khz , and the coupling strength between the qd and nr is @xmath105 . for mfs , there are no experimental values for the lifetime of the mfs and the coupling strength between the exciton and mfs in the recent literature . however , according to a few recent experimental reports @xcite , it is reasonable to assume that the lifetime of the mfs is @xmath106 mhz . since the coupling strength between the qd and nearby mfs is dependent on their distance , we also expect the coupling strength @xmath107 ghz via adjusting the distance between the hybrid qd - nr system and the iron chains . figure 2(a ) shows the coherent optical properties of the qd as functions of probe - exciton detuning @xmath108 at the detuning of the exciton frequency and the pump frequency @xmath109 , i.e. , the absorption ( @xmath110 ) and dissipation ( @xmath111 ) properties of the qd without considering any coupling ( @xmath112 ) , which indicates the normal absorption and dissipation of the qd , respectively . turning on the qd - nr coupling ( @xmath105 ) and without considering the qd - mf coupling ( @xmath113 ) , two sharp peaks will appear in both the absorption and dissipation spectra as shown in fig . 2(b ) . from the curves , we find that the two sharp peaks at both sides of the spectra just correspond to the vibrational frequency of the nr . the physical origin of this result is due to mechanically induced coherent population oscillation , which makes quantum interference between the resonator and the beat of the two optical fields via the qd when the probe - pump detuning is equal to the nr frequency @xcite . this reveals that if fixing the pump field on - resonance with the exciton and scan through the frequency spectrum , the two sharp peaks can obtain immediately in the coherent optical spectra , which also indicates a scheme to measure the frequency of the nr . this phenomenon stems from the quantum interference between the vibration nr and the beat of the two optical fields via the exciton when probe - pump detuning @xmath114 is adjusted equal to the frequency of the nr . therefore , the qd - nr coupling play a key role in the hybrid system , and if we ignore the coupling ( @xmath115 ) , the above phenomenon will disappear completely as shown in fig . compared with fig.2(b ) , in fig.2(c ) , we consider the qd coupled with the nearby mf @xmath18 without taking the qd - nr coupling into account , i.e. the condition of @xmath115 and @xmath107 ghz . as the mfs appear in the ends of iron chains and coupled to the qd , both the probe absorption ( the blue curve ) and dissipation ( the green curve ) spectra will present an remarkable signature of mfs under @xmath116 ghz . the physical origin of this result is due to the qd - mf coherent interaction and we can interpret this physical phenomenon with dressed state between the exciton and mfs . the qd coupled to the nearby mf will induce the upper level of the state @xmath117 to split into @xmath118 and @xmath119 ( @xmath120 denotes the number states of the mfs ) . the left peak in the coherent optical spectra signifies the transition from @xmath1 to @xmath121 while the right peak is due to the transition of @xmath122 to @xmath119 chenhj . to determine this signature is the true mfs rather than the normal electrons that couple with the qd , we have used a tight binding hamiltonian to describe the electrons in whole iron chains , the numerical results indicate the signals in the absorption and dissipation spectra are the true mfs signature @xcite . if we consider both the two kinds coupling , i.e. the qd - nr coupling ( @xmath105 ) and qd - mfs coupling ( @xmath107 ghz ) as shown in fig . 2(d ) , not only the two sharp peaks locate at the nr frequency induced by its vibration , i.e. two peaks are at @xmath123 ghz ( @xmath124ghz ) , there is also mfs signal appear at @xmath125 ghz ( @xmath116 ghz ) induced by the qd - mf coupling . in fig . 2(c ) , we only consider the situation of @xmath34 . in fact , if the iron chains length @xmath29 is much larger than the pb superconducting coherent length @xmath30 , @xmath31 will approach zero . therefore , it is necessary to consider the conditions of @xmath126 and @xmath33 , and we define them as coupled mfs mode ( @xmath32 ) and uncoupled mfs mode ( @xmath33 ) , respectively . figure 3(a ) and figure 3(b ) show the absorption and dissipation spectra as a function of detuning @xmath127 with qd - mf coupling constants @xmath128 ghz under @xmath34 and @xmath33 , respectively . compared with the coupled mfs mode , the uncoupled qd - mf hamiltonian will reduce to @xmath129 which is analogous j - c hamiltonian of standard model under @xmath33 , and the probe absorption spectrum ( the blue curve ) shows a symmetric splitting as the qd - mf coupling strength @xmath107 ghz which is different from of coupled mfs mode presenting unsymmetric splitting due to a detuning @xmath130 ghz . therefore , our results reveal that the signals in the coherent optical spectra is a real signature of mf , and the optical detection scheme can work at both the coupled majorana edge states and the uncoupled majorana edge states . in fig . 3(c ) , we further make a comparison of the probe absorption spectrum under the coupled mfs mode ( @xmath34 ) and uncoupled mfs mode ( @xmath131 ) . it is obvious that the probe absorption spectrum display the analogous phenomenon of electromagnetically induced transparency ( eit ) @xcite under both the two conditions . the dip in the probe absorption spectrum goes to zero at @xmath132 and @xmath133 ghz with @xmath33 and @xmath34 , respectively , which means the input probe field is transmitted to the coupled system without absorption . such a phenomenon is attributed to the destructive quantum interference effect between the majorana modes and the beat of the two optical fields via the qd . if the beat frequency of two lasers @xmath57 is close to the resonance frequency of mfs , the majorana mode starts to oscillate coherently , which results in stokes - like ( @xmath134 ) and anti - stokes - like ( @xmath135 ) scattering of light from the qd . the stokes - like scattering is strongly suppressed because it is highly off - resonant with the exciton frequency . however , the anti - stokes - like field can interfere with the near - resonant probe field and thus modify the probe field spectrum . here the majorana modes play a vital role in this coupled system , and we can refer the above phenomenon as majorana modes induced transparency , which is analogous with eit in atomic systems @xcite . on the other hand , we can propose a means to determine the qd - mf coupling strength @xmath35 via measuring the distance of the two peaks with increasing the qd - mf coupling strength in the probe absorption spectrum . figure 3(d ) indicates the peak - splitting width as a function of the qd - mf coupling strength @xmath35 under the condition of the coupled mfs mode ( @xmath136 ) and the uncoupled mfs mode ( @xmath33 ) which follows a nearly linear relationship . it is obvious that the two lines ( the uncoupled mfs and the coupled mfs mode ) have a slight deviation . however , the deviation becomes slighter with increasing coupling strength . therefore , it is essential to enhance the coupling strength for a clear peak splitting via adjusting the distance between the qd and the nearby mfs . in this case the coupling strength can obtain immediately by directly measuring the distance of the two peaks in the probe absorption spectrum . as shown in fig . 2(d ) , there are not only two sharp peaks locate at the nr frequency induced by its vibration but also the mfs signal appear at @xmath137 induced by the qd - mf coupling in the probe absorption spectrum ( the blue curve ) under the two kinds coupling . in fig . 4(a ) , we further consider switching the detuning @xmath116 ghz to @xmath138 ghz at small exciton - pump detuning @xmath139 ghz . it is obvious that the resonance amplification process ( 1 ) and the resonance absorption process ( 2 ) in the probe absorption spectrum without considering the qd - mf coupling ( the blue curve , @xmath113 ) will accordingly transform into the the resonance absorption process ( 3 ) and the resonance amplification process ( 4 ) due to the qd - mf coupling ( the green curve , @xmath140 ghz ) . return to fig . 1(a ) , there are two kinds of coupling which are qd - nr coupling and qd - mf coupling in the hybrid system . for the qd - nr system , the two sharp peaks in the probe absorption corresponding to the resonance amplification ( 1 ) and absorption process ( 2 ) can be elaborated with dressed states @xmath141 , @xmath142 , @xmath143 , @xmath144 ( @xmath145 denotes the number state of the resonance mode ) , and the two sharp peaks indicate the transition between the dressed states @xcite . however , once mfs appear in the ends of iron chains and coupled to the qd , the ground state @xmath146 and the exciton state @xmath2 of the qd will also modify by the number states of the mfs @xmath120 and induce the majorana dressed states @xmath147 , @xmath148 , @xmath118 , @xmath149 . with increasing the qd - mf coupling , the majorana dressed states will affect the amplification ( 1 ) and absorption process ( 2 ) significantly , and even realize the inversion between the absorption ( 3 ) and amplification ( 4 ) process due to the qd - mf coherent interaction ( the green curve ) . to illustrate the advantage of the nr in the hybrid system , we introduce the exciton resonance spectrum to investigate the role of nr in the coupled qd - nr , which is benefited for readout the exciton states of qd . in fig . 4(b ) , we adjust the detuning @xmath116 ghz to @xmath150 ghz , therefore , the location of the two sideband peaks induced by the qd - mf coupling coincides with the two sharp peaks induced by the vibration of nr , thus the nr is resonant with the coupled qd - mf system and makes the coherent interaction of qd - mf more strong . figure 4(b ) shows the exciton resonance spectrum of the probe field as a function of the probe detuning @xmath151 with the detuning @xmath139 ghz under the coupled mfs mode @xmath34 . the black and red curves correspond to @xmath115 and @xmath152 for the qd - mf coupling @xmath153 ghz , respectively . it is obvious that the role of nr is to narrow and to increase the exciton resonance spectrum . in this case , the nr behaves as a phonon cavity will enhance the sensitivity for detecting mfs . we have proposed an all - optical means to detect the existence of mfs in iron chains on the superconducting pb surface with a hybrid qd - nr system . the signals in the coherent optical spectra indicate the possible majorana signature , which provides another supplement for detecting mfs . due to the vibration of nr , the exciton resonance spectrum becomes much more significant and then enhances the detection sensitivity of mfs . in addition , the qd - mf coupling in our system is a little feeble , while ref . [ 35 ] presents a strong qd - mf coupling and the coupling strength can reach kilohertz , which is beneficial for mfs detection . on the other hand , if we consider embedding a metal nanoparticle - quantum dot ( mnp - qd ) complex chenhj , lijj in the nr , the surface plasmon induced by the mnp will enhance the coherent optical property of qd , which may be robust for probing mfs . the concept proposed here , based on the quantum interference between the nr and the beat of the two optical fields , is the first all - optical means to probe mfs . this coupled system will provide a platform for applications in all - optically controlled topological quantum computing based on mfs . the authors gratefully acknowledge support from the national natural science foundation of china ( no.11574206 , no.10974133 , no.11274230 , no.61272153 , no.61572035 , no.51502005 , and no.11404005 ) , the key foundation for young talents in college of anhui province ( no . 2013sqrl026zd ) , and the foundation for phd in anhui university of science and technology . i. yeo , p. l. de assis , a. gloppe , e. dupont - ferrier , p. verlot , n. s. malik , e. dupuy , j. claudon , j. m. grard , a. auffves , g. nogues , s. seidelin , j. p. poizat , o. arcizet , and m. richard , nat . nanotechnol . * 9 * , 106 ( 2014 ) . fig.1 sketch of the proposed setup for optically detecting majorana fermions ( mfs ) . ( a ) the energy - level diagram of a qd coupled to mfs and nr , which includes two kinds coupling , i.e. the qd - mf coupling ( the dotted frame ) and the qd - nr coupling ( the dashed frame ) . ( b ) the iron chains on the superconducting pb surface , and a pair of mfs appear in the ends of the iron chains . the nearby mf is coupled to ( c ) a qd embedded in a nanomechanical resonator ( nr ) with optical pump - probe technology . fig.2 the absorption ( the blue curve ) and dispersion ( the green curve ) spectra of probe field as a function of the probe detuning @xmath127 under different conditions . ( a ) without considering any coupling , i.e. , @xmath115 and @xmath113 . ( b ) the qd - nr coupling strength is @xmath105 and @xmath113 . ( c ) the qd - mf coupling strength is @xmath107 ghz and @xmath115 . ( d ) considering both the qd - nr coupling and qd - mf coupling , i.e. , @xmath105 and @xmath154 ghz . the parameters used are @xmath94 ghz , @xmath95 ghz , @xmath155 khz , @xmath124 ghz , @xmath156 mhz , @xmath157(ghz)@xmath158 , @xmath116 ghz , and @xmath109 . fig.3 ( a ) and ( b ) show the probe absorption ( the blue curve ) and dispersion ( the green curve ) spectra with qd - mf coupling strengths @xmath107 ghz under @xmath34 and @xmath33 , respectively . ( c ) the probe absorption spectrum under @xmath34 ( the green curve ) and @xmath131 ( the blue curve ) , respectively . ( d ) the linear relationship between the distance of peak splitting and the coupling strength of qd - mf @xmath159 . the other parameters used are the same as in fig.2 . fig.4 ( a ) the probe absorption spectrum as a function of the probe detuning @xmath160 with considering ( the blue curve , @xmath153 ghz ) and without considering ( the green curve , @xmath113 ) the qd - mf coupling under the qd - nr coupling strength @xmath105 . ( b ) the exciton resonance spectrum as a function of @xmath127 with @xmath115 and @xmath105 at the qd - mf coupling strength @xmath153 ghz . @xmath150 ghz , @xmath139 ghz , @xmath161(ghz)@xmath158 , the other parameters used are the same as fig.2 . under different conditions . ( a ) without considering any coupling , i.e. , @xmath115 and @xmath162 . ( b ) the qd - nr coupling strength is @xmath105 and @xmath163 . ( c ) the qd - mf coupling strength is @xmath164 ghz and @xmath115 . ( d ) considering both the qd - nr coupling and qd - mf coupling , i.e. , @xmath105 and @xmath164 ghz . the parameters used are @xmath165 ghz , @xmath166 ghz , @xmath167 khz , @xmath168 ghz , @xmath169 mhz , @xmath170(ghz)@xmath158 , @xmath116 ghz , and @xmath109.,width=453 ] ghz under @xmath171 and @xmath172 , respectively . ( c ) the probe absorption spectrum under @xmath173 ( the green curve ) and @xmath174 ( the blue curve ) , respectively . ( d ) the linear relationship between the distance of peak splitting and the coupling strength of qd - mf @xmath175 . the other parameters used are the same as in fig.2.,width=453 ] with considering ( the blue curve , @xmath176 ghz ) and without considering ( the green curve , @xmath162 ) the qd - mf coupling under the qd - nr coupling strength @xmath105 . ( b ) the exciton resonance spectrum as a function of @xmath127 with @xmath115 and @xmath152 at the qd - mf coupling strength @xmath177 ghz . @xmath138 ghz , @xmath139 ghz , @xmath178(ghz)@xmath158 , the other parameters used are the same as fig.2.,width=453 ]
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motivated by a recent experiment [ nadj - perge et al . , science 346 ,
602 ( 2014 ) ] providing evidence for majorana zero modes in iron chains on the superconducting pb surface , in the present work , we theoretically propose an all - optical scheme to detect majorana fermions , which is very different from the current tunneling measurement based on electrical means .
the optical detection proposal consists of a quantum dot embedded in a nanomechanical resonator with optical pump - probe technology . with the optical means ,
the signal in the coherent optical spectrum presents a distinct signature for the existence of majorana fermions in the end of iron chains .
further , the vibration of the nanomechanical resonator behaving as a phonon cavity will enhance the exciton resonance spectrum , which makes the majorana fermions more sensitive to be detectable .
this optical scheme affords a potential supplement for detection of majorana fermions and supports to use majorana fermions in fe chains as qubits for potential applications in quantum computing devices .
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properties of rotating spinor bose - einstein condensates attract a lot of attention now . first examples of these systems with hyperfine spin @xmath0 were found in optically trapped @xmath1na @xcite . vortex phase diagram of spinor condensates is very rich , since the order parameter has three components in @xmath0 case and five components in @xmath2 case . topological excitations in spinor condensates were studied theoretically in a large number of articles see , e.g. , refs mizushima1,isoshima1,reijnders , ueda1,pogosov . at the same time , an interest is now growing to temperature effects in atomic condensates . @xcite study theoretically the berezinskii - kosterlitz - thouless ( bkt ) transition associated with the proliferation of thermally - excited vortex - antivortex pairs . for instance , in ref . @xcite it was shown that in quasi two - dimensional condensates bkt transition can occur at rather low temperatures , @xmath3 , at number of particles in the system @xmath4 . recently , some signatures of possible bkt phase were also found close to the critical temperature @xmath5 in experimental work bktexper , where condensates in optical lattice have been studied . finally , experimental evidence for the bkt transition in trapped condensates was reported in ref . @xcite deal with the thermal fluctuations of positions of vortices in rotated scalar condensates . note that , according to the mermin - wagner - hohenberg theorem , bose - einstein condensation is not possible in 2d homogeneous systems . however , application of the trapping potential leads to the macroscopic occupation of the ground state of bose gas . the aim of the present paper is to study the effect of thermal fluctuations in rotated quasi two - dimensional _ spinor _ condensates . these systems have a specific degree of freedom , associated with the relative angle among different components of the order parameter corresponding to different hyperfine state . in other words , this angle determines coherence among components of the order parameter . also it influences a transverse magnetization of the condensate . in this paper , we focus on thermal fluctuations of this angle . note that experimentally , at present time , it is possible to study the condensate phase @xcite , see also ref @xcite . in addition , recently , a new and nondestructive method for measuring the local magnetization of the condensate was proposed and successfully applied in ref . @xcite . we show that the relative angle among hyperfine components of the order parameter in 2d case can experience strong thermal fluctuation even at low temperatures . the reason is the weakness of the spin energy of the system as compared to interactions in density channel . also fluctuations of this angle lead to significant relative fluctuations of the local transverse magnetization of the condensate , which are much larger in the antiferromagnetic case than in the ferromagnetic one . this paper is organized as follows . in section ii , we give a basic formulation of the problem . in section ii , we discuss our main results for the fluctuations of angle and spin textures . we conclude in section iii . we consider harmonically - trapped quasi 2d bose - einstein condensate with spin @xmath0 . the trapping potential is given by @xmath6 where @xmath7 is a trapping frequency , @xmath8 is the mass of the atom , and @xmath9 is the radial coordinate . the system is rotated with the angular velocity @xmath10 , well below the critical rotation speed @xmath11 , and the number of atoms in the cloud is @xmath12 . in this paper , we restrict ourselves on the range of temperatures much smaller than @xmath13 . therefore , we can neglect a noncondensate contribution to the free energy of the cloud . the total energy of the system in this approximation coincides with the energy of the condensate . for the number of condensed particles , we use the ideal gas result:@xmath14 . \label{2}\]]at the same time , @xmath15 where @xmath16 is a riemann zeta function , @xmath17 . ( 2 ) and ( 3 ) remain accurate even for the case of interacting particles @xcite . we also introduce a dimensionless temperature @xmath18 . since we are considering low temperatures , @xmath19 , temperature dependence of condensed particles number can be neglected , @xmath20 . the order parameter in the @xmath0 condensate has three components @xmath21 @xmath22 . the free energy of the system can be written as machida , ho@xmath23 , \label{4}\end{aligned}\]]where the integration is performed over the system area , repeated indices are summed , @xmath24 @xmath25 is the angular momentum operator , which can be expressed in a matrix form through the usual pauli matrices , @xmath26 is the one - body hamiltonian , given by@xmath27 constants @xmath28 and @xmath29 characterize interactions in density and spin channels and are given by @xmath30 @xmath31 where @xmath32 and @xmath33 are scattering lengths for atoms with total spin 0 and 2 , and @xmath34 is the concentration of atoms in longitudinal direction . in real spinor condensates , @xmath35 , since @xmath36 . typically , @xmath37 , and this ratio can be tuned . in this paper , we study the case of relatively dilute condensate and take @xmath38 . we will consider different values of @xmath12 but at fixed value of interaction parameter @xmath28 . this is possible , since , in the case of a single layer cloud , we can always tune the trapping frequency in the longitudinal direction keeping @xmath28 constant . to ensure the regime of quasi - two - dimensionality , we can also tune @xmath7 . in this case , we have to change the rotation speed to keep the dimensionless rotation speed the same , and the themperature to fix dimensionless @xmath39 . in real atomic condensates , @xmath40 is approximately several nanometers . the most realistic value of @xmath12 for this @xmath28 is close to @xmath41 , and to illustrate the effect of @xmath12 we will consider the following range : @xmath42 . the total magnetization of the condensate is fixed : @xmath43 magnetization @xmath44 is normalized in terms of @xmath12 and maximum of @xmath45 is equal to 1 . one has also to take into account the normalization condition for the order parameter : @xmath46 the spatial profiles of all the components of the order parameter in the equilibrium can be found from the condition of minimum of energy ( 4 ) . it is also convenient to introduce the longitudinal @xmath47 and transverse @xmath48 local magnetizations of the condensate : @xmath49 @xmath50 the spin energy in this case can be represented as @xmath51 in this paper , we restrict ourselves only to the case of axially - symmetric phases , when moduli of all the components of the order parameter are independent on the azimuthal angle and depend only on the radial coordinate @xmath52 . note that equilibrium vortex phases in this situation were studied in refs . @xcite for the spin @xmath0 condensate and in ref . @xcite for @xmath2 system . for the axially - symmetric phases , each component of the order parameter can be represented as @xmath53where @xmath54 is a polar angle , @xmath55 is a winding number , and @xmath56 is a relative phase . we will denote such phases as ( @xmath57 , @xmath58 , @xmath59 ) . as it was shown in ref . @xcite , an axial symmetry of the solution implies that winding numbers satisfy the relation : @xmath60 . in this case , according to eqs . ( 11 ) and ( 12 ) , the spin energy depends on relative angle @xmath61 it is important to note that only a spin contribution to the total energy ( 4 ) depends on phases @xmath62 via the spin - mixing term . for the stationary state , which is a local minimum of gross - pitaevskii functional ( 4 ) , the value of @xmath63 is determined by the sign of interaction constant in a spin channel @xmath64 . for positive @xmath64 ( antiferromagnetic case ) , a minimum of @xmath65 is attained at @xmath66 , whereas for negative @xmath67 ( ferromagnetic case ) @xmath68 . according to the results of ref . @xcite , for the antiferromagnetic state ( @xmath69 ) , phases ( -1 , 0 , 1 ) and ( 1 , 1 , 1 ) are energetically favorable in the region of small and moderate values of magnetization @xmath44 . phase ( -1 , 0 , 1 ) is realized at low rotation frequencies @xmath70 and ( 1 , 1 , 1 ) at higher @xmath70 . in ref . @xcite , it was shown that ( 0 , 1 , 2 ) state is favorable in the ferromagnetic case ( @xmath71 ) in a region of moderate values of @xmath70 and @xmath72 in these phases , all three hypefine states are populated . fluctuations of @xmath63 have a sense only in this case , since the @xmath63-dependent part of the energy is equal to zero identically , if one of the components of the order parameter is zero . in this paper , we will concentrate on these three vortex states , since they are appropriate candidates for the illustration of the effect of thermal fluctuations . note that in homogeneous spin-1 condensate atoms populate only two or one hyperfine state(s ) ; they can populate three states only if the system is trapped and experiences rotation , which generates vortices . an important feature of real atomic spinor bose - einstein condensates is a weakness of the spin interactions comparing to the interaction in density channel ( @xmath73 ) . at the same time , the coherence among the different components of the order parameter ( angle @xmath63 ) is fully determined by the spin interaction . angle @xmath63 also influences the transverse magnetization of the condensate , as seen from eq . ( note that a longitudinal component of magnetization is independent on @xmath63 . smallness of @xmath64 comparing to @xmath28 leads to the fact that thermal fluctuations of relative angle @xmath63 become significant at much lower temperatures than fluctuations of the density of particles . therefore , at relatively low temperatures , one can assume that the moduli of all the components of the order parameter remain fixed ( that can be also checked numerically ) , whereas @xmath63 is fluctuating . for the case of small fluctuations of @xmath63 , one can use a harmonic approximation and represent the deviation of the energy of the system from the equilibrium , @xmath74 , as a quadratic function in terms of the deviation of angle @xmath63 from the equilibrium @xmath75 : @xmath76where @xmath77 . under these assumptions , the average square of the deviation of @xmath63 from the equilibrium is given by @xmath78 integrals in eq . ( 15 ) can be calculated analytically . after taking into account eq . ( 3 ) , we get:@xmath79we also introduce a quantity @xmath80 , which can be considered as an average deviation of angle @xmath63 from the equilibrium . we see that @xmath81 depends on dimensionless temperature @xmath82 , the number of particles @xmath12 and integral @xmath83 . for a given vortex phase , @xmath83 is also a function of total magnetization @xmath44 . it is important to emphasize that the scaling relation ( 16 ) has a sense only if @xmath28 is independent of @xmath12 , as discussed above . in order to calculate @xmath83 , we use a variational method , which was previously applied by us in ref . @xcite to evaluate energies of various axially - symmetric vortex phases in spin @xmath2 condensate . in this approach , each component of the order parameter is modeled by a trial function and values of variational parameters are found from the condition of minimum of total energy . in fig . 1 we plot calculated dependence of @xmath81 ( measured in degrees ) as a function of the number of particles in the system for different vortex phases at @xmath84 and @xmath85 . this value of @xmath28 is close to typical experimental ones ( @xmath86 nm , @xmath87 nm@xmath88 ) , see also calculations of ref . we assume that , for ( -1 , 0 , 1 ) and ( 1 , 1 , 1 ) states , @xmath89 and @xmath90 , whereas for ( 0 , 1 , 2 ) , @xmath91 and @xmath92 . note that @xmath93 for the particular phase is independent on @xmath70 , since @xmath83 has the same property . we see that even for quite low temperatures , @xmath94 can be rather large and the coherence among different components of the order parameter is practically destroyed . for smaller value of @xmath95 , fluctuations of @xmath63 are , of course , even stronger . to illustrate the effect of temperature , in the inset to fig . 1 , we show the dependence of @xmath81 on @xmath96 for ( 0 , 1 , 2)-phase ( curve 1 ) , ( 1 , 1 , 1 ) phase ( curve 2 ) , and ( -1 , 0 , 1 ) phase ( curve 3 ) at fixed number of atoms @xmath97 , @xmath98 for the first curve and @xmath99 for two others . note that @xmath81 is almost independent on total magnetization @xmath44 of the condensate . as we already pointed out , fluctuations of @xmath63 lead to that of @xmath48 . in a harmonic approximation , one can express the average deviation of @xmath100 from the equilibrium @xmath101 through the deviation of @xmath102 : @xmath103 where @xmath104 for the antiferromagnetic case and @xmath105 for the ferromagnetic one . if in the antiferromagnetic state the total magnetization is not large , @xmath106 , one can expect that @xmath107 , and , therefore , even small fluctuations of @xmath63 lead to strong relative fluctuations of @xmath108 . at the same time , for ferromagnetic case , @xmath105 in this equation , and relative fluctuations of @xmath100 are much smaller . we have calculated @xmath109 for different vortex phases and our calculations revealed that @xmath110 is almost independent on radial coordinate @xmath9 for vortex phases ( -1 , 0 , 1 ) and ( 1 , 1 , 1 ) . this is due to the fact that @xmath111 for these states , therefore , @xmath112 is nearly proportional to @xmath113 , and , according to eq . ( 15 ) , @xmath114 should only slightly depend on @xmath9 . in fig . 2 we present @xmath115 as a function of total magnetization of the condensate for ( -1 , 0 , 1 ) state at @xmath84 , @xmath89 ( antiferromagnetic case ) , and @xmath116 . we see that relative fluctuations of transverse magnetization can be significant even at low temperature . value of @xmath115 decreases with increase of @xmath44 . this result is natural , since condensate becomes more polarized with growing @xmath44 . an absolute value of @xmath109 also remains sizable . although value of fractional quantity @xmath115 is growing with decreasing of @xmath44 , the value of @xmath108 itself becomes smaller . therefore , we found that the most appropriate values of @xmath44 to observe fluctuations of transverse magnetization is around @xmath117 , where both @xmath118 and @xmath108 are high : @xmath119 , whereas @xmath48 is comparable to the longitudinal magnetization @xmath47 in the fully polarized state at @xmath120 , where it should be easily detectable experimentally . value of @xmath118 depends also on the vortex phase ; we found that in ( 1 , 1 , 1 ) state it is even much larger than in ( -1 , 0 , 1 ) state . also we have calculated @xmath109 for the ferromagnetic ( 0 , 1 , 2 ) phase . as can be expected , in this case , relative fluctuations of @xmath121 are much weaker . physically , this is because @xmath121 is proportional to the ferromagnetic order parameter @xcite , which is responsible for the ferromagnetic ordering . therefore , one can expect that in the ferromagnetic phase this order parameter is more robust with respect to thermal fluctuations , than in the antiferromagnetic one . in addition , average deviation of @xmath121 from the equilibrium is negative and its modulus is growing with increase of @xmath44 , in contrast to the antiferromagnetic system . thermal fluctuations should also be important in the case of @xmath2 condensate , where there are two interaction constants in spin channel and two characteristic angles . therefore , one can expect more complicated behavior , as compared to @xmath0 condensate . for instance , in homogeneous @xmath2 system , a cyclic state can have a lowest energy ; in this case atoms populate three hyperfine states , and the spin energy depends on the coherence among them . fluctuation problem for this system was analyzed in ref . a new method to create such entangled states in spin-1 condensate was recently applied experimentally in ref . @xcite , where a microwave energy was injected to the system . as a result , particles redistribute from spin @xmath122 state to spin @xmath123 and @xmath124 states , and all three magnetic sublevels become populated . the spin - mixing dynamics in @xmath0 condensate was studied theoretically in ref . @xcite . note that in eq . ( 14 ) we have assumed that fluctuating @xmath63 is spatially independent that is not true in general case . however , spatial gradients of @xmath102 give some additional contribution to the kinetic energy of the system , which is much larger than the spin energy . therefore , gradients of @xmath102 result in rather large increase of total energy , and we can neglect them for the trapped system , at least for our range of parameters . in other words , healing length for @xmath63 far exceeds the thomas - fermi radius of the system , and , therefore , although @xmath63 is fluctuating inside the cloud , it remains nearly constant @xcite , except of the surface layer , where the density of particles is low . thermal fluctuations of @xmath63 should be also noticeable in three - dimensional condensates at low and moderate temperatures . in general , the dependences of the number of condensed particles on the reduced temperature and critical temperature on the total number of atoms for 3d case are similar to that in 2d system , which are described by eqs . ( 2 ) and ( 3 ) . the main difference is the powers of @xmath39 and @xmath12 in the right hand sides of eqs . ( 2 ) and ( 3 ) that are @xmath125 and @xmath126 ( @xmath127 ) , respectively . however , in this case one has to take accurately into account the possibility of long wave length fluctuations of @xmath102 in longitudinal direction and formation of kinks @xcite . in this paper , we have studied the effect of thermal fluctuations on the coherence among different components of the order parameter in quasi 2d rotating @xmath0 bose - einstein condensate , when all three hyperfine states are populated . different axially - symmetric vortex phases were considered . we have shown that the deviation of the relative phase @xmath128 from the equilibrium can be very significant even at low temperatures , much smaller than @xmath5 . fluctuations of relative angle induce sizable fluctuations of the spin texture , namely , local transverse magnetization of the condensate . we have shown that these fluctuations are much more pronounced in antiferromagnetic case than in the ferromagnetic one . the recently proposed in ref . @xcite direct and nondestuctive method for the imaging of spinor bec spatial magnetization ( or some of its modification ) can be applied for the experimental study of the thermal fluctuations of spin textures , since it enables multiple - shot imaging and one can directly observe the dynamics of a single sample . authors acknowledge useful discussions with t. k. ghosh and t. mizushima . w. v. pogosov is supported by the japan society for the promotion of science . j. stenger , d. m. stamper - kurn , h. j. miesner , a. p. chikkatur , and w. ketterle , nature * 396 * , 345 ( 1999 ) . t. mizushima , k. machida , and t. kita , phys . lett . * 89 * , 030401 ( 2002 ) . t. isoshima and k. machida , phys . a * 66 * , 023602 ( 2002 ) . j. w. reijnders , f. j. van lankvelt , k. schoutens , and n. read , phys . lett . * 89 * , 120401 ( 2002 ) . k. kasamatsu , m. tsubota , and m. ueda , int . j. of mod . b * 19 * , 1835 ( 2005 ) . w. v. pogosov , r. kawate , t. mizushima , and k. machida , phys . a * 72 * , 063605 ( 2005 ) . a. trombettoni , a. smerzi , and p. sodano , new j. phys . * 7 * , 57 ( 2005 ) . t. p. simula , m. d lee , and d. a. w. hutchinson , phil . mag . ( in press ) , cond - mat/0412512 . m. holzmann , g. baym , j. p. blaizot , and f. laloe , cond - mat/0508131 . t. p. simula and p. b. blakie , phys . * 96 * , 020404 ( 2006 ) . d. schumayer and d. a. w. hutchinson , cond - mat/0601500 . s. stock , z. hadzibabic , b. battelier , m. cheneau , and j. dalibard , phys . 95 * , 190403 ( 2005 ) . z. hadzibabic , p. kruger , m. cheneau , b. battelier , and j. dalibard , to be published in nature ; cond - mat/0605291 . y. castin , z. hadzibabic , s. stock , j. dalibard , and s. stringari , cond - mat/0511330 . w. v. pogosov and k. machida , cond - mat/0601604 . y. j. wang , d. z. anderson , v. m. bright , e. a. cornell , q. diot , t. kishimoto , m. prentiss , r. a. saravanan , s. r. segal , and s. wu , phys . * 94 * , 090405 ( 2005 ) . m. h. wheeler , k. m. mertes , j. d. erwin , and d. s. hall , phys . * 93 * , 170402 ( 2004 ) . m. saba , t. a. pasquini , c. sanner , y. shin , w. ketterle , and d. e. pritchard , science , * 307 * , 1945 ( 2005 ) . m. s. chang , q. qin , w. zhang , l. you , and m. s. chapman , nature physics * 1 * , 111 ( 2005 ) . j. m. higbie , l. e. sadler , s. inouye , a. p. chikkatur , s. r. leslie , k. l. moore , v. savalli , and d. m. stamper - kurn , phys . lett . * 95 * , 050401 ( 2005 ) . c. gies , b. p. van zyl , s. a. morgan , and d. a. w. hutchinson , phys . a * 69 * , 023616 ( 2004 ) . t. ohmi and k. machida , j. phys . jpn . * 67 * , 1822 ( 1998 ) . t. l. ho , phys . lett . * 81 * , 742 ( 1998 ) . w. v. pogosov and k. machida , cond - mat/0604505 . h. pu , c. k. law , s. raghavan , j. h. eberly , and n. p. bigelow , phys . a * 60 * , 1463 ( 1999 ) . * fig . 1 . * dependences of @xmath81 ( in degrees ) on the number of particles in the system for different vortex phases at fixed value of interaction constant @xmath85 ( see in the text ) and @xmath84 . in the ( -1 , 0 , 1 ) phase , @xmath89 , @xmath90 ; in the ( 1 , 1 , 1 ) state , @xmath129 , @xmath90 ; in the ( 0 , 1 , 2 ) state , @xmath91 , @xmath130 . inset shows @xmath81 as a function of temperature for ( 0 , 1 , 2 ) phase ( curve 1 ) , ( 1 , 1 , 1 ) phase ( curve 2 ) , and ( -1 , 0 , 1 ) phase ( curve 3 ) at the same values of @xmath44 , @xmath28 and @xmath39 . number of atoms is @xmath116 , interaction constants are @xmath98 for the first curve and @xmath131 for two others .
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we consider the effect of thermal fluctuations on rotating spinor @xmath0 condensates in axially - symmetric vortex phases , when all the three hyperfine states are populated .
we show that the relative phase among different components of the order parameter can fluctuate strongly due to the weakness of the interaction in the spin channel .
these fluctuations can be significant even at low temperatures
. fluctuations of relative phase lead to significant fluctuations of the local transverse magnetization of the condensate .
we demonstrate that these fluctuations are much more pronounced for the antiferromagnetic state than for the ferromagnetic one .
| 7,235 | 139 |
in our single source model ( updated version is in @xcite ) we explained the knee as the effect of a local , recent supernova , the remnant from which accelerated mainly oxygen and iron . these nuclei form the intensity peaks which perturb the total background intensity . the comprehensive analysis of the world s data gives as our datum the plots given in the figure 1 ; these are deviations from the running mean for both the energy spectrum mostly from cherenkov data and the summarised electron size spectrum . it is against these datum plots that our comparison will be made . in the present work we endeavour to push the subject forward by examining a number of aspects . they are examined , as follows : + ( i ) can we decide whether the solar system is inside the supernova shock or outside it ? + ( ii ) is the identification of oxygen and iron in the peaks correct ? + ( iii ) can both the peaks be due to protons rather than nuclei ? in view of claims from a few experiments ( dice , blanca ) that the mean mass is low in the pev region , it is wise to examine this possibility . the appreciation that the frequency of sn in the local region of the interstellar medium ( ism ) has been higher than the galactic average , over the past million years , has improved the prospects for the ssm being valid @xcite and thereby increases the probability that we are close to the surface of a remnant . it is doubtlessly possible for particles to escape from an snr shock and propagate ahead . such a situation has been considered in the berezhko - model. the problem concerns uncertainties in the diffusion coefficient for the ism ; however , estimates have been made @xcite and figure 1 shows the result for the sun being outside the shock at the distance of 1.5@xmath0 for the center of snr ( @xmath0 is the radius of the remnant ) . it is seen that the result does not fit well the datum points at all . the model tested must be rejected in its given form . it is possible to restore it by taking an energy spectrum of more nearly the form for the inside snr location or at the position outside , but very close to the shell . the corresponding cureves are shown in figure 1 by full lines . a tolerable astrophysical case could be made for helium and oxygen rather than oxygen and iron , and the direct measurements at lower energies than the knee region do not really rule it out . figure 2 shows the @xmath1-values for the corresponding spectra . the separation of the he and o peaks is a little greater than for o and fe ( 8/2 compared with 26/8 ) and this causes the he , o pattern to be displaced somewhat . although the fit to the datum points is not as good as for o , fe , the he , o combination can not be ruled out on the basis of the @xmath1-plots alone . the absence of the preferred - by - us nuclei between the two peaks is a worry , though ( incertion of carbon does not help to fill the gap between two peaks ) . the fe peak would then be expected at log(@xmath2 ) = 1.1 . calculations have been made for the case of two proton peaks , the proton spectra having been taken to be the standard interior - to - the snr form . the result is also shown in figure 2 . an interesting situation develops here . although it is possible to tune either the energy spectrum or the size spectrum to fit the @xmath1-results , it is not possible to choose an energy spectrum which fits both . this arises because of the sensitivity of the number of electrons at the detection level to the primary mass . in figure 2 the separation of the proton peaks in the energy spectrum was chosen such that the @xmath1-distribution for shower size was a reasonable fit to the data . however , the separation of the peaks in the energy spectrum necessary for the shower size fit is less than that for o , fe by 0.15 ; the result is that after the necessary binning ( 0.2 in @xmath3 units ) for the energy spectrum there is no agreement there . it is evident from the foregoing that the two - proton peak model is unacceptable . this result cast doubt on the analyses of eas data which conclude that the mean primary mass is low ( @xmath4 ) in the pev region . as mentioned already , it is our view that some , at least , of the models used in the mass analyses are inappropriate for the interactions of nuclei , particularly for the production and longitudinal development of the electromagnetic component . it is interesting to know , in connection with mean mass estimates , that the recent work using the tibet eas array @xcite has given strong support for the result - favoured by us - in which the average cosmic ray mass increases with energy . in fact , their mass is even higher than ours : @xmath5 , compared with our 2.4 , at 1 pev , and 3.3 , compared with 3.0 at 10 pev . equally significant is the fact that the sharpness of the iron component that they need to fit the overall data is quite considerable : @xmath6 = 1.4 . it will be remembered that straightforward galactic diffusion - the conventional model - gives @xmath7 for any one mass component and @xmath8 for the whole spectrum @xcite . returning to the question of our location with respect to the snr it seems difficult to account for the @xmath1-distribution if we are some distance outside the shell , unless the diffusion coefficient for cosmic ray propagation in the ism is almost energy - independent . we appear to be inside , or only just outside . finally , concerning the nature of the peaks : o , fe or he , o , it is difficult to rule out the latter from the @xmath1-plots alone , although the lack of an iron peak is surprising . however , there is some evidence from the tunka-25 cherenkov experiment for a further peak at roughly the correct energy for the third ( fe ) peak @xcite . there is also a hint of a peak in kascade spectrum , which is observed at an even higher energy than in tunka-25 @xcite . most other experiments - but not all - do not have the sensitivity to detect a further peak so the situation here is still open . we still prefer our original suggestion , viz . that the peaks are due to o and fe , and their shape is the consequence of the sharp cut - off in the energy spectrum of particles accelerated by snr . the main reason for the preference is the fact that o and fe spectra extrapolate and fit direct measurements of those components rather well @xcite and there are good astrophysical reasons favouring these nuclei . the single source model , with its explanation of the knee in the cosmic ray energy spectrum in terms of particles ( probably principally nuclei of oxygen and iron ) from a recent , local sn , has been examined further . it is true that the identity of the nuclei is not completely secure and it is just possible that rather than o , fe , the combination is he , o : however , we still prefer the original explanation . the question of the nature of the particles responsible for the knee is , therefore , still somewhat uncertain ; however , that there is structure in the spectrum , indicative of a single source , seems to be rather secure . 99 erlykin a.d . , wolfendale a.w . , 2001 , g : nucl . part . phys . , * 27 * , 1005 erlykin a.d . , wolfendale a.w . , 2001 , j. phys . g : nucl . part . * 27 * , 941 erlykin a.d . , wolfendale a.w . , 2001 , g : nucl . part . * 27 * , 959 berezhko e.g. et al . , 1996 , jetp , * 82*,1 berezhko e.g. , 1999 ( private communication ) amenomori m. et al . 2000a , phys . d , * 62 * , 112002 ; 2000b , phys . d , * 62 * , 072007 erlykin a.d . , wolfendale a.w . , 1997 , j. phys . g : nucl . part . phys . , * 23 * , 979 ; 1998a , astropart * 8 * , 265 ; 1998b , j. phys . g : nucl . part . * 9 * , 213 budnev n. et al . 2002 , 18 ecrs , moscow ( to be published ) schatz g. , 2002 , astropart . phys . , * 17 * , 13 erlykin a.d . 1998 , astropart . * 8 * , 283
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an analysis is made of the masses and spectral features for cosmic rays in the pev region , insofar as they have a bearing on the problem of the interaction of cosmic ray particles . in our single source model
we identified two peaks seen in a summary of the world s data on primary spectra , and claimed that they are probably due to oxygen and iron nuclei from a local , recent supernova . in the present work
we examine other possible mass assignments .
we conclude that of the other possibilities only helium and oxygen ( instead of o and fe ) has much chance of success ; the original suggestion is still preferred , however . concerning our location with respect to the snr shell , the analysis suggests that we are close to it - probably just inside .
| 2,301 | 185 |
millisecond pulsars ( msps ) have for some time been known to exhibit exceptional rotational stability , with decade long observations providing timing measurements with accuracies similar to atomic clocks ( e.g. ) . such stability lends itself well to the pursuit of a wide range of scientific goals , e.g. observations of the pulsar psr b1913 + 16 showed a loss of energy at a rate consistent with that predicted for gravitational waves @xcite , whilst the double pulsar system psr j0737 - 3039a / b has provided precise measurements of several ` post keplerian ' parameters allowing for additional stringent tests of general relativity @xcite . for a detailed review of pulsar timing refer to e.g. @xcite . in brief , the arrival times of pulses ( toas ) for a particular pulsar will be recorded by an observatory in a series of discrete observations over a period of time . these arrival times must all be transformed into a common frame of reference , the solar system barycenter , in order to correct for the motion of the earth . a model for the pulsar can then be fitted to the toas ; this characterises the properties of the pulsar s orbital motion , as well as its timing properties such as its orbital frequency and spin down . this is most commonly carried out using the tempo2 pulsar - timing packages @xcite , or more recently , the bayesian pulsar timing package temponest @xcite . when performing this fitting process , both tempo2 and temponest assume purely gaussian statistics in the properties of the uncorrelated noise . in realistic datasets , however , this assumption is not necessarily correct . if the underlying probability density function ( pdf ) for the noise is not gaussian , for example , if there is an excess of outliers relative to a gaussian distribution , modifiers to the toa error bars that scale their size are used to find the best approximation to a gaussian distribution . this can be performed using a single modifier for a given receiving system determined across an entire dataset , or as in the ` fixdata ' plugin for tempo2 @xcite , where the modifier is determined separately for a series of short time lags . while the latter of these two approaches can better account for a non - gaussian distribution in the uncorrelated noise , it does so at the expense of a potentially large number of additional free parameters , and ultimately does not address the core issue , that the underlying distribution is not gaussian . both approaches then have the direct consequence of decreasing the precision with which one can estimate the timing parameters , and any other signals of interest , such as intrinsic red spin noise due to rotational irregularities in the neutron star @xcite or correlated noise due to a stochastic gravitational wave background ( gwb ) generated by , for example , coalescing black holes ( e.g. @xcite ) . indeed , currently all published limits on the signals induced by a gwb have been obtained under the assumption that the statistics of the toa errors are gaussian ( see e.g. @xcite ) . in this paper we introduce a method of performing a robust bayesian analysis of non - gaussianity present in pulsar timing data , simultaneously with the pulsar timing model , and additional stochastic parameters such as those describing the red noise , and dispersion measure variations . the parameters used to define the presence of non - gaussianity are zero for gaussian processes , giving a simple method of defining the strength of non - gaussian behaviour . in section [ section : bayes ] we will describe the basic principles of our bayesian approach to data analysis , giving a brief overview of how it may be used to perform model selection , and introduce multinest . in sections [ section : nongausslike ] and [ section : toy ] we introduce the non - gaussian likelihood we will use in our pulsar timing analysis , and apply it to a simple toy problem . in section [ section : pulsarnongaussian ] we then extend this likelihood to the subject of pulsar timing , and apply it to both simulated and real data in sections [ section : pulsarsims ] and [ section : realdata ] respectively , before finally offering some concluding remarks in section [ section : conclusion ] . this research is the result of the common effort to directly detect gravitational waves using pulsar timing , known as the european pulsar timing array ( epta ) @xcite . given a set of data @xmath1 , bayesian inference provides a consistent approach to the estimation of a set of parameters @xmath2 in a model or hypothesis @xmath3 . in particular , bayes theorem states that : @xmath4 where @xmath5 is the posterior probability distribution of the parameters , @xmath6 is the likelihood , @xmath7 is the prior probability distribution , and @xmath8 is the bayesian evidence . since the evidence is independent of the parameters @xmath2 it is typically ignored when one is only interested in performing parameter estimation . in this case inferences are obtained by taking samples from the ( unnormalised ) posterior using , for example , standard markov chain monte carlo ( mcmc ) sampling methods . for model selection , however , the evidence is key , and is defined simply as the factor required to normalise the posterior over @xmath2 : @xmath9 where @xmath10 is the dimensionality of the parameter space . as the evidence is just the average of the likelihood over the prior , it will be larger for a simpler model with a compact parameter space if more of that parameter space is likely . more complex models where large areas of parameter space have low likelihood values will have a smaller evidence even if the likelihood function is very highly peaked , unless they are significantly better at explaining the data . thus , the evidence automatically implements occam s razor . the question of model selection between two models @xmath11 and @xmath12 can be answered via the model selection ratio @xmath13 , commonly referred to as the ` bayes factor ' : @xmath14 where @xmath15 is the a priori probability ratio for the two models , which in this work we will set to unity but occasionally requires further consideration . the bayes factor then allows us to obtain the probability of one model compared the other simply as : @xmath16 in practice when performing bayesian analysis we do not work with the likelihood , but the log likelihood . in this case the quantity of interest is the log bayes factor , which is simply the difference in the log evidence for the two models . for example , a difference in the log evidence of 3 for two competing models gives a bayes factor of @xmath17 , which in turn gives a probability of @xmath18 . we use the difference in the log evidence in sections [ section : pulsarsims ] and [ section : realdata ] to perform model selection between our gaussian and non - gaussian models . while many techniques exist for calculating the evidence , such as thermodynamic integration @xcite , it remains a challenging task both numerically and computationally , with evidence evaluation at least an order - of - magnitude more costly than parameter estimation . nested sampling @xcite is an approach designed to make the calculation of the evidence more efficient , and also produces posterior inferences as a by - product . the multinest algorithm ( @xcite ) builds upon this nested sampling framework , and provides an efficient means of sampling from posteriors that may contain multiple modes and/or large ( curving ) degeneracies , and also calculates the evidence . since its release multinest has been used successfully in a wide range of astrophysical problems , including inferring the properties of a potential stochastic gravitational wave background in pulsar timing array data @xcite , and is also used in the bayesian pulsar timing package temponest . this technique has greatly reduced the computational cost of bayesian parameter estimation and model selection , and is employed in this paper . in this section we will outline the method adopted for including non - gaussian behaviour in our analysis . we use the approach developed in @xcite , which is based on the energy eigenmode wavefunctions of a simple harmonic oscillator . we will describe this in brief below in order to aid future discussion . we begin by considering our data , the vector @xmath19 of length @xmath20 , as the sum of some signal @xmath21 and noise @xmath22 such that : @xmath23 we can then construct the likelihood that the residuals after subtracting our model signal from the data follows an uncorrelated gaussian distribution of width @xmath24 as : @xmath25,\ ] ] with @xmath26 the diagonal noise covariance matrix for the residuals , such that @xmath27 , and @xmath28 the determinant of @xmath26 . we now extend this to the general case in order to allow for non - gaussian distributions by modelling our pdf as the sum of a set of gaussians , modified by hermite polynomials @xmath29 ( see e.g. @xcite for previous uses of hermite polynomials in describing departures from gaussianity ) , defined as : @xmath30 therefore , for a general random variable @xmath31 the pdf for fluctuations in @xmath31 can be written : @xmath32\left|\sum_{n=0}^{\infty}\alpha_nc_nh_n\left(\frac{x}{\sqrt{2}\sigma}\right)\right|^2\ ] ] with @xmath33 free parameters that describe the relative contributions of each term to the sum , and @xmath34 is a normalization factor . equation [ eq : prx ] forms a complete set of pdfs , normalised such that : @xmath35c_nh_n\left(\frac{x}{\sqrt{2}\sigma}\right)c_mh_m\left(\frac{x}{\sqrt{2}\sigma}\right ) = \delta_{mn},\ ] ] with @xmath36 the kronecker delta , where the ground state , @xmath11 , reproduces a standard gaussian pdf , and any non - gaussianity in the distribution of @xmath31 will be reflected in non - zero values for the coefficients @xmath33 associated with higher order states . the only constraint we must place on the values of the amplitudes @xmath37 is : @xmath38 with @xmath39 the maximum number of coefficients to be included in the model for the pdf . this is performed most simply by setting : @xmath40 we can therefore rewrite eq . [ eq : gausslike ] in this more general form as : @xmath41\nonumber \\ & \times&\prod_{i=1}^{n_d}\left|\sum_{n=0}^{n_{\mathrm{max}}}\alpha_nc_nh_n\left(\frac{d_i - s_i}{\sqrt{2}\sigma}\right)\right|^2.\end{aligned}\ ] ] the advantage of this method is that one may use a finite set of non - zero @xmath33 to model the non - gaussianity , without mathematical inconsistency . any truncation of the series still yields a proper distribution , in contrast to the more commonly used edgeworth expansion ( e.g. @xcite ) . before applying the formalism described in section [ section : nongausslike ] to the practice of pulsar timing , we first demonstrate its use in a toy problem . here our data vector @xmath19 contains 10000 points drawn from a non - gaussian distribution obtained using eq . [ eq : prx ] , with parameters listed in table [ table : toyparams ] . .parameters used to generate non - gaussian noise in a simple toy problem . [ cols="^,^,^ " , ] [ table : logevidence ] table [ table : logevidence ] lists the log evidence values for different sets of non - gaussian coefficients , normalised such that the log evidence for no additional coefficients ( i.e. assuming gaussian statistics ) is 0 . we see that there is a significant increase in the log evidence ( @xmath42 39 ) when including even just two coefficients , indicating definitive support for their inclusion in the model . as the number increases the rise in evidence increases , reaching a maximum with 4 included coefficients . given the timing model , red noise and dispersion measure variation solutions that were subtracted from the data were obtained from a gaussian analysis , we will however still include coefficients up to and including @xmath43 in the full analysis . given the large dimensionality of the problem this analysis can not be carried out using multinest . as such we make use of the guided hamiltonian sampler used previously in pulsar timing analysis in @xcite . this sampler makes use of both gradient information in the likelihood , and also the hessian in order to efficiently sample from large parameter spaces . table [ table:0437 ] lists the timing model parameter estimates and their nominal standard deviations for both the gaussian and non - gaussian analysis . in all cases we find the parameter estimates and their uncertainties to be consistent between both methods . in fig . [ figure:0437noise ] ( top ) we show the one and two - dimensional marginalised posterior distributions for the red noise and dispersion measure variation power law amplitudes and spectral indicies for the non - gaussian ( left ) and gaussian ( right ) analysis . both are also extremely consistent with one another , however when overlaying the two sets of 1-dimensional posterior distributions for each of the 4 parameters separately ( bottom 4 panels ) some differences become apparent between the non - gaussian ( blue dashed lines ) and gaussian ( red solid lines ) analysis . in particular the dispersion measure variation power law parameter estimates show a slight shift towards higher amplitudes and shallower spectral indices in the non - gaussian case . despite these similarities in the timing and stochastic parameter estimates between the gaussian and non - gaussian analysis , fig . [ figure:0437pdf ] indicates a definitive detection of non - gaussianity in the dataset , in agreement with the difference in the log evidence for the noise only analysis . in the top plot we show the one and two - dimensional marginalised posterior distributions for the 5 non - gaussian coefficients fit in the analysis of j0437@xmath04715 . vertical lines are included at 0 where visible in the plots , however , except for @xmath44 all the coefficients are inconsistent with this value . in the bottom plot we then show the set of equally weighted pdfs obtained from the non - gaussian analysis ( black lines ) setting @xmath45 . in addition we over plot the mean of the distribution ( red line ) and a unit gaussian ( blue line ) all of which have been normalised to have a sum of 1 . the difference between the gaussian and non - gaussian pdfs is clear , with a larger probability for both small ( @xmath46 ) and larger ( @xmath47 ) deviations than given by the gaussian pdf . that such a significant detection of non - gaussianity does not lead to larger changes in the parameter estimates can potentially be attributed to a frequency dependence on the significance of the @xmath48 parameters . in @xcite the 10 cm j0437@xmath04715 data was found to be describable through gaussian statistics alone . this would suggest that the non - gaussianity we detect exists primarily at low frequencies . in figure [ figure:0437normres ] we show the normalised residuals from fig . [ figure:0437noise ] separated into its 10 cm , 20 cm and 50 cm components , along with histograms for each wavelength . here the increase in non - gaussian behaviour can clearly be seen as the wavelength increases . given the lowest frequencies have the greatest degree of non - gaussianity it is less surprising that there is little impact on the timing or red spin noise parameters , as the low frequency data contributes the least to these parts of the model . the low frequencies do , however , contribute greatly to the constraints on dispersion measure variations , and it is here we see the greatest difference between the gaussian and non - gaussian models . lcc + pulsar name & j0437@xmath04715 + mjd range & 50191.055619.2 + data span ( yr ) & 14.86 + number of toas & 5052 + + model parameter & non gaussian & gaussian + right ascension , @xmath48 ( rad ) & 1.20979650940(10 ) & 1.20979650943(11 ) + declination , @xmath49 ( rad ) & @xmath00.82471224153(8 ) & @xmath00.82471224154(8 ) + pulse frequency , @xmath50 ( s@xmath51 ) & 173.6879458121850(3 ) & 173.6879458121849(4 ) + first derivative of pulse frequency , @xmath52 ( s@xmath53 ) & @xmath01.728365(4)@xmath54 & @xmath01.728365(4)@xmath54 + dispersion measure , dm ( @xmath55pc ) & 2.64462(11 ) & 2.64461(11 ) + first derivative of dispersion measure , @xmath56 ( @xmath55pcyr@xmath51 ) & @xmath06(6)@xmath57 & @xmath07(7)@xmath57 + dm2 ( @xmath55 pc yr@xmath53 ) & @xmath01(2)@xmath58 & @xmath01(2)@xmath58 + proper motion in right ascension , @xmath59 ( masyr@xmath51 ) & 121.439(3 ) & 121.441(3 ) + proper motion in declination , @xmath60 ( masyr@xmath51 ) & @xmath071.474(3 ) & @xmath071.474(3 ) + parallax , @xmath61 ( mas ) & 6.4(2 ) & 6.3(2 ) + orbital period , @xmath62 ( d ) & 5.7410462(3 ) & 5.7410461(3 ) + epoch of periastron , @xmath63 ( mjd ) & 54530.1722(3 ) & 54530.1721(3 ) + projected semi - major axis of orbit , @xmath31 ( lt - s ) & 3.36671463(8 ) & 3.36671464(8 ) + longitude of periastron , @xmath64 ( deg ) & 1.35(2 ) & 1.36(2 ) + orbital eccentricity , @xmath65 & 1.91800(14)@xmath57 & 1.91796(15)@xmath57 + first derivative of orbital period , @xmath66 & 3.724(6)@xmath67 & 3.724(6)@xmath67 + first derivative of @xmath31 , @xmath68 ( @xmath69 ) & 1(2)@xmath54 & 1(2)@xmath54 + periastron advance , @xmath70 ( deg / yr ) & 0.0150(12 ) & 0.0150(13 ) + companion mass , @xmath71 ( @xmath72 ) & 0.223(14 ) & 0.223(15 ) + longitude of ascending node , @xmath73 ( degrees ) & 208.0(12 ) & 208.3(13 ) + orbital inclination angle , @xmath74 ( degrees ) & 137.1(8 ) & 137.3(8 ) + [ table:0437 ] + + @xmath75{ngnoise.pdf } & \includegraphics[width=100mm]{gaussnoise.pdf } \\ \hspace{-1.5 cm } \includegraphics[width=100mm]{redamp.pdf } & \hspace{-1.5 cm } \includegraphics[width=100mm]{redspec.pdf } \\ \hspace{-1.5 cm } \includegraphics[width=100mm]{dmamp.pdf } & \hspace{-1.5 cm } \includegraphics[width=100mm]{dmspec.pdf } \\ % \vspace{-1.5 cm } \end{array}$ ] @xmath76{ngterms.pdf } \\ \includegraphics[width=120mm]{ngpdfs.png } \\ % \end{array}$ ] @xmath77{j043710cmnormres_scissored.pdf } & \includegraphics[width=85 mm , height=65mm]{10cmnormreshistogram.png } \\ \includegraphics[width=80mm]{j043720cmnormres_scissored.pdf } & \includegraphics[width=85 mm , height=65mm]{20cmnormreshistogram.png } \\ \includegraphics[width=80mm]{j043750cmnormres_scissored.pdf } & \includegraphics[width=85 mm , height=65mm]{50cmnormreshistogram.png } \\ % \end{array}$ ] in this paper we have introduced a method of performing a robust bayesian analysis of non - gaussianity present in the residuals in pulsar timing analysis , simultaneously with the pulsar timing model , and additional stochastic parameters such as those describing the red noise , and dispersion measure variations present in the data . deviations from gaussianity are described using a set of parameters @xmath37 that act to modify the probability density of the noise , such that @xmath78 describes gaussian noise , and any non zero values provide support for non - gaussian behaviour . the advantage of this method is that one may use a finite set of non - zero @xmath79 to model the non - gaussianity , without mathematical inconsistency . any truncation of the series still yields a proper distribution , in contrast to the more commonly used edgeworth expansion ( e.g. @xcite ) . we applied this method to two simulated datasets . in simulation one the noise was drawn from a non - gaussian distribution , and in simulation 2 it was purely gaussian . in simulation 1 , the effect of the non - gaussianity was to introduce a higher proportion of outliers relative to a gaussian distribution . this resulted in an overestimation of the toa uncertainties when assuming a gaussian likelihood , and decreased the precision with which the timing model parameters could be extracted compared to an analysis that correctly incorporated the non - gaussian behaviour on the noise . in the second case we showed that the parameter estimates of the timing model parameters of interest were consistent when including , or not , the @xmath37 parameters , as is to be expected when the noise is gaussian . we then applied this method to the publicly available parkes pulsar timing array ( ppta ) data release 1 dataset for the binary pulsar j0437@xmath04715 . we detect a significant non - gaussian component in the non - thermal component of the uncorrelated noise , however as the non - gaussianity is most dominant in the lowest frequency data the impact on the timing precision in the pulsar is minimal , with only the parameter estimates of the power law dispersion measure variations being visible changed between the gaussian and non - gaussian analysis . janssen g. h. , stappers b. w. , kramer m. , purver m. , jessner a. , cognard i. , 2008 , in bassa c.,wang z. , cumming a. , kaspiv.m . , eds , aip conf . proc . 983 , 40 years of pulsars : millisecond pulsars , magnetars and more . phys . , new york , p. 633 skilling j. , 2004 , in fischer r. , preuss r. , von toussaint u. , eds , aip conf . 735 , bayesian inference and maximum entropy methods in science and engineering . , new york , p. 395
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we introduce a method for performing a robust bayesian analysis of non - gaussianity present in pulsar timing data , simultaneously with the pulsar timing model , and additional stochastic parameters such as those describing red spin noise and dispersion measure variations .
the parameters used to define the presence of non - gaussianity are zero for gaussian processes , giving a simple method of defining the strength of non - gaussian behaviour .
we use simulations to show that assuming gaussian statistics when the noise in the data is drawn from a non - gaussian distribution can significantly increase the uncertainties associated with the pulsar timing model parameters .
we then apply the method to the publicly available 15 year parkes pulsar timing array data release 1 dataset for the binary pulsar j0437@xmath04715 . in this analysis we present a significant detection of non - gaussianity in the uncorrelated non - thermal noise , but we find that it does not yet impact the timing model or stochastic parameter estimates significantly compared to analysis performed assuming gaussian statistics .
the methods presented are , however , shown to be of immediate practical use for current european pulsar timing array ( epta ) and international pulsar timing array ( ipta ) datasets .
[ firstpage ] methods : data analysis , pulsars : general , pulsars : individual
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the dense knots that populate the closest bright planetary nebula ngc 7293 ( the helix nebula ) must play an important role in mass loss from highly evolved intermediate mass stars and therefore in the nature of enrichment of the interstellar medium ( ism ) by these stars . it is likely that similar dense condensations are ubiquitous among the planetary nebulae ( odell et al . 2002 ) as the closest five planetary nebulae show similar or related structures . they are an important component of the mass lost by their host stars , for the characteristic mass of individual knots has been reported as @xmath4 ( from co emission , huggins et al . 2002 ) , @xmath5 ( from the dust optical depth determination by meaburn et al . ( 1992 ) , adjusted for the improved distance ) , and about @xmath6 m@xmath7 ( odell & burkert 1997 , again from the dust optical depth but with better spatial resolution ) , and their number has been variously estimated to be from 3500 ( odell & handron 1996 ) from optical observations to much larger numbers ( 23,000 meixner et al . 2005 , henceforth mx05 ; 20,00040,000 hora et al . 2006 , henceforth h06 ) from infrared imaging . therefore , these condensations contain a significant fraction to a majority of all the material ejected . it is an extremely important point to understand if the ism is being seeded by these knots and if they survive long enough to be important in the general properties of the ism and also the process of formation of new stars . to understand those late phases , long after the knots have escaped the ionizing environment of their central stars , one must understand their characteristics soon after their formation - which is the subject of this study . there has been a burst of interest in the helix nebula and its knots beginning with the lower resolution groundbased study of meaburn et al . ( 1992 ) and the hubble space telescope ( hst ) images at better than 0.1 resolution ( odell & handron 1996 , odell & burkert 1997 ) in the optical window . the entire nebula has been imaged in the h@xmath1 _ v_=1 - 0 s(1 ) 2.12 @xmath0 m line at scales and resolutions of about 4 ( speck et al . 2002 ) , and 1.7/pixel ( h06 ) , while huggins et al . 2002 ) have studied one small region at 1.2 resolution , and the nic3 detector of the nicmos instrument of the hst has been used by meixner et al . ( 2004 , mx05 ) to sample several outer regions at about 0.2 resolution . a lower resolution ( 2 ) study in the longer wavelength 0 - 0 rovibrational lines has imaged the entire nebula with the spitzer space telescope ( h06 ) , extending a similar investigation by cox et al . ( 1998 , henceforth cox98 ) at 6/pixel with the infrared space observatory . radio observations of the co ( huggins et al . 2002 , young et al . 1999 ) and h i ( rodrguez et al . 2002 ) emission have even lower spatial resolution , but , the high spectral resolution allows one to see emission from individual knots . the three dimensional model for the helix nebula has also evolved during this time . we now know that the inner part of the nebula is a thick disk of 500 diameter seen at an angle of about 23 from the plane of the sky ( odell et al . 2004 , henceforth omm04 ) . this disk has a central core of high ionization material traced by he ii emission ( 4686 ) , and a series of progressively lower ionization zones until its ionization front is reached . the more easily visible lower ionization portions of the inner - disk form the inner - ring of the nebula . there are polar plumes of material perpendicular to this inner disk extending out to at least 940 ( omm04 ) to both the northwest and southeast . there is an apparent irregular outer - ring which meaburn et al . ( 2005 , henceforth m05 ) argue is a thin layer of material on the surface of the perpendicular plumes , whereas omm04 ) and odell ( 2005 ) argue that this is due to a larger ring lying almost perpendicular to the inner disk . the nature of the knots has attracted considerable attention . odell & burkert ( 1997 ) determined the properties using hst wfpc2 emission line images in h@xmath8 , [ n ii ] , and [ o iii ] , while odell et al . ( 2000 , henceforth ohb00 ) analyzed hst slitless spectra of the bright knot 378 - 801 in h@xmath8 and [ n ii ] , an investigation extended in a study ( odell et al , henceforth ohf05 ) with better slitless images in the same lines and also the [ o i ] line at 6300 . we will adopt the position based designation system described in odell & burkert ( 1997 ) and the trigonometric parallax distance of 219 pc from harris et al . the object 378 - 801 is the best studied of the knots and the primary target for the program reported upon in this paper . at 219 pc distance from the sun , the 1.5 chord of the bright cusp surrounding the neutral central core of 378 - 801 is @xmath9 cm . odell & burkert ( 1997 ) estimate that the peak density in the ionized cusp is about 1200 @xmath10 and the central density of the core , derived from the optical depth in dust , is @xmath11 @xmath10 , a number similar to the h@xmath1 density of @xmath1210@xmath13 @xmath10 necessary to produce the thermalized population distribution found for the j states within the @xmath14 levels of the electronic ( x @xmath15 ) ground state by cox98 . cox98 determined that two sample regions of knots were close to a population distribution of about 900 k , a similar result is found by an analysis ( 4.2 ) of new observations ( h06 ) of different regions of knots . as was argued in odell & handron ( 1996 ) , the knots are neutral condensations ionized on the side facing the central star . lpez - martn et al . ( 2001 ) have shown that the early apparent discrepancy between the observed and predicted surface brightness of the bright cusps is resolved once one considers the dynamic nature of the flow from the cusp ionization front , which depresses the recombination emission from the ionized gas . the central cores are molecular , being visible in co ( huggins et al . 2002 ) , and producing the multiple velocity components one sees in the low spatial resolution - high velocity resolution co studies ( young et al . the h@xmath1 emission is produced in a thin layer of material immediately behind the ionized cusp ( huggins et al . ohb00 show that the optical structure within the ionized cusp can only be explained if the material is heated on a timescale that is longer than the time for cool material to flow from the ionization front across the width of the cusp . this slow heating rate means that forbidden lines are seen only further away from the ionization front ( because these require energetic electrons to cause their collisional excitation ) whereas recombination lines like h@xmath8 arise preferentially from the cool regions closer to the ionization front . a succession of papers ( burkert & odell 1998 ; ohb00 , ohf05 ) attempting to simultaneously account for both the ionization and flow of material have produced a general model where cool material in the molecular central core flows towards the ionization front , is slowly heated , and upon passing through the ionization front is more rapidly heated and accelerated . at first examination the most recent models look satisfactory . however , the models fail because the theoretically expected zone of 900 k gas has much too low a column density to account for the observed surface brightness in h@xmath1 ( ohf05 ) . these h@xmath1 observations must be telling us about something overlooked in previous models . as we discuss in 4.4 , this missing ingredient seems to be that the earlier models were for a static structure , whereas the knots are actually in a state of active flow . this paper reports on work intended to clarify the nature of the knots in the helix nebula . new observations with all three of the imaging instruments on the hst were made and are described in 2 , then analyzed in 3 . a new and more refined theoretical model is presented in 4 , while these and other recent observations and models are discussed in 5 . in this investigation we draw on both our own new observations made with the hst and published observations made with a variety of telescopes . these observations range from the optical through the infrared . the new hst observations were made during eight orbits over the period 2006 may 22 - 24 as program go 10628 ( co - author odell as principlal investigator ) . the helix nebula is sufficiently large that we could simultaneously observe it with the three operating imaging instruments , the wfpc2 ( holtzman et al . 1995 ) , the nicmos ( thompson et al . 1998 ) , and the acs ( gonzaga , s. et al . 2005 ) . by careful selection of the pointing and orientation of the spacecraft , we were able to sample three regions that are useful for understanding the knots and their structure . in each case the new observations are either unique or of substantially longer exposure time than previous similar observations . the placement of the fields of view are shown in figure 1 . the nicmos observations were made with the nic3 camera ( 256x256 pixels , each about 0.2/pixel ) , with eight exposures in both the f212n ( isolating the 2.12 @xmath0 m h@xmath1 line and the underlying continuum ) and the f215n ( isolating primarily the underlying continuum as there are no strong lines in this region ) filters . each of the sixteen exposures was 1280 s duration . the pointing was changed in a four position pattern , with steps of 5 . the on - the - fly standard processing image products were our starting point . these images were combined using iraf and tasks from the hst data processing stsdas package provided by the space telescope science institute ( stsci ) . the method of calibration used in ohf05 was adopted , where the nebular and instrumental continuum was subtracted using the signal from the f215n filter . the resultant image is shown in figure 2 along with aligned wfpc2 optical emission line images from programs go 5086 and 5311 . the knot 378 - 801 is located in the low central region of each image and details of its bright cusp and tail are analyzed in 3.1 and 3.2 . it is notable that all of the h@xmath1 cusps have corresponding h@xmath8 and [ n ii ] counterparts and that all of the h@xmath8 and [ n ii ] cusps have h@xmath1 counterparts on these well exposed , high resolution images . a field to the southsoutheast of the central star and falling into the outer - ring portion of the helix nebula was imaged with the acs camera . eight exposures of nominally 1200 s each were made in both the f658n filter ( which passes the h@xmath8 and [ n ii ] 6583 lines equally well ) and the f502n filter ( dominated by the [ o iii ] 5007 line ) . the field of view shown in figure 1 overlaps only slightly with the acs mosaic built up during the go 9700 survey with the same filter pairs ( omm04 ) . the signal to noise ratio was much higher than in the go 9700 survey since that study used total exposures of about 850 s in both of filters . the images in the four offset pointings were combined using tasks within the stsdas package . no attempt at absolute calibration was made because of the f658n filter requiring additional observations with the f660n filter , which is dominated by [ n ii ] ( odell 2004 ) . the resulting image for f658n is shown in figure 3 . originally of 0.05 pixels , it has been averaged into 2x2 samples in order to increase the signal to noise ratio . since the finest features are larger than the size of the resultant pixels ( 0.1 ) no loss of detail was incurred . the f502n image is much lower signal and is not presented here , although portions of it are reproduced and discussed in 3.4 . sixteen exposures of 1100 s each were made with the wfpc2 f469n filter that isolates the he ii recombination line at 4686 . the images from the four pointings were combined using the stsdas package of _ dither _ tasks and the results are shown in figure 4 . for comparison , a matching section of the acs mosaic that was derived as part of program go 9700 ( omm04 ) is also shown . there are three recent papers that contain observations pertinent to our discussion of the nature of the knots in the helix nebula . these include two space infrared observations and one groundbased study . in a recent paper h06 present the results of an extensive observational study of the helix nebula . they have imaged the object out to the northeast - arc ( omm04 ) with the irac camera ( fazio et al . 2004 ) in four broad filters centered on 3.6 @xmath0 m , 4.5 @xmath0 m , 5.8 @xmath0 m , and 8.0 @xmath0 m . these filters are dominated by emission from rovibrational lines within the @xmath14 ground electronic state , although an atomic continnum must be present in addition to a few collisionally excited forbidden and recombination emission lines . the resolution of these images is about 2 , so that one can not resolve structure within the cusps , but one can see structure along radial lines passing through the bright cusps and their much longer tails . spectra were obtained with the irs ( houck et al . 2004 ) at three positions , two falling in the outer - ring at locations north and southwest of the central star and the third location ( used for background subtraction ) falling directly on the fainter northeast - arc feature . they present calibrated fluxes for rovibrational lines of h@xmath1 from 0 - 0 s(7 ) at 5.51 @xmath0 m out to 0 - 0 s(1 ) at 17.0 @xmath0 m . they also present a groundbased image in a filter centered on the 2.12 @xmath0 m h@xmath1 line which is of comparable spatial resolution but wider field of view that the speck et al . ( 2002 ) study . this image is not flux calibrated but appears to go fainter than the speck et al . ( 2002 ) image . in the program go 9700 study that produced a continuous mosaic of acs images mx05 also obtained parallel nic3 images in the f212n filter at six science positions and one sky position . each position was actually a double exposure of two pointings , which allowed a slight overlap of the nic3 fields . the total exposure at each position was about 750 s and double that in the regions of overlap . the method of calibration was different as only f212n images were obtained ( the few f175w images were not useful for calibration of the f212n images ) . it was assumed that the sky images included all the background signal that need to be subtracted , which means that it did not subtract nebular continuum . the nic3 images are under - sampled ( the pixel size of 0.2 is about the same as the telescope s resolution at this wavelength ) as in our new observations of the field around 378 - 801 . when comparing the data , one should note that the maximum effective exposure time is about 1500 s for the go 9700 f212n images , while in our study it is 10,240 s. as discussed in 3.4 , our acs field overlaps with much of the mx05 position 2 field , allowing a more meaningful comparison of optical and infrared images of this region than was possible in the mx05 study . these new images of the helix nebula with three different hst cameras provide new information on a number of subjects related to the nature and formation of the knots . the nicmos h@xmath1 image allows a discussion of the cusp and tail structure , the he ii image places constraints on the location of the knots to the southwest , and the acs images allow a more complete comparison of early h@xmath1 images to the south and southeast with comparable resolution optical emission line images . the new nic3 h@xmath1 2.12 @xmath0 m line observations were scaled to the same pixel size as the wfpc2 ( 0.0996/pixel ) using bilinear interpolation and carefully aligned . although this region has been imaged numerous times in this line , this is clearly the best image in terms of both its resolution and signal . the three primary optical line images and the new h@xmath1 image are shown in figure 2 . in order to make a quantitative analysis , the images were rotated and a sample three pixels wide along the axis of the cusp - core - tail was made . the results are shown in figure 5 , where the peak value of the h@xmath1 2.12 @xmath0 m and [ nii ] emission is normalized to unity as are the outer portions of the [ oiii ] profiles . 2.12 @xmath0 m infrared line . the 2.12 @xmath0 m image has lower resolution ( about two pixels ) and a comparison of the appearance in the different lines is discussed in the text . ] the optical line results are similar to those found by odell & burkert ( 1997 ) , burkert & odell ( 1998 ) , odell , henney , & burkert ( 2000 ) and ohf05 in that the ionization occurring furthest from the knots ionization front ( [ o iii ] ) is weak and extended , this intrinsic low brightness allows one to see the core of the knot in silhouette against the background nebular emission in this line . although the extinction peaks in the core of the knot , it extends to about 5 from the bright cusp . the [ n ii ] emission is strong and displaced ( 0.05 ) away from the ionization front with respect to the h@xmath8 emission . one sees extinction in h@xmath8 from the core out to almost the same distance as in [ o iii ] . the lack of apparent extinction in [ n ii ] must be due to there being relatively more emission in that line in the sheath of ionized gas surrounding the shadow of the knot . the full width at half maximum ( fwhm ) of the h@xmath8 and the [ n ii ] images are 0.48 and since the fwhm of the stars in the field of view is 0.25 , quadratic subtraction of this instrumental component leaves an intrinsic line width in those emission lines of 0.41 . the fwhm of the h@xmath1 line is 0.61 and the nearby star s is 0.42 , leaving an intrinsic fwhm for h@xmath1 of 0.44 with a peak displaced 0.11 towards the core of the knot from the h@xmath8 peak . the fwhm corresponds to a length of @xmath16 cm and the displacement to @xmath17 cm . the ionized line characteristics are similiar to those found in the previous studies , but we have added here the important characteristic of the small but certain displacement of the broader h@xmath1 . the earlier h@xmath1studies lacked the resolution to determine this characteristic , or in the case of mx05 , lacked the high resolution optical lines necessary for the comparison . the calibrated peak surface brightness in the cusp of 378 - 801 is @xmath18 @xmath19 in the h@xmath1 2.12 @xmath0 m line . this agrees well with the peak value of @xmath20 @xmath19 found by huggins et al . ( 2002 ) , where they used the calibration of speck et al . ( 2002 ) and utilized a spatial resolution that would not have recognized the narrowness of the peak . the well defined tail in 378 - 801 is primarily formed by a shadow in the ionizing lyman continuum ( lyc ) radiation cast by the optically thick core , with illumination occurring by diffuse ( recombination ) lyc photons and direct radiation grazing the edge of the core . the first order theory describing this situation was presented by cant et al . ( 1998 ) and applied to the tails of the helix nebula and the shadows behind the orion nebula proplyds soon after ( odell 2000 ) . ohf05 discussed the structure in tail of 378 - 801 within the light of this theory and its next order refinements ( wood et al . 2004 ) , but were unable to explain the details of what was being seen . h@xmath1 in the tails was first detected by walsh & ageorges ( 2003 ) and we are able to establish with our new observations where this emission arises . we present in figure 6 results from traces across the tail of 378 - 801 extending from 3.1 to 6.0 behind the peak of the cusp in h@xmath8 . this region does not extend as far as the partially obscured knot lying on the east side of the tail with its cusp 8.3 beyond 378 - 801 s h@xmath8 cusp . one sees that there is a well defined signature of a limb brightened sheath in both h@xmath1 2.12 @xmath0 m and h@xmath8 , with the peaks of the h@xmath1 emission occuring inside the h@xmath8 , as expected if the h@xmath8 is associated with a local ionization front . since the gas ionized by diffuse radiation should be much cooler than the directly illuminated nebular gas , the h@xmath8 emissivity would be high and the ionized sheath is well defined . these observations establish that conditions in the tail do allow an ionization front to form , while cant et al . ( 1998 ) and odell ( 2000 ) had expected that the shadowed region may be fully ionized . it is not surprising that no [ o iii ] emission is seen , rather , that the dust in the tail , concentrated to the middle of the tail ( as noted by ohf05 ) , causes extinction of the background nebular light . the apparent quandary , noted in ohf05 , is that the [ n ii ] emission appears to come from inside the ionized sheath of the tail . we already noted that the h@xmath8 and h@xmath1 structure indicates that an ionization boundary occurs at the edge of the radiation shadow , so there is an apparent contradiction in finding ionized nitrogen emission originating inside an ionization front . this contradiction is removed by comparison of the [ n ii ] emission and the optical depth , as determined by the [ o iii ] image . figure 7 shows a comparison of the dust optical depth and the [ n ii ] intensity . the similarity of the distribution of the optical depth and the [ n ii ] brightness argues that the [ n ii ] is actually nebular or cusp light scattered by the only marginally optically thick ( peak value @xmath21=0.2 column of dust ) . this feature is easy to see because the expected low electron temperature of the sheath s ionization front would suppress the collisionally excited [ n ii ] . if our intepretation of [ n ii ] is correct , then there should be a similar component of scattered h@xmath8 radiation . this may be what somewhat fills - in the region between the two limb - brightened components ( in addition to the low level of surface brightness expected when examining a thin shell ) . in the cusp [ n ii ] is stronger than h@xmath8 emission whereas in this part of the nebula the opposite is true . this point argues that much of the scattered [ n ii ] emission arises from the nearby bright cusp , rather than the surrounding nebula . the similar distribution of each argues that the [ n ii ] emission is caused by scattering of nebular [ n ii ] emission , as discussed in the text . ] the distribution along the tail core is discussed in detail in ohf05 ( their 4.1.2 ) and only a few comments need be added . the question is complex because the knots seem to originate near the nebular ionization front , then are shaped by the radiation field as the ionization front expands beyond them ( odell et al . this means that material in the shadowed region will have never seen direct ionizing radiation and could represent pre - knot material from the planetary nebula s photon dominated region ( pdr ) . the other source of tail material could be neutral gas accelerated outwards by the rocket effect ( e.g. mellema et al . 1998 ) . unfortunately , the high velocity resolution study of co by huggins et al . ( 2002 ) does not really illuminate the question . their angular resolution was a gaussian beam of @xmath22 , with the long axis aligned almost along the axis of the tail of 378 - 801 . since there was a strong co component coming from the partially shadowed knot lying 8.3 beyond 378 - 801 s h@xmath8 bright cusp , this means that there is a not a clear resolution of the co emission from the core of 378 - 801 and the partially shadowed knot . as a result , one can not hope to interpret the small differences in the position of the peak emission at different velocities as core - tail differences . this could be done with higher spatial resolution co observations . figure 4 shows our deep heii images alongside the same field covered at comparable resolution with the acs in h@xmath8+[n ii ] and [ o iii ] . a detailed comparison of the two images indicates that there is no case of a heii feature corresponding to a h@xmath8+[n ii ] or [ o iii ] feature , nor any heii only features . the part of the wfpc2 field closest to the central star is 144 distance . the profile of the heii core of the central disk , to which the knots in this part of the nebula belong ( omm04 ) derived from a wide field of view heii image ( odell 1998 ) shows that the core is down almost to 50% of its peak emission at this distance . if any knots actually occur within the nebula s heii core , we would expect that in the simplest knot models , we would see a heii cusp outside the [ o iii ] zone of each knot and this is not the case . however , the basic model is not that simple . the detailed models of ohb00 and ohf05 show that the normal progression of ionization states in the cusp are preserved , that is , closest to the ionization front there is an he@xmath23+h@xmath24 zone , outside of which there is a he@xmath25+h@xmath24 zone , and outside of that a he@xmath26+h@xmath24 . in a nearly constant electron temperature nebula the innermost zone is best traced by the [ n ii ] emission , the next zone by the [ o iii ] emission , and the outermost zone by the heii emission . things are not so simple in the case of the knots . as the gas flows through the cusp it is heated only slowly , so that the collisionally excited [ n ii ] emission peaks further out , where the gas is hotter , more than making up for the lower fraction of n@xmath27 ions . by the time that the second zone is reached the density has dropped significantly and the [ o iii ] emission is broad and weak . one would expect to find a heii zone associated with a knot only if the knot lies within the nebula s heii core . this heii zone would be quite weak because the density of the knot s gas would have been greatly decreased this far out . moreover , the gas has probably nearly reached the temperature of the nebula and these higher temperatures suppress the emission of this recombination line . this means that it will be hard to actually detect by their heii emission any knots within the heii core of the nebula . probably the strongest evidence that no knots exist in the heii core lies in the fact that we do nt see any objects in extinction in any of the observed emission lines . as noted in 2.1.2 , our acs field overlapped with one of the double - pointings made with nic3 in f212n as part of program go 9700 ( mx05 ) . mx05 compared their f212n images with the corresponding five fields in omm04 , where the resolution was about 1and groundbased images were used because the hst acs mosaic did not extend out this far . this factor of five difference in resolution made it difficult to draw firm conclusions about differences and similarities of appearance . they did , however , conclude that even their short ( 750 s ) overlapping double exposures were sufficient to establish that the knot cusps were more visible in h@xmath1 . in 2.1.1 we showed that in the vicinity of 378 - 801 that the knots are equally visible in both the ionization cusps and the h@xmath1 cusps . in figure 8 we see a comparison of the go 9700 f212n ( h@xmath1 ) images with our new acs images . there is an excellent correlation of appearance , although the contrast of the h@xmath1 emission above the essentially zero nebular background is higher than in f658n ( h@xmath8+[n ii ] ) and as usual the knots are only easily seen in the f502n ( [ o iii ] ) when the knot is in the foreground and can be seen in extinction against the background nebular emission . after considering the flexibility of display of the high signal to noise h@xmath8+[n ii ] images , it is difficult to support the conclusion that h@xmath1 2.12 @xmath0 m images are a better way of searching for knot cusps , except for any regions where there is high obscuration . through the new h@xmath1 2.12 @xmath0 m nic3 images in the current program ( go 10628 ) and the earlier mx05 study with shorter exposures , there is a larger and hopefully representative sample of resolved h@xmath1 cusps over a wide range of stellar distances ( @xmath28 ) . to look for systematic differences , we have identified isolated cusps in each of the fields available . we selected the three closest knots in the go 10628 field , three in both of mx05 s positions one and two , and two in mx05 s positions three and four , no isolated cusps being available in mx05 s position five . in each case we measured the surface brightness at the peak of the h@xmath1 cusp , the approximate chord across the knot s center , the width of the h@xmath1 cusp , and determined @xmath28 . the go 10628 and mx05 positions 1 - 4 had average @xmath28 values of 139 , 290 , 278 , 375 , and 464 respectively . the average surface brightnesses ( in @xmath19 ) of the cusp peaks were @xmath29 , @xmath30 , @xmath31 , @xmath32 , and @xmath33 . the average cusp widths were 0.6 , 0.3 , 0.5 , 0.8 , and 1.1 . the average chord values were 1.5 , 2.5 , 2.2 , 2.4 , and 2.7 . the most pronounced change in knot characteristic is the cusp peak surface brightness , dropping about linearly with @xmath28 , with the cusp width growing slowly and the chord width more steadily with @xmath28 . the physical interpretation of these patterns awaits a better understanding of individual knots . it should be pointed out that the relative physical distances from the stars is likely to be increasing more rapidly than the relative values of @xmath28 . this is because in the 3-d model of the helix nebula in omm04 , the main disk is inclined at an angle about 23 out of the plane of the sky , with the northwest side closer to the observer , while the outer - disk is inclined about 53 out of the plane of the sky , with the southsoutheast side closer to the observer . objects associated with the inner - disk would have a distance multiplication factor of 1.09 and those in the outer - disk a multiplication factor of 1.66 . the objects in the go 10628 nic3 field are almost certainly associated with the inner - disk . mx05 s positions 1 and 2 could belong with either system ( accurate radial velocities would determine this ) and their positions 3 and 4 are almost certainly associated with the outer - disk . figure 2 demonstrates the remarkable similarity of appearance of the knots in h@xmath8 and in our f212n ( 2.12 @xmath0 m ) images and we discussed the quantitative properties of 378 - 801 in 3.1 and 3.2 . we have investigated the similarities of the knots by selecting the nine objects ( including 378 - 801 ) within the nic3 field of view that are sufficiently isolated to allow a good background subtraction . the peak surface brightness in each cusp was derived in both 2.12 @xmath0 m and h@xmath8 for a sample 3 wfpc2 pixels wide ( 0.3 ) . the peak surface brightness in the tail of each object was determined in a sample 11 wfpc2 pixels wide ( 1.1 ) across the tail , the closer end of the sample being 20 wfpc2 pixels ( 2.0 ) displaced from the tip of the bright cusp . we used the 2.12 @xmath0 m calibration described in 2.1.1 and the h@xmath8 calibration of odell & doi ( 1999 ) , expressing the surface brightnesses in units of photons @xmath34 s@xmath35 sr@xmath35 . the average cusp surface brightness ratio in 2.12 @xmath0m / h@xmath8 was 5.5@xmath361.0 . the average surface brightness ratio for the tail as compared with the cusp was 0.23@xmath360.08 in 2.12 @xmath0 m and 0.17@xmath360.05 in h@xmath8 . this means that in the cusp the h@xmath1 2.12 @xmath0 m line alone is putting out more than five times as many photons as in h@xmath8 and that the contrast between the tail and cusp may be slightly higher ( the ratio is lower ) in h@xmath8 than in 2.12 @xmath0 m . the remarkable similarity of the knots in a recombination line that follows photoionization of atomic hydrogen and emission in a molecular line heretofore assumed to be the result of radiative pumping between electronic states is discussed in 4.5 . our understanding of the physics of the knots has evolved with a better understanding of a model that satisfactorily explains the knots . the initial descrepancy between the cusp surface brightness and the simplest photoionization model ( odell & handron 1996 ) was resolved by lpez - martn et al . ( 2001 ) when it was shown that the advection - dominated nature of the flow through the knot ionization fronts leads to a total rate of recombinations in the ionized gas that is significantly below what is predicted from naive models of ionization equilibrium . the peculiar photoionization structure of h@xmath8 and [ n ii ] emission can also be understood in similar terms the heating timescale of the ionized gas is comparable with the dynamic timescales for flow away from the knot surface , leading to resolvable temperature gradients , which strongly affect the relative distribution of recombination line and collisional line emission . the most refined model is that of ohf05 , which included both the effects of the radiation field and also the hydrodynamic expansion of the knot s ionization front . it is probably accurate to say that the photoionized portions of the knots are now adequately understood , or at least that the models are broadly consistent with the best observations . the structure in the tails is only beginning to be understood . to the first order , the tails are the effects of radiation shadows in the dominant ionizing species , the lyc photons ( cant et al . 1998 , odell 2000 ) . with this paper ( 3.2 ) we have now determined that the well observed tails are ionization bounded , with h@xmath1 sheaths inside the zone of ionized gas that occurs at the edge of the lyc shadow . the inner part of the tail is dense enough in dust to scatter surrounding nebular light , although the origin of this material as arising from the original process that forms the knots or as material that is moving back from the knot remains uncertain . the greatest quandary surrounds the explanation of the h@xmath1 zone that is observed immediately inside the ionized cusps of the knots . the approximate location of this zone of observed h@xmath1 is qualitatively where one would expect it . for reasons given below , it is almost certainly not excited by shocks . other models ( e.g. natta & hollenbach 1998 ) argue that the heating is by absorption of soft x - rays and others that the excitation mechanism is probably fluorescence , where non - ionizing photons from the stellar continuum excite molecules to the b @xmath37 and c @xmath38 electronic states , which then decay , producing the populations of the ground electronic state that give rise to the observed infrared lines . within the core of the knot the density is sufficiently high and the temperature sufficiently low that multiple heavier molecules are formed and the observed co is simply the most easily observed abundant tracer of these heavy molecules . an alternative method of exciting the h@xmath1 molecules is by shocks . at first this idea seems attractive because planetary nebulae as a class are known to possess high velocity stellar winds and large scale mass flows with sufficient energy to excite the low lying energy states of h@xmath1 that give rise to the observed infrared lines . cox98 first pointed out that the lack of a stellar wind ( cerruti - sola & perinotto 1985 ) rules out excitation by wind - driven shocks . a more complete assessment of shocks as the exciting source was given in ohf05 ( their 4.3.2 ) , where it is shown that although h@xmath1 is heated sufficiently immediately behind a transient shock this shock would quickly move through the knot and up the tail . the well defined location of the h@xmath1 emission zones immediately behind the ionized cusps and the ionized sheath of the tail strongly argues that we are dealing with a quasi - stationary process , rather than something quite dynamic , like shocks . h06 base their interpretation of h@xmath1 emission on the assumption of shock excitation . their assumption is based on the weakness of the h@xmath1 1 - 1 s(7 ) line , stating that the radiative models they use predict strong emission in that line , which they do not observe in their spectra . the shortcomings of that criterion for determining that the excitation comes from shocks , rather than radiative processes is discussed below in 4.1 . as we show in 4.2 , the h06 spectra also argue for a high excitation temperature , as found by cox98 . the h06 shock interpretation of the relative population distribution of the h@xmath1 energy states used six free parameters as it required three different shock velocities , each with a different relative intensity . this means that one ca nt use the population distribution to confirm that method of excitation . a key element of understanding the h@xmath1 emission is the excitation temperature of the gas . cox98 used spectra of the h@xmath1 0 - 0 s(2 ) to s(7 ) lines to derive the population of their upper states and found that their two sampled regions matched an excitation temperature of 900@xmath3650 k. we show below ( 4.2 ) that the new spectra of h06 of the h@xmath1 0 - 0 s(1 ) to s(7 ) lines in two additional regions agree with the results of cox98 and support the idea that the h@xmath1 emission comes from gas that is much hotter than the 50 k conditions expected ( ohf05 ) in the core of the knots . ohf05 demonstrated that even their most detailed pdr models could not explain the high surface brightness in h@xmath1 of the knot cusps , an argument first made by cox98 from more general considerations . the argument reduces to the fact that the observed surface brightness in h@xmath1 2.12 @xmath0 m radiation is too high to be explained by the column density of 900 k h@xmath1 that is predicted . ohf05 did not have high resolution h@xmath1 2.12 @xmath0 m images of their sample knot ( 378 - 801 ) and a comparison using the results of the new observations reported here are given in 4.5 . several papers , including the recent mx05 study , have reported that the surface brightnesses are compatible with earlier the theoretical models of natta & hollenbach ( 1998 ) in spite of the fact that those authors point out that the knots do not adhere to their general model and would have a higher surface brightness . a more complete critique of earlier claims of agreement of theory and observations is given in ohf05 ( 4.3.3 ) . in this section we present the total flux from the central star and nebula in 4.1 , establish that the knots commonly have high excitation temperatures ( 4.2 ) , show that the absence of strong 1 - 1 s(7 ) emission is not a strong argument for shock excitation of the h@xmath1 ( 4.3 ) , compare the recent data on h@xmath1 emission with the best models ( 4.4 ) , determine that there is no evidence for radial features extending into the middle of the nebula ( 4.5 ) , and critique a recent paper that argues for the tails being formed primarily by hydrodynamic processes in 4.6 . the emission from the nebula is in at least quasi - equilibrium with radiation from the central star . this means that the relative fluxes in various nebular and cusp emission lines and in the stellar continuum impose important constraints that must be observed by the correct model for the cusp h@xmath1 emission . the stellar continuum has been well defined down to 1200 by bohlin et al . ( 1982 ) , who conclude that the star has a luminosity of @xmath39 ( corrected to the trigonometric parallax distance ) and an effective temperature of 123,000 k. in the long wavelength end of the continuum , the flux per wavelength interval is very close to @xmath40 , as expected when one looks at much lower energies than where the peak emission occurs . this total luminosity corresponds to a flux at the earth of @xmath41 @xmath42 . natta & hollenbach ( 1998 ) argue that the h@xmath1 is heated by x - rays of greater than 100 ev because only these high energy photons would penetrate the ionization boundary . there are two emitters in the high energy end of the spectrum , the central star and a high temperature component of about ( 10@xmath43 k ) ( leahy et al . 1994 , leahy et al . 1996 , guerrero et al . the central star emission in the 0.1 - 2.0 kev range is @xmath44 @xmath42 and the emission from the 10@xmath43 k component is @xmath45 ( leahy et al . 1994 ) . the wavelength range for the fluorescent pumping mechanism is from about 912 to 1100 as determined by the minimum energy for exciting the lyman bands and the cutoff imposed by the lyc absorption of hydrogen . extrapolating the continuum from the slightly longer wavelengths that have been observed gives a total flux in this interval of @xmath46 @xmath42 . if the heating is due to photons above the ionization threshold for neutral hydrogen , then the calculated flux for a 123,000 k blackbody of 120 l@xmath47 in the interval from 13.6 ev through 100 ev is @xmath48 @xmath42 , representing the largest amount of power coming from the central star . the observed and predicted properties of the star s flux are summarized in table 1 . [ tab : fluxes ] .nominal fluxes at earth from the helix central star@xmath49 [ cols= " < , > , < , < , < " , ] [ tab : gary ] h06 argues that the weakness of the 1 - 1 s(7 ) line shows that the h@xmath50 emission must be shock excited rather than photo excited . since their subsequent interpretation of the observed features of the nebula are based on this assumption , it merits critical examination . h06 state , without presenting detailed proof , that the 1 - 1 s(7 ) line is normally strong " under photo excitation . they did not detect this line , but they did detect the nearby 0 - 0 s(7 ) line . therefore , we base our discussion on the flux ratio of 1 - 1 s(7 ) to the 0 - 0 s(7 ) line . examination of their spectra ( h06 figure 9 ) indicates that the flux ratio must be less than about 0.1 . the 1 - 1 s(7 ) appears to be present in the cox98 spectra at a level giving a line ratio of about 0.1 . figure 10 shows the results for a series of calculations in which isothermal clouds with a range of temperatures and densities was exposed to the radiation field of the central star . the central star was approximated as a blackbody and the x - ray continua as a series of free - free emitters with the published luminosities and temperatures . the total continuum was attenuated by an effective column density of 10@xmath51 @xmath34 to approximate the extinction of the ionizing radiation by the h@xmath27 region . typical planetary nebula abundances and ism grains were assumed . the clouds had a thickness of @xmath52 cm . the figure shows the 1 - 1 s(7)/0 - 0 s(7 ) line ratio . the predicted ratio is generally quite small for the values of the h@xmath1density and temperature considered , approaching the upper limit that we identify above only at combinations of low h@xmath1 density and temperature . cox98 argue that the total density must exceed 10@xmath13 @xmath10 in order for collisions to dominate and produce the closely single - temperature population distribution . this would be the density to combine with the derived temperature of 900 k for comparison with the s(7 ) line ratio , if the emitting zone were dominantly molecular hydrogen . in that case the predicted line ratio is much lower than the upper limit of the observations and one concludes that the observed weakness of the 0 - 0 s(7 ) line is not a useful indicator of the excitation mechanism as assumed by h06 . ohf05 derived a density of the molecular hydrogen from observations of the 2.12 @xmath0 m line , finding a value of @xmath53 @xmath10 . however , they assume that the population of the level producing the transition was in lte at 900 k , which seems to be the case as the upper state producing the 2.12 @xmath0 m line falls right on the population distribution for this temperature . using this density would also still indicate that the s(7 ) line ratio is not a useful indicator of the excitation mechanism . however , the challenge remains of explaining the unexpected combination of temperature and density . most of the existing calculations of h@xmath50 population distributions and resulting emission spectra have been done for the case of a pdr near an hii region ( black & van dishoeck 1987 ) . the stellar radiation field of an o or b star peaks near the wavelengths that excite the electronic transitions of h@xmath1 , about 1000 . the main effect of illumination by this continuum is absorption into excited electronic states which then decay into excited vibration - rotation levels within the ground electronic state of h@xmath1 . a very non - thermal distribution is produced by these electronic photo excitations , as h06 points out . however , the cox98 and h06 spectra discussed in the preceding section show that the population distribution within lower levels of h@xmath1 is well matched by a thermal distribution at about 900k . the non - thermal population distribution produced by o - star photo - excitation is simply not seen . this is not surprising since the environment is so dissimilar from that near a main sequence o or b star . the stellar radiation field of the helix nebula central star peaks at much shorter wavelengths than that of an o star , and 4.1.2 shows that the observed stellar continuum in the 912 - 1100 interval is too small to account for the luminosity of the h@xmath1 lines by photo - excitation , since the excitation of the fluorescent lines uses but a small fraction of the total energy in the 912 - 1100 interval . we agree with h06 , that the pure rotational h@xmath1lines are not photon pumped , but not for the reasons they give . they are far too bright to be produced by photo - excitation by the current stellar continuum . we established in 4 that the pdr s of the knots almost certainly can not be powered by shocks . another energy source is needed . in 4.1.1 and 4.1.2 we examined the energy budget for the helix nebula , establishing that only the central star radiation more energetic than 13.6 ev ( the extreme ultraviolet radiation or euv ) has enough power to drive the large total flux of h@xmath1 emission that is observed , thus expanding upon and quantifying the conclusions in cox98 . this means that the soft x - ray heating processes and 912 - 1100 ( fuv ) photo - excitation mechanisms ( natta & hollenbach 1998 ) do not explain the helix observations . the absence of a stellar wind and other time - scale considerations have already established ( ohf05 ) that shocks can not be powering the h@xmath1 emission and in 4.3 we showed why the justification of h06 for a shocks interpretation is incorrect . although phillips ( 2006 ) established a loose correlation between soft x - ray flux and h@xmath1 emission , this correlation is probably secondary rather than primary , as there is insufficient x - ray emission to power the h@xmath1 emission ; but , the stars that are strong h@xmath1 emitters have high temperatures , like the central star in the helix nebula . this means that these stars share the property of the euv radiation being dominant over fuv radiation . clearly a new process is required . based on this observational foundation , we identify a new state of equilibrium that may be common , but has not previously been identified . a new mechanism utilizing the euv radiation is briefly described here and will be elaborated upon in a future publication . the ionized flows from the knots are _ advection dominated _ , meaning that recombinations are relatively unimportant ( henney 2001 ) . as a result , neither the fuv or x - ray models ( hollenbach & tielens 1997 , natta & hollenbach 1998 ) is relevant to the dissociation fronts in the helix knots . instead , the dissociation front merges with the ionization front ( bertoldi & draine 1996 ) and the dissociation of h@xmath50 in this merged front is controlled by the _ ionizing _ euv . the fact that neutral hydrogen 21 cm is not observed in the inner region of the helix nebula ( rodrguez et al . 2002 ) where the optically bright knots are found supports this model and the appearance of 21 cm emission from the more distant and fainter outer - ring knots indicates that a neutral hydrogen zone is only present there . the low ionization parameter found in the helix knots leads to substantial deviations from ionization and thermal equilibrium since the dynamical time is shorter than the ionization and heating times . the effects of this upon the emission from the ionized gas was discussed at length in ohf05 . the dissociation of h@xmath50 in such a front is predominantly due to chemical reactions with ionized species such as o@xmath54 , and is therefore a strong function of the ionization fraction , which is determined by the absorption of euv radiation . the radiation field is largely determined by the opacity in the fraction of hydrogen that is neutral , the key element being its determination of the amount of o@xmath54 . it is the reaction of o@xmath54 with h@xmath1 that destroys the h@xmath1 rather than the much slower rate of photo - dissociation of h@xmath1 . this is essentially a one - way process , with h@xmath1 entering the zone from the cold molecular core and being converted directly to h@xmath27 . this transition zone is heated by the photo - ionization of neutral hydrogen and can be quite broad and the preliminary models indicate that it can produce warm h@xmath1 column densities of about 10@xmath55 @xmath34 , as required by the observations ( ohf05 ) . to the best of our knowledge , no models of such euv - dominated dissociation regions have been calculated . we are calculating detailed models of such regions , which will be reported upon in a future paper and restrict ourselves here to this brief description . it is likely that the same process determines the emission seen from the sheath of the tails in 2.12 @xmath0 m . the first - order theory for shadowed columns behind optically thick knots was presented by cant et al . they illustrated that the shadowed regions are illuminated by lyc photons emitted from recombining hydrogen , that this radiation was closer to 13.6 ev than the ionizing stellar radiation , and that the flux density of these diffuse lyc photons was about 0.15 that of the direct lyc flux from the central star . these diffuse lyc photons are almost certainly the source of the heating of the h@xmath1 in the tails as the surface brightness in 2.12 @xmath0 m is about 0.23@xmath360.08 that in the directly illuminated bright cusp ( 3.6 ) . the implausibility of shocks is also true here and the shortfall of energy from the fuv radiation is even greater than in the cusps because having a strong diffuse fuv radiation field would demand a large optical depth in dust for the nebula as a whole , which is not indicated by its emission line spectrum . a separate detailed h@xmath1 model is required for the tail because the illuminating fuv will be of lower energy and the density much lower than in the bright cusp . in a recent paper , meaburn et al . ( 2005 ) presented the analysis of images made in the center of the helix nebula in h@xmath8+[n ii ] in 1992 ( technical details described in meaburn et al . they present a high contrast rendering of the image ( their figure 10 ) and argue that radial `` spokes '' can be seen to faintly continue inside the boundary of the `` cometary globules '' to within about 30 from the central star . an arguably superior image of the region is available in the cerro tololo interamerican observatory 4-m mosaic images made in a similar filter ( h@xmath8+[nii ] ) and resolution , with a pixel scale ( 0.26/pixel ) . the individual exposures were 300 s and in the central region , where the fields of the four different pointings overlap , the effective exposure was 1200 s. a straightforward examination of this image using various levels of brightness and contrast did not reveal the features posited by meaburn et al . ( odell 2005 ) . we have now more intensively examined the same images , by employing median filters of 20x20 pixels and 40x40 pixels and dividing the original images by these , a technique previously employed ( omm04 ) to enhance the visibility of radial features in the outer parts of the nebula . the results are shown in figure 11 in negative depiction or the highly stretch range of intensities of 0.97 - 1.03 . one sees no indication of radial features extending inside the radius at which the knots disappear . we argue that our images are a better test of such features since one can see numerous stars and galaxies that do not appear in the meaburn et al . ( 2005 ) image . this non - detection of such features , even only slightly inside the position of the easily visible bright cusp knots is a strong argument that this boundary indicates where knots were first formed and does not represent a boundary where knots have been destroyed . an attractive model for generating the initial irregularities that develop into the knots is presented in the calculations of garca - segura et al . ( 2006 ) , who argue that these should arise at the boundary of shocked material as the initial fast - wind phase of the nebula ends , which is likely to be at about this position within the nebula . in a series of papers , dyson and collaborators have developed a model for the structure of the helix knots based on the hydrodynamic interaction that results when ionized gas is injected into a subsonic stream that flows past the injection source ( dyson et al . 1993 , 2006 ) . although the injected ionized gas is assumed to arise from an ionization front on the head of a neutral globule , the radiation transfer and ionization process is not explicitly included in the models . this makes it very difficult to make meaningful comparison between the results of these models and observations of the helix nebula . our new h@xmath1 observations clearly show ( 3.2 ) that the limb - brightened edges of the knot tails correspond to an ionization front . additionally , the width of the tail is equal to the width of the bright cusp at the head of the knot and its conic projection ( odell 2000 ) . this would seem to conclusively establish that radiation shadowing , rather than hydrodynamic interactions , is the _ primary _ determinant of the structure of the tails . in the stream - source model , although tail widths are predicted to be of the same order as the width of the injection source , there is no reason to expect them to be equal , unless the parameters are fine - tuned . the kinematic arguments given in support of the stream - source model ( meaburn et al . 2006 ) also do not stand up to close scrutiny . they are based on the ground - based echelle spectroscopic observations of meaburn et al . ( 1998 ) , which seem to show an acceleration of gas along the tail of the knot 378801 . however , a comparison with the much higher resolution _ hst _ observations ( e.g. , odell et al . 2005 , and also the image available at http://hubblesite.org/gallery/album/entire_collection/pr199613b ) . clearly shows that the sample regions used in that observational study all correspond to independent knots that are merely projected onto the tail of 378801 . therefore , the observed variation in velocity does not represent an acceleration along the tail , but simply a tendency towards higher velocities in knots that are farther from the central star , which has already been noted from co observations ( young et al . we have been able to use existing and new observations to reach a number of important conclusions about the knots in the helix nebula . there is sufficient energy to power the nebula s h@xmath1 emission only in extreme ultraviolet radiation from the central star with energies @xmath5613.6 ev , thus eliminating photo - excitation by the 912 - 1100 and x - ray flux that has been assumed in previous general models . there is no evidence from infrared emission lines for shock excitation of the knots h@xmath1 emission , the lack of a driving stellar wind and previous arguments of time - scale having come to the same conclusion . spectrophotometry of multiple lines in four sample regions and the total nebular flux ratio in the h@xmath1 0 - 0 s(5 ) and 2.12 @xmath0 m lines indicates that the h@xmath1 emitting zones are all about 988@xmath36119 k , closely resembling lte . the 2.12 @xmath0 m emission from individual knots falls immediately inside the ionized gas zone traced by h@xmath8 emission . this is true for both the bright cusps and their fainter tails , the latter establishing that the tails are primarily ionizing radiation shadows , rather than the result of purely hydrodynamic processes . the advection dominated nature of the knot cusps means that there is no extended neutral hydrogen zone between the cold molecular knot core and the ionized gas layer . this zone of irradiation of h@xmath1 by euv photons is probably the region producing the observed hot gas in the cusps on the star - facing side of the molecular knots and the shadowed regions of the tails . no evidence was found for knots within the he ii core nor were earlier claims verified of linear features extending nearly in to the central star , arguing that the knots have only been created outside the high ionization core . anton koekemoer of the space telescope science institute provided valuable guidance in the use of the drizzle package of tasks , making possible the smooth combination of our multiple observations of the same fields . we are grateful to angela speck for providing the calibrated 2.12 @xmath0 m image from her speck et al . ( 2002 ) paper and to pierre cox for his comments on the h@xmath1 physical processes . cro thanks the centro de radioastronoma y astrofsica , unam , mexico for generously supporting a two - week visit in february 2006 , during which initial work for this paper was carried out and to grant go 10628 from the space telescope science institute . gjf s work was supported in part by grant ar-10653 from the space telescope science institute , nasa grant nng05gd81 g , and nsf grant ast 0607028 . wjh acknowledges financial support from dgapa - unam , mexico , project in112006 . bertoldi , f. , & draine , b. t. 1996 , apj , 458 , 222 black . j. h. , & van dishoek , e. f. 1987 , apj , 322 , 412 bohlin , r. c. , harrington , j. p. , & stecher , t. p. 1982 , apj , 252 , 635 burkert , a. , & odell , c. r. 1998 , apj , 503 , 792 cant , j , raga , a. , steffen , w. , & shapiro , p. r. 1998 , apj , 5002 , 695 cerruti - sola , m. , & perinotto , m. 1985 , apj , 291 , 237 cox , p. , boulanger , f. , huggins , p. j. , tielens , a. g. g. m. , forveille , t. , bachiller , r. , cesarsky , d. , jones , a. p. , young , k. , roelfsema , p. r. , & cernicharo , j. 1998 , apj , 495 , l23 ( cox98 ) dabrowski , i. 1984 , canadian j. phys . , 62 , 1639 dyson , j. e. , hartquist , t. w. , & biro , s. , 1993 , mnras , 261 , 430 dyson , j. e. , pittard , j. m. , meaburn , j. , & falle , s. a. e. g. 2006 , a&a , 457 , 561 fazio , g. et al . 2004 , apjs , 154 , 10 garca - segura , g. , lpez , j. a. , steffen , w. , meaburn , j. , & manchado , a. 2006 , apj , in press gonzaga , s. , et al . 2005 , acs instrument handbook , version 6.0 , ( baltimore : stsci ) guerrero , m. a. , chu , y .- h . , gruendl , r. a. , williams , r. m. , & kaler , j. b. 2001 , apj , 553 , l55 harris , h. c. , dahn , c. c. , canzian , b. , guetter , h. h. , leggett , s. k. , levine , s. e. , luginbuhl , c. b. , monet , a. k. b. , monet , d. g. , pier , j. r. , stone , r. c. , tilleman , t. , vrba , f. j. , & walker , r. l. 2007 , aj , in press henney , w. j. 2001 , rmxac , 10 , 57 hollenbach , d. j. , & tielens , a. g. g. m. 1997 , araa , 35 , 179 hora , j. l. , latter , w. b. , smith , h. a. , & marengo , m. 2006 , apj , in press ( h06 ) holtzman , j. a. , burrows , c. j. , castertano . s. , hester , j. j. , trauger , j. t. , watson , a. m. , & worthey , g. 1995 , pasp , 107 , 1065 houck , j. r. et al . 2004 , apjs , 154 , 18 huggins , p. j. , forveille , t. , bachiller , r. , cox , p. , ageorges , n. , & walsh , j. r. 2002 , apj , 573 , l55 . lpez - martn , l. , raga , a. c. , melleman , g. , henney , w. j. , & cant , j. 2001 , apj , 548 , 288 leahy , d. a. , zhang , c. y. , & kwok , s. 1994 , apj , 422 , 205 leahy , d. a. , zhang , c. y. , volk , k. , & kwok , s. 1996 , apj , 466 , 352 meaburn , j. , boumis , p. , lpez , j. a. , harman , d. j. , bryce , m. , redman , m. p. , & mavromatakis , f. 2005 , mnras , 360 , 963 m05 meaburn , j. , clayton , c. a. , bryce , m. , walsh , j. r. , holloway , a. j. , & steffen , w. 1998 , mnras , 294 , 201 meaburn , j. , walsh , j. r. , clegg , r. e. s. , walton , n. a. , & taylor , d. 1992 , mnras , 255 , 177 meixner , m. , mccullough , p. , hartman , j. , odell , c. r. , & speck , a. k. 2004 , in asp conf . 313 , asymmetrical planetary nebulae iii : winds , structure , and the thunderbird , ed . m. meixner et al . 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we use new hubble space telescope and archived images to clarify the nature of the ubiquitous knots in the helix nebula , which are variously estimated to contain a significant to majority fraction of the material ejected by its central star .
we employ published far infrared spectrophotometry and existing 2.12 @xmath0 m images to establish that the population distribution of the lowest ro - vibrational states of h@xmath1 is close to the distribution of a gas in local thermodynamic equilibrium ( lte ) at @xmath2 k. in addition , we present calculations that show that the weakness of the h@xmath1 0 - 0 s(7 ) line is not a reason for making the unlikely - to - be true assumption that h@xmath1 emission is caused by shock excitation .
we derive a total flux from the nebula in h@xmath1 lines and compare this with the power available from the central star for producing this radiation .
we establish that neither soft x - rays nor 9121100 radiation has enough energy to power the h@xmath1 radiation , only the stellar extreme ultraviolet radiation shortward of 912 does .
advection of material from the cold regions of the knots produces an extensive zone where both atomic and molecular hydrogen are found , allowing the h@xmath1 to directly be heated by lyman continuum radiation , thus providing a mechanism that will probably explain the excitation temperature and surface brightness of the 2.12 @xmath0 m cusps and tails .
new images of the knot 378 - 801 in the h@xmath1 2.12 @xmath0 m line reveal that the 2.12 @xmath0 m cusp lies immediately inside the ionized atomic gas zone .
this property is shared by material in the tail region .
the h@xmath1 2.12 @xmath0 m emission of the cusp confirms previous assumptions , while the tail s property firmly establishes that the tail " structure is an ionization bounded radiation shadow behind the optically thick core of the knot .
the new 2.12 @xmath0 m image together with archived hubble images is used to establish a pattern of decreasing surface brightness and increasing size of the knots with increasing stellar distance .
although the contrast against the background is greater in 2.12 @xmath0 m than in the optical lines , the higher resolution and signal of optical images remains the most powerful technique for searching for knots .
a unique new image of a transitional region of the nebula s inner disk in the heii 4686 line fails to show any emission from knots that might have been found in the he@xmath3 core of the nebula .
we also re - examined high signal - to - noise ratio ground - based telescope images of this same inner region and found no evidence of structures that could be related to knots .
| 19,908 | 718 |
galaxy clusters provide an independent means of examining any viable model of cosmic structure formation through the growth of structure and by the form of their equilibrium mass profiles , complementing cosmic microwave background and galaxy clustering observations . a consistent framework of structure formation requires that most of the matter in the universe is in the hitherto unknown form of dark matter , of an unknown nature , and that most of the energy filling the universe today is in the form of a mysterious `` dark energy '' , characterized by a negative pressure . this model actually requires that the expansion rate of the universe has recently changed sign and is currently accelerating . clusters play a direct role in testing cosmological models , providing several independent checks of any viable cosmology , including the current consensus @xmath11 cold dark matter ( @xmath11cdm ) model . a spectacular example has been recently provided from detailed lensing and x - ray observations of the `` bullet cluster '' ( aka , ie0657 - 56 ; * ? ? ? * ; * ? ? ? * ) , which is a consequence of a high - speed collision between two cluster components with a mass ratio of the order of @xmath12 @xcite , displaying a prominent bow shock preceding a cool bullet lying between the two clusters , implying these clusters passed through each other recently @xcite . here the bullet system reveals lensing mass contours that follow the bimodal distribution of cluster members , demonstrating that the bulk of the dark matter is relatively collisionless as galaxies @xcite , as also shown by a comprehensive analysis of galaxy and dark - matter dynamics for a1689 @xcite . other cases of merging systems show that in general displacement of the hot gas relative to the dark matter is related to interaction @xcite . for dynamically - relaxed clusters , the form of the equilibrium mass profile reflects closely the distribution of dark matter ( see * ? ? ? * ) which , unlike galaxies , does not suffer from halo compression by adiabatic contraction of cooled gas . the majority of baryons in clusters are in the form of hot , diffuse x - ray emitting gas , and represents only a minor fraction of the total lensing mass near the centers of clusters @xcite . the predicted navarro - frenk - white profile ( hereafter , nfw ; * ? ? ? * ; * ? ? ? * ) derived from simulations based on collisionless , cold ( non - relativistic ) dark matter has a continuously - declining logarithmic gradient @xmath13 towards the center of mass , much shallower than the isothermal case ( @xmath14 ) within the characteristic scale radius , @xmath15 ( @xmath16 for cluster - sized halos ) . a useful index of the degree of concentration , @xmath17 , compares the virial radius , @xmath1 , to @xmath15 of the nfw profile , @xmath18 . this has been confirmed thoroughly with higher resolution simulations @xcite , with some intrinsic variation related to the individual assembly history of a cluster @xcite . gravitational lensing observations are underway to provide reliable and representative cluster mass profiles to test this since the first careful measurements showed that the nfw profile provides a good fit to the entire mass profile when weak and strong lensing are combined . other well studied clusters with similarly high quality data are also in good agreement providing strong support for the cdm scenario ( e.g. , * ? ? ? interestingly these studies reveal that although the dark matter is consistent with being cold , the predicted profile concentration of the standard @xmath11cdm model falls short of some lensing results ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this observed tendency for higher proportion of mass to lie at smaller radius in projection is also indicated by the generally large einstein radii determined from strong lensing of well studied clusters @xcite finding a substantial discrepancy with the predictions despite careful accounting for potential selection biases inherent to lensing @xcite . these observations could suggest either substantial mass projected along the line of sight , perhaps in part due to halo triaxiality @xcite , or a large overconcentration of mass ; the latter could imply modification within the context of the cdm family of models . the abundance of massive clusters is very sensitive to the amplitude of the initial mass power spectrum @xcite representing the most massive objects to have collapsed under their own gravity , and confirmed by @xmath19-body simulations of hubble volumes @xcite . such calculations predict for example that the single most massive cluster to be found in the universe is expected to be with @xmath20 out to @xmath21 ( see figure 5 of * ? ? ? * ) , similar to the most massive known clusters detected locally @xcite . is currently the most massive known cluster measured reliably by lensing , @xmath22 . ] at higher redshifts this comparison becomes more sensitive to the cosmological model , with an order of magnitude decline in the abundance of @xmath23 clusters at @xmath24 compared to the present @xcite . hence , the existence of such massive clusters like xmmuj2235 - 25 at @xmath25 @xcite , from lensing work , begins to motivate alternative ideas such as departures from gaussian initial density fluctuation spectrum , or higher levels of dark energy in the past @xcite , although some non - gaussian models can be ruled out by using the cosmic x - ray background measurements @xcite . the main attraction of gravitational lensing in the cluster regime ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) is the model - free determination of mass profiles allowed over a wide range of radius when the complementary effects of strong and weak lensing are combined @xcite . in practice , the quality of data required challenges with few facilities , which are able to generate data of sufficient precision to provide a significant detection of the weak lensing signal on an individual cluster basis . in this paper we aim to pursue in greater depth the utility of massive clusters for defining highest - precision mass profiles by combining all lensing information available in the cluster regime . in particular , we shall make a full use of the magnification information , afforded by measuring spatial variations in the surface number density of faint background galaxies @xcite , which we show here is readily detectable in the high - quality images we have obtained for this purpose . in our earlier work , particularly on a1689 and cl0024 + 17 , the magnification information determined from the background counts was used only as a consistency check of distortion measurements @xcite . recently , we have shown in @xcite how to overcome the intrinsic clustering of background galaxies , which otherwise perturbs locally the magnification signal , and how to combine the two independent weak - lensing data sets to improve the quality of two - dimensional mass reconstruction using regularized maximum - likelihood techniques . here we further explore new statistical methods designed to obtain an optimal combination of the complementary lens distortion and magnification effects . our aim here is to develop and apply techniques to a sample of five well - studied , high - mass clusters with @xmath26 , a1689 , a1703 , a370 , cl0024 + 17 , and rxj1347 - 11 , for examining the underlying mass profiles extracted from the combined weak - lensing data sets . we then add to this detailed strong - lensing information for the inner @xmath27kpc region of these clusters , for which we have identified many new sets of multiple images from advanced camera for surveys ( acs ) observations @xcite with the _ hubble space telescope _ ( _ hst _ ) , to derive improved inner mass profiles for a full determination of the entire mass profiles of the five well - studied clusters . the paper is organized as follows . we briefly summarize in [ sec : basis ] the basis of cluster weak gravitational lensing . in [ sec : method ] we present our comprehensive lensing method in a bayesian framework for a direct reconstruction of the projected cluster mass profile from combined weak - lensing shape distortion and magnification bias measurements . in [ sec : results ] we apply our method to subaru weak - lensing observations of five massive clusters to derive projected mass profiles to beyond the cluster virial radius ; we also combine our new weak - lensing mass profiles with inner strong - lensing based information from _ hst_/acs observations to make a full determination of the entire cluster mass profiles . finally , summary and discussions are given in [ sec : discussion ] . throughout this paper , e use the ab magnitude system , and adopt a concordance @xmath11cdm cosmology with @xmath28 , @xmath29 , and @xmath30 . errors represent a confidence level of @xmath31 ( @xmath32 ) unless otherwise stated . the deformation of the image for a background source can be described by the jacobian matrix @xmath33 ( @xmath34 ) of the lens mapping . the real , symmetric jacobian @xmath35 can be decomposed as @xmath36 , where @xmath37 is kronecker s delta , @xmath38 is the lensing convergence , and @xmath39 is the trace - free , symmetric shear matrix , @xmath40 with @xmath41 being the components of spin-2 complex gravitational shear @xmath42 . in the strict weak lensing limit where @xmath43 , @xmath39 induces a quadrupole anisotropy of the background image , which can be observed from ellipticities of background galaxy images @xcite . the local area distortion due to gravitational lensing , or magnification , is given by the inverse jacobian determinant , @xmath44 where we assume subcritical lensing , i.e. , @xmath45 . the lens magnification @xmath46 can be measured from characteristic variations in the number density of background galaxies ( * ? ? ? * see also [ subsec : magbias ] ) . the lensing convergence @xmath38 is a weighted projection of the matter density contrast along the line of sight ( e.g. , * ? ? ? * ) . for gravitational lensing in the cluster regime ( e.g. , * ) , @xmath38 is expressed as @xmath47 , namely the projected mass density @xmath48 in units of the critical surface mass density for gravitational lensing , defined as @xmath49,\end{aligned}\ ] ] where @xmath50 , @xmath51 , and @xmath52 are the proper angular diameter distances from the observer to the source , from the observer to the deflecting lens , and from the lens to the source , respectively , and @xmath53 is the mean distance ratio averaged over the population of source galaxies in the cluster field . in general , the observable quantity for quadrupole weak lensing is not the gravitational shear @xmath54 but the complex _ reduced _ shear ( see [ subsec : gt ] ) , @xmath55 in the subcritical regime where @xmath56 ( or @xmath57 in the negative parity region with @xmath58 ) . the reduced shear @xmath59 is invariant under the following global linear transformation : @xmath60 with an arbitrary scalar constant @xmath61 . this transformation is equivalent to scaling the jacobian matrix @xmath62 with @xmath63 , @xmath64 , and hence leaves the critical curves @xmath65 invariant . furthermore , the curve @xmath66 , on which the gravitational distortions disappear , is left invariant under the transformation ( [ eq : invtrans ] ) . this mass - sheet degeneracy can be unambiguously broken by measuring the magnification effects ( see [ subsec : magbias ] ) , because the magnification @xmath46 transforms under the invariance transformation ( [ eq : invtrans ] ) as @xmath67 alternatively , the constant @xmath63 can be determined such that the mean @xmath38 averaged over the outermost cluster region vanishes , if a sufficiently wide sky coverage is available . such that the enclosed mass within a certain aperture is consistent with cluster mass estimates from some other observations ( e.g. , * ? ? ? in this section we develop a bayesian method to reconstruct the projected cluster mass profile @xmath68 from observable lens distortion and magnification profiles , without assuming particular functional forms for the mass distribution , i.e. , in a model - independent fashion . although the methodology here is presented for the analysis of individual clusters , it can be readily generalized for a statistical analysis using stacked lensing profiles of a sample of clusters . in [ subsec : stack ] , alternatively , we provide a method to stack reconstructed projected mass profiles of individual clusters to obtain an ensemble - averaged profile . the observable quadrupole distortion of an object due to gravitational lensing is described by the spin-2 reduced shear , @xmath69 ( equation [ [ eq : redshear ] ] ) , which is coordinate dependent . for a given cluster center on the sky , one can form coordinate - independent quantities , the tangential distortion @xmath70 and the @xmath71 rotated component , from linear combinations of the distortion coefficients as @xmath72 and @xmath73 , with @xmath74 being the position angle of an object with respect to the cluster center . in the strict weak - lensing limit , the azimuthally - averaged tangential distortion profile @xmath75 satisfies the following identity ( e.g. , * ? ? ? * ) : @xmath76 , where @xmath77 is the azimuthal average of @xmath78 at radius @xmath79 , and @xmath80 is the mean convergence interior to radius @xmath79 . with the assumption of quasi - circular symmetry in the projected mass distribution ( see * ? ? ? * ) , the tangential distortion is expressed as @xmath81 in the nonlinear but subcritical ( @xmath45 ) regime . to the first order of @xmath38 . see 3.4 of @xcite for details . ] in the absence of higher order effects , weak lensing only induces curl - free tangential distortions , while the azimuthal averaged @xmath82 component is expected to vanish . in practice , the presence of @xmath82 modes can be used to check for systematic errors . from shape measurements of background galaxies , we calculate the weighted average of @xmath83 in a set of @xmath19 radial bands ( @xmath84 ) as @xmath85 where @xmath86 is the center of the @xmath87th radial band of @xmath88 $ ] , the index @xmath89 runs over all of the objects located within the @xmath87th annulus , @xmath90 is the tangential distortion of the @xmath89th object , and @xmath91 is the statistical weight ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) for the @xmath89th object , given by @xmath92 with @xmath93 being the variance for the shear estimate of the @xmath89th galaxy and @xmath94 being the softening constant variance ( e.g. , * ? ? ? * ) . in our analysis , we choose @xmath95 , which is a typical value of the mean rms @xmath96 over the background sample ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we use the continuous limit of the area - weighted band center for @xmath86 ( see equation [ [ eq : medianr ] ] ) . we perform a bootstrap error analysis to assess the uncertainty @xmath97 in the tangential distortion profile @xmath98 . lensing magnification , @xmath99 , influences the observed surface density of background sources , expanding the area of sky , and enhancing the observed flux of background sources @xcite . the former effect reduces the effective observing area in the source plane , decreasing the number of background sources per solid angle ; on the other hand , the latter effect amplifies the flux of background sources , increasing the number of sources above the limiting flux . the net effect is known as magnification bias , and depends on the intrinsic slope of the luminosity function of background sources . the number counts for a given magnitude cutoff @xmath100 , approximated locally as a power - law cut with slope @xmath101 ( @xmath102 ) , are modified in the presence of lensing as @xmath103 @xcite , where @xmath104 is the unlensed counts , and @xmath46 is the magnification , @xmath105 . in the strict weak lensing limit , the magnification bias is @xmath106 . for the number counts to measure magnification , we use a sample of _ red _ background galaxies , for which the intrinsic count slope @xmath107 at faint magnitudes is relatively flat , @xmath108 , so that a net count depletion results @xcite . on the other hand , the faint blue background population tends to have a steeper intrinsic count slope close to the lensing invariant slope ( @xmath109 ) . the count - in - cell statistic @xmath110 is measured from a flux - limited sample of red background galaxies on a regular grid of equal - area cells , each with a solid angle of @xmath111 . note that a practical difficulty of the magnification bias measurement is contamination due to the intrinsic clustering of background galaxies , which locally can be larger than the lensing - induced signal in a given cell . in order to obtain a clean measure of the lensing signal , such intrinsic clustering needs to be downweighted ( e.g. , * ? ? ? * ; * ? ? ? * ) . for a mass profile analysis , we calculate the mean number density @xmath112 of the red background sample as a function of radius from the cluster center , by azimuthally averaging @xmath113 , using the same radial bins ( @xmath84 ) as done for the distortion measurement . the lens magnification bias is expressed in terms of the number density of background galaxies as @xmath114 with @xmath115 being the unlensed mean surface number density of background galaxies . the normalization and slope parameters ( @xmath116 ) can be estimated from the source counts in cluster outskirts using wide - field imaging data ( [ subsec : back ] ) . in practice , we adopt the following prescription : * a positive tail of @xmath117 cells is excluded in each annulus to remove inherent small scale clustering of the background @xcite . * each grid cell is weighted by the fraction of its area lying within the respective annular bins @xcite . * the uncertainty @xmath118 in @xmath119 includes not only the poisson contribution but also the variance due to variations of the counts along the azimuthal direction , i.e. , contributions from the intrinsic clustering of background galaxies @xcite . * the cell size @xmath120 can be as large as the typical radial band width for a mass profile analysis , which can cause an additional variance due to poisson and sampling errors . we thus average over a set of radial profiles obtained using different girds offset with respect to each other by half a grid spacing in each direction . * the masking effect due to bright cluster galaxies , bright foreground objects , and saturated pixels is properly taken into account and corrected for @xcite . in our analysis , we use method b of appendix [ appendix : mask ] developed in this work . the relation between distortion and convergence is nonlocal , and the convergence derived from distortion data alone suffers from a mass - sheet degeneracy ( [ sec : basis ] ) . however , by combining the distortion and magnification measurements the convergence can be obtained unambiguously with the correct mass normalization . here we aim to derive a discrete convergence profile from observable lens distortion and magnification profiles ( see [ subsec : gt ] and [ subsec : magbias ] ) within a bayesian statistical framework , allowing for a full parameter - space extraction of model and calibration parameters . a proper bayesian statistical analysis is of particular importance to explore the entire parameter space and investigate the parameter degeneracies , arising in part from the mass - sheet degeneracy . in this framework , we sample from the posterior probability density function ( pdf ) of the underlying signal @xmath121 given the data @xmath122 , @xmath123 . expectation values of any statistic of the signal @xmath121 shall converge to the expectation values of the a posteriori marginalized pdf , @xmath123 . in our problem , the signal @xmath121 is a vector containing the discrete convergence profile , @xmath124 @xmath125 , and the average convergence within the inner radial boundary @xmath126 of the weak lensing data , @xmath127 , so that @xmath128 , being specified by @xmath129 parameters . the bayes theorem states that @xmath130 where @xmath131 is the likelihood of the data given the model ( @xmath121 ) , and @xmath132 is the prior probability distribution for the model parameters . we combine complementary and independent weak lensing information of tangential distortion and magnification bias to constrain the underlying cluster mass distribution @xmath133 . the total likelihood function @xmath134 of the model @xmath121 for combined weak lensing observations is given as a product of the two separate likelihoods , @xmath135 , where @xmath136 and @xmath137 are the likelihood functions for distortion and magnification , respectively . the log - likelihood for the tangential distortion is given as @xmath138 ^ 2}{\sigma_{+.i}^2 } + { \rm const}.,\ ] ] where @xmath139 is the theoretical prediction for the observed distortion @xmath140 , and the errors @xmath141 ( @xmath84 ) due primarily to the variance of the intrinsic source ellipticity distribution can be conservatively estimated from the data using bootstrap techniques ( [ subsec : gt ] ) . similarly , the log - likelihood function for the magnification bias is given as @xmath142 ^ 2}{\sigma_{\mu.i}^2 } + { \rm const}.,\ ] ] where @xmath143 is the theoretical prediction for the observed counts @xmath144 , and the errors @xmath145 include both contributions from poisson errors in the counts , @xmath146 , and contamination due to intrinsic clustering of red background galaxies , @xmath147 , as discussed in [ subsec : magbias ] : @xmath148 the lensing observables @xmath140 and @xmath144 ( @xmath84 ) can be readily expressed by the given model parameters @xmath121 , as shown in appendices [ appendix : avkappa ] and [ appendix : lprof ] . for each parameter of the model @xmath121 , we consider a simple flat prior with a lower bound of @xmath149 , that is , @xmath150 additionally , we account for the calibration uncertainty in the observational parameters , i.e. , the normalization and slope parameters @xmath151 of the background counts and the relative lensing depth @xmath152 due to population - to - population variations between the background samples used for the magnification and distortion measurements ( see appendix [ appendix : lprof ] ) . cccccc a1689 & 0.183 & @xmath153 & @xmath154 & 0.82 & 1 + a1703 & 0.281 & @xmath155 & @xmath156 & 0.78 & 2,3 + a370 & 0.375 & @xmath157 & @xmath158 & 0.60 & 2,3 + cl0024 + 17 & 0.395 & @xmath157 & @xmath158 & 0.80 & 4 + rxj1347 - 11 & 0.451 & @xmath159 & @xmath158 & 0.76 & 2,3 + ccc a1689 & @xmath160 @xmath161 & 1,2 + a1703 & @xmath162 @xmath163 & 3 , 4 , 5 + a370 & @xmath164 @xmath165 & 6 , 7 + cl0024 + 17 & @xmath166 @xmath167 & 8 + rxj1347 - 11 & @xmath168 @xmath169 & 7 , 9 , 10 + c|ccc|ccc|c a1689 & 8.8 & @xmath170 & @xmath171 & 12.0 & @xmath170 & @xmath171 & @xmath172 + a1703 & 10.0 & @xmath173 & @xmath174 & 6.9 & @xmath175 & @xmath176 & @xmath177 + a370 & 16.7 & @xmath178 & @xmath179 & 21.6 & @xmath180 & @xmath181 & @xmath182 + cl0024 + 17 & 17.2 & @xmath183 & @xmath184 & 17.8 & @xmath185 & @xmath186 & @xmath187 + rxj1347 - 11 & 6.4 & @xmath188 & @xmath189 & 7.7 & @xmath190 & @xmath191 & @xmath192 + cccc a1689 & @xmath193 & @xmath194 & @xmath195 + a1703 & @xmath196 & @xmath197 & @xmath198 + a370 & @xmath199 & @xmath200 & @xmath201 + cl0024 + 17 & @xmath202 & @xmath203 & @xmath204 + rxj1347 - 11 & @xmath199 & @xmath205 & @xmath206 + @xmath207{f1a.eps } & \includegraphics[width=45mm , angle=0]{f1b.eps } & \includegraphics[width=45mm , angle=0]{f1c.eps } \end{array } $ ] @xmath208{f1d.eps } & \includegraphics[width=45mm , angle=0]{f1e.eps } \end{array } $ ] @xmath209{f2a.eps } & \includegraphics[width=20mm , angle=270]{f2b.eps } & \includegraphics[width=20mm , angle=270]{f2c.eps } & \includegraphics[width=20mm , angle=270]{f2d.eps } & \includegraphics[width=20mm , angle=270]{f2e.eps } & \includegraphics[width=20mm , angle=270]{f2f.eps}\\ \end{array } $ ] @xmath209{f2g.eps } & \includegraphics[width=20mm , angle=270]{f2h.eps } & \includegraphics[width=20mm , angle=270]{f2i.eps } & \includegraphics[width=20mm , angle=270]{f2j.eps } & \includegraphics[width=20mm , angle=270]{f2k.eps } & \includegraphics[width=20mm , angle=270]{f2l.eps}\\ \end{array } $ ] @xmath210{f3a.eps } & \includegraphics[width=40mm , angle=270]{f3b.eps } & \includegraphics[width=40mm , angle=270]{f3c.eps } \\ \end{array } $ ] @xmath211{f3d.eps } & \includegraphics[width=40mm , angle=270]{f3e.eps}\\ \end{array } $ ] in this section we apply our bayesian mass reconstruction method to a sample of five well - studied high - mass clusters ( @xmath26 ) at intermediate redshifts , a1689 ( @xmath212 ) , a1703 ( @xmath213 ) , a370 ( @xmath214 ) , cl0024 + 17 ( @xmath215 ) , and rxj1347 - 11 ( @xmath216 ) , observed with the wide - field camera ( @xmath217 ; see * ? ? ? * ) on the 8.2 m subaru telescope . table [ tab : sample ] gives a summary of cluster observations . the clusters were observed deeply in several optical passbands , with exposures in the range 200010000s per passband , with seeing in the detection images ranging from @xmath218 to @xmath219 ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? these massive clusters are known as strong lensing clusters , displaying prominent strong - lensing features and large einstein radii of @xmath220 ( e.g. , for a fiducial source redshift @xmath221 ; * ? ? ? * ; * ? ? ? for the clusters , the central mass distributions have been recovered in detail by our strong - lens modeling @xcite . here the models were constrained by a number of multiply - lensed images identified previously in very deep multi - color imaging with _ hst_/acs ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? table [ tab : rein ] gives a summary of the einstein radii of the five clusters as constrained from our detailed strong lens modeling with the acs observations . here we will add to our high - quality spin-2 shape measurements @xcite the independent magnification information based on deep multi - band imaging with subaru , in order to achieve the maximum possible lensing precision . full details of the subaru observations and weak - lensing shape analysis of these clusters were presented in a series of our papers ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * see also table [ tab : sample ] ) . the level of shear calibration bias with our implementation of the ksb+ method @xcite has been assessed by @xcite using simulated subaru suprime - cam images ( m. oguri 2010 , in private communication ; * ? ? ? we find , typically , @xmath222 of the shear calibration bias , and @xmath223 of the residual shear offset which is about 1 order of magnitude smaller than the typical distortion signal in cluster outskirts . this level of calibration bias is subdominant compared to the statistical uncertainty ( @xmath224 ) due to the intrinsic scatter in galaxy shapes , and to the dilution effect which can lead to an underestimation of the true signal for @xmath225 by a factor of 25 ( see figure 1 of * ? ? ? a careful background selection is critical for a weak - lensing analysis so that unlensed cluster members and foreground galaxies do not dilute the true lensing signal of the background @xcite . we use undiluted samples of background galaxies derived in our previous lensing work , as summarized in table [ tab : back ] . when deep multi - color photometry is available in our cluster fields , we use the background selection method of ( * ? ? ? * see also umetsu et al . 2010 ) to define blue and red background samples ( a1703 , a370 , cl0024 + 17 , rxj1347 - 11 ) , which relies on empirical correlations for galaxies in color - color space derived from the deep subaru photometry , by reference to the deep photometric - redshift survey in the cosmos field @xcite . otherwise , we use the color - magnitude selection method @xcite to define a sample of red galaxies ( a1689 ) whose colors are redder than the red sequence of cluster e / s0 galaxies . these red galaxies are expected to lie in the background by virtue of @xmath226-corrections which are greater than the red cluster sequence galaxies , as convincingly demonstrated spectroscopically by @xcite . apparent magnitude cuts are applied in the reddest band available for each cluster to avoid incompleteness near the detection limit . a flux - limited sample of red background galaxies is used for the magnification analysis ( see [ subsec : magbias ] ) . for the distortion analysis , we use a full composite sample of red and blue ( if available ) background galaxies , where the galaxies used are well resolved to make reliable shape measurements ( see , e.g. , * ? ? ? table [ tab : back ] lists for respective color samples the mean surface number density @xmath227 of background galaxies , the effective source redshift , @xmath228 , , corresponding to the mean depth @xmath229 , is defined as @xmath230 . for details , see 3.4 of @xcite . ] and the mean distance ratio @xmath229 averaged over the source redshift distribution . we also quote in table [ tab : back ] the values of the relative mean depth , @xmath231 , between the background samples used for the magnification and distortion measurements ( see [ subsubsec : prior ] and appendix [ appendix : lprof ] ) . we adopt a conservative uncertainty of @xmath232 in the relative mean depth @xmath152 for all of the clusters based on our previous work @xcite . the conversion from the observed counts into magnification depends on the normalization @xmath115 and the slope parameter @xmath107 of the unlensed number counts , which can be reliably estimated thanks to the wide field of view of subaru / suprime - cam ( see [ subsec : massrec ] ) . table [ tab : magbias ] lists the magnitude cuts ( @xmath100 ) and unlensed count parameters ( @xmath116 ) as measured from the red background counts in outermost annular regions in cluster outskirts ( @xmath233 ) . following the methodology outlined in [ subsec : gt ] and [ subsec : magbias ] ( see also appendix [ appendix : mask ] ) , we derive lens distortion and magnification profiles of five massive clusters from subaru observations . in order to obtain meaningful radial profiles , one must carefully define the center of the cluster . it is often assumed that the cluster mass centroid coincides with the position of the brightest cluster galaxy ( bcg ) , whereas the bcgs can be offset from the mass centroids of the corresponding dark matter halos @xcite . here we utilize our detailed mass maps in the cluster cores recovered from strong - lens modeling of acs observations ( [ subsec : sample ] ) , providing an independent mass - centroid determination . we find that for these five clusters there is only a small offset of typically @xmath234 ( @xmath235kpc@xmath236 at the highest cluster redshift of our study , @xmath237 ) between the bcg and the dark - matter center of mass ( see also 4.2 of * ? ? ? * ) , often implied by other massive bright galaxies in the vicinity of the bcg . this level of cluster centering offset is substantially small as compared to the typical inner radial boundary of weak lensing observations , @xmath238 . in the present work , we therefore simply assume that the cluster mass centroid coincides with the location of the bcg , which is adopted as the cluster center in our one - dimensional profile analysis . the lensing profiles were calculated in @xmath19 discrete radial bins over the range of radii @xmath239 $ ] , with a constant logarithmic radial spacing of @xmath240 , where the inner radial boundary @xmath241 is taken such that @xmath242 ( see table [ tab : rein ] ) . the typical inner boundary is @xmath243 for cluster weak lensing . the outer radial boundary @xmath244 was chosen to be sufficiently larger than the typical virial radius of high mass clusters with @xmath245 , @xmath246mpc , but sufficiently small ( @xmath247 ) with respect to the size of the suprime - cam s field - of - view so as to ensure accurate psf ( point spread function ) anisotropy correction . the number of radial bins @xmath19 was determined for each cluster such that the per - pixel detection signal - to - noise ratio ( @xmath248 ) is of the order of unity , which is optimal for an inversion problem . the radial binning scheme is summarized in table [ tab : data ] . cccccc a1689 & @xmath249 & 11 & 13.8 & 8.8 & 17.8 + a1703 & @xmath250 & 10 & 9.9 & 7.1 & 12.7 + a370 & @xmath251 & 14 & 17.3 & 11.0 & 23.8 + cl0024 + 17 & @xmath252 & 12 & 13.0 & 8.5 & 18.5 + rxj1347 - 11 & @xmath253 & 11 & 9.7 & 5.7 & 12.6 in figure [ fig : data ] we show the resulting distortion and magnification profiles for our sample of five massive lensing clusters . in all the clusters , a strong depletion of the red galaxy counts is seen in the central , high - density region of the cluster , and clearly detected out to several arcminutes from the cluster center . the statistical significance of the detection of the depletion signal is in the range @xmath254@xmath255 ( see table [ tab : data ] ) . the detection significance of the tangential distortion derived from the full background sample ranges from @xmath256 to @xmath257 , and is better than the red counts . the @xmath82-component is consistent with a null signal in most of radial bins , indicating the reliability of our weak - lensing analysis . the magnification measurements with and without the masking correction are roughly consistent with each other . typically , the masking area is negligible ( a few % ) at large radii , and increases up to @xmath258 of the sky close to the cluster center ( see appendix [ appendix : mask ] ) . to test the consistency between our distortion and depletion measurements , we calculate the depletion of the counts expected for the best - fitting nfw profile derived from the distortion measurements ( figure [ fig : data ] ) , normalized to the observed density @xmath115 ( table [ tab : magbias ] ) . this comparison shows clear consistency between two independent lensing observables with different systematics ( see 5.5 of umetsu & broadhurst 2008 ) , which strongly supports the reliability and accuracy of our weak - lensing analysis ( see also supplemental material presented in appendix [ appendix : supplement ] ) . c|cccc|ccccc a1689 & @xmath259 & @xmath260 & @xmath261 & @xmath262 & @xmath263 & @xmath264 & @xmath265 & @xmath266 & @xmath267 + a1703 & @xmath268 & @xmath269 & @xmath270 & @xmath271 & @xmath272 & @xmath273 & @xmath274 & @xmath275 & @xmath276 + a370 & @xmath277 & @xmath278 & @xmath279 & @xmath280 & @xmath281 & @xmath282 & @xmath283 & @xmath284 & @xmath285 + cl0024 + 17 & @xmath286 & @xmath287 & @xmath288 & @xmath289 & @xmath290 & @xmath291 & @xmath292 & @xmath293 & @xmath294 + rxj1347 - 11 & @xmath295 & @xmath296 & @xmath297 & @xmath298 & @xmath299 & @xmath300 & @xmath301 & @xmath302 & @xmath303 we use a markov chain monte carlo ( mcmc ) approach with metropolis - hastings sampling to reconstruct the discrete cluster mass profile @xmath304 within a bayesian statistical framework ( [ subsec : bayesian ] ) . we largely follow the sampling procedure outlined in @xcite , but employ the gelman - rubin @xmath305 statistic @xcite as a simple but reasonable convergence criterion of generated chains . once convergence to a stationary distribution is achieved , we run a long final chain of @xmath306 steps , which adequately samples the underlying posterior probability distribution . for all of the parameters , the number of iterations required for convergence is much less than our final chain length . note , only the final chain is used for our parameter estimations and error analysis . we estimate the location of the maximum a posteriori probability for each model parameter using the bisection method in conjunction with bootstrap techniques . the covariance matrix @xmath307 ( @xmath308 ) for the discrete mass profile @xmath121 is estimated from the mcmc samples . as an example , we show in figure [ fig : post ] one - dimensional marginalized posterior pdfs for the mass profile @xmath121 of a1689 @xmath309 . the results are marginalized over all other parameters , including the observational parameters @xmath310 . the resulting posterior distributions are all clearly single - peaked , and approximately gaussian for most of the parameters . it is clearly evident that the mass - sheet degeneracy is broken thanks to the inclusion of magnification information on the local area distortion . excluding magnification data , on the other hand , strongly modifies the gaussian shape of the marginalized posterior pdfs , producing long non - gaussian tails and broadening the distribution function , resulting in large errors for the reconstructed mass profile . the improvement here from adding the magnification measurements is significant , @xmath10 in terms of cluster mass profile measurements ( see table [ tab : data ] ) . figure [ fig : kplot ] shows the lensing convergence profiles @xmath311 , for the five clusters reconstructed using our bayesian method from combined subaru distortion and magnification data . also shown for comparison are independent reconstructions from the same tangential distortion data ( but without the magnification data combined ) using the one - dimensional method of ( * ? ? ? * see also umetsu et al . 2010 ) based on the nonlinear extension of aperture mass densitometry , which employs an outer boundary condition on the mean convergence in the outermost radial bin , @xmath312 . here @xmath313 for an isolated nfw halo can be negligibly small if the outermost radii are taken as large as the cluster virial radius ( see * ? ? ? this method has been applied successfully to subaru weak lensing observations of massive clusters including a1689 @xcite , a1703 @xcite , and cl0024 + 17 @xcite . our results with different combinations of lensing measurements and boundary conditions , having different systematics , are in agreement with each other . this consistency clearly demonstrates that our results are robust and insensitive to the choice of boundary condition as well as to systematic errors in the lensing measurements , such as the shear calibration error , as found by @xcite . unlike the distortion effect , the magnification bias due to the local area distortion falls off sharply with increasing distance from the cluster center . we find from the reconstructed mass profiles that the lens convergence at large radii of @xmath314 $ ] is of the order @xmath315 . the expected level of the depletion signal in the weak - lensing limit is @xmath316 for a _ maximally_-depleted sample with @xmath317 , indicating a depletion signal of @xmath318 in the cluster outskirts where we have estimated the unlensed background counts , @xmath115 . this level of signal is smaller than the fractional uncertainties in estimated unlensed counts @xmath115 of @xmath319 , thus consistent with the assumption . note that the calibration uncertainties in our observational parameters @xmath310 have been properly taken into account and marginalized over in our bayesian analysis . in the presence of magnification , one probes the number counts at an effectively fainter limiting magnitude : @xmath320 . the level of magnification is on average small in the weak - lensing regime but for the innermost bin reaches a factor of 2 to 4 depending on the cluster . here we use the count slope at the fainter effective limit ( @xmath321 ) when making the magnification estimate , to be self - consistent . in our analysis we have implicitly assumed that the power - law behavior ( equation [ [ eq : magbias ] ] ) persists down to @xmath322 mag fainter than @xmath100 where the count slope may be shallower . for a given level of count depletion , an overestimation of the count slope could lead to an overestimation of the magnification , thus biasing the resulting mass profile . however , the number count slope for our data flattens only slowly with depth varying from @xmath108 to @xmath323 from a limit of @xmath324 to @xmath325 , so that this introduces a small correction of only typically 8%11% for the most magnified bins ( @xmath326 ) . in fact , we have found a good consistency between the purely shear - based results and the results based on the combined distortion and magnification data ( see figure [ fig : kplot ] ) . to quantify and characterize the cluster mass distribution , we compare the reconstructed @xmath38 profile with the physically and observationally motivated nfw model . here we consider a generalized parametrization of the nfw model of the following form @xcite : @xmath327 which has an arbitrary power - law shaped central cusp , @xmath328 , and an asymptotic outer slope of @xmath329 . this reduces to the nfw model for @xmath330 . we refer to the profile given by equation ( [ eq : gnfw ] ) as the generalized nfw ( gnfw , hereafter ) profile . it is useful to introduce the radius @xmath331 at which the logarithmic slope of the density is isothermal , i.e. , @xmath14 . for the gnfw profile , @xmath332 , and thus the corresponding concentration parameter reduces to @xmath333 . we specify the gnfw model with the central cusp slope , @xmath334 , the halo virial mass , @xmath335 , and the concentration , @xmath336 . we employ the radial dependence of the gnfw lensing profiles given by @xcite . we first fix the central cusp slope to @xmath330 ( nfw ) , and constrain @xmath337 from @xmath338 fitting to the discrete cluster mass profile @xmath304 reconstructed from the combined weak - lensing distortion and magnification measurements . the @xmath338 function for weak lensing is defined by @xmath339 c^{-1}_{ij } \big [ s_j-\hat{s}_j(m_{\rm vir},c_{\rm vir } ) \big],\ ] ] where @xmath340 is the nfw model prediction for the discrete mass profile @xmath121 . the resulting constraints on the nfw model parameters and the predicted einstein radius @xmath341 are shown in table [ tab : nfw ] . for all the cases , the best - fit nfw model from weak lensing properly reproduces the observed location of the einstein radius , consistent with the independent strong - lensing observations ( see table [ tab : rein ] ) . in table [ tab : data ] we quote the values of the total detection @xmath342 in the reconstructed mass profile @xmath121 based on the combined distortion and magnification data . @xmath210{f4a.eps } & \includegraphics[width=40mm , angle=270]{f4b.eps } & \includegraphics[width=40mm , angle=270]{f4c.eps } \\ \end{array } $ ] @xmath211{f4d.eps } & \includegraphics[width=40mm , angle=270]{f4e.eps}\\ \end{array } $ ] c|cc|cc|cc|cc a1689 & @xmath343 & @xmath344 & @xmath345 & @xmath346 & @xmath347 & @xmath348 & @xmath349 & @xmath350 + a1703 & @xmath351 & @xmath352 & @xmath353 & @xmath354 & @xmath355 & @xmath356 & @xmath357 & @xmath358 + a370 & @xmath359 & @xmath360 & @xmath361 & @xmath362 & @xmath363 & @xmath364 & @xmath365 & @xmath366 + cl0024 + 17 & @xmath367 & @xmath368 & @xmath369 & @xmath370 & @xmath371 & @xmath372 & @xmath373 & @xmath374 + rxj1347 - 11 & @xmath375 & @xmath376 & @xmath377 & @xmath378 & @xmath379 & @xmath380 & @xmath381 & @xmath382 our comprehensive bayesian analysis of the weak lensing distortion and magnification of background galaxies allows us to recover the mass normalization , given as the mean convergence @xmath383 within the innermost measurement radius @xmath241 ( @xmath384 ) , without employing inner strong lensing information . we use the non - parametric deprojection method of @xcite to derive for each cluster three - dimensional virial quantities ( @xmath385 ) and values of @xmath386 within a sphere of a fixed mean interior overdensity @xmath387 with respect to the critical density @xmath388 of the universe at the cluster redshift @xmath389 . we first deproject the two - dimensional mass profiles obtained in [ subsec : massrec ] and derive non - parametric three - dimensional mass profiles @xmath390 simply assuming spherical symmetry , following the method introduced by @xcite . this method is based on the fact that the surface - mass density @xmath391 is related to the three - dimensional mass density @xmath392 by an abel integral transform ; or equivalently , one finds that the three - dimensional mass @xmath390 out to spherical radius @xmath393 is written in terms of @xmath391 as @xmath394 where @xmath395 @xcite , ) , which can be readily avoided by a suitable coordinate transformation . ] and the first term of the right - hand side can be obtained as @xmath396 with @xmath397 . the errors are estimated from monte carlo simulations based on the full covariance matrix of the lensing convergence profile ( for details , see * ? ? ? * ) . in figure [ fig : m23 ] we show the non - parametric cumulative projected ( blue - hatched ) and spherical ( gray - shaded ) mass profiles , @xmath398 and @xmath399 , separately for the five clusters . table [ tab : m3d ] gives a summary of the spherical mass estimates @xmath400 corresponding to @xmath401 , and 2500 from our non - parametric deprojection analysis , where @xmath402 is the virial overdensity of the spherical collapse model evaluated at the cluster redshift @xmath389 . overall , we find a good agreement between the virial mass estimates from the parametric and non - parametric deprojection approaches ( see [ subsec : wl+sl ] for the results of gnfw fits to the combined weak and strong lensing data ) . our virial mass estimate of a1689 is obtained as @xmath403 from the combined distortion and magnification profiles , consistent with the results of @xcite , who combined strong lensing , weak lensing distortion and magnification data in a full two - dimensional analysis , and derived @xmath404 , where this @xmath32 error includes both statistical and systematic uncertainties ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? ( the errors represent only the statistical one ) from an entropy - regularized maximum - likelihood combination of subaru distortion and magnification data sets , in excellent agreement with the one dimensional results in this work . ] this is also in good agreement with the results from the recent high - resolution lensing observations by @xcite , @xmath405 . these lensing results are consistent with careful dynamical work by @xcite , who obtained a virial mass estimate of @xmath406 for a1689 . as found early by @xcite , our comprehensive weak - lensing analysis implies a370 is the most massive cluster now known , @xmath407 . our virial mass estimate is slightly higher than that derived in our earlier weak - lensing work combined with the inner einstein radius information , @xmath408 @xcite , where the difference is primarily due to our improved background selection and depth estimate presented in @xcite . for rxj1347 - 11 , we find its nfw virial mass is slightly overestimated compared to the non - parametric estimate of @xmath409 due to the projection of subclumps associated with the large scale structure around the cluster ( see * ? ? ? our non - parametric mass estimates are in good agreement with independent x - ray , dynamical and lensing analyses @xcite . for a1703 and cl0024 + 17 , our new mass estimates are fully consistent with our recent weak - lensing results derived from the subaru distortion data alone ( * ? ? ? * ; * ? ? ? * see also [ subsec : massrec ] ) . the statistical precision of lensing constraints can be further improved by stacking the signal from an ensemble of clusters with respect to their centers , providing average properties of cluster mass profiles . as discussed by @xcite , this stacking analysis has several important advantages . a notable advantage of the stacking analysis is that the resulting average profile is insensitive to the inherent asphericity and substructure ( in projection ) of individual cluster mass distributions , as well as to uncorrelated large - scale structure projected along the same line of sight . consequently , the statistical precision can be boosted by stacking together a number of clusters , especially on small angular scales ( see * ? ? ? to do this , we first define a new set of radial bands in which the mass profiles of individual clusters are re - evaluated for a stacking analysis . here we scale each cluster mass profile according to the cluster virial radius @xmath1 obtained by our non - parametric method ( see [ subsec : rvir ] ) . for each cluster , we construct an @xmath410 _ projection _ matrix @xmath411 that projects the mass profile @xmath412 ( @xmath84 ) of the cluster onto the new radial bands scaled in units of @xmath1 ( @xmath413 ) . assuming a constant density in each radial band , the projection matrix @xmath411 is uniquely specified by the conservation of mass . with this projection matrix , the mass profile in the new basis is written as @xmath414 accordingly , the error covariance matrix in the new basis is @xmath415 with the mass profiles of individual clusters on a common basis , we can stack the clusters to produce an averaged mass profile . here we re - evaluate the mass profiles of the individual clusters in @xmath416 logarithmically - spaced radial bins over the range of radii @xmath417r_{\rm vir}$ ] . since the noise in different clusters is uncorrelated , the mass profiles of individual clusters can be co - added according to ( e.g. , * ? ? ? * ) @xmath418 where the index @xmath419 runs over all of the clusters , and @xmath420 is the inverse critical surface mass density for the @xmath419th cluster , @xmath421 . the error covariance matrix for the stacked mass profile @xmath422 is obtained as @xmath423 where the index @xmath419 runs over all of the clusters . we show in the top panel of figure [ fig : stack ] the resulting model - independent average mass profile @xmath424 with its statistical @xmath32 uncertainty as a function of the scaled projected radius @xmath425 , obtained by stacking the five clusters using equations ( [ eq : stack ] ) and ( [ eq : covar_stack ] ) ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? we note that the effect of different cluster redshifts has been taken into account by proper error propagation in terms of the lensing efficiency functions ( @xmath420 ) of individual clusters to average over . the stacked mass profile exhibits a fairly smooth radial trend , and is detected at a high significance level of @xmath2 out to @xmath426 . in the bottom panel of figure [ fig : stack ] , we plot the logarithmic density slope @xmath427 of the stacked mass profile as a function of the scaled projected radius along with nfw model predictions with @xmath428 and @xmath429 . the logarithmic gradient of the average profile shows a slight steepening trend with increasing radius in projection , consistent with nfw profiles with @xmath430 . finally , we quote model - independent constraints on the average logarithmic density slope to be @xmath5 at @xmath431 and @xmath7 at @xmath432 for the five clusters . the subaru data allow the weak lensing profiles of individual clusters to be accurately measured in several independent radial bins in the subcritical regime ( @xmath433 ) . the projected mass profile can be unambiguously recovered on an individual cluster basis from combined weak - lensing shape distortion and magnification bias measurements . here we combine our weak - lensing profiles with detailed strong - lensing information for the inner @xmath27kpc region of these clusters , for which we have identified many new sets of multiple images from deep _ hst_/acs observations @xcite , for a full determination of the entire mass profiles of the five well - studied clusters . figure [ fig : fullmass ] shows a sample of joint mass profiles for our five clusters recovered over two decades of radius ranging from 10kpc@xmath236 to @xmath434kpc@xmath236 . note in this comparison we have excluded the central weak - lensing bin @xmath435 and the strong - lensing data points at radii overlapping with the subaru data . in each case , the weak and strong lensing are in excellent agreement where the data overlap ( typically around @xmath436kpc@xmath437 ) , and the joint mass profiles form well - defined radial profiles with a continuously - steepening radial trend from the central region to beyond the virial radius ( @xmath438 ) . our high - quality lensing data , covering the entire cluster , allow us to place useful constraints on the gnfw structure parameters ( [ subsec : lprof ] ) , namely , the central cusp slope @xmath334 as well as the nfw virial mass and concentration parameters . using our full lensing constraints , we obtain for each cluster the best - fit gnfw model as summarized in table [ tab : nfw ] . for the halo mass and concentration parameters ( @xmath439 ) , we find good agreement between the results with and without the inner strong - lensing profile combined ( for cl0024 + 17 , see also * ? ? ? our joint mass profiles for the entire cluster region are consistent with a generalized form of the nfw density profile with modest variations in the central cusp slope ( @xmath440 ) , except for the ongoing merger rxj1347 - 11 ( see * ? ? ? * ) , for which we find an unacceptable fit with a reduced @xmath338 of @xmath441 for @xmath442 degrees of freedom , due to various local deviations from the model , especially in the inner mass profile which is tightly constrained by strong lensing . the best - fit value of @xmath334 derived for the relaxed cluster a1703 is @xmath443 , being consistent with nfw ( @xmath330 ) , which is in excellent agreement with independent lensing results by @xcite , @xmath444 , and @xcite , @xmath445 . we have developed a new method for a direct reconstruction of the projected mass profile of galaxy clusters from combined weak - lensing distortion and magnification measurements ( [ subsec : gt ] and [ subsec : magbias ] ) within a bayesian statistical framework , which allows for a full parameter - space extraction of the underlying signal . this method applies to the full range of radius outside the einstein radius , where nonlinearity between the surface mass density and the observables extends to a radius of a few arcminutes , and recovers the absolute mass normalization . a proper bayesian statistical analysis is essential to explore the entire parameter space and investigate the parameter degeneracies ( [ subsec : bayesian ] ) , arising from the mass - sheet degeneracy ( [ sec : basis ] ) . this method can be readily generalized for a statistical analysis using stacked lensing profiles of a sample of clusters . we have applied our comprehensive lensing method to a sample of five high - mass clusters for which detailed strong - lensing information is readily available from _ hst_/acs observations . the deep subaru multi - band photometry , in conjunction with our background - selection techniques @xcite , allows for a secure selection of uncontaminated blue and red background populations . in each cluster , a strong depletion of the red galaxy counts has been detected at a significance level of @xmath254@xmath255 ( table [ tab : data ] ) . a comparison shows clear consistency between two independent lensing observables with different systematics , ensuring the reliability of our weak - lensing analysis ( figure [ fig : data ] ) . the combination of independent subaru distortion and magnification data breaks the mass - sheet degeneracy , as examined by our bayesian statistical analysis ( figure [ fig : post ] ) . excluding magnification data , on the other hand , strongly modifies the gaussian shape of the marginalized posterior pdfs , producing long non - gaussian tails and broadening the distribution function , resulting in large reconstruction errors . the improvement here from adding the magnification measurements is significant , @xmath10 in terms of cluster mass profile measurements ( table [ tab : data ] ) . we have formed a model - independent mass profile from stacking the clusters , which is detected at @xmath2 out to beyond the virial radius , @xmath3 . we found that the projected logarithmic slope , @xmath446 , steepens from @xmath5 at @xmath6 to @xmath7 at @xmath8 , consistent with nfw profiles with @xmath430 . we also obtained for each cluster inner strong - lensing based mass profiles from deep _ hst_/acs observations , which we have shown overlap well with the outer subaru - based profiles and together are well described by a generalized navarro - frenk - white profile , except for the ongoing merger rxj1347 - 11 @xcite , with modest variations in the central cusp slope ( @xmath440 ) , perhaps related to the dynamical state of the cluster . these high - mass lensing clusters with large einstein radii appear to be centrally concentrated in projection , as found in several other well studied massive clusters from careful lensing work . an accurate characterization of the observed sample is crucial for any cluster - based cosmological tests . it has been suggested that clusters selected with giant arcs represent a highly biased population in the context of @xmath11cdm . for those clusters with large einstein radii ( say , @xmath447 ) , a large statistical bias of about @xmath448 is derived from @xmath19-body simulations @xcite , representing the most triaxial cases . the mean level of mass concentration inferred from our weak lensing analysis is high , @xmath449 ( simply ignoring the mass and redshift dependencies ) , for our sample with @xmath450 , as compared to the @xmath11cdm predictions , @xmath451 , evaluated at the median redshift of our sample , @xmath452 . this represents an overall discrepancy of @xmath453 with respect to the predictions , without taking into account the effects of projection bias . this apparent discrepancy is also evident when the weak - lensing mass profiles are combined with the inner strong - lensing information from deep acs observations ( [ subsec : wl+sl ] ) . applying a bias correction of @xmath454 ( see * ? ? ? * ; * ? ? ? * ) , the discrepancy in @xmath455 is reduced to the @xmath456 level . another possible source of systematic errors in the mass profile determination is the cluster off - centering effect ( [ subsec : lprof ] ) . the effect is essentially the smoothing of the central lensing signal ( see * ? ? ? * ) , which flattens the recovered convergence profile below the offset scale , and therefore reduces the derived mass concentration ( @xmath457 ) and cusp slope ( @xmath334 ) parameters . for our sample of clusters , we found a cluster centering offset of typically @xmath458 using our detailed strong - lensing models . this level of centering offset ( @xmath459kpc@xmath236 ) is much smaller than the estimated values for the inner characteristic radius of our clusters , @xmath460 , and hence may not significantly affect our concentration measurements . however , this could potentially lead to an underestimation of the central cusp slope @xmath334 . our results are consistent with previous lensing work which similarly detected a concentration excess in the lensing - based measurements for strong - lensing clusters @xcite . our findings could imply , for these clusters , either substantial mass projected along the line of sight , due in part to intrinsic halo triaxiality , or a higher - than - expected concentration of dark matter . nevertheless , the overall level of uncertainties may be too large to robustly test the cdm predictions with the present sample . the forthcoming space telescope cluster survey , clash , will provide a definitive derivation of mass profiles for a larger , lensing - unbiased sample of relaxed x - ray clusters ( @xmath461kev ) , combining high - resolution panchromatic space imaging with deep , wide - field subaru weak - lensing observations , for a definitive determination of the representative mass profile of massive clusters . we thank the anonymous referee for a careful reading of the manuscript and and for providing invaluable comments . we are very grateful for discussions with doron lemze , sandor molnar , masahiro takada , masayuki tanaka , nobuhiro okabe , and sherry suyu , whose comments were very helpful . k.u . acknowledges yuji chinone for helpful comments on mcmc techniques . we thank nick kaiser for making the imcat package publicly available . the work is partially supported by the national science council of taiwan under the grant nsc97 - 2112-m-001 - 020-my3 . for the cluster magnification analysis , the masking effect due to bright cluster galaxies , bright foreground objects , and saturated objects has to be properly taken into account and corrected for @xcite . here we describe two different approaches for estimating the effect of masking in the magnification bias measurement . in this method ( method a ) , we describe each masking object as an ellipse specified by its structure parameters , such as the size , axis ratio , and position angle . sextractor provides such useful estimates of object properties ( magnitude , size , axis ratio , position angle , and so on ) for calculating masking areas and the resulting correction . in practice , we account for the masking of observed sky by excluding a generous area @xmath462 around each masking object , where @xmath463 and @xmath464 are defined as @xmath465 times the major ( a_image ) and minor axes ( b_image ) computed from sextractor , corresponding roughly to the isophotal detection limit ( see @xcite and @xcite ) . we then calculate the correction factor for this masking effect as a function of radius from the cluster center , and renormalize the number density accordingly . note that the actual choice of @xmath466 is insensitive to the resulting magnification measurement @xmath467 @xcite , as this effect will cancel out when taking a ratio of the local surface density @xmath468 to the unlensed mean number density @xmath115 ( see 5.5.3 of umetsu & broadhurst 2008 ) . however , this method is not suitable for estimating masking areas by saturated stars and stellar trails , which can not be described by simple ellipses . for this case , we manually place a rectangular region mask around each saturated object to exclude a proper area . this approach has been commonly adopted in previous studies of cluster magnification bias @xcite . in the top panel of figure [ fig : mask ] , we show as an example the masked region in a1689 based on the subaru @xmath154-band photometry , as obtained using method a ( see also umetsu & broadhurst 2008 ) . the gray area shows bright masking objects ( @xmath469 ab mag ) , each described as an ellipse . the black rectangular regions are masked areas for saturated stars and bright stellar trails . we show in figure [ fig : fmask ] the level of mask area correction in a1689 as a function of radius from the cluster center . the masking area is negligible ( a few % ) at large radii , and increases up to @xmath470 of the sky close to the cluster center . typically , the level of masking correction is found to be sufficiently small ( @xmath471 ) , so that the uncertainty in the masking correction is of the second order , and hence negligible . here we describe an alternative method ( method b ) for the masking correction which we have developed in this work . this method has practical advantages in comparison to method a : that is , it is easy to implement and can be fully automated once the configuration of analysis parameters is tuned , especially useful for large scale sky surveys such as the subaru hsc survey . in this approach , we use a pixelized flag image to exclude masking regions , namely those connected pixels that belong to bright foreground objects , bright cluster galaxies , saturated stars , and stellar trails . this can be done using sextractor s check - image output with @xmath472 . we tune sextractor configuration s detection parameters in order to optimize the detection for a particular object size and brightness . in our analysis of subaru images , the per - pixel detection threshold above the local sky background is set to @xmath453 ( @xmath473 ) , and the minimum number of connected pixels above threshold is set to @xmath474 ( @xmath475 ) with @xmath476 sampling . the bottom panel of figure [ fig : mask ] shows the distribution of masked regions in a1689 as obtained using method b. figure [ fig : fmask ] compares the masking correction factor in a1689 obtained using methods a and b , shown as a function of radius from the cluster center . it is shown that the two methods give consistent results in terms of both shape and amplitude of the radial profile of the correction factor . in this appendix , we aim to derive a discrete expression for the mean interior convergence @xmath477 using the azimuthally - averaged convergence @xmath77 . in the continuous limit , the mean convergence @xmath478 interior to radius @xmath79 can be expressed in terms of @xmath77 as @xmath479 for a given set of @xmath129 annular radii @xmath480 @xmath481 , defining @xmath19 radial bands in the range @xmath482 , a discretized estimator for @xmath483 can be written in the following way : @xmath484 with @xmath485 and @xmath486 being the area - weighted center of the @xmath87th annulus defined by @xmath487 and @xmath488 ; in the continuous limit , we have @xmath489 we derive expressions for the binned tangential distortion @xmath83 and magnification @xmath46 in terms of the binned convergence @xmath38 , using the following relations : @xmath490 where both the quantities depend on the mean convergence @xmath491 interior to the radius @xmath86 , which is the center of the @xmath87th radial band of @xmath88 $ ] ( see appendix [ appendix : avkappa ] ) . by assuming a constant density in each radial band and by noting that @xmath86 is the _ median _ radius of the @xmath87th radial band , @xmath492 can be well approximated by @xmath493 where @xmath494 and @xmath495 can be computed using the formulae given in appendices [ appendix : avkappa ] and [ appendix : lprof ] . if we are to combine background samples of different populations ( @xmath463 and @xmath464 ) and different lensing depths @xmath229 , we should take into account population - to - population variations in the lensing depth , @xmath496 , which can be measured from deep multi - band imaging , such as the 30-band cosmos database @xcite . the tangential distortion @xmath497 for population ( @xmath464 ) can be expressed in terms of the convergence @xmath498 for population ( @xmath463 ) as @xmath499 similarly , the magnification @xmath500 for population ( @xmath464 ) can be expressed in terms of @xmath501 . in this supplemental material section , we compare observed tangential - distortion ( shear ) @xmath70 and count - depletion ( magnification ) @xmath468 radial profiles with their respective reconstructed profiles , taking a370 at @xmath502 ( figure [ fig : a370 ] ) and cl0024 + 17 at @xmath503 ( figure [ fig : cl0024 ] ) as examples . we note that this supplemental material is added for the arxiv version only , and is not included in the published apj version ( umetsu et al . 2011 , apj , 729 , 127 ) .
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we directly construct model - independent mass profiles of galaxy clusters from combined weak - lensing distortion and magnification measurements within a bayesian statistical framework , which allows for a full parameter - space extraction of the underlying signal .
this method applies to the full range of radius outside the einstein radius , and recovers the absolute mass normalization .
we apply our method to deep subaru imaging of five high - mass ( @xmath0 ) clusters , a1689 , a1703 , a370 , cl0024 + 17 , and rxj1347 - 11 , to obtain accurate profiles to beyond the virial radius ( @xmath1 ) . for each cluster
the lens distortion and magnification data are shown to be consistent with each other , and the total signal - to - noise ratio of the combined measurements ranges from 13 to 24 per cluster .
we form a model - independent mass profile from stacking the clusters , which is detected at @xmath2 out to @xmath3 . the projected logarithmic slope @xmath4 steepens from @xmath5 at @xmath6 to @xmath7 at @xmath8 .
we also derive for each cluster inner strong - lensing based mass profiles from deep advanced camera for surveys observations with the _ hubble space telescope _ , which we show overlap well with the outer subaru - based profiles and together are well described by a generalized form of the navarro - frenk - white profile , except for the ongoing merger rxj1347 - 11 , with modest variations in the central cusp slope ( @xmath9 ) .
the improvement here from adding the magnification measurements is significant , @xmath10 in terms of cluster mass profile measurements , compared with the lensing distortion signal .
| 20,273 | 443 |
according to the principle of emergence , the rich properties and the many different forms of materials originate from the different ways in which the atoms are ordered in the materials . landau symmetry - breaking theory provides a general understanding of those different orders and resulting rich states of matter.@xcite it points out that different orders really correspond to different symmetries in the organizations of the constituent atoms . as a material changes from one order to another order ( i.e. , as the material undergoes a phase transition ) , what happens is that the symmetry of the organization of the atoms changes . for a long time , we believed that landau symmetry - breaking theory describes all possible orders in materials , and all possible ( continuous ) phase transitions . however , in last twenty years , it has become more and more clear that landau symmetry - breaking theory does not describe all possible orders . after the discovery of high @xmath0 superconductors in 1986,@xcite some theorists believed that quantum spin liquids play a key role in understanding high @xmath0 superconductors@xcite and started to introduce various spin liquids.@xcite despite the success of landau symmetry - breaking theory in describing all kinds of states , the theory can not explain and does not even allow the existence of spin liquids . this leads many theorists to doubt the very existence of spin liquids . in 1987 , in an attempt to explain high temperature superconductivity , an infrared stable spin liquid chiral spin state was discovered,@xcite which was shown to be perturbatively stable and exist as quantum phase of matter ( at least in a large @xmath1 limit ) . at first , not believing landau symmetry - breaking theory fails to describe spin liquids , people still wanted to use symmetry - breaking to describe the chiral spin state . they identified the chiral spin state as a state that breaks the time reversal and parity symmetries , but not the spin rotation symmetry.@xcite however , it was quickly realized that there are many different chiral spin states that have exactly the same symmetry , so symmetry alone is not enough to characterize different chiral spin states . this means that the chiral spin states contain a new kind of order that is beyond symmetry description.@xcite this new kind of order was named@xcite topological order . the key to identify ( and define ) new orders is to identify new universal properties that are beyond the local order parameters and long - range correlations used in the landau symmetry breaking theory . indeed , new quantum numbers , such as ground state degeneracy@xcite , the non - abelian berry s phase of degenerate ground states@xcite and edge excitations@xcite , were introduced to characterize ( and define ) the different topological orders in chiral spin states . recently , it was shown that topological orders can also be characterized by topological entanglement entropy.@xcite more importantly , those quantities were shown to be universal ( robust against any local perturbation of the hamiltonian ) for chiral spin states.@xcite the existence of those universal properties establishes the existence of topological order in chiral spin states . near the end of 1980 s , the existence of chiral spin states as a theoretical possibility , as well as their many amazing properties , such as fractional statistics,@xcite spin - charge separation,@xcite chiral gapless edge excitations,@xcite were established reliably , at least in the large @xmath1-limit introduced in . even non - abelian chiral spin states can be established reliably in the large @xmath1-limit.@xcite however , it took about 10 years to establish the existence of a chiral spin state reliably without using large @xmath1-limit ( based on an exactly soluble model on honeycomb lattice).@xcite soon after the introduction of chiral spin states , experiments indicated that high - temperature superconductors do not break the time reversal and parity symmetries . so chiral spin states do not describe high - temperature superconductors . thus the theory of topological order became a theory with no experimental realization . however , the similarity between chiral spin states and fractional quantum hall ( fqh ) states allows one to use the theory of topological order to describe different fqh states.@xcite just like chiral spin states , different fqh states all have the same symmetry and are beyond the landau symmetry - breaking description . also like chiral spin states , fqh states have ground state degeneracies@xcite that depend on the topology of the space.@xcite those ground state degeneracies are shown to be robust against any perturbations . thus , the different orders in different quantum hall states can be described by topological orders , and the topological order does have experimental realizations . the topology dependent ground state degeneracy , that signal the presence of topological order , is an amazing phenomenon . in fqh states , the correlation of any local operators are short ranged . this seems to imply that fqh states are `` short sighted '' and they can not know the topology of space which is a global and long - distance property . however , the fact that ground state degeneracy does depend on the topology of space implies that fqh states are not `` short sighted '' and they do find a way to know the global and long - distance structure of space . so , despite the short - range correlations of any local operators , the fqh states must contain certain hidden long - range correlation . but what is this hidden long - range correlation ? this will be one of the main topic of this paper . since high @xmath0 superconductors do not break the time reversal and parity symmetries , nor any other lattice symmetries , some people concentrated on finding spin liquids that respect all those symmetries and hoping one of those symmetric spin liquids hold the key to understand high @xmath0 superconductors . between 1987 and 1992 , many symmetric spin liquids were introduced and studied.@xcite the excitations in some of constructed spin liquids have a finite energy gap , while in others there is no energy gap . those symmetric spin liquids do not break any symmetry and , by definition , are beyond landau symmetry - breaking description . by construction , topological order only describes the organization of particles or spins in a gapped quantum state . so the theory of topological order only applies to gapped spin liquids . indeed , we find that the gapped spin liquids do contain nontrivial topological orders@xcite ( as signified by their topology dependent and robust ground state degeneracies ) and are described by topological quantum field theory ( such as @xmath2 gauge theory ) at low energies . one of the simplest topologically ordered spin liquids is the @xmath2 spin liquid which was first introduced in 1991.@xcite the existence of @xmath2 spin liquid as a theoretical possibility , as well as its many amazing properties , such as spin - charge separation,@xcite fractional mutual statistics,@xcite topologically protected ground state degeneracy,@xcite were established reliably ( at least in the large @xmath1-limit introduced in ) . later , kitaev introduced the famous toric code model which establishes the existence of the @xmath2 spin liquid reliably without using large @xmath1-limit.@xcite the topologically protected degeneracy of the @xmath2 spin liquid was used to perform fault - tolerant quantum computation . the study of high @xmath0 superconductors also leads to many gapless spin liquids . the stability and the existence of those gapless spin liquid were in more doubts than their gapped counterparts . but careful analysis in certain large @xmath1 limit do suggest that stable gapless spin liquids can exist.@xcite if we do believe in the existence of gapless spin liquids , then the next question is how to describe the orders ( the organizations of spins ) in those gapless spin liquids . if gapped quantum state can contain new type of orders that are beyond landau s symmetry breaking description , it is natural to expect that gapless quantum states can also contain new type of orders . but how to show the existence of new orders in gapless states ? just like topological order , the key to identify new orders is to identify new universal properties that are beyond landau symmetry description . clearly we can no longer use the ground state degeneracy to establish the existence of new orders in gapless states . to show the existence of new orders in gapless states , a new universal quantity projective symmetry group ( psg ) was introduced.@xcite it was argued that ( some ) psgs are robust against any local perturbations of the hamiltonian that do not change the symmetry of the hamiltonian.@xcite so through psg , we establish the existence of new orders even in gapless states . the new orders are called quantum order to indicate that the new orders are related to patterns of quantum entanglement in the many - body ground state.@xcite what is missed in landau theory so that it fails to describe those new orders ? what is the new feature in the organization of particles / spins so that the resulting order can not be described by symmetry ? to answer those questions , let us consider a simple quantum system which can be described with landau theory the transverse field ising model in two dimensions : @xmath3 , where @xmath4 , @xmath5 , and @xmath6 are the pauli matrices on site @xmath7 . in @xmath8 limit , the ground state of the system is an equal - weight superposition of all possible spin - up and spin - down states : @xmath9 , where @xmath10 label a particular spin - up ( @xmath11 ) and spin - down ( @xmath12 ) configuration . in the @xmath13 limit , the system has two degenerate ground states @xmath14 and @xmath15 . the transverse field ising model has a @xmath16 symmetry . the ground state @xmath17 respect such a symmetry while the ground state @xmath18 ( or @xmath19 ) break the symmetry . thus the small @xmath20 ground state @xmath17 and the small @xmath21 ground state @xmath18 describe different phases since they have different symmetries . we note that @xmath18 is the exact ground state of the transverse field ising model with @xmath22 . the state has no quantum entanglement since @xmath18 is a direct product of local states : @xmath23 where @xmath24 is an up - spin state at site @xmath7 . the state @xmath17 is the exact ground state of the transverse field ising model with @xmath25 . it is also a state with no quantum entanglement : @xmath26 where @xmath27 is a state with spin in @xmath28-direction at site @xmath7 . we see that the states ( or phases ) described by landau symmetry breaking theory has no quantum entanglement at least in the @xmath25 or @xmath22 limits . in the @xmath29 and @xmath30 limits , the two ground states of the two limits still represent two phases with different symmetries . however , in this case , the ground states are not unentangled . on the other hand since the ground states in the @xmath25 or @xmath22 limits have finite energy gaps and short - range correlations , a small @xmath20 or a small @xmath21 can only modify the states locally . thus we expect the ground states have only short - range entanglement . the above example , if generalized to other symmetry breaking states , suggests the following conjecture : _ _ if a gapped quantum ground state is described by landau symmetry breaking theory , then it has short - range quantum entanglement . _ _ the direct - product states and short - range entangled states only represent a small subset of all possible quantum many - body states . thus , according to the point of view of the above conjecture , we see that what is missed by landau symmetry breaking theory is long - range quantum entanglement . it is this long - range quantum entanglement that makes a state to have nontrivial topological / quantum order . however , mathematically speaking , the above conjecture is a null statement since the meaning of short - range quantum entanglement is not defined . in the following , we will try to find a more precise description ( or definition ) of short - range and long - range quantum entanglement . we will start with a careful discussion of quantum phases and quantum phase transitions . to give a precise definition of quantum phases , let us consider a local quantum system whose hamiltonian has a smooth dependence on a parameter @xmath31 : @xmath32 . the ground state average of a local operator @xmath33 of the system , @xmath34 , naturally also depend on @xmath31 . if the function @xmath34 , in the limit of infinite system size , has a singularity at @xmath35 for some local operators @xmath33 , then the system described by @xmath32 has a quantum phase transition at @xmath35 . after defining phase transition , we can define when two quantum ground states belong to the same phase : let @xmath36 be the ground state of @xmath37 and @xmath38 be the ground state of @xmath39 . if we can find a smooth path @xmath32 , @xmath40 that connect the two hamiltonian @xmath37 and @xmath39 such that there is no phase transition along the path , then the two quantum ground states @xmath36 and @xmath38 belong to the same phase . we note that `` connected by a smooth path '' define an equivalence relation between quantum states . a quantum phase is an equivalence class of such equivalence relation . such an equivalence class is called an universality class . if @xmath36 is the ground state of @xmath37 and in the limit of infinite system size all excitations above @xmath36 have a gap , then for small enough @xmath31 , we believe that the systems described by @xmath32 are also gapped.@xcite in this case , we can show that , the ground state of @xmath32 , @xmath41 , is in the same phase as @xmath36 , for small enough @xmath31 , the average @xmath34 is a smooth function of @xmath31 near @xmath42 for any local operator @xmath33.@xcite after scaling @xmath31 to @xmath43 , we find that : @xmath44 $ , then there is no phase transition along the path $ g$. } } \ ] ] in other words , for gapped system , a quantum phase transition can happen only when energy gap closes@xcite ( see fig . [ trans ] ) . is finite for all @xmath31 in the segment @xmath45 $ ] , show that the algebra of local operators for different @xmath31 s are isomorphic to each other at low energies , using a quasi - adiabatic continuation of quantum states . ] here , we would like to assume that the reverse is also true : a closing of the energy gap for a gapped state always induces a phase transition . or more precisely @xmath46 $ , and $ |\phi(0)\>$ and $ |\phi(1)\>$ are ground states of $ h(0)$ and $ h(1)$ respectively . } } \ ] ] the above two boxed statements imply that two gapped quantum states are in the same phase @xmath47 if and only if they can be connected by an adiabatic evolution that does not close the energy gap . given two states , @xmath36 and @xmath38 , determining the existence of such a gapped adiabatic connection can be hard . we would like to have a more operationally practical equivalence relation between states in the same phase . here we would like to show that _ two gapped states @xmath36 and @xmath38 are in the same phase , if and only if they are related by a local unitary ( lu ) evolution . _ we define a local unitary(lu ) evolution as an unitary operation generated by time evolution of a local hamiltonian for a finite time . that is , @xmath48 |\phi(0)\>\end{aligned}\ ] ] where @xmath49 is the path - ordering operator and @xmath50 is a sum of local hermitian operators . note that @xmath51 is in general different from the adiabatic path @xmath32 that connects the two states . first , assume that two states @xmath36 and @xmath38 are in the same phase , therefore we can find a gapped adiabatic path @xmath32 between the states . the existence of a gap prevents the system to be excited to higher energy levels and leads to a local unitary evolution , the quasi - adiabatic continuation as defined in , that maps from one state to the other . that is , @xmath52\ ] ] the exact form of @xmath51 is given in and will be discussed in more detail in appendix . in the hamiltonian . ( a , b ) for gapped system , a quantum phase transition can happen only when energy gap closes . ( a ) describes a first order quantum phase transition ( caused by level crossing ) . ( b ) describes a continuous quantum phase transition which has a continuum of gapless excitations at the transition point . ( c ) and ( d ) can not happen for generic states . a gapped system may have ground state degeneracy , where the energy splitting between the ground states vanishes when system size @xmath53 : @xmath54 . the energy gap @xmath55 between ground and excited states on the other hand remains finite as @xmath53 . ] on the other hand , the reverse is also true : _ if two gapped states @xmath36 and @xmath38 are related by a local unitary evolution , then they are in the same phase . _ since @xmath36 and @xmath38 are related by a local unitary evolution , we have @xmath56 |\phi(0)\ > $ ] . let us introduce @xmath57 .\end{aligned}\ ] ] assume @xmath36 is a ground state of @xmath37 , then @xmath58 is a ground state of @xmath59 . if @xmath60 remains local and gapped for all @xmath61 $ ] , then we have found an adiabatic connection between @xmath36 and @xmath38 . first , let us show that @xmath60 is a local hamiltonian . since @xmath62 is a local hamiltonian , it has a form @xmath63 where @xmath64 only acts on a cluster whose size is @xmath65 . @xmath65 is called the range of interaction of @xmath62 . we see that @xmath60 has a form @xmath66 , where @xmath67 . to show that @xmath68 only acts on a cluster of a finite size , we note that for a local system described by @xmath51 , the propagation velocities of its excitations have a maximum value @xmath69 . since @xmath70 can be viewed as the time evolution of @xmath64 by @xmath71 from @xmath72 to @xmath73 , we find that @xmath68 only acts on a cluster of size @xmath74,@xcite where @xmath75 is the range of interaction of @xmath76 . thus @xmath60 are indeed local hamiltonian . if @xmath62 has a finite energy gap , then @xmath60 also have a finite energy gap for any @xmath77 . as @xmath77 goes for @xmath78 to @xmath43 , the ground state of the local hamiltonians , @xmath60 , goes from @xmath36 to @xmath38 . thus the two states @xmath36 and @xmath38 belong to the same phase . this completes our argument that states related by a local unitary evolution belong to the same phase . thus through the above discussion , we show that @xmath79 a more detailed and more rigorous discussion of this equivalence relation is given in appendix a , where exact bounds on locality and transformation error is given . the relation defines an equivalence relation between @xmath36 and @xmath38 . the equivalence classes of such an equivalence relation represent different quantum phases . so the above result implies that the equivalence classes of the lu evolutions are the universality classes of quantum phases for gapped states . using the lu evolution , we can obtain a more precise description ( or definition ) of short - range entanglement : @xmath80 if a state can not be transformed into an unentangled state through a lu evolution , then the state has long - range entanglement . we also see that @xmath81 thus all states with short - range entanglement belong to the same phase . the local unitary evolutions we consider here do not have any symmetry . if we require certain symmetry of the local unitary evolutions , states with short - range entanglement may belong to different symmetry breaking phases , which will be discussed in section [ symtop ] . since a direct - product state is a state with trivial topological order , we see that a state with a short - range entanglement also has a trivial topological order . this leads us to conclude that a non - trivial topological order is related to long - range entanglement . since two gapped states related by a lu evolution belong to the same phase , thus two gapped states related by a local unitary evolution have the same topological order . in other words , @xmath82 or more pictorially , topological order is a pattern of long - range entanglement . in , it was shown that the `` topologically non - trivial '' ground states , such as the toric code,@xcite can not be changed into a `` topologically trivial '' state such as a product state by any unitary locality - preserving operator . in other words , those `` topologically non - trivial '' ground states have long - range entanglement . . the green shading represents a causal structure . ] the lu evolutions introduced here is closely related to _ quantum circuits with finite depth_. to define quantum circuits , let us introduce piece - wise local unitary operators . a piece - wise local unitary operator has a form @xmath83 where @xmath84 is a set of unitary operators that act on non overlapping regions . the size of each region is less than some finite number @xmath85 . the unitary operator @xmath86 defined in this way is called a piece - wise local unitary operator with range @xmath85 . a quantum circuit with depth @xmath87 is given by the product of @xmath87 piece - wise local unitary operators ( see fig . [ qc ] ) : @xmath88 . in quantum information theory , it is known that finite time unitary evolution with local hamiltonian ( lu evolution defined before ) can be simulated with constant depth quantum circuit and vice - verse . therefore , the equivalence relation can be equivalently stated in terms of constant depth quantum circuits : @xmath89 where @xmath87 is a constant independent of system size . because of their equivalence , we will use the term `` local unitary transformation '' to refer to both local unitary evolution and constant depth quantum circuit in general . the lu transformation defined through lu evolution is more general . it can be easily generalized to study topological orders and quantum phases with symmetries ( see section [ symtop]).@xcite the quantum circuit has a more clear and simple causal structure . however , the quantum circuit approach breaks the translation symmetry . so it is more suitable for studying quantum phases that do not have translation symmetry . in fact , people have been using quantum circuits to classify many - body quantum states which correspond to quantum phases of matter . in , the local unitary transformations described by quantum circuits was used to define a renormalization group transformations for states and establish an equivalence relation in which states are equivalent if they are connected by a local unitary transformation . such an approach was used to classify 1d matrix product states . in , the local unitary transformations with disentanglers was used to perform a renormalization group transformations for states , which give rise to the multi - scale entanglement renormalization ansatz ( mera ) in one and higher dimensions . the disentanglers and the isometries in mera can be used to study quantum phases and quantum phase transitions in one and higher dimensions . later in this paper , we will use the quantum circuit description of lu transformations to classify 2d topological orders through classifying the fixed - point lu transformations . in the above discussions , we have defined phases without any symmetry consideration . the @xmath51 or @xmath86 in the lu transformation does not need to have any symmetry and can be sum / product of any local operators . in this case , two hamiltonians with an adiabatic connection are in the same phase even if they may have different symmetries . also , all states with short - range entanglement belong to the same phase ( under the lu transformations that do not have any symmetry ) . on the other hand , we can consider only hamiltonians @xmath62 with certain symmetries and define phases as the equivalent classes of symmetric local unitary transformations : @xmath90 where @xmath51 or @xmath91 has the same symmetries as @xmath62 . we note that the symmetric local unitary transformation in the form @xmath92 always connect to the identity transformation continuously . this may not be the case for the transformation in the form @xmath91 . to rule out that possibility , we define symmetric local unitary transformations as those that connect to the identity transformation continuously . the equivalent classes of the symmetric lu transformations have very different structures compared to those of lu transformations without symmetry . each equivalent class of the symmetric lu transformations is smaller and there are more kinds of classes , in general . without any symmetry . ( b ) the possible phases for a hamiltonian @xmath93 with some symmetries . the shaded regions in ( a ) and ( b ) represent the phases with short range entanglement ( those ground states can be transformed into a direct product state via a generic lu transformations that do not have any symmetry . ) ] fig . [ topsymm ] compares the structure of phases for a system without any symmetry and a system with some symmetry in more detail . for a system without any symmetry , all the short - range - entangled ( sre ) states ( those ground states can be transformed into a direct product state via a generic lu transformations that do not have any symmetry ) are in the same phase ( sre in fig . [ topsymm](a ) ) . on the other hand , long range entanglement ( lre ) can have many different patterns that give rise to different topological phases ( lre 1 and lre 2 in fig . [ topsymm](a ) ) . the different topological orders usually give rise to quasiparticles with different fractional statistics and fractional charges . for a system with some symmetries , the phase structure can be much more complicated . the short - range - entangled states no longer belong to the same phase , since the equivalence relation is described by more special symmetric lu transformations : + ( a ) states with short range entanglement belong to different equivalent classes of the symmetric lu transformations if they have different broken symmetries . they correspond to the symmetry - breaking ( sb ) short - range - entanglement phases sb - sre 1 and sb - sre 2 in fig . [ topsymm](b ) . they are landau s symmetry breaking states . + ( b ) states with short range entanglement can belong to different equivalent classes of the symmetric lu transformations even if they do not break any symmetry of the system . ( in this case , they have the same symmetry . ) they correspond to the symmetric ( sy ) short - range - entangled phases sy - sre 1 and sy - sre 2 in fig . [ topsymm](b ) . we say those states have symmetry protected topological orders . haldane phase@xcite and @xmath94 phase of spin-1 chain are examples of states with the same symmetry which belong to two different equivalent classes of symmetric lu transformations ( with parity symmetry).@xcite band and topological insulators@xcite are other examples of states that have the same symmetry and at the same time belong to two different equivalent classes of symmetric lu transformations ( with time reversal symmetry ) . also , for a system with some symmetries , the long - range - entangled states are divided into more classes ( more phases ) : + ( c ) symmetry breaking and long range entanglement can appear together in a state , such as sb - lre 1 , sb - lre 2 , in fig . [ topsymm](b ) . the topological superconducting states are examples of such phases.@xcite + ( d ) long - range - entangled states that do not break any symmetry can also belong to different phases such as the symmetric long - range - entanglement phases sy - lre 1 , sy - lre 2 , in fig . [ topsymm](b ) . the many different @xmath2 symmetric spin liquids with spin rotation , translation , and time - reversal symmetries are examples of those phases.@xcite some time - reversal symmetric topological orders were also called topological mott - insulators or fractionalized topological insulators.@xcite after defining topological order as the equivalent classes of many - body wave functions under lu transformations , we like to ask : how to describe ( or label ) the different equivalent classes ( the different topological orders or patterns of long - range entanglement ) ? one simple way to do so is to use the full wave function which completely describe the different topological orders . but the full wave functions contain a lot of non - universal short - range entanglement . as a result , such a labeling scheme is a very inefficient many - to - one labeling scheme of topological orders . to find a more efficient or even one - to - one labeling scheme , we need to remove the non - universal short - range entanglement . into a direct - product state , if and only if the state @xmath95 has no long - range quantum entanglement . here , a dot represents a site with physical degrees of freedom . a vertical line carries an index that label the different physical states on a site . the presence of horizontal lines between dots represents quantum entanglement . ] into a direct product . removing / adding the degrees of freedom in the form of direct product defines an additional equivalence relation that defines the topological order ( or classes of long - range entanglement ) . ] as the first application of the notion of lu transformation , we would like to describe a wave function renormalization group flow introduced in , . the wave function renormalization can remove the short - range entanglement and simplify the wave function . in , the wave function renormalization for string - net states is generated by the following two basic moves @xmath96{xikljo } \epm = & \delta_{ij } \phi \bpm \includegraphics[height=0.3in]{xi0 } \epm \\ \label{fmv } \phi \bpm \includegraphics[height=0.3in]{xijklmxo } \epm = & \sum_{n } f^{jim^*}_{lk^*n^ * } \phi \bpm \includegraphics[height=0.3in]{xijklnxo } \epm\end{aligned}\ ] ] ( note that the definition of the f - tensor in is slightly different from the definition in this paper . ) the two basic moves can generate a generic wave function renormalization which can reduce the string - net wave functions to very simple forms.@xcite later in , the wave function renormalization for generic states was discussed in a more general setting , and was called mera . the two basic string - net moves and correspond to the isometry and the disentangler in mera respectively . in the mera approach , the isometries and the disentanglers are applied in a layered fashion , while in the string - net approach , the two basic moves can be applied arbitrarily . in this section , we will follow the mera setup to describe the wave function renormalization . later in this paper , we will follow the string - net setup to study the fixed - point wave functions . note that we can use a lu transformation @xmath97 to transform some degrees of freedom in a state into direct product ( see fig . [ entre ] ) . we then remove those degrees of freedom in the form of direct product . such a procedure does not change the topological order . the reverse process of adding degrees of freedom in the form of direct product also does not change the topological order . we call the local transformation in fig . [ entre ] that changes the degrees of freedom a generalized local unitary ( glu ) transformation . it is clear that a generalized local unitary transformation inside a region @xmath98 does not change the reduced density matrix @xmath99 for the region @xmath98 . this is the reason why we say that ( generalized ) local unitary transformations can not change long - range entanglement and topological order . similarly , the addition or removal of decoupled degrees of freedom to or from the hamiltonian , @xmath100 , will not change the phase of the hamiltonian ( the ground states of @xmath62 and @xmath101 are in the same phase ) , if those degrees of freedom form a direct product state ( the ground state of @xmath102 is a direct product state ) . let us define the glu transformation @xmath97 more carefully and in a more general setting . consider a state @xmath95 . let @xmath99 be the reduced density matrix of @xmath95 in region @xmath98 . @xmath99 may act in a subspace of the total hilbert space @xmath103 in region a , which is called the support space @xmath104 of region @xmath98 . the dimension @xmath105 of @xmath104 is called support dimension of region @xmath98 . now the hilbert space @xmath103 in region a can be written as @xmath106 . let @xmath107 , @xmath108 be a basis of this support space @xmath104 , @xmath107 , @xmath109 be a basis of @xmath110 , where @xmath111 is the dimension of @xmath103 , and @xmath112 , @xmath113 be a basis of @xmath103 . we can introduce a lu transformation @xmath114 which rotates the basis @xmath112 to @xmath107 . we note that in the new basis , the wave function only has non - zero amplitudes on the first @xmath105 basis vectors . thus , in the new basis @xmath107 , we can reduce the range of the label @xmath115 from @xmath116 $ ] to @xmath117 $ ] without losing any information . this motivates us to introduce the glu transformation as a rotation from @xmath112 , @xmath113 to @xmath118 , @xmath108 . the rectangular matrix @xmath97 is given by @xmath119 . we also regard the inverse of @xmath97 , @xmath120 , as a glu transformation . a lu transformation is viewed as a special case of glu transformation where the degrees of freedom are not changed . clearly @xmath121 and @xmath122 are two projectors . the action of @xmath123 does not change the state @xmath95 ( see fig . [ glut](b ) ) . we note that despite the reduction of the degrees of freedom , a glu transformation defines an equivalent relation . two states related by a glu transformation belong to the same phase . the renormalization flow induced by the glu transformations always flows within the same phase . acts in region a of a state @xmath95 , which reduces the degree freedom in region a to those contained only in the support space of @xmath95 in region a. ( b ) @xmath121 is a projector that does not change the state @xmath95 . ] as an application of the wave function renormalization , in this section , we will study the structure of fixed - point wave functions under the wave function renormalization , which will lead to a classification of topological order ( without any symmetry ) . we note that as wave functions flow to a fixed point , the glu transformations in each step of the renormalization also flow to a fixed point . so instead of studying fixed - point wave functions , here , we will study the fixed - point glu transformations . for this purpose , we need to fix the renormalization scheme . in the following , we will discuss a renormalization scheme motivated by the string - net wave function.@xcite after we specify a proper wave function renormalization scheme , then the fixed - point wave function is simply the wave function whose `` form '' does not change under the wave function renormalization . since those fixed - point glu transformations do not change the fixed - point wave function , their actions on the fixed - point wave function do not depend on the order of the actions . this allows us to obtain many conditions that glu transformations must satisfy . from those conditions , we can determine the forms of allowed fixed - point glu transformations . this leads to a general description and a classification of topological orders and their corresponding fixed - point wave functions . the renormalization scheme that we will discuss was first used in to characterize the scale invariant string - net wave function . it is also used in to simplify the string - net state in a region , which allows us to calculate the entanglement entropy of the string - net state exactly . a similar approach was used in to show quantum - double / string - net wave function to be a fixed - point wave function and its connection to 2d mera.@xcite in the following , we will generalize those discussions by not starting with string - net wave functions . we just try to construct local unitary transformations at a fixed point . we will see that the fixed - point conditions on the glu transformations lead to a mathematical structure that is similar to the tensor category theory the mathematical framework behind the string - net states . since the wave function renormalization may change the lattice structure , we will consider quantum states defined on a generic trivalence graph @xmath124 : each edge has @xmath125 states , labeled by @xmath126 ( see fig . [ nijkv ] ) . we assume that the index @xmath115 on the edge admits an one - to - one mapping @xmath127 : @xmath128 that satisfies @xmath129 . as a result , the edges of the graph are oriented . the mapping @xmath128 corresponds to the reverse of the orientation of the edge ( see fig . [ llstar ] ) . each vertex also has physical states , labeled by @xmath130 ( see fig . [ nijkv ] ) . each labeled graph ( see fig . [ nijkv ] ) corresponds to a state and all the labeled graphs form an orthonormal basis . our fixed - point state is a superposition of those basis states : @xmath131{strnet}\emm \right ) \left |\bmm \includegraphics[scale=0.2]{strnet}\emm\right \>$ ] . here we will make an important assumption about the fixed - point wave function . we will assume that the fixed - point wave function is `` topological '' : two labeled graphs have the same amplitude if the two labeled graphs can be deformed into each other continuously on the plane without the vertices crossing the links . for example @xmath132{gs1}\epm = \psi_\text{fix}\bpm \includegraphics[scale=0.25]{gs2}\epm $ ] . due to such an assumption , the topological orders studied in this paper may not be most general . we also assume that all the fixed - point states on each different graphs to have the same `` form '' . this assumption is motivated by the fact that during wave function renormalization , we transform a state on one graph to a state on a different graph . the `` fixed - point '' means that the wave functions on those different graphs are all determined by the same collection of the rules , which defines the meaning of having the same `` form '' . however , the wave function for a given graph can have different total phases if the wave function is calculated by applying the rules in different orders . those rules are noting but the fixed - point glu transformations . . there are @xmath125 states on each edge which are labeled by @xmath133 . there are @xmath134 states on each vertex which are labeled by @xmath130 . ] corresponds to the reverse of the orientation of the edge ] . if the three edges of a vertex are in the states @xmath115 , @xmath135 , and @xmath136 respectively , then the vertex has @xmath137 states , labeled by @xmath138 . note the orientation of the edges are point towards to vertex . also note that @xmath139 runs anti - clockwise . ] before describing the wave function renormalization , let us examine the structure of entanglement of a fixed - point wave function . first , let us consider a fixed - point wave function @xmath140 on a graph . we examine the wave function on a patch of the graph , for example , @xmath141{f1g}\emm$ ] . the fixed - point wave function @xmath142{f1g}\epm$ ] ( only the relevant part of the graph is drawn ) can be viewed as a function of @xmath143 : @xmath144{f1 g } \epm$ ] if we fix @xmath145 and the indices on other part of the graph . ( here the indices on other part of the graph is summarized by @xmath146 . ) as we vary the indices @xmath146 on other part of graph ( still keeping @xmath115 , @xmath135 , @xmath136 , and @xmath85 fixed ) , the wave function of @xmath143 , @xmath147 , may change . all those @xmath147 form a linear space of dimension @xmath148 . @xmath149 is an important concept that will appear later . we note that the two vertices @xmath150 and @xmath151 and the link @xmath152 form a region surrounded by the links @xmath145 . so we will call the dimension-@xmath148 space the support space @xmath153 and @xmath148 the support dimension for the state @xmath140 on the region surrounded by the fixed boundary state @xmath145 . similarly , we can define @xmath154 as the support dimension of the @xmath142{ijkal}\epm$ ] on a region bounded by links @xmath155 . since the region contains only a single vertex @xmath150 with @xmath134 physics states , we have @xmath156 . we can use a local unitary transformation on the vertex to reduce the range of @xmath150 to @xmath157 where @xmath158 . in the rest of this paper , we will implement such a reduction . so , the number of physical states on a vertex depend on the physical states of the edges that connect to the vertex . if the three edges of a vertex are in the states @xmath115 , @xmath135 , and @xmath136 respectively , then the vertex has @xmath137 states , labeled by @xmath138 ( see fig . [ nijk ] ) . here we assume that @xmath159 we note that in the fixed - point wave function @xmath142{f1g}\epm$ ] , the number of choices of @xmath143 is @xmath160 . thus the support dimension @xmath148 satisfies @xmath161 . here we will make an important assumption the saturation assumption : _ the fixed - point wave function saturate the inequality : _ @xmath162 we will see that the entanglement structure described by such a saturation assumption is invariant under the wave function renormalization . our wave function renormalization scheme contains two types of renormalization . the first type of renormalization does not change the degrees of freedom and corresponds to a local unitary transformation . it corresponds to locally deform the graph @xmath163{f1 g } \emm$ ] to @xmath163{f2 g } \emm$ ] . ( the parts that are not drawn are the same . ) the fixed - point wave function on the new graph is given by @xmath142{f2g}\epm$ ] . again , such a wave function can be viewed as a function of @xmath164 : @xmath165{f2 g } \epm$ ] if we fix @xmath145 and the indices on other part of the graph . the support dimension of the state @xmath142{f2g}\epm$ ] on the region surrounded by @xmath145 is @xmath166 . again @xmath167 , where @xmath168 is number of choices of @xmath164 . the saturation assumption implies that @xmath169 . the two fixed - point wave functions @xmath142{f1g}\epm$ ] and @xmath142{f2g}\epm$ ] are related via a local unitary transformation . thus @xmath170 , which implies @xmath171 we express such an unitary transformation as @xmath172 or graphically as @xmath173{f1 g } \epm \simeq \sum_{n=0}^n \sum_{\chi=1}^{n_{kjn^ * } } \sum_{\del=1}^{n_{nil^ * } } f^{ijm,\al\bt}_{kln,\chi\del } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2 g } \epm .\end{aligned}\ ] ] where @xmath174 means equal up to a constant phase factor . ( note that the total phase of the wave function is unphysical . ) we will call such a wave function renormalization step a f - move . we would like to remark that relates two wave functions on two graphs @xmath175 and @xmath176 which only differ by a local reconnection . we can choose the phase of @xmath177 to make @xmath174 into @xmath178 : @xmath179{f1 g } \epm = \sum_{n=0}^n \sum_{\chi=1}^{n_{kjn^ * } } \sum_{\del=1}^{n_{nil^ * } } f^{ijm,\al\bt}_{kln,\chi\del } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2 g } \epm .\end{aligned}\ ] ] but such choice of phase only works for a particular pair of graphs @xmath175 and @xmath176 . to use @xmath177 to relate all pair of states that only differ by a local reconnection , in general , we may have a phase ambiguity , with the value of phase depend on the pair of graphs . so can only be a relation up to a total phase factor . for fixed @xmath115 , @xmath135 , @xmath136 , and @xmath85 , the matrix @xmath180 with matrix elements @xmath181 is a @xmath182 dimensional matrix ( see ) . the mapping @xmath183 generated by the matrix @xmath180 is unitary . since , as we change @xmath146 , @xmath184 and @xmath147 span two @xmath182 dimensional spaces . thus @xmath180 is a @xmath185 unitary matrix @xmath186 where @xmath187 when @xmath188 , @xmath189 , @xmath190 , and @xmath191 otherwise . we can deform @xmath192{f1 g } \emm$ ] to @xmath192{f1ga } \emm$ ] , and @xmath192{f2 g } \emm$ ] to @xmath192{f2ga } \emm$ ] . we see that @xmath193{f1ga } \epm \simeq \sum_{n=0}^n \sum_{\chi,\del } f^{kl^*m^*,\bt\al}_{ij^*n^*,\del\chi } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2ga } \epm .\end{aligned}\ ] ] eqn . ( [ ihwave ] ) and eqn . ( [ ihwavea ] ) relate the same pair of graphs , and thus @xmath194 ( where we have used the condition @xmath195 . ) using the relation , we can rewrite as @xmath196{f2 g } \epm \simeq \sum_{m=0}^n \sum_{\al,\bt } ( f^{ijm,\al\bt}_{kln,\chi\del})^ * \phi_\text{fix } \bpm \includegraphics[scale=.40]{f1 g } \epm .\end{aligned}\ ] ] we can also express @xmath197{f2 g } \epm $ ] as @xmath198{f2 g } \epm \simeq \sum_{m=0}^n \sum_{\al,\bt } f^{jkn,\chi\del}_{l^*i^*m^*,\bt\al } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f1 g } \epm\end{aligned}\ ] ] using the relabeled . so we have @xmath199 since the total phase of the wave function is unphysical , the total phase of @xmath177 can be chosen arbitrarily . we can choose the total phase of @xmath177 to make @xmath200 if we apply twice , we reproduce . thus is not independent and can be dropped . from the graphic representation , we note that @xmath201 when @xmath202 or @xmath203 , the left - hand - side of is always zero . thus @xmath204 when @xmath202 or @xmath205 . when @xmath206 or @xmath207 , wave function on the right - hand - side of is always zero . so we can choose @xmath204 when @xmath206 or @xmath207 . the f - move maps the wave functions on two different graphs through a local unitary transformation . since we can locally transform one graph to another graph through different paths , the f - move must satisfy certain self consistent condition . for example the graph @xmath208{pent1 g } \emm $ ] can be transformed to @xmath209{pent3 g } \emm $ ] through two different paths ; one contains two steps of local transformations and the other contains three steps of local transformations as described by . the two paths lead to the following relations between the wave functions : @xmath210{pent1 g } \epm & \simeq \sum_{q,\del,\eps } f^{mkn,\bt\chi}_{lpq,\del\eps } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent2 g } \epm \simeq \sum_{q,\del,\eps;s,\phi,\ga } f^{mkn,\bt\chi}_{lpq,\del\eps } f^{ijm,\al\eps}_{qps,\phi\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent3 g } \epm , \end{aligned}\ ] ] @xmath211{pent1 g } \epm & \simeq \sum_{t,\eta,\vphi } f^{ijm,\al\bt}_{knt,\eta\vphi } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent4 g } \epm \simeq \sum_{t,\eta,\vphi;s,\ka,\ga } f^{ijm,\al\bt}_{knt,\eta\vphi } f^{itn,\vphi\chi}_{lps,\ka\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent5 g } \epm \nonumber\\ & \simeq \sum_{t,\eta,\ka;\vphi;s,\ka,\ga;q,\del,\phi } f^{ijm,\al\bt}_{knt,\eta\vphi } f^{itn,\vphi\chi}_{lps,\ka\ga } f^{jkt,\eta\ka}_{lsq,\del\phi } \phi_\text{fix } \bpm \includegraphics[scale=.40]{pent3 g } \epm .\end{aligned}\ ] ] the consistence of the above two relations leads a condition on the @xmath212 tensor . to obtain such a condition , let us fix @xmath115 , @xmath135 , @xmath136 , @xmath85 , @xmath213 , and view @xmath214{pent1 g } \epm$ ] as a function of @xmath215 : @xmath216{pent1 g } \epm$ ] . as we vary indices on other part of graph , we obtain different wave functions @xmath217 which form a dimension @xmath218 space . in other words , @xmath218 is the support dimension of the state @xmath140 on the region @xmath215 with boundary state @xmath219 fixed ( see the discussion in section [ entstru ] ) . since the number of choices of @xmath215 is @xmath220 , we have @xmath221 . here we require a similar saturation condition as in : @xmath222 similarly , the number of choices of @xmath223 in @xmath214{pent3 g } \epm $ ] is also @xmath224 . here we again assume @xmath225 , where @xmath226 is the support dimension of @xmath214{pent3 g } \epm $ ] on the region bounded by @xmath219 . so the two relations and can be viewed as two relations between a pair of vectors in the two @xmath218 dimensional vector spaces . as we vary indices on other part of graph ( still keeping @xmath219 fixed ) , each vector in the pair can span the full @xmath218 dimensional vector space . so the validity of the two relations and implies that @xmath227 which is a generalization of the famous pentagon identity ( due to the extra constant phase factor @xmath228 ) . we will call such a relation projective pentagon identity . the projective pentagon identity is a set of nonlinear equations satisfied by the rank-10 tensor @xmath177 and @xmath229 . the above consistency relation is equivalent to the requirement that the local unitary transformations described by on different paths all commute with each other up to a total phase factor . the second type of wave function renormalization does change the degrees of freedom and corresponds to a generalized local unitary transformation . one way to implement the second type renormalization is to reduce @xmath208{4to1a } \emm $ ] to @xmath192{4to1b } \emm $ ] ( the part of the graph that is not drawn is unchanged ) : @xmath230{4to1a } \epm \simeq \sum_{\la=1}^{n_{ijk } } f ^{abc,\al\bt\ga } _ { ijk,\la } \phi_\text{fix } \bpm \includegraphics[scale=.40]{4to1b } \epm .\end{aligned}\ ] ] but we can define a simpler second type renormalization , by noting that @xmath163{4to1a } \emm$ ] can be reduced to @xmath163{tpolij } \emm$ ] via the first type of renormalization steps ( see fig . [ tritpol ] ) , which are local unitary transformations . in the simplified second type renormalization , we want to reduce @xmath163{tpolij } \emm$ ] to @xmath163{iline } \emm$ ] , so that we still have a trivalence graph . this requires that the support dimension @xmath231 of the fixed - point wave function @xmath232{tpolij } \epm $ ] is given by @xmath233 this implies that @xmath234{tpolij } \epm \simeq \del_{ii ' } \phi_\text{fix } \bpm \includegraphics[scale=.40]{tpol } \epm .\end{aligned}\ ] ] the simplified second type renormalization can now be written as ( since @xmath235 ) @xmath236{tpol } \epm \simeq p_i^{kj,\al\bt } \phi_\text{fix } \bpm \includegraphics[scale=.40]{iline } \epm .\end{aligned}\ ] ] we will call such a wave function renormalization step a p - move.@xcite here @xmath237 satisfies @xmath238 and @xmath239 the condition ensures that the two wave functions on the two sides of have the same normalization . we note that the number of choices for the four indices @xmath240 in @xmath241 must be equal or greater than @xmath43 : @xmath242 notice that @xmath243{tpol } \epm \simeq \sum_{m,\la,\ga } f^{jj^*k,\bt\al}_{i^*i^*m^*,\la\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{buble } \epm \nonumber\\ & \simeq \sum_{m,\la,\ga , l,\nu,\mu } f^{jj^*k,\bt\al}_{i^*i^*m^*,\la\ga } f^{i^*mj,\la\ga}_{m^*i^*l,\nu\mu } \phi_\text{fix } \bpm \includegraphics[scale=.40]{tpolud } \epm\end{aligned}\ ] ] using and its variation @xmath244{tpolud } \epm \simeq p_{i^*}^{lm,\mu\nu } \phi_\text{fix } \bpm \includegraphics[scale=.40]{iline } \epm .\end{aligned}\ ] ] we can rewrite as @xmath245 which is a condition on @xmath241 . more conditions on @xmath177 and @xmath241 can be obtained by noticing that @xmath234{f1gp } \epm \simeq \sum_{n=0}^n \sum_{\chi,\del } f^{ijm,\al\bt}_{kln,\chi\del } \phi_\text{fix } \bpm \includegraphics[scale=.40]{f2gp } \epm , \end{aligned}\ ] ] which implies that @xmath246{ikl1 } \epm \nonumber\\ & \simeq \sum_{n,\chi,\del } f^{ijm,\al\bt}_{kln,\chi\del } p^{jp,\chi\eta}_{k^ * } \del_{kn } \phi_\text{fix } \bpm \includegraphics[scale=.40]{ikl2 } \epm .\end{aligned}\ ] ] we find @xmath247 in the last section , we discussed the conditions that a fixed - point glu transformation @xmath248 must satisfy . after finding a fixed - point glu transformation @xmath248 , in this section , we are going to discuss how to calculate the corresponding fixed - point wave function @xmath140 from the solved fixed - point glu transformation @xmath248 . first we note that , using the two types of wave function renormalization introduced above , we can reduce any graph to @xmath192{oi } \emm$ ] . so , once we know @xmath214{oi } \epm$ ] , we can reconstruct the full fixed - point wave function @xmath140 on any connected graph . let us assume that @xmath234{oi } \epm = a^i = a^{i^*}\end{aligned}\ ] ] here @xmath249 satisfy @xmath250 the condition @xmath251 is simply the normalization condition of the wave function . the condition @xmath252 come from the fact that the graph @xmath163{oi } \emm$ ] can be deformed into the graph @xmath192{oia } \emm$ ] on a sphere . to find the conditions that determine @xmath249 , let us first consider the fixed - point wave function where the index on a link is @xmath115 : @xmath253{iga1 } \epm$ ] , where @xmath146 are indices on other part of graph . we note that the graph @xmath163{iga1 } \emm$ ] can be deformed into the graph @xmath192{iga2 } \emm$ ] on a sphere . thus @xmath197{iga1 } \epm= \phi_\text{fix}\bpm \includegraphics[scale=.35]{iga2 } \epm$ ] . using the f - moves and the p - moves , we can reduce @xmath192{iga2 } \emm$ ] to @xmath192{oi } \emm$ ] : @xmath254{iga1 } \epm= \phi_\text{fix}\bpm \includegraphics[scale=.40]{iga2 } \epm \simeq f(i,\ga ) \phi_\text{fix } \bpm \includegraphics[scale=.40]{oi } \epm\end{aligned}\ ] ] we see that @xmath255{oi } \epm = a^i\neq 0\end{aligned}\ ] ] for all @xmath115 . otherwise , any wave function with @xmath115-link will be zero . to find more conditions on @xmath249 , we note that @xmath234{ooijm } \epm \simeq p^{mj,\ga\la}_i \phi_\text{fix } \bpm \includegraphics[scale=.40]{oi } \epm = p^{mj,\ga\la}_ia^i .\end{aligned}\ ] ] by rotating @xmath192{ooijm } \emm$ ] by 180@xmath256 , we can show that @xmath257 or @xmath258 we also note that @xmath234{thetaijk } \epm & \simeq \sum_{m,\la,\ga } f^{ijk^*,\al\bt}_{j^*im,\la\ga } \phi_\text{fix } \bpm \includegraphics[scale=.40]{ooijm } \epm .\end{aligned}\ ] ] this allows us to show @xmath259 where @xmath260{thetaijk } \epm \nonumber\\ & \phi^\th_{ikj,\al\bt } = \e^{\imth \th_{a2 } } \phi^\th_{kji,\al\bt } , \nonumber\\ & \phi^\th_{ikj,\al\bt}=0 , \text { if } n_{ikj}=0.\end{aligned}\ ] ] the condition @xmath261 comes from the fact that the graph @xmath192{thetaijk } \emm$ ] and the graph @xmath192{thetakij } \emm$ ] can be deformed into each other on a sphere . also , for any given @xmath262 that satisfy @xmath263 , the wave function @xmath264{ijkga } \epm$ ] must be non - zero for some @xmath146 , where @xmath146 represents indices on other part of the graph . then after some f - moves and p - moves , we can reduce @xmath192{ijkga } \emm$ ] to @xmath192{thetakij } \emm$ ] . so , for any given @xmath262 that satisfy @xmath263 , @xmath214{thetakij } \epm$ ] is non - zero for some @xmath151 . since such a statement is true for any choices of basis on the vertex @xmath150 , we find that for any given @xmath262 that satisfy @xmath263 and for any non - zero vector @xmath265 , @xmath266{thetakij } \epm$ ] is non - zero for some @xmath151 . this means that the matrix @xmath267 is invertible , where @xmath267 is a matrix whose elements are given by @xmath268{thetakij } \epm $ ] . let us define @xmath269{thetakij } \epm \big ] = \det(m_{kji})$ ] , we find that @xmath270{thetakij } \epm \big ] = \det [ \phi^\th_{kji,\al\bt } ] \neq 0.\end{aligned}\ ] ] the above also implies that @xmath271 the conditions eqns . ( [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] ) allow us to determine @xmath249 ( and @xmath272 ) . from , we see that relation @xmath197{thetaijk } \epm = \phi_\text{fix}\bpm \includegraphics[scale=.35]{thetakij } \epm$ ] leads to some equations for @xmath273 , @xmath241 , and @xmath249 . more equations for @xmath273 , @xmath241 , and @xmath249 , can be obtained by using the relations @xmath274{tetra1 } \epm & = \phi_\text{fix } \bpm \includegraphics[scale=.40]{tetra2 } \epm \nonumber\\ & = \phi_\text{fix } \bpm \includegraphics[scale=.40]{tetra3 } \epm , \end{aligned}\ ] ] from the tetrahedron rotation symmetry@xcite and @xmath275{tetra1 } \epm \simeq \sum_{\ga\la } f^{ijm^*,\al\bt}_{kln,\ga\la } \phi_\text{fix } \bpm \includegraphics[scale=.40]{sq00 } \epm \nonumber\\ & \simeq \sum_{\ga\la , p\si\eps } f^{ijm^*,\al\bt}_{kln,\ga\la } f^{ln^*i^*,\del\la}_{nlp^*,\si\eps } \phi_\text{fix } \bpm \includegraphics[scale=.40]{osqo } \epm \nonumber\\ & \simeq \sum_{\ga\la , p\si\eps } f^{ijm^*,\al\bt}_{kln,\ga\la } f^{ln^*i^*,\del\la}_{nlp^*,\si\eps } p^{pl^*,\si\eps}_{n } \phi_\text{fix } \bpm \includegraphics[scale=.40]{sqo } \epm \nonumber\\ & = \sum_{\ga\la , p\si\eps } f^{ijm^*,\al\bt}_{kln,\ga\la } f^{ln^*i^*,\del\la}_{nlp^*,\si\eps } p^{pl^*,\si\eps}_{n } \phi^\th_{kjn^*,\ga\chi } .\end{aligned}\ ] ] it is not clear if and will lead to new independent equations or not . in the following discussions , we will not include and . we find that , at least for simple cases , the equations without and are enough to completely determine the solutions . to summarize , the conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equations whose variables are @xmath137 , @xmath273 , @xmath241 , @xmath249 , and @xmath276 . finding @xmath137 , @xmath273 , @xmath241 , and @xmath249 that satisfy such a set of non - linear equations corresponds to finding a fixed - point glu transformation that has a non - trivial fixed - point wave function . so the solutions @xmath277 give us a characterization of topological orders . this may lead to a classification of topological order from the local unitary transformation point of view . in this section , let us find some simple solutions of the fixed - point conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) for the fixed - point glu transformations @xmath278 and the fixed - point wave function @xmath249 . formally , the solutions of the fixed - point conditions are not isolated . they are parameterized by several continuous phase factors . in this section , we will discuss the origin of those phase factors . we will see that those different phase factors do not correspond to different states of matter ( different equivalence classes of glu transformations ) . so after removing those unimportant phase factors , the solutions of the fixed - point conditions are isolated ( at least for the simple examples studied here ) . we notice that , apart from two normalization conditions , all of the fixed - point conditions are linear in @xmath241 and @xmath249 . thus if @xmath279 is a solution , then @xmath280 is also a solution . however , the two phase factors @xmath281 do not lead to different fixed - point wave functions , since they only affect the total phase of the wave function and are unphysical . thus the total phases of @xmath282 and @xmath249 can be adjusted . we can use this degree of freedom to set , say , @xmath283 and @xmath284 . similarly the total phase of @xmath273 is also unphysical and can be adjusted . we have used this degree of freedom to reduce to . but this does not totally fix the total phase of @xmath273 . the transformation @xmath285 does not affect . we can use such a transformation to set the real part of a non - zero component of @xmath273 to be positive . the above three phase factors are unphysical . however , the fixed - point solutions may also contain phase factors that do correspond to different fixed - point wave functions . for example , the local unitary transformation @xmath286 does not affect the fusion rule @xmath137 , where @xmath287 is the number of links with @xmath288-state and @xmath289-state . such a local unitary transformation changes @xmath279 and generates a continuous family of the fixed - point wave functions parameterized by @xmath290 . those wave functions are related by local unitary transformations that continuously connect to identity . thus , those fixed - point wave functions all belong to the same phase . similarly , we can consider the following local unitary transformation @xmath291 that acts on each vertex with states @xmath292 on the three edges connecting to the vertex . such a local unitary transformation also does not affect the fusion rule @xmath137 . the new local unitary transformation changes @xmath279 and generates a continuous family of the fixed - point wave functions parameterized by the unitary matrix @xmath293 . again , those fixed - point wave functions all belong to the same phase . in the following , we will study some simple solutions of the fixed - point conditions . we find that , for those examples , the solutions have no addition continuous parameter apart from the phase factors discussed above . this suggests that the solutions of the fixed - point conditions correspond to isolated zero - temperature phases . let us first consider a system where there are only two states @xmath295 and @xmath296 on each link of the graph . we choose @xmath297 and the simplest fusion rule that satisfies , , is @xmath298 since @xmath299 , there is no states on the vertices . so the indices @xmath300 labeling the states on a vertex can be suppressed . the above fusion rule corresponds to the fusion rule for the @xmath294 loop state discussed in . so we will call the corresponding graphic state @xmath294 loop state . due to the relation , the different components of the tensor @xmath301 are not independent . there are only four independent potentially non - zero components which are denoted as @xmath302, ... ,@xmath303 : @xmath304 \fbox \ilnkaa \jlnkaa \klnkaa \llnkaa \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{0 } \nonumber\\ f^{000}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkbb \llnkbb \mlnkaa \nlnkbb \end{tikzpicture}\emm & = ( f^{011}_{100 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkbb \llnkaa \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^*=(f^{101}_{010 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkaa \llnkbb \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^ * \nonumber\\ & = f^{110}_{001 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkaa \llnkaa \mlnkaa \nlnkbb \end{tikzpicture}\emm = f_{1 } \nonumber\\ f^{011}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkaa \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = ( f^{101}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkbb \llnkaa \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f_{2 } \nonumber\\ f^{110}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{3}\end{aligned}\ ] ] we note that @xmath301 in relates wave functions on two graphs . in the above we have drawn the two related graphs after the @xmath212 tensor , where the first graph following @xmath212 corresponds to the graph on the left - hand side of and the second graph corresponds to the graph on the right - hand side of . the doted line corresponds to the @xmath295-state on the link and the solid line corresponds to the @xmath296-state on the link . there are four potentially non - zero components in @xmath282 , which are denoted by @xmath305, ... ,@xmath306 : @xmath307 we can adjust the total phases of @xmath308 and @xmath249 to make @xmath309 and @xmath310 . we can also use the local unitary transformation @xmath311 with @xmath312 to make @xmath313 , since the @xmath212 s described by @xmath314 in are the only @xmath212 s that change the number of @xmath296-links . the fixed - point conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equations on the ten variables @xmath315 , @xmath308 , and @xmath249 . many of the non - linear equations are dependent or even equivalent . using a computer algebraic system , we simplify the set of non - linear equations . the simplified equations are ( after making the phase choice described above ) @xmath316 where @xmath317 . the above simplified equations can be solved exactly . we find two solutions parameterized by @xmath317 : @xmath318 we also find @xmath319 the @xmath320 fixed - point state corresponds to the @xmath2 loop condensed state whose low energy effective field theory is the @xmath2 gauge theory.@xcite we call such a state , simply , the @xmath2 state . the @xmath321 fixed - point state corresponds to the double - semion state whose low energy effective field theory is the @xmath322 chern - simons gauge theory@xcite @xmath323 to obtain another class of simple solutions , we modify the fusion rule to @xmath324 while keeping everything the same . the above @xmath137 also satisfies , , and . the new fusion rule corresponds to the fusion rule for the @xmath294 string - net state discussed in . so we will call the corresponding graphic state @xmath294 string - net state . again , due to the relation , the different components of the tensor @xmath301 are not independent . now there are seven independent potentially non - zero components which are denoted as @xmath302, ... ,@xmath325 : @xmath326 \fbox \ilnkaa \jlnkaa \klnkaa \llnkaa \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{0 } \nonumber\\ f^{000}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkbb \llnkbb \mlnkaa \nlnkbb \end{tikzpicture}\emm & = ( f^{011}_{100 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkbb \llnkaa \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^*=(f^{101}_{010 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkaa \llnkbb \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^ * \nonumber\\ & = f^{110}_{001 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkaa \llnkaa \mlnkaa \nlnkbb \end{tikzpicture}\emm = f_{1 } \nonumber\\ f^{011}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkaa \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = ( f^{101}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkbb \llnkaa \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f_{2 } \nonumber\\ f^{011}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbb \klnkbb \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = ( f^{101}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkaa \klnkbb \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f^{111}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkaa \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm \nonumber\\ & = ( f^{111}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkaa \mlnkbb \nlnkbb \end{tikzpicture}\emm ) ^*=f_{3 } \nonumber\\ f^{110}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{4 } \nonumber\\ f^{110}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkaa \nlnkbb \end{tikzpicture}\emm & = ( f^{111}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkbb \nlnkaa \end{tikzpicture}\emm ) ^*=f_{5 } \nonumber\\ f^{111}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbb \jlnkbb \klnkbb \llnkbb \mlnkbb \nlnkbb \end{tikzpicture}\emm & = f_{6}\end{aligned}\ ] ] note that @xmath212 s described by @xmath314 and @xmath327 are the only @xmath212 s that change the number of @xmath296-links and the number of @xmath328-vertices . so we can use the local unitary transformation @xmath329 to make @xmath314 and @xmath327 to be positive real numbers . ( here @xmath330 is the total number of @xmath296-links and @xmath331 is the total number of @xmath328-vertices . ) we also use the freedom of adjusting the total sign of @xmath301 to make re@xmath332 . there are five potentially non - zero components in @xmath282 , which are denoted by @xmath305, ... ,@xmath333 : @xmath334 we use the freedom of adjusting the total phase of @xmath282 to make @xmath305 to be a positive number . we can also use the freedom of adjusting the total phase of @xmath249 to make @xmath335 to be a positive number . the fixed - point conditions ( [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equation on the variables @xmath315 , @xmath308 , and @xmath249 , which can be simplified . the simplified equations have the following form @xmath336 let @xmath337 be the positive solution of @xmath338 : @xmath339 . we see that @xmath340 . the above can be written as @xmath341 we also find @xmath319 the fixed - point state corresponds to the @xmath294 string - net condensed state@xcite whose low energy effective field theory is the doubled @xmath342 chern - simons gauge theory.@xcite the above simple examples correspond to non - orientable string - net states . here we will give an example of orientable string - net state . we choose @xmath343 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 the above @xmath137 satisfies , , and . due to the relation , the different components of the tensor @xmath301 are not independent . there are eight independent potentially non - zero components which are denoted as @xmath302, ... ,@xmath349 : @xmath326 \fbox \ilnkaa \jlnkaa \klnkaa \llnkaa \mlnkaa \nlnkaa \end{tikzpicture}\emm & = f_{0 } \nonumber\\ f^{000}_{111 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkbc \llnkbc \mlnkaa \nlnkbc \end{tikzpicture}\emm & = ( f^{011}_{200 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbc \klnkcb \llnkaa \mlnkbc \nlnkaa \end{tikzpicture}\emm ) ^*=f^{120}_{002 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkcb \klnkaa \llnkaa \mlnkaa \nlnkcb \end{tikzpicture}\emm \nonumber\\ & = ( f^{202}_{020 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkaa \klnkaa \llnkcb \mlnkcb \nlnkaa \end{tikzpicture}\emm ) ^*=f_{1 } \nonumber\\ f^{000}_{222 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkaa \klnkcb \llnkcb \mlnkaa \nlnkcb \end{tikzpicture}\emm & = ( f^{022}_{100 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkcb \klnkbc \llnkaa \mlnkcb \nlnkaa \end{tikzpicture}\emm ) ^*=(f^{101}_{010 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkaa \klnkaa \llnkbc \mlnkbc \nlnkaa \end{tikzpicture}\emm ) ^*\nonumber\\ & = f^{210}_{001 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkbc \klnkaa \llnkaa \mlnkaa \nlnkbc \end{tikzpicture}\emm = f_{2 } \nonumber\\ f^{011}_{011 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbc \klnkaa \llnkbc \mlnkbc \nlnkbc \end{tikzpicture}\emm & = f^{022}_{022 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkcb \klnkaa \llnkcb \mlnkcb \nlnkcb \end{tikzpicture}\emm = ( f^{101}_{202 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkaa \klnkcb \llnkaa \mlnkbc \nlnkcb \end{tikzpicture}\emm ) ^*\nonumber\\ & = ( f^{202}_{101 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkaa \klnkbc \llnkaa \mlnkcb \nlnkbc \end{tikzpicture}\emm ) ^*=f_{3 } \nonumber\\ f^{011}_{122 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkbc \klnkbc \llnkcb \mlnkbc \nlnkcb \end{tikzpicture}\emm & = ( f^{101}_{121 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkaa \klnkbc \llnkcb \mlnkbc \nlnkbc \end{tikzpicture}\emm ) ^*=f^{112}_{021 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkbc \klnkaa \llnkcb \mlnkcb \nlnkbc \end{tikzpicture}\emm \nonumber\\ & = ( f^{112}_{102 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkbc \klnkbc \llnkaa \mlnkcb \nlnkcb \end{tikzpicture}\emm ) ^*=f_{4 } \nonumber\\ f^{022}_{211 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkaa \jlnkcb \klnkcb \llnkbc \mlnkcb \nlnkbc \end{tikzpicture}\emm & = ( f^{202}_{212 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkaa \klnkcb \llnkbc \mlnkcb \nlnkcb \end{tikzpicture}\emm ) ^*=f^{221}_{012 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkcb \klnkaa \llnkbc \mlnkbc \nlnkcb \end{tikzpicture}\emm \nonumber\\ & = ( f^{221}_{201 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkcb \klnkcb \llnkaa \mlnkbc \nlnkbc \end{tikzpicture}\emm ) ^*=f_{5 } \nonumber\\ f^{112}_{210 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkbc \klnkcb \llnkbc \mlnkcb \nlnkaa \end{tikzpicture}\emm & = ( f^{120}_{221 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkcb \klnkcb \llnkcb \mlnkaa \nlnkbc \end{tikzpicture}\emm ) ^*=(f^{210}_{112 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkbc \klnkbc \llnkbc \mlnkaa \nlnkcb \end{tikzpicture}\emm ) ^ * \nonumber\\ & = f^{221}_{120 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkcb \klnkbc \llnkcb \mlnkbc \nlnkaa \end{tikzpicture}\emm = f_{6 } \nonumber\\ f^{120}_{110 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkbc \jlnkcb \klnkbc \llnkbc \mlnkaa \nlnkaa \end{tikzpicture}\emm & = ( f^{210}_{220 } \bmm\begin{tikzpicture}[scale=0.26 ] \fbox \ilnkcb \jlnkbc \klnkcb \llnkcb \mlnkaa \nlnkaa \end{tikzpicture}\emm ) ^*=f_{7}\end{aligned}\ ] ] there are nine potentially non - zero components in @xmath282 , which are denoted by @xmath305, ... ,@xmath350 : @xmath351 using the transformations discussed in section [ fixphase ] , we can fix the phases of @xmath314 , @xmath352 , @xmath325 , and @xmath305 to make them positive . the fixed - point conditions ( [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) form a set of non - linear equation on the variables @xmath315 , @xmath308 , and @xmath249 , which can be solved exactly . after fixing the phases using the transformations discussed in section [ fixphase ] , we find only one solution @xmath353 we also find @xmath319 the fixed - point state corresponds to the @xmath344 string - net condensed state@xcite whose low energy effective field theory is the @xmath322 chern - simons gauge theory@xcite @xmath354 which is the @xmath344 gauge theory . we note that all the above simple solutions also satisfy the standard pentagon identity , although we solved the weaker projective pentagon identity . it is not clear if we can find non - trivial solutions that do not satisfy the standard pentagon identity . there are several ways to define time reversal operation for the graphic states . the simplest one is given by @xmath355 where @xmath146 represents the labels on the vertices and links which are not changed under @xmath356 . ( this corresponds to the situation where the different states on the links and the vertices are realized by different occupations of scalar bosons . ) for such a time reversal transformation , @xmath357 and the real solutions of the fixed - point conditions ( [ nnnn ] , [ di ] , [ nnstar ] , [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] , [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) give us a classification of time reversal invariant topological orders in local spin systems . note that the time reversal invariant topological orders are equivalent class of local orthogonal transformations that connect to the identity transformation continuously . different real solutions @xmath358 of the fixed - point conditions do not always correspond to different time reversal invariant topological orders . the solutions differ by some unimportant phase factors ( which are @xmath359 signs ) correspond to the same topological order . to understand the above result , we notice that , from the structure of the fixed - point conditions , if @xmath279 is a solution , then @xmath360 is also a solution , where @xmath361 , @xmath362 , and @xmath363 . however , the three phase factors @xmath364 , @xmath365 , and @xmath366 do not lead to different fixed - point wave functions , since they only affect the total phase of the wave function and are unphysical . on the other hand , the fixed - point solutions may also contain phase factors that do correspond to different fixed - point wave functions . for example , the local orthogonal transformation @xmath367 does not affect the fusion rule @xmath137 , where @xmath287 is the number of links with @xmath288-state and @xmath289-state . such a local orthogonal transformation changes @xmath279 and generates a discrete family of the fixed - point wave functions . similarly , we can consider the following local orthogonal transformation @xmath368 that acts on each vertex with states @xmath292 on the three edges connecting to the vertex . such a local orthogonal transformation also does not affect the fusion rule @xmath137 . the new local orthogonal transformation changes @xmath279 and generates a family of the fixed - point wave functions parameterized by the orthogonal matrix @xmath369 . now the question is that do those solutions related by local orthogonal transformations have the same time reversal invariant topological order or not . we know that two gapped wave functions have the same time reversal invariant topological order if and only if they can be connected by local orthogonal transformation that _ connects to identity continuously_. it is well known that an orthogonal matrix whose determinant is @xmath370 does not connect to identity . thus it appears that local orthogonal transformations some times can generate different time reversal invariant topological orders . however , when we use the equivalent classes of local orthogonal transformations to define time reversal invariant topological orders , we not only assume the local orthogonal transformations to connect to identity continuously , we also assume that we can expand the local hilbert spaces ( say by increasing the range of the indices @xmath115 and @xmath150 that label the states on the edges and the vertices ) . the local orthogonal transformations can act on those enlarged hilbert spaces and can connect to identity in those enlarged hilbert spaces . even when a local orthogonal transformation can not be deformed into identity in the original hilbert space , it can always be deformed into identity continuously in an enlarged hilbert space . thus two real wave functions related by a local orthogonal transformation always have the same time reversal invariant topological order . for example , an orthogonal matrix @xmath371 that acts on states @xmath295 and @xmath296 does not connect to identity within the space of two by two orthogonal matrices . however , we can embed the above orthogonal matrix into a three by three orthogonal matrix that acts on @xmath295 , @xmath296 , and @xmath372 : @xmath373 . such a three by three orthogonal matrix does connect to identity within the space of three by three orthogonal matrices . this completes our argument that all local orthogonal transformations can connect to identity continuously at least in an enlarged hilbert space . so , after factoring out the unimportant phase factors discussed above , the real solutions of the fixed - point conditions may uniquely correspond to time reversal invariant topological orders . the four types of real solutions discussed in the last section are examples of four different time reversal invariant topological orders . for the @xmath294 loop states and the @xmath294 string - net state , we only have two states on each link . in this case , we can treat the two states as the two states of an electron spin . the time reversal transformation now becomes @xmath374 on each link . for such a time reversal transformation @xmath375 . the two @xmath294 loop states and the @xmath294 string - net state are not invariant under such an time - reversal transformation . once the fixed - point states have been identified and the labeling of topological orders has being found , we then face the next important issue : given a generic ground state wave function of a system , how to identify the topological orders in the state ? in other words , how to calculate the data @xmath376 that characterize the topological orders from a generic wave function ? one way to address the above issue is to have a general renormalization procedure which flows other states in the same phase to the simple fixed - point state so that we can identify topological order from the resulting fixed - point state . that is , we want to find a local unitary transformation which removes local entanglement and gets rid of unnecessary degrees of freedom from the state . how to find the appropriate unitary transformation for a specific state is then the central problem in this renormalization procedure . such a procedure for one dimensional tensor product states(tps ) ( also called matrix product states ) has been given in . here we will propose a method to renormalize two dimensional tps , where nontrivial topological orders emerge . the basic idea is to use the glu transformation discussed in section [ wvrg ] . note that , through the glu transformation , we can reduce the number of labels in a region a to the minimal value without loosing any quantum information ( see fig . [ glut ] ) . this is because the glu transformation is a lossless projection into the support space of the state in the region a. by performing such glu transformations on _ overlapping _ regions repeatedly ( see fig . [ qc]a ) , we can reduce a generic wave function to the simple fixed - point form discussed above . it should be noted that any state reducible in this way can be represented as a mera @xcite . in the following , we will present this renormalization procedure for two dimensional tps where we find a method to calculate the proper glu transformations . the tensor product states are many - body entangled quantum states described with local tensors . by making use of the entanglement information contained in the local tensors , we are able to come up with an efficient algorithm to renormalize two dimensional tps . this algorithm can be very useful in the study of quantum phases . due to the efficiency in representation , tps has found wide application as variational ansatz states in the studies of quantum many - body systems @xcite . suppose that in a variational study we have found a set of tensors which describe the ground state of a two dimensional many - body hamiltonian and want to determine the phase this state belongs to . we can apply our renormalization algorithm to this tensor product state , which removes local entanglement and flows the state to its fixed point . by identifying the kind of order present in the fixed - point state , we can obtain the phase information for the original state . in this section , we will give a detailed description of the algorithm and in the next section we will present its application to some simple ( but nontrivial ) cases . the states we are concerned with have translational symmetry and can be described with a translational invariant tensor network . to be specific , we discuss states on a hexagonal lattice . generalization to other regular lattices is straight - forward . consider a two - dimensional spin model on a hexagonal lattice with one spin ( or one qudit ) living at each vertex . the hilbert space of each spin is @xmath377-dimensional . the state can be represented by assigning to every vertex a set of tensors @xmath378 , where @xmath115 labels the local physical dimension and takes value from @xmath43 to @xmath377 . @xmath379 are inner indices along the three directions in the hexagonal lattice respectively . the dimension of the inner indices is @xmath380 . note that the figures in this note are all sideviews with inner indices in the horizontal plane and the physical indices pointing in the vertical direction , if not specified otherwise . representing a 2d quantum state on hexagonal lattice . @xmath115 is the physical index , @xmath379 are inner indices . right : a tensor product state where each vertex is associated with a tensor . the inner indices of the neighboring tensors connect according to the underlying hexagonal lattice.,width=336 ] the wave function is given in terms of these tensors by @xmath381 where @xmath382 denotes tensor contraction of all the connected inner indices on the links of the hexagonal lattice . a renormalization procedure of quantum states is composed of local unitary transformations and isometry maps such that the state flows along the path @xmath383 , @xmath384 , @xmath385, ... and finally towards a fixed point @xmath386 . with the tensor product representation , flow of states corresponds to a flow of tensors @xmath387 , @xmath388 , @xmath389 ... we will give the detailed procedure of how the tensors are mapped from one step to the next in the following section . in one round of renormalization , we start from tensor @xmath390 , do some operation to it which corresponds to local unitary transformations on the state , and map @xmath390 to @xmath391 . the whole procedure can be broken into two parts : the f - move and the p move , in accordance with the two steps introduced in the previous section . in the f - move , we take a @xmath392{f1}\emm$ ] configuration in the tensor network and map it to a @xmath393{f2}\emm$ ] configuration by doing a local unitary operation . we will see that the tensor product representation of a state leads to a natural way of choosing an appropriate unitary operation for the renormalization of the state . in order to do so , first we define the double tensor @xmath394 of tensor @xmath395 as @xmath396 graphically the double tensor @xmath394 is represented by two layers of tensor @xmath395 with the physical indices connected . represented as two layers of tensor @xmath395 with the physical indices contracted . the gray layer is the lower layer.,width=144 ] the tensor @xmath395 giving rise to the same double tensor @xmath394 is not unique . any tensor @xmath397 which differs from @xmath395 by an unitary transformation @xmath97 on physical index @xmath115 gives the same @xmath394 as @xmath97 and @xmath398 cancels out in the contraction of @xmath115 . on the other hand , an unitary transformation on @xmath115 is the only degree of freedom possible , i.e. any @xmath397 which gives rise to the same @xmath394 as @xmath395 differs from @xmath395 by a unitary on @xmath115 . therefore , in the process of turning a tensor @xmath395 into a double tensor @xmath394 and then split it again into a different tensor @xmath397 , we apply a non - trivial local unitary operation on the corresponding state . a well designed way of splitting the double tensor will give us the appropriate unitary transformation we need , as we show below . and @xmath399 on neighboring sites into a single double tensor @xmath394 ( 2)splitting double tensor @xmath394 into tensor @xmath400 ( 3 ) svd decomposition of tensor @xmath400 into tensors @xmath401 and @xmath402.,width=336 ] f - move has the following steps . first , construct double tensors for two neighboring sites on the lattice and combine them into a single double tensor with @xmath403 inner indices . @xmath404 note that with respect to the bipartition of indices @xmath405 and @xmath406 , @xmath394 is hermitian @xmath407 and positive semidefinite . therefore it has a spectral decomposition with positive eigenvalues @xmath408 . the corresponding eigenvectors are @xmath409 @xmath410 this spectral decomposition lead to a special way of decomposing double tensor @xmath394 into tensors . define a rank @xmath403 tensor @xmath400(as shown in fig . [ fig : f - move ] after step 2 ) as follows : @xmath411 @xmath400 has four inner indices @xmath412 of dimension @xmath380 and four physical indices @xmath413 also of dimension @xmath380 which are in the direction of @xmath412 respectively . as @xmath409 form an orthonormal set , it is easy to check that @xmath400 gives rise to double tensor @xmath394 . going from @xmath414 and @xmath415 to @xmath400 , we have implemented a local unitary transformation on the physical degrees of freedom on the two sites , so that in @xmath400 the physical indices and the inner indices represent the same configuration . in some sense , we are keeping only the physical degrees of freedom necessary for entanglement with the rest of the system while getting rid of those that are only entangled within this local region . now we do a singular value decomposition of tensor @xmath400 in the direction orthogonal to the link between @xmath414 and @xmath415 and @xmath400 is decomposed into tensors @xmath401 and @xmath402 . @xmath416 this step completes the f - move . ideally , this step should be done exactly so we are only applying local unitary operations to the state . numerically , we keep some large but finite cutoff dimension for the svd step , so this step is approximate . on a hexagonal lattice , we do f - move on the chosen neighboring pairs of sites ( dash - circled in fig . [ fig : rg_hex ] ) , so that the tensor network is changed into a configuration shown by thick dark lines in fig . [ fig : rg_hex ] . physical indices are omitted from this figure . now by grouping together the three tensors that meet at a triangle , we can map the tensor network back into a hexagonal lattice , with @xmath417 the number of sites in the original lattice . this is achieved by the p - move introduced in the next section . to form a new tensor @xmath388 on one site of the renormalized hexagonal lattice . ( 2)constructing the double tensor @xmath418 from @xmath388 so that we can start to do f - moves again.,width=336 ] now we contract the three tensors that meet at a triangle together to form a new tensor in the renormalized lattice as shown in the first step in fig . [ fig : p - move ] @xmath419 where @xmath420 is the physical index of the new tensor which includes all the physical indices of @xmath421 : @xmath422 . note that in the contraction , only inner indices are contracted and the physical indices are simply group together . constructing the double tensor @xmath394 from @xmath395 , we get the renormalized double tensor on the new hexagonal lattice which is in the same form as @xmath423,@xmath399 and we can go back again and do the f - move . one problem with the above renormalization algorithm is that , instead of having one isolated fixed - point tensor for each phase , the algorithm has a continuous family of fixed points which all correspond to the same phase . consider a tensor with structure shown in fig . [ fig : cdl ] . are entangled within each group but not between the groups.,width=115 ] the tensor is a tensor product of three parts which include indices @xmath424 respectively . it can be shown that this structure remains invariant under the renormalization flow . therefore , any tensor of this structure is a fixed point of our renormalization flow . however , it is easy to see that the state it represents is a tensor product of loops around each plaquette , which can be disentangled locally into a trivial product state . therefore , the states all have only short - range entanglement and correspond to the topologically trivial phase . the trivial phase has then a continuous family of fixed - point tensors . this situation is very similar to that discussed in . we will keep the terminology and call such a tensor a corner double line tensor . not only does corner double line tensor complicate the situation in the trivial phase , it leads to a continuous family of fixed points in every phase . it can be checked that the tensor product of a corner double line with any other fixed - point tensor is still a fixed - point tensor . the states they correspond to differ only by small loops around each plaquette and represent the same topological order . therefore any single fixed - point tensor gets complicated into a continuous class of fixed - point tensors . in practical application of the renormalization algorithm , in order to identify the topological order of the fixed - point tensor , we need to get rid of such corner double line structures . due to their simple structure , this can always be done , as discussed in the next section . now we present some examples where our algorithm is used to determine the phase of a tensor product state . the algorithm can be applied both to symmetry breaking phases and topological ordered phases . in the study of symmetry breaking / topological ordered phases , suppose that we have obtained some tensor product description of the ground state of the system hamiltonian . we can then apply our algorithm to the tensors , flow them to the fixed point , and see whether they represent a state in the symmetry breaking phase / topological ordered phase or a trivial phase . for system with symmetry / topological order related to gauge symmetry , it is very important to keep the symmetry / gauge symmetry in the variational approach to ground state and search within the set of tensors that have this symmetry / gauge symmetry@xcite . the resulting tensor will be invariant under such symmetries / gauge symmetries , but the state they correspond to may have different orders . in the symmetry breaking case , the state could have this symmetry or could spontaneous break it . in the topological ordered phase , the state could have nontrivial topological order or be just trivial . our algorithm can then be applied to decide which is the case . in order to correctly determine the phase for such symmetric tensors , it is crucial that we maintain the symmetry / gauge symmetry of the tensor throughout our renormalization process . we will discuss in detail two cases : the ising symmetry breaking phase and the @xmath2 topological ordered phase . for simplicity of discussion and to demonstrate the generality of our renormalization scheme , we will first introduce the square lattice version of the algorithm . all subsequent applications are carried out on square lattice . ( algorithm on a hexagonal lattice would give qualitatively similar result , though quantitatively they might differ , e.g. on the position of critical point . ) tensor product states on a square lattice are represented with one tensor @xmath425 on each vertex , where @xmath115 is the physical index and @xmath426 are the four inner indices in the up , down , left , right direction respectively . we will assume translational invariance and require the tensor to be the same on every vertex . the renormalization procedure is be a local unitary transformation on the state which flows the form of the tensor until it reaches the fixed point . it is implemented in the following steps . first , we form the double tensor @xmath394 from tensor @xmath395 @xmath427 into tensor @xmath428 ( 2 ) svd decomposition of @xmath428 in two different directions , resulting in tensors @xmath401 , @xmath402 and @xmath0 , @xmath429 respectively.,width=240 ] then do the spectral decomposition of positive operator @xmath394 into @xmath430 and form a new tensor @xmath431 @xmath432 @xmath85,@xmath152,@xmath433,@xmath434 are physical indices in the up , down , left , right directions respectively . this is illustrated in step 1 of fig . [ fig : rg_sq_1 ] . this step is very similar to the second step in the f - move on hexagonal lattice . next we do svd decomposition of tensor @xmath428 . for vertices in sublattice @xmath98 we decompose between the up - right and down - left direction as shown in step 2 of fig . [ fig : rg_sq_1 ] . for vertices in sublattice @xmath21 we decompose between the up - left and down - right direction as shown in step 3 of fig . [ fig : rg_sq_1 ] . @xmath435 has been applied . physical indices of the tensors are not drawn here.,width=115 ] after the decomposition , the original lattice ( gray lines in fig . [ fig : rg_sq_2 ] ) is transformed into the configuration shown by thick dark lines in fig . [ fig : rg_sq_2 ] . physical indices are omitted from this figure . if we now shrink the small squares , we get a tensor product state on a renormalized square lattice . fig . [ fig : rg_sq_3 ] shows how this is done . in step 1 , we contract the four tensors that meet at a small square @xmath436 where @xmath420 stands for the combination of all physical indices @xmath437 , @xmath438 . in step 2 , we construct a double tensor @xmath394 form tensor @xmath395 and completes one round of renormalization . now we can go back to step 1 in fig . [ fig : rg_sq_1 ] and flow the tensor further . now we are ready to discuss two particular examples , the ising symmetry breaking phase and the @xmath2 topological ordered phase , to demonstrate how our algorithm can be used to determine the phase of a tensor product state . a typical example for symmetry breaking phase transition is the transverse field ising model . consider a square lattice with one spin @xmath439 on each site . the transverse field ising model is @xmath440 where @xmath441 are nearest neighbor sites . the hamiltonian is invariant under spin flip transformation @xmath442 for any @xmath443 . when @xmath444 , the ground state spontaneously breaks this symmetry into either the all spin up state @xmath445 or the all spin down state @xmath446 . in this case any global superposition @xmath447 , the ground state has all spin polarized in the @xmath448 direction ( @xmath449 ) and does not break this symmetry . in the variational study of this system , we can require that the variational ground state always have this symmetry , regardless if the system is in the symmetry breaking phase or not . then we will find for @xmath444 the ground state to be @xmath450 . such a global superposition represents the spontaneous symmetry breaking . for @xmath451 , we will find the ground state to be @xmath449 and does not break the symmetry . for @xmath452 , we will need to decide which of the previous two cases it belongs to . we can first find a tensor product representation of an approximate ground state which is symmetric under the spin flip transformation , then apply the renormalization algorithm to find the fixed point and decide which phase the state belongs to . below we will assume a simple form of tensor and demonstrate how the algorithm works . suppose that the tensors obtained from the variational study @xmath425 , where @xmath115,@xmath453,@xmath454,@xmath455,@xmath456 can be @xmath78 or @xmath43 , takes the following form @xmath457 @xmath458 is a parameter between @xmath78 and @xmath43 . under an @xmath448 operation to the physical index , the tensor is changed to @xmath400 @xmath459 @xmath400 can be mapped back to @xmath395 by switching the @xmath78,@xmath43 label for the four inner indices @xmath426 . such a change of basis for the inner indices does not change the contraction result of the tensor and hence the state that is represented . therefore , the state is invariant under the spin flip transformation @xmath460 and we will say that the tensor has this symmetry also . when @xmath461 , the tensor represents state @xmath450 , which corresponds to the spontaneous symmetry breaking phase . we note that the @xmath461 tensor is a direct sum of dimension-1 tensors . such a direct - sum structure corresponds to spontaneous symmetry breaking , as discussed in detail in . when @xmath462 , the tensor represents state @xmath449 which corresponds to the symmetric phase . when @xmath463 , there must be a phase transition between the two phases however , as @xmath458 goes from @xmath78 to @xmath43 , the tensor varies smoothly with well defined symmetry . it is hard to identify the phase transition point . now we can apply our algorithm to the tensor . first , we notice that at @xmath461 or @xmath43 , the tensor is a fixed point for our algorithm . next , we find that for @xmath464 , the tensor flows to the form with @xmath461 , while for @xmath465 , it flows to the form with @xmath462 . therefore , we can clearly identify the phase a state belongs to using this algorithm and find the phase transition point . note that in our algorithm , we explicitly keep the spin flip symmetry in the tensor . that is , after each renormalization step , we make sure that the renormalized tensor is invariant under spin flip operations up to change of basis for the inner indices . if the symmetry is not carefully preserved , we will not be able to tell the two phases apart . obtained by taking the ratio of the contraction value of the double tensor in two different ways . @xmath466 is invariant under change of scale , basis transformation and corner double line structures of the double tensor and can be used to distinguish different fixed point tensors . for clarity , only one layer of the double tensor is shown . the other layer connects in exactly the same way.,width=240 ] we also need to mention that for arbitrary @xmath458 , the fixed point that the tensor flows to can be different from the tensor at @xmath461 or @xmath43 by a corner double line structure . we need to get rid of the corner double line structure in the result to identify the real fixed point . this is possible by carefully examining the fixed point structure . another way to distinguish the different fixed points without worrying about corner double lines is to calculate some quantities from the fixed - point tensors that are invariant with the addition of corner double lines . we also want the quantity to be invariant under some trivial changes to the fixed point , such as a change in scale @xmath467 or the change of basis for physical and inner indices . one such quantity is given by the ratio of @xmath468 and @xmath469 defined as @xmath470 , and @xmath471 . [ fig : x2x1 ] gives a graphical representation of these two quantities . in this figure , only one layer of the double tensor is shown . the other layer connects in the exactly the same way . it is easy to verify that @xmath466 is invariant under the change of scale , basis transformation and corner double lines . for tensors under the renormalization flow . as the number of rg steps increases , the transition in @xmath466 becomes sharper and finally approaches a step function at fixed point . the critical point is at @xmath472.,width=355 ] we calculate @xmath466 along the renormalization flow . the result is shown in fig . [ fig : x1_ising ] . at the @xmath461 fixed point , @xmath473 while at @xmath462 , @xmath474 . as we increase the number of renormalization steps , the transition between the two fixed points becomes sharper and finally approaches a step function with critical point at @xmath472 . tensors with @xmath475 belongs to the symmetry breaking phase while tensors with @xmath476 belongs to the symmetric phase . the algorithm can also be used to study topological order of quantum states . in this section , we will demonstrate how the algorithm works with @xmath2 topological order . consider again a square lattice but now with one spin @xmath439 per each link . a simple hamiltonian on this lattice with @xmath2 topological order can be defined as @xmath477 where @xmath213 means plaquettes and @xmath478 is all the spin @xmath439s around the plaquette and @xmath479 means vertices and @xmath480 is all the spin @xmath439s connected to the vertex . the ground state wave function of this hamiltonian is a fixed - point wave function and corresponds to the @xmath294 loop state with @xmath320 as discussed in the previous section . the ground state wave function has a simple tensor product representation . for simplicity of discussion we split every spin @xmath439 into two and associate every vertex with four spins . the tensor @xmath481 has four physical indices @xmath482 and three inner indices @xmath483 . @xmath484 it can be checked that @xmath485 is a fixed - point tensor of our algorithm . this tensor has a @xmath2 gauge symmetry . if we apply @xmath486 operation to all the inner indices , where @xmath486 maps @xmath78 to @xmath78 and @xmath43 to @xmath370 , the tensor remains invariant as only even configurations of the inner indices are nonzero in the tensor . consider then the following set of tensor parameterized by @xmath31 @xmath487 at @xmath488 , this is exactly @xmath485 and the corresponding state has topological order . at @xmath42 , the tensor represents a product state of all @xmath78 and we denote the tensor as @xmath489 . at some critical point in @xmath31 , the state must go through a phase transition . this set of tensors are all invariant under gauge transformation @xmath490 on their inner indices and the tensor seems to vary smoothly with @xmath31 . one way to detect the phase transition is to apply our algorithm . we find that , at @xmath491 , the tensors flow to @xmath485 , while at @xmath492 , the tensors flow to @xmath489 . we determine @xmath35 to be between @xmath493 . as this model is mathematically equivalent to two dimensional classical ising model where the transition point is known to great accuracy , we compare our result to that result and find our result to be within @xmath494 accuracy ( @xmath495 ) . again in the renormalization algorithm , we need to carefully preserve the @xmath2 gauge symmetry of the tensor so that we can correctly determine the phase of the states . the fixed - point tensor structure might also be complicated by corner double line structures , but it is always possible to identify and get rid of them . similarly , we can calculate the invariance quantity @xmath466 to distinguish the two fixed points . @xmath474 for @xmath485 while @xmath473 for @xmath489 . the result is plotted in fig . [ fig : x1_z2 ] and we can see that the transition in @xmath466 approaches a step function after a large number of steps of rg , i.e. at the fixed point . the critical point is at @xmath496 . for @xmath492 , the tensor belongs to the trivial phase , while for @xmath491 , the tensor belongs to the @xmath2 topological ordered phase . for tensors under the renormalization flow . as the number of rg steps increases , the transition in @xmath466 becomes sharper and finally approaches a step function at fixed point . the critical point is at @xmath496.,width=355 ] our algorithm can also be used to demonstrate the stability of topological order against local perturbation . as is shown in ref . , local perturbations to the @xmath2 hamiltonian correspond to variations in tensor that do not break the @xmath2 gauge symmetry . we picked tensors in the neighborhood of @xmath485 which preserve this gauge symmetry randomly and applied our renormalization algorithm ( gauge symmetry is kept throughout the renormalization process ) . we find that as long as the variation is small enough , the tensor flows back to @xmath485 , up to a corner double line structure . this result shows that the @xmath2 topological ordered phase is stable against local perturbations . in this paper , we discuss a defining relation between local unitary transformation and quantum phases . we argue that two gapped states are related by a local unitary transformation if and only if the two states belong to the same quantum phase . we can use the equivalent classes of local unitary transformations to define `` patterns of long range entanglement '' . so the patterns of long range entanglement correspond to universality classes of quantum phases , and are the essence of topological orders.@xcite as an application of this point of view of quantum phases and topological order , we use the generalized local unitary transformations to generate a wave function renormalization , where the wave functions flow within the same universality class of a quantum phase ( or the same equivalent class of the local unitary transformations ) . in other words , the renormalization flow of a wave function does not change its topological order . such a wave function renormalization allows us to classify topological orders , by classifying the fixed - point wave functions and the associated fixed - point local unitary transformations . first , we find that the fixed - point local unitary transformations are described by the data @xmath497 that satisfy @xmath498 @xmath499 @xmath500 [ see eqns . ( [ nnnn ] , [ di ] , [ nnstar ] [ 2ffstar ] , [ 1fstarf ] , [ penid ] , [ pffp ] , [ pfp ] ) ] . from the data @xmath501 we can further find out the fixed - point wave function by solving the following equations for @xmath502 : @xmath503 [ see eqns . ( [ anonz ] , [ aiaistar ] , [ papa ] , [ phipa ] , [ phiikj ] , [ phinonz ] ) ] . the combined data @xmath504 that satisfy the conditions , , and classify a large class of topological orders . we see that the problem of classifying a large class of topological orders becomes the problem of solving a set of non - linear algebraic equations , , , and . the combined data @xmath504 that satisfy the conditions , , , and also classify a large class of time reversal invariant topological orders , if we restrict ourselves to real solutions . the solutions related by local orthogonal transformations all belong to the same phase , since the local orthogonal transformations always connect to identity if we enlarge the hilbert space . we like to point out that we can not claim that the solutions @xmath505 classify all topological orders since we have assumed that the fixed - point local unitary transformations are described by tensors of finite dimensions . it appears that chiral topological orders , such as quantum hall states , are described by tensors of infinite dimensions . we note that the data @xmath504 just characterize different fixed - point wave functions . it is not guaranteed that the different data will represent different topological orders . however , for the simple solutions discussed in this paper , they all coincide with string - net states , where the topological properties , such as the ground state degeneracy , number of quasiparticle types , the quasiparticle statistics _ etc _ , were calculated from the data . from those topological properties , we know that those different simple solutions represent different topological orders . also @xmath506 are real for the simple solutions discussed here . thus they also represent topological orders with time reversal symmetry . ( certainly , at the same time , they represent stable topological orders even without time reversal symmetry . ) we also like to point out that our description of fixed - point local unitary transformations is very similar to the description of string - net states . however , the conditions , , , and on the data appear to be weaker than ( or equivalent to ) those@xcite on the string - net data . so the fixed - point wave functions discussed in this paper may include all the string - net states ( in 2d ) . last , we present a wave function renormalization scheme , based on the glu transformations for generic tps . such a wave function renormalization always flows within the same phase ( or within the same equivalence class of lu transformations ) . it allows us to determine which phase a generic tps belongs to by studying the resulting fixed - point wave functions . we demonstrated the effectiveness of our method for both symmetry breaking phases and topological ordered phases . we find that we can even use tensors that do not break symmetry to describe spontaneous symmetry breaking states : if a state described by a symmetric tensor has a spontaneous symmetry breaking , the symmetric tensor will flow to a fixed - point tensor that has a form of direct sum . we would like to thank i. chuang , m. hastings , m. levin , f. verstraete , z .- h . wang , y .- s . wu , s. bravyi for some very helpful discussions . xgw is supported by nsf grant no . zcg is supported in part by the nsf grant no . nsfphy05 - 51164 . in section [ qphucl ] and section [ tolren ] , we argued about the equivalence relation between gapped quantum ground states in the same phase . we concluded that two states are in the same phase if and only if they can be connected by local unitary evolution or constant depth quantum circuit : @xmath507 |\phi(0)\>\\ |\phi(1)\ > \sim |\phi(0)\ > & \text{\ iff\ } & |\phi(1)\ > = u^m_{circ } |\phi(0)\>\end{aligned}\ ] ] now we want to make these arguments more rigorous , by stating clearly what is proved and what is conjectured , and by giving precise definition of two states being the same , the locality of operators , etc . we will show the equivalence in the following steps : ( all these discussions can be generalized to the case where the system has certain symmetries . @xmath51 and @xmath508 used in the equivalence relation will then have the same symmetry as the system hamiltonian @xmath62 . ) first , according to the definition in section [ qphucl ] , two states @xmath36 and @xmath38 are in the same phase if we can find a family of local hamiltonians @xmath32 , @xmath509 $ ] with @xmath41 being its ground state such that the ground state average of any local operator @xmath33 , @xmath510 changes smoothly from @xmath42 to @xmath488 . here we allow a more general notion of locality for the hamiltonian @xcite and require @xmath32 to be a sum of local operators @xmath511 : @xmath512 where @xmath511 is a hermitian operator defined on a compact region @xmath486 . @xmath513 sums over a set @xmath514 of regions . the set @xmath514 contains regions that differ by translations . the set @xmath514 also contains regions with different sizes . however , @xmath511 approaches zero exponentially as the size of the region @xmath486 approaches infinity . or more precisely , for all sites @xmath515 in the lattice @xmath516 for some positive constant @xmath517 . here sums over all regions in the set @xmath514 that cover the site @xmath515 , @xmath519 denotes operator norm , @xmath520 is the cardinality of @xmath486 , and @xmath521 is the diameter of @xmath486 . therefore , instead of being exactly zero outside of a finite region , the interaction terms can have an exponentially decaying tail . if @xmath36 and @xmath38 are gapped ground states of @xmath37 and @xmath39 , then we assume that for all @xmath510 to be smooth , @xmath32 must remain gapped for all @xmath31 . if @xmath32 closes gap for some @xmath35 , then there must exist a local operator @xmath33 such that @xmath510 has a singularity at @xmath35 . we call the gapped @xmath32 an adiabatic connection between two states in the same phase . let @xmath32 be a differentiable family of local hamiltonians and @xmath41 be its ground state . if the excitation gap above @xmath41 is larger than some finite value @xmath522 for all @xmath31 , then we can define @xmath523 , such that @xmath524 |\phi(0)\>$ ] . where @xmath525 is a function which has the following properties : 1 . the fourier transform of @xmath525 , @xmath526 is equal to @xmath527 for @xmath528 . @xmath529 is infinitely differentiable . 3 . @xmath530 . under this definition , @xmath51 is local ( almost ) and satisfies @xmath531 \right \| \leq h'(\text{dist}(z , b ) ) \left | z \right \vert \left \vert \t h_z(g ) \right \| \left \vert o_b \right \| \label{local_th}\ ] ] where @xmath532 is any operator supported on site @xmath21 , @xmath533 is the smallest distance between @xmath21 and any site in @xmath486 , and @xmath534 is a function which decays faster than any negative power of @xmath85 . @xmath535 $ ] denotes commutator of two operators . this is called the quasi - adiabatic continuation of states . therefore , we can show that if @xmath36 and @xmath38 are in the same phase , then we can find a local ( as defined in ) hamiltonian @xmath536 such that @xmath524 |\phi(0)\>$ ] . in other words , states in the same phase are equivalent under local unitary evolution . note that here we map @xmath36 exactly to @xmath38 . there is another version of quasi - adiabatic continuation @xcite , where the mapping is approximate . in that case , @xmath536 can be defined to have only exponentially small tail outside of a finite region instead of a tail which decays faster than any negative power . @xmath537 |\phi(0)\>$ ] will not be exactly the same as @xmath38 , but any local measurement on them will give approximately the same result . next we want to show that the reverse is also true . suppose that @xmath36 is the gapped ground state of a local hamiltonian @xmath37 , @xmath538 and each @xmath539 is supported on a finite region @xmath486 . apply a local unitary evolution generated by @xmath540 to @xmath36 and take it to @xmath41 , @xmath541 , @xmath542 , where @xmath543 $ ] . @xmath41 is then ground state of @xmath544 . under unitary transformation @xmath545 the spectrum of @xmath37 does nt change , therefore @xmath32 remains gapped . to show that @xmath32 also remains local , we use the lieb - robinson bound derived in , which gives @xmath546 \right \| & = & \left \vert \left[u_gh_z(0)u_g^{\dagger } , o_b\right ] \right \|\\ & \leq & h(\text{dist}(z , b ) ) \left |z\right \vert \left \vert h_z(0 ) \right \| \left \vert o_b \right \|\end{aligned}\ ] ] where @xmath547 decays faster than any negative power of @xmath85 . therefore , @xmath511 remains local up to a tail which decays faster than any negative power . @xmath32 then forms a local gapped adiabatic connection between @xmath36 and @xmath38 . to detect for phase transition , we must check whether the ground state average value of any local operator @xmath33 , @xmath510 , has a singularity or not . @xmath548 . using the lieb - robinson bound given in , we find @xmath549 remains local ( in the sense of ) and evolves smoothly with @xmath31 . therefore , @xmath510 changes smoothly . in fact , because @xmath32 is differentiable , the derivative of @xmath510 to any order always exists . therefore , there is no singularity in the ground state average value of any local operator @xmath33 and @xmath36 and @xmath38 are in the same phase . we have hence shown that states connected with local unitary evolution are in the same phase . this completes the equivalence relation stated in . the definition of locality is slightly different in different cases , so there is still some gap in the equivalence relation . however , we believe that the equivalence relation should be valid with a slight generalization of the definition of locality . lastly , we want to show that the equivalence relation is still valid if we use constant depth quantum circuit instead of local unitary evolution . this is true because we can always simulate a local unitary evolution using a constant depth quantum circuit and vice - verse . to simulate a local unitary evolution of the form @xmath537 $ ] , first divide the total time into small segments @xmath550 such that @xmath551 \simeq e^{-i\delta t h(m\delta t)}$ ] . the set of local operators @xmath553 can always be divided into a finite number of subsets @xmath554 , @xmath555 ... such that elements in the same subset commute with each other . then we can do trotter expansion and approximate @xmath556 as @xmath557 , where @xmath558 , @xmath559 ... each term in @xmath560 , @xmath561 commute , therefore they can be implement as a piece - wise local unitary operator . putting these piece - wise local unitary operators together , we have a quantum circuit which simulates the local unitary evolution . the depth of circuit is proportional to @xmath562 . it can be shown that to achieve a simulation with constant error , a constant depth circuit would suffice@xcite . on the other hand , to simulate a constant depth quantum circuit @xmath563 , where @xmath564 with a local unitary evolution , we can define the time dependent hamiltonian as @xmath565 , such that for @xmath566 , @xmath567 . it is easy to check that @xmath568=u_{circ}$ ] . the simulation time needed is @xmath569 . we can always choose a finite @xmath550 such that @xmath570 is finite and the evolution time is finite .
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two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states . on the other hand ,
gapped ground states remain within the same phase under local unitary transformations .
therefore , local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems .
since local unitary transformations can remove local entanglement , the above equivalence / universality classes correspond to pattern of long range entanglement , which is the essence of topological order .
the local unitary transformation also allows us to define a wave function renormalization scheme , under which a wave function can flow to a simpler one within the same equivalence / universality class . using such a setup , we find conditions on the possible fixed - point wave functions where the local unitary transformations have _ finite _ dimensions .
the solutions of the conditions allow us to classify this type of topological orders , which generalize the string - net classification of topological orders .
we also describe an algorithm of wave function renormalization induced by local unitary transformations .
the algorithm allows us to calculate the flow of tensor - product wave functions which are not at the fixed points .
this will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor - product state .
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all known normal nuclei are made of the two nucleons , the proton and the neutron . besides those two lightest baryons there exist still a couple of other stable ( but weakly decaying ) baryons , the hyperons . up to now the inclusion of multiple units of strangeness in nuclei remains experimentally as theoretically rather largely unexplored . this lack of investigation reflects the experimental task in producing nuclei containing ( weakly decaying ) strange baryons , which is conventionally limited by replacing one neutron ( or at maximum two ) by a strange @xmath0-particle in scattering experiments with pions or kaons . there exists nowadays a broad knowledge about single hypernuclei , i.e. nuclei , where one nucleon is substituted by a @xmath1 ( or @xmath2 ) by means of the exchange reaction @xmath3 . over the last two decades a rich phenomology has resulted for such hypernuclei . however , there exist more or less no experimental insight how more than one hyperon behave inside a nuclei . the technical problem is to create within a tiny moment , smaller than the decay time of a hyperon , enough hyperons and then to bring them together with nucleons to form any potential multihypernucleus . by employing a relativistic shell model calculation , which gives a rather excellent description of normal nuclei and single @xmath1-hypernuclei , it was found that such configurations might exist as ( small ) bound multihypernuclei ( memo - metastable exotic multihypernuclear object ) @xcite . strange matter could also be realized in a completely different picture . indeed , this second and much more speculative possibility was raised by physicists much earlier . the fundamental theory of strong interactions , quantum chromodynamics , does not forbid the principle existence of ` larger ' hadronic particles , so called multiquark states . today only the mesons and baryons are known in nature . however , there could exist states with more than three quarks . going further with this speculation one comes to the conclusion that only multiquark states with nearly the same number of up , down and strange quarks might exist as ( meta-)stable configurations @xcite . such a very speculative form of strange matter is called strange quark matter . ( ultra-)relativistic heavy ion collisions provide the only ( earth based ) source for the formation of either strangelets ( small lumps of strange quark matter ) or multi - hypernuclear objects , consisting of nucleons , @xmath1 s and @xmath4 s , as dozens of hyperons are produced in a single central event . in principle , strangelets can be produced via two different scenarios : by a condensation out of a quark - gluon plasma or by a coalescence of hyperons out of the created hot and dense fireball . for the former scenario it is essential that within the phase transition of the deconfined matter to hadronic particles the _ net _ strangeness ( counting a surplus of strange over antistrange quarks ) is getting enriched in the plasma phase . this distillation ( or separation ) of strangeness , i.e. the possible conglomeration of net strangeness , has been predicted to occur for a first order phase transition of a baryonrich qgp @xcite . in particular , if the strangelet does exist in principle , it has to be regarded as a cold , stable and bound manifestation of that phase being a remnant or ` ash ' of the originally hot qgp - state . on the other hand a further necessary request for the possible condensation is that the initially hot plasma phase has to cool down considerably during the ongoing phase transition . within our present knowledge of the phase transition such a behaviour can neither be unambiqously shown to happen nor be excluded . in section 2 we briefly summarize the reasons for the ( possible ) existence of this novel and exotic states . in section 3 the mechanism of strangeness distillation and the possible production of small strange matter states are reviewed . we conclude this section by discussing the detection possibilities of small and finite strangelets with respect to their lifetimes against strong or weak hadronic decays . in section 4 we finally sketch on how the physics of strange matter can affect the physical picture of dense neutron stars and the issue of baryonic dark matter . the first speculation about the possible existence of collapsed nuclei was given by bodmer @xcite . he argued that another form of baryonic matter might be more stable than ordinary nuclei . indeed it was speculated there both on the possible existence of hyperonic matter with baryons as colorless constituents or strange quark matter with quarks as major constituents . the paper , however , lacked detailed calculation as the mit bag model or walecka model were only available a few years later . let us now briefly summarize how a stable or metastable strangelet might look like @xcite : think of bulk objects , containing a large number of quarks @xmath5 , so - called multiquark droplets . multiquark states consisting only of u- and d - quarks must have a mass larger than ordinary nuclei , otherwise normal nuclei would be unstable . however , the situation is different for droplets of sqm , which would contain approximately the same amount of u- , d- and s - quarks . speculations on the stability of strangelets are based on the following observations : ( 1 ) the ( weak ) decay of a s - quark into a d - quark could be suppressed or forbidden because the lowest single particle states are occupied . ( 2 ) the strange quark mass can be lower than the fermi energy of the u- or d - quark in such a dense quark droplet . opening a new flavour degree of freedom therefore tends to lower the fermi energy and hence also the mass per baryon of the strangelet . sqm may then appear as a nearly neutral state . if the mass of a strangelet is smaller than the mass of the corresponding ordinary nucleus with the same baryon number , the strangelet would be absolutely stable and thus be the true groundstate of nuclear matter @xcite . within the mit bag model very low phenomenological bag parameters @xmath6 mev have to be employed for modeling such a possibility . on the other hand , it is also conceivable that the mass per baryon of a strange droplet is lower than the mass of the strange @xmath1- baryon , but larger than the nucleon mass ( see fig . the droplet is then in a metastable state , it can not decay spontanously into @xmath1 s . for bag parameters b@xmath7 lower than 190 mev strange quark droplets can only decay via weak interactions . for larger b - values strangelets are instable . due to the strong finite size effects @xcite and the wider range of the employed model parameters , smaller strangelets are much more likely to be metastable than being absolutely stable . also the possible influence of color magnetic and color electric potentials has been necglected in these calculations due to their complicated group structure . for very light strangelets with baryon number @xmath8 it turns out @xcite that such contributions are in fact repulsive , making the possible candidates less stable ( with the well - known exception of the h dibaryon ) . a classification scheme for metastable combinations of nucleons and hyperons exhibits that combinations of nucleons , @xmath0 s and @xmath9 s are favoured compared to combinations with @xmath10 s due to their q - values in vacuum @xcite . an example is shown in fig . 2 , where the single particle levels of a strange nucleus consisting of two of each proton , neutron , @xmath0 , @xmath11 , @xmath12 are plotted . note that each baryon sits in the 1s - state . the reaction @xmath13 can not induce a strong decay because the two @xmath0 s sitting in the 1s - level cause the produced @xmath0 s to escape in vacuum . but this is energetically unfavoured resulting in an overall metastable compound system . the calculation was carried out in a relativistic mean field model taking care of the nucleon - nucleon and nucleon - hyperon interaction . an extension of this model also implements the scarce information about the hyperon - hyperon interaction . this is done by introducing two new meson fields into the theory , @xmath14 and @xmath15 , which couple to strange baryons only @xcite . the binding energy for memos will be moderately enhanced ( on a scale of nuclear binding energies ) due to this additional interactions . indeed , binding energies of @xmath16 mev and more have been found with a net strangeness fraction of @xmath17 . even negatively charged strange nuclear systems are possible without loosing stability . the lightest stable object of this type is likely to be @xmath18 . the global properties of a strangelet or a memo are likely to be identical : a similar small charge @xmath19 and nearly the same average baryon density @xmath20 . in turn , this would suggest that a nearly neutral and heavy candiate could not unambiguosly be considered as a strangelet . in principle , to distinguish experimentally between both one has in addition to resolve the mass @xmath21 very accurately . a memo is only bound in the order of @xmath22 mev whereas the strangelet may be bound from @xmath23 mev ( which is , of course , speculation ) . the resembling microscopic structure gives raise to the speculation that both states might have a strong overlap and correlation . the memo would decay into a strangelet , if the latter is energetically more favourable . multiple collisions in the reaction of two bombarding nuclei ensure that the interacting system starts to equilibrate which might be suited to search for the most interesting collective effects . in particular all exotic objects need for their formation _ large strange particle numbers , high degree of equilibration _ and _ large densities . _ in order to get this kind of states one should therefore use high energies to produce enough strangeness and energy density , and heavy nuclei to gain as much equilibration as possible . this situation is now achieved at brookhaven ags using au+au and at cern sps using pb+pb . it may then be possible to produce some of the lightest multistrange objects in the laboratory . in the following we want to sketch why the production of sqm clusters , if they do exist in principle , is likely , if a baryon rich and hot deconfined qgp is created in such collisions . the net strangeness of the qgp is zero from the onset , although an equal , however large , number of strange and antistrange quarks has been produced . however , there is a physical mechanism which separates the strange quarks from their antiparticles @xcite ( see fig . it is ` simple ' for the antistrange quarks to materialize in kaons k(@xmath24 ) because of the lots of light quarks as compared to the s - quarks which could only move into the suppressed antikaons ( @xmath25 ) or the heavy hyperons . hence , during a near equilibrium phase transition a large antistrangeness builds up in the hadron matter while the qgp retains a large net strangeness excess . for modeling the evolution of an initially hot fireball a two phase equilibrium description between the hadron gas and the qgp was combined with nonequilibrium radiation by incorporating the rapid freeze - out of hadrons from the hadron phase surrounding the qgp droplet during a first order phase transition ( last reference of @xcite ) . two scenarios may describe the evolution to the final state : the quark droplet may remain unstable until the strange quarks have clustered into @xmath1-particles and other strange hadrons to carry away the strangeness and the plasma has completely vanished into standard particles . this scenario is customarily accepted . however , if sqm exists at low temperatures in configurations having a mass per baryon lower than the mass of the @xmath1-particle , the hot sqm droplet would remain at the phase transition boundary much longer . as shown in @xcite , producing strange baryons like @xmath0 particles is energetically more expensive and therefore less likely than producing sqm like strangelets . towards the end of the evolution only baryons are allowed to escape from the droplet , since at this point all of the antiquarks are gone . the baryons will be mostly nucleons , since the hyperons are heavier and require more energy for formation . these nucleons remove energy but they do not carry away any strange quarks , so the ratio of strange to nonstrange quarks increases further , refining the distillation of strangeness . the hot strange matter in fact might cool down to cold lumps of size @xmath26 , depending on the original baryon content of the plasma . 3b gives an impression how the hadronisation proceeds for a large bag constant ( @xmath27 mev no strangelet in the groundstate ) and a small bag constant ( @xmath28 mev ) . for the large bag constant the system hadronizes completely in @xmath29 8 @xmath30 , which is customarily expected and thus not too surprising . yet , a strong increase of the net strangeness of the system is found in both situations , and the plasma drop reaches a strangeness fraction of @xmath31 when the volume becomes small . indeed , for the small bag constant , however , a _ cold _ strangelet emerges from the expansion and evaporation process with an approximate baryon number of @xmath32 17 , a radius of @xmath33 , and a net strangeness fraction of @xmath34 , i.e. a charge to baryon ratio @xmath35 ! if a baryon rich qgp is temporarily created it will assemble the strange quarks during the ongoing evolution and expansion . it might either lead to the formation of a strangelet , or it will decay mainly into hyperons in its late stage . the strangelet would be a remnant or ` ash ' of the deconfined quark matter phase and thus would provide a signal for the formation of a qgp @xcite . as emphasized already in the introduction , for the possible condensation of small pieces of strange quark matter out of an initially hot qgp with no net strangeness two mechanisms have to be realized by nature : ( 1 ) strangeness distillation and ( 2 ) ongoing cooling ( e.g. by pion evaporation ) during the phase transition resulting in a loss of internal heat . the strangeness distillation during the hadronization of a baryonrich qgp is a very intuitive process . it works for all model parameters like e.g. also much higher bag constants @xcite . in a recent study the above described model including particle evaporation from the outer surface has been combined within 1-dimensional longitudinal hydrodynamical expansion @xcite with initial conditions expected at cern - sps energies and future rhic energies . again , a strong accumulation of net strangeness in the plasma phase has been found during the ongoing phase transition . still , the mechanism by which a qgp state is converted into hadrons is a major uncertainty in the different descriptions . the hadronisation transition has often been described by geometric and statistical models , where the matter is assumed to be in partial or complete equilibrium during the whole expansion phase . a fully microscopic and numerical model of the hadronization of confined and color - neutral hadrons out of a deconfined plasma has very recently been realized within the friedberg - lee model by traxler et al . exploratory studies of a baryonrich system within this model have shown in fact that the strangeness distillation is a genuine feature of the hadronisation . it might , however , be very difficult to experimentally verify that this distillation indeed happens during the transition . we remark that the separation mechanism might be probed by density interferometry with hyperons or kaons @xcite . the hadrons with negative strangeness , the @xmath1 , @xmath36 and @xmath37 , are expected to be produced mainly at the last stage of the phase transition when the size of the quark phase volume has become quite small . the realization of the second mechanism by nature , i.e. the necessary ongoing cooling for final possible strangelet condensation , is not as obvious . it was pointed out already in the second ref . of @xcite that the strangelet formation can only go hand in hand with strong cooling rather than reheating . besides the expansion of the system additional pion and nucleon evaporation should help to allow for a possible , yet necessary cooling . on the other hand one realizes from fig . 3b that depending on the bag parameter employed there might be ( moderate to strong ) cooling in the one and reheating in the other case . in all the numerous and recent studies within thermal hadronic models one tries to find common parameters of the fireball by trying to fit the ratios of measured hadronic abundancies within a few parameters . within the above model of a rapidly disintegrating qgp we had repeated these kind of analysis @xcite and were forced to assume a high bag constant of @xmath38 mev in order to find a similar good agreement like earlier investigations . then , according to the model , no real cooling would emerge during the phase transition and thus also no strangelet may condense . the temperature will drop , if the specific entropy per baryon in the hadron phase exceeds that in the quark phase , @xmath39 , otherwise the temperature has to slightly increase during the transition @xcite . the first is the case ( within the model ) when the bag constant @xmath40 is small or moderate and allows for the existence of ( meta-)stable strange quark matter states . although this intimate connection between cooling of the plasma phase and the existence of strange quark matter is intriguing , it might be valid only within the simple parametrisation of the quark gluon plasma phase within a bag model description . ultimately , whether @xmath41 is larger or smaller than @xmath42 at finite , nonvanishing chemical potentials might theoretically only be proven rigorously by lattice gauge calculations in the future , as also the principle existence of ( meta-)stable strange quark matter . more conservative estimates of production of small multistrange objects ( either strangelets or memos ) without the need of a temporarily present intermediate qgp phase are based on coalescence models being put forward by dover and coworkers @xcite . such estimates yield very small production rates , for instance @xmath43 events per central au+au collisions at 11.7 agev for the lightest bound @xmath9 system @xmath44he . very simple coalescence estimates give production probabilities of strange clusters of the order of @xmath45 , where @xmath46 denotes the strangeness and @xmath47 the baryon number of the cluster . for a maximum sensitivity of @xmath48 only strangelets or memos with baryon numbers of @xmath49 are expected to be seen . as has been seen recently by the e864-collaboration @xcite the penalty factor @xmath50 for an additional unit of baryon number at ags energies in central collisions is in fact very small , @xmath51 , implying that the formation of exotic objects by coalescence is even less favorable . 4 shows calculated thermal multiplicities of various hypermatter clusters for central pb+pb collisions at sps energies @xcite . these numbers should serve as an upper limit for the production via coalescence . the ( thermal ) penalty factor suppresses the abundances of heavy clusters : metastable hypermatter can only be produced with a probability @xmath52 for @xmath53 ( e.g. a \{@xmath54 } object ) . hence , only exotic objects with very low mass number are expected to be ( possibly ) seen at the ags or at cern . it is important to note that these objects are a new form of matter , not a specific new particle . the strange droplets produced in these reactions do not come in the form of a single type of particle . many different sizes of droplets may be produced , spanning a range in mass , charge , and strangeness content . the experimental task of finding the new form of matter is therefore challenging . here any detected particle having an unusual charge to mass ratio is a potential strange matter candidate . employing tof - techniques in present settings to reveal the velocity and thus the charge to mass ratio , the experimental setup sets a natural time scale of @xmath55 ns @xcite . so , an important question we finally have to adress are the lifetimes of these objects . the lifetime of a memo should be similar to the @xmath1 s lifetime , i.e. @xmath56 sec . however , all the present experiments are unable to observe metastable hyperclusters due to the required lifetimes @xmath57s . on the other hand , if a produced strangelet is absolutely stable , the only energetically possible decay mode is the rather slow weak leptonic decay ( @xmath58 , @xmath59 ) , which will turn the strangelet to its minimum value in energy . it is much more likely , if at all , however , that a small strangelet is a weakly decaying metastable state . the situation turns out to be even more complicated : strangelets will not be in their ground state when being produced in a heavy ion collision . suppose a strangelet is created out of the hot and dense matter with some arbitrary strangeness , charge and baryon number . strong interactions and the distillation process will alter the composition of a strangelet by particle evaporation on a timescale of a few hundred fm / c after the collision . the strangelet , if surviving this , will cool down until it reaches the domain of weak interactions . weak hadronic decay by hadron emission takes place on a timescale being estimated between @xmath60 s. as a conservative quideline one might even also consider the life - time of the @xmath1-particle . strangelets stable against further strong decay channels but decaying by the weak hadronic decay will be dubbed as short - lived candidates . strangelets stable against weak hadronic decay will then still be subject to the weak leptonic decay occuring on a timescale of @xmath61 s. they are dubbed as long - lived candidates . obviously , the present experiments are probably only capable to look for the long - lived candidates . following the qualitative and old ideas of chin and kerman @xcite , we had carried out a detailed analysis for small short - lived and long - lived strangelet candidates @xcite , where shell effects are quite pronounced and immportant . one qualitative outcome is that all candiates are likely to be negative , especially the long - lived candiadates . at first sight this might appear counter - intuitive . a closer inspection reveals that strong nucleon decay will drive a strangelet to a higher net strangeness content @xmath62 and carries away positive charge . consider a system of bulk sqm in the ground - state with a finite total strangeness of e.g. @xmath63 ( compare fig . 1 , @xmath64 mev ) . one would now naively argue that strange quark matter with @xmath65=0.4 is the ground - state of the system , because the energy per baryon of this state is lower than that of the hyperonic state . however , the total energy per baryon can be lowered by additional @xmath6650 mev by assembling the non - strange quarks into pure nucleonic degrees of freedom ( i.e. ` @xmath67-particle or nucleon emission ' ) , leaving the strange quarks in a strange matter droplet , its strangeness fraction enriched to @xmath68 slightly _ above _ its minimum value . this strong nucleon decay will stop to happen for energetical reasons at the so called tangent point ( compare fig.1 ) . finite size corrections , i.e. strong shell effects , on the other side , of course , will change this picture quantitatively . a similar reasoning shows that the weak nucleon decay again will change the candidates to become even more negative . 5 shows the result of this investigation for long - lived candidates which are stable against weak hadronic decay modes @xcite . different bag parameters have been chosen in order to get some feeling for the different possibilities . besides the _ neutral _ so called quark - alpha state @xcite with @xmath69 ( @xmath70 ) a valley of stability appears at @xmath71 for long - lived , negative candidates . for bag parameters of @xmath72 mev no long - lived candidates have been found at all . still there exist a much richer spectrum of shortlived strangelets or memos with a lifetime of the order of the @xmath1 or somewhat below @xcite . it might well be that only metastable strange clusters with @xmath66 cm flight path seem to have a chance of being created . future experiments geared to proof the ( non)existence of strangelets therefore should clearly cover such short lifetimes . an open geometry detectional device will be needed , which clearly is a challenging task due to the large background of charged hadrons at the target in violent events with high multiplicities . originally ( strange ) quark matter in bulk was thought to exist only in the interiour of neutron stars where the pressure is high enough that the neutron matter melts into its quark substructure @xcite . at least in the cores of neutron stars ( where the density rises up to the order of @xmath73 times normal nuclear density ) it is not very likely that matter consists of individual hadrons . on the other hand it is also known that the pure ` neutron ' matter is not really a nuclear matter state made solely out of neutrons , but at least at higher densities consists also of a considerable amount of protons as well as hyperons . indeed , it was shown by glendenning that hyperons @xcite appear at a moderate density of about @xmath74 times normal nuclear matter density . these new species influence the properties of the equation of state of matter and the global properties of neutron stars . there may be so many hyperons in the neutron star that the whole object is more appropriately dubbed a giant hypernucleus . the gross structure of a neutron star like its mass m and radius r is influenced by the composition of its stellar material . this holds especially in the case of the existence of strangeness bearing `` exotic '' components like hyperons or strange quark matter which may significantly change the characteristic mass - radius ( mr ) relation of the star . for example hyperons considerably soften the eos and reduce the maximum mass of a neutron star . @xmath75 besides the speculative possibility of nearly pure sqm stars , which would be the case , if sqm is absolutely stable in bulk , it is more likely that in the interiour of the star a phase transition to hadronic matter will take place . the ` neutron ' star would then have the form of a so called hybrid star , i.e. a star which is made of baryonic matter in the outer region , but with a quark matter core in the deep interiour . the deconfinement phase transition from hadronic matter to the sqm phase is constructed according to the requirement of global charge neutrality between both phases @xcite . the pressure of both phases spans up a two dimensional surface over the plane of the relevant chemical potentials of the baryons ( @xmath76 ) and the electrons ( @xmath77 ) . this situation is depicted in fig . one finds that the pressure varies smoothly and continuously with the proportion of both phases in equilibrium , which , in turn will lead to a mixed phase of finite radial extent ( of several kilometers ) inside the star @xcite . 7 shows the resulting mr relations for various situations and qcd coupling constants @xmath78 employed within the effective mass bag model discussed in @xcite . the left hand side ( @xmath79 ) shows the pure sqm star results , which turn out to be rather compact objects . the situation changes completely for the hybrid stars , depicted on the right hand side of fig.[rm1 ] ( @xmath80 ) . their mr relation approaches the curve of the pure hadron star ( denoted by h ) if the phase transition occurs deep insight the star . accordingly it will be difficult to judge from measured properties of pulsars to really disentangle the possible interiour structure of neutron stars. for possible consequences which might be observable , like e.g. timing structure of pulsar spin - down , higher spin rates , or stronger cooling rates , especially relevant for young neutron stars , we want to refer to @xcite . witten raised the intriguing possibility that strange quark matter might in principle also be absolutely stable and may also provide an explanation for cold ( baryonic ) dark matter in the universe @xcite . if being stable and nearly neutral , it could exist at all possible sizes @xcite , as the small coulomb energy is not sufficient for a break up into smaller pieces @xcite . it might thus span as a charge neutral state the empty ` nuclear desert ' @xcite within the range from @xmath81 up to sizes of neutron stars @xmath82 . in witten s prospective scenario hot strange quark matter nuggets could have condensed out of the deconfined phase during the phase transition in the expanding and cooling early universe @xcite . these would carry most of the tiny surplus in baryon number of the whole universe . the baryon number would remain inside if the heat and entropy of the nugetts is carried away mainly by neutrino emission instead of mesons and especially baryons . if an absolutely stable groundstate would exist the hot nuggets would further cool and instead of suffering a complete hadronisation they might settle into these new states and hence could resolve the dark matter problem . since then the idea of absolute stability has stimulated a lot of work on potential consequences in astrophysics @xcite . today it is fair to say that there is still no real evidence for absolutely stable sqm to be a possible dark matter candiate . there had been early conjectures @xcite that escpecially smaller glumps of sqm could not have survived the ongoing and still hot epoch of the cosmological evolution , but would have in fact been completely evaporated into single hadrons . hence , particle physicists were led to look for other possible ( and maybe even more exotic ) explanations on the issue of dark matter . on the other side , in a very recent ( and speculative ) study @xcite the issue of stable quark matter was taken up again . there it is speculated that a ( stable ) pre quark - matter phase might in fact also explain the formation of galaxies clusters , the formation of massive black holes in the galactic centres , and also might be a source of the @xmath83-ray bursts . we refer the interested reader to @xcite . + * acknowledgements : * thanks go to c. dover , a. dumitru , a. gal , l. gerland , p. koch - steinheimer , d. rischke , j. schaffner - bielich , k. schertler , c. spieles , h. stcker , m. thoma and c. traxler for their help and collaboration . 99 for a general review see : c. greiner and j. schaffner , _ int . j. mod . phys._*e5 * ( 1996 ) 239 ; c. greiner and j. schaffner , physics of strange matter , nucl - th/9801062 ; + for an introduction see : h. crawford and c. greiner , ` the search for strange matter ' , _ scientific american _ , vol . 1 , january 1994 , p. 58 j. schaffner , c. greiner and h. stcker , _ phys . c _ * 46 * , 322 ( 1992 ) e. witten , _ phys . d _ * 30 * , 272 ( 1984 ) ; e. farhi and r. l. jaffe , _ phys . d _ * 30 * , 2379 ( 1984 ) c. greiner , p. koch and h. stcker , _ phys . lett . _ * 58 * , 1825 ( 1987 ) ; c. greiner , d.h . rischke , h. stcker and p. koch , _ phys . d _ * 38 * , 2797 ( 1988 ) ; c. greiner and h. stcker , _ phys . d _ * 44 * , 3517 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contribution to these proceedings ; g. van buren , contribution to these proceedings ; m. munoz , contribution to these proceedings j. sandweiss , _ nucl . b _ ( proc . suppl . ) * 24b * , 234 ( 1991 ) ; + f.s . rotondo , proc . quark matter ` 96 , _ nucl . a _ * 610 * , 297c ( 1996 ) h.j . crawford et al . , _ nucl . b _ ( proc . * 24b * , 251 ( 1991 ) f. dittus , proc . quark matter ` 95 , _ nucl . phys . a _ * 590 * , 347c ( 1995 ) ; r. klingenberg , proc . quark matter ` 96 , _ nucl . phys . a _ * 610 * , 306c ( 1996 ) s. a. chin and a. k. kerman , _ phys lett . _ * 43 * , 1292 ( 1979 ) j. schaffner , c. greiner , a. diener and h. stcker , _ phys . * c 55 * , 3038 ( 1997 ) ; j. schaffner , strangelets and strange quark matter ' , nucl - th/971104 f. c. michel , _ phys . rev . lett . _ * 60 * , 677 ( 1988 ) for a detailed survey on the astrophysical implications see : ` strange quark matter in physics and astrophysics ' , edited by j. madsen and p. haensel , _ nucl . phys . b _ ( 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relativistic heavy ion collisions offer the possibility to produce exotic metastable states of nuclear matter containing ( roughly ) equal number of strangeness compared to the content in baryon number .
the reasoning of both their stability and existence , the possible distillation of strangeness necessary for their formation and the chances for their detection are reviewed . in the later respect
emphasize is put on the properties of small lumps of strange quark matter with respect to their stability against strong or weak hadronic decays .
in addition , implications in astrophysics like the properties of neutron stars and the issue of baryonic dark matter will be discussed .
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type ia supernovae ( sne ia ) have been the tool that made possible the discovery of the acceleration of the expansion of the universe ( riess et al . 1998 ; perlmutter et al . 1999 ) , and they are now providing new insights on the cosmic component , dubbed `` dark energy '' , thus revealed . however , in contrast with their key role as cosmological probes , and after more than 50 years of supernova research , the nature of their progenitors remains elusive . as far back as 1960 , it was established that type i supernovae ( in fact , the now denominated sne ia , or thermonuclear supernovae ) should result from the ignition of degenerate nuclear fuel in stellar material ( hoyle & fowler 1960 ) . the absence of hydrogen in the spectra of the sne ia almost immediately suggested that they were due to thermonuclear explosions of white dwarfs ( wds ) . isolated white dwarfs were once thought to be possible progenitors ( finzi & wolf 1967 ) , but soon discarded due to incompatibility with basic results from stellar evolution . instead , accretion of matter from a close companion star in a binary system , by a previously formed c+o white dwarf with a mass close to the chandrasekhar mass , provides a viable mechanism to induce the explosion ( wheeler & hansen 1971 ) . two main competing production channels are still under discussion nowadays . one possible path is the so called single degenerate ( sd ) channel , where a c+o white dwarf grows in mass by accretion from a non degenerate stellar companion : a main sequence star , a subgiant , a helium star , a red giant , or an agb star ( whelan & iben 1973 ; nomoto 1982 ) . another possible path is the double degenerate ( dd ) channel ( webbink 1984 ; iben & tutukov 1984 ) , where two wds merge due to the loss of angular momentum by gravitational radiation . the merging could produce the collapse of the white dwarf ( saio & nomoto 1985 ) , or it can produce a larger c+o white dwarf configuration that then explodes ( pakmor et al . 2012 ) . in the decade of the 90 s , the variety amongst sne ia was discovered , ranging from events such as sn 1991bg to those as sn 1991 t , through normal sne ia ( see filippenko 1997a , b ; branch et al . 2007 ; leibundgut 2011 ) . such diversity was made amenable for cosmology when the correlation of the luminosity at the maximum of the light curve of each sn ia with its rate of decline was parameterized ( phillips 1993 , 1999 ; riess , press & kirshner 1995 ; perlmutter et al . it became clear , then , that sne ia could be used as distance indicators in cosmology , and that led to the aforementioned discovery . yet , the first decade of the new century has brought new surprises : super chandrasekhar supernovae , as well as extremly faint ones ( see below ) . neither of them are useful for cosmology , although they are not a severe nuisance there , since they can be easily identified , and eliminated from the large samples of sne ia collected for cosmological probes . also , various teams have started to measure supernova rates at a wide variety of redshifts . the idea of using sne ia rates to discover the nature of the progenitor systems has now become an active line of research . finally , high resolution spectroscopic observations of sn have yielded the surprising result of time varying absorptions , which indicate the existence of outflows in the circumstellar medium surrounding some sn , and points to possible nova activity previous to the explosion . an intriguing c ii feature has been identifieed , close to the si ii line typical of sne ia , and that has led to thinking in two different directions : either the thermonuclear flame does not burn the outermost layers of the white dwarf , or maybe c is a signature of the merged white dwarf companion of the sn . there are also better estimates of the maximum h mass that could be present in the envelopes of the pre sne , if the explosions were triggered by accretion from a non degenerate companion . there is continued failure to detect h from the radio emission of the sne ia , and there could be constraints from the x ray emission as well . the task of searching for the companion star in galactic supernovae has already given some definite results , and there are , now , simulations of the impact of the sn ejecta on the companion star that can be compared with the observations . in the following sections , we present and discuss those new results . in section 2 we briefly review the different models proposed to explain the sn ia phenomenon . section 3 examines how the delay time distribution ( dtd ) constrains the possible sn ia progenitors . in section 4 we discuss the carbon and oxygen absorption features seen , in recent years , in the spectra of sn ia at early times , while section 5 deals with the emission features at late times . section 6 discusses the variable blueshifted sodium feature seen in some sne ia . the x ray constraints are presented in section 7 , and the radio constraints in section 8 . in section 9 we report the limits on the luminosities of the companions of sne ia obtained from pre explosion images . section 10 deals with the detection of companions throught the early light curves of sne ia . section 11 reviews the direct searches for surviving companions , in the galaxy and in the large magellanic cloud . section 12 deals with the identification of possible candidates to sne ia through reconstruction of the orbital evolution of diverse close binary systems containing white dwarfs . section 13 addresses the important problem of the outliers from the peak brightness decline rate of the light curve relationship used to make these sne calibrated candles for cosmology . section 14 deals with the bulk of sne ia used for cosmology . we summarize the current state of affairs in the last section . an ideally complete model of a type ia supernova should start from the formation and subsequent evolution of the binary system assumed to originate it , include possible common envelope episodes , mass tranfer stages ( generally nonconservative ) , especially those immediately leading to the explosion , rotational states of the two stars involved , and finally the ignition process and its development into a full thermonuclear explosion ( hydrodynamics and nucleosynthesis ) . from that , light curves and spectra of the emitted light should be computed , for the different stages of the outburst ( extending until the nebular phase ) . the characteristics of the resulting remnants should be predicted as well . in addition , the frequency of the explosions corresponding to the model has to be estimated , for different galactic environments . the whole programme involves very diverse domains of expertise , so the progress has been disperse . those models were first proposed by webbink ( 1984 ) and by iben & tutukov ( 1984 ) . the most favored scenario , in the latter work , started from binaries with component masses in the range 59 @xmath0 . they experienced two common envelope stages and ended as a pair of c+o white dwarfs , with masses in the range 0.71 @xmath0 , separated by distances 0.20.35 @xmath1 and orbiting each other with periods @xmath2 between 12 min and 14 hrs . the system then losses angular momentum by emission of gravitational waves and the two white dwarfs merge on a time scale ranging from 10@xmath3 to 10@xmath4 yr ( merging of binaries due to the emission of gravitational radiation had already been considered by tutukov & yungelson 1979 ) . the merging would occur through disruption of the less massive component of the system . that component fills its roche lobe first ( larger radius and smaller roche lobe ) . the mass transfer , then , should have a runaway character , since the more mass the white dwarf losses , the larger its radius becomes ( and the smaller its roche lobe ) . the material of the disrupted white dwarf would form a thick disk around the more massive one , which would accrete mass from it until reaching the chandrasekhar limit and explosively ignite c at its centre . with the caveat that the effects on the orbit of the two common envelope phases could only be roughly approximated , iben & tutukov ( 1984 ) found that the rate of such mergings might , alone , account for the galactic sn ia rate . in the preceding scenario , it was assumed that the only effect of the accretion of matter by the more massive white dwarf , from the debris of its companion , should be growth up to the chandrasekhar mass . that was soon challenged by saio & nomoto ( 1985 ) , who argued that a very fast mass transfer ( @xmath5 ) would produce an off centre c flash . a c burning front would then propagate ( nonexplosively ) down to the centre of the white dwarf , changing the chemical composition from c+o to o+ne+mg along the way . electron captures on mg and ne would subsequently make the chandrasekhar mass smaller than the white dwarf mass , and gravitational collapse would ensue . the outcome should thus be the formation of a neutron star , rather than a sn ia explosion . nomoto & iben ( 1985 ) further concluded that the off centre c ignition would always occur unless the mass accretion rate were less than one fifth of the eddington limit for an isolated white dwarf . later hydrodynamic simulations ( benz et al . 1990 ; guerrero et al . 2004 ) have confirmed that a heavy accretion disk is formed around the most massive white dwarf . whether a sn ia ensues would depend on the mass accretion rate of the white dwarf from the disk , that being determined by the viscosity of the latter : a sufficiently low viscosity would allow the white dwarf to grow while avoiding the off centre , non explosive ignition of c ( mochkovich , guerrero & segretain 1997 ) . the issue remains open . another effect of mass acccretion from a massive disk should be the gain of angular momentum by the white dwarf . its consequences have been studied by piersanti et al . ( 2003a , b ) and tornamb & piersanti ( 2013 ) . they find that rotation can stabilize the white dwarf against contraction , even when its mass becomes larger than the chandrasekhar mass . later , loss of angular momentum by emission of gravitational waves allows contraction , until the conditions for explosive c ignition are reached at the centre . recently , lebanon , soker & garcia berro ( 2014 ) find constraints on the classical dd type of progenitor from the disk originated matter around the exploding wd , as this matter will be shocked by the sn ejecta and end up in a radiation implying a larger progenitor radius than observed . a different approach is based on the violent merging of a pair of white dwarfs ( pakmor et al . another possibility is the direct collision of a pair of white dwarfs . such collisions could take place in environments of high star number density , like globular clusters or the galactic centre ( benz et al . 1989 ) , and also in triple star systems where a close pair of white dwarfs have their orbits perturbed by the third , more distant star ( thompson 2011 ; katz & dong 2012 ; kushnir et al . 2013 ; dong et al . 2014 ) . in the hydrodynamic simulations of pakmor et al . ( 2010 ) , the merging of two equal mass ( @xmath6 ) white dwarfs produced a subluminous sn ia . further modeling ( pakmor et al . 2011 ) has shown that violent mergings where the primary mass is @xmath6 can give rise to subluminous sne ia for unequal masses too , provided that the mass ratio is more than about 0.8 . later on , pakmor et al . ( 2012 ) have found that the violent merging of two c+o white dwarfs , with masses 0.9 m@xmath7 and 1.1 m@xmath7 ( assumed to induce the detonation of the c+o mixture ) can account for normal sne ia as well . that has been very recently confirmed by garca senz et al . ( 2013 ) , who find that the amount of @xmath8ni produced in the collisions ranges from 0.1 m@xmath7 to 1.1 m@xmath7 , thus covering from subluminous through normal up to overluminous sne ia . they argue , however , that given the distribution of white dwarf masses , mostly subluminous events should arise . on the other hand , ruiter et al . ( 2013 ) find that the brightness distribution of the explosions produced by violent mergers matches the shape of that observed for sne ia ( although the issue depends on the occurrence of a particular phase of mass accretion during binary evolution ) . very recently , kromer et al . ( 2013a ) have successfully explained the narrow emission lines of [ o i ] in the late time spectra of sn 2010lp ( a subluminous sn ia ) by the violent merger of two c+o white dwarfs , with masses of 0.9 and 0.76 m@xmath7 . even more recently , moll et al . ( 2013 ) have found that prompt detonations following the merging of two white dwarfs can not only reproduce both common and overluminous sne ia , but also the width luminosity relation on which the use of these supernovae as cosmological distance indicators is based . a key point , though , is whether there are or not enough close binary white dwarf binaries to account for the rate of occurrence of sne ia . one approach to this problem is to look for such systems in the galaxy . napiwotzki et al . ( 2007 ) have reported the results of a systematic radial velocity survey for double degenerate binaries as potential progenitors of type ia supernovae . more than 1000 wds and pre white dwarfs were observed with the vlt . the frequency of he wds is much higher than that of c+o wds , and they regard he wd donors as a possible important channel for sne ia . recently , badenes & maoz ( 2012 ) , using multi epoch spectroscopy of @xmath9 4000 white dwarfs from the sloan digital sky survey , have determined the white dwarf merger rate per unit stellar mass in our galaxy . they find that the total rate might well account for the sn ia rate in the milky way and galaxies of the same type , but that the rate of merging of pairs of white dwarfs with a total mass above the chandrasekhar mass is only @xmath91/14 of the total rate . so , unless sub chandrasekhar mergers can produce sne ia , the double degenerate channel should be , at most , a minor contributor to the sn ia phenomenon . sn ia models based on the non violent merging of two white dwarfs , with a total mass below the chandrasekhar mass , involve a c+o plus a he white dwarf . the he accreted by the more massive ( c+o ) white dwarf detonates and the shock wave thus generated can either induce the detonation of the c+o layers immediately below or converge near the centre and produce a c+o detonation there ( sim et . al . 2012 ) . these authors find that the light curves and spectra of such explosions do match those observed in sne ia . besides , ruiter et al . ( 2011 ) had calculated that if those double detonation models were able to produce explosions similar to sne ia , then the sub chandrasekhar explosions would account for a substantial fraction , at least , of the observed sn ia rate . they also found that the double white dwarf channel involving a c+o plus a he white dwarf should have a distribution of delay times ( between formation of the binary and the sn explosion ) spanning from 800 myr up to the hubble time . the channel leading to a sn ia explosion via mass accretion , by a c+o white dwarf , from a non degenerate binary companion , was first modelled by whelan & iben ( 1973 ) , although the idea , as we have seen , already appears in wheeler & hansen ( 1971 ) : a primary with a mass of 1.83 @xmath0 , plus a secondary with @xmath10 , initially form a system with an orbital period between 5 and 9 years . the primary then evolves and becomes a c+o white dwarf , with a mass close to the chandrasekhar mass . the secondary , after @xmath9 10@xmath4 yr , becomes an agb star , fills its roche lobe , and transfers mass to the white dwarf , which then reaches the chandrasekhar mass and explodes . in this first model , the mass of the secondary was chosen to explain the occurrence of sne ia in elliptical galaxies , long after star formation has stopped . subsequently , a variety of initial binary systems , in which mass accretion by the white dwarf can take place at different evolutionary stages of its companion ( main sequence , subgiant , red giant , agb ) , and either from roche lobe overflow or from a stellar wind , have been proposed . also , mass loss by the two components of the binary can take place more than once , as well as mass transfer ( either conservative or non conservative ) . the possibility that the companion might have lost its hydrogen rich envelope and become a helium star , at the time of mass tranfer to the white dwarf , has also been considered . a problem common to both the single degenerate models and to the double degenerate models , in which the conditions for explosive c ignition , close to the centre of the c+o white dwarf , are reached when the white dwarf mass has grown to the chandrasekhar mass ( or when rotational support has been lost , for masses above such limit ) , is that thermonuclear burning propagation should be subsonic at first ( deflagration ) and , at some point , become supersonic ( detonation ) . if only deflagrations were involved , the explosions could just produce a subclass of subluminous sne ia ( fink et al . on the other hand , pure detonations of chandrasekhar mass white dwarfs would burn the c+o mixture to fe peak elements entirely , in stark contrast with the observations , that show intermediate mass elements at and around maximum light . a self consistent modeling of the deflagration / detonation transition still remains elusive , although steady progress is being made ( woosley 2007 ; aspden , bell & woosley 2010 ; schmidt et al . 2010 ; woosley , kerstein & aspden 2011 ) . there is another problem that only concerns single degenerate models of sne ia : the possible explosive ignition of the material ( hydrogen or helium ) accumulated on the surface of the accreting c+o white dwarf , before the latter reaches the chandrasekhar mass . also , in the case of hydrogen , a high rate of accretion may not lead to explosion but to the formation of an extended envelope , which would be ejected by interaction with the mass donor star , that terminating the accretion process . accumulation of hydrogen on the surface of a white dwarf at a low rate should produce nova like explosions , in which all the accreted material ( and maybe even more ) would be expelled ( nomoto 1982 ; livio & truran 1992 ) . the lower limit on the mass accretion rate , to avoid explosions and strong flashes , is uncertain and depends on the mass of the white dwarf , but a value @xmath11 is often given , although kercek , hillebrandt & truran ( 1999 ) , in 3d simulations , obtained steady hydrogen burning for rates as low as @xmath12 ( see discussion in hillebrandt & niemeyer 2000 ) . even if hydrogen is burned steadily or in weak flashes , the resulting helium layer should explosively burn , in a detonation , if the rate of accumulation of helium were @xmath13 ( although this is only true for white dwarf masses @xmath14 ) . the same applies to the direct accretion of helium . lower rates would thus be excluded from the production of chandrasekhar mass explosions , but they could lead to sub chandrasekhar , edge lit explosions , instead . the explosion of helium shells in accreting white dwarfs as a type ia sn mechanism was first proposed by taam ( 1980 ) and nomoto ( 1982a ) ( the `` double detonation '' model ) , and numerically studied , in one dimensional ( 1d ) numerical simulations , by woosley , taam & weaver ( 1986 ) , and by livne ( 1990 ) . the earliest 2d simulations were made by livne & glasner ( 1991 ) . such explosions were further investigated by woosley & weaver ( 1994 ) , and by livne & arnett ( 1995 ) . the `` robustness '' of the double detonation model has been recently checked in 2d and 3d simulations by moll & woosley ( 2013 ) , whilst the conditions for producing helium detonations have been examined by woosley & kasen ( 2011 ) . the first evidence of a sub chandrasekhar explosion was found by ruiz lapuente et al . ( 1993 ) ( later confirmed by mazzali et al . 1997 ) , from modeling the spectra of sn 1991bg . whether double detonation models produce explosions characteristically similar to those of sne ia remains an open question , but ruiter et al . ( 2011 ) find that the helium star channel would have delay times @xmath15 500 myr ( `` prompt explosions '' ) , while the double white dwarf channel ( c+o plus he white dwarf ) would have longer delay times , as mentioned above . successful accretion of hydrogen , leading to growth of a c+o white dwarf up to the chandrasekhar mass through steady shell hydrogen and helium burning , thus requires high accretion rates . that can be achieved for binary stellar companions at different stages ot their evolution , but if hydrogen accumulates faster than it can be burned , a red giant like structure should form , soon engulfing the companion and leading to a common envelope stage . a solution to this problem was found by hachisu , kato & nomoto ( 1996 ) : when the mass accretion rate exceeds the maximum rate at which hydrogen can be burned , there is no static envelope solution , and the excess material is blown off in a wind . the optically thick wind solution had been previously found to explain the light curves of nova outbursts by kato & hachisu ( 1994 ) . white dwarfs accreting mass at high rates should emit large amounts of radiation in the x ray band , and they should appear as luminous supersoft x ray sources ( van den heuvel et al . 1992 ; di stefano & nelson 1996 ; yungelson et al . 1996 ; kahabka & van den heuvel 1997 ; li & van den heuvel 1997 ; orio 2006 ) . however , as we will see in section 7 , that might be in conflict with the observed x ray emission of elliptical galaxies and galaxy bulges . another possible channel involving a c+o white dwarf plus a non degenerate companion star is the core degenerate scenario recently advocated by soker ( 2013 ) : a chandrasekhar or super chandrasekhar white dwarf is formed from the merging of a white dwarf with the hot , more massive core of an agb star . the initial white dwarf is disrupted and its material accreted by the agb core , that leading to the formation of a rapidly spinning , more massive white dwarf . the delay time till explosion would then be given by the spin down time of the new white dwarf . mergings of a white dwarf with the core of an agb star , following a common envelope episode , had also been considered as a production mechanism of sne ia by sparks & stecher ( 1974 ) and by livio & riess ( 2003 ) . soker et al . ( 2013 , 2014 ) have proposed the core degenerate scenario to explain the characteristics of two very different sne : ptf11kx and sn 2011fe . in the preceding , we have not dealt with the especifics of the hydrodynamic modeling of the different types of explosions surveyed , which has reached unprecedented standards of realism ( see hillebrandt et al . they are being matched by 3d models of the spectra that should arise from the explosions ( baron , hauschildt & chen 2009 ) . the delay time distribution ( dtd ) is given by the time evolution of the sn rate that would follow an instantaneous burst of star formation . it is related to the observed rate @xmath16 by : @xmath17 where @xmath18 is the dtd , @xmath19 is the star formation rate , and @xmath20 and @xmath21 are in the sn rest frame ( see , for instance , ruiz lapuente & canal 1998 ) . early research on sne ia rates and dtds indicated the need of a two component model at least : one that could be fitted with a short dtd population ( @xmath22 yr ) and another one with a long dtd population ( 34 gyr ) ( scannapieco & bildsten 2005 ; mannucci et al . 2006 ; brandt et al . recent studies ( maoz & mannucci 2012 ) suggest that the dtd peaks at the shortest times , as a function @xmath23 ( see also mannucci et al . 2005 , and oemler & tinsley 1979 ) . such a function characterizes the dominance of double degenerate mergings . as already mentioned in the previous section , badenes & maoz ( 2012 ) find that the total merger rate of white dwarf pairs ( for sub chandrasekhar wds ) is similar to the observed sne ia rate ( see also ruiter et al . 2011 ) . the sne ia searches in clusters of galaxies at high redshifts ( barbary et al . 2012 ) have indicated a @xmath24 behavior of the rates , with @xmath25 @xmath26 -1.41@xmath27 , which also favors the wd merging channel . such best fit value is consistent with measurements of the late dtd in field galaxies ( totani et al . most predictions for the sd scenario show a steeper late time dtd ( greggio 2005 ; ruiter et al . 2009 ; mennekens et al . 2010 ) , whith @xmath25 ranging from @xmath25 @xmath26 -1.6 ( greggio 2005 ) to @xmath25 @xmath15 -3 ( mennekens et al . 2010 ) , depending on the details of the scenario and on how the binary evolution is calculated ( see , however , hachisu et al . 2008 , and pritchet et al . recently , bonaparte et al . ( 2013 ) have computed the cosmic sn ia rate , for several cosmic star formation rates and progenitor models , and compared it with the observational data . no firm conclusions can be derived , concerning the sn ia progenitors , but the existence of prompt sne ia , exploding within the first @xmath28 yr after the corresponding systems are formed , is required , althought their fraction should not exceed 1520% of the total , to be consistent with the chemical evolution of the galaxies ( see , for an updated view of this subject , the review by maoz , mannucci & nelemans 2014 ) . also , from the analysis of the environments of 90 hubble flow sne ia discovered by the _ nearby supernova factory _ , rigault et al . ( 2013 ) find evidence of two distinct populations with different ages : one associated with current star formation and another one corresponding to passively evolving environments . if the sne ia explosions would come from white dwarf mergings , one would expect to see carbon and oxygen in the early time spectra , coming from the surrounding clumps of this material , originated by the merging . with the access to large numbers of sne ia , the _ sn factory _ has found c ii absorption lines in the spectra of several of them ( thomas et al . these authors estimate that 22@xmath29 @xmath30 of sne ia exhibit c ii signatures as late as 5 days before maximum light . in some cases one can treat them as spherically symmetric absorptions , in others as `` carbon blobs '' . in the context of explosions from the merging of two c+o white dwarfs , the presence of this photospheric carbon at high velocities seems justified . parrent et al . ( 2011 ) have studied both the spherically symmetric c ii absorption and the non spherically symmetric cases . altavilla et al . ( 2007 ) found evidence of a c blob in the normal sn ia sn 2004dt . folatelli et al ( 2012 ) have also found evidence of unburnt carbon in sne ia from the _ carnegie supernova project _ , in 30@xmath30 of the objects ( see figure 1 ) . in the case of the normal sn 2011fe , in the m101 galaxy , the availability of early spectra has allowed to see both c ii and o i emissions ( parrent et al . 2012 ; nugent et al . the absorption of o i appears at higher velocities , which suggests that sn 2011fe may have had an appreciable amount of unburned oxygen within the outer layers of the ejecta . mazzali et al . ( 2014 ) , from modeling of the spectral evolution , find that the high velocity tail of the ejecta differs from the predictions of both deflagration and delayed detonation models of the explosion . 6580 and @xmath317234 lines , and a matching synthetic spectrum from a model with carbon ( dashed line ) . a synthetic spectrum from a model without carbon is shown in dotted lines . ( see folatelli et al . ( courtesy of gaston folatelli . aas . reproduced with permission ) . ] nomoto , kamiya & nakasato ( 2013 ) suggest that unburnt c inside the ejecta might just come from asymmetric sne ia explosions , in a single degenerate model , for instance , if off center ignitions take place ( maeda et al . however , it is also possible that unburnt c would be left inside the ejecta in explosions coming from mergings of c+o wds , even from violent ones . such carbon would not have been ignited by the detonation that takes place in such mergings ( hicken et al . 2007 ; pakmor et al . 2010 ; hillebrandt et al . 2013 ) . in that case , we would have a minimum of @xmath930 @xmath30 of sne ia coming from double degenerate systems . it is important to study through spectropolarimetry ( wang and wheeler 2008 ; hflich 1991 ; jeffery 1989 , 1990 ) the asymmetry of the c ii and o i features . thus far , line asymmetries have been seen in some sne ia , such as sn 2004dt ( wang and wheeler 2008 ) . the merging models of c+o wds need testing in the nebular phase . as discussed in rpke et al ( 2012 ) , several properties of the merging models could be observable in that phase . in general , the model of ( initial ) deflagration of a wd when reaching the chandrasekhar mass leads to a high degree of neutronization , since burning occurs at densities higher than @xmath32 . such neutronization can be seen , for instance , from the nebular emission of ni ii ( coming from the stable isotope @xmath33ni ) . in the violent merging of two white dwarfs , instead , the burning occurs at peak densities below @xmath32 ( rpke et al . 2012 ) , and nebular emission of stable ni should not be so prominent . in fact , the nebular emission of the decay products of @xmath8ni is expected to be asymmetric , both in the single degenerate model and in the non violent merging of two c+o wds , the asymmetry arising from the material produced by a deflagration initiated off centre , when the wd reaches the chandrasekhar mass . the ni produced later , in the detonation phase , would be more symmetrically distributed , instead . those asymmetries of the nebular emission by the products of @xmath8ni decay have already been seen , according to maeda et al . ( 2010a , b ) . maeda et al . ( 2010a ) assert that the nebular spectra do reveal ignitions offset from the centre of the wd , and that this is a generic feature of sne ia . a point made by rpke et al . ( 2012 ) is that , in the violent merging of c+o wds , the detonation of the secondary wd , at low densities , should introduce copious amounts of oxygen in the innermost ejecta . that could give rise to visible [ o i ] @xmath34 6300 , 6364 . this has been seen in sn 2010lp ( kromer et al . 2013a ; taubenberger et al . 2013a ) , and it gives strong support to that scenario . a different question ( ruiz - lapuente 1996 ) is that , at lower densities , the [ fe ii ] and [ fe iii ] lines are less collisionally excited and become weaker than what is seen in the data . this was shown for the sub chandrasekhar edge lit detonations of livne & arnett ( 1995 ) which gave a very poor fit to the observations , while chandraskehar models with 0.6 m@xmath35 of @xmath8ni fit very well the data of normal sne ia . the fit to the nebular data for the new class of sub chandrasekhar explosion models remains to be tested , since the result just mentioned came from the models of he detonations in sub chandrasekhar wds available in the 90 s ( ruiz lapuente 1996 , hereafter r96 ; see figure 2 ) . on the other hand , in the models based on chandrasekhar mass wds resulting from accretion of h from a non degenerate star , one would expect to see emission of h at @xmath316563 . such emission is not seen in normal sne ia ( leonard 2007 ) . hydrodynamic simulations indicate that seeing the h@xmath25 emission is related to the mixing of h with other elements . in a hydrodynamic simulation for a main sequence companion to the sn ( liu et al . 2012 ) , up to 19@xmath30 of the total mass of the companion star is stripped by the impact of the sn ejecta , and those debris should mix with the most slowly moving layers of the ejecta . there have been a few sne ia with h@xmath25 in emission . such are , for instance , the cases of sn 2002ic ( hamuy et al . 2003 ) , of sn 2005gj ( aldering et al . 2006 ) , and of ptf11kx ( dilday et al . 2012 ) . we will address this point in section 13.2 . as first seen by patat et al . ( 2007 ) , some sne ia have variable na i d absorption lines , significantly blueshifted with respect to the absorption features of other elements . this result has been confirmed by simon et al . ( 2009 ) , sternberg et al . ( 2011 ) , and foley et al . ( 2012 ) ( see figure 3 ) . the general interpretation is that a significant fraction of sne ia progenitor systems have outflows of material previous to the explosion . the na i d lines arise from the ionized circumstellar medium ( csm ) . foley et al . ( 2012 ) find a correlation with higher velocity ejecta in the sne ia that show blueshifted na i d line profiles . they suggest the possibility that progenitor systems with strong outflows tend to have more kinetic energy per unit mass than those with weak or no outflows . patat et al . ( 2013 ) have shown that the sn 2011fe was surrounded by a `` clean '' environment , and there is a lack of time variable blueshifted absorption features . they found sn 2011fe consistent with the progenitor being a binary system with a main sequence or even a degenerate star . nugent et al . ( 2011 ) found that the exploding star was likely a c+o wd and , from the lack of an early shock , that the companion was most likely a main sequence star or there is no surviving companion . ( 2011 ) , from pre explosion images , also exclude companions more evolved that subgiants ( see section 9 ) . bloom et al . ( 2012 ) also find that only degeneracy supported compact objects wds and neutron stars are viable as the primary star . with few caveats , they also restrict the companion ( secondary ) star radius to @xmath36 , that excluding roche lobe overflowing red giant and main sequence companions to high significance . it would be interesting to see if , within the sample of sne ia showing c ii and o i absorptions , there are cases of outflows . it is tempting to think that the supernovae with variable na i d features are connected to nova precursors and the ones showing c and o in the outermost layers are connected with mergings of wds , instead . recently , sternberg et al . ( 2013 ) have found that 18% of the sne ia events show time variable na i d features associated with circumstellar material . one might tentatively associate them with recurrent novae . recurrent novae are binaries harboring a wd close to the chandrasekhar mass . classical novae are the outcome of unstable thermonuclear burning in accreting white dwarfs . those white dwarfs have typical masses @xmath37 . if the accretion rate and white dwarf mass are high , the time between flashes can become short enough that the recurrence can be observed . due to the large accretion rate and insignificant mass loss by ejection , it has been proposed that in recurrent novae ( rne ) the white dwarfs may grow to the chandrasekhar mass and give rise to sne ia ( starrfield et al . 1988 ; schaefer 2010 ) . indeed , rne can be the progenitors of a part of the sne ia that show variable circumstellar na absorptions . amongst recurrent novae , there are those of the u sco type , where the donor is a main sequence or subgiant star , and those of the rs oph type , where the donor is a red giant . the supernova ptf11kx , classified as a sn ia csm for its interaction with the circumstellar medium , is believed to have a symbiotic recurrent nova as its progenitor system ( dilday et al . 2012 ) , although it has also been attributed to a violent prompt merger of a white dwarf with the core of a massive agb star by soker et al . ( 2013 ) , who also argue that the mass of the shell surrounding ptf11kx is too high to have been produced by a recurrent nova . they estimate the hydrogen mass in the shell to be @xmath38 . also , l , yungelson & han ( 2006 ) have shown that symbiotic novae are unlikely sne ia progenitors , due to their low efficiency in hydrogen accumulation . sahman et al . ( 2013 ) suggest that the recurrent nova cl aql will become a sn ia within 10 myr . they find that the mass of the white dwarf is 1.00 @xmath39 0.14 m@xmath7 , and the mass of the companion is 2.32 @xmath39 0.19 m@xmath7 . the radius of the latter is 2.07@xmath39 0.06 r@xmath7 . they estimate that the secondary is a slightly evolved a type star , and suggest that the system is rapidly evolving into a supersoft x ray source . patat ( 2011 ) sees , in the variable sodium lines of some sne ia , a possible connection with recurrent novae . recently , soraisam & gilfanov ( 2014 ) have compared the nova statistics for m31 with the sne ia rates . they find that significant mass accumulation , in the unstable burning regime , is only possible for wds with masses below 1.25 @xmath0 . more massive wds do not significantly accumulate mass . thus , the final stage of mass growth can not occur at low mass accretion rates , when the burning is unstable . therefore , to be sne ia progenitor candidates , the systems should go into the stable burning regime in the final phases . systems consisting of a mass accreting white dwarf and a roche lobe filling , more massive , slightly evolved main sequence or subgiant star , steadily burning h , should appear as luminous supersoft x ray sources ( see section 2 ) . with varying mass accretion rates , however , they can also burn h unstably , at times , and then appear as recurrent novae of the u sco type . nomoto et al . ( 2002 ) ( see also hachisu et al . 1999 ) propose an scenario in which a c+o white dwarf is formed from a red giant star with a helium core of @xmath40 . following a first common envelope episode , a helium star results and then evolves to form a c+o white dwarf of @xmath41 . a part of the helium envelope would have been transferred to the main sequence companion . the white dwarf would thus accrete and burn a mixture of h and helium . depending on the mass accretion rate , a wind might be blown from the surface of the white dwarf . it should not be optically thick enough to absorb all the x ray emission , but it could , however , absorb a part of the soft x rays ( hachisu , kato & nomoto 2010 ; see next section ) . it will be intersting to see whether variable circumstellar material would be observed in this scenario . the two different channels to sn ia explosions , the single degenerate path and the double degenerate one , lead to very different predictions for the x ray emission ( gilfanov & bogdn 2010 ) . whereas no strong x ray emission is expected , prior to explosion , in the merger scenario , in the single degenerate scenario the white dwarf that accretes mass from a non degenerate companion becomes a source of x rays for about 10@xmath42 yr before the explosion . if the growth in mass of the white dwarf is due to accretion of hydrogen , followed by steady burning of hydrogen into helium , one expects a thermonuclear luminosity @xmath43 where @xmath44 is the energy per unit mass released by hydrogen burning , @xmath45 is the hydrogen mass fraction in the accreted material , and @xmath46 is the mass accretion rate . for standard values . @xmath47 erg s@xmath48 , which is more than one order of magnitude larger than the gravitational energy released by accretion , @xmath49 ( @xmath50 and @xmath51 being the mass and radius of the white dwarf ) . that sustains a surface temperature of the white dwarf : @xmath52 such sources are observed in the milky way and nearby galaxies , and they are known ( see section 2.2 ) as supersoft sources ( van den heuvel et al . 1992 ; kahabka & van den heuvel 1997 ) . gilfanov & bogdn ( 2010 ) report that the observed x ray flux from six nearby elliptical galaxies and galaxy bulges is a factor @xmath93050 less than predicted by the accretion scenario , based upon an estimate of the supernova rate . they conclude that no more than @xmath9 5% of type ia supernovae in early type galaxies can be produced by white dwarfs accreting hydrogen in binary systems . hachisu , kato & nomoto ( 2010 ) , however , suggest that there is , in fact , no inconsistency , since symbiotic supersoft sources have fluxes @xmath53 erg s@xmath48 in the 0.30.7 kev range . there is also uncertainty in theoretically deriving the x ray luminosity of the supersoft sources , due to the still rough atmosphere models of mass accreting wds and to the neglect of absorption of the soft x rays by the cool wind material from the companion star . on the other hand , x ray emission can inform us about the circumstellar medium around the sne ia ( badenes et al . 2007 ) . in that work , the authors disfavor optically thick accretion winds from the wd surface . such winds would produce large cavities in the interstellar medium ( ism ) . the fundamental properties of the seven supernova remnants ( snrs ) of type ia of their sample ( sn 1885 , kepler , tycho , sn 1006 , 0509 - 67.5 , 0519 - 69.0 , and n103b ) are incompatible with snr models expanding inside such cavities . in general , the search for x ray emission at the time of the supernova outburst has also provided probes of the circumstellar medium , which so far is considered to be of low density ( hughes et al . recently , rusell & immler ( 2012 ) have examined 53 sne ia observed with the _ swift _ x ray telescope , and their upper limit to the x - ray emission gives further evidence that the companion stars in sne ia are neither massive nor evolved ( post main sequence ) , due to the corresponding limit on wind mass loss rate inferred . they can not rule out , instead , main sequence star companions , with mass loss rates @xmath54 . a double white dwarf system is also permitted , due to the lack of circumstellar interaction and hence lack of x rays there . the tightest constraint on the progenitor of a sn ia , coming from x ray emission , is that for sn 2011fe ( horesh et al . 2012 ; margutti et al . 2012 ) . the x ray observations yield an upper limit 2@xmath5510@xmath56 m@xmath7 yr@xmath48 to the mass outflow ( assuming a wind velocity @xmath57 = 100 km s@xmath48 ) . as we have seen in section 2 , accretion at a rate @xmath9 10@xmath58 m@xmath7 yr@xmath48 is thought to be necessary for stable accretion and nuclear burning on the surface of a white dwarf ( nomoto 1982 ) . supersoft sources can achieve those rates ( kahabka & van den heuvel 1997 ) , although the rates can also be either lower or higher , in these sources . horesh et al . ( 2012 ) analyse the models of interaction of the wind with the circumstellar material and conclude that the data from sn 2011fe can rule out a symbiotic system , but not a main sequence or subgiant mass donor . the same conclusion is reached from their analysis of the radio emission ( see next section ) . margutti et al . ( 2012 ) also discard symbiotic systems , as well as roche lobe overflowing subgiants and main sequence secondary stars if @xmath59 1% of the transferred mass is lost at the lagrangian points . the nearby sn 2014j has provided a new opportunity to test the presence of x - ray emission from the pre explosion x ray images ( nielsen et al . according to these authors , the upper limits from the _ chandra _ x ray observatory do exclude a classical super soft source as the progenitor . near the chandrasekhar mass , the effective temperature corresponding to the stable nuclear burning on the wd surface exceeds 100 ev . for this temperature , the 3@xmath60 upper limit on the bolometric luminosity is @xmath61 , assuming a column density of hydrogen @xmath62 and a black body spectrum . that confidently excludes a classical super soft source during the final stages of the mass accumulation by the progenitor . due to the large absorption , the _ chandra _ upper limits are less constraining at lower temperatures . they do not exclude , therefore , less conventional progenitors , e.g. a wd enshrouded in an optically thick envelope or wind . deep x ray observations of the post explosion environment ( margutti et al . 2014 ) now rule out single degenerate progenitors with steady mass loss until the time of the explosion ( the maximum mass spilled by the system should be @xmath63 1% ) , and do only allow recurrent novae with a recurrence time @xmath15 300 yrs , stars where the mass loss ceases before the explosion , or double wd systems . from a different angle , there have been , from the start , great expectations to detect hard x ray and @xmath64ray photons from sn 2014j ( isern et al . 2013 ; the & burrows 2014 ) , since the supernova was close enough to be detected by _ integral _ and _ nustar_. it has , indeed , been detected by _ integral _ ( churazov et al . the line flux suggests that 0.62 @xmath39 0.13 @xmath0 of radioactive @xmath8ni have been synthesized in the core . the mass of the ejecta ( from the continuum emission ) would be @xmath9 1.4 @xmath0 and composed of roughly equal fractions of iron group and intermediate mass elements . there is thus agreement with the model of the explosion of a chandrasekhar mass wd . diehl et al . ( 2014 ) find that about 0.06 m@xmath7 of @xmath8ni should be at the outskirts of the ejecta . this has suggested that he accreted by the white dwarf could have exploded in the external layers and triggered the central ignition . the lack of radio emission from type ia supernovae has been useful in discarding one type of single degenerate path as a major contributor : sne ia from symbiotic systems . in symbiotic systems , the white dwarf accretes mass from the wind of a giant or agb companion . the wind accretion should produce radio emission when the sn ejecta interact with the circumstellar environment created by such systems . panagia et al . ( 2006 ) set upper limits on mass loss rates of @xmath9 10@xmath58 m@xmath7 yr@xmath48 . hancock et al ( 2011 ) suggest upper limits , to the average mass loss rate of the companion by stellar wind , of @xmath65 . these authors say that such limit is inconsistent with sne ia in which the accretion comes from intermediate or high mass companions . instead , a main sequence star having fast winds ( @xmath66 10 km @xmath48 ) could remain undetected , even with much higher mass loss rates . the nearby supernova sn 2011fe has made possible the most sensitive radio study of a sn ia made up to now ( chomiuk et al . the data set direct constraints to the density of the surrounding medium at radii @xmath67 cm , that implying an upper limit on the mass loss rate from the progenitor system of @xmath68 ( assuming a wind speed of 100 km s@xmath48 ) , or expansion inside a uniform csm with density @xmath69 6 @xmath70 . drawing from the observed properties of non conservative mass transfer in accreting white dwarfs , they use the limits on the density of the circumstellar environment to exclude a good fraction of the parameter space of possible progenitor systems of sn 2011fe . a symbiotic progenitor system can be ruled out , as well as any other system characterized by a high mass transfer rate onto the white dwarf which could give rise to optically thick accretion winds . assuming that a small fraction , @xmath9 1% of the mass transferred , is lost from the progenitor system , they can also eliminate much of the parameter space occupied by potential progenitors such as recurrent novae or , alternatively , progenitors undergoing stable nuclear burning . they eliminate , therefore , for sn 2011fe , a large fraction of the parameter space associated with popular single degenerate progenitor models , leaving only a limited region , mostly inhabited by some double degenerate systems , as well as by exotic single degenerates in which a sufficient time delay takes place between mass accretion and sn explosion . the even closer sn 2014j has also been observed in radio with the vla , without any detection ( chandler & marvil 2014 ) . this points out to a surrounding medium of low density as well . it has been possible to put constraints on the progenitors from pre explosion images in other galaxies . this endeavour has been of particular interest for sn 2011fe , since it exploded in the galaxy m101 , at 6.4 mpc only . another nine sne ia with preexisting _ hst _ data on their host galaxies have also been close enough ( within 25 mpc ) to search for the progenitors . it has only been possible to set upper limits which rule out normal stars with initial masses larger than 6 m@xmath7 at the tip of the agb branch , young post agb stars with initial masses larger than 4 m@xmath7 , and post red giant stars with initial masses above 9 m@xmath7 ( li et al . 2011a ) . the case of sn 2011fe arose great expectations , since the sn was much closer than in previous occasions . there was , however , no object seen at the location of the supernova in pre explosion images , down to magnitude 27.4 ( in the acs / f435w band ) ( li et al . the conclusion of the analysis is that , for sn 2011fe , the red giant progenitor is excluded , while a subgiant or a main sequence companion star still are possible progenitors , from the imaging approach . very recently , the nearby supernova sn 2014j in m82 has been tested in the same way . this supernova is only at 3.5 mpc . rosa , greggio & botticella ( 2014 ) have analyzed deep archival _ wfc3/ir images of m82 in the f110w and f160w filters , taken in jan . 2010 , and used them in an attempt to identify a progenitor for the sn , by registering the _ hst _ images with images of the sn taken on jan . as in the sn 2011fe case , they can again exclude a red giant companion . the limits are consistent with the companion being ( if not another wd ) a subgiant or a main sequence star . goobar et al ( 2014 ) have also explored the _ hst _ images of the explosion region . the observational limits , however are not as constraining here as in the case of sn 2011fe . cm ( green lines ) ; a 6 m@xmath7 ms companion at @xmath71 cm ( blue lines ) ; a 2 m@xmath7 ms companion at @xmath72 cm ( red lines ) , and the lack of companion ( black lines ) . ( courtesy of dan kasen . aas . reproduced with permission ) . ] according to kasen ( 2010 ) , the impact of the supernova debris on the companion produces a bright x ray ( 0.12 kev ) burst lasting from minutes to hours . the diffusion of this x ray emission gives rise to a longer lasting optical / uv emisson which exceeds the radioactively powered emission from the supernova for the first few days after the explosion . this effect can be seen in figure 4 . the signatures are prominent for viewing angles looking down upon the shocked region , which should be about a 10% of the times . kasen ( 2010 ) concludes that the current optical and uv data do effectively constrain the red giant companion channel , disfavoring it . zheng et al . ( 2014 ) ( see their figure 3 ) , have presented very early light curve data from sn 2014j . when comparing them with the predictions of kasen ( 2010 ) , the non companion case appears favored . high z searches might provide the tightest constraint on the sn progenitors . goldhaber et al . ( 2001 ) show , in their figure 1 , the light curves of 35 high z sne ia found by the _ supernova cosmology project ( scp)_. these data , as well as more recent ones ( conley et al . 2006 ; hayden et al . 2010 ; bianco et al . 2012 ) do not show any evidence for a companion in the early light curves of high z sne ia samples . hayden et al . ( 2010 ) , from a simulation of the shock interaction with a companion , rule stars with masses larger that 6 m@xmath7 and also disfavour red giant companions . another method to identify the progenitors of the sne ia was proposed by ruiz lapuente ( 1997 ) : to inspect the stars within the innermost regions of the galactic sne ia remnants in search of the mass donor star ( to either find it or to show its absence ) , in the area where it should still remain after the explosion , moving with a peculiar velocity gained from the orbital velocity in the binary system before the explosion . a given star , to be candidate to donor in a sn ia explosion , should be at the distance of the remnant , moving with enhanced velocity , and maybe also show signs of contamination by the iron peak rich part of the supernova ejecta . a subgiant named tycho g was found to be a likely candidate companion for sn 1572 ( ruiz lapuente et al . 2004 ) , since it is close to centre of the snr , at a distance compatible with that of the remnant , and it is in a region where stars follow the rotational pattern of the galaxy , but it has a radial velocity well above the 20 to 40 km s@xmath48 , typical at the distance of sn 1572 . it also has a high proper motion . a chemical analysis of the star showed enhancement of ni in the surface ( gonzlez hernndez et al . 2009 ) , suggesting contamination by the supernova ejecta . that was disputed by kerzendorf et al . ( 2009 , 2013 ) , who argued that all those characteristics might just correspond to a chance interloper . based on greatly improved proper motion measurements and a more refined chemical analysis , bedin et al . ( 2014 ) , however , have shown that the probability of having found such an interloper at random is extremely low . schaefer & pagnotta ( 2012 ) have looked as well for a companion star in a snr of the lmc ( snr 050967.5 ) and found no star that could have been the mass donor in the progenitor system . their result points to the supernova having resulted from merging of two white dwarfs . in addition , edwards , pagnotta & schaefer ( 2012 ) have examined the innermost area of the remnant snr 051969.0 , also in the lmc , and eliminated red giants , subgiants , and he stars as possible companions of the sn . gonzlez hernndez et al . ( 2012 ) ( see also kerzendorf et al . 2012 ) have inspected the remnant of the galactic sn 1006 , determining distances and chemical abundances for all candidate stars within the innermost 27% of the area of the remnant . the lack of detection of any viable candidate star rules out red giant and subgiant stars , as well as any star brighter than m@xmath73 @xmath9 + 4.9 ( approximately equal , or slightly less than the solar luminosity ) . the key point , from the theoretical point of view , is that all groups that have simulated the impact of the ejecta of a supernova on its companion star ( marietta , burrows & fryxell 2000 ; pakmor et al . 2008 ; pan , ricker & taam 2012a ; liu et al . 2012 , 2013 ) consistently find that the companion survives the explosion . this important conclusion is the basis for the observational searches . there has been debate on whether the surviving companion of a wd plus main sequence system , or a wd plus subgiant system , would show rapid rotation after the explosion ( gonzlez hernndez et al . 2009 ; kerzendorf et al . 2009 , 2013 ; bedin et al . hydrodynamic simulations by pan , ricker & taam ( 2012a ) , for main sequence companions , show that they would lose about half of their initial angular momentum , their rotational velocity dropping to a quarter of the original rotational velocity . the simulations by liu et al . ( 2012 ) , also for a companion on the main sequence , equally show that its rotational velocity can be significantly reduced by the effects of the impact of the sn ejecta , falling to a 32%14% of its pre explosion value , due to remotion of 55%89% of the initial angular momentum , taken away by the material stripped during the interaction with the supernova ejecta . it is easy to see ( fig . 1 of marietta , burrows & fryxell 2000 , for instance ) that , in the case of a 1.1 m@xmath7 subgiant , remotion of @xmath74 by the impact of the ejecta means reducing the radius of the star , immediately after the impact , to about 1/3 of its previous value only ( in front of about 1/2 in the main sequence case ) , so the drop in rotational veocity must be correspondly larger . in their simulations , pan , ricker & taam ( 2012a ) find that the contamination with ni in the companion star , from the passage of the sn ejecta , is of @xmath75 , for a main sequence star , and of @xmath76 for a red giant . another point concerns the luminosity to be expected , for the surviving companions of recent sne ia . podsiadlowski ( 2003 ) found that , in the case of a subgiant , the star , 10@xmath7710@xmath78 yr after the explosion , might be either significantly overluminous or underluminous , that depending on the amount of heating and the amount of mass stripped , as well as on the previous binary mass transfer . more recently , shappee , kochanek & stanek ( 2013 ) have claimed that , in the case of a main sequence companion ( and maximizing the heating ) , the object should remain significantly overluminous for the above time lapse , but the more realistic simulations of pan , ricker & taam ( 2012b ) , also for a main sequence star , predict luminosities much closer to that of tycho g , @xmath79 yr after the explosion . in the case of a subgiant , a larger fraction of the material should be directly stripped by the shock wave generated by the impact of the sn ejecta , and there should be less heating of the fraction of the envelope that remains bound . in all the preceding considerations , it has been implicitly assumed that there is no significant delay between the accretion phase that brings the white dwarf to the chandrasekhar mass and the sn explosion . that has been questioned by di stefano , voss & claeys ( 2011 ) , who propose a model in which the c+o white dwarf , spun up by accretion of matter and angular momentum , is able to sustain a mass above the chandrasekhar mass , and only reaches the conditions for explosive c burning when it has lost enough angular momentum , on a time scale that may be long enough to allow the companion star to evolve to the white dwarf stage . also , overcoming the chandrasekhar mass limit would allow exhaustion of the envelope of the companion star , only its compact core remaining at the end of the mass transfer phase . based on that , di stefano & kilic ( 2012 ) argue that the lack of evidence of ex companion star in the above mentioned snr 050967.5 does not mean that such companion does not exist , since it could have become a c+o or a he white dwarf by the time of the explosion . the problem of the time scale of spin down of the primary white dwarf has been very recently addressed by meng & podsiadlowski ( 2013 ) , who obtain an upper limit of a few 10@xmath42 yr . such times would still allow a companion star to become dimmer than the upper limit set by schaefer & pagnotta ( 2012 ) , according to di stefano and kilic ( 2012 ) . another remnant being now studied is the kepler snr ( sn 1604 ) . from the lack of bright stars in the field , kerzendorf et al . ( 2014 ) have ruled out red giants as possible companions of sn 1604 . vlt observations with flames of the stars in more than 20% of the inner core of the snr have now been granted ( ruiz lapuente et al . . it will be very interesting to see what high resolution spectra reveal . other fairly symmetrical galactic snr are planned to undergo a similar scrutiny another approach to dilucidate the progenitors of sne ia is to reconstruct the orbit of systems containing a wd . along these lines , the tight binary system cd30@xmath80 11223 has been found to consist of a c+o white dwarf plus a hot helium star ( geier et al . the system turns out to be a progenitor candidate for the double detonation sn ia scenario ( see section 2.2 ) . wang & han ( 2012 ) have studied this kind of possible progenitor system ( see also wang et al . a c+o white dwarf is first formed , from the initially more massive star , and the system then is in a close orbit . the mass donor later reaches roche lobe overflow and becomes a he star , but evolves , after exhaustion of the central he , to the red giant stage . the system thus becomes a c+o wd plus a he red giant . the mechanism can also work with the he star still staying in the main sequence phase . fink , hillebrandt & rpke ( 2007 ) reproduce , in a 3d simulation of a double detonation , these explosions . they find that the he detonation in a shell succesfully gives rise to a second detonation in the c+o core . in the outcome , @xmath8ni masses about 0.400.45 m@xmath7 are produced , with rapidly expanding @xmath8ni in the outer layers . they note , however , the lack of observations of this type of explosion ( sn 1991 t could resemble it , but the core contained 0.8 m@xmath7 of @xmath8ni ) . a c+o wd plus a he donor could , instead , be the progenitor of the so called `` type .ia '' supernovae . we do not include these in our physical diagram for sne ia , however , since their likely he features at maximum ( kasiwal et al . 2010 , and references therein ) rule them out as possible sne ia ( we only consider , in this paper , the types of explosions , either total or partial , that by their features at maximum can be regarded to be such ) . from the many systematic searches made at various redshifts , it has been possible to identify sne ia that fall well away from the phillips ( 1993 , 1999 ) relationship between the peak luminosiy and the rate of decline of the light curve . such relationship was traditionally related to the amount of @xmath8ni synthesized in chandrasekhar mass models , since its variation not only correlated with that of the maximum luminosity , but it also produced opacity variation in the envelope of the sn , which resulted in slower decline rates of the light curve for larger ni masses . in such view ( hflich & khokhlov 1996 ; pinto & eastman 2000 ; bravo et al . 2009 ) , only variations among chandrasekhar mass wd explosions were the cause of the relationship . a new proposal , which completely changes the explanation , is to assume that the phillips relationship results from variation in the viewing angle of the family of detonations of merging sub chandrasekhar explosions ( moll et al . 2013 ) . that marks a new turn in the search for the physical basis of a relationship that is crucial for cosmology , and it thus calls for further investigation . since this is relevant for cosmology , we mention that the method of determination of h@xmath81 using nebular spectra of sne ia ( ruiz lapuente 1996 ) favored a value of 68 km @xmath39 7 ( stat ) @xmath39 1 ( updated systematic error ) km s@xmath48 mpc@xmath48 , in good agreement with the latest results from the _ planck _ satellite ( ade et al . 2013 ) . the light curves of sne ia ( hflich & khokhlov 1996 ) favored 67 @xmath39 9 km s@xmath48 mpc@xmath48 ( also in agreement with _ planck _ ) . riess et al . ( 2011 ) , within the _ program , had found @xmath82 km s@xmath48 mpc@xmath48 , in tension with the _ planck _ result ( but see , more recently , riees 2014 ) . a few discoveries of highly luminous ( @xmath83 -20.4 ) sne ia suggest the existence of super chandrasekhar mass explosions ( howell et al . these supernovae show very slowly evolving si ii @xmath31 6355 absorption velocity , and they can also show a plateau in their blue light curve ( scalzo et . al . 2012 ) . the very large mass of @xmath8ni needed to explain those events can plausibly be produced by the collision of two white dwarfs ( raskin et al . 2010 ) or by accretion on a rapidly rotating c+o wd . the _ sn factory _ has tried to evaluate the percentage of super chandrasekhar sne ia , and they suggest about 2% ( aldering 2011 ) . a super chandraskhar wd can be formed if it is supported by rapid rotation ( hachisu 1986 ; see also section 10 ) , and the rotating wd is more massive than a non rotating wd with the same central density ( yoon & langer 2005 ) . for a given central density , the density profile is shallower for those more massive wds , and therefore the mass contained within the density range for @xmath8ni production is larger . also , for the same central density , a flame produces more @xmath8ni due to less pre expansion ahead of the propagating flame . in the case of merging , one can have tamped detonations ( howell et al . those tamped detonations in rapidly rotating wds can synthesize amounts of @xmath8ni as high as 1.62 m@xmath7 . taubenberger et al . ( 2011 ) have estimated that the total mass of the wd , in the case of sn 2009dc , was @xmath84 , and the ejected @xmath8ni mass was @xmath85 . the fact that some of those events show c ii absorption features in their spectra reinforces the hypothesis that they come from mergings of two wds . on the other hand , from modeling ( hillebrandt et al . 2013 ) of the violent wd merger scenario ( pakmor et al . 2010 ) , it seems unlikely that these supernovae would come from violent double degenerate mergers . the @xmath8ni mass produced only depends on the mass of the primary wd , there . since exploding c+o wds , in the violent merger model , usually have masses well below 1.3 m@xmath7 ( ruiter et al . 2013 ) , that limits the amount of @xmath8ni produced in the explosion to @xmath86 only . however , recent analyses of the nebular spectra of super chandrasekhar events ( taubenberger et al . 2013a ) , the case of sn 2009dc in particular , indicate that those outbursts can be explained by a merger of two massive c+o white dwarfs , producing @xmath9 1 m@xmath7 of @xmath8ni and @xmath9 2 m@xmath7 of ejecta . that would come from the explosion of a chandrasekhar mass white dwarf , enshrouded by 0.60.7 m@xmath7 of c+o rich material . 6355 feature typical of the sne ia class , as well as prominent fe iii absorption features at @xmath34 4200 and 4900 . in panel b , a comparison of the spectrum of sn 2002ic at the epoch + 6 days from maximum with a spectrum of sn 1991 t , obtained at the + 4 days epoch , shows similarity , except for the h@xmath25 emission , which is not present in sn 1991 t . ( courtesy of mario hamuy . nature publishing group . reproduced with permission ) . ] there is a fraction of supernovae which show narrow hydrogen emission lines . they were first noticed by hamuy et al . ( 2003 ) , in sn 2002ic ( see figure 3 ) . such sn have been labelled in various ways , until recently being dubbed sne ia cms ( silverman et al . 2013 ) . the existence of the sn ia csm class of objects seems to indicate that at least some sne ia do arise from the sd channel , since a h rich csm can form during the evolution of sd systems . hamuy et al . ( 2003 ) suggest that sn 2002ic could have arised from a binary system containing a c+o white dwarf plus a massive ( 37 @xmath0 ) agb star , where the total mass loss in h can reach a few solar masses , since their analysis of the narrow component of h@xmath25 implies a high mass loss rate of @xmath87 ( see figure 5 ) . the accreted mass would come from the wind of the agb star , partially captured by the wd . despite the mass loss rate being so high , however , the supernova has not been detected in radio . dilday et al ( 2012 ) find that the supernova ptf11kx is of type ia , and suggest a symbiotic nova progenitor ( see , however , soker et al . its late time spectrum confirms that it is , indeed , a sn ia . a time series of high resolution spectra of this supernova reveals a complex circumstellar environment , with multiple shells similar to those ejected by nova rs ophiuchi . dilday et al . ( 2012 ) found , from the _ palomar transient factory ( ptf ) _ , that the sn ia csm are about 0.11% of all sne ia . this is more or less consistent with the theoretical expectations for the fraction of sne ia from the symbiotic progenitor channel : between 1 and 30% ( han et al . 2004 ; l et al . 2009 ) . concerning other typical characteristics , sne ia csm have peak absolute magnitudes in the range 21.3 @xmath88 19 mag , with relatively long rise times of @xmath8940 days . they do not emit neither in radio nor at x ray wavelengths . sn 1991bg came as a suprise , being a subluminous sn ia , one order of magnitude fainter than normal sne ia ( filippenko et al . 1992 ; leibundgut et al . 1993 ; ruiz lapuente et al . the amount of @xmath8ni synthesized was only about 0.07 @xmath0 ( ruiz lapuente et al . 1993 ) . it is clearly out of the phillips relation . later , there were many more discoveries of this type , and one could start to think of a sn 1991bg class . li et al . ( 2011b ) quantify this class as 15% of all sne ia . pakmor et al . ( 2011 ) suggest that violent mergers of wds with a primary of 0.9 m@xmath7 reproduce very well the 1991bg like sn . indeed , the simulated optical light curves fit well the data ( hillebrandt et al . 2013 ) . very recently , the presence of [ o i ] @xmath346300 , 6364 emission in the nebular spectrum of sn 2010lp , suggesting that oxygen is distributed in a non spherical region close to the centre of the sn ejecta , has also been interpreted as the result of a violent merger ( taubenberger et al . 2013b ) . these events do exhibit unusually strong ca features at nebular phases , while they look as spectroscopically normal sne ia at maximum . their distribution within their host galaxies shows great similarity with that of the sne ia and indicate old progenitor systems ( lyman et al . according to these authors , they are consistent with helium shell detonations on low mass c+o white dwarfs . the objects display low peak luminosities , fast photometric evolution , high ejecta velocities , strong ca emission lines , and they are located in the extreme outskirts of their host galaxies ( kasliwal et al . ; see figure 6 ) . foley et al . ( 2013 ) identify a new subclass of supernovae called type iax . they have low maximum velocities ( 2000 @xmath90 8000 km @xmath48 ) , and typically low peak magnitudes ( 14.2 @xmath91 18.9 mag ) . in fact , this is the same family of sne ia identified by li et al . ( 2003 ) and dubbed sn 2002cx like sne ia . foley et al . ( 2013 ) find that this subclass comprises 31@xmath92% . of all sne ia . white et al ( 2014 ) , however , reduce it to 5.6@xmath93 . given the large uncertainty , in figure 9 we have plotted an average of the two estimates . those sn2002cx ( or sn iax ) events exhibit iron rich spectra at early phases , like sn 1991 t ( see ruiz lapuente et al . 1991 for the iron rich spectra of sn 1991 t ) , a luminosity as low as events like sn 1991bg ( i.e. 2 mag below normal type ia ) , and expansion velocities roughly half those of normal sne ia . this subclass has a small @xmath8ni production , as seen at late phases . they move fast into the nebular phase , that giving evidence of the small total mass ejected . a survey of the models able to produce such kind of explosions suggests objects made of a 0.6 m@xmath7 c+o wd , with a layer of @xmath94 of he on top , which undergo a he detonation ( li et al . 2003 ; wang , justham & han 2013 ) . the origin of the subclass is still under debate , however , because it is quite inhomogeneous ( narayan et al . white et al . ( 2014 ) divide it in two subclasses : the `` sn 2002cx like '' and the `` sn 2002 es like '' sne ia . the former tend to appear in later type or more irregular hosts , have more varied and generically dimmer luminosities , longer rise times , and they lack a ti ii through in their spectra , when compared to the latter . the frequency of sn 2002cx events , as compared with normal and sn 1991bg like events , has also been estimated by perets et al . variants of sn 2002cx events are seen in faint supernovae such as sn 2008 ha ( foley et al . 2009 , 2010 ) . this last supernova is the faintest member of its subclass . its late time photometry is consistent with the production of just a few times 10@xmath95 m@xmath7 of @xmath8ni , similar to the estimates from the early light curve ( foley et al . the small ejecta and @xmath8ni masses are consistent with a failed deflagration of a wd , that did not disrupt the progenitor ( jordan et al . 2012 ; kromer et al . there is still another model , proposed earlier , for this particular type iax supernova : it is the fallback of a core collapse supernova ( moriya et al . 2010 ) . this model , then , does not treat sn 2008 ha as a thermonuclear supernova , but as the collapse of a c+o star of 13m@xmath7 ( model 13co2 in their paper ) . the boundary between the fallback region and the ejecta is determined by whether the velocity of the region exceeds the escape velocity or not . lyman et al . ( 2013 ) , argue that the host environments and morphologies point to a generally younger population for this subclass . a model which synthesizes 0.003 m@xmath96 of @xmath8ni and ejects 0.074 m@xmath7 of material seems to reproduce the spectra and light curve of sn 2008 ha . foley et al . ( 2013 ) find the environment of some of these type iax sn to be typical of old progenitor systems . they discuss that the most significant reason for their classification of sn 2008 ha as a sn ia is the presence of signatures of the products of thermonuclear processing of c+o , in particular that of sulfur lines , with an intensity that is only typical of sne ia ( foley et al . 2009 ) . very recently , foley et al . ( 2014 ) report the possible detection of the stellar donor of sn 2008 ha in images from the _ different possibilities for the progenitor remain open , though the age is constrained to be @xmath15 80 myr . thus , while type iax sne , fall well below the phillips relation , some of these events should be regarded as subluminous sne ia , which might be linked either to he detonations in a shell or to failed deflagrations . others might originate in core the diversity of the events ( see figures 7 , 8) suggests that several mechanisms take place within the sample of sne iax , explaining the properties of the different observed events . @xmath97 @xmath98 diagram using the cfa3 sample of 185 sne ia ( hicken et al . the solid line is the phillips et al . ( 1999 ) relation . the fundamental relation developed by phillips ( 1993 ) was modified in phillips et al . the figure is adapted from hillebrandt et al . ( 2013 ) , and we use the same sources for superluminous and 91bg - like subluminous events . the diversity of type iax is taken from narayan et al . the area of the ca rich transients is marked on the diagram , following kasliwal et al . the data on the luminous fast evolving sne ia come from perets et al . the proportion in nature of the different events is tentatively given in figure 9 . ] perets et al . ( 2010 , 2011a , b ; poznanski et al . 2010 ) have identified a new subclass of sne ia : rapidly declining sne , such as sn 1885a , 1939c , and 2002bj , that , unlike the iax subtype , are not faint . sn 2002bj had a short rise time ( @xmath15 7 days ) and @xmath99 , which is a post maximum decline much faster than that of the bulk of sne ia ( @xmath100 ) , and even faster than that of sn 1991bg like events ( @xmath101 ) . however , it reached a peak magnitude @xmath102 , which is not faint . sn 1885a and 1939c arose in old environments , which points to also old wds as their progenitors . in the case of sn 2002bj , the ejected mass appears to be low ( @xmath103 ) , that being consistent with the estimated mass of the sn 1885a remnant . perets et al . ( 2011b ) evaluate the frequency of the sne in this subclass as being at least 12% of the global sne ia rate . shell detonations have been suggested as the explosion mechanism ( perets et al . 2010 , 2011a ; poznanski et al . 2010 ) , but its consistency with the observations still remains an open question . it has been presented in section 6 that a significant fraction ( 18% ) of the population of sne ia show time variable na i d features , other features being variable as well . those supernovae are normal sne ia , as the ones that we use for cosmology . their characteristics point to recurrent novae ( patat 2011 ) or to systems with significant outflows prior to explosion , as progenitors . another chunk of the normal ( cosmological ) sne ia likely comes from double degenerate systems ( @xmath9 30% ) , as suggested by the unburnt c and o in the outermost layers ( see section 4 ) . one might ask : how can cosmological sne ia have two different origins ? the reason is that in both cases the whole c+o wd is burnt , giving rise to the light curve whose variation in peak magnitude is due to the amount of @xmath8ni synthesized in the explosion , and the line opacity of those events modulates their rate of decline to give @xmath104 - 0.663[\delta m_{\rm 15}(b ) -1.1]^{2}$ ] , where @xmath105 is the absolute blue magnitude , at maximum , of a sn ia with @xmath106 ( phillips et al . this relation extends the possibility of using sne ia with @xmath107 high , since they are seen to make a continuum with the slower ones . scalzo et al . ( 2014 ) derive , from the bolometric light curves of a sample of sne ia obtained by the _ snfactory _ , a range of both ejected masses and @xmath8ni masses . ejected masses would range , for normal sne ia , from 0.9 m@xmath7 for the fastest decliners to 1.4 m@xmath7 for the slowest ones . the 91t like sne ia eject masses in excess of 1.4 m@xmath7 and produce @xmath8ni masses around 0.81.0 m@xmath7 . the middle part of the sample is occupied by sne ia with chandrasekhar mass ejecta and producing 0.6 m@xmath7 of @xmath8ni ( as derived from the nebular spectra of sne ia in r96 ) . it has been argued ( domnguez et al . 2001 ) that evolutionary effects in the maximum rate of decline relation could make difficult the use of sn ia at very high @xmath108 . branch et al . ( 2001 ) propose an strategy to test evolutionary effects , in the prospect of a space mission devoted to the study of dark energy by means of high@xmath108 sne ia . their proposed plan , now completed for large samples of nearby sne ia , is to study all possible evolutionary effects in sne ia samples at low z. such effects have now been evaluated from big samples , and it has been shown that they do not to interfere with our understanding of dark energy ( ruiz lapuente 2007 ) . the sne ia at @xmath108 = 1.914 ( jones et al . 2013 ) , at @xmath108 = 1.71 ( rubin et al . 2013 ) , and at @xmath108= 1.55 ( rodney et al . 2012 ) look normal and show no evolutionary effects . concerning cosmology , it has been seen that these highest - z observations do suppport the @xmath109 model of our universe ( see also conley et al . 2011 ; sullivan et al . 2011 ) . we have learned a lot about normal ( cosmological ) sne ia from the two recent nearby sne ia : sn 2011fe and sn 2014j . for the first one , the _ hst _ deep images have allowed to discard many single degenerate scenarios ( li et al . 2011a , b ) . sn 2011fe showed unburnt c and o in the very early spectra . one possible explanation is that it came from a double degenerate progenitor ( see nomoto , kamiya & nakasato 2013 for an alternative explanation of the unburnt c and o material ) . for sn 2014j , it has been possible to rule out red giants as the companions of the c+o wd that exploded . it continues to be safe to exclude sn 1991bg events , sne iax , sne ia - csm and super chandrasekhar sne ia ( ca rich transients as well ) from the cosmological samples , as they lay outside the brightness decline rate relation used in cosmology . they might have been responsible ( in particular sne iax and sn 1991bg events ) for some outliers present in the early samples ( such as the first _ _ sne ia ) , for which we lacked explanation at those epochs of the cosmologically motivated sne ia searches . dust absorption in the host galaxy is an important source of systematic error in sn cosmology . a way to minimize its effect on distance determinations is to measure the properties of the sn in the rest frame infrared . it has been demonstrated ( wood vasey et al . 2008 ; kirshner 2010 ; barone nugent et al . 2012 ) that sne ia are better standard candles in the infrared than in the optical wavelengths . the infrared sample of nearby sne ia is steadily growing , now ( friedman et al . 2014 ) . as we have seen , the last decade has brought considerable progress in the still far from closed search for the progenitor systems of the sne ia . a new picture has emerged , where single degenerate progenitors would now make a much narrower channel than it was thought to be in the 90 s . the highlight , here , is that the companions to sne ia , at the time of explosion , are very unlikely to be red giants or supergiants , as well as massive main sequence or subgiant stars . this result comes from searches for companion stars in sne ia remnants , in our galaxy and in the lmc ; by looking at pre explosion images of nearby sne ia , like sn 2011fe ; from x ray surveys of sne ia made with _ swift _ ; from radio observations of nearby sne ia , and from confrontation of the early light curves of sne ia with theoretical predictions . on the other hand , the much expanded surveys for transients , at different wavelengths , have found from very dim explosions to overluminous outbursts . multidimensional modeling of different types of thermonuclear explosions has reached a new level of realism , and they are confronted with observations uncovering the different layers of the exploding objects with unprecedented detail . we still lack , however , the identification of a sne ia with the dismissal of some previously observed object . two different channels ( sd and dd ) leading to sn ia explosions still appear to be required , although the balance has now shifted towards the double degenerate channel , where violent mergings or violent collisions of white dwarfs appear as a promising mechanism . the initial conditions leading to such collisions are still unclear , however . for subluminous sne ia of the sne iax type , or ca rich transients , edge lit he detonations that might not disrupt the underlying c+o wd , or failed deflagrations in the outer shells , seem able to account for the observations and for the rates of the explosions . from all the preceding , we can reach some tentative conclusions about the fraction of supernovae arising from different kinds of progenitor systems and their explosion mechanisms ( see figure 9 ) . @xmath110 the supernovae that show unburnt carbon make 30% of the sne ia class . if we add to that the super chandrasekhar explosions ( 2@xmath30 ) , we can infer that around 32% of thermonuclear supernovae should arise from the merging of two white dwarfs . tamped detonations in rapidly rotating white dwarfs may be the explosion mechanism , in the super chandrasekhar case , although mergings of massive white dwarfs could also account for these events . @xmath110 between 31@xmath92 @xmath30 and 5.6@xmath93 of all sne ia do belong to the iax subtype , likely arising either from failed deflagrations or from surface detonations of low mass white dwarfs . the large error bars come from the uncertainty on the selection effects that work against detection of subluminous events . there are indications that this subclass of sne ia with low ejection velocities do , in fact , split into two subclasses . further research in this are is needed . @xmath110 there is a 15@xmath30 of sne ia of the 91bg type , which are tentatively associated with violent mergings of white dwarfs where the primary is of about 0.9 m@xmath7 . detonation of the c+o mixture should then occur . @xmath110 the fraction of explosions resulting from the accretion of hydrogen from a red giant star seems to be very low , as metioned before . the sne coming from a symbiotic progenitor should only make 0.11@xmath30 of all sne ia . @xmath110 about 12% of events have luminosities falling well in the middle of the bulk of sne ia , but they rise fast and do decline very fast after maximum ( the fraction migh be larger , due to selection effects ) . their origin remains unclear . @xmath110 the remaining fraction of sne ia , a 21 @xmath30 , could come from systems made of a c+o white dwarf plus a main sequence or subgiant star , and they could appear , before the explosion , as supersoft x ray sources or / and recurrent novae . in the sn of this and the former group ( as well as in those coming from non violent dd mergings ) , the explosion should start as a ( generally off centre ) deflagration , to become a detonation in reaching lower density layers . given the large sample of peculiar sne ia , it is worth obtaining spectral sequences , when using sn for cosmology . such project would be made possible by dedicated space missions and by ground based programs ( wood vasey 2010 ) . i would like to thank gaston folatelli , mario hamuy , dan kasen , mansi kasliwal , gautham narayan , and josh simon , for their kind permissions to use their figures . my thanks also go to hagai perets , lev yungelson , zhengwei liu , alexander tutukov , thomas tauris , enrique garca berro , joe lyman , raffaella margutti , todd thompson , mickael rigault , markus kromer , dan kasen , boaz katz , mario hamuy , michael wood vasey , and marat gilfanov , for valuable comments , suggestions and criticisms to the first draft of this paper . the useful input from an anonymous referee is also acknowledged . this work has been supported by grant aya201236353 , from the ministerio de economa y competitividad of spain . napiwotzki , r. , karl , c.a . , nelemans , g. , et al . 2007 , in _ european workshop on white dwarfs _ , asp conference series , 372 , ed . r. napiwotzki & m.r . burleigh ( astron pacific , san francisco ) , p. 387 nomoto , k. , uenishi , t. , kobayashi , c. , umeda , h. , ohkubo , t. , hachisu , i. , & kato , m. 2002 , in _ from twilight to highlight : the physics of supernovae _ w. hillebrandt & b. leibundgut ( springer , berlin ) , p. 115
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although type ia supernovae ( sne ia ) are a major tool in cosmology and play a key role in the chemical evolution of galaxies , the nature of their progenitor systems ( apart from the fact that they must content at least one white dwarf , that explodes ) remains largely unknown . in the last decade ,
considerable efforts have been made , both observationally and theoretically , to solve this problem .
observations have , however , revealed a previously ususpected variety of events , ranging from very underluminous outbursts to clearly overluminous ones , and spanning a range well outside the peak luminosity decline rate of the light curve relationship , used to make calibrated candles of the sne ia . on the theoretical side ,
new explosion scenarios , such as violent mergings of pairs of white dwarfs , have been explored .
we review those recent developments , emphasizing the new observational findings , but also trying to tie them to the different scenarios and explosion mechanisms proposed thus far .
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the tangent space to an integral projective variety @xmath0 of dimension @xmath1 in a smooth point @xmath2 , named @xmath3 , is always of dimension @xmath1 . it is no longer true for the osculating spaces . for instance , as it was pointed out by togliatti in @xcite , the osculating space @xmath4 , in a general point @xmath2 , of the rational surface @xmath5 defined by @xmath6 is of projective dimension @xmath7 instead of @xmath8 . indeed there is a non trivial linear relation between the partial derivatives of order @xmath9 of @xmath10 at @xmath2 that define @xmath4 . this relation is usually called a _ laplace equation _ of order @xmath9 . more generally , we will say that @xmath5 satisfies a laplace equation of order @xmath11 when its @xmath11-th osculating space @xmath12 in a general point @xmath13 is of dimension less than the expected one , that is @xmath14 . the study of the surfaces satisfying a laplace equation was developed in the last century by togliatti @xcite and terracini @xcite . togliatti @xcite gave a complete classification of the rational surfaces embedded by linear systems of plane cubics and satisfying a laplace equation of order two . in the paper @xcite , perkinson gives a complete classification of smooth toric surfaces ( theorem 3.2 ) and threefolds ( theorem 3.5 ) embedded by a monomial linear system and satisfying a laplace equation of any order . very recently miro - roig , mezzetti and ottaviani @xcite have established a nice link between rational varieties ( i.e. projections of veronese varieties ) satisfying a laplace equation and artinian graded rings @xmath15 such that the multiplication by a general linear form has not maximal rank in a degree @xmath16 . on the contrary , when the rank of the multiplication map is maximal in any degree , the ring is said to have the _ weak lefschetz property _ ( briefly wlp ) . the same type of problems arises when we consider the multiplication by powers @xmath17 ( @xmath18 ) of a general linear form @xmath19 . indeed , if the rank of the multiplication map by @xmath17 is maximal for any @xmath20 and any degree , the ring is said to have the _ strong lefschetz property _ ( briefly slp ) . + these properties are so called after stanley s seminal work : the hard lefschetz theorem is used to prove that the ring @xmath21}{(x_0^{d_0},\ldots , x_n^{d_n})}$ ] has the slp ( * ? ? ? * theorem 2.4 ) . from this example one can ask if the artinian complete intersection rings have the wlp . actually @xmath22}{(f_0,f_1,f_2)}$ ] has the wlp ( first proved in @xcite and then also in @xcite ) but it is still not known for more than three variables . many other questions derive from this first example . + for more details about known results and some open problems we refer to @xcite . let @xmath23 be an artinian ideal generated by the @xmath24 forms @xmath25 , all of the same degree @xmath26 , and @xmath27 be the _ syzygy bundle _ associated to @xmath28 and defined in the following way : @xmath29 for shortness we will denote @xmath30 and , forgetting the twist by @xmath26 , in all the rest of this text we call it the syzygy bundle . as in @xcite , many papers about the lefschetz properties involve the _ syzygy bundle_. indeed , in ( * ? ? ? * proposition 2.1 ) , brenner and kaid prove that the graded piece of degree @xmath31 of the artinian ring @xmath32}{(f_0,\ldots , f_r)}$ ] is @xmath33 . in [ @xcite , thm . 3.2 ] the authors characterize the failure of the wlp ( in degree @xmath34 , i.e. for the map @xmath35 ) when @xmath36 by the non injectivity of the restricted map @xmath37 on a general hyperplane @xmath19 . let us say , in few words , what we are doing in this paper and how it is organized . first of all we recall some definitions , basic facts and we propose a conjecture ( section [ s1 ] ) . in section [ s2 ] we extend to the slp the characterization of failure of the wlp given in @xcite . then we translate the failure of the wlp and slp in terms of existence of special singular hypersurfaces ( section [ s3 ] ) . it allows us to give an answer to three unsolved questions in @xcite . in section [ s4 ] we construct examples of artinian rings failing the wlp and the slp by producing the appropriate singular hypersurfaces . in the last section we relate the problem of slp at the range 2 to the topic of line arrangements ( section [ s5 ] ) . let us now give more details about the different sections of this paper . in section [ s2 ] , more precisely in theorem [ p1 ] , we characterize the failure of the slp by the non maximality of the induced map on sections @xmath38 the geometric consequences of this link are explained in section [ s3 ] ( see theorem [ th1bis ] ) . the non injectivity is translated in terms of the number of laplace equations and the non surjectivity is related , via apolarity , to the existence of special singular hypersurfaces . then we give propositions [ pr54 - 1 ] , [ pr54 - 2 ] and [ pr54 - 3 ] that solve three problems posed in ( * ? ? ? * problem 5.4 and conjecture 5.13 ) . in section [ s4 ] we produce many examples of ideals ( monomial and non monomial ) that fail the wlp and the slp . the failure of the wlp is studied for monomial ideals generated in degree @xmath7 on @xmath39 ( theorem [ th3 ] ) , in degree @xmath8 on @xmath39 ( proposition [ th4 ] ) , in degree @xmath7 on @xmath40 ( proposition [ d4 m ] ) ; the failure of the slp is studied for monomial ideals generated in degree @xmath7 ( proposition [ d4mslp ] ) ; finally , we propose a method to produce non monomial ideals that fail the slp at any range ( proposition [ nmslp ] ) . in the last section lefschetz properties and line arrangements are linked . the theory of line arrangements , more generally of hyperplane arrangements , is an old and deep subject that concerns combinatorics , topology and algebraic geometry . one can say that it began with jakob steiner ( in the first volume of crelles s journal , 1826 ) who determined in how many regions a real plane is divided by a finite number of lines . it is relevant also with sylvester - gallai s amazing problem . hyperplane arrangements come back in a modern presentation in arnold s fundamental work @xcite on the cohomology ring of @xmath41 ( where @xmath42 is the union of the hyperplanes of the arrangement ) . for a large part of mathematicians working on arrangements , it culminates today with the terao conjecture ( see the last section of this paper or directly @xcite ) . this conjecture concerns particularly the derivation sheaf ( also called logarithmic sheaf ) associated to the arrangement . in this paper we recall the conjecture . in proposition [ th5 ] we prove that the failure of the slp at the range 2 of some ideals is equivalent to the unstability of the associated derivation sheaves . thanks to the important literature on arrangements , we find artinian ideals that fail the slp . for instance the coxeter arrangement , called b3 , gives an original ideal that fails the slp at the range 2 in a non trivial way ( see proposition [ b3 ] ) . we finish by a reformulation of terao s conjecture in terms of slp . the ground field is @xmath43 . + the dual @xmath44 of a vector space @xmath45 is denoted by @xmath46 . + the dimension of the vector space @xmath47 is denoted by @xmath48 where @xmath1 is clearly known in the context . + the vector space generated by the set @xmath49 is @xmath50 . + the join variety of @xmath11 projective varieties @xmath51 is denoted by @xmath52 ( see @xcite for the definition of join variety ) . + the fundamental points @xmath53 in @xmath54 are denoted by @xmath55 . + we often write in the same way a projective hyperplane and the linear form defining it ; we use in general the notation @xmath56 on @xmath54 and the notation @xmath57 on @xmath39 for hyperplanes . + the ideal sheaf of a point @xmath2 is @xmath58 . let @xmath59=\bigoplus r_t$ ] be the graded polynomial ring in @xmath60 variables over @xmath43 . the dimension of the vector space @xmath61 is @xmath48 . + let @xmath62 be a graded artinian algebra , defined by the ideal @xmath28 . note that @xmath63 is finite dimensional over @xmath43 . the artinian algebra @xmath63 ( or the artinian ideal @xmath28 ) has the weak lefschetz property ( wlp ) if there exists a linear form @xmath19 such that the homomorphism induced by the multiplication by @xmath19 , @xmath64 has maximal rank ( i.e. is injective or surjective ) for all @xmath16 . the artinian algebra @xmath63 ( or the artinian ideal @xmath28 ) has the strong lefschetz property ( slp ) if there exists a linear form @xmath19 such that @xmath65 has maximal rank ( i.e. is injective or surjective ) for all @xmath16 and @xmath20 . * it is clear that the slp for @xmath66 corresponds to the wlp . * actually , it can be proved that if a lefschetz element exists , then there is an open set of such elements , so that one can call general linear form such an element . * we will often be interested in artinian rings @xmath63 that fail the slp ( or wlp ) , i.e. when for any linear form @xmath19 there exist @xmath16 and @xmath20 such that the multiplication map @xmath65 has not maximal rank . in that case we will say that @xmath63 ( or @xmath28 ) fails the slp at range @xmath20 and degree @xmath16 . when @xmath66 we will say simply that @xmath63 fails the wlp in degree @xmath16 . one of the main examples comes from togliatti s result ( see for instance @xcite , example 3.1 ) : the ideal @xmath67 fails the wlp in degree @xmath9 . there are many ways to prove it . one of them comes from the polarity on the rational normal cubic curve . it leads to a generalization that gives one of the few known non toric examples . ( ( * ? ? ? * theorem 3.1 ) ) let @xmath68 be an integer and @xmath69 be non concurrent linear forms on @xmath39 . then the ideal @xmath70 fails the wlp in degree @xmath71 . indeed on the general line @xmath72 the @xmath73 forms of degree @xmath74 become dependent thanks to the polarity on the rational normal curve of degree @xmath74 . we propose the following conjecture . for @xmath75 it is again togliatti s result . let @xmath76 be non concurrent linear forms on @xmath39 and @xmath77 be a form of degree @xmath74 on @xmath39 . then the ideal @xmath78 fails the wlp in degree @xmath71 if and only if @xmath79 in ( * ? ? ? * proposition 2.3 ) , the failure of the wlp in degree @xmath34 is related to the restriction of the syzygy bundle to a general hyperplane . here we extend this relationship to the slp situation at any range and in many degrees , by using the syzygy bundle method originated in @xcite . [ p1 ] let @xmath80 be an artinian ideal generated by homogeneous forms of degree @xmath26 and @xmath81 the syzygy bundle defined by the exact sequence @xmath82 where @xmath83 let @xmath16 be a non - negative integer such that @xmath84 and @xmath20 be an integer such that @xmath18 . then @xmath28 fails the slp at the range @xmath20 in degree @xmath85 if and only if the induced homomorphism on sections ( denoted by @xmath86 ) @xmath87 has not maximal rank for a general linear form @xmath19 . the theorem is not true if @xmath88 i.e. if there exists a syzygy of degree @xmath16 among @xmath89 . in @xcite the authors consider the injectivity of the map @xmath86 for @xmath90 and for @xmath91 . in that case , since the forms @xmath92 are the generators of @xmath28 , we have of course @xmath93 . in ( * ? ? ? * proposition 2.1 ) the authors proved that @xmath94 for any @xmath95 . let us consider the canonical exact sequence @xmath96 we obtain a long exact sequence of cohomology @xmath97 let us assume first that @xmath98 . then we have always @xmath99 and it gives a shorter exact sequence : @xmath100 moreover , since @xmath101 , we have also @xmath102 . then by tensoring the exact sequence defining the bundle @xmath81 by @xmath103 and taking the long cohomology exact sequence , we find : @xmath104 since the kernel and cokernel of both maps , @xmath105 and @xmath106 are the same , the theorem is proved for @xmath101 . if @xmath107 , let us introduce the number @xmath108 . this number is equal to @xmath109 and we have a long exact sequence : @xmath110 let us consider now the long exact sequence : @xmath111 since @xmath112 ( and @xmath113 ) , it remains a shorter exact sequence @xmath114 as before , since the kernel and cokernel of both maps are the same , the theorem is proved . let us introduce the numbers @xmath115 , @xmath116 the following corollary is a didactic reformulation of the theorem above . assume that there is no syzygy of degree @xmath16 among the @xmath92 s . then @xmath28 fails the slp at the range @xmath20 in degree @xmath117 if and only if one of the two following equivalent conditions occurs : * @xmath118 , * @xmath119 . in the next section we translate this corollary in geometric terms . we recall that the @xmath11-th osculating space @xmath120 to a @xmath1-dimensional complex projective variety @xmath121 at @xmath2 is the subspace of @xmath122 spanned by @xmath2 and by all the derivative points of degree less than or equal to @xmath11 of a local parametrization of @xmath5 , evaluated at @xmath2 . of course , for @xmath123 we get the tangent space @xmath124 . a @xmath1-dimensional variety @xmath121 whose @xmath11-th osculating space at a general point has dimension @xmath125 is said to satisfy @xmath126 independent laplace equations of order @xmath11 . we will say , for shortness , that the _ number _ of laplace equations is @xmath126 . if @xmath127 , then there are always @xmath128 linear relations between the partial derivatives . these relations are trivial laplace equations of order s. we will not consider them in the following , so when we write there is a laplace equation of order @xmath11 we understand a non - trivial laplace equation of order @xmath11 . let us briefly explain now the link with projections of @xmath129 . let @xmath130 be a complex vector space of linear forms of dimension @xmath60 such that @xmath131 . we consider the veronese embedding : @xmath132 & \mapsto & [ l^{t } ] . \end{array}\ ] ] the image @xmath129 is called the veronese @xmath1-fold of order @xmath133 . at the point @xmath134\in v_t({\mathbb p}^n)$ ] , the @xmath11-th osculating space , @xmath135 , is the space of degree @xmath26 forms possessing a factorization @xmath136 where @xmath137 is a form of degree @xmath11 ( * ? ? ? * theorem 1.3 ) . it is identified with @xmath138 . let us think about the projective duality in terms of derivations ( it is in fact the so - called apolarity , see @xcite ) . a canonical basis of @xmath139 is given by the @xmath140 derivations : @xmath141 let @xmath142 an ideal generated by @xmath24 forms of degree @xmath26 . note that @xmath89 are points in @xmath143 . we denote by @xmath144 the vector subspace of @xmath145 generated by the @xmath89 and by @xmath146 , for any @xmath147 . let us introduce the orthogonal vector space to @xmath148 @xmath149 it gives an exact sequence of vector spaces @xmath150 and the corresponding projection map @xmath151 of course one can identify @xmath152 and write the decomposition @xmath153 in the following two situations , the vector space @xmath154 is easy to describe : 1 . when @xmath144 is generated by @xmath24 monomials of degree @xmath26 , @xmath155 is generated by the remaining @xmath156 monomials . 2 . when @xmath157 where @xmath158\in { \mathbb p}(r_{1}^*)$ ] , @xmath155 is generated by degree @xmath26 polynomials that vanish at the points @xmath159\in { \mathbb p}(r_{1}).$ ] it is well known that the tangent spaces to the veronese varieties can be interpreted as singular hypersurfaces . more precisely a hyperplane containing the tangent space @xmath160}v_t({\mathbb p}^n)$ ] corresponds in the dual space @xmath161 to a hypersurface of degree @xmath133 that is singular at the point @xmath162 $ ] . more generally a hyperplane containing the @xmath11-th ( @xmath163 ) osculating space @xmath160}^{s}v_t({\mathbb p}^n)$ ] corresponds to a hypersurface of degree @xmath133 and multiplicity @xmath164 at the point @xmath162 $ ] ( see for instance @xcite ) . thus the dual variety of @xmath129 is the discriminant variety that parametrizes the singular hypersurfaces of degree @xmath133 when the @xmath11-th osculating variety of @xmath129 parametrizes the hypersurfaces of degree @xmath133 with a point of multiplicity @xmath165 . we propose now an extended version of the `` main '' theorem of @xcite ( to be precise theorem 3.2 ) . [ th1bis ] let @xmath80 be an artinian ideal generated by @xmath24 homogeneous polynomials of degree @xmath26 . let @xmath166 be integers such that @xmath167 , @xmath18 . assume that there is no syzygy of degree @xmath16 among the @xmath92 s . the following conditions are equivalent : 1 . the ideal @xmath28 fails the slp at the range @xmath20 in degree @xmath85 . 2 . there exist @xmath168 , with @xmath169 , independent vectors @xmath170 and @xmath171 forms @xmath172 such that @xmath173 for a general linear form @xmath19 of @xmath54 . the @xmath1-dimensional variety @xmath174 satisfies @xmath169 laplace equations of order @xmath117 . [ item_iv_thm ] for any @xmath175 , @xmath176 , with @xmath177 . the equivalence @xmath178 is proved in theorem [ p1 ] . since @xmath28 is generated in degree @xmath26 , the map @xmath179 is surjective and the relation @xmath180 is equivalent to @xmath181}^{d+i - k}v_{d+i}({\mathbb p}^n)\neq \emptyset$ ] . more generally the number of independent relations @xmath173 is the dimension of the kernel of the map @xmath86 i.e. the dimension of @xmath182 ; this number of independent relations , written in a geometric way , is @xmath183}^{d+i - k}v_{d+i}({\mathbb p}^n)]+1,\,\,\ , ( \delta \ge 0)\ ] ] where the projective dimension is @xmath184 if the intersection is empty . the number is the number of ( non trivial ) laplace equations . indeed , the dimension of the @xmath185-th osculating space to @xmath174 is @xmath186 since the @xmath185-th osculating space to @xmath187 meets the center of projection along a @xmath188 . in other words , the @xmath1-dimensional variety @xmath174 satisfies @xmath126 laplace equations and @xmath189 is equivalent to @xmath190 . the image by @xmath191 of the @xmath185-th osculating space to the veronese @xmath187 in a general point has codimension @xmath192 in @xmath193 . the codimension corresponds to the number of hyperplanes in @xmath193 containing the osculating space to @xmath174 . these hyperplanes are images by @xmath191 of hyperplanes in @xmath194 containing @xmath195 and the @xmath185-th osculating plane to @xmath187 at the point @xmath196 $ ] . in the dual setting it means that these hyperplanes are forms of degree @xmath31 in @xmath197 with multiplicity @xmath198 at @xmath162 $ ] . it proves that @xmath189 is equivalent to @xmath199 . to summarize , the number of laplace equations is @xmath200 and @xmath201 1 . [ cor_cones ] let us explain the geometric meaning of theorem [ th1bis ] [ item_iv_thm ] in a simple case : if @xmath202 , then [ item_iv_thm ] means that @xmath28 fails the slp at the range @xmath20 in degree @xmath117 if and only if there exists at any point @xmath203 a hypersurface of degree @xmath31 with multiplicity @xmath204 at @xmath205 given by a form in @xmath206 . 2 . let @xmath207 where @xmath208 are general linear forms . the vector space @xmath197 , where @xmath209 is the vector space of the forms of degree @xmath31 vanishing in @xmath24 points @xmath210 $ ] with multiplicity @xmath211 . in other words @xmath212 ) ( see ( * ? ? ? * corollary 3 ) ) . geometrically it can be described as @xmath213}^{i}v_{d+i}({\mathbb p}^n ) , \cdots , t_{[l_r^{d+i}]}^{i}v_{d+i}({\mathbb p}^n)).$ ] 3 . by the theorem above , when @xmath214 , the ideal @xmath207 fails the slp at the range @xmath20 in degree @xmath215 if and only if the following intersection is not empty : @xmath216 } v_{d+i}({\mathbb p}^n ) , \cdots , t^i_{[l_r^{d+i}]}v_{d+i}({\mathbb p}^n ) ) \cap \ , t^{d+i - k}_{[l^{d+i}]}v_{d+i}({\mathbb p}^n).\ ] ] 4 . here we focus the attention also on the number @xmath126 of laplace equations satisfied by @xmath217 . the geometric meaning of this number was highlighted by terracini @xcite for laplace equations of order @xmath9 and recently for any order by @xcite , where a classification of varieties satisfying many laplace equations is given . the characterization of the failure of the slp by the existence of ad - hoc singular hypersurfaces allows us to answer , in the three following propositions , some questions posed by migliore and nagel . let us recall their questions : ( * ? ? ? * problem 5.4 ) let @xmath218 for a general linear form @xmath19 . @xmath219 fails the wlp , for @xmath220 . there are some natural questions arising from this example : 1 . [ problem1 ] prove the failure of the wlp in previous example for all @xmath221 . 2 . what happens for mixed powers ? 3 . [ problem3 ] what happens for almost complete intersections , that is , for @xmath222 powers of general linear forms in @xmath24 variables when @xmath223 ? ( * ? ? ? * conjecture 5.13 ) let @xmath224 be general linear forms and @xmath225 1 . [ conj1 ] if @xmath226 and@xmath227 then @xmath219 fails the wlp . 2 . if @xmath228 then @xmath219 fails the wlp if and only if @xmath229 . we prove [ problem1 ] of ( * ? ? ? * problem 5.4 ) in proposition [ pr54 - 1 ] , [ problem3 ] of ( * ? ? ? * problem 5.4 ) , for @xmath230 and @xmath231 , in proposition [ pr54 - 2 ] and [ conj1 ] of ( * ? ? ? * conjecture 5.13 ) in proposition [ pr54 - 3 ] . since all these results concern powers of linear forms , let us first verify that the hypothesis on the global syzygy in theorem [ th1bis ] is not restrictive . [ lem - syz ] let @xmath28 be the ideal @xmath232 where the @xmath233 are linear forms and @xmath234 . let @xmath81 be its syzygy bundle . then @xmath235 one direction is obvious . let us assume that @xmath236 and that there exists a relation @xmath237 with @xmath238 forms of @xmath239 . both hypotheses imply that the projective space @xmath240}^{i}v_{d+i}({\mathbb p}^n ) , \cdots , t_{[l_r^{d+i}]}^{i}v_{d+i}({\mathbb p}^n))$ ] has dimension strictly less than the expected one . since the linear forms are general , it implies that the algebraic closure of @xmath241}^{i}v_{d+i}({\mathbb p}^n)$ ] has not the expected dimension . it contradicts the lemma 3.3 in @xcite . proposition [ pr54 - 2 ] is already proved in ( * ? ? ? * lemma 4.8 ) and also in ( * ? ? ? * theorem 4.2 ( ii ) ) . we propose here a new proof based on the existence of a singular hypersurface characterizing the failure of the slp . let us mention that , on @xmath39 a hypersurface of degree @xmath31 with a point of multiplicity @xmath31 is simply an union of lines ( as , for instance , in theorem [ th3 ] and proposition [ th4 ] ) , but on @xmath54 , with @xmath101 , a hypersurface of degree @xmath31 with a point of multiplicity @xmath31 is more generally a cone over a hypersurface in the hyperplane at infinity . this is the key argument in the proofs of the three following propositions . [ pr54 - 1 ] let @xmath242 be an integer such that @xmath243 . then the ideal @xmath244 fails the wlp in degree @xmath245 . of course it is equivalent to say that @xmath246 fails the wlp in degree @xmath245 for @xmath247 general linear forms . let us consider the syzygy bundle associated to the linear system @xmath248 since @xmath249 lemma [ lem - syz ] implies @xmath250 let @xmath19 be a linear form . when @xmath243 we have @xmath251 . according to theorem [ th1bis ] the failure of the wlp in degree @xmath245 is equivalent to the existence of a surface with multiplicity @xmath252 in the points @xmath253 and @xmath254 and multiplicity @xmath255 at a moving point @xmath205 . the five concurrent lines in @xmath205 passing through @xmath256 belong to a quadric cone with equation @xmath257 ( the cone over the conic at infinity through the five points ) . since @xmath258 the hypersurface @xmath259 has the desired properties . in @xmath54 there is always a quadric through @xmath260 points in general position . then given any general point @xmath261 , there is a quadratic cone with a vertex at @xmath205 and passing through @xmath260 fixed points in general position . then we prove , [ pr54 - 2 ] in the following cases the ideal @xmath262 fails the wlp in degree @xmath245 : * @xmath263 and @xmath264 , * @xmath231 and @xmath265 , * @xmath266 and @xmath267 . let us consider the syzygy bundle associated to the linear system @xmath268 let @xmath19 a linear form . then the inequality @xmath269 is true if and only if @xmath242 and @xmath1 are one of the possibilities stated in the theorem . in all these cases we have @xmath270 , and by lemma [ lem - syz ] , @xmath271 . according to theorem [ th1bis ] the failure of the wlp is equivalent to the existence of a hypersurface with multiplicity @xmath252 in the points @xmath159 $ ] and multiplicity @xmath255 at the moving point @xmath205 . the lines through @xmath205 and @xmath159 $ ] belong to a quadratic cone with equation @xmath257 ( the cone over the quadric at infinity through the points ) . since @xmath258 the hypersurface @xmath259 has the desired properties . [ pr54 - 3 ] the ideal @xmath272 fails the wlp in degree @xmath273 where @xmath274 be general linear forms on @xmath275 . since @xmath276 lemma [ lem - syz ] implies @xmath277 we have to prove that , on a general hyperplane @xmath19 , the cokernel of @xmath278 has dimension strictly greater than @xmath279 the dimension of this cokernel is the dimension of the quartics with a quadruple point @xmath162 $ ] and @xmath280 double points . we consider on the hyperplane at infinity the vector space @xmath45 of quadrics through the images of the @xmath280 points @xmath281 , \ldots , [ l_8^{\vee}]$ ] . it has dimension @xmath282 . let @xmath283 be a basis of this space of quadrics . then the vector space @xmath284 of quartics generated by the products @xmath285 has dimension @xmath286 and all these quartics are singular in the @xmath280 points . in @xmath275 the independent quartic cones over these quartics belong to the cokernel . in the next section , we propose many examples of ideals failing the wlp or the slp by producing ad - hoc singular hypersurfaces . in their nice paper about osculating spaces of veronese surfaces , lanteri and mallavibarena remark that the equation of the curve given by three concurrent lines depends only on six monomials instead of seven . more precisely let us consider a cubic with a triple point at @xmath287 passing through @xmath288 , @xmath289 and @xmath290 . its equation is @xmath291 and it depends only on the monomials @xmath292 . so there is a non zero form in @xmath293 that is triple at a general point . in this way they explain the togliatti surprising phenomena ( ( * ? ? ? * theorem 4.1 ) , @xcite and @xcite ) . we apply this idea in our context . recall that in the monomial case being artinian to the ideal @xmath28 means that it contains the forms @xmath294 . let us consider the @xmath295 fundamental points @xmath55 and let us assume that the number @xmath24 of monomials generating @xmath28 is chosen such that @xmath296 for @xmath297 fixed integers . then , as it is noted in item [ cor_cones ] of remarks after theorem [ th1bis ] , the ideal @xmath28 fails the slp at the range @xmath20 in degree @xmath117 if and only if there exists at any point @xmath205 a hypersurface of degree @xmath31 with multiplicity @xmath298 at @xmath205 given by a form in @xmath206 . we have to write this equation with a number of monomials as small as possible . then the orthogonal space becomes bigger and we will cover all the possible choices . first of all we describe exhaustively the monomial ideals @xmath299 $ ] of degree @xmath7 that do not verify the wlp . [ th3 ] up to permutation of variables the monomial ideals generated by five quartic forms in @xmath300 $ ] that fail the wlp in degree @xmath273 are the following * @xmath301 , * @xmath302 . geometrically it is evident that the first ideal @xmath303 fails the wlp . indeed under the veronese map , a linear form @xmath19 becomes a rational normal curve of degree four that defines a projective space @xmath304 and , modulo @xmath19 , the restricted monomials @xmath305 can be interpreted as points of this @xmath304 . then the tangent line to the rational quartic curve at the point @xmath306 $ ] contains the two points @xmath307 $ ] and @xmath308 $ ] . this line meets the plane @xmath309 in one point ; it implies that @xmath310 for the second ideal , it is not evident to see that the line @xmath311 always ( for any restriction ) meets the plane @xmath309 . let us consider the points @xmath288 , @xmath289 and @xmath290 and the degree @xmath7 curves with a quadruple point in @xmath287 passing through these three points . these curves are product of four lines : @xmath312 expanding @xmath77 explicitly in the coordinates @xmath313 , we see that the forms @xmath314 are missing and that twelve monomials appear to write its equation . since we want only ten monomials , we have to remove two . the following possibilities occur : * @xmath315 ( or equivalently by permutation of variables @xmath316 $ ] or @xmath317 , @xmath318 and @xmath319 $ ] ) then the remaining linear system is @xmath320 it corresponds to the first case i.e. to the ideal @xmath321 . * @xmath322 and @xmath318 but @xmath323 ( or equivalentely by permutation of variables @xmath324 $ ] or @xmath325 $ ] ) then the remaining linear system is @xmath326 it corresponds to the second case i.e. to the ideal @xmath327 . the quartic curve with multiplicity four in @xmath287 consists , in the first case , of two lines and a double line that are concurrent ; in the second case of four concurrent lines in harmonic division . we do not apply the same technique to describe exhaustively the monomial ideals @xmath328 $ ] of degree @xmath8 that do not verify the wlp because the computations become too tricky . but we can give some cases by geometric arguments . [ th4 ] the following monomial ideals * @xmath329 , * @xmath330 , where @xmath331 is any monomial , * @xmath332 , fail the @xmath333 in degree @xmath7 . under the veronese map , a linear form @xmath19 becomes a rational normal curve of degree five that defines a projective space @xmath334 and , modulo @xmath19 , the restricted monomials @xmath305 can be interpreted as points of this @xmath334 . then the tangent line to the rational quintic curve at the point @xmath335 $ ] contains the two points @xmath336 $ ] and @xmath337 $ ] . this line meets the plane @xmath338 in one point ; it implies that @xmath339 in the same way the osculating plane at @xmath335 $ ] i.e. @xmath340 meets the plane @xmath338 in one point . in the last case , the geometric argument is not so evident . let us set @xmath341 and @xmath342 . then the equation of the product of the five concurrent lines is @xmath343 for any point @xmath344 we choose @xmath345 such that @xmath346 and @xmath347 . then the equation depends only on @xmath348 monomials and the remaining monomials are @xmath349 we describe now some monomial ideals in @xmath350 $ ] , generated in degree @xmath273 , that do not verify the wlp . [ th3 - 1 ] the monomial ideals @xmath351 where the forms @xmath352 are chosen among one of the following sets of monomials : * @xmath353 ( case ( a1 ) ) * @xmath354 ( case ( a2 ) ) * @xmath355 ( case ( a3 ) ) * @xmath356 ( case ( a4 ) ) * @xmath357 ( case ( b1 ) ) fail the wlp in degree @xmath9 . we do not know if under permutation of variables the description above is exhaustive or not . the singular cubic that we are considering here are union of concurrent planes and not all the cubic cones . we look for a surface of degree @xmath273 with multiplicity @xmath273 at a general point @xmath344 that passes through the points @xmath358 such that its equation depends only on the remaining monomials in @xmath359 . such a cubic surface is a cone over a cubic curve . here , instead of a general cubic cone we consider only three concurrent planes . since these @xmath273 planes have to pass through @xmath360 and @xmath361 it remains only , after a simple verification , the following cases : 0.3 0.3 + 0.3 0.3 + 0.3 * ( a1 ) the equation of the cubic is @xmath362 * ( a2 ) the equation of the cubic is @xmath363 * ( a3 ) the equation of the cubic is @xmath364 where at any point @xmath365 the function @xmath366 verifies @xmath367 . * ( a4 ) the equation of the cubic is @xmath368 . * ( b1 ) the equation of the cubic is @xmath369 if we want @xmath370 to be of dimension @xmath371 ( for instance @xmath372 ) we need @xmath373 independent cubics with a triple point . so , to get the failure of the wlp , we need @xmath373 independent cubics with a triple point . let us recover with our method two linear systems of eight cubic forms ( the complete classification is already done in ( * ? ? ? * theorem 4.10 ) ) that fail the wlp in degree @xmath9 . the following monomial ideals * @xmath374 * @xmath375 fail the @xmath333 in degree @xmath9 . the ideals @xmath28 and @xmath376 correspond respectively to the cases @xmath189 and @xmath377 in ( * ? ? ? * theorem 4.10 ) . let us consider the following three forms defining singular cubics passing through the fundamental points and a general point @xmath365 : @xmath378 they are particular cases of type @xmath379 in the proof of proposition [ th3 - 1 ] . they are linearly independent and can be written with twelve monomials . then it remains only @xmath280 forms for @xmath370 : @xmath380 let us consider the following three forms defining singular cubics passing through the basis points and the general point @xmath365 : @xmath381 they are cases of type @xmath382 in the proof of proposition [ th3 - 1 ] . they are linearly independent and can be written with twelve monomials : @xmath383 it remains only @xmath384 of course the same argument ( concurrent planes or hyperplanes ) can be used in degree or dimension bigger than @xmath273 . for instance let us give a set of monomial ideals in @xmath350 $ ] , generated in degree @xmath7 that fail the wlp . [ d4 m ] let @xmath385 be eleven monomials chosen among @xmath386 then the ideal @xmath387 fails the wlp in degree @xmath273 . at any point @xmath388 an equation of a surface of degree @xmath7 with multiplicity @xmath7 at @xmath205 that passes through the points @xmath389 is given by @xmath390 we conclude this section with an example that fails the slp at the range @xmath9 . [ d4mslp ] the ideal @xmath391 $ ] fails the slp at the range @xmath9 in degree @xmath9 . let @xmath288 , @xmath289 , @xmath290 and @xmath392 be four points . we consider the quartic curve consisting of the union of the four lines @xmath393 and @xmath394 . it is a quartic passing through @xmath395 and triple at @xmath392 . it depends on the six monomials @xmath396 then it remains @xmath397 monomials @xmath398 the associated syzygy bundle @xmath81 verifies @xmath399 for a general linear form @xmath19 . it proves that @xmath400 fails the slp at the range @xmath9 in degree @xmath9 . let us study now the interesting case @xmath401 where @xmath402 is a finite set of distinct points in @xmath403 of length @xmath404 and @xmath405 its ideal sheaf . the set @xmath402 corresponds by projective duality to a set of @xmath404 distinct lines in @xmath39 defined by linear forms @xmath406 . we will now consider the ideal @xmath407 generated by @xmath408 . we have @xmath409 . [ nmslp ] let @xmath18 , @xmath410 and @xmath411 a finite set of @xmath24 distinct points in @xmath403 where @xmath57 are linear forms on @xmath39 . assume that there exists a subset @xmath412 , of length @xmath413 , contained in a curve @xmath414 of degree @xmath415 . then the ideal @xmath416 fails the slp at the range @xmath20 in degree @xmath417 . the union of @xmath414 and @xmath418 concurrent lines at a point @xmath2 passing through the remaining points @xmath419 , is a non zero section of @xmath420 . by theorem [ th1bis ] it proves that @xmath28 fails the slp at the range @xmath20 in degree @xmath417 . with this method it is always possible to find systems of any degree that fail the slp by exhibiting a curve of degree @xmath26 with multiplicity @xmath421 at a general point @xmath2 . but one can find some set of points for which these special curves do not split as product of lines ( see proposition [ b3 ] in the next section ) . a line arrangement is a collection of distinct lines in the projective plane . arrangement of lines or more generally arrangement of hyperplanes is a famous and classical topic that has been studied by many authors for a very long time ( see @xcite or @xcite for a good introduction ) . let us denote by @xmath422 the equation of the union of lines of the considered arrangement . another classical object associated to the arrangement is the vector bundle @xmath423 defined as the kernel of the jacobian map : @xmath424 the bundle @xmath425 is called _ derivation bundle _ ( sometimes logarithmic bundle ) of the line arrangement ( see @xcite and @xcite for an introduction to derivation bundles ) . the splitting of @xmath427 over a line @xmath430 is related to the existence of curves ( with a given degree @xmath431 ) passing through @xmath402 that are multiple ( with multiplicity @xmath432 ) at @xmath433 . more precisely , in our context it implies the following characterization of unstability . we recall that a rank two vector bundle @xmath440 on @xmath441 is _ unstable _ if and only if its splitting @xmath442 on a general line @xmath72 verifies @xmath443 . this characterization is a consequence of the grauert - mlich theorem , see @xcite . [ th5 ] let @xmath444 $ ] be an artinian ideal generated by @xmath445 polynomials @xmath446 where @xmath57 are distinct linear forms in @xmath39 . let @xmath447 be the corresponding set of points in @xmath403 . then the following conditions are equivalent : the failure of the slp at the range @xmath9 in degree @xmath449 is equivalent to the existence at a general point @xmath450 of a curve of degree @xmath26 with multiplicity @xmath34 at @xmath450 belonging to @xmath451 . by the lemma [ linksd ] it is equivalent to the following splitting @xmath452 on a general line @xmath72 . in other words the failure of the slp is equivalent to have a non balanced decomposition and according to grauert - mlich theorem it is equivalent to unstability . let us give now an ideal generated by non monomials quartic forms that fails the slp at the range @xmath9 . it comes from a line arrangement , called b3 arrangement ( see ( * ? ? ? * pages 13 , 25 and 287 ) ) , such that the associated derivation bundle is unstable ( in fact even decomposed ) . the existence of a quartic curve with a general triple point is the key argument . but contrary to the previous examples , this quartic is irreducible and consequently not obtainable by proposition [ nmslp ] . consider the set @xmath402 of the nine dual points of the linear forms @xmath454 . let @xmath28 be the artinian ideal @xmath455 and @xmath81 its syzygy bundle . the derivation bundle of the arrangement is @xmath456 ( it is free with exponents @xmath457 ; see @xcite for a proof ) . then , according to the lemma [ linksd ] there is at any point @xmath2 a degree @xmath7 curve with multiplicity @xmath273 at @xmath2 passing through @xmath402 . in other words , by theorem [ th1bis ] , @xmath28 fails the slp at the range @xmath9 and degree @xmath9 . if @xmath461 is odd we can add to @xmath402 one point @xmath2 in general position with respect to @xmath402 and we can prove in the same way that @xmath464 fails the slp at the range @xmath9 and degree @xmath465 . let us denote by @xmath466 the dual set of points of @xmath467 . since there exists at any general point @xmath450 a curve of degree @xmath431 passing through @xmath468 , the lemma [ linksd ] implies that @xmath427 is unstable and proposition [ th5 ] implies that @xmath28 fails the slp at the range @xmath9 and degree @xmath463 . one of the main conjecture about hyperplane arrangements ( still open also for line arrangements ) is terao s conjecture . it concerns the free arrangements . the conjecture says that freeness depends only on the combinatorics of the arrangement . let us recall that the combinatorics of the arrangement @xmath469 is determined by an incidence graph . its vertices are the lines @xmath470 and the points @xmath471 . its edges are joining @xmath470 to @xmath472 when @xmath473 . we refer again to @xcite for a good introduction to the subject . terao s conjecture is valid not only for line arrangement but more generally for hyperplane arrangements . let us consider a free arrangement @xmath474 with exponents @xmath428 ( @xmath475 ) and let us denote by @xmath476 its dual set of points . we assume that terao s conjecture is not true i.e , that there exists a non free arrangement @xmath469 with the same combinatorics of @xmath477 . let us add @xmath478 points @xmath479 in general position to @xmath476 in order to form @xmath480 and to th dual set @xmath402 of@xmath467 to form @xmath481 . then the length of both sets of points is @xmath482 . on the general line @xmath72 we have @xmath483 when , since @xmath402 is not free , we have a less balanced decomposition for @xmath427 ( this affirmation is proved in @xcite ) : @xmath484 it implies that @xmath485 then adding @xmath478 lines passing through @xmath450 and the @xmath478 added points we obtain @xmath486 , @xmath487 and @xmath488 the bundle @xmath489 is balanced with splitting @xmath490 and @xmath491 then @xmath489 is semistable and @xmath492 is unstable . in other words the ideal @xmath493 fails the slp at the range @xmath9 and degree @xmath494 when @xmath495 has the slp at the range @xmath9 and degree @xmath494 . let @xmath496 a set of points of length @xmath482 in @xmath403 such that the ideal @xmath497 has the slp at the range @xmath9 and degree @xmath494 . assume that @xmath498 has the same combinatorics of @xmath476 . then @xmath499 has the slp at the range @xmath9 and degree @xmath494 . , the cohomology ring of colored braid group . zametki _ , 5(2 ) : 227231 , 1969 . , osculating varieties of veronese varieties and their higher secant varieties . _ canad . j. math . _ 59 , no . 3 , 488502 ( 2007 ) . , syzygy bundles on @xmath39 and the weak lefschetz property . _ illinois j. math . _ , 51:12991308 , 2007 . , les arrangements dhyperplans : un chapitre de gomtrie combinatoire bourbaki seminar , vol . 1980/81 , p. 122 , lecture notes in math . , 901 , springer , berlin - new york , 1981 . , on varieties with higher osculating defect , _ arxiv:1204.4399_. to appear on revista matemtica iberoamericana , 29 ( 4 ) 2013 . , _ classical algebraic geometry : a modern view_. to be published by cambridge univ . press . , bounding cohomology groups of vector bundles on @xmath500 , _ math . _ , 246(3):251270 , 1979/80 . , inverse system of a symbolic power , i , _ j. of algebra _ 174:10801090 , 1995 . , on a theorem of togliatti . j. _ 2(4):379397 , 2002 . , hyperplane arrangements of torelli type . _ arxiv:1011.4611 _ , november 2010 . to appear on compositio math . , _ algebraic geometry , a first course_. volume 133 of _ graduate texts in math . _ springer verlag , 1992 . , the weak and strong lefschetz properties for artinian k - algebras . _ j. algebra _ , 262:99126 , 2003 . , inverse systems , gelfand - tsetlin patterns and the weak lefschetz property _ journal of the london mathematical society _ 84:712730 , 2011 . , togliatti systems . _ osaka j. math . _ , 43(1):112 , 2006 . , osculatory behavior and second dual varieties of del pezzo surfaces . _ adv . in geom . _ , 2(4):345363 , 2002 . , on the weak lefschetz property for powers of linear forms . _ algebra and number theory _ , 4:487526 , 2012 . , laplace equations and the weak lefschetz property , canad . j. math . 65(2013 ) , 634 - 654 . , a tour of the weak and strong lefschetz properties . _ arxiv:1109.5718 _ , september 2011 . to appear on journal of commutative algebra . , _ vector bundles on complex projective spaces_. progress in mathematics , vol . 3 , birkhuser , boston , mass . , 1980 . , _ arrangement of hyperplanes _ , volume 300 of _ grundlerhen der mathematischen wissenschaften [ fundamental principles of mathematical sciences ] . _ springer - verlag , berlin , 1992 . , inflections of toric varieties . , 48:483516 , 2000 . , theory of logarithmic differential forms and logarithmic vector fields . _ j. fac . tokyo sect . _ , 27(2):265291 , 1980 . , elementary modifications and line configurations in @xmath39 . _ comment . _ 78(3):447462 , 2003 . , weyl groups , the hard lefschetz theorem , and the sperner property . _ siam j. alg . disc . meth . _ 1(2):168184 , 1980 . , sulle @xmath501 che rappresentano pi di @xmath502 equazioni di laplace linearmente indipendenti . , 33:176186 , 1912 . , alcune osservazioni sulle superficie razionali che rappresentano equazioni di laplace . _ ann . mat . pura appl . _ 25(4):325339 , 1946 . , varits de type togliatti . _ c.r.a.s . _ 343(6):411414 , 2006 . , fibrs logarithmiques sur le plan projectif . toulouse math . _ 16(2):385395 , 2007 .
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in a recent paper @xcite miro - roig , mezzetti and ottaviani highlight the link between rational varieties satisfying a laplace equation and artinian ideals failing the weak lefschetz property . continuing their work
we extend this link to the more general situation of artinian ideals failing the strong lefschetz property .
we characterize the failure of the slp ( which includes wlp ) by the existence of special singular hypersurfaces ( cones for wlp ) .
this characterization allows us to solve three problems posed in @xcite and to give new examples of ideals failing the slp .
finally , line arrangements are related to artinian ideals and the unstability of the associated derivation bundle is linked to the failure of the slp .
moreover we reformulate the so - called terao s conjecture for free line arrangements in terms of artinian ideals failing the slp .
| 15,753 | 235 |
observations by the infrared astronomy satellite ( @xmath0 ) led to the discovery of a class of galaxies with enormous far - ir luminosities . subsequent observations over a large range of wavelengths have shown that these objects , called ulig for ultraluminous infrared galaxies , have 1 ) bolometric luminosities and space densities comparable to those of optical quasars ( sanders et al . 1988 ) ; 2 ) a broad range in host galaxy spectral type , including starburst galaxies , seyfert i and ii , radio galaxies , and quasars ; 3 ) morphologies often suggestive of recent interactions or merging ( carico et al . 1990 ; leech et al.1994 ; rigopoulou et al . 1999 ) ; and 4 ) large amounts of molecular gas concentrated in small ( @xmath91 kpc ) central regions ( e.g. scoville et al . 1989 ; solomon et al . 1997 ) . understanding the nature of the prime energy source in ulig has proven difficult ( e.g. smith , lonsdale , & lonsdale 1998 ) . many of the observed characteristics indicate that very strong starbursts could be the culprit . alternatively , an active galactic nucleus ( agn ) may power the ulig ( e.g. lonsdale , smith , & lonsdale 1993 ) . the very high luminosities suggest an evolutionary connection between ulig and quasars , wherein a dust - enshrouded central massive black hole is gradually revealed as the appearance of the object changes from ulig to quasar ( sanders et al . 1988 ) . much effort has been expended in trying to determine the primary source of energy starbursts or agn driving the large fir luminosities . the recent studies using iso indicate that the vast majority of the power comes from starbursts in @xmath10 of the observed systems ( genzel et al . 1998 ; lutz et al . 1998 ) . rigopoulou et al . ( 1999 ) present the results of an expanded version of the mid - ir spectroscopic survey first reported by genzel et al . using iso to observe 62 ulig at @xmath11 , they measured the line to continuum ratio of the 7.7 @xmath1 m polycyclic aromatic hydrocarbon ( pah ) feature to differentiate between starburst and agn as the dominant source of the large fir luminosity . pah features have been shown to be strong in starburst galaxies and weak in agn ( moorwood 1986 ; roche et al . rigopoulou et al . confirmed the results of genzel et al . ( 1998 ) , and also found , based on near - ir imaging , that approximately 2/3 of their sample have double nuclei and nearly all the objects show signs of interactions . for a recent review of ulig see sanders & mirabel ( 1996 ) . ulig are also of great interest for studies of early star formation in the building of galaxies . recent sub - mm observations suggest that objects similar to ulig may contain a significant fraction of the star formation at high redshifts ( e.g. lilly et al . but so far most studies have found ulig only in the nearby universe . sanders et al . ( 1988 ) initially studied a group of 10 objects at @xmath12 . previously published systematic surveys have found objects mostly at @xmath13 ( leech et al . 1994 ; clements et al . 1996a , 1996b ) . a few high redshifts objects have been found , all of which turn out to contain hidden agn . these include fsc 15307 + 3252 at @xmath14 ( cutri et al . 1994 ) and fsc 10214 + 4724 at @xmath15 ( rowan - robinson et al.1991 ) . the former object was found to exhibit a highly polarized continuum , indicating the presence of a buried quasar ( hines et al.1995 ) while the latter was found to be lensed ( eisenhardt et al.1996 ) and also shows signs of containing a hidden agn ( lawrence et al . 1993 ; elston et al . 1994 ; goodrich et al . 1996 ) . further progress in this field has been hampered by the lack of identified ulig at moderately high redshifts . no new deep far - ir survey will become available prior to the launch of _ sirtf _ , which will be capable of studying ulig in detail at high redshifts . so , the @xmath0 database remains the primary source of targets for finding high redshift ulig . radio observations provide a relatively unbiased method for extracting fir galaxies from the @xmath0 faint source catalog ( fsc ; moshir et al . 1992 ) because radio continuum emission is relatively unaffected by extinction in dense gas and dust . such fir / radio samples are ideal for detailed investigations of the complex relationships between the interstellar media , starbursts , and possible agn in ulig . for example , a sample of radio - loud objects was constructed by cross - correlating the @xmath0 fsc with the texas 365 mhz radio catalog ( txfs ; dey & van breugel 1990 ) . subsequent optical identifications and spectroscopy showed that the txfs objects tend to be distant agn . so a radio - quiet sample , extracted from the fsc , should be an excellent means of finding ulig without agn i.e . powered by starbursts at interesting cosmological distances . in this paper , we report on such a sample : we describe the sample selection process and discuss the near - ir imaging . we defer a detailed analysis of the radio properties and optical spectroscopy to future papers . we have used two large area surveys in the radio and far - ir , which we briefly describe here , to select ulig candidates . in the radio , we have used the first ( faint images of the radio sky at twenty cm ; becker , white , & helfand 1995 ) . using the vla , this project is surveying @xmath16 steradians down to a 5@xmath17 limit of 1 mjy with 5 arcsec resolution and subarcsec positional accuracy . one of the problems with finding distant ulig using @xmath0 is that there are many faint galaxies visible in a deep optical image within the relatively large error ellipse of an fir source . the high resolution and good positional information of first offer an excellent means of choosing the best of the many optical candidates on which to spend valuable large telescope time getting redshifts . we used the second version of the catalog ( released 1995 october 16 ) , which samples 2925 degrees@xmath18 in two regions of sky in the north ( @xmath19 ra(j2000 ) @xmath20 , @xmath21 dec(j2000 ) @xmath22 ) and south ( @xmath23 ra(j2000 ) @xmath24 , @xmath25 dec(j2000 ) @xmath26 ) galactic caps . in the far - ir we have used the @xmath0 fsc ( moshir et al . 1992 ) which resulted from the faint source survey ( fss ) . relative to the @xmath0 point source catalog , the fss achieved better sensitivity by point - source filtering the detector data streams and then coadding those data before finding sources . at 60 ( the band used for defining our candidates ) , the fsc covers the sky at @xmath27 and has a reliability ( integrated over all signal - to - noise ratios ) of @xmath28 . the limiting 60 @xmath1 m flux density of the fsc is approximately 0.2 jy , where the signal - to - noise ratio ( snr ) is @xmath75 . the fss also resulted in the faint source reject file which contains extracted sources not in the fsc with an snr above 3.0 . we used the fsr , in addition to the fsc , with part of first to increase the number of targets in the fall sky . the @xmath0 fsc was positionally cross - correlated with the second version of the first catalog , with the requirements that an fsc source must have a real 60 @xmath1 m detection ( @xmath29 ) and that it be within 60 arcsec of the first source . the 60 @xmath1 m band was chosen because it is more reliable than the 100 @xmath1 m band and samples close to the wavelength peak of the ulig power . the resulting first - fsc ( ff ) catalog contains 2328 matches . to increase the available objects in the fall sky , we also performed a positional match of the fsr with the south galactic cap portion of first , which yielded an additional 176 matches . the 20 cm and 60 @xmath1 m flux densities for this sample of 2504 sources are plotted in figure [ f20v60 ] . the majority of the ff sources fall along the well - known radio - fir correlation ( condon et al . 1991 ) , extending from nearby starburst galaxies to much fainter fir / radio flux levels . the surface density of such objects is approximately 1 degree@xmath30 down to the @xmath31 limits of 1 mjy at 20 cm and @xmath70.2 jy at 60 @xmath1 m . we generated optical finding charts using the digitized sky surveys , available from the space telescope science institute , for all 2504 matches . the radio source position and the fsc error ellipse were overlaid on these charts . visual inspection of these finding charts was carried out to select optically faint targets for further study , with the expectation that such targets would be distant ulig . approximately 150 targets , which will carry the designation ff along with the usual coordinate naming scheme , were selected in this manner . a strict cutoff in optical magnitude was not employed , and we make no attempt to construct a sample which has a well - defined limiting magnitude in the optical . in practice , the magnitude of the targets selected for optical imaging and spectroscopy depended on the observing conditions , i.e. some targets which are not visually faint on the dss image were observed during cloudy conditions . while the first and fsc catalogs do have well - defined flux limits , our sample was not constructed in order to be complete to a chosen flux level in either the radio nor the far - ir bands . the main goal of the survey is simply to find high - redshift ulig . it is worth noting that our target list would include objects with observed characteristics in the radio , optical , and far - ir similar to those of fsc10214 + 4724 ( which itself lies outside of the first area that we used and so can not fall into our catalog ) . we have not found any ulig at redshifts as great as that of fsc10214 + 4724 in the @xmath73000 degree@xmath18 surveyed . during several runs from march 1996 to april 1999 , the kast spectrograph ( miller & stone 1994 ) at the shane 3 m telescope of lick observatory was used to obtain optical images and spectroscopy of the candidate ulig from our ff catalog . the observing procedure typically consisted of taking two 300 s images in the @xmath32 band , identifying the optical counterpart of the ff source in these data , and immediately following up with slit spectroscopy of the optical object . because the resolution and positional accuracy of first are high , it was usually clear which optical object coincided with the radio source . the fwhm of the seeing in the images was usually in the range 1.52.0 arcsec . standard stars were observed in imaging mode when conditions were photometric . however , because much of the data were obtained during non - photometric conditions , @xmath32 magnitudes will not be presented here for the sample . unless the source morphology demanded a particular value , the position angle of the slit was set to the parallactic angle . the object was dithered along the slit by @xmath33 arcsec between two exposures to aid in fringe subtraction . optical spectra of 1200 - 6000s duration were obtained of the optical source using the 300 line mm@xmath34 grating in the red - side spectrograph , which provides @xmath74.6 pixel@xmath34 resolution from 507010590 , and a 452/3306 grism in the blue - side spectrograph which provides @xmath72.5 pixel@xmath34 resolution from 30005900 . the slit width was set at 2 arcsec . the images and spectra were reduced using standard techniques . near - infrared images were obtained of the targets for which redshifts had been determined in order to better ascertain the morphologies of the galaxies . @xmath35 images were obtained for nearly all identified targets with nsfcam at the irtf 3 m telescope in 1998 august and 1999 february . additional observations of 3 targets were obtained in service mode in september 1999 . nsfcam was used in its 0.3 arcsec pixel@xmath34 mode which provides a 77@xmath3677 arcsec field . typical total exposure times per object were 960s ; more distant objects were observed for twice this period . conditions were photometric with seeing averaging 0.9 arcsec . observations of standard stars from the persson et al . ( 1998 ) list were obtained and used to calibrate the images onto the california institute of technology ( cit ) system , which is defined in elias et al . the data were reduced using standard techniques . five targets were observed in the @xmath5 band using gemini ( mclean et al . 1993 ) at the shane 3 m telescope on 1998 october 7 . gemini has 0.68 arcsec pixels which give it a 174 arcsec field . objects were observed for 1080 s each in photometric conditions with seeing of @xmath71.2 arcsec . the data were reduced using standard techniques and calibrated onto the cit system using observations of ukirt faint source standards ( casali & hawarden 1992 ) . two distant targets were imaged at the keck i telescope with nirc ( matthews & soifer 1994 ) in 1998 april . ff1106 + 3201 was observed in the @xmath5 band for 16 minutes and ff1614 + 3234 was observed for 32 minutes in the @xmath37 band . both objects were observed in clear conditions with @xmath70.5 arcsec seeing . these data were reduced using standard techniques and calibrated onto the cit system using observations of ukirt faint source standards ( casali & hawarden 1992 ) . we attempted spectroscopic observations of approximately 150 iras / first candidates , of which 116 yielded redshift information . the 108 with infrared imaging are listed in table 1 ; the 8 sources with redshifts but lacking infrared images are not considered further . the sources which did not provide useful spectra were usually observed in poor conditions ; the reasons for their lack of redshifts were not because of having intrinsically challenging spectra . the object names in table 1 are based on the first radio position . the source in the first catalog would have the name given by the object s coordinates shown in our table 1 , in the format first j_hhmmss.s+ddmmss _ where the coordinates are truncated , not rounded . in the @xmath0 fsc , the fir source name is different from that implied by our ff name , so we have included the fsc source name as a column in table 1 . the z designation in the fsc name means that the fir source is from the fsr catalog . the typical resolution of the spectroscopy was @xmath38 ( fwhm ) at @xmath39 , implying typical uncertainties of @xmath40 in redshift . redshifts were determined from the spectra after identifying probable emission lines and continuum features . in practice the features most often used were the [ o ii]@xmath413727 , [ o iii]@xmath414959,5007 , and h@xmath42 lines , and the d4000 break . the vast majority of the spectra have the emission lines characteristic of star formation ; very few show any signs , such as high ionization lines , of an agn . four sample spectra , covering a range in redshift , signal to noise , and spectral type , are shown in figure [ spex ] . a more detailed analysis of the optical spectra is deferred to a later paper . the @xmath35 images are displayed for each object , along with the optical finding chart from the dss , in figure [ optir ] . photometry of the ff objects was obtained from the @xmath35 images . in table 1 , the @xmath5 magnitudes within 3 arcsec diameter apertures , centered on the peak of the near - ir emission , are given for each object . the 5 @xmath17 detection limit in most of the images is @xmath43 so the limiting factor in the uncertainty of the photometry is not the signal to noise , since most objects have magnitudes some 34 magnitudes brighter than the detection limit , but rather systematics in the zeropoint . we estimate that the uncertainty in the zeropoint is @xmath44 mag . for most objects the 3 arcsec diameter aperture contains @xmath45 of the total light . the morphologies of the objects tend to show signs of galaxy interactions , including tidal tails , multiple nuclei , and disturbed outer envelopes . approximately 2/3 of the sample show such features , while 1/3 of the sample appear to be normal galaxies . a brief description of the near - ir morphology for each ff is included in table 1 . one of the major advantages of using first in our survey is the high accuracy of its positional information . the coordinates listed in table 1 are those of the radio source as given by the first catalog , which has an absolute astrometric uncertainty of @xmath71 arcsec . the 20 cm vla images of all objects listed in table 1 were extracted from the first database . the radio morphologies were classified by visual inspection of these cutout images , and by consulting the deconvolved sizes listed in the first catalog . the 20 cm morphological information is given for each ff source in table 1 . the 20 cm flux densities listed in table 1 have typical uncertainties of 10% at the 2 mjy level . improved @xmath0 flux densities were obtained for all objects in table 1 with the addscan utility at ipac . in addition to the 60 @xmath1 m band used to construct our ff catalog , data at 12 @xmath1 m , 25 @xmath1 m , and 100 @xmath1 m was searched for detections . almost none of the objects in table 1 were detected at either of the shorter two wavelengths , so no information is included from these wavebands in table 1 . many detections were obtained in the 100 @xmath1 m data ; these are included where available in table 1 , and one @xmath17 upper limits are indicated in parentheses . the uncertainty in the typical 60 @xmath1 m measurement in the sample is @xmath46 , and @xmath47 in the 100 @xmath1 m band where detected . the 60 @xmath1 m and 100 @xmath1 m flux densities were used to calculate the far - infrared luminosity , as defined by sanders & mirabel ( 1996 ) : l(40500 @xmath1 m ) @xmath48 $ ] , where @xmath49 is the luminosity distance in mpc , @xmath50 $ ] , and @xmath51 . throughout this paper we use h@xmath52 km s@xmath34 mpc@xmath34 and @xmath53 with @xmath54 . when 100 @xmath1 m detections could not be obtained with addscan , the 1@xmath17 limiting flux densities were used in the calculation of the l@xmath55 . the l@xmath55 are given in table 1 , and are plotted in figure [ p20vlfir ] . the uncertainty in the l@xmath55 is dominated by a combination of the typical @xmath56 flux measurement errors and the @xmath47 uncertainty in the scaling factor c which accounts for the extrapolated flux longward of the 100 @xmath1 m band . the reliability of the optical identification with the radio source for the objects in table 1 is very high . the optical / radio source association with the fsc far - infrared source is less certain , because of the relatively large positional uncertainty of the @xmath0 detections . but there are at least four reasons to believe that the identified optical / radio sources are indeed the fsc sources as well . first , in all cases , the first position is within twice the 1 @xmath17 error ellipse of the fsc source . second , the optical spectra show emission lines typical of star - forming galaxies , as expected for most far - ir luminous objects . third , in the cases where both 60 and 100 @xmath1 m detections were obtained , the @xmath0 flux ratios are typical of fir luminous galaxies ( soifer et al . 1987 ) . finally , in nearly all cases the fir / radio flux ratio lies on the well - established correlation as seen in figure [ f20v60 ] . there are at least two possible ways that the wrong association is being made . first , a galaxy optically brighter than the identified ff source could lie just outside the @xmath0 error ellipse and still be the source of the iras detection . but in our sample there are no such galaxies which are also detected by first , as would be expected if such objects were the true source of the fir emission . second , a faint radio source could be missed by the first survey that would coincide with the @xmath57 source . using the rms given in the first catalog for each object , lower limits to the resulting @xmath58 ratios for objects beneath the @xmath59 detection limits are found to be @xmath7500 - 700 , far greater than the standard ratio for star - forming galaxies . finally its worth noting that on average one expects to find only 0.035 first sources within the typical iras error ellipse area , so the probability of a random radio source being associated with an fsc source is low . in our sample of 108 identified ff objects , approximately 4 could be due to radio sources unassociated with the far - ir source . although we defer the scientific analysis of the new sample to future contributions , we will briefly compare some basic global properties of our ff sample to those of other similarly large samples of high redshift ulig which have been previously published , e.g. leech et al . ( 1994 ) and clements et al . our new sample has ulig at higher average redshift ( @xmath60 ) than that of leech et al . ( @xmath61 ) and that of clements et al . ( @xmath62 ) . as for the interaction rate , on which there has been some disagreement in the literature ( sanders & mirabel 1996 ) , our result that 2/3 of the sample shows signs of galaxy interactions is in agreement with leech et al . but not clements et al . , who found that @xmath63 of their sample were interacting systems . finally , the rate of agn - type optical spectra in our sample , which is only @xmath64 , is somewhat less than the @xmath65 found by clements et al . we do see a higher incidence of agn - type spectra at the highest l@xmath55 as has been noted previously by several studies ; for a review of this topic see sanders & mirabel ( 1996 ) . we have constructed a new survey of ulig using a match of the @xmath0 fsc with the second version of the first catalog , which covered nearly 3000 degrees@xmath18 . by choosing for further study only optically faint matches from the dss which also fall on the radio - fir flux correlation , we have attempted to find high redshift ulig which are powered primarily by starbursts . optical images and spectra were obtained of 108 such targets , which were found to lie in the redshift range @xmath66 ; a redshift histogram is shown in figure [ zhist ] . nearly all of these targets have @xmath67 greater than @xmath68 , and have a higher average redshift of @xmath69 than in other recent ulig surveys . near - ir imaging shows that while more than the majority of objects show clear signs of galaxy interactions , nearly 1/3 appear to be normal at arcsec resolution in the @xmath5 band . with this sample , we intend to examine the nature of ulig evolution in future contributions . the authors thank the staff of lick observatory for their help in obtaining the optical data , and bill vacca and dave griep at the irtf for their assistance in obtaining the near - ir data , and for conducting servicing observing for three of the sample objects . we have made use of the online facilities provided by ipac . we also thank bob becker for providing us with the first catalog and for help in using its contents . finally we thank rob kennicutt for a speedy referee report . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag - w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . the plates were processed into the compressed digital form with permission of these institutions . the work by sas , wvb , and cdb at igpp / llnl was performed under the auspices of the u.s . department of energy under contract w-7405-eng-48 to the university of california . the work by ds was supported by igpp grants 98-ap017 and 99-ap026 .
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we present a new sample of distant ultraluminous infrared galaxies .
the sample was selected from a positional cross
correlation of the @xmath0 faint source catalog with the first database .
objects from this set were selected for spectroscopy by virtue of following the well - known star - forming galaxy correlation between 1.4 ghz and 60 @xmath1 m flux , and by being optically faint on the poss . optical identification and spectroscopy
were obtained for 108 targets at the lick observatory 3 m telescope .
most objects show spectra typical of starburst galaxies , and do not show the high ionization lines of active galactic nuclei .
the redshift distribution covers @xmath2 , with 13 objects at @xmath3 and an average redshift of @xmath4 .
@xmath5-band images were obtained at the irtf , lick , and keck observatories in sub - arcsec seeing of all optically identified targets .
about 2/3 of the objects appear to be interacting galaxies , while the other 1/3 appear to be normal .
nearly all the identified objects have far - ir luminosities greater than @xmath6 , and @xmath725 % have @xmath8 .
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we consider a particular class of exact solutions of einstein s equations which describe impulsive gravitational or matter waves in a de sitter or an anti - de sitter background . one class of such solutions has recently been derived by hotta and tanaka @xcite and analysed in more detail elsewhere @xcite . this was initially obtained by boosting the source of the schwarzschild(anti-)de sitter solution in the limit in which its speed approaches that of light while its mass is reduced to zero in an appropriate way . in a de sitter background , the resulting solution describes a spherical impulsive gravitational wave generated by two null particles propagating in opposite directions . in an anti - de sitter background which contains closed timelike lines , the impulsive wave is located on a hyperboloidal surface at any time and the source is a single null particle with propagates from one side of the universe to the other and then returns in an endless cycle . in this paper we investigate a more general class of such solutions . the global structure of the space - times and the shape of the impulsive wave surfaces are exactly as summarised above and described in detail in @xcite . here we consider a wider range of possible sources . we present an interesting class of impulsive gravitational waves that are also generated by null particles , but these particles in general can have an arbitrary multipole structure . the space - times are conformal to the impulsive limit of a family of type n solutions of kundt s class @xcite . when the cosmological constant is negative , the solutions given here can be related to the impulsive limit of a class of solutions previously given by siklos @xcite . it may be noted that a family of impulsive spherical gravitational waves have also been obtained by hogan @xcite . these are particular ( impulsive ) cases of the robinson trautman family of solutions with a cosmological constant . they will be discussed further elsewhere and are not related to the solutions given here . as is well known , the de sitter and anti - de sitter space - times can naturally be represented as four - dimensional hyperboloids embedded in five - dimensional minkowski spaces . impulsive waves can easily be introduced into these space - times using this formalism . this is done is section 2 in which the form of the solution is constructed explicitly and the nature of its source is described . appropriate coordinate systems for the separate cases of de sitter and anti - de sitter backgrounds are described respectively in sections 3 and 4 together with a discussion of the geometrical properties of the waves . their relation to previously known solutions is indicated in section 5 . we wish to consider impulsive waves in a de sitter or an anti - de sitter background . in these cases , the background can be represented as a four - dimensional hyperboloid @xmath0 embedded in a five - dimensional minkowski space - time @xmath1 where @xmath2 for a cosmological constant @xmath3 , @xmath4 for a de sitter background ( @xmath5 ) , and @xmath6 for an anti - de sitter background ( @xmath7 ) in which there are two timelike coordinates @xmath8 and @xmath9 . let us now consider a plane impulsive wave in this 5-dimensional minkowski background . without loss of generality , we may consider this to be located on the null hypersurface given by @xmath10 so that the surface has constant curvature . for @xmath4 , the impulsive wave is a 2-sphere in the 5-dimensional minkowski space at any time @xmath8 . alternatively , for @xmath6 , it is a 2-dimensional hyperboloid . the geometry of these surfaces has been described in detail elsewhere @xcite using various natural coordinate systems . in this five - dimensional notation , we consider the class of complete space - times that contain an impulsive wave on this background and that can be represented in the form @xmath11 where @xmath12 is determined on the wave surface ( [ e2.5 ] ) . thus , @xmath13 must be a function of two parameters which span the surface . an appropriate parameterisation of this surface is given by @xmath14 where @xmath15 when @xmath4 and @xmath16 when @xmath6 . in terms of these parameters , it can be shown that the function @xmath17 must satisfy the linear partial differential equation @xmath18 where @xmath19 represents the source of the wave . it is a remarkable fact that this equation arises in such a similar form for both de sitter and anti - de sitter backgrounds . this equation will be derived separately for both cases in the following sections . it may immediately be observed that a solution of ( [ e2.8 ] ) of the form @xmath20 const . represents a uniform distribution of null matter over the impulsive surface . this may always be added to any other non - trivial solution . however , from now on we will only consider solutions which are vacuum everywhere except for some possible isolated sources . let us now consider solutions that can be separated in the form @xmath21 where @xmath22 is a real constant . since @xmath23 is a periodic coordinate it follows that , for continuous solutions ( except possibly at the poles @xmath24 ) , @xmath22 must be a non - negative integer . for a vacuum solution with this condition , ( [ e2.8 ] ) reduces to an associated legendre equation @xmath25 this has the general solution @xmath26 where @xmath27 and @xmath28 are associated legendre functions of the first and second kind of degree 1 , and @xmath29 and @xmath30 are arbitrary constants . the only possible nonsingular solutions involve the associated legendre functions of the first kind . these are nonzero here only for @xmath31 , and the solutions are given by @xmath32 or any linear combination of them . it may then be observed that the second of the above expressions can be obtained from the first by a simple `` rotation '' of the coordinates on the wave surface ( [ e2.5 ] ) , so that they are essentially the same solution . we can thus restrict attention to the space - time ( [ e2.6 ] ) with @xmath33 . it can then be shown that this case is conformally flat . the impulsive component in ( [ e2.6 ] ) can be removed by the discontinuous linear transformation @xmath34 where @xmath35 , @xmath36 and @xmath37 is the heaviside step function . ( this does not introduce impulsive components into the weyl tensor . ) thus , these nonsingular solutions represent only the de sitter or anti - de sitter backgrounds in different coordinates . in these backgrounds , there is no equivalent to the plane impulsive gravitational wave ( for @xmath38 ) for which the weyl tensor has constant components over the wave surface . it now follows that the only nontrivial solution of ( [ e2.9 ] ) involves the legendre functions of the second kind . these necessarily have singularities at @xmath24 which may correspond to poles at which the sources of the impulsive wave may be located . summing over all possible modes , a general real solution is obtained in the form @xmath39 \label{e2.10}\ ] ] where @xmath30 and @xmath40 are real constants representing the arbitrary amplitude and phase of each component . it may be recalled that the associated legendre functions of the second kind are generated by the relation @xmath41 where @xmath42 . the first few of these functions are given by @xmath43 which have been expressed in forms that are applicable for real @xmath44 for both @xmath45 and @xmath46 . the first of these terms ( @xmath47 ) gives the simplest ( axially symmetric ) solution @xmath48 . % \label{e2.12}\ ] ] in fact this is exactly the solution found by hotta and tanaka @xcite ( with @xmath49 ) which was obtained by boosting the source of the schwarzschild(anti-)de sitter space - time to the ultrarelativistic limit . in this case , the singularities correspond to sources represented by two delta functions @xmath50.\ ] ] let us now consider the further terms for arbitrary @xmath22 . from the definition of @xmath51 given in ( [ e2.10 ] ) and the identity ( [ ea.7 ] ) from the appendix , it can be shown that @xmath52\cos[m(\phi-\phi_m ) ] \nonumber % \label{e2.17 } \end{aligned}\ ] ] where @xmath53 is the @xmath54 derivative of the delta function . comparing this with ( [ e2.8 ] ) , it can be seen that each of the components @xmath55 corresponds to sources at @xmath24 given by @xmath56\cos[m(\phi-\phi_m ) ] . \label{e2.18}\ ] ] these components describe point sources with an @xmath22-pole structure . they have the appropriate dependence on @xmath44 as the @xmath54 derivative of the delta function , together with the appropriate periodic dependence on @xmath23 . the multipole character of first three of these modes is clearly illustrated in fig . 1 . ( it may be noted that similar multipole sources can generate impulsive _ pp_-waves in space - times with @xmath38 @xcite . ) , @xmath57 and @xmath58 near the singular point representing one of the sources of the impulsive waves . ] finally we observe that the solution ( [ e2.10 ] ) represents a general solution containing point sources which are arbitrary combinations of @xmath22-poles @xmath59 the constants @xmath30 represent the strength of each @xmath22-pole and @xmath40 its orientation . when the cosmological constant is positive , it is most convenient to work with the global coordinate system given by @xmath60 in which @xmath61 $ ] , @xmath62 $ ] . in these coordinates it can be seen that the impulsive wave is localised on the surface given by @xmath63 . thus , on the impulsive null hypersurface @xmath64 the coordinates ( [ e3.1 ] ) are identical to those of ( [ e2.7 ] ) with the identity @xmath65 . in this case the line element ( [ e2.6 ] ) with the solution ( [ e2.10 ] ) takes the form @xmath66 now , in order to justify the equation ( [ e2.8 ] ) , we may adapt the approach of dray and t hooft @xcite . in deriving an exact solution for a spherical impulsive wave in a schwarzschild space - time , they have given field equations for such a wave in a more general class of backgrounds . these also apply in a space - time with a positive cosmological constant . using a line element of the form @xmath67 -g(u , v)(\d\theta^2+\sin^2\theta\d\phi^2 ) \label{e3.3}\ ] ] and requiring that @xmath68 and @xmath69 on the null hypersurface @xmath70 , the field equations given in @xcite and @xcite reduce to the single equation on the impulse @xmath71 where @xmath72 is the laplacian on the sphere and the source of the wave is given by @xmath73 . now , putting @xmath74 and @xmath75 $ ] , the line element ( [ e3.2 ] ) can be transformed to the form ( [ e3.3 ] ) where @xmath76 and @xmath77 are given by @xmath78 with these , ( [ e3.4 ] ) takes the explicit form @xmath79 making the substitution @xmath80 , the laplacian on a sphere becomes @xmath81 + ( 1-z^2)^{-1}\partial_\phi\partial_\phi\ ] ] with which ( [ e3.5 ] ) takes the form ( [ e2.8 ] ) which is thus established for this case . we now consider the explicit solutions ( [ e2.10 ] ) given by @xmath82 . \label{e3.6}\ ] ] the simplest case in which @xmath83 for @xmath84 , is exactly the solution @xmath85 obtained by hotta and tanaka @xcite as described elsewhere @xcite . it represents a spherical impulsive wave in a de sitter background generated by two null particles moving in opposite directions . the particles are situated at the poles @xmath86 and @xmath87 . using these global coordinates , it can be seen that the impulsive wave is located on the cosmological horizon of a de sitter space - time . ( this is analogous to the solution given in @xcite in which the impulsive wave is located on the horizon of a schwarzschild space - time . ) moreover , since @xmath64 on the wave , it can be seen from ( [ e3.2 ] ) that at any time the area of the spherical wavefront spanned by @xmath88 and @xmath23 is a constant equal to @xmath89 . in fact it describes a spherical impulsive wave propagating from the north pole to the south pole in a closed form of the de sitter universe which contracts to a minimum size and then re - expands as described in @xcite and @xcite . the general solution ( [ e3.6 ] ) of the space - time ( [ e3.2 ] ) can be seen to represent a similar wave generated by two null particles with arbitrary multipole structure . the first few higher multipole terms are given simply by @xmath90 \cos(\phi-\phi_1 ) \nonumber \\ h_2&=&{2\over\sin^2\theta}\cos[2(\phi-\phi_2 ) ] \nonumber \\ h_3&=&-8{\cos\theta\over\sin^3\theta}\cos[3(\phi-\phi_3 ) ] . \nonumber \end{aligned}\ ] ] it has been argued above that the area of the spherical wavefront spanned by @xmath88 and @xmath23 is a constant . therefore this particular wave is non - expanding ( with the background either expanding or contracting through it ) . in view of this property , we would expect that this solution can be related to a particular ( impulsive ) case of the generalised class of kundt waves with non - vanishing cosmological constant @xmath91 presented by garca daz and plebaski @xcite . this has also been described by ozsvth , robinson and rzga @xcite as their class @xmath92 . adapting the coordinate system of @xcite , the line element for this class of solutions can be given in the form @xmath93\d\tilde u^2 \label{e22}\ ] ] where @xmath94 and @xmath95 is required to satisfy the equation @xmath96 it may be observed that this class is conformal to kundt s class of type n vacuum solutions with vanishing cosmological constant @xcite . in fact it can be shown @xcite that this is the only class of vacuum solutions that are conformal to kundt s class of type n with @xmath3 zero . since this is just the de sitter space - time when @xmath97 and @xmath5 , we can concentrate here on the case of an impulsive wave in which @xmath98 . now performing the transformation @xmath99 where @xmath100 in this case , the line element ( [ e22 ] ) becomes @xmath101 this can be seen to be exactly the solution ( [ e2.6 ] ) in which the de sitter background in the five - dimensional form ( [ e2.1 ] ) is parameterised by @xmath102 where , for consistency with ( [ e2.7 ] ) we only need to consider the impulsive wave located on @xmath103 . in addition , the field equation ( [ e23 ] ) is identical to ( [ e2.8 ] ) . finally , we may note that the left hand side of equation ( [ e3.5 ] ) or ( [ e23 ] ) is just the laplacian over the sphere plus two operating on a function . it follows that the solutions described above can be rotated arbitrarily over the sphere . since the equation is linear , solutions can therefore be constructed which contain an arbitrary number of pairs of arbitrary multipole particles distributed arbitrarily over the impulsive spherical wave . however , the impulsive wave is unique it is a sphere of constant surface area equal to @xmath89 . when the cosmological constant is negative , it is most convenient to introduce the global coordinate system given by @xmath104 in which @xmath105 , @xmath106 and @xmath107 . although this coordinate system is unconventional , it is particularly convenient for our purposes here . in these coordinates it can be seen that the impulsive wave is localised on the surface given by @xmath108 . thus , on the impulsive null hypersurface @xmath109 , the coordinates ( [ e4.1 ] ) are identical to those of ( [ e2.7 ] ) with the identity @xmath110 . in this case the general line element ( [ e2.6 ] ) for an impulsive wave in an anti - de sitter background takes the form @xmath111 it may immediately be observed that these coordinates are naturally adapted such that the impulsive wave is given by @xmath109 , and that the wave surface of a constant negative curvature which is spanned by the parameters @xmath112 and @xmath23 do not vary with time . the geometrical properties of these waves have been described elsewhere @xcite using different coordinate systems . basically , the impulsive wave is hyperboloidal and is generated by a single null particle moving in an anti - de sitter background which contains closed timelike geodesics . the particle propagates from one side of the universe to the other and then returns in an endless cycle . the wave propagating in one direction is obtained by the parameterisation @xmath113 as above , while propagation in the opposite direction can be parameterised by changing the signs of @xmath114 , @xmath115 and @xmath9 in ( [ e4.1 ] ) which is equivalent to putting @xmath116 . it is also convenient to reparameterise the wave surfaces by introducing an alternative global coordinate system in which @xmath117 then , also putting @xmath118 and @xmath119 , ( [ e4.2 ] ) becomes @xmath120 in these coordinates , the parameterisation of ( [ e2.1 ] ) with @xmath6 is given by @xmath121 it may immediately be observed that ( [ e4.4 ] ) is conformal to an impulsive _ pp_-wave . in fact it is the impulsive member of a family of solutions described by siklos @xcite which include the only vacuum space - times that are conformal to _ pp_-waves . in this work , siklos found a specific family of exact type n solutions ( including possible pure radiation ) with a negative cosmological constant given by @xmath122 provided @xmath123 satisfies the equation @xmath124 where @xmath125 . since the left hand side does not depend on @xmath126 explicitly , an arbitrary wave profile may be assumed and the solutions considered here simply correspond to the impulsive case in which @xmath127 . putting @xmath128 equation ( [ e4.6 ] ) can be written as @xmath129 which , using the coordinates @xmath113 and @xmath23 given by ( [ e4.3 ] ) , may be confirmed to be exactly of the form ( [ e2.8 ] ) . equation ( [ e2.8 ] ) is thus justified also for the case of a negative cosmological constant . having established the equation ( [ e2.8 ] ) in this case , we now express the explicit solutions ( [ e2.10 ] ) in the form @xmath130 . \label{e4.8}\ ] ] this now clearly represents an impulsive gravitational wave on a null hyperboloidal surface generated by a single null particle of arbitrary multipole structure located at the point @xmath131 on the surface . as in the previous section , we may finally note that the field equation is linear and includes the laplacian over a hyperboloidal surface of constant negative curvature . it therefore again follows that solutions can be constructed which contain an arbitrary number of arbitrary multipole particles distributed arbitrarily over the impulsive wave surface . in his paper @xcite , siklos has also shown that , for the vacuum case ( except for some possible point sources ) , the general solutions of ( [ e4.6 ] ) for @xmath123 is of the form @xmath132 where @xmath133 is an arbitrary function of @xmath134 and @xmath125 ( holomorphic in @xmath135 ) . for space - times that are conformal to kundt waves for both positive and negative cosmological constant , ozsvth , robinson and rzga @xcite have presented the equivalent explicit vacuum solution to their equation ( [ e23 ] ) which also involves an arbitrary function which is holomorphic in @xmath136 which is related to @xmath135 by @xmath137 using ( [ e5.1 ] ) and also ( [ e4.3 ] ) with @xmath113 , a general vacuum solution of ( [ e2.8 ] ) can be written as @xmath138 where @xmath139 . in terms of the coordinates @xmath44 and @xmath23 , @xmath135 is given by @xmath140 the explicit solutions described in the current paper may easily be represented in this form . for these cases , the siklos function @xmath123 may be expressed as @xmath141 , where @xmath142 corresponds to the distinct @xmath22-pole modes @xmath55 . for completeness , we may now identify the expressions corresponding to the first few modes described above . for @xmath47 , the monopole solution for @xmath143 is equivalent to @xmath144\ ] ] which corresponds to @xmath145 when @xmath146 , the dipole solution for @xmath57 is equivalent to @xmath147[(x - a)^2+y^2]}\right ] \nonumber \\ & & \hskip13pc \times \big[(a^2-x^2-y^2)\cos\phi_1 + 2ay\sin\phi_1\big ] \nonumber \end{aligned}\ ] ] which corresponds to @xmath148 \log{\zeta+a\over\zeta - a}.\ ] ] when @xmath149 , the quadrupole solution for @xmath58 is equivalent to @xmath150\cos2\phi_2 + 4ay(a^2-x^2-y^2)\sin2\phi_2 \over[(x+a)^2+y^2]^2[(x - a)^2+y^2]^2}\ ] ] which corresponds to @xmath151.\ ] ] jp was supported by a visiting fellowship from the royal society and , in part , by the grant gacr-202/96/0206 of the czech republic and the grant gauk-230/96 of the charles university . 99 hotta m and tanaka m 1993 _ class . quantum grav . _ * 10 * , 307 . podolsk j and griffiths j b 1997 _ phys . d _ * 56 * to appear in oct . kramer d , stephani h , maccallum m a h and herlt e 1980 _ exact solutions of einstein s field equations _ , cambridge university press , chapter 27 . siklos s t c 1985 _ galaxies , axisymmetric systems and relativity _ m a h maccallum , cambridge university press , 247 . hogan p a 1992 _ phys . lett . a _ * 171 * , 21 . griffiths j b and podolsk j 1997 _ phys . lett . a _ to appear . dray t and t hooft g 1985 _ nucl . b _ * 253 * , 173 . garca daz a and plebaski j f 1981 _ j. math . * 22 * 2655 . ozsvth i , robinson i and rzga k 1985 _ j. math . * 26 * 1755 . podolsk j 1993 _ phd thesis _ , charles university , prague . it is well known that , at least in the range @xmath152 $ ] , any function can be expressed as a sum of legendre polynomials . in particular , using the identity @xmath153 , it can be shown that @xmath154 . \label{ea.1}\ ] ] it also follows immediately from the closure property of the set of legendre polynomials that @xmath155 the associated legendre functions are generated by the relations @xmath156 by differentiating ( [ ea.1 ] ) @xmath22 times and multiplying by @xmath157 , it can be shown that @xmath158 . \label{ea.4}\ ] ] now , let us introduce the operator @xmath159 . then , applying the identity @xmath160p_j^m(z)=0 $ ] to ( [ ea.4 ] ) , it can immediately be seen that @xmath161 . % \label{ea.5}\ ] ] using the definition of @xmath162 in ( [ ea.3 ] ) , this becomes @xmath163 % \label{ea.6}\ ] ] and , from ( [ ea.2 ] ) , we finally obtain that @xmath164 \label{ea.7}\ ] ] where @xmath53 is the @xmath54 derivative of the delta function .
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we describe a class of impulsive gravitational waves which propagate either in a de sitter or an anti - de sitter background .
they are conformal to impulsive waves of kundt s class . in a background with positive cosmological constant they are spherical ( but non - expanding ) waves generated by pairs of particles with arbitrary multipole structure propagating in opposite directions .
when the cosmological constant is negative , they are hyperboloidal waves generated by a null particle of the same type . in this case
, they are included in the impulsive limit of a class of solutions described by siklos that are conformal to _ pp_-waves .
pacs class 04.20.jb , 04.30.nk running title : _ impulsive waves in ( anti-)de sitter space - time _
| 6,839 | 209 |
type ia supernovae ( sne ia ) serve as distance indicators used to measure the expansion history of the universe . although supernovae are not perfect standard candles , the peak absolute magnitude of an individual event can be inferred from observed multi - band light curves and a redshift using trained empirical relations . sn ia optical light curves have homogeneous time evolution , which allowed them to be described by a template . the relationship between light - curve decline rates and their correlation with absolute magnitude was noted by @xcite and further developed by @xcite , and was confirmed with the supernovae observed by the calan / tololo survey @xcite . an observed - color parameter was added to the modeling of multi - band light curves . today there is a suite of models that parameterize supernova light - curve shapes and colors , which are used to standardize absolute magnitudes to within a seemingly random @xmath7@xmath8 mag dispersion . the host galaxy conveys information about the supernova progenitor environment . although they do not describe an individual star , the host mass , specific star formation rate , and metallicity provide an expectation of the progenitor initial conditions that can be related to peak absolute magnitude . dependence of light - curve parameters and hubble residuals ( inferred magnitudes from light curves minus those expected from the cosmological distance - redshift relation , or hubble law ) on global host - galaxy properties has been sought . @xcite showed and @xcite confirmed that the light - curve shape parameter is correlated with host - galaxy morphology . @xcite find that hubble residuals depend on host mass . @xcite find a similar dependence on metallicity while @xcite find a dependence on both metallicity and specific star formation rate ( ssfr ) . * hereafter c13b ) perform such an analysis on the supernovae of the nearby supernova factory ( snfactory , * ? ? ? supernova distances are derived using linear magnitude corrections based on light - curve shape and color parameters from salt2 fits to snfactory synthetic photometry , using the procedure described in ; in this article these linearly - corrected distances are referred to as `` salt2 '' distances . host mass , ssfr , and metallicity are derived from photometric and spectroscopic observations of the associated galaxies ( * ? ? ? * hereafter c13a ) . their findings are consistent with previous studies ; when splitting the sn ia sample by host mass , ssfr , and metallicity at @xmath9 , @xmath10 , and @xmath11 respectively , they find that sne ia in high - mass ( low - ssfr , high - metallicity ) hosts are on average @xmath12 mag ( @xmath13 mag , @xmath14 mag ) brighter than those in low - mass ( high - ssfr , low - metallicity ) hosts after brightness corrections based on the salt2 light - curve shape and color brightness corrections . the hubble residuals depend on the model used to determine absolute magnitude . although there is the expectation that the progenitor variability tracked by host - galaxy parameters must also be directly manifest within the supernova signal itself , it appears not to be captured by the light - curve models used and the associated standardization in the cited work . the sdss - ii supernova survey , using samples divided by passive and star - forming hosts , finds hubble residual biases between both salt2- and mlcs2k2-determined distances @xcite : indication that the bias from the two light - curve fitters share a common source . the two parameters of one model are highly correlated with the two parameters of the other @xcite , which brings to question whether a third light - curve parameter associated with host properties is not being captured by salt2 or mlcs2k2 . although there are searches for such a third light - curve parameter associated with hubble residual bias ( e.g.@xcite who test whether heterogeneity in light - curve rise times can account for the sdss - ii result ) , as of yet no such parameter has been found . * hereafter k13 ) expand the optical light - curve parameterization by characterizing light curves through the probability distribution function of a gaussian process for the regressed values at phases @xmath15 to 35 in one - day intervals relative to peak , rather than the parameters of a best - fit model . the relationship between the k13 light - curve parameters and light - curve shapes can be seen in figure 4 of k13 , and are described briefly here . the effect of the @xmath16 parameter on the light curve is relatively phase - independent and is increasingly stronger in bluer bands , very similar to the behavior of host - galaxy dust and the color parameters of other fitters . the @xmath4 parameter affects the light - curve width and color around peak , similar to the stretch ( @xmath17 ) and @xmath18 parameters of salt2 and mlcs . the @xmath5 parameter affects peak colors in a fashion inconsistent with dust ( @xmath19 , @xmath20 , @xmath21 are positively correlated ) , controls the near - uv light curve width , and influences the light - curve decline 20 to 30-days after peak brightness . the @xmath22 parameter most notably affects peak color and the light - curve shape through all phases of the @xmath23 band . the k13 light curve parameters capture light - curve diversity distinct from those of salt2 ; figure 10 shows plots of salt2 versus k13 light - curve parameters . the absolute magnitude at peak @xmath24-band brightness is taken to be an unknown function of a set of 15 light - curve parameters ; after modeling the function as a gaussian process and training , the absolute magnitude can be determined to a dispersion as low as 0.09 mag . the larger number of light - curve parameters ( and their principal component compression ) that reduce the dispersion may be sensitive to the pertinent information encoded in the host - galaxy parameters . in this article we look for dependence of the k13 light - curve parameters and hubble residuals on host - galaxy parameters . k13 use a supernova dataset analyzed in c13b , so the absence of correlations between hubble residuals and host - galaxy parameters could provide positive evidence for the improved performance of the k13 method with respect to supernova environmental biases . our sample starts with the 119 sne ia from k13 . these objects have spectrophotometric data sets obtained by the snfactory with the supernova integral field spectrograph ( snifs , * ? ? ? * ) that had been fully processed as of early 2011 . the instrument has a fully filled @xmath25 spectroscopic field of view subdivided into a grid of @xmath26 spatial elements and a dual - channel spectrograph covering 32005200 and 510010000 . the median signal - to - noise within 2.5-days of peak brightness is 10.2 per 2.4 bin . the observing strategy provides observations of each supernova every 2 to 4 days up to @xmath27 days after maximum light . the spatial information provided by snifs enables photometric flux extraction for each spectral resolution element , making the reduced data spectrophotometric . flux - calibrated multi - band light curves are composed of synthetic photometry calculated by integrating spectra over the transmission function of observer optical @xmath28 bands . we are interested in the light - curve parameters and hubble residuals of the supernovae analyzed in k13 ( the same studied by c13b ) . the objective of k13 was to calibrate absolute magnitudes at peak brightness based on light - curve shapes and colors . the goal of this article is to examine possible biases in hubble residuals , not to minimize the dispersion as in k13 ; we therefore use a different sample selection increasing the number of supernovae used in this analysis to improve the statistics . the supernova subsamples considered in our analysis are summarized in table [ samples : tab ] . our * full * sample requires @xmath29 points over all bands with @xmath30 : this ensures temporal coverage to inform the regression of the multi - band light curves from roughly ten days before to forty days after @xmath24 maximum light . a @xmath31 requirement reduces sensitivity of hubble residuals on the cosmological parameters @xmath32 and @xmath33 . in k13 , sn # 13 was identified as having one spectrum produce synthetic photometry discrepant with the rest of the light curve ( see their figure 3 for the effect ) . for consistency with the k13 we take the very conservative step of rejecting any supernova with at least one synthetic photometry point @xmath34 away from the mean regressed light curve predicted by the gaussian process , even though such spectra already had been internally flagged as having processing errors . only six objects ( each with one outlying point ) , sne # 8 , # 14 , # 85 , # 98 , # 99 , and # 111 are thus culled . clc initial & snfactory data fully processed by early 2011 & 119 + full & @xmath35 photometric points with @xmath36 & + & @xmath37 & + & removal of sn # 13 identified in k13 as having spurious photometry & + & no photometry @xmath34 from regressed mean & 103 + early & in full sample & + & photometry at least 2 days before @xmath24 maximum & 64 + the above requirements leave a full sample of 103 supernovae . to ensure that biases are not introduced through extrapolation in the gaussian process regression , we define the * early * sample as the subset of 64 supernovae from the full sample with data at least two days before maximum brightness . in k13 , the calibration of absolute magnitudes is trained for the specific restframe wavelengths probed by the observation , and several filter sets were considered . the dispersion in hubble residuals of the validation sample of 43 sne was smallest for the absolute - magnitude calibration for the @xmath38-band of the blueshifted dark energy survey @xmath28 filter set , i.e. the rest - frame bands of a supernova at redshift @xmath39 covered by standard @xmath28 , which amount to supernova - frame effective wavelengths of 378 , 497 , 602 , and 708 nm . these filters are annotated with a superscript @xmath40 . as our objective is to see how well sn ia distances could be calibrated without restricting ourselves to a specific observer filter system , we use this particular training to look for the dependence of light - curve parameters and hubble residuals on host - galaxy properties . the hubble residuals are calculated as the difference between the true and inferred peak magnitudes of the gaussian - process model from table 5 in k13 ; the true magnitude being that regressed by the light - curve model , the inferred being that regressed from light - curve shape and color parameters . they perform four distinct analyses with different training and validation sets such that each supernova was in a validation set once and only once . for each supernova , the tabulated absolute magnitude is the one inferred while it was part of the validation set ; in this work the hubble residuals of all supernovae are treated collectively even though they were not analyzed with the same trained model . the statistics of interest in this article can not benefit and in fact suffer from the overtraining of light - curve and absolute - magnitude models that can occur in the training process of k13 . not all supernovae were used in the trainings of the k13 model ; `` training '' subsets were used to specify the supernova model , which in turn were used to predict absolute magnitudes of independent `` validation '' subsets . each absolute magnitude used in the current article was determined while the supernova was in the validation set , i.e. the training did not include the supernova itself . any overtraining of the training set thus appears as an added source of error in the absolute magnitudes . in the present article , the signals of interest are deviations of hubble residuals that are correlated with host - galaxy properties ; no host - galaxy property information was used in the determination of the absolute magnitudes . the early sample is a fair sample with which to examine the k13 analysis and training : supernovae lacking rise - time data extends the gaussian process regression outside of its expected range of applicability as peak magnitudes would then be extrapolated rather than interpolated . the light curves in extrapolated regions revert toward the mean function ( the updated template of @xcite in the analysis of k13 ) , and have uncertainties bounded by the gaussian process kernel model rather than data . for this reason , the k13 training sample was restricted to supernovae with data earlier than 2 days before maximum . we look for a correlation between light - curve parameters and the global host - galaxy properties of c13a in the early sample . note that each property has a different galaxy subset for which there is a measurement ; the metallicity measurement in particular requires emission lines and no contamination from agn emission . figure [ hosts : fig ] shows scatter plots from one of the light - curve trainings . the @xmath16 light curve parameter is associated with color and exhibits similar trends as the salt2 @xmath41 parameter shown in c13b ; larger ( redder ) values trending with higher mass and metallicity hosts as could be explained by host - galaxy dust absorption or a color - metallicity dependence . the @xmath4 parameter is correlated with salt2 @xmath17 and a smaller range of @xmath4 values in low - mass and low - metallicity hosts is apparent . the @xmath5 parameter has a larger color shift compared to @xmath4 , broadens the post - maximum light curve , and shifts the height and phase of the secondary maximum in @xmath42 and @xmath43 . there are no low @xmath5 values in low - metallicity hosts . the @xmath22 parameter is anti - correlated with @xmath17 , and is associated with variability in uv light - curve brightness and shape as well as the red secondary maximum . the set of extreme positive @xmath22 values are associated with high - mass galaxies . we use the results of c13 who apply a salt2 fit and make magnitude corrections on synthetic photometry using observer - frame non - overlapping boxcar filter functions whose wavelength ranges correspond to johnson - cousins @xmath24 , @xmath44 , and @xmath45 . the k13 hubble residuals are plotted against the salt2 residuals in figure [ resres : fig ] , with the early sample distinguished with filled symbols . qualitatively , most of the solid points with salt2 hubble residual less than @xmath46 have k13 residuals that are closer to zero ( they lie above the @xmath47 curve ; also compare the skewness from the negative sides of the histograms in figure [ resres : fig ] . ) the positive correlation between the residuals shows that the two methods are subject to shared `` intrinsic dispersion '' that is irreducible for both . over the entire early sample there is no significant bias between the supernova distances determined by k13 and salt2 : the distribution of the difference between their hubble residuals has mean @xmath48 mag . however , biases do appear when the early sample is divided by global host - galaxy property . for each host parameter , the population is split into two samples at boundaries established by previous studies : at @xmath49 for host - galaxy mass , @xmath50 for specific star formation rate , and @xmath51 for metallicity . not all hosts have measurements of these properties ; for the early sample the number of supernovae in the low- and high - bin subsets are @xmath52 for mass , @xmath53 for ssfr , and @xmath54 for metallicity . for the full sample the numbers are @xmath55 for mass , @xmath56 for ssfr , and @xmath57 for metallicity . the k13 minus salt2 hubble residuals in subsets defined by these splits are given in table [ k13_salt_offsets : tab ] . the significance of the differences are at @xmath58 , and @xmath59 for mass , specific star formation rate , and metallicity respectively . ccccccc @xmath60 & @xmath61 & @xmath62 & @xmath63 & @xmath64 & @xmath65 & @xmath66 + @xmath67 & @xmath68 & @xmath69 & @xmath70 & @xmath71 & @xmath72 & @xmath73 + the biases in k13 and salt2 hubble residuals imply that they should produce different hubble residual steps . the mean hubble residual and intrinsic magnitude dispersion are fit for each subset defined by host - galaxy parameters and any significant signal in the mean of the less - than sample minus the greater - than sample ( i.e. the hubble residual step ) is an indicator of population dependence . a non - zero slope of a linear fit to the data also indicates dependence . ccccccc @xmath40 & @xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 & @xmath79 + @xmath80 & @xmath81 & @xmath82 & @xmath83 & @xmath84 & @xmath85 & @xmath86 + @xmath87 & @xmath88 & @xmath89 & @xmath90 & @xmath91 & @xmath92 & @xmath93 + salt2 & @xmath94 & @xmath95 & @xmath96 & @xmath97 & @xmath98 & @xmath99 + @xmath40 & @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 & @xmath105 + salt2 & @xmath106 & @xmath95 & @xmath107 & @xmath97 & @xmath108 & @xmath99 + salt2 & @xmath109 & @xmath110 & @xmath111 & @xmath112 & @xmath113 & @xmath114 + for the k13 4-band regressed approach , the hubble residuals and slopes are shown as a function of host galaxy mass , specific star formation rate , and metallicity in figure [ corr : fig ] . the hubble residual steps and slopes are listed in the early/@xmath40 row of table [ offsets : tab ] . the signals for step or slope have significance @xmath115 for all cases except for the slope in ssfr that has a @xmath116 significance . we test for the presence of significant residual correlated structure that may be not be captured in the simple step and slope statistics by assuming an underlying covariance in the hubble residuals described as @xmath117 and fitting for @xmath118 and @xmath119 , where @xmath120 is a normalization parameter , @xmath119 the correlation length , @xmath121 is the matrn function , and @xmath118 is the separation in the log galaxy parameter . probability distribution functions of the covariance parameters are built from the best fits of multiple realizations of sets of points generated based on the data and their uncertainties . in all cases , @xmath122 falls within less than 1 standard deviation of the mean ; the data are consistent with having no correlation . to examine the influence of wavelength coverage , the light - curve to absolute - magnitude relation is retrained as in k13 using the same filter set but omitting the extreme bands , i.e. for @xmath80-data and @xmath87-data only . the steps and slopes are listed as `` early/@xmath80 '' and `` early/@xmath87 '' in table [ offsets : tab ] : no significant steps are seen in either of these trainings but non - zero slopes ( at @xmath123 ) are found with ssfr for both . for the @xmath87 set in particular , the significance and size of the slope exceeds that of salt2 . the salt2 ( v2.2.0 ) hubble residual steps and slopes are listed as the `` early / salt2 '' entry in table [ offsets : tab ] . the significance of the mass step is reduced to @xmath124 . splits in the other host - galaxy properties show similar weakening of the significance of the step in the two samples . differences in the hubble residual steps from different samples or light - curve analyses are listed for select pairs in table [ offsets2:tab ] . the steps of the various filter sets are calculated using the same supernovae and data , so their differences and uncertainties are not taken directly from table [ offsets : tab ] but are recalculated independently . a relevant case is the effect of excluding the blueshifted @xmath125-band at 378 nm ; as noted above the @xmath87 fit has a comparatively larger and significant slope with ssfr than those of other runs . cccc @xmath126 & @xmath127 & @xmath128 & @xmath129 + @xmath130 & @xmath131 & @xmath132 & @xmath133 + @xmath134 & @xmath135 & @xmath136 & @xmath137 + @xmath40-salt2 & @xmath138 & @xmath139 & @xmath140 + @xmath40-salt2 & @xmath141 & @xmath142 & @xmath143 + the host - galaxy - property values used to separate the sample come from published work that searched for systematic biases . the objective of this article is to use k13 hubble residuals to probe existing positive detections of bias residuals , not to again mine for systematics . nevertheless , we check whether a different bias is evident in our sample . figure [ bound : fig ] shows hubble - residual steps for a range of parameter boundaries ; there are no @xmath144 steps . neither the k13- nor salt2-determined distances of the early sample show significant @xmath123 hubble residual steps with respect to host - galaxy properties . we now consider how this subset behaves relative to larger subsets with relaxed criteria for the first phase of snifs data . this bridges the results of the early sample with c13b , who use the same data and salt2 fits from the same parent supernova set to find a step at @xmath123 with host mass , consistent with previous studies @xcite . our recalculation of salt2 hubble residual steps for the c13b sample are given in table [ offsets : tab ] as `` c13b / salt2 '' ; the results are consistent with c13b to 0.003 mag , the slight differences being due to our fitting of separate ( rather than joint ) intrinsic dispersions for each population . the principal feature that distinguishes the early sample is the requirement for data at least @xmath145 days before peak . we here examine the effect of changing the earliest phase required in the light curves . while the k13 analysis is expected to be biased with underestimated uncertainties for supernovae without data coverage at peak , this does provide useful insight when bridging salt2 results of the early sample with that of the full sample . figure [ firstdate : fig ] shows the hubble residual steps with respect to host mass as a function of the required phase of first observation , and the number of low - mass and high - mass that enter each calculation . we first consider the asymptotic limit where the first phase must be @xmath146 days after peak : this unconstraining requirement accommodates all 103 supernovae in the full sample . table [ k13_salt_offsets : tab ] shows that like the early sample , biases between k13 and salt2 hubble residuals in the full sample only appear after spliting by host - galaxy property . k13 and salt2 hubble residual steps for the full sample are given in the `` full '' rows of table [ offsets : tab ] . this larger set does have a positive salt2 mass - step detection at @xmath147 ; the k13 result has a less significant step detection at @xmath148 . the steps calculated from median hubble residuals are 0.061 mag and @xmath149 mag for salt2 and k13 respectively , within @xmath150 of the weighted - mean results . similar trends are seen with ssfr and metallicity . note that the full sample , with a slightly smaller sample , gives results similar to c13b . the results of the early sample are ( marginally at @xmath151 ) consistent with those of the full sample . the complement @xmath152 sample has a step of @xmath153 , so the difference in the steps of the two samples is @xmath154 mag . the step of the early sample is within @xmath155 of the full sample using salt2 . the null hubble - residual step evolves into a positive signal with a slight increase in signal and slight decrease of noise . a bias in the salt2 determination of corrected peak brightness between the early and full samples could be introduced when including or omitting rise - time data . the reapplication of salt2 on artificial supernovae in the early sample created by removal of early data shifts magnitudes by @xmath156 mag and @xmath157 for low- and high - mass hosts respectively and a new mass step of @xmath158 ; this does not account for the 0.048 mag difference between the early and full samples . samples defined by a more stringent first - phase requirement than the early sample have a small number and a fraction lower than the asymptotic limit of high - mass hosted supernovae . we have performed an extensive search for physical sources of bias between observing start and host - mass : none have been found . statistics examined include the first - phase versus local surface brightness , supernova magnitude and color at first observation , the salt2 @xmath17 light - curve shape parameter , and source of supernova discovery . for k13 , the step in the early sample is within @xmath159 of that from the full sample . the k13 median steps are consistent with zero over all first - phase requirements . the criterion for data at least @xmath145 days before peak for the early sample is based on the selection of k13 , fixed before any association with host - galaxy properties was made . figure [ firstdate : fig ] shows that relaxing the criterion to include additional supernovae with coverage before peak does not affect the hubble residual step . three supernovae with k13 hubble residuals @xmath160 mag from high - mass hosts with first data phases between maximum and 3-days after maximum , are responsible for the disparity between mean and median and the increase in hubble residual step for both k13 and salt2 . the differences between the k13 and salt2 mass steps are calculated and shown in table [ offsets2:tab ] for the early and full samples . the correlation between k13 and salt2 hubble residuals discussed in [ analysis : sec ] must be accounted for : the quadratic sum of the two step uncertainties and the simple by - eye comparison of the error bars in figure [ resres : fig ] overestimate uncertainty . the difference in the k13 and salt2 mass steps for the early sample is at @xmath148 and for the late sample @xmath161 , suggesting that the lower mass step of k13 is significant . the probabilities that the magnitude of the mass step is less for the k13 distances compared to salt2 distances are 0.954 and 0.998 for the early and full samples respectively . the differing salt2 hubble residual steps of the early and late samples may be more efficiently attributed to some underlying light - curve parameter that is not captured by salt2 and that is not uniformly represented in the two samples ; candidate parameters include those identified in the gaussian process analysis . in a manner similar to the splitting of samples based on host - galaxy parameters , we now calculate salt2 hubble residual steps dividing the full sample by each of the first four k13 parameters ; results are given in table [ offsetsk13:tab ] for samples split by @xmath16 through @xmath22 in turn . the hubble residual steps are significant ( @xmath144 ) in samples divided by @xmath4 and @xmath5 . hubble residuals as a function of these parameters are shown in figure [ saltpca : fig ] , which also shows as a point of reference and contrast k13 hubble residuals as a function of these parameters . the most significant step in k13 hubble residuals for the same four parameters is @xmath162 . recall that as noted in [ analysis : sec ] and seen in figures 69 of k13 , the k13 parameters do capture diversity in the uv and the secondary maximum at redder wavelengths that are not captured in the magnitude corrections using the salt2 parameters @xmath41 and @xmath17 . cccc @xmath163 & @xmath164 & @xmath165 & @xmath166 absolute magnitudes for a subset of nearby supernova factory objects determined using a novel regression method and tabulated in k13 are used to measure the dependence of hubble residuals on global host - galaxy properties . hubble residual steps with host mass , specific star formation rate , and metallicity are @xmath0 , @xmath167 , and @xmath168 mag respectively : all consistent with zero step . the salt2 hubble residual step for the same subset is @xmath94 mag , consistent with a null step at @xmath159 . as a point of comparison with a previous positive detection , @xcite use an independent supernova sample and color stretch magnitude corrections to measure a mass step of 0.11 mag at @xmath169 . this article s and their measurements are consistent at the @xmath170 level using k13 , and at @xmath150 using salt2 hubble residuals . for an expanded sample of supernovae with light curves unlike those in the k13 training , the hubble residual step with mass is @xmath100 mag , which is consistent with a null step at the 8% level . in contrast the color stretch corrected magnitude step is @xmath106 mag , consistent with the null step at only 0.15% . the difference between the k13 and salt2 mass steps for the full sample has an almost @xmath171 significance . the analysis in this article differs with c13b in sample selection . expansion of the sample to include supernovae with later first data asymptotically gives significant @xmath172 ) salt2 hubble residual steps with host - mass but not for k13 hubble residuals . to examine the difference between the early and full samples , we calculate the difference between the salt2 mass steps of the early and @xmath152 samples to be @xmath173 , a difference this large has a 4.2% chance of random occurrence . the early and late samples do have different populations of objects , for example the early sample has a higher ratio of low - mass hosts ; however , the difference in relative fraction of high- to low - mass hosts is not statistically significant . we conclude that there is an evolution from null to positive - signal salt2 hubble residual steps going from early to full samples , but that more statistics is required to determine whether the samples are inherently drawn from different parent distributions . k13 hubble residual steps also evolve with sample , but asymptote at @xmath151 for the full sample . the analysis in this article also differs with c13b in the inference of supernova absolute magnitude . there is salt2 hubble residual dependence on both the k13 @xmath4 and @xmath5 parameters at the @xmath144 level though not meeting the @xmath171 standard for a positive detection ) that does not appear in k13 hubble residuals . we conclude that multi - band light curves do convey information about supernova absolute magnitude that is not captured in salt v.2.2.0 . the sample of supernova used to train salt2 may have a different supernova population than that of the snfactory data , and its performance on the results of the ensemble average is expected to improve with retraining using a subset of the snfactory sample . the gp analysis has an advantage in that it nulls out population effects by having training and validation sets drawn from a common sample . nevertheless , if salt2 did capture the diversity of light curves no bias would arise due to differing training and validation populations assuming that systematic changes in absolute magnitude are accompanied with changes in the time - evolving sed . the continued study of correlations between host - galaxy properties and hubble residuals is of importance even as tools that measure distances improve . correlations serve as important diagnostics of potential biases related to progenitor population that are not captured by light - curve parameters or absolute magnitude inference . better accuracy can be achieved from local host - galaxy properties extracted from the region around the supernova @xcite , which can give a more accurate view of the progenitor environment compared to global properties . physically , there is the expectation that multi - band light curves with broad wavelength coverage transmit information about the progenitor system . due to limitations in current light - curve fitters , some of this information has only been noted indirectly through host - galaxy properties . this article shows that the novel light - curve analysis of k13 may be capturing more aspects of sn ia diversity . application of the method to more supernovae will provide better statistics to test this hypothesis . we thank dan birchall for observing assistance , the technical and scientific staffs of the palomar observatory , the high performance wireless radio network ( hpwren ) , the national energy research scientific computing center ( nersc ) , and the university of hawaii 2.2 m telescope . we wish to recognize and acknowledge the significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . nc acknowledges support from the lyon institute of origins under grant anr-10-labx-66 . this work was supported by the director , office of science , office of high energy physics , of the u.s . department of energy under contract no . de - ac02 - 05ch11231 ; by a grant from the gordon & betty moore foundation ; in france by support from cnrs / in2p3 , cnrs / insu , and pnc , and by french state funds managed by the anr within the investissements davenir program under reference anr11idex000402 ; and in germany by the dfg through trr33 `` the dark universe . '' nersc is supported by the director , office of science , office of advanced scientific computing research , of the u.s . department of energy under contract no . de - ac02 - 05ch11231 . hpwren is funded by national science foundation grant number ani-0087344 , and the university of california , san diego .
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@xcite [ k13 ] introduced a new methodology for determining peak - brightness absolute magnitudes of type ia supernovae from multi - band light curves .
we examine the relation between their parameterization of light curves and hubble residuals , based on photometry synthesized from the nearby supernova factory spectrophotometric time series , with global host - galaxy properties .
the k13 hubble residual step with host mass is @xmath0 mag for a supernova subsample with data coverage corresponding to the k13 training ; at @xmath1 , the step is not significant and lower than previous measurements .
relaxing the data coverage requirement the hubble residual step with host mass is @xmath2 mag for the larger sample ; a calculation using the modes of the distributions , less sensitive to outliers , yields a step of 0.019 mag .
the analysis of this article uses k13 inferred luminosities , as distinguished from previous works that use magnitude corrections as a function of salt2 color and stretch parameters : steps at @xmath3 significance are found in salt2 hubble residuals in samples split by the values of their k13 @xmath4 and @xmath5 light - curve parameters .
@xmath4 affects the light - curve width and color around peak ( similar to the @xmath6 and stretch parameters ) , and @xmath5 affects colors , the near - uv light - curve width , and the light - curve decline 20 to 30 days after peak brightness
. the novel light - curve analysis , increased parameter set , and magnitude corrections of k13 may be capturing features of sn ia diversity arising from progenitor stellar evolution .
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supergiant fast x ray transients ( sfxts ) are a new class of high mass x ray binaries ( hmxbs ) discovered by ( e.g. * ? ? ? * ) that are associated with ob supergiant stars via optical spectroscopy . in the x rays they display outbursts significantly shorter than those of typical be / x ray binaries characterized by bright flares with peak luminosities of 10@xmath510@xmath6 erg s@xmath7 which last a few hours ( as observed by ; * ? ? ? * ; * ? ? ? as their quiescence is characterized by a luminosity of @xmath8 erg s@xmath7 ( e.g. * ? ? ? * ; * ? ? ? * ) , their dynamic range is of 35 orders of magnitude . while in outburst , their hard x ray spectra resemble those of hmxbs hosting accreting neutron stars , with hard power laws below 10kev combined with high energy cut - offs at @xmath930 kev , sometimes strongly absorbed at soft energies @xcite . so , even if pulse periods have only been measured for a few sfxts , it is tempting to assume that all sfxts might host a neutron star . the mechanism producing the outbursts is still being debated , and it is probably related to either the properties of the wind from the supergiant companion @xcite or to the presence of a centrifugal or magnetic barrier @xcite . was discovered during _ asca _ observations of the scutum arm region performed on 1994 april 12 , and 1999 october 34 as a flaring source which exhibited flux increases by a factor of 10 ( up to @xmath10 erg @xmath3 s@xmath7 ) with rising times on the order of 1hr @xcite , a strong absorption @xmath11 @xmath3 , and coherent pulsations with a period of @xmath12s . a _ chandra _ observation on 2004 may 12 , which provided the coordinates refined to arcsecond accuracy [ ra(j2000@xmath13 , dec(j2000@xmath14 @xmath15 , @xcite ] , found the source at a much fainter level ( @xmath16 erg @xmath3 s@xmath7 ) , and with a spectrum that was fit with an absorbed power - law model [ @xmath17 , @xmath18 @xmath3 ] . a newly discovered source , igr j18410@xmath00535 , was observed to flare by on 2004 october 8 @xcite , as it reached @xmath20mcrab in the 2060kev energy range ( integrated over 1700s ) and 20 mcrab in the 60200kev range . the source was also detected in the 2060kev energy range in subsequent observations , at a flux roughly half that of the initial peak . @xcite identified igr j18410@xmath00535 as . @xcite established that the ir counterpart was 2mass 18410043@xmath00535465 , a reddened star with a weak double - peaked h@xmath21 emission line , initially classified as a be star , which @xcite later reclassified as b1 ib type star ; this corroborated the evidence that is a member of the sfxt class , as proposed by @xcite . @xcite presented the first broad - band spectrum of this source , obtained with ( ibis@xmath22jem - x ) , that they fit with an absorbed power - law with @xmath23 , @xmath24 @xmath3 . in 2007 @xmath25 @xcite observed the outburst of the periodic sfxt igr j11215@xmath05952 @xcite , which allowed us to discover that the accretion phase during the bright outbursts lasts much longer than a few hours , as seen by lower - sensitivity instruments . this is contrary to what was initially thought at the time of the discovery of this new class of sources . between 2007 october 26 and 2008 november 15 , was observed by @xmath25 as part of a sample of 4 sfxts which included igr j16479@xmath04514 , xte j1739302 , and . the main aims were to characterize their long - term behavior , to determine the properties of their quiescent state , to monitor the onset of the outbursts and to measure the outburst recurrence period and duration @xcite . approximately two observations per week were collected with the x ray telescope ( xrt , * ? ? ? * ) and the uv / optical telescope ( uvot , * ? ? ? * ) . during such an intense and sensitive monitoring , was the only sfxt that did not go through a bright outburst , although several on - board burst alert telescope ( bat , * ? ? ? * ) detections have been recorded @xcite . in this paper we report on the observations of the first outburst of observed by @xmath25 on 2010 june 5 and we compare its properties with those of the prototype of the sfxt class , , which went into a bright outburst on 2010 march 04 . and count s@xmath7 detector@xmath7 , respectively . the empty circles correspond to bat in event mode ( s / n@xmath26 ) , filled circles to bat survey mode data . , width=321 ] ax j1841.0@xmath00536 triggered the @xmath25/bat on 2010 june 5 at 17:23:30 ut ( trigger 423958 , * ? ? ? * ; * ? ? ? this is the first outburst of ax j1841.0@xmath00536 detected by the bat for which @xmath25 performed a slew , thus allowing broad - band data collection . the source was detected in a 1344s bat image trigger , during a pre - planned observation , and there is an indication that the source was already in outburst before this observation began and well after it ended . the xrt began observing the field rather late , at 17:51:50 ut ( @xmath27s ) , after the very long bat image trigger . the automated target ( at , sequences 00423958000 - 001 ) observations lasted for several orbits , until @xmath28ks after the trigger ) . follow - up target of opportunity ( too ) observations for a total of 10.8ks were obtained ( sequences 00030988093101 ) . the data cover the first 11d after the beginning of the outburst . the sfxt prototype triggered the bat on 2010 march 04 at 23:13:54 ut ( trigger 414875 , * ? ? ? @xmath25 executed an immediate slew , so that the narrow - field instruments ( nfi ) started observing it about 395s after the trigger . the at ran for @xmath29ks and was followed by one too observation ( 00035056149 ) for @xmath30ks until the source went into moon constraint . for the 2010 march 4 outburst of . , width=321 ] the xrt data were processed with standard procedures ( xrtpipeline v0.12.4 ) , filtering and screening criteria by using ftools in the heasoft package ( v.6.9 ) , as fully described in e.g. @xcite . we used the latest spectral redistribution matrices ( 20100930 ) . the bat data were analysed using the standard bat analysis software within ftools . mask - tagged bat light curves were created in several energy bands ( see * ? ? ? * for further details ) . survey data products , in the form of detector plane histograms ( dph ) , are available , and were also analysed with the standard batsurvey software . all quoted uncertainties are given at 90% confidence level for one interesting parameter unless otherwise stated . [ cols="<,<,<,<,<,<,>,<,<,<,<,<,<,<,<",options="header " , ] @xmath31 is absorbing column density ( @xmath32 @xmath3 ) ; @xmath33 is 210kev observed flux ( @xmath34 erg @xmath3 s@xmath7 ) ; @xmath35 is 210kev luminosity ( @xmath36 erg s@xmath7 ) . pow : absorbed powerlaw . cpl : cutoff powerlaw , energy cutoff e@xmath37 ( kev ) . hct : absorbed powerlaw , high energy cutoff e@xmath37 ( kev ) , e - folding energy e@xmath38 ( kev ) . figure [ axj1841fig : lcv_campaign ] ( left ) shows the _ swift_/xrt 0.210kev light curve of throughout our 2008 monitoring program @xcite background - subtracted and corrected for pile - up , psf losses , and vignetting . all data in one observation ( 12ks , typically ) were generally grouped as one point , except for the june 5 outburst , which shows up as a vertical line on the adopted scale [ fig . [ axj1841fig : lcv_campaign ] ( right ) ] . the observed dynamical range of this source in the xrt band is @xmath39 , considering as the lowest point a 3@xmath40 upper limit obtained on mjd 54420 at @xmath41 counts s@xmath7 , and the highest point the peak of the june 5 outburst . [ axj1841fig : lcv_allbands ] shows the detailed light curves during the brightest part ( first orbit ) of the outburst in several energy bands . the bat light curve is rather flat and weak , but significant signal is found at the lower energies ( 1550kev ) . for the timing analysis we converted the event arrival times to the solar system barycentre with the task barycorr and the _ chandra _ position @xcite . we note that the xrt pc - mode readout frequency slightly undersamples the source period of @xmath424.7 s @xcite with respect to its nyquist frequency , which would guarantee an unambiguous reconstruction of the signal . timing searches were conducted in various time intervals and energy ranges around @xmath424.7s , employing two methods : a fast - folding algorithm and an unbinned @xmath43 test @xcite . in both cases the searches were inconclusive and we could not set meaningful upper limits on the pulsed fraction , because of the red noise and the scalloping due to the poor sampling . for the 2010 june 5 outburst of , we extracted the mean spectrum of the brightest x - ray emission ( obs . 00423958000 , @xmath441708 to 2390s ) and performed a fit in the 0.310kev band of the data , which were rebinned with a minimum of 20 counts bin@xmath7 to allow @xmath45 fitting . an absorbed power - law model yielded a column of @xmath46 @xmath3 , a photon index @xmath47 ( @xmath48 , 17 degrees of freedom , dof ) , and a 210kev unabsorbed flux of @xmath49 erg @xmath3 s@xmath7 . as there is no strict overlap between the bat event data and the xrt data ( see fig . [ axj1841fig : lcv_allbands ] ) , we extracted one spectrum from the bat event file of the whole observation 00423958000 ( ` event ' ) , and one from the xrt event file in the same observation . there are , however , bat survey data available in the interval @xmath50@xmath51s , and we extracted one ( ` survey ' ) spectrum in this restricted interval . we performed joint fits in the 0.310kev and 14100kev energy bands for xrt and bat , respectively . a factor was included to allow for normalization uncertainties between the two spectra . the broad - band fits performed with the bat ` survey ' spectrum yield consistent results with the ones performed with the bat ` event ' spectrum , albeit with more unconstrained parameters due to the much more limited statistics . therefore in table [ axj1841:tab : broadspec ] we report the fits performed with the bat ` event ' spectrum . a simple absorbed power - law model is clearly an inadequate representation of the broad band spectrum ( @xmath52 for 29 dof ) , so we also considered other models typically used to describe the x ray emission from accreting pulsars in hmxbs , such as an absorbed power - law model with an exponential cutoff ( cutoffpl in xspec ) and an absorbed power - law model with a high energy cut - off ( highecut ) . the latter models provide a significantly more satisfactory fit of the broad - band emission , resulting in a hard powerlaw - like spectrum below 10kev , with a roll over of the higher energies when simultaneous xrt and bat data fits are performed . figure [ axj1841:fig : meanspec ] shows the fits for the highecut model . table [ axj1841:tab : broadspec ] reports the average 210kev luminosities . two estimates of the distance are available , from ( * ? ? ? * @xmath53kpc ) and ( * ? ? ? * @xmath54kpc ) , so we assumed 5 kpc , which is consistent with both . figure [ axj1841fig:17544lcv_allbands ] shows the first orbit of the xrt and bat light curves of the 2010 march 04 outburst of . the xrt count rate reaches a peak exceeding 25 counts s@xmath7 , then decreases to about 0.5 counts s@xmath7 and increases again up to about 20 counts s@xmath7 at the end of the first orbit of observations . the xrt / wt spectrum ( @xmath55@xmath56s ) , extracted with a grade 0 selection to mitigate residual calibration uncertainties at low energies , was fit with an absorbed power law resulting in @xmath57 , @xmath58 @xmath3 ( @xmath59 for 125 dof ) , and @xmath60 erg @xmath3 s@xmath7 ( unabsorbed ) . the xrt / pc spectrum ( @xmath61 to @xmath62s ) yields @xmath63 @xmath3 , @xmath64 ( @xmath65 , 17 dof ) and @xmath66 erg @xmath3 s@xmath7 ( unabsorbed ) . we extracted a bat spectrum strictly simultaneous with the xrt one and fitted them ( 0.310kev , 1450kev ) with the same models as adopted for . the results are in table [ axj1841:tab : broadspec ] , while figure [ axj1841:fig:17544meanspec ] shows the fits for the highecut model . for the luminosity calculation we adopted a distance of 3.6 kpc @xcite . for the 2010 march 4 outburst of . , width=283 ] in this paper we report our analysis of the 2010 june 5 outburst of and the 2010 march 04 outburst of the sfxt prototype . while in the first case , the image trigger was a very long one and nfi data could be collected only @xmath67s after the trigger , when the source was relatively dim , in the second case , the slew occurred immediately after the trigger , while was still very bright . figure [ axj1841fig : best_sfxts ] ( panels e and g ) shows the full light curves of the outbursts of and as they were observed by @xmath25 for 11 and 2 days after the trigger , respectively . the xrt light curve shows a decreasing trend from the initial bright flare from a maximum of @xmath68 counts s@xmath7 down to @xmath69 counts s@xmath7 during the first day , with several flares superimposed , hence yielding a dynamic range of approximately 900 during this outburst . then , after three days , the source count rate rose again and reached @xmath70 counts s@xmath7 . we estimate that the observed dynamical range of this source in the xrt band , considering the historical data we collected during our monitoring campaign ( * ? ? ? * ; * ? ? ? * see fig . [ axj1841fig : lcv_campaign ] ) is @xmath39 , hence placing it well in the customary range for sfxts . the outburst of has similar characteristics to the one observed on 2008 march 31 , as the xrt light curve shows a peak at about 25 counts s@xmath7 , decreases to about 0.5 counts s@xmath7 and then increases again up to about 20 counts s@xmath7 at the end of the first orbit ( fig . [ axj1841fig:17544lcv_allbands ] ) . this behaviour was previously observed in and , most notably , in igr j08408@xmath04503 @xcite and sax j1818.6@xmath01703 @xcite , so that this multiple - peak structure of the light curve could be considered a defining characteristic of the sfxt class and it is likely due to inhomogeneities within the accretion flow ( e.g. * ? ? ? figure [ axj1841fig : best_sfxts ] compares the light curves of and with the outbursts of sfxts as observed during our monitoring campaigns with @xmath25 . the most complete set of x ray observations of an outburst of a sfxt is the one of the periodic sfxt igr j11215@xmath05952 @xcite , which was surprisingly long . we now know that such a length of the outburst ( hence the length of the accetion phase ) is a common characteristic of the whole sample of sfxts followed by @xmath25 , and in this respect fits right in , as its outburst lasted several days . 5952 did not trigger the bat , so it is referred to mjd 54139.94 ) . points denote detections , triangles 3@xmath40 upper limits . red data points ( panels e , g ) refer to observations presented here for the first time , while grey points to data presented elsewhere . where no data are plotted , no @xmath25 data were collected . vertical dashed lines mark time intervals equal to 1 day , up to a week . references : igr j084084503 ( 2008 - 07 - 05 , * ? ? ? * panel a ) ; igr j11215@xmath05952 ( 2007 - 02 - 09 , * ? ? ? * panel b ) ; igr j16479@xmath04514 ( 2005 - 08 - 30 , * ? ? ? * panel c ) ; xte j1739@xmath0302 ( 2008 - 08 - 13 , * ? ? ? * panel d ) ; sax j1818.6@xmath01703 ( 2009 - 05 - 06 , * ? ? ? * panel f ) . panels e and g report the 2010 - 03 - 04 outburst of igr j17544@xmath02619 and the 2010 - 06 - 05 outburst of ax j1841.0@xmath00536 , respectively ( this work ) . , width=321 ] we have presented the broad - band ( 0.3100kev ) simultaneous spectroscopy of . this allows us to make a comparison with the findings on the other sfxts that were observed in the same fashion . the soft x ray spectral properties observed during this flare are generally consistent with those observed with _ asca _ during the 1999 flare ( * ? ? ? * @xmath11 @xmath3 , @xmath71 ) . as was observed relatively late after the trigger , no meaningful information can be derived on variability of the soft spectral parameters during the outburst , such as the absorbing column density . however , we note that the value of @xmath72 in outburst follows the same trend of ` harder when brighter ' as reported in table 4 of @xcite , which was based on out - of - outburst emission . for the joint bat@xmath22xrt spectrum during the 2010 june 5 outburst , an absorbed power - law model is an inadequate description , and more curvy models are required . we considered an absorbed power - law model with an exponential cutoff and an absorbed power - law model with a high energy cut - off , models typically used to describe the x ray emission from accreting neutron stars in hmxbs . we obtained a good fit of the 0.3100kev spectrum , characterized by high absorption @xmath73 @xmath3 , a hard power law below 10kev , and a high energy cutoff . these properties of are reminiscent of those of the prototypes of the sfxt class , [ whose data we have presented here and in @xcite ] , and xte j1739@xmath0302 ( @xcite ) . although no statistically significant pulsations were found in the present data , is one of the 4 sfxts with known pulse period @xcite , @xmath74s , the others being igr j11215@xmath05952 ( 186.78@xmath750.3s , * ? ? ? * ) , igr j16465@xmath04507 ( 228@xmath756s , * ? ? ? * ) , and igr j18483@xmath00311 ( 21.0526@xmath750.0005s , * ? ? ? * ) . while lacking the detection of cyclotron lines , which would yield a direct measurement of the magnetic field @xmath76 of the neutron star , an indirect estimate can be obtained by considering the highecut fit to the broad - band spectrum of in outburst . our value of the high energy cutoff @xmath77kev , although loosely constrained , yields a @xmath7810@xmath79 g @xcite this value for @xmath76 , which is indeed similar to the one derived for the prototype of the sfxt class , is inconsistent with a magnetar nature of . in conclusion , we have shown how ax j1841.0@xmath00536 nicely fits in the sfxt class , based on the observed properties of during the 2010 june 5 outburst : a large dynamical range in x ray luminosity , the similarity of the light curve length and shape to those of the prototype of the class , and the x ray broad - band spectrum , which we show here for the first time down to 0.3kev , thus constraining both the absorption and the cutoff energy . we thank the @xmath25 team duty scientists and science planners and the remainder of the @xmath25 xrt and bat teams , s. barthelmy in particular , for their invaluable help and support . this work was supported in italy by contract asi - inaf i/009/10/0 , at psu by nasa contract nas5 - 00136 . pe acknowledges financial support from the autonomous region of sardinia through a research grant under the program po sardegna fse 20072013 , l.r . 7/2007 `` promoting scientific research and innovation technology in sardinia '' . lllllll source & sequence & obs / mode & start time ( ut ) & end time ( ut ) & exposure & time since trigger + & & & ( yyyy - mm - dd hh : mm : ss ) & ( yyyy - mm - dd hh : mm : ss ) & ( s ) & ( s ) + ax j1841.0@xmath00536 & 00423958000 & bat / evt & 2010 - 06 - 05 17:15:13 & 2010 - 06 - 05 19:01:13 & 1515 & -502 + & 00423958000 & xrt / pc & 2010 - 06 - 05 17:52:02 & 2010 - 06 - 05 18:03:24 & 682 & 1708 + & 00423958001 & xrt / pc & 2010 - 06 - 05 19:02:59 & 2010 - 06 - 06 09:46:49 & 6111 & 5965 + & 00030988093 & xrt / pc & 2010 - 06 - 07 21:07:22 & 2010 - 06 - 07 21:23:58 & 969 & 186228 + & 00030988094 & xrt / pc & 2010 - 06 - 08 14:49:26 & 2010 - 06 - 08 15:05:56 & 977 & 249952 + & 00030988095 & xrt / pc & 2010 - 06 - 09 16:18:40 & 2010 - 06 - 09 16:38:56 & 1212 & 341705 + & 00030988096 & xrt / pc & 2010 - 06 - 10 19:47:57 & 2010 - 06 - 10 21:33:56 & 1414 & 440663 + & 00030988097 & xrt / pc & 2010 - 06 - 11 13:10:31 & 2010 - 06 - 11 13:35:56 & 1503 & 503217 + & 00030988098 & xrt / pc & 2010 - 06 - 13 13:41:14 & 2010 - 06 - 13 15:23:58 & 848 & 677859 + & 00030988099 & xrt / pc & 2010 - 06 - 14 04:09:44 & 2010 - 06 - 14 05:58:57 & 1554 & 729969 + & 00030988100 & xrt / pc & 2010 - 06 - 15 02:39:58 & 2010 - 06 - 15 04:22:57 & 1184 & 810983 + & 00030988101 & xrt / pc & 2010 - 06 - 16 03:55:52 & 2010 - 06 - 16 04:14:58 & 1110 & 901937 + igr j17544@xmath02619 & 00414875000 & bat / evt & 2010 - 03 - 04 23:10:00 & 2010 - 03 - 04 23:30:02 & 1202 & -239 + & 00414875000 & xrt / wt & 2010 - 03 - 04 23:20:40 & 2010 - 03 - 05 00:47:11 & 421 & 401 + & 00414875000 & xrt / pc & 2010 - 03 - 04 23:27:59 & 2010 - 03 - 05 00:49:12 & 1520 & 840 + & 00414875001 & xrt / wt & 2010 - 03 - 05 01:21:58 & 2010 - 03 - 05 08:48:39 & 783 & 7679 + & 00414875001 & xrt / pc & 2010 - 03 - 05 01:22:03 & 2010 - 03 - 05 09:02:56 & 2502 & 7684 + & 00035056149 & xrt / pc & 2010 - 03 - 06 00:59:19 & 2010 - 03 - 06 01:33:50 & 801 & 92721 +
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_ swift _ observed an outburst from the supergiant fast x ray transient ( sfxt ) ax j1841.0@xmath00536 on 2010 june 5 , and followed it with xrt for 11 days .
the x ray light curve shows an initial flare followed by a decay and subsequent increase , as often seen in other sfxts , and a dynamical range of @xmath1 .
our observations allow us to analyse the simultaneous broad - band ( 0.3100kev ) spectrum of this source , for the first time down to 0.3kev , which can be fitted well with models usually adopted to describe the emission from accreting neutron stars in high - mass x
ray binaries , and is characterized by a high absorption ( @xmath2 @xmath3 ) , a flat power law ( @xmath4 ) , and a high energy cutoff .
all of these properties resemble those of the prototype of the class , igr j17544@xmath02619 , which underwent an outburst on 2010 march 4 , whose observations we also discuss .
we show how well ax j1841.0@xmath00536 fits in the sfxt class , based on its observed properties during the 2010 outburst , its large dynamical range in x ray luminosity , the similarity of the light curve ( length and shape ) to those of the other sfxts observed by _
swift _ , and the x ray broad - band spectral properties . [ firstpage ] x - rays : binaries - x - rays : individual : ax j1841.0@xmath00536 - x - rays : individual : igr j17544@xmath02619 facility : _ swift _
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a submanifold @xmath0 in some symplectic manifold @xmath1 is called lagrangian if @xmath2 and @xmath3 . a simple example is given by the zero section @xmath4 in the cotangent bundle of a smooth manifold @xmath5 , and this is universal in the sense that a neighborhood of any lagrangian embedding of a closed @xmath5 into some symplectic manifold is symplectomorphic to a neighborhood of @xmath6 . lagrangian submanifolds play a fundamental role in symplectic geometry and topology , as many constructions and objects can be recast in this form . in fact , already in a 1980 lecture ( cf . @xcite ) , a. weinstein formulated the `` symplectic creed '' : _ everything is a lagrangian submanifold . _ today , lagrangian submanifolds ( sometimes decorated with additional structures ) are for example studied as objects of the _ fukaya category _ , which plays a fundamental role in kontsevich s formulation of homological mirror symmetry . rather than delving into such general theories , i want to concentrate here on a quite simple , and in fact basic , question : _ which closed , oriented @xmath7-manifolds admit a lagrangian embedding into the standard symplectic space @xmath8 , with @xmath9 ? _ an excellent introduction to this question , containing a discussion of some of the relevant classical algebraic topology , as well as early results obtained by holomorphic curve methods , is @xcite , which i will quote freely . for @xmath10 there is not much to say , since @xmath11 is the only connected closed 1-manifold , and the lagrangian condition @xmath12 is trivial in this case . in general , a necessary condition for an oriented closed manifold @xmath13 to admit a lagrangian embedding into @xmath14 is that its euler characteristic @xmath15 should vanish . this is because the self - intersection number of any submanifold of @xmath14 is clearly zero , but it is also equal to the euler characteristic of the normal bundle , which for lagrangian submanifolds is isomorphic to the cotangent bundle . so for @xmath16 , the only orientable closed manifold that could have a lagrangian embedding into @xmath17 is @xmath18 , and it embeds e.g. as the product of one circle in each @xmath19-factor . for non - orientable closed surfaces @xmath20 , classical algebraic topology implies that a necessary condition for the existence of a lagrangian embedding is that @xmath21 is divisible by 4 , and a beautiful construction by givental @xcite shows that for strictly negative euler characteristic this is also sufficient . the embedding question was only recently completely answered , when shevchishin showed that the klein bottle does not have a lagrangian embedding into @xmath17 ( @xcite , see also @xcite for an alternative argument by nemirovski ) . already for @xmath22 , elementary algebraic topology does not tell us much . it was one of the many important results in gromov s landmark paper @xcite to show that there are no exact lagrangian embeddings into @xmath23 , in the sense that any global primitive @xmath24 of the symplectic form @xmath25 has to restrict to a non - exact closed 1-form on the lagrangian submanifold @xmath26 . this in particular rules out @xmath27 , but of course there are plenty of closed orientable 3-manifolds with @xmath28 . all of this and more is discussed in @xcite . the goal of this chapter is to show how knowledge about string topology can be applied to give a far - reaching refinement of gromov s result . in particular , i aim to present the overall strategy for proving the following result : ( fukaya)[thm : fukayamain ] let @xmath5 be a compact , orientable , aspherical spin manifold of dimension @xmath7 which admits an embedding as a lagrangian submanifold of @xmath14 . then a finite covering space @xmath29 of @xmath5 is homotopy equivalent to a product @xmath30 for some closed @xmath31-manifold @xmath32 . moreover , @xmath33 is the centralizer of some element @xmath34 which has maslov class equal to 2 and positive symplectic area . the assertion about the maslov class is known as _ audin s conjecture _ , and was originally asked for tori in @xmath14 , see @xcite . the spin condition is a technical assumption ( it is needed to make the relevant moduli spaces of holomorphic disks orientable ) , and i expect that it can be removed by reformulating the argument somewhat . the asphericity assumption ( meaning that all higher homotopy groups of @xmath5 vanish ) enters the proof in a fairly transparent way , and one can imagine various replacements . as a corollary , we obtain the following more precise statement in dimension 3 . ( fukaya)[cor : fukaya1 ] if the closed , orientable , prime 3-manifold @xmath5 admits a lagrangian embedding into @xmath35 , then @xmath5 is diffeomorphic to a product @xmath36 of the circle with a closed , orientable surface . the fact that the product @xmath36 does embed as a lagrangian submanifold into @xmath35 follows from an elementary construction , see e.g. @xcite . basically , one starts from an isotropic embedding of @xmath20 into @xmath35 , e.g. by embedding it into the lagrangian subspace @xmath37 . then one uses the fact that a small neighborhood necessarily is symplectomorphic to a neighborhood of the zero section in @xmath38 , the direct sum of the cotangent bundle with a trivial symplectic vector bundle of rank 2 , to embed the product @xmath36 by taking the product of the zero section in @xmath39 with a standard small @xmath40 . the above statements are special cases of a more general result discovered by kenji fukaya , and first described in @xcite , see also @xcite . as with most results involving @xmath41-holomorphic curves , the underlying idea can be traced back to misha gromov s foundational paper @xcite . his proof of the fact that there are no exact compact lagrangian submanifolds of @xmath14 contains an important seed for fukaya s arguments . therefore , after discussing some aspects of moduli spaces of holomorphic disks in the next section , i begin section [ sec : main ] by sketching gromov s argument , followed by the discussion of an instructive example and the statement of fukaya s refinement ( theorem [ thm : blackbox ] ) . in a nutshell , fukaya s important observation was that the compactification of the moduli spaces of holomorphic disks with boundary on a lagrangian submanifold @xmath42 can be expressed in terms of string topology operations , in particular the loop bracket ( and possibly also its higher analogues at the chain level ) . after an interlude section on basic properties of l@xmath43 algebras , i describe how theorem [ thm : fukayamain ] is a fairly straightforward consequence of theorem [ thm : blackbox ] . corollary [ cor : fukaya1 ] will follow by using specific facts from 3-dimensional topology . before finishing with a guide to the literature , i discuss a few further small observations . at the moment of this writing , full proofs of the above theorems are not available yet . one would like to construct a chain model @xmath44 for the free loop space of the manifold @xmath5 with the loop bracket operation @xmath45 ( and higher operations as necessary ) , such that @xmath46 is an l@xmath43-algebra . moreover when this model is considered with coefficients in a suitable novikov ring , the compactified moduli spaces of holomorphic disks should give rise to an element in this chain complex satisfying the maurer - cartan equation in this l@xmath43-algebra . so the principal problem is that the apparent freedom one has in building the chain model is severely constrained by the need to make it fit with the analysis of holomorphic disks , which in particular involves delicate transversality and gluing issues . in this text , i will largely ignore the technical difficulties , by stating the key results as black boxes . the main tool in our study of lagrangian embeddings @xmath47 will be moduli spaces of holomorphic disks with boundary on @xmath5 . before introducing the relevant spaces , i will briefly review some standard notions . the first of these is the _ maslov index _ of a loop of lagrangian subspaces in @xmath14 . the lagrangian grassmannian @xmath48 is defined as the space of all lagrangian subspaces of @xmath49 . standard symplectic linear algebra shows that a real @xmath7-dimensional subspace @xmath50 is lagrangian if and only if it is orthogonal with respect to the standard euclidean inner product to @xmath51 , its rotation by @xmath52 . moreover , any orthonormal basis of a lagrangian subspace is a unitary basis of @xmath14 , and conversely the real linear span of a unitary frame is a lagrangian subspace . since @xmath53 acts transitively on the unitary frames generating the same lagrangian subspace , one concludes that @xmath48 can be identified with @xmath54 . the map @xmath55 which associates to a unitary matrix the square of its determinant induces a well - defined map @xmath56 , and for any loop @xmath57 we define _ its maslov index _ as @xmath58 clearly @xmath59 so defined is an invariant of the free homotopy class of @xmath60 . moreover , one can check that @xmath61 induces an isomorphism @xmath62 . now given a lagrangian immersion @xmath63 , any loop @xmath64 determines a loop in @xmath48 , simply by taking the loop of tangent spaces of @xmath5 at the image points of @xmath60 . in this way , we get a maslov index @xmath65 discussions of the maslov index can be found in various texts , see e.g. @xcite , @xcite or @xcite . in particular , it is easy to see that for a lagrangian immersion of an oriented manifold @xmath5 , the maslov index takes values in @xmath66 . another useful notion is the _ area _ or _ energy _ of a disk @xmath67 with boundary on a lagrangian submanifold @xmath68 . it is defined as @xmath69 and one easily checks that it only depends on the free relative homotopy class of @xmath70 . indeed , just observe that given a homotopy @xmath71 \times d , [ 0,1 ] \times { { \partial}}d ) \to ( { { \mathbb{c}}}^n , l)$ ] with @xmath72 and @xmath73 , we have @xmath74 \times { { \partial}}d } h^*\omega = 0\ ] ] by the assumption that @xmath5 is a lagrangian submanifold . in particular , @xmath75 descends to a homomorphism @xmath76 . finally , in order to discuss holomorphic curves , we introduce the relevant spaces of almost complex structures . generally , an _ almost complex structure _ @xmath41 on some manifold @xmath77 is an automorphism of the tangent bundle @xmath78 with @xmath79 . if @xmath77 carries a symplectic form @xmath80 , then an almost complex structure @xmath41 on @xmath77 is said to be _ tamed by _ @xmath80 , if @xmath81 for all nonzero tangent vectors @xmath82 . in other words , @xmath80 is a positive area form on each 1-dimensional @xmath41-complex subspace of each tangent space . it is a standard result that these @xmath80-tamed almost complex structures form a contractible ( in particular non - empty ) space . given such a tamed @xmath41 , one can define a riemannian metric on @xmath77 by setting @xmath83 one can also define the notion of a @xmath41-holomorphic map @xmath84 from some riemann surface @xmath85 to @xmath77 , simply by asking that it satisfy the usual cauchy - riemann equations with respect to the given @xmath41 : @xmath86 in local conformal coordinates @xmath87 on the riemann surface this can be written equivalently as @xmath88 in riemannian geometry , the @xmath89-energy of a map @xmath90 is defined as @xmath91 where @xmath61 is any volume form on @xmath20 and @xmath92 denotes the operator norm of @xmath93 with respect to the metric @xmath94 on @xmath20 and the metric @xmath95 on @xmath77 . the integrand turns out to be independent of the choice of @xmath61 , as any scaling factor also appears in the operator norm with opposite exponent . now the importance of the taming condition stems from the following crucial fact : suppose @xmath96 is @xmath41-holomorphic , and we have chosen local conformal coordinates @xmath97 on @xmath20 . then at @xmath98 we have @xmath99 so that @xmath100 in particular , for @xmath41-holomorphic disks @xmath101 , the energy as defined above , which was a purely topological quantity , is the same as the usual @xmath89-energy of the map @xmath70 . choose an almost complex structure @xmath41 tamed by the standard symplectic structure @xmath25 on @xmath14 . given a relative homotopy class @xmath102 , we consider the set @xmath103 = a \in \pi_2({{\mathbb{c}}}^n , l)\}.\ ] ] the real 2-dimensional group @xmath104 of biholomorphisms of the disk fixing @xmath105 acts on @xmath106 by precomposition , and the quotient @xmath107 is called the moduli space of holomorphic disks in the class @xmath108 . the reader should take note that in the literature moduli spaces of holomorphic curves are almost universally denoted by @xmath109 , usually with some decoration , and the precise meaning of the symbol should very carefully be checked in each case . the equation defining @xmath106 is elliptic , and so the linearization is a fredholm operator . the index theorem for holomorphic curves , as discussed for example in ( * ? ? ? * appendix c ) , implies that the index of this operator , and hence the expected dimension of the moduli space @xmath106 , is @xmath110 . the assignment @xmath111 can be viewed as a section of a suitable banach space bundle . under favourable circumstances this section can be arranged to be transverse to the zero section , in which case its zero set @xmath112 is a manifold of the expected dimension . in general , this is a serious technical difficulty beyond the scope of the present discussion , and resolving it requires substantial work . note that , for @xmath113 , the action of @xmath114 on @xmath106 is free . so , assuming that we can arrange transversality , we conclude that @xmath115 is a smooth manifold of dimension @xmath116 as shown by examples in @xcite , one generally needs the spin condition on @xmath5 to be able to orient the moduli spaces @xmath115 . for several reasons the individual spaces @xmath115 tend to not be very useful for proving anything interesting about @xmath5 ( with the notable exception of theorem [ thm : gromov - notexact ] ) . the first is that these spaces strongly depend on @xmath41 , as simple examples show . [ ex : nosol ] consider @xmath117 and the relative homotopy class @xmath118 which has degree 1 in the first factor and degree 0 in the second . given any real number @xmath119 , consider the almost complex structure @xmath120 on @xmath17 given by the matrix @xmath121 if @xmath122 ranges in some interval @xmath123 , then all the @xmath120 are tamed by the symplectic form @xmath124 on @xmath125 . now for the standard split complex structure @xmath126 on @xmath17 , the moduli space @xmath127 is clearly non - empty , since for any @xmath128 the map @xmath129 defines an element in it . in general , given any @xmath120-holomorphic map @xmath70 in the class @xmath108 , note that the projection @xmath130 onto the first @xmath19-factor is holomorphic in the usual sense , and since it has degree 1 it will be a biholomorphism . so , precomposing with a suitable element of @xmath114 , we may assume that this projection @xmath130 is just a rotation by @xmath131 . a short computation shows that in this case for @xmath70 to be @xmath120-holomorphic , it is necessary and sufficient that @xmath132 now if @xmath133 is a solution of this equation , then @xmath134 so for @xmath135 the moduli space @xmath136 is empty . the phenomenon discussed in this example is a crucial ingredient in the proof below of gromov s theorem [ thm : gromov - notexact ] . a second , related complication is the failure of compactness , a simple instance of which is described in example [ ex : noncompact ] below . both of these apparent problems can be overcome by considering the collection of all @xmath137 at once . in the next subsection , i briefly discuss the compactness issue . gromov s compactness theorem , in its modern formulation , asserts that every sequence of holomorphic curves of fixed topology and uniformly bounded energy , whose images lie within a compact subset of the target , has a subsequence converging in a suitable sense to a limiting `` stable curve '' . a proof in text book form for holomorphic spheres was given by mcduff and salamon @xcite , and an exposition of gromov s proof for the higher genus case was written up by hummel @xcite . the case of curves with boundary is for example discussed by liu as part of her ph.d . thesis @xcite . for holomorphic disks with lagrangian boundary conditions the precise statement and proof were worked out in u. frauenfelder s diploma thesis , published as @xcite . here , i will describe the statement under the simplifying assumption that the target symplectic manifold @xmath1 is exact , i.e. @xmath138 , as in the case of my main example @xmath139 . this assumption in particular implies that there are no non - constant holomorphic spheres in @xmath140 , and so the possible limiting configurations are much more restricted than in the general case . basically , all the stable curves arising as gromov limits will be stable trees of disks , as explained presently . recall that a _ tree _ is a ( finite ) set @xmath141 together with an edge relation @xmath142 satisfying the following conditions : * ( symmetric ) * if @xmath143 then @xmath144 . * ( antireflexive ) * if @xmath143 then @xmath145 . * ( connected ) * for all @xmath146 with @xmath147 there exist @xmath148 with @xmath149 , @xmath150 and such that @xmath151 for all @xmath152 . * ( no cycles ) * if @xmath153 satisfy @xmath151 and @xmath154 for all @xmath155 then @xmath156 . one usually draws trees by drawing the corresponding 1-dimensional cw - complexes , which have one vertex for each @xmath157 and one ( unoriented ) 1-cell connecting @xmath122 and @xmath158 whenever @xmath143 ( and @xmath144 ) . it follows from these axioms that the cw - complex corresponding to a tree in this way is connected and contractible . moreover , deleting an edge connecting two vertices @xmath146 splits the cw - complex into two connected components , and we denote the subset of vertices in the component of @xmath158 by @xmath159 . @xmath122 [ br ] at 32 45 @xmath158 [ l ] at 86 39 @xmath159 [ tl ] at 133 19 and @xmath158 , and the corresponding subtree @xmath159.,title="fig : " ] a map @xmath160 is called a _ tree homomorphism _ if for each @xmath161 the preimage @xmath162 is a tree , and moreover @xmath143 implies that either @xmath163 or @xmath164 . a bijective tree homomorphism is called _ tree isomorphism_. @xmath122 [ l ] at 480 311 @xmath158 [ l ] at 560 284 @xmath60 [ l ] at 639 269 @xmath165 [ l ] at 565 127 @xmath166 [ l ] at 76 347 @xmath167 [ l ] at 297 270 @xmath168 [ l ] at 303 182 @xmath169 [ l ] at 327 82 next we fix an exact symplectic manifold @xmath1 with a compact lagrangian submanifold @xmath170 and an almost complex structure @xmath41 tamed by @xmath80 . given this data , we define a _ stable tree of holomorphic disks with one boundary marked point _ to be a tuple @xmath171 modelled over a tree @xmath172 , consisting of a collection of holomorphic maps @xmath173 indexed by @xmath174 , a collection of nodal points @xmath175 indexed by directed edges @xmath143 , and a marked point @xmath176 labelled by @xmath177 , subject to the following conditions : 1 . if @xmath143 then @xmath178 . 2 . for each @xmath157 , the _ special boundary points associated with the vertex @xmath122 _ , namely the points @xmath179 for different @xmath180 with @xmath143 , together with @xmath181 for @xmath182 , are all pairwise distinct . 3 . if @xmath183 is constant , then the cardinality of the set @xmath184 of all special points is at least 3 . as usual , two such stable trees of disks @xmath185 and @xmath186 modelled on trees @xmath141 and @xmath187 , respectively , are called _ equivalent _ if there exist a tree isomorphism @xmath188 and a collection of mbius transformations @xmath189 such that @xmath190 for all @xmath191 with @xmath143 . define the _ energy _ of a stable tree of holomorphic disks as the sum @xmath192 now we can give the relevant notion of convergence . let @xmath1 be an exact symplectic manifold , let @xmath170 be a compact lagrangian submanifold , and let @xmath41 be an @xmath80-tame almost complex structure on @xmath140 . a sequence @xmath193 of holomorphic disks with one boundary marked point @xmath194 is said to _ gromov converge _ to a stable tree of disks @xmath195 with one boundary marked point if there exist sequences of elements @xmath196 indexed by @xmath157 such that the following statements hold : 1 . the sequence @xmath197 converges to @xmath181 . 2 . for every @xmath157 , the sequence of maps @xmath198 converges to @xmath183 uniformly on compact subsets of @xmath199 . if @xmath143 then @xmath200 converges to @xmath179 uniformly on compact subsets of @xmath201 . 4 . if @xmath143 then @xmath202 intuitively , one can imagine the mbius transformations @xmath203 as microscopes , focusing on subregions in the domain to detect some piece of the limiting map . item ( 3 ) in the definition then says that different microscopes really capture different phenomena . item ( 4 ) is a way of phrasing that the limit captures all the essential pieces . in particular , together with ( 2 ) it implies that @xmath204 now gromov s compactness theorem for disks can be stated as follows . let @xmath1 be an exact symplectic manifold , let @xmath170 be a compact lagrangian submanifold , and let @xmath41 be an @xmath80-tame almost complex structure on @xmath140 . suppose @xmath193 is a sequence of holomorphic disks with bounded energy such that all images are contained in some compact subset of @xmath140 . then @xmath205 has a gromov convergent subsequence . moreover , for a convergent sequence @xmath205 the limit is unique up to equivalence . a careful exposition of the proof of this theorem , without the simplifying assumption that @xmath140 is exact , can be found in @xcite . rather than going into details here , i will illustrate the phenomenon by considering a specific example . [ ex : noncompact ] the situation is interesting already for holomorphic disks with boundary on @xmath40 with respect to the standard complex structure @xmath126 . by the maximum principle , these are necessarily maps @xmath206 . they exist for all non - negative degrees , and the degree @xmath207 of the map equals the degree of its restriction to the boundary circle . it turns out that the maslov index of the class @xmath208 is @xmath209 . applying the general index formula , one finds that @xmath210 for @xmath211 , this index equals 1 . indeed , the group of holomorphic degree 1 maps , i.e. holomorphic automorphisms of @xmath212 is 3-dimensional ( it can be identified with @xmath213 ) . in fact , any element of this group can be written uniquely as a composition @xmath214 , where @xmath215 is a rotation given by @xmath216 and @xmath217 is a map fixing @xmath218 . since we only consider the moduli space of maps up to the equivalence relation of precomposing with an element of @xmath114 , the moduli space can be identified with @xmath11 by recording the image of @xmath105 under any map in the equivalence class . in particular , the moduli space @xmath219 is compact . next consider a map of higher degree @xmath220 . again any such map can be written as a composition @xmath214 , where @xmath221 is a product ( not composition ! ) of @xmath207 maps of degree @xmath222 , each fixing @xmath223 , and @xmath215 is a rotation . the map @xmath221 is characterized completely in terms of its zeros , counted with multiplicities . in fact , if these zeros are @xmath224 , , @xmath225 , then we have @xmath226 by precomposing with an appropriate @xmath227 , we may always arrange that one of the zeros , say @xmath225 , equals 0 . the coordinates of the other zeros give @xmath228 free local parameters for @xmath221 , and together with the rotation parameter for @xmath215 we get @xmath229 as predicted by the dimension formula above . the moduli space @xmath230 of degree @xmath220 self - maps of the disk @xmath212 is noncompact . in fact , consider representative maps @xmath231 of a sequence of points in the moduli space and orderings of the zeros @xmath232 of @xmath231 such that @xmath233 ( as above , this can be arranged by precomposing a given representative with some element of @xmath114 , which does not change the equivalence class ) . after passing to a subsequence , we get convergence of the @xmath234 to some limiting @xmath235 for each @xmath236 . the formula above still makes sense in the limit , but the `` zeros '' @xmath237 which lie on the circle @xmath11 contribute a trivial factor of @xmath222 to the product . so if there are @xmath238 of these `` phantom '' zeros , the naive limiting map @xmath239 will have degree @xmath240 and so it is not an element of @xmath241 . notice that in the above discussion , we made two arbitrary choices : a choice of ordering @xmath234 of the zeros of @xmath231 , and the choice to always reparametrize so that @xmath233 . suppose for definiteness that with these choices we have @xmath242 . then there are unique maps @xmath243 such that @xmath244 , and so we get a different sequence of representatives @xmath245 of the same divergent sequence of points in the moduli space @xmath241 . just as above we get a , generally different , limiting map @xmath246 for a suitable subsequence . note that in this reparametrization , we will have @xmath247 , and so @xmath246 has degree @xmath248 . if we would analyse the situation fully , we would recover , for a suitable subsequence , the existence of finitely many sequences of mbius transformations @xmath249 , such that the reparametrized maps @xmath250 converge to some limiting map @xmath251 of degree @xmath252 in such a way that @xmath253 . in addition , these maps fit together and form a disk tree as described in the compactness theorem above . very roughly , the compactness theorem asserts that one can compactify a given space @xmath115 by adding pieces built out of moduli spaces @xmath254 with @xmath255 . this compactification is often denoted by @xmath256 . it admits an obvious stratification , where the stratum of codimension @xmath257 corresponds to stable trees of disks modelled on trees with exactly @xmath257 ( unoriented ) edges . indeed , the heuristic dimension count proceeds as follows . denote by @xmath258 the number of special points on the disk associated to @xmath157 and by @xmath259 its relative homotopy class . note that @xmath260 , where @xmath257 is the number of edges of the tree , since we had one marked point to start with and each edge gives rise to two nodal points . the formal dimension of the moduli space of disks associated to the vertex @xmath261 is @xmath262 requiring that the nodal points corresponding to an edge in @xmath141 are mapped to the same point in @xmath5 gives @xmath263 constraints . putting these together , we find that the formal total dimension equals @xmath264 _ if _ , for a given @xmath41 , all the moduli spaces appearing in the compactification @xmath256 were transversely cut out , one could hope to prove a _ gluing theorem _ , asserting that in fact the compactified moduli space is a manifold with boundary and corners . this is generally too much to ask . in @xcite , fukaya , oh , ohta and ono describe a procedure to put a so - called _ kuranishi structure _ on the compactified moduli spaces . without going into details , this roughly means that these spaces admit fundamental chains _ that make them function as if they were manifolds with corners_. theorem [ thm : blackbox ] below should be understood in this sense . presumably , the ongoing polyfold project of hofer , wysocki and zehnder ( cf . @xcite ) will eventually lead to an alternative approach to the problem of putting enough structure on the compactified moduli space to prove a statement like theorem [ thm : blackbox ] . as already mentioned , it is instructive to review the proof for the following well - known theorem of gromov . [ thm : gromov - notexact ] if a compact manifold @xmath5 admits a lagrangian embedding into @xmath14 , then @xmath265 . ( sketch ) i sketch the proof of this theorem given in ( * ? ? ? * section 9.2 ) , slightly rephrasing the end of the argument in order to make the relation to the following discussion even more apparent . fix a lagrangian embedding @xmath47 , and choose a vector @xmath266 with @xmath267 . consider the set @xmath268 \times d \times { { \mathbb{c}}}^n)$ ] of hamiltonian functions such that @xmath269 the idea is to consider , for a fixed @xmath270 , the moduli space @xmath271 of maps @xmath272 satisfying the following conditions : * @xmath273 for some @xmath274 $ ] , and * the relative homotopy class @xmath275 \in \pi_2({{\mathbb{c}}}^n , l)$ ] vanishes . so for @xmath276 , we are considering holomorphic disks with boundary on the lagrangian @xmath5 , and since the relative homotopy class vanishes , these are precisely the constant maps . note that we do not divide out any automorphisms here , since for positive @xmath24 these have no reason to preserve the solution space to the equation . one can prove that for fixed small @xmath277 , there is still a compact @xmath7-dimensional family of solutions . in fact , for generic choice of the hamiltonian @xmath278 , standard transversality techniques show that @xmath271 is a smooth @xmath279-dimensional manifold , whose boundary consists of those elements with @xmath280 . on the other hand , a straightforward computation as in example [ ex : nosol ] shows that , for our choice of @xmath108 , there are no solutions to the equation with @xmath281 . so if @xmath271 was compact , it would give a smooth cobordism from @xmath5 to the empty set . now consider the evaluation map @xmath282 from what we said above , the boundary of this @xmath283-chain in the free loop space of @xmath5 is the cycle of constant loops @xmath284 \in c_n(\lambda l)$ ] . since this cycle is nontrivial in homology , @xmath271 can not be compact . it follows from elliptic regularity theory ( cf . * theorem 4.1.1 ) ) that if compactness fails , there is a sequence @xmath285 such that @xmath286 as @xmath287 . appropriately rescaling such a sequence and applying removal of singularities as in @xcite , one finds either a nonconstant holomorphic sphere or a nonconstant holomorphic disk with boundary on @xmath5 . since @xmath14 does not contain nonconstant holomorphic spheres ( such spheres would be contractible and have positive energy , contradicting stokes theorem ) , the only possibility is the existence of some nonconstant holomorphic disk @xmath288 . now the standard symplectic form @xmath289 is positive on all complex lines in @xmath14 , and hence on all the tangent planes to the image of @xmath82 , so we have @xmath290 moreover , @xmath291 , where e.g. @xmath292 , and so stokes theorem implies that @xmath293 on the other hand , the lagrangian condition states that @xmath294 vanishes pointwise when restricted to @xmath5 . combining these observations , it follows that @xmath295 is a closed 1-form representing a nonzero class in @xmath296 , and this proves the theorem . what was the essence of the proof of gromov s theorem ? basically , the point is that the space @xmath271 has a single boundary component , corresponding to the space of constant disks , and hence for topological reasons it can not be compact . analysing the breakdown of compactness , we found holomorphic disks . the elements of @xmath271 do not appear to be holomorphic curves , due to the nonzero right hand side @xmath297 of the equation . however , they can in fact be viewed as holomorphic maps into @xmath298 with respect to a family of almost complex structures which have an off - diagonal term built out of this right hand side , projecting holomorphically and with degree 1 to the disk . this basic phenomenon was already present in example [ ex : nosol ] . the stable map compactification in this particular case is given by bubble trees of disks with exactly one main component , satisfying the original equation , and all other bubbles being strictly holomorphic ( in the graph picture just mentioned , each of these has constant projection to the disk ) . fukaya s insight was to see that the new boundary can be described in terms of string topology operations . namely , for each @xmath102 , consider the compactified moduli space @xmath299 of holomorphic disks in the relative homotopy class @xmath108 . similarly , denote by @xmath300 the space of solutions @xmath301 to the equation @xmath302\ ] ] in the relative homotopy class @xmath102 . pretending as always that transversality holds , this space is a manifold of dimension @xmath303 and we denote its compactification by @xmath304 . assuming the analysis can be made to work , both @xmath305 and @xmath304 can be thought of as chains on @xmath306 , simply by associating to each map of the disk its restriction to the boundary circle . in fact , for @xmath307 there is a slight ambiguity , since its elements are only well - defined up to precomposition by @xmath308 . but one can easily get around this point , for example by replacing the actual map by a parametrization proportional to arc length . to arrive at a clean statement , i will introduce some further notation . suppose we are given a suitable model @xmath44 for the chains on the free loop space of @xmath5 with coefficients in @xmath309 , such that for each @xmath310 the compactified spaces @xmath305 and @xmath304 , with their respective evaluation maps to @xmath306 , define elements in it . note that @xmath306 is a disjoint union over its connected components @xmath311 , which can be identified with conjugacy classes @xmath60 of elements of @xmath312 . it is convenient to introduce a new complex @xmath313 whose underlying vector space is @xmath44 , but with grading shifted according to the maslov index , i.e. an element in @xmath314 will have degree @xmath315 in @xmath313 . the complex @xmath313 comes with a filtration by the symplectic area as follows . it is shown in ( * ? ? ? 4.1.4 ) that the infimum @xmath316 of the symplectic energy is strictly positive . since each loop on @xmath5 bounds a disk in @xmath14 , we can view the energy as a map @xmath317 . the energy of @xmath318 is now defined as @xmath319 , where @xmath320 is the energy of the free homotopy class of the loops parametrized by the chain @xmath321 ( assumed to have connected domain of definition ) . then the filtration @xmath322 on @xmath313 is given by @xmath323 now consider the completion @xmath324 of @xmath313 with respect to this filtration . this means that an element in @xmath324 will be a possibly infinite sum @xmath325 of chains @xmath326 , provided that for each @xmath327 there are only a finite number of summands satisfying @xmath328 . with this definition , gromov compactness and our grading convention imply that @xmath329 is a well - defined element of @xmath324 of degree @xmath330 . in fact , it is contained in the submodule @xmath331 of chains with strictly positive area . similarly , @xmath332 is an element of @xmath324 of degree @xmath279 , as follows by applying the analogue of to the graph of elements of @xmath300 in @xmath333 . as explained at the end of section 2 , the compactification of @xmath305 is obtained by adding lower dimensional strata built as fiber products of other such moduli spaces along evaluation maps . in particular , the codimension 1 pieces @xmath334 are build from configurations of two holomorphic disks for which suitable boundary points are mapped to the same point in @xmath5 . a more careful analysis reveals that , on the level of boundary values of the holomorphic maps , these configurations correspond to loop brackets @xmath335 with @xmath336 . [ ex:2torus ] the mechanism just described can be seen in example [ ex : noncompact ] , but maybe it is slightly easier to visualize for the standard lagrangian torus @xmath337 . any class @xmath338 is characterized by two integers @xmath339 giving the degrees of the projections to the two coordinate disks . for a moduli space @xmath340 with respect to the standard complex structure @xmath126 on @xmath17 to be nontrivial we need @xmath341 . leaving aside the constant maps , the simplest moduli spaces @xmath342 and @xmath343 are compact , and in fact one can identify both with @xmath344 . indeed , the equivalence classes of the maps @xmath345 given by @xmath346 for @xmath347 represent all elements in @xmath342 , and similarly the maps @xmath348 represent all elements in @xmath343 . note that because of the symmetries in the problem , in this particularly simple example the evaluation maps at @xmath222 are submersions , so that the geometric definition of the loop bracket @xmath349 can be used . to illustrate the discussion above , we want to argue that @xmath350.\end{aligned}\ ] ] every element of the space @xmath351 is a map @xmath352 of the form @xmath353 with @xmath354 and @xmath355 . the space @xmath356 is obtained as the quotient by the diagonal action of @xmath114 , so the equivalence class of @xmath70 as above is alternatively represented by both @xmath357 or @xmath358 , where @xmath359 is uniquely associated with the equivalence class of @xmath70 . now consider a sequence @xmath360 with @xmath224 and @xmath361 fixed but @xmath362 and @xmath363 varying . assume that the projection of the sequence to @xmath356 leaves every compact subset , meaning that the corresponding sequence @xmath364 in the above notation does the same . elementary considerations now show that for a suitable subsequence @xmath365 there will be a point @xmath366 such that 1 . @xmath367 uniformly on compact subsets of @xmath368 and @xmath369 uniformly on compact subsets of @xmath370 , or 2 . @xmath371 uniformly on compact subsets of @xmath370 and @xmath372 uniformly on compact subsets of @xmath368 . in both cases , the corresponding subsequences @xmath373 and @xmath374 converge to elements @xmath375 and @xmath376 , respectively , and the pair @xmath377 represents a boundary point of @xmath378 . one checks that as @xmath224 and @xmath361 and the sequence @xmath365 vary , one obtains all boundary points from this construction . in case ( i ) , @xmath379 and @xmath380 , which is the unique intersection point of @xmath381 and @xmath382 , and in case ( ii ) the roles are reversed . in particular , the boundary loops of the two limit disks concatenate to represent points in the loop bracket @xmath383 . we now return to the general discussion . similarly to the case of @xmath307 , the codimension 1 stratum for @xmath304 corresponds to stable maps consisting of one component satisfying the perturbed equation and one holomorphic disk , and so it is described by the loop brackets of the form @xmath384 . depending on the precise technical implementation , the gluing along lower dimensional boundary strata might actually introduce more terms , corresponding to higher operations . the main technical assertions which should come out of such an implementation can be formulated as the following theorem . to get a cleaner statement , i have chosen to state it in slightly stronger form than is strictly necessary . the concept of a filtered l@xmath43 algebra which appears in the statement is discussed in detail in the following section , where i also give some algebraic perspective on the equations and . [ thm : blackbox ] let @xmath47 be a closed , oriented , spin lagrangian submanifold . then on the filtered , degree - shifted chain complex @xmath324 associated to a suitable chain model @xmath44 for the free loop space @xmath306 there exists a filtered l@xmath43-algebra structure @xmath385 of degree @xmath386 , whose bracket on homology coincides with the loop bracket of string topology , and such that 1 . the union of moduli spaces @xmath307 gives rise to an element @xmath387 of degree @xmath330 satisfying @xmath388 2 . the union of moduli spaces @xmath389 gives rise to an element @xmath390 of degree @xmath279 satisfying @xmath391,\ ] ] where @xmath284\in c_n(\lambda l , { { \mathbb{q}}})$ ] denotes the chain of constant loops . i will treat this theorem as a black box , and deduce the main results in the introduction from it by using abstract algebraic arguments and some 3-manifold topology . for a graded vector space @xmath392 we denote by @xmath393 $ ] the vector space with grading shifted by @xmath7 , i.e. @xmath393_d = c_{d+n}$ ] . on the @xmath257-fold tensor product @xmath394 , we consider two actions of the permutation group @xmath395 . in the first one , a permutation @xmath396 acts on some tensor product of elements @xmath397 of pure degrees @xmath398 via @xmath399 with @xmath400 . the quotient is the @xmath257th symmetric power @xmath401 of @xmath402 , whose decomposable elements we write as @xmath403 . the second action is the first one twisted by the sign representation , @xmath404 the quotient is the @xmath257th exterior power @xmath405 of @xmath402 , whose elements are usually denoted by @xmath406 . with these definitions , for an element @xmath407 of odd degree we have @xmath408 , but @xmath409 . an l@xmath43 algebra of degree 0 consists of a graded vector space @xmath402 and a sequence of multilinear operations @xmath410 of degree @xmath411 satisfying the sequence of quadratic relations @xmath412 for each @xmath413 . ( the signs are made explicit below . ) more generally , an l@xmath43 algebra structure of degree @xmath207 on @xmath402 is defined to be an l@xmath43 structure of degree @xmath414 on @xmath415 $ ] . if the vector space @xmath402 of an l@xmath43 algebra of degree 0 is concentrated in degree 0 , then for degree reasons the only possibly nontrivial operation is @xmath45 , and the relation for @xmath416 turns out to be the jacobi identity for @xmath45 , so we recover lie algebras as a special case . if @xmath417 for @xmath418 , then we recover the definition of a dg lie algebra . indeed , the first relation reads @xmath419 . the second relation shows that @xmath420 is a derivation of @xmath45 , and the third relation is again the jacobi identity . in general , the jacobi identity holds `` up to homotopy '' given by @xmath421 , so it always holds for the induced bracket on @xmath422 . to make the signs in the quadratic relations as well as other signs below explicit , it is useful to give an alternative description . first observe the graded linear isomorphism @xmath423 & \to s^k(c[-1])\\ c_1 \wedge \dots \wedge c_k & \mapsto ( -1)^{\sum ( k - i)|c_i| } c_1 \cdots c_k,\end{aligned}\ ] ] where @xmath398 denotes the degree in @xmath402 . next introduce operations @xmath424)\to c[-1]$ ] as @xmath425 and note that with this degree shift these are all of degree @xmath426 . set @xmath427):= \oplus_{k\geq 1 } s^k(c[-1])$ ] and observe that each of these operations can be extended to a map @xmath428 ) \to s(c[-1])$ ] defined as @xmath429 where @xmath430 is the sign introduced above finally , one defines @xmath431 ) \to s(c[-1])$ ] . then the quadratic relations ( with the correct signs ) are equivalent to the single equation @xmath432 the above passage from the operations @xmath433 on @xmath402 to the operation @xmath434 on @xmath427)$ ] is called the _ bar construction_. conceptually , one views @xmath427)$ ] as a coalgebra via the comultiplication @xmath435 ) \to s(c[-1 ] ) \otimes s(c[-1])$ ] given by @xmath436 this map has the coassociativity property @xmath437 and it also turns out to be cocommutative in the sense that @xmath438 , where @xmath439)\otimes s(c[-1 ] ) \to s(c[-1 ] ) \otimes s(c[-1])$ ] is the signed permutation of the two factors . then @xmath440 is the unique way to extend @xmath441 as a coderivation , i.e. as a map satisfying the co - leibniz rule @xmath442 conversely , one can prove that any coderivation @xmath443 ) \to s(c[-1])$ ] is completely determined by its _ linear part _ @xmath444 ) \to c[-1]$ ] . so @xmath434 is the unique coderivation of degree @xmath426 on @xmath427)$ ] such that the restriction of its linear part to @xmath445)$ ] equals @xmath441 . it is also easy to see that the commutator @xmath446:= d_1 \circ d_2 - ( -1)^{|d_1||d_2| } d_2 \circ d_1 $ ] of two homogeneous coderivations is a coderivation , and so in our example above @xmath447 $ ] has this property . these remarks explain why the relation is equivalent to the sequence of relations , since this sequence is obtained by restricting the linear part of @xmath448 to @xmath445)$ ] for each @xmath449 ( and precomposing with @xmath450 ) . so in summary , an l@xmath43 structure on a graded vector space @xmath402 is the same as a coderivation of square zero on the symmetric tensor coalgebra @xmath427)$ ] . i have adopted homological conventions here , whereas often in the literature one finds cohomological conventions , where the @xmath433 have degrees @xmath451 , and in the bar construction one shifts degrees by @xmath222 instead of @xmath426 . given two l@xmath43 algebras @xmath452 and @xmath453 , a morphism from @xmath313 to @xmath454 consists of a sequence of maps @xmath455 of degrees @xmath456 satisfying the sequence of relations @xmath457 for @xmath413 . again , to state the signs correctly , it is useful to pass to the associated maps @xmath458 ) \to c'[-1]$ ] of degree @xmath414 given by @xmath459 . any such collection of linear maps determines a unique morphism of coalgebras @xmath460 ) \to s(c'[-1])$ ] , given by @xmath461 the fact that @xmath462 is a morphism of l@xmath43 algebras can now be stated equivalently ( including the correct signs ) as @xmath463 the first important result about l@xmath43 algebras asserts that the structure of an l@xmath43 algebra can be transferred from a complex @xmath402 to its homology with respect to @xmath420 , without the loss of any essential information . more precisely , it is formulated as follows . [ thm : hpt ] suppose @xmath452 is an l@xmath43 algebra over a field of characterstic 0 . then there exists an l@xmath43 algebra structure @xmath464 on the homology which is homotopy equivalent to @xmath313 . here a _ homotopy equivalence _ between l@xmath43 algebras is the essentially obvious generalization of the classical notion . in particular , it is an l@xmath43 morphism which induces an isomorphism in the homology of the underlying complexes . it is a theorem that every such map admits a homotopy inverse . for detailed definitions and a proof of these assertions , including the theorem , see e.g. @xcite . the construction of the homotopy equivalence starts with a linear homotopy equivalence @xmath465 given by choosing a cycle in each homology class , which has a homotopy inverse @xmath466 given by projection along a complement of the image of @xmath467 . one sets @xmath468 and @xmath469 , and constructs the higher maps @xmath470 and operations @xmath471 , @xmath472 simultaneously by induction . the fact that the homologies of the two complexes agree is used to prove that all relevant obstructions vanish . now let @xmath452 be an l@xmath43 algebra . suppose that @xmath402 is the completion of some complex @xmath473 with respect to a doubly infinite filtration @xmath474 with @xmath475 , so that elements of @xmath402 are ( possibly infinite ) sums of elements of @xmath473 of the form @xmath476 denote the induced filtration on @xmath402 by @xmath477 . the l@xmath43 structure on @xmath402 is called _ filtered _ if @xmath478 in the following discussion , it is convenient to denote by @xmath479 the image of an element under the identity map @xmath480 $ ] of degree @xmath481 . an element @xmath482 of degree @xmath426 satisfying the equation @xmath483\ ] ] is called a _ maurer - cartan element of @xmath313_. since @xmath484 , the left hand side of this equation is indeed a well - defined element of @xmath313 . note that is an instance of this equation . also , this equation is equivalent to @xmath485 moreover , an easy calculation yields if @xmath486 is a maurer - cartan element of @xmath313 and @xmath487 is arbitrary , then @xmath488 in particular , the map @xmath489 \to c[-1]$ ] given by @xmath490 is a differential . finally , we describe what happens to equations and under morphisms . [ prop : pushforward ] suppose @xmath491 is a morphism between l@xmath43 algebras @xmath313 and @xmath492 preserving filtrations as above . 1 . if @xmath493 is a maurer - cartan element for @xmath313 , then @xmath494 with @xmath495 is a maurer - cartan element for @xmath492 . 2 . if @xmath493 is a maurer - cartan element and @xmath496 satisfy @xmath497 then the elements @xmath498 , @xmath499 and @xmath500 with @xmath501 satisfy @xmath502 to prove the first assertion , just observe that for a maurer - cartan element @xmath486 one has @xmath503 where the equality @xmath504 follows directly from the definitions . to prove the second assertion , one first checks that for any elements @xmath505 $ ] @xmath506 using this , we compute @xmath507 on the other hand , @xmath508 looking back at theorem [ thm : blackbox ] , we see that it asserts that the holomorphic disks with boundary on the lagrangian submanifold give rise to a maurer - cartan element @xmath122 in the l@xmath43 structure on @xmath324 such that with respect to the twisted differential the element @xmath284 \in \widehat{{\mathcal{c}}}$ ] becomes exact . since @xmath284 $ ] is never exact with respect to the ordinary boundary operator @xmath509 , this tells us that both the maurer - cartan element @xmath122 and at least one of the operations @xmath433 with @xmath510 must be nontrivial , since otherwise the twisted differential coincides with the untwisted boundary operator @xmath511 . this observation lies at the core of fukaya s proof of theorem [ thm : fukayamain ] . before i discuss that , i will state two purely topological facts that will turn out to be useful . [ lem : top1 ] let @xmath64 be a loop , and denote by @xmath512 the centralizer of @xmath60 , i.e. the set of all elements commuting with @xmath60 . let @xmath513 be a connected covering of @xmath5 associated to the subgroup @xmath514 , and let @xmath515 be a lift of @xmath60 . then the projection @xmath516 induces a homeomorphism @xmath517 between the components of @xmath518 and @xmath60 in the respective free loop spaces . since @xmath513 is a covering , any free homotopy @xmath519 \times s^1 \to l$ ] with @xmath520 admits a ( unique ) lift @xmath521 to @xmath29 with @xmath522 , and so in particular @xmath523 is the image of @xmath524 under the map @xmath525 induced by the projection . this proves surjectivity of @xmath525 . to prove injectivity , assume that @xmath526 . note that our two lifts @xmath527 and @xmath528 of @xmath165 are related by a deck transformation , i.e. by the action of some homeomorphism @xmath529 satisfying @xmath530 . if @xmath531 is a free homotopy from @xmath518 to @xmath532 , then @xmath533 is a homotopy from @xmath534 to @xmath528 . since by assumption @xmath535 is also freely homotopic to @xmath518 , we conclude that if @xmath525 is not injective , then @xmath60 has at least two preimages , namely @xmath518 and @xmath536 . now suppose @xmath518 and @xmath536 are freely homotopic for some deck transformation @xmath537 , so that they are both preimages of @xmath60 under @xmath525 . a free homotopy @xmath521 from @xmath518 to @xmath536 can be reinterpreted as a based homotopy from @xmath518 to @xmath538 , where @xmath539 \times \{1\}}$ ] is the path travelled by the base point under the homotopy . note that @xmath540 projects to a closed loop in @xmath5 representing @xmath541 . in particular , the projection of the homotopy @xmath521 yields that @xmath542 but @xmath543 was chosen to be the centralizer of @xmath60 in @xmath312 , so @xmath544 . in other words , this implies that @xmath540 was a closed loop and so @xmath545 , i.e. any two preimages of @xmath60 coincide . together with the previous observation this shows that @xmath525 is injective , completing the proof of the lemma . [ lem : top2 ] in the situation of the previous lemma , assume moreover that @xmath5 ( and so @xmath29 as well ) is aspherical . then evaluation at the base point @xmath546 is a homotopy equivalence . the fiber of the map @xmath547 at @xmath548 is the space @xmath549 of loops which are based at @xmath548 and _ freely _ homotopic to @xmath515 . as in the previous proof , we observe that any free homotopy between @xmath550 and @xmath515 can be reinterpreted as a based homotopy between @xmath165 and @xmath551 . but @xmath515 is central in @xmath552 , so that @xmath553 is _ based _ homotopic to @xmath518 . so we conclude that in fact @xmath554 is the component of @xmath515 in the based loop space of @xmath29 , which is contractible since @xmath29 is aspherical . so @xmath555 is a fibration with contractible fibers , and hence a homotopy equivalence . [ cor : top3 ] if @xmath5 is an aspherical manifold , then every component of the free loop space @xmath306 has the homotopy type of a cw complex of dimension at most @xmath556 . @xmath557 after these preliminaries , i come to the proof of the main theorem . ( of theorem [ thm : fukayamain ] ) recall that the maurer - cartan element @xmath558 is built from the moduli spaces @xmath559 , which have geometric dimensions @xmath560 and the element @xmath561 is built from the spaces @xmath562 with geometric dimensions @xmath563 denote by @xmath564 the homology with respect to the usual boundary operator of @xmath324 . recall that we denote by @xmath565 the image of @xmath566 under the degree shift @xmath567 $ ] . according to theorem [ thm : hpt ] , the l@xmath43 structure on @xmath324 pushes forward to an l@xmath43 structure on @xmath564 under a homomorphism @xmath568 , and by proposition [ prop : pushforward ] , this homomorphism maps the elements @xmath122 , @xmath158 and @xmath284 $ ] in @xmath324 to elements @xmath569 , @xmath570 and @xmath284 $ ] in @xmath564 satisfying the equation @xmath571}.\ ] ] writing @xmath572 the part of this equation corresponding to the trivial relative homotopy class can be written more explicitly as @xmath573}.\ ] ] since the homomorphism between the l@xmath43 structures preserves degrees , the geometric degrees of @xmath574 and @xmath575 are @xmath576 and @xmath577 , respectively . by the assumption that @xmath5 is aspherical , corollary [ cor : top3 ] implies that the homology @xmath564 is concentrated in geometric degrees @xmath578 . combining this observation with and the fact that the maslov index is even for orientable lagrangian submanifolds @xmath5 , we find that for the term @xmath579 to be nonzero we must have @xmath580 the first equation immediately implies that @xmath61 is not identically zero . moreover , if @xmath581 for all @xmath582 , it follows that @xmath583 , again contradicting the first equation . thus we conclude that some @xmath584 with @xmath585 must be nonzero , implying that the corresponding moduli space is nonempty . so @xmath586 is represented by a holomorphic disk , and hence must have positive symplectic energy . set @xmath587 and let @xmath588 denote the centralizer of @xmath60 . notice that we have a short exact sequence @xmath589 in which the last map admits an inverse sending @xmath222 to @xmath60 . it follows that the map @xmath590 , defined by @xmath591 , is an isomorphism ( @xmath592 is indeed a group homomorphism because @xmath60 commutes with all elements of @xmath402 ) . since @xmath5 is a @xmath593 , the covering space @xmath29 of @xmath5 with @xmath594 is a @xmath595 , so it is homotopy equivalent to @xmath30 for a @xmath596 space @xmath32 . to complete the proof of the theorem , it remains to show that @xmath32 is closed or , equivalently , that @xmath597 is a finite covering space . note that the class @xmath586 with @xmath598 had the property that @xmath585 , and moreover @xmath574 is a nonzero element of geometric degree @xmath7 in @xmath564 . so the homology in degree @xmath7 of @xmath599 must be nonzero . but combining lemma [ lem : top1 ] and lemma [ lem : top2 ] , we see that @xmath311 is homotopy equivalent to the @xmath7-manifold @xmath600 . the nonvanishing of its top - dimensional homology now implies that @xmath29 is closed , which in turns means that @xmath597 is a finite covering space . the more precise statement in dimension 3 can be proven with some specific results from 3-dimensional topology . i wish to thank k. fukaya , k. honda and s. maillot for helpful correspondence , which lead to the following proof of corollary [ cor : fukaya1 ] . ( of corollary [ cor : fukaya1 ] ) let @xmath5 be a compact , orientable , prime 3-manifold . it is well - known ( see e.g. @xcite ) that either @xmath601 or @xmath5 is irreducible , meaning that every embedded two - sphere in @xmath5 bounds a ball in @xmath5 . if an irreducible 3-manifold @xmath5 admits a lagrangian embedding into @xmath35 , then by gromov s theorem [ thm : gromov - notexact ] it has infinite first homology , and hence infinite fundamental group , and so its universal cover @xmath29 is non - compact . moreover , by the sphere theorem ( see @xcite ) , an irreducible 3-manifold has trivial second homotopy group . it follows that @xmath602 for @xmath413 , and so by hurewicz s theorem @xmath603 for @xmath449 , implying that @xmath5 itself is aspherical . now by theorem [ thm : fukayamain ] , a finite cover of @xmath5 is homotopy equivalent to @xmath36 for some closed oriented surface @xmath20 , and a result of waldhausen ( * ? ? ? * corollary 6.5 ) implies that this homotopy equivalence can be improved to a homeomorphism . recall from the proof above that the fundamental group @xmath402 of the cover arises as the centralizer of an element @xmath604 with @xmath605 . now i will argue that in fact @xmath60 is central in @xmath312 , so that the covering projection is actually a homeomorphism . indeed , consider the exact sequence @xmath606 where @xmath607 . since the centralizer @xmath402 of @xmath60 is of finite index , @xmath608 is of finite index in @xmath609 . from the above proof of the theorem , we see that @xmath610 is finitely generated ( it is the fundamental group of @xmath20 ) , so @xmath609 is also finitely generated . then by stallings fibration theorem ( @xcite , see also ( * ? ? ? * theorem 11.6 ) ) , we deduce that @xmath609 is the fundamental group of a compact surface @xmath611 , which under our current assumptions must be closed of genus at least 1 . now the proof concludes with the following observation : any automorphism @xmath612 of the fundamental group @xmath609 of a closed oriented surface which is trivial on some finite index subgroup @xmath610 is trivial . if the surface is a sphere there is nothing to prove , so we consider the case that the genus of the surface is at least 1 . one knows that the fundamental group @xmath609 of a closed surface has no torsion . let @xmath613 be given and consider the infinite cyclic subgroup @xmath614 generated by @xmath537 . the subgroup @xmath615 contains the finite index subgroup @xmath616 , on which @xmath612 acts trivially . but any automorphism of an infinite cyclic group fixing some nontrivial subgroup must be the identity , so @xmath617 . since this applies to any @xmath613 , the lemma is proven . applying the lemma to the action of @xmath60 on @xmath609 by conjugation , which clearly fixes all elements of @xmath618 , we finally conclude that @xmath60 is central in @xmath312 , so that @xmath619 , which finishes the proof of the theorem . above , i have presented fukaya s elegant arguments leading to some substantial new results about lagrangian submanifolds in @xmath14 . at first glance , it seems that string topology is really essential to the approach . however , on further inspection , one discovers that there may be a way to avoid it almost entirely . the basic idea is the following . by viterbo s theorem ( see @xcite and chapter chap : viterbo ) , the homology of the free loop space can be described in symplectic terms as the symplectic homology of the cotangent bundle . this theory can be defined in more general situations , for example for exact symplectic manifolds with contact - type boundary . moreover , for _ exact _ codimension 0 embeddings @xmath620 one has restriction maps @xmath621 . it seems reasonable to expect ( and is the subject of current work ) that every algebraic structure that exists on the homology of the free loop space can also be defined on symplectic homology in general , even if the underlying domain is not a cotangent bundle . in the exact case , the restriction homomorphism should respect all these structures . but even more should be true . in the case of a non - exact embedding @xmath620 , there will be a maurer - cartan element in @xmath622 such that after twisting all the structures by this maurer - cartan element we get a morphism from @xmath623 to the twisted version @xmath624 . this expectation is consistent with ( and gives one of several possible conceptual explanations for ) the results of fukaya for lagrangians in @xmath14 . indeed , with @xmath625 being a small neighborhood of the zero section in the cotangent bundle of @xmath5 and @xmath626 , we are exactly in the situation just described . what i have argued in earlier sections is that , after twisting by a maurer - cartan element coming from the embedding , the unit @xmath284 \in h_*(\lambda l)$ ] with respect to the loop product has become exact , which by a standard argument will force the twisted homology to vanish completely . this is good news , because only in this case can we even expect to have a morphism from @xmath627 to this ring . the prediction is that this morphism can indeed be defined in a suitable chain version of the theory . once the above argument has been made to work , it extends the applicability of fukaya s approach in several directions . notice that string topology only enters indirectly , via viterbo s isomorphism . as long as the algebraic operations can be defined and the morphism associated to a codimension zero embedding respects them , one does not even need to know that the operations on symplectic homology are the same as those in string topology ( although this is of course expected to be true ) . moreover , one can study non - exact codimension 0 embeddings of general exact symplectic manifolds with contact boundary by this method . the basic source for this chapter are of course fukaya s papers @xcite . versions of theorem [ thm : fukayamain ] under additional assumptions , like monotonicity of the lagrangian submanifold , are much easier to achieve , see e.g. @xcite and the references therein . for an introduction to symplectic topology the book @xcite is recommended . it covers a lot more than is necessary to understand the problem discussed here , and it gives some hints why lagrangian submanifolds are so central in symplectic topology . to learn something more specific about lagrangian embeddings and immersions , the excellent survey @xcite is still the best place to start . recently , new results have appeared which suggest that the problem in higher dimensions is more flexible than previously expected @xcite . a chain complex @xmath628 for the free loop space on which the loop bracket is fully defined , and which therefore might serve in the implementation of theorem [ thm : blackbox ] , has recently been proposed by irie @xcite . finally , the reader who really wants to appreciate the discussion in this chapter needs to know quite a bit about holomorphic curves . one good source which thoroughly covers a lot of the basics , including a version of gromov compactness and a complete proof of gromov s theorem [ thm : gromov - notexact ] , is @xcite . many aspects of the theory are also covered in the earlier book @xcite . with these as a guide , the monumental @xcite will hopefully look less daunting . k. fukaya , _ application of floer homology of langrangian submanifolds to symplectic topology _ , in : paul biran et al . ( eds . ) , morse theoretic methods in nonlinear analysis and in symplectic topology , springer , 2006 , 231-276 . c .- c . m. liu , _ moduli of @xmath41-holomorphic curves with lagrangian boundary conditions and open gromov - witten invariants for an @xmath11-equivariant pair _ , ph.d . thesis , harvard 2002 , arxiv : math.sg/0210257 d. mcduff and d. salamon , _ @xmath41-holomorphic curves and symplectic topology _ , american mathematical society colloquium publications , 52 . american mathematical society , providence , ri , 2004 . xii+669 pp . isbn : 0 - 8218 - 3485 - 1 s. yu . nemirovski , _ homology class of a lagrangian klein bottle _ , isvestiya math . 73 , 2009 , no . 4 , 689698 . v. v. shevchishin , _ lagrangian embeddings of the klein bottle and combinatorial properties of mapping class groups _ , isvestiya math . 73 , 2009 , no . 4 , 797859
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in this chapter i discuss some applications of string topology to the study of lagrangian embeddings into symplectic manifolds , as discovered by fukaya @xcite .
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first evidence of the violation of time reversal symmetry has been found in the kaon system @xcite . despite strong efforts no other signal of violation of time reversal symmetry has been found to date . however , by now , studying time reversal symmetry has become a corner stone of the search for physics beyond the standard model of elementary particles @xcite . some alternatives or extensions of the standard model are due to dynamical symmetry breaking , multi higgs models , spontaneous symmetry breaking , grand unified theories ( e.g. so(10 ) ) , extended gauge groups ( leading e.g. to right - handed bosons @xmath3 in left - right symmetric models ) , super symmetric ( susy ) theories , etc . , each implying specific ways of @xmath4 violation . for a recent review of models relevant in the context of @xmath4 violation see e.g. @xcite , and refs . therein . these theories `` beyond '' the standard model are formulated in terms of quarks and leptons whereas nuclear low energy tests of @xmath4 involve hadronic degrees of freedom ( mesons and nucleons ) @xcite . to extract hadronic degrees of freedom from observables one may introduce effective @xmath1odd nucleon nucleon potentials @xcite , or more specific @xmath1odd mesonic exchange potentials @xcite . as in the context of @xmath0-violation see e.g. @xcite , these potentials have been proven quite useful to treat the nuclear structure part involved and to extract effective @xmath1odd hadronic coupling constants @xcite . in turn they allow to compare the sensitivity of different experiments , which has been done recently in ref . however , in order to compare upper bounds on a more fundamental level of @xmath1odd interactions , it is necessary to relate hadronic degrees of freedom to quark degrees of freedom in some way . this step is hampered by the absence of a complete solution of quantum chromo dynamics ( qcd ) at the energies considered here . in many cases a rough estimate in the context of time reversal violation may be sufficient , and , in the simplest case , factors arising from hadronic structure may be neglected . in the context of @xmath0odd time reversal violation e.g. concepts such as pcac and current algebra @xcite have been utilized to improve the evaluation of hadronic structure effects . in the @xmath0even case , which is considered here , this approach is not applicable ( no goldstone bosons involved here ) . however , it may be useful to utilize quark models specifically designed for and quite successful in describing the low energy sector . in fact , experimental precision tests still continue to make progress and so theorists face a renewed challenge to translate these experimental constrains to a more fundamental interaction level . the purpose of the present paper is to give estimates on hadronic matrix elements that arise when relating quark operators to the effective hadronic parameterizations of the @xmath0even @xmath1odd interaction . these are the charge @xmath2 type exchange and the axial vector type exchange nucleon nucleon interaction @xcite . they will shortly be outlined in the next section . the ansatz to calculate @xmath5 matrix elements from the quark structure is described in section iii . the last section gives the result for different types of quark models and a conclusion . for completeness , note that in general also @xmath1-odd and @xmath0-odd interactions are possible , and in fact most of the simple extensions of the standard model mentioned above give rise to such type of @xmath1violation . parameterized as one boson exchanges they lead e.g. to effective pion exchange potentials that are essentially long range , see @xcite . limits on @xmath0odd @xmath1odd interactions are rather strongly bound by electric dipole moment measurements , in particular by that of the neutron @xcite . in contrast bounds on @xmath0even @xmath1odd interactions are rather weak . note , also that despite theoretical considerations @xcite new experiments testing generic @xmath1odd @xmath0even observables have been suggested ; for the present status see e.g. refs . due to the moderate energies involved in nuclear physics tests of time reversal symmetry , hadronic degrees of freedom are useful and may be reasonable to analyze and to compare different types of experiments . for a recent discussion see ref . @xcite . in the following only @xmath1-odd and @xmath0-_even _ interactions will be considered . they may be parameterized in terms of effective one boson exchange potentials . due to the behavior under @xmath6 , @xmath0 , and @xmath1 symmetry transformations , see e.g. @xcite , two basic contributions are possible then : a charged @xmath2 type exchange @xcite and an axial vector exchange @xcite . the effective @xmath2 type @xmath1odd interaction is @xmath6odd due to the phase appearing in the isospin sector and is only possible for charged @xmath2 exchange . it has been suggested by simonius and wyler , who used the tensor part to parameterize the interaction @xcite , @xmath7 there is some question of whether to choose an `` anomalous '' coupling @xcite , viz . @xmath8 . the numerical value of @xmath9 is usually taken to be @xmath10 close to the strong interaction case @xcite . we shall see in the following that it is not unreasonable to introduce such a factor since in may be related to `` nucleonic structure effects '' , which are not of @xmath1 violating origin ( similar to nuclear structure effects that are also treated separately ) . combining the @xmath1odd vertex with the appropriate @xmath1even vertex leads to the following effective @xmath1odd @xmath0even one boson exchange @xmath5 interaction , @xmath11 where @xmath12 , and @xmath13 , and @xmath14 is the strong coupling constant , as e.g. provided by the bonn potential @xcite . the axial vector type interaction has been suggested by @xcite . unlike the @xmath2type interaction the isospin dependence is not restricted and may be isoscalar , vector , and/or tensor type . the effective lagrangian for the @xmath15 coupling for example is given by @xmath16 combined with the appropriate @xmath1even vertex this leads to an effective axial vector type exchange @xmath5 potential @xcite , @xmath17 the bounds on the @xmath1odd coupling strengths arising from various experiments have been discussed in ref . a more recent bound not included there is from an improved analysis @xcite of the @xmath18fe @xmath19decay experiment @xcite . bounds are in the order of 10% if derived from generic @xmath1odd @xmath0even observables and slightly more than an order of magnitude smaller , if related to the electric dipole moments @xcite . to complete this section , note that although possible , two boson exchanges have not been considered up to now . we now turn to effective @xmath1-odd @xmath0-even quark operators . the simplest operator that leads to an effective @xmath1odd @xmath0even vector type vertex @xmath20 analogous to the @xmath2type interaction eq . ( [ eqn : vnn ] ) is @xmath21 here @xmath22 , @xmath23 denote flavored quark fields . again the flavor dependence is responsible for @xmath6 , viz . @xmath1violation due to the phase dependence . a tensor term has not been introduced for simplicity . on the basis of eq . ( [ quarkv ] ) such a term will arise in a natural way in the effective hadronic @xmath24 @xmath1-odd lagrangian through the quark structure effects as will be explained below . the second generic quark operator utilizing axial vector bilinear operators is given by @xcite , @xmath25 with the on shell equivalent @xmath26 in order to recover eqs . ( [ eqn : vnn ] ) and ( [ eqn : ann ] ) we utilize the constituent quark model . this model has been rather successful and valuable in reproducing gross features of low energy phenomenae , such as mass spectra , form factors , coupling constants , magnetic moments etc . , see e.g. @xcite . to relate quark operators to effective hadronic operators we utilize the fock space representation of hadrons in terms of constituent quarks , viz . @xmath27 since there is no low energy solution of qcd , the evaluation of the matrix elements of the l.h.s of eq . ( [ fock ] ) needs further consideration . in general , the same problem arises in the context of strong interactions . an extensive overview of the different approaches to tackle the problem in this case has been given by ref . @xcite . here we follow the ideas first formulated in ref . @xcite , and extensively studied for different quark models in @xcite . the resulting strong interaction potential is a generic hybrid model connecting quark degrees of freedom with effective meson nucleon degrees of freedom . the basic idea is summarized in the following . suppose the two nucleons overlap , and two quarks are sufficiently close together . this situation is depicted in figure [ fig : qqnn]a ) . then , to begin with , the matrix elements may be evaluated without introducing any mesonic fields . in terms of the constituent quark model @xmath28 excitations are neglected ( or partially parameterized in the constituent quark mass ) . only at larger distances of the nucleons , mesons are essential and may appear as @xmath28 correlations on the nonperturbative qcd vacuum @xcite that might be the physical vacuum of the low energy regime @xcite . however , the appearance of mesons is disconnected from the problem of @xmath1odd force . therefore , in the following we assume that the hadronization mechanism is the _ same for both @xmath1odd and the usual @xmath1even strong interaction _ and investigate the relative strength of the @xmath1odd matrix elements to the @xmath1even matrix elements . this is done in the framework of the virginia potential that assumes a quark pairing mechanism to generate effective meson nucleon coupling constants @xcite . to illustrate the quality of this ansatz table [ tab : virginia ] shows the resulting coupling constants using different quark models compared to the values of a recent version of the bonn potential @xcite . in this framework we utilize the factorization approximation to evaluate the matrix element of eq . ( [ fock ] ) , see figure [ fig : qqnn]b ) @xmath29 to demonstrate the calculation , we use the simple constituent quark model . this model is supplemented by an explicit lower component , which already occurs implicitly in the dirac magnetic moments @xcite and in the two body pair current through electromagnetic gauge invariance @xcite . this way a treatment of relativistic effects has been introduced , see e.g. @xcite and ref . therein . the integration of the internal degrees of freedom reads , @xmath30\ ] ] with @xmath31 the three quark wave internal function . in the rest system the space part of the wave function is given by @xcite @xmath32\ ] ] where numerical values may be chosen as @xmath33 , and @xmath34 . the coordinates are normalized lovelace coordinates , viz . @xmath2 for the pair and @xmath35 for the odd quark . integration is done in the breit system . the symbol @xmath36 denotes either one of the vertices in ( [ quark ] ) . evaluation for the different type of operators leads to the following expressions ( using the isospin formalism for @xmath37 quarks ) @xmath38 the factors arising are related to the quark structure of nucleons . note , that in eq . ( [ axial ] ) one recognizes the well known coupling of the axial vector current ( for @xmath39 ) , viz . @xmath40 of the nonrelativistic constituent quark model , besides factors arising from relativistic corrections due to the lower dirac component . the latter reduce the value of @xmath41 close to the experimental one @xcite . in eq . ( [ tensor ] ) a tensor coupling appears , which belongs to the @xmath2 type @xmath1odd exchange . indeed due to the quark structure factors its relative strength is larger than the first term of the r.h.s . of eq . ( [ tensor ] ) , and therefore preferable in an ansatz of a @xmath2type @xmath1odd force as done by simonius and wyler @xcite . the factor in front the tensor term may be interpreted as `` anomalous '' coupling . it appears in analogy to the electromagnetic interaction , where the pauli term of the electromagnetic photon nucleon interaction can be recovered from a pure dirac coupling on quark level . this has been explained and shown in ref . the resulting relation between quark and hadronic @xmath1odd coupling strength on the basis of the constituent quark model is , for @xmath2 type exchange @xmath42 and for axial type of exchange @xmath43 here the expression for isoscalar and isovector are the same . similar results may be obtained using different types of quark models . in the context of the virginia potential those studied are the mit bag model and a relativistic model with linear confining potential , see ref . these are used here in the same way as demonstrated for the constituent quark model in the previous section . the values for the quark structure effects evaluated using typical quark model parameters of low energy phenomenology are given in table [ tab : values ] . in fact , due to the symmetries inherent in the quark pairing mechanism ( viz . the virginia potential ) it is possible to arrive at the following relations between the coupling constants , viz . @xmath44 this equation shows that the factors appearing in the @xmath45 type exchange may be related to the anomalous coupling @xmath46 . so , inclusion of @xmath9 might give a bound closer to the more basic quark degrees of freedom . in conclusion , provided the hadronization process does not substantially differ for @xmath1odd and @xmath1even interactions , the factors arising reflect the _ nucleon _ structure effects . the origin of the structure factors are due to the spin , isospin structure and the different mass scales ( i.e. @xmath47 vs. @xmath48 ) . these have also been essential in deriving the relative strength of the strong coupling constants as given in table [ tab : virginia ] . the author gratefully acknowledges support by the national institute for nuclear theory at the university of washington , seattle , during his stay on the int program `` physics beyond the standard model at low and intermediate energies '' . this work has been supported by deutsche forschungsgemeinschaft be 1092/4 - 1 . p. herczeg , nucl . phys . * 75 * ( 1966 ) 655 r. bryan , a. gersten , phys . * 26 * ( 1971 ) 1000 , * 27 * ( 1971 ) 1102(e ) e.c.g . sudarshan , proc . a305 * , ( 1968 ) 319 m. simonius , phys . lett . * 58b * ( 1975 ) 147 m. simonius , d. wyler , nucl a286 * ( 1977 ) 182 w.c . haxton , e.m . henley , phys . * 51 * ( 1983 ) 1937 p. herczeg , in _ tests of time reversal invariance in neutron physics _ , eds , n.r . roberson et al . , ( world scientific publishing , singapore , 1987 ) p. 24 . e.g. adelberger , w.c . haxton ann . nucl . part . * 35 * 501 ( 1985 ) . i.s . towner and a.c . hayes , phys . rev . * c 49 * , 2391 ( 1994 ) . gudkov , x .- he , and b.h.j . mckellar , phys . * c47 * , 2365 ( 1993 ) . m. beyer , nucl . a493 * 335 ( 1989 ) . m. beyer , phys . * c48 * 906 ( 1993 ) . w.c . haxton and a. hoering , nucl . phys . * a 560 * , 469 ( 1993 ) . haxton , a. hring , m.j . musolf , phys . d50 3422 ( 1994 ) . crewther , p. di vecchia , g. veneziano , e. witten , phys . lett . * 88b * 123 ( 1979 ) . altarev et al . , phys . lett . * b276 * 242 ( 1992 ) . smith , phys . b234 * 191 ( 1990 ) . pendlebury , nucl . phys . * a546 * 359c ( 1992 ) . montanet et al . ( particle data group ) , phys.rev . * d50 * 1173 ( 1994 ) . p. herczeg , private communication : p. herzceg , j. kambor , m. simonius , and d. wyler to be published . j. engel , p. frampton , and r. springer , preprint nucl - th/9505026 ( 1995 ) . c. itzykson and j .- b . zuber , _ quantum field theory _ ( mcgraw hill international , singapore , 1980 ) . p. herczeg , private communication . ressell , j. engel , p. vogel , preprint map-189 , nucl - th/9512013 ( 1995 ) . cheung , h.e . henrikson , f. boehm , phys . rev . * c16 * , 2381 ( 1977 ) . khriplovich , nucl . phys . * b352 * 382 ( 1991 ) . see e.g. f.e . close , _ an introduction to quarks and partons _ ( academic press , 1979 ) ; a. le yaouanc , l. oliver , o. pene , j .- c . raynal , _ hadron transitions in the quark model _ ( gordon and breach , 1988 ) ; r.k . bhaduri , _ models of the nucleon : from quarks to soliton _ ( addison - wesley , 1988 , lecture notes and supplements in physics , 22 ) . f. myhrer , rev . 60 629 ( 1988 ) h.j . weber , z. phys . * a297 * ( 1980 ) 261 , * a301 * , 141 ( 1981 ) , phys . rev . * c26 * , 2333 ( 1982 ) m. bozoian and h.j . weber , phys . * c28 * 811 ( 1983 ) m. beyer , h.j . weber , phys . * 146b * ( 1984 ) 383 ; phys . rev . * c35 * ( 1987 ) 14 m.a . shifman , a.i . vainshtein , and v.i . zakharov , nucl . * b147 * , 385 and 448 ( 1980 ) for a recent discussion of constituent quarks see : d. diakonov , gatchina preprint , nucl - th/960323 . r. machleidt , k. holinde and ch . elster , phys . rep . * 149 * , 1 ( 1987 ) n. isgur and g. karl , phys * d18 * , 4187 ( 1978 ) ; * d 19 * , 2653 ( 1979 ) . m. beyer , d. drechsel , and m.m . giannini , phys . 122 b * , 1 ( 1983 ) h.j . weber , m. weyrauch phys . * c32 * , 1342 ( 1985 ) m. beyer and s.k . singh , phys . * b 160 * , 26 ( 1985 ) ; z. phys . * c31 * , 421 ( 1986 ) @xmath49 & 3.8 & [ -0.5 ] & 9.85 & [ -0.49]&10.6 & [ 0.0]\\ \rho(763 ) & ( 1^{--},1 ^ -)&0.7 & [ 2.2 ] & 0.42&[3.2 ] & 1.10 & [ 1.53 ] & 0.41 & [ 6.1]\\ f_1(1285 ) & ( 1^{++},0^+)&0.5 & [ -1.5 ] & 0.22 & & 0.59 & [ 0 ] & & \\ a_1(119 ) & ( 1^{++},1 ^ -)&1.0 & [ -1.5 ] & 0.6 & & 1.64 & [ 0 ] & & \\ \hline\hline \end{array}\ ] ]
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tests of time reversal symmetry at low and medium energies may be analyzed in the framework of effective hadronic interactions . here , we consider the quark structure of hadrons to make a connection to the more fundamental degrees of freedom .
it turns out that for @xmath0even @xmath1odd interactions hadronic matrix elements evaluated in terms of quark models give rise to factors of 2 to 5 . also , it is possible to relate the strength of the anomalous part of the effective @xmath2 type @xmath1odd @xmath0even tensor coupling to quark structure effects .
| 5,870 | 150 |
in this work we consider the mechanical response of an amorphous solid quenched at zero temperature . our formalism permits dealing explicitly with finite size systems : it rests on the idea that , during a quench at zero temperature , any finite size system relaxes toward one of many local minima in the potential energy landscape . @xcite being at zero temperature , the system is then prescribed to lie at this minimum at all times . small external perturbations are then expected to induce continuous changes in the local minimum . large external perturbations may induce the vanishing of the local minimum occupied by the system : this vanishing occurs when the basin of attraction of this minimum reduces to a single point , that is when the minimum collides with at least one saddle point . @xcite the difference between small , continuous changes of the local configuration and catastrophic events is exemplified figure [ fig : strain ] where the response of an amorphous system is shown as a function of shear . the parameter @xmath0 measures the total shear deformation from a quenched state . in this picture , continuous segments are associated with shear - induced changes of local minima and discontinuities to their vanishing . after a minimum has disappeared , the system , coupled to a zero temperature thermal bath , relaxes in search of a new minimum in the potential energy landscape . @xcite the separation of continuous changes of the local minimum and catastrophic events is , of course , not limited to shear deformation . in a more general context , suppose that we denote by @xmath0 the amplitude of any external drive applied to the system . suppose that the system lies at a minimum which , for @xmath1 , is far from any catastrophe . let us denote @xmath2 the amplitude of a perturbation of the external drive around @xmath3 . if we were to slowly increase the amplitude of the perturbation from 0 , the system would smoothly follow a trajectory @xmath4 in configuration space for all amplitudes @xmath5 such that the basin of the local minimum remains non - vanishing . if the hessian is non - degenerate , this trajectory @xmath6 is unique . if we now stop the perturbation at some @xmath7 then revert the drive down to @xmath8 , the system would simply follow the trajectory @xmath4 from @xmath9 to @xmath3 , backward . in this sense we will say that the continuous segments are microscopically reversible . when the system is driven quasi - statically along these continuous segments , no energy is dissipated . the reason is that at each point along these segments , the system is at mechanical equilibrium : the force applied to any particle is exactly zero and zero forces do no work . the quasi - static response corresponds to the thermodynamically reversible , elastic , part of the mechanical response . we will see , however , that energy is dissipated when the system is driven at finite deformation rates along these continuous segments . this dissipation results from the fact that finite deformation rates induce non - zero forces which dissipate energy via the coupling with the zero temperature thermal bath . we shall let underline and double - underlines respectively indicate vectors and tensors referred to a fixed cartesian system @xmath10 . we shall also use the convention that greek indices refer to cartesian components of vectors or tensors , while roman indices refer to the particle numbers . bold type denotes fields which are defined on every particle in the material : @xmath11 dots and double dots indicate matrix products and summation convention is always applied on repeated ( greek and latin ) indices . by convention , we also write : @xmath12 the superscript @xmath13 indicates the transpose of a matrix , @xmath14 its inverse , and @xmath15 the inverse of its transpose . the identity matrix is denoted @xmath16 , and its components @xmath17 . in this work , utilizing the formalism underlying the andersen - parrinello - rahman theory @xcite we shall focus on the situation where a system of particles is contained in a periodic simulation cell . the formalism we describe can easily be adapted to study the deformation of a material confined between walls : to do so , it is sufficient to embed the whole system confined particles plus walls in the cell and mandate that the particles constituting the walls affinely follow its deformation . the shape of the simulation cell ( which is , by construction , a parallelepiped ) is represented by the set of @xmath18 or @xmath19 bravais vectors : @xmath20 or @xmath21 ; its volume is @xmath22 . we consider a system of @xmath23 particles , with positions @xmath24 in real space . the interaction potential @xmath25 depends on the positions of the particles but also on the shape of the simulation cell which enforces boundary conditions . `` macroscopic '' deformations of the sample are performed by changing the bravais vectors . since we are concerned with variations of the local minimum around some reference configuration , we will often use a _ reference _ configuration @xmath26 of the cell and compare it with a _ current _ value @xmath27 . following ray and rahman , @xcite we introduce a transformation of particle coordinates which maps any vector @xmath28 onto a cubic reference cell : @xmath29 with @xmath30 $ ] . if we change the cell coordinate from @xmath26 to @xmath27 and require that all particles affinely follow the deformation of the cell , any particle at point @xmath31 is mapped onto @xmath32 . we denote @xmath33 which , in the usual language of elasticity , is the deformation gradient tensor . @xcite once the reference frame @xmath26 is specified , any configuration @xmath34 of the system can be parameterized by a pair @xmath35 , with the convention that : @xmath36 . in this parameterization , changes in @xmath37 correspond to affine transformations of all the particles following the cell shape , while changes in @xmath38 correspond to the non - affine part of the displacement of the particles . at zero temperature , an infinitesimal deformation of the system is often performed in two steps . first , starting from a local minimum @xmath39 at @xmath26 , the particle coordinates affinely follow the change of the cell coordinate from @xmath26 to @xmath27 . the @xmath39 remain constant . the real - space position of particle @xmath40 is thus mapped from @xmath38 onto @xmath41 . second , the particles are allowed to relax to the nearest equilibrium position , @xmath27 being fixed . they reach new positions @xmath42 which differ in general from @xmath43 . the non - affine part of the deformation is then characterized by the displacements as viewed in the reference frame , @xmath44 . for small displacements , the particles continuously follow changes of local minima . the real space positions of the particles at equilibrium are thus a continuous function of @xmath27 ( on some interval of strains ) , and we could denote these equilibria as @xmath45 . likewise , the continuous changes of local minima are most readily studied by monitoring changes in the _ reference _ co - ordinates : @xmath46 . by definition , elastic constants are second order derivatives of the energy with respect to strain . since strain is characterized by second order tensors , the elastic constants are fourth order tensors . before presenting the details of the tensorial formalism , however , it seemed pedagogically sounder to us to first consider the simple situation where the shape of the cell can be parameterized by a single degree of freedom . suppose then that we prescribe the tensor @xmath47 as a function of a scalar parameter @xmath0 . for varying @xmath0 , so long as the local minimum does not vanish , the system follows a continuous trajectory in configuration space as illustrated on figure [ fig : strain ] . given a reference cell @xmath48 at @xmath3 , the energy functional can be written either as a function of @xmath49 and @xmath0 : @xmath50 ; or as a function of @xmath51 and @xmath0 : @xmath52 . we introduce the notation @xmath53 to emphasize that contrarily to @xmath54this function is defined after a choice of reference cell with bravais matrix @xmath26 . changing @xmath0 for fixed @xmath39 corresponds to performing an affine strain of the whole system the particles and the boundary . when particles are constrained to follow deformation - induced changes of a local minimum , their real - space positions @xmath55 and the corresponding non - affine displacements @xmath56 are now one - parameter functions of @xmath0 . an equation of motion for the non - affine displacement fields derives from a straightforward application of the implicit function theorem : the trajectory is specified by the condition that the system is always at mechanical equilibrium : @xmath57 note that the second equality is a property of the point derivatives of any observable of the form , @xmath58 : @xmath59 and is derived in appendix a. differentiating the condition @xmath60 once with respect to @xmath0 leads to : @xmath61 where the symbol @xmath62 is introduced to indicate derivatives which are taken under the constraints of mechanical equilibrium . this equation is formally valid for all @xmath63 , even though it will primarily be used in the limit @xmath64 ( i.e. @xmath65 ) . using equation ( [ eqn : deriv : point ] ) , we see that in the limit @xmath64 , equation ( [ eqn : motion : gamma ] ) involves the hessian : @xmath66 and the field of virtual forces , @xmath67 in semi - condensed notation , equation ( [ eqn : motion : gamma ] ) reads : @xmath68 in order to solve equation ( [ eqn : motion : gamma:0 ] ) we need to take care to eliminate the zero modes of the hessian . generically there are @xmath69 zero modes for a system in @xmath69 dimension : those correspond to translation invariance ; their existence indicates that a solution to equation ( [ eqn : motion : gamma:0 ] ) can only be defined up to global translations of the particles . there are no zero modes associated with rotations because the geometry of the simulation cell ( a parallelepiped ) breaks the invariance of the problem under global rotations of the particles ( at fixed cell boundaries ) . the zero modes can thus be eliminated in the standard way by subtracting off the projection onto uniform translations from any particular solution . up to this invariance , the solution of ( [ eqn : motion : gamma:0 ] ) reads : @xmath70 the numerical practice which consist in , first affinely deforming the simulation cell , then letting the system relax toward the displaced minimum , provides a direct physical interpretation for this equation . @xmath71 is the field of forces which would result from an elementary affine deformation of all the particles . equation ( [ eqn : dr:0 ] ) shows that the non - affine displacements are just the linear response of the system to these extra forces . examples for the fields @xmath71 and @xmath72 in simple ( left ) and @xmath73 ( right ) as defined in the text for a typical atomistic system . the deformation mode is simple shear . note the random character of @xmath74 and the strong correlations in @xmath73 , title="fig : " ] ( left ) and @xmath73 ( right ) as defined in the text for a typical atomistic system . the deformation mode is simple shear . note the random character of @xmath74 and the strong correlations in @xmath73 , title="fig : " ] shear and pure compression are shown figure [ fig : fields ] and [ fig : fields : c ] . ( the details of the numerical simulation will be given in section 1.5 . ) we observe that the short - range randomness of the vector @xmath75 contrasts with the large vortex - like structures displayed by the non - affine `` velocity '' fields @xmath76 . we will later use the apparent disorder of the field @xmath77 to construct a statistical treatment of the contributions of these non - affine displacement to the elastic constants . for now , let us move on to study how these non - affine fields enter microscopic equations for the elastic constants . ( left ) and @xmath73 ( right ) in pure compression . , title="fig : " ] ( left ) and @xmath73 ( right ) in pure compression . , title="fig : " ] upon deformation , the first and second derivative of the energy @xmath78 are related to the components of the stress and the elastic constant respectively . we will later enter a full tensorial derivation of the equations to specify this relation . for now , we can simply differentiate @xmath79 with respect to @xmath0 . at first order , we obtain a projection of the stress : @xmath80 where the second equality holds because of mechanical equilibrium . the partial derivative which appears at the rightmost side of these equalities corresponds to the projection of stress which would be inferred from the assumption that all the particles undergo affine displacements . since the total derivative of the energy as a function of @xmath0 ( while enforcing the condition of mechanical equilibrium ) is identical to the partial derivative , it means that non - affine displacements do not contribute to the _ first _ derivative of energy with respect to @xmath0 , _ i.e. _ the projected stress . the situation , however , is different when the second derivative of the energy is considered . indeed , we have : @xmath81 the first term in the right hand side of this equation , @xmath82 , corresponds to energy changes associated with strictly affine displacements of the particles . it is the born approximation for the second derivative of the energy with respect to @xmath0 . the last term introduces a correction to the born approximation which results from the existence of non - affine displacements . since @xmath83 is positive definite , the corrective term is negative : non - affine displacements reduce the second order derivative of the energy from its born approximation . this is another statement of the idea that non - affine displacement are associated with further minimization of the energy of the system from a tentative homogeneous deformation . this property holds for any mode of deformation @xmath47 . it means , for example , that the non - affine corrections to the shear and compression moduli must be negative . in the preceding section , we considered the non - affine displacements in response to some prescribed mode of deformation parameterized by a scalar parameter , @xmath0 . we now proceed to study the tensorial form of the above equations : derivatives of the energy functional with respect to the components of the strain tensor will provide analytical expressions for stress and elastic constants . the calculations can be simplified by noting that the number of independent components of the strain tensor is reduced by symmetry . in particular , a useful symmetry property holds under the general assumption that @xcite the interaction potential can be written as a function of the full set of distances between pairs of interacting particles : @xmath84 , where the index @xmath85 runs overs all pairs of particles , and for each pair , @xmath86 modulo the periodic boundary conditions . ( note that for a periodic simulation cell , the distance between two points depends on their positions relative to the boundaries of the cell : the way distances are calculated thus depends on the shape of the cell , hence , on @xmath27 . ) we stress that the ability to write the energy functional in this way is not restricted to potentials involving only pair interactions : it also holds for _ e.g. _ three body interactions which depends on bond angles , as used for silicon @xcite , or embedded atom potentials @xcite , as used for metals . this formalism does not apply , however , to situations when rotational degrees of freedom must be taken into account as , for instance , in cosserat approaches to granular materials @xcite or to model anisotropic objects like nematics . @xcite we expect a similar formal treatment to be possible in these situations , but it will require using a slightly more general formalism that we do not wish to address here . let us consider a reference configuration @xmath26 and a current configuration @xmath27 . suppose that @xmath87 and @xmath31 are the difference between the positions of two particles in both these systems of coordinates . they are related by @xmath88 , whence : @xmath89 with the green - saint venant strain tensor : @xcite @xmath90 we want to write the energy functional after the change of variable where any configuration @xmath34 of the system is mapped onto @xmath91 . under the general assumption that the potential energy can be written as a function of the distances @xmath92 only , equation ( [ eqn : stretch ] ) can be used to write : @xmath93 where it appears that the energy of the system is a function of @xmath94 and @xmath95 only . @xcite the notation @xmath53 is introduced here to emphasize that this functional of @xmath94 and @xmath95 is defined after a choice of reference cell with bravais matrix @xmath26 ( whereas the functional @xmath25 does not depend on this choice ) . a choice of @xmath26 being made , however , the energy functional depends on the current cell coordinates only _ via _ the symmetric tensor @xmath96 , but not _ via _ the whole tensor @xmath27 . we recover the separation of coordinates : changing @xmath96 for fixed @xmath39 corresponds to performing an affine strain of the whole system the particles and the boundary ; changing @xmath39 in the reference cell corresponds to performing non - affine displacements of the particles . we are now in a position to derive equations of motion for non - affine displacement fields in a fully tensorial form . in any configuration , the force acting on particle @xmath40 is : @xmath97 and mechanical equilibrium reads : @xmath98 the derivatives are taken here for fixed cell coordinates . the equation of motion for @xmath39 , is now obtained by differentiation of ( [ eqn : mechanical ] ) with respect to the components of @xmath96 : @xmath99 with the usual summation convention of repeated ( greek and latin ) indices . as above , the symbol @xmath62 denotes partial derivatives with respect to some tensorial components while enforcing mechanical equilibrium ( [ eqn : mechanical ] ) . in the limit @xmath100 , equation ( [ eqn : motion ] ) involves the hessian : @xmath101 and the field of third order tensors , @xmath102 in semi - condensed notation , equation ( [ eqn : motion ] ) reads : @xmath103 provided standard caution in the inversion of the hessian , we then get , up to translation modes : @xmath104 which is the non - affine displacement field in response to elementary deformation along @xmath105 . to get further insight into the correction term in equation ( [ eqn : c : xi ] ) , and on the physical interpretation of @xmath77 , let us write : @xmath106 this equality comes after differentiation of equation ( [ eqn : force ] ) with respect to @xmath96 and use of mechanical equilibrium ( equation ( [ eqn : mechanical ] ) ) . as in our pedagogical example of one - parameter strain , the field @xmath107 , can be interpreted as the forces which would result from an elementary affine displacement of all particles in the strain direction @xmath105 . ( more specifically , @xmath108 , with no summation on the greek indices , is the force resulting from a small affine transformation by @xmath109 ; @xmath107 is the tangent direction to the changes of forces upon affine transformations . ) we see that @xmath110 can be seen as the fluctuation of the force @xmath111 in response to an elementary strain . this interpretation emphasizes the random character of field @xmath107 : local forces are , by definition zero at mechanical equilibrium ; however , their variation under an elementary strain depends on the configuration of the particles with which particle @xmath40 interacts ; in particular , @xmath110 should be zero if the conformation of particles surrounding @xmath40 is symmetric which is the case of a bravais crystal . @xmath107 is thus a measure of the local asymmetry of particle configurations . equation ( [ eqn : dr ] ) further states that the non - affine displacement is nothing but the linear response of the particles to these extra forces . let us now derive microscopic equations for the stress and elastic constants . because the total potential energy is a function of @xmath96 and @xmath51 only it is sufficient to characterize the elastic response of a material using the derivatives of the potential energy with respect to the components of @xmath96 . we note , however , that the usual definition of the cauchy stress and elastic stiffnesses involves derivatives with respect to the components of the deformation gradient tensor @xmath37 . derivatives with respect to the components of @xmath96 provide the so - called thermodynamic tension and elastic constants ( or also thermodynamic stiffnesses ) . the definitions of thermodynamic tension , cauchy stress , elastic constants , elastic stiffnesses and their relation are given in appendix a. by definition , the thermodynamic tension is the derivative of the energy functional with respect to the components of @xmath96 : @xmath112 since @xmath96 is a symmetric tensor , we can conclude without further examination that thermodynamic tension is also symmetric . using mechanical equilibrium , we next have : @xmath113 as in our pedagogical example of one - parameter strain , we see here that the thermodynamic tension is equal to the partial derivative of @xmath53 ( here , with respect to @xmath96 ) . this partial derivative corresponds to changes in energy during a strictly affine displacement of the particles : we see that the existence of non - affine displacement fields does not appear in the expression for the thermodynamic tension . expression ( [ eqn : xi : tension ] ) is valid for any finite strain : this will allow us to later differentiate this expression a second time and obtain elastic constants . often , though , we will use it in the limit @xmath100 . note that this expression for the thermodynamic tension provides another interpretation of the field @xmath77 . we can write : @xmath114 from the definitions of @xmath115 and @xmath77 . this expression is to compare with equation ( [ eqn : xi : cauchy ] ) . we see that @xmath110 can be seen as a fluctuation of stress ( or tension ) in response to an elementary displacement of particle @xmath40 . this interpretation was noted in @xcite . the elastic constants are second derivatives of the energy functional with respect to the green - saint venant strain tensor : @xmath116 the derivative is here `` total '' since this is the second derivative of the energy following deformation - induce changes of a minimum . as we can permute the order of derivatives , elastic constants verify @xmath117 , and since @xmath118 is symmetric , @xmath119 . to obtain an analytical expression for the elastic constants , it suffices to differentiate expression ( [ eqn : xi : tension ] ) once and take the limit @xmath100 afterwards : @xmath120 using equation ( [ eqn : dr ] ) and the definition of @xmath121 , it then comes : @xmath122 we recognize the first term in equation ( [ eqn : c : xi ] ) as the born approximation @xmath123 . the contraction of the inverse of the hessian on components of @xmath121 provides the correction terms . here , we illustrate these ideas with numerical simulations of a two - dimensional bidisperse mixture of particles interacting through a shifted lennard - jones potential . @xcite particle sizes @xmath125 and a number ratio @xmath126 are used to prevent crystallization . the system is prepared via an initial quench from an infinite temperature state . further details regarding the numerical protocols will be found in our forthcoming dedicated numerical study @xcite . the interaction potential is pair - wise additive : @xmath127 . in this case , formal expressions for the fields @xmath110 and the born approximation for the elastic constants can be related directly to the derivatives of @xmath128 . these expressions ( ( [ eqn : xi : micro : pair ] ) and ( [ eqn : born : micro : pair ] ) ) are derived in appendix b. we just recall here equation ( [ eqn : born : micro : pair ] ) : @xmath129 where we have introduced the normalized vector between pairs of particles : @xmath130 , as well as the bond tensions and stiffnesses : @xmath131 the material is expected to be isotropic so that the elastic constants take only two independent values under permutations of indices : @xmath132 , which define the lam constants , @xmath133 and @xmath134 . using equation ( [ eqn : born : micro : pair:0 ] ) we observe that the born term is identical for the two lam constants @xmath133 and @xmath134 . introducing the notation @xmath135 , we find : @xmath136 to calculate the lam constants , it suffices to consider two modes of deformation and the associated non - affine fields . we thus consider @xmath137 and @xmath138 which are associated with pure shear and pure compression respectively . plots of @xmath139 and @xmath140 , and their corresponding displacement fields were given on figure [ fig : fields ] and [ fig : fields : c ] respectively . pure shear grants access to the sum : @xmath141 . pure compression grants access to the sum : @xmath142 , where @xmath143 is in two dimensions the compression modulus . with the fields @xmath139 and @xmath140 , we can directly construct the non - affine correction to @xmath134 : @xmath144 ; and to @xmath145 : @xmath146 . we recall that since the hessian is positive definite , these corrections are necessarily negative . we thus have in all generality that for an isotropic material , @xmath147 and @xmath148 . using these fields , we find in this sample , for the shear and compression moduli : @xmath149 , @xmath150 , whence @xmath151 and @xmath152 , @xmath153 , whence @xmath154 . all moduli are reported in dimensionless stress units . these values compare to the highest density systems studied by tanguy et al @xcite . we note that the correction term to the compression modulus appears to be small . this can be understood to arise from a simple property . let us consider the case when the potential is pairwise additive , and is homogeneous in the sense that the force between each pair of particles is a homogeneous function of their distance : @xmath155 . a pure compression corresponds to a global scaling of all distances : if the forces are homogeneous , a global scaling of the distances preserves a state of mechanical equilibrium . in other word , whenever the forces are homogeneous functions of the distances , the fluctuating force field associated with pure compression , @xmath156 , vanishes identically . hence , the correction to the born approximation vanishes for the compression modulus : @xmath157 . for a compressed lennard - jones system , if the total energy is dominated by the pairs of particles which are at close distance , the interaction is thus dominated by the repulsive power - law divergence of the potential . since this power - law leads to forces which are homogeneous functions of the distances , this repulsive part of the lennard - jones potential leads to vanishing corrections to the born approximation . as a result the overall amplitude of the field @xmath140 is small , hence the small correction to @xmath145 . in particular , this is consistent with the data from tanguy et al @xcite and explains why the corrections induce important changes in both @xmath133 and @xmath134 but smaller changes in their sum . in expression ( [ eqn : c : xi ] ) , the correction to elastic constants involves a contraction of the hessian on the fields @xmath121 . since the hessian is positive definite we are perturbing around a local minimum it acts as a scalar product in expressions such as : @xmath158 . this form of the corrections to elasticity suggests that further insight may be gained from a normal mode decomposition of the fields @xmath107 . we denote the eigenvectors of the hessian by @xmath159 and the associated eigenvalues by @xmath160 . we introduce a particle mass to correctly scale eigenvalues and eigenfrequencies and write @xmath161 . ( in our numerical simulations , the mass is taken to be unity . ) the vector @xmath107 is written as : @xmath162 with @xmath163 . using this decomposition and equation ( [ eqn : dr ] ) , the non - affine displacement fields read : @xmath164 the non - affine contribution to elasticity reads : @xmath165 as we saw from figures [ fig : fields ] and [ fig : fields : c ] , the fields @xmath77 display a very random character . the randomness of the fields @xmath71 can be further assessed by studying fluctuations of the weights of their normal mode decomposition . to examine this question , we report , in figure [ fig : xiscatter ] , all values of @xmath166 , with @xmath167 corresponding to pure shear deformation and pure compression . these quantities are obtained by calculating the fields @xmath168 and performing a full diagonalization of the hessian . we see that within a frequency shell @xmath169 $ ] , the values of @xmath166 are scattered , so that they can indeed be interpreted as random variables . on the scatter plot , it is apparent that the distributions of @xmath166 are symmetric with respect to the horizontal axis , hence that the @xmath166 s have zero mean : this property is enforced by symmetry since each of the eigenvectors of the hessian is defined up to a sign convention . the width of the distributions of @xmath166 seem to depend smoothly on @xmath170 and increase with @xmath170 for the most part . at high frequencies the scatter plots thin out because there are fewer and fewer eigenvectors to sample the density of states close to its upper cut - off . on the basis of this observation , we make the assumption that the random fields , @xmath107 , and in particular their projections on normal modes , @xmath171 , are self - averaging quantities . we thus introduce the correlators on frequency shells : @xmath172}\ ] ] in this definition , the average is performed for all the projections of @xmath121 and @xmath107 on eigenvectors with eigenfrequency @xmath169 $ ] . the assumption that @xmath171 are self - averaging means that the quantities @xmath173 are expected to converge toward well - defined functions of @xmath170 in the thermodynamic limit : either when a large ensemble of systems of a given size is considered , or when one single system of a large size is considered . from their definition , we observe that functions @xmath173 verify the same symmetries as the elastic constants , namely : @xmath174 in the thermodynamic limit , we can finally rewrite equation ( [ eqn : c : xi ] ) as : @xmath175 where @xmath176 is the density of states is , somewhat unconventionally , normalized to the number of degrees of freedom per unit volume . ] . this equation is a sum rule : it relates elastic constants to the integral of a correlator between microscopic fields . versus the corresponding eigenfrequencies of the hessian ( with @xmath177 in shear ( top ) and @xmath178 in compression ( bottom ) ) . @xmath166 is the projection of the field @xmath179 onto the eigenmode @xmath180 with frequency @xmath181 . this set of data has been obtained using one typical system of size @xmath182 . each point makes a contribution to the non - affine corrections to the elastic constants as described in the text . ] like the elastic constants , @xmath183 are tensorial quantities . therefore , if the material is isotropic given the symmetry property ( [ eqn : gamma : sym]) these functions can take only two independent values corresponding to the two lam constants @xmath133 and @xmath134 : @xmath184 in this case , the two correlators , @xmath185 and @xmath186 provide a complete description of the non - affine corrections to the lam constants according to equation ( [ eqn : sum ] ) . to illustrate this discussion , we return to our two - dimensional numerical study of two modes of deformation , simple shear and pure compression . this protocol suffices to measure the two lam constants of our system . the vector fields @xmath139 and @xmath140 grant direct access to the quantities @xmath187 for @xmath167 . elementary algebra , similar to the calculations of section 1.5 , shows that these quantities also verify : @xmath188 and @xmath189 . we report , in figure [ fig : dos ] , $ ] , as described in the text . results were obtained as ensemble averages over 20 systems of sizes @xmath190 with @xmath191 : 20 circles ( black ) , 25 squares ( red ) , and 30 diamonds ( green ) . convergence of the curves demonstrates that finite size effects are small . ( a ) density of states . ( b , c ) magnitude of `` noise '' field , @xmath192 , for simple shear ( b ) and compression ( c ) . ( d , e ) net contribution to elasticity , @xmath193 , of each frequency band for shear ( d ) and compression ( e ) . , title="fig : " ] + $ ] , as described in the text . results were obtained as ensemble averages over 20 systems of sizes @xmath190 with @xmath191 : 20 circles ( black ) , 25 squares ( red ) , and 30 diamonds ( green ) . convergence of the curves demonstrates that finite size effects are small . ( a ) density of states . ( b , c ) magnitude of `` noise '' field , @xmath192 , for simple shear ( b ) and compression ( c ) . ( d , e ) net contribution to elasticity , @xmath193 , of each frequency band for shear ( d ) and compression ( e ) . , title="fig : " ] $ ] , as described in the text . results were obtained as ensemble averages over 20 systems of sizes @xmath190 with @xmath191 : 20 circles ( black ) , 25 squares ( red ) , and 30 diamonds ( green ) . convergence of the curves demonstrates that finite size effects are small . ( a ) density of states . ( b , c ) magnitude of `` noise '' field , @xmath192 , for simple shear ( b ) and compression ( c ) . ( d , e ) net contribution to elasticity , @xmath193 , of each frequency band for shear ( d ) and compression ( e ) . , title="fig : " ] $ ] , as described in the text . results were obtained as ensemble averages over 20 systems of sizes @xmath190 with @xmath191 : 20 circles ( black ) , 25 squares ( red ) , and 30 diamonds ( green ) . convergence of the curves demonstrates that finite size effects are small . ( a ) density of states . ( b , c ) magnitude of `` noise '' field , @xmath192 , for simple shear ( b ) and compression ( c ) . ( d , e ) net contribution to elasticity , @xmath193 , of each frequency band for shear ( d ) and compression ( e ) . , title="fig : " ] $ ] , as described in the text . results were obtained as ensemble averages over 20 systems of sizes @xmath190 with @xmath191 : 20 circles ( black ) , 25 squares ( red ) , and 30 diamonds ( green ) . convergence of the curves demonstrates that finite size effects are small . ( a ) density of states . ( b , c ) magnitude of `` noise '' field , @xmath192 , for simple shear ( b ) and compression ( c ) . ( d , e ) net contribution to elasticity , @xmath193 , of each frequency band for shear ( d ) and compression ( e ) . , title="fig : " ] the density of states @xmath194 , the correlators @xmath195 , @xmath196 , and the quantities @xmath197 ( with @xmath198 ) which measure the total contribution of the shell @xmath199 $ ] to the non - affine corrections . three system sizes have been used in this plot : @xmath200 and @xmath201 , where @xmath191 is the length of the square simulation cell in dimensionless units . for each size , an ensemble of 20 system was used to provide statistical accuracy . the functions @xmath202 and @xmath203 measure the variance of the distributions of @xmath204 and @xmath205 respectively , which appear on the scatter plot of figure [ fig : xiscatter ] . we see that that these functions are roughly increasing as was suggested from the observation of the scatter plot . important fluctuations of @xmath186 and @xmath206 at high frequencies result from the lack of statistical representation in the region where the density of states vanishes and where the scatter plots thin out . for their main part , however , the curves lie on top of one another for the three system sizes we have used : this is evidence that the continuous limit has already been reached . this observation is consistent with the numerics of tanguy et al @xcite since the values of the elastic constants they measured did not vary much for the different sizes they studied , from @xmath207 to @xmath208 . we believe that the good convergence of these measurements results from the decorrelation of the fluctuating forces @xmath209 at very short distances : because the correlation length is short , the finite sizes of the systems considered here provide enough sampling of the distribution of @xmath209 . the density of states , and the @xmath210 s exhibit rather complicated functional forms . in contrast , it is striking to us that for simple shear and pure compression @xmath211 seems rather featureless ; particularly in comparison with both the density of states and the functions @xmath212 . we have observed this for different numerical models and different densities . @xcite the monotonous decay of @xmath213 and @xmath214 seems to indicate that a simple physical mechanism like a transfer of elasticity from low to high frequency phonons governs the non - affine corrections . this reminds us of an analysis by radjai and roux of quasi - static deformation of granular materials @xcite in which the authors suggested that the non - affine velocity field could exhibit statistical properties similar to turbulence . this sound like an appealing suggestion , although , at present , we have no evidence to further test the existence of turbulent - like energy transfers from large to small scales . in the above formalism , we assumed that deformation was very slowly forced upon a piece of material , so that the relaxation of the system toward an inherent structure was always faster than the changes induced by the external drive . this guaranteed that the system followed an energy minimum in configuration space . our analysis , however , is essentially based on a second order expansion of the energy functional : it assumes small displacements , but may , in principle , apply to situations when the timescale of the external drive becomes relevant . we consider here oscillatory perturbations of small amplitude , and assume that the system remains in the linear response regime around some energy minimum , far away from any catastrophic event . we expect this situation to be of relevance to questions such as _ e.g. _ acoustic damping in granular materials @xcite or viscoelasticity in dense emulsions and foams @xcite . near a minimum , but not exactly _ at _ mechanical equilibrium , the particles are subjected to non - zero forces : an oscillatory external forcing at finite rate thus works on the system of particles . energy is injected into the system and is lost in dissipative mechanisms at the microscopic scale ( friction between grains or viscous dissipation between bubbles ) . we will follow the usual practice of characterizing energy transfers in the linear response regime using the components of the complex stress response , that is the storage and loss moduli , @xmath215 and @xmath216 . @xcite we insist that we focus here on features of @xmath215 and @xmath216 that arise merely from the existence of non - affine displacement fields _ around a single minimum_. in an experiment think , for example , of a foam it is unclear whether sufficiently small strain amplitudes can be achieved so as to stay within the basin of attraction of a single equilibrium configuration . thus measured values of @xmath215 and @xmath216 would also receive contributions from plasticity in addition to the microscopic visco - elasticity studied here : transitions between distinct inherent structures would become a relevant mechanism of energy dissipation @xcite . it s only after a careful study of the contribution of linear response that we will be in a position to characterize the relative importance of these various dissipative mechanisms . the present work is a preliminary step in this direction . in order to work at finite strain rates , we need to use newton s equations coupled with the deformation of our simulation cell . in order to prevent the system from heating up , a microscopic mechanism must be specified for energy dissipation . dissipative mechanisms vary from system to system . in granular packings , dissipation involves the deformation of asperities at contact between grains . in foams , dissipation involves fluid flow inside a liquid phase : it is modeled by durian @xcite by viscous terms involving velocity differences between neighboring bubbles . in numerical simulations of structural glasses and supercooled liquids , the coupling with an external bath is modeled either by a nos - hoover @xcite thermostat , by use of a lagrange multiplier @xcite , or by a berendsen thermostat @xcite . it thus appears , after examination of these different systems , that energy dissipation often arises as some sort of viscous damping at the microscopic level . we will thus limit our present discussion to the situation where dissipation enters via a viscous drag force applied on individual particles . below , we give some rule - of - thumb estimates for what the value of this viscosity might be in the case of dense emulsions . in the sllod algorithm , the viscous term is generally taken to be proportional to the peculiar velocity , @xcite that is the velocity @xmath217 of the particle minus its velocity corresponding to the affine flow @xmath218 . similarly , in foams or granular materials , the viscous damping does not depend on the velocity in real space but on the difference between the particles velocity and their surrounding environment . we thus first write the equations of motion as : @xmath219 in term of @xmath39 , it follows that : @xmath220 we see here that a term @xmath221 arises which depends on the position of the particle in the simulation cell . another term of similar flavor would be @xmath222 : it does not appear in equation ( [ eqn : motion : r00 ] ) because it has already been eliminated by the above assumption that the viscous dissipation applies to the _ peculiar _ velocities . such terms introduce a spatially dependent coupling between particular motion and the cell coordinates : they can not be allowed to enter the equations of motion if we want to enforce translation invariance . this was originally noted by andersen @xcite and later by ray and rahman @xcite . within the formalism of souza and martins @xcite , an equation of motion with no spatially dependent terms can be derived systematically from a lagrangian . it relies on the idea that the distance traveled by a particle should be calculated while removing a component of rigid rotation . we do not wish to detail this formalism here , but simply recall that adding appropriate terms in the lagrangian of the system ( particles plus simulation cell ) permits eliminating space - dependent terms from equation ( [ eqn : motion : r00 ] ) . following these authors , we eliminate the term @xmath221 and focus on translation invariant equations of motion for an athermal system in sllod dynamics . these equations read : @xmath223 we want to account for the dynamics of the system when it is submitted to small amounts of strain @xmath224 . to do so , we write a perturbative expansion of equation ( [ eqn : motion : r0 ] ) around a known equilibrium state @xmath39 . this expansion is written in terms of the displacements @xmath225 , which are also of order @xmath226 . at first order in @xmath226 , the perturbed equations of motion read : @xmath227 using the definition of @xmath228 and @xmath229 ( equations ( [ eqn : h:2 ] ) and ( [ eqn : xi:2 ] ) ) it follows that : @xmath230 this equation governs the linear response of our system around a single minimum . starting from this equation , various limits allow us to recover more usual expressions . for example , taking @xmath231 and canceling the term on the right hand side yields the equation governing the vibration modes of the system ; canceling @xmath232 and @xmath233 yields the equations which governs quasi - static response and defines the non - affine displacement fields . to solve equation ( [ eqn : lin ] ) , we perform two transformations : a fourier transform of the time domain , and a normal mode decomposition of the displacement field . the fourier components @xmath234 are the responses to perturbations of the form @xmath235 . they are further decomposed onto normal modes as : @xmath236 and @xmath237 . with these notations , we find : @xmath238 which is solved by : @xmath239 the perturbed stress is then obtained by writing at first order in @xmath226 : @xmath240 using ( [ eqn : x : sol ] ) , the complex stress response in the frequency domain thus reads : @xmath241 it can thus be recast in the form : @xmath242 with : @xmath243 taking the thermodynamic limit , we can furthermore introduce the functions @xmath173 defined in equation ( [ eqn : gamma ] ) . equation ( [ eqn : g ] ) can then be written as an integral in frequency domain : @xmath244 it is then an easy task to extract the real and imaginary part of the stress response . observe that we recover the born approximation and the true elastic constants in the high and low frequency limits respectively . the response moduli are often studied in the overdamped limit , when the kinetic energy remains small compared to the elastic energy . @xcite in this case , the term @xmath245 is negligible before @xmath246 and equation ( [ eqn : moduli ] ) can be given a simpler form . to do so , we introduce the timescales : @xmath247 and write equation ( [ eqn : moduli ] ) as : @xmath248 where @xmath249 the complex modulus @xmath250 is thus naturally written as a sum of maxwell elements . by definition , @xcite the function @xmath251 is the relaxation spectrum of the system . it is the product of the density of states with the correlator @xmath183 . the specific forms of this relation associated with a given experimental geometry pure shear or pure compression derive trivially from these tensorial expressions . : 20 circles ( black ) , 25 squares ( red ) , and 30 diamonds ( green ) . ] we plot the shear relaxation spectrum for our model system in figure [ fig : relaxationspectrum ] , again for each of the three ensembles of systems with lengths 20,25 , and 30 , demonstrating the system size independence of the results . since the true microscopic mechanisms of dissipation in real structural glasses remain somewhat poorly understood , we focus here on the case of a dense emulsion or foam , where the system is overdamped by construction . to make a semi - quantitative comparison with experiments on such systems , we need to give a rough estimate for the values of the coefficient of viscosity , @xmath233 , and the overall stiffness scale , @xmath252 , in our model . a truly quantitative connection with experiment is somewhat beyond the scope of the present work and will be explored in our dedicated numerical study @xcite . we should also remind the reader that the data presented in this work for illustrative purposes is taken from a compressed system with lennard - jones interactions ... clearly not sufficient for a fully quantitative comparison . however , the hope is that there is a good deal of qualitative similarity in the structure of the hessian matrix for these various classes of disordered systems , and we proceed to dimensionalize @xmath252 and @xmath233 in order to make a semi - quantitative comparison with experiments on packings of emulsified droplets . a series of experiments , simulations , and theoretical works on such a system was carried out by mason and coworkers @xcite . the surface tension of the droplets was found to be about @xmath253 , which , for well - compressed systems , should set the average scale of the interparticle stiffnesses . for crystalline systems , the scale of interparticle stiffness is characterized by the maximum eigenvalue of the system ( corresponding to eigenvectors which are plane waves with periodicity equal to one reciprocal lattice vector ) , thus we scale the eigenvalues of the hessian matrix of our system such that the maximum eigenvalues are roughly @xmath254 . those maximum eigenvalues of the hessian matrix , as can be read off from the density of states , are about 5000 in our dimensionless units of stiffness , so we must have @xmath255}{[l]}=\frac{10}{5000}=2.10^{-3}{\text{dyne}/\text{cm}}$ ] . this sets the scales of the second derivatives of the potential . the order of magnitude of elastic constants then comes out by dividing @xmath256 by the average interparticule distance , that is by the diameter of droplets in the case of an emulsion . taking a droplet radius of a micron , this sets the scale of elastic constants as @xmath257=10~\text{dyne}/\text{cm}^2 $ ] . our measurement of elastic constants in the range of 100 , thus corresponds to values of order @xmath258 . following durian @xcite , we ought to choose a value for @xmath233 such that the drag force , @xmath259 , between two droplets sliding past each other with relative velocity @xmath260 generates a stress in the film which is equal to the strain rate in the film times the effective viscosity , so we must have : @xmath261 where @xmath262 is the typical lateral extent of the film ( which , following durian , we take to be of the order of the droplet radius itself ) , and @xmath263 is the width of the film gap . we can now dimensionalize our units of time , @xmath264=\nu/\bar{\lambda}$ ] , using the estimate of liu _ et . al . _ for the effective viscosity in the film at the droplet interface of @xmath265cp , a droplet radius of a micron , and a film thickness of a nanometer , yielding : @xmath264=5\text{s}$ ] . from the plot of the relaxation spectrum in figure [ fig : relaxationspectrum ] , we expect the viscoelastic effects and the crossover to affine deformation to appear in the range of frequencies from around 1hz up to about 1khz in these systems . in this article , we have studied molecular displacements associated with deformation - induced , continuous changes of a minimum in the potential energy landscape of an amorphous solid . for small amounts of deformation , the trajectory followed by the particles is smooth , and the tangent displacements analogous to a velocity , with strain playing the role of an effective time involve affine and non - affine fields . both fields enter at first order in the equation of motion of the particles during quasi - static deformation . the non - affine fields can be calculated by inverting the hessian on a fluctuating force field @xmath266 , which is formally a third order tensor . for every component @xmath267 of the deformation , @xmath121 is the field of forces resulting on every particle from the _ affine _ motion of its neighborhood . the non - affine displacement fields and the corrections they induce on elastic constants , can be expressed solely in terms of the hessian and of various tensorial components of the field @xmath121 . we next observe that the field @xmath121 is weakly correlated in space , so that it can essentially be considered as a random vector field . this randomness is confirmed by the scattered values for the projections of @xmath121 on the eigenmodes of the hessian . the normal - mode analysis of the field @xmath121 grants access to the contribution of each frequency shell to non - affine corrections to elasticity . our numerical calculations indicate that the correlation function @xmath268}$ ] converges toward a smooth function in the thermodynamic limit : this means that the contribution of each frequency shell to non - affine corrections to elasticity is self - averaging . in our view , the existence of a well - defined limit for the elastic properties of amorphous structures , as observed by tanguy _ et al _ , rest on this self - averaging property . we moreover observed that the contribution of frequency shells seems to be a rather simple function of the frequency , as opposed to both @xmath173 and the density of states . this simple form of corrections to elasticity suggests that an elementary mechanism of transfer of energy between frequency shells may be at work and determines corrections to elastic constants . this is reminiscent of the observation by radjai and roux of turbulent like features of non - affine displacement fields , @xcite although we have not determined whether turbulent - like scalings arise in our non - affine fields . a better understanding of the mechanism underlying this transfer through frequency shells should in principle permit construction of approximations for the elastic constants of amorphous materials . after studying elastic properties of an amorphous solid in response to quasi - static deformation , we considered the case when the deformation rate is finite . this required us to introduce both a molecular mass and a damping term to provide equations of motions which correspond to a molecular system in contact with a bath at low or zero temperature . we have shown that the visco - elastic response of the solid can be written in the form of a sum of elementary damped oscillators . the frequency spectrum that is the distribution of timescales of elementary vibration modes is directly related to the function @xmath173 . it thus appears , that a broad spectrum of timescales arise at linear order solely from the structure of any particular energy minimum in a high dimensional configuration space . the system remains solid since it resides in the neighborhood of a local minimum in configuration space at all times : this is different from the usual idea that visco - elastic response is associated with transitions between minima in the energy landscape @xcite . in a real system , this type of dissipation may occur simultaneously with other mechanisms : energy transport via phonons , dissipation resulting from anharmonic terms , or transitions between various energy minima ( _ i.e. _ true plasticity ) . the estimation of these various contributions and their interplay seem to us an exciting direction for future research . it will require the construction of models , close to the experiments , where various dissipative mechanisms can be estimated quantitatively . finally , we stress that the corrections to elastic constants involve @xmath269 factors as in equation ( [ eqn : sum ] ) : the convergence of the integral in equation ( [ eqn : sum ] ) thus depends on the low frequency behavior of the functions @xmath183 and of the density of states . this becomes a problem when the system develops low frequency eigenmodes which have a non - zero scalar product with the field @xmath266 : a quick inspection of equation ( [ eqn : munormal ] ) indeed shows that such localized low frequency phonons would lead to diverging terms in the non - affine corrections to the born approximation . we can identify two situations when this occurs : ( i ) a localized eigenvector with a vanishing frequency is involved whenever the local minimum occupied by the system reaches a catastrophe . @xcite this property enabled us to obtain universal scalings for the elastic moduli close to a shear induced catastrophe , @xcite and observe the divergence of the non - affine corrections at these points . ( ii ) eigenmodes seem to drift toward the low frequency part of the spectrum when the system approaches the unjamming point of liu and nagel . @xcite a divergence of the non - affine corrections to elasticity could thus arise around this point . in this case , we expect the divergence of the non - affine corrections to elasticity to control the unjamming transition around the @xmath270 point . al received support from the william . m. keck foundation , the mrsec program of the nsf under award no . dmr00 - 80034 , the james . s. mcdonnell foundation , nsf grant no . dmr-9813752 , the lucile packard foundation , the mitsubishi corporation , and the national science foundation under grant no . phy99 - 07949 . cm was supported under the auspices of the u.s . department of energy by the university of california , lawrence livermore national laboratory under contract no . w-7405-eng-48 and would like to acknowledge the guidance and support of v. v. bulatov and j. s. langer and the hospitality of the llnl university relations program . stresses are first derivatives of an energy functional with respect to strains . @xcite since there are different definitions of the strain tensors ( @xmath96 or @xmath37 or @xmath271 or @xmath272 ) , there are different ways to take this derivative . moreover , this derivative can be lagrangian or eulerian . we thus have various choices , which leads to a number of possible definitions . two definitions of the stress are most important : the thermodynamic tension is the lagrangian derivative with respect to @xmath96 ; the true stress is the eulerian derivative with respect to the deformation gradient tensor @xmath37 . let us consider some energy functional , parameterized by the cell coordinates : @xmath273 . this functional can represent different objects : for example , it can be the energy @xmath274 for fixed @xmath39 : its strain derivatives were denoted as partial derivatives in the previous discussion ; it can also be the energy @xmath275 , provided constraints which enforce a relation @xmath276 . the formalism developed here does not depend on any specific definition of this energy functional but only on the existence of some function @xmath273 . the value of @xmath277 in a given reference configuration is denoted @xmath278 . we saw the important role played by the green - saint venant strain tensor @xmath96 : it accounts for the mapping of distances after an affine transformation ( see equation ( [ eqn : stretch ] ) ) . this property permits writing the strain - dependence of the energy functional _ via _ @xmath96 only , whenever the energy is a function of the set of distances @xmath92 between the particles . @xcite by definition , the thermodynamic tension @xcite is conjugate to the green - saint venant strain tensor : @xmath279 where @xmath280 is the volume of the reference cell . the thermodynamic tension @xmath281 is defined after a choice of a reference configuration : it is a lagrangian derivative . it identifies , in the continuum limit , with the second piola - kirchhoff stress tensor . @xcite this object is known , in general , not to have a simple mechanical interpretation : it can not be interpreted as the tensor generating forces on surface elements . the cauchy stress corresponds to the usual definition of a stress : it gives surface forces when contracted on a vector normal to a surface which is prescribed in the physical space . it can be shown that the cauchy stress can be written as a derivative of the energy with respect to the deformation gradient tensor : @xmath282 unlike the second piola - kirchoff stress ( i.e. the thermodynamic tension ) the cauchy stress does not depend on the choice of a reference state . this equality holds in the limit @xmath283 : in this sense it is an eulerian derivative of the energy . as we will show below , the second piola - kirchoff stress becomes identical to the cauchy stress in the limit where the current and reference configuration are identified . in order to relate these two definitions of the stress tensor , we need to be able to switch between @xmath96- and @xmath37-derivatives . from the definition ( [ eqn : epsilon ] ) of the green - saint venant strain tensor , infinitesimal displacements read : @xmath284 since @xmath118 is symmetric , in a @xmath69-dimensional problem , it has only @xmath285 independent components , while @xmath37 has @xmath286 . the rotational degrees of freedom are taken into account by considering the antisymmetric infinitesimal displacements : @xcite @xmath287 which is the generator of infinitesimal rotations , and has @xmath288 independent components . from equations ( [ eqn : depsilon ] ) and ( [ eqn : domega ] ) , it comes : @xmath289 using this relation , the differential form of an arbitrary function of @xmath37 , @xmath290 , reads : @xmath291 using the property that the contraction of a symmetric with an antisymmetric tensor vanishes and the fact that @xmath96 is symmetric , we see that the derivative of a function @xmath292 with respect to @xmath96 is : @xmath293\ ] ] incidentally , we also find from the preceding calculation that iff @xmath292 only depends on @xmath96 , hence does not vary under infinitesimal rotations , it verifies : @xmath294 with these formulae in hand , let us come back to the definition ( [ eqn : cauchy : def ] ) of the cauchy stress . taking the limit @xmath283 ( or @xmath100 ) , in equation ( [ eqn : deriv : sym ] ) and looking at equation ( [ eqn : cauchy : def ] ) we see that @xmath295 is symmetric . using equation ( [ eqn : deriv : eta ] ) , it appears that it equals the thermodynamic tension _ in this limit _ , yet in this limit only . we emphasize here that the symmetry of the stress tensor results directly from the invariance of the interaction potential under elementary rotations : the energy functional is expected to be a function of @xmath296 and @xmath96 only : its derivatives with respect to the components of @xmath297 vanish . this is not , however , equivalent to the energy functional being invariant under global rotations ( the bravais cell being a parallelepiped is not ) . we note , however , that in general @xmath298 ( evaluated off the reference configuration ) need not be symmetric : this may explain the observation of non - symmetric stresses in @xcite . barron and klein @xcite provide the following relation between the cauchy stress and the thermodynamic tension : @xmath299 this relation holds for finite deformations . it can be obtained , following these authors , after a taylor expansion of the energy versus the different strain tensors . we provide here a different derivation which relies on an interesting property concerning the transport of derivatives . as usual , a reference configuration @xmath26 is given and the system is strained to a current configuration @xmath27 . in order to define the cauchy stress around the configuration @xmath27 , we will need to consider @xmath27 as a new reference configuration @xmath300 and also consider a new current configuration @xmath301 . we thus have three different sets of cell coordinates : @xmath26 , @xmath302 and @xmath301 . we need to define strain tensors between each pair of these configuration : the deformation gradient tensors are denoted @xmath303 , @xmath304 and @xmath305 . these transformations are illustrated schematically in figure [ fig : framescartoon ] . likewise , the green - saint venant tensors are denoted : @xmath306 , @xmath307 and @xmath96 . we have the property : @xmath308 . for any arbitrary function @xmath290 , and for @xmath37 , we can write : @xmath309 whence , @xmath310 using this and equation ( [ eqn : deriv : eta ] ) it is then an easy check that : @xmath311 this relation simply means that the tensorial derivative with respect to @xmath96 transforms as a tensor under a change of reference configuration . we just saw in equation ( [ eqn : deriv : f ] ) that this is not true of the derivative with respect to @xmath37 . going back to the definition of the cauchy stress ( equation ( [ eqn : cauchy : def ] ) ) and taking the limit @xmath312 , we have @xmath313 and using equation ( [ eqn : deriv : transport ] ) in the limit @xmath314 ( or @xmath315 ) , we recover equation ( [ eqn : cauchy ] ) . it is useful to `` specialize '' the previous expressions for the situations when the energy , or any observable can be parameterized as @xmath316 . we first have : @xmath317 the factor 1/2 results from the fact that each pair is counted twice with the convention of implicit summation over repeated indices . plugging equation ( [ eqn : deriv : f : micro ] ) ( and its transpose ) into equation ( [ eqn : deriv : eta ] ) we immediately obtain , @xmath318 we see from equation ( [ eqn : deriv : transport ] ) that this expression consist of a reference - independent formula , which is transported backward in the reference configuration as in equation ( [ eqn : deriv : transport ] ) . specializing the functional @xmath292 to @xmath319 , we can now provide an expression for the cauchy stress tensor , using either equation ( [ eqn : cauchy ] ) and ( [ eqn : deriv : eta : micro ] ) or alternatively ( [ eqn : cauchy : def ] ) and ( [ eqn : deriv : f : micro ] ) . it reads : @xmath320 this is a generalization of the kirkwood formula to the case of an arbitrary n - body interaction potential . note that specialization to systems with pairwise interactions immediately yields the standard expression . specializing the function @xmath292 to @xmath321 , we can furthermore obtain a useful formula . indeed , in the limit @xmath283 , equation ( [ eqn : deriv : eta : micro ] ) reduces to : @xmath322 by definition , elastic _ constants _ are second derivatives of the energy functional with respect to the green - saint venant strain tensor : @xmath323 in this expression , the energy functional @xmath277 can be any given function of the tensor , @xmath118 . for example , it can be the energy functional of an atomistic system @xmath324 for fixed positions of the particles in a reference configuration . derivatives of such a functional were denoted as partial derivatives of @xmath53 throughout the text . but @xmath277 could also be defined as the energy of an atomistic system following deformation induced changes of a given minimum . in this case the partial derivatives of @xmath277 would correspond to total derivatives for the energy functional @xmath53 . the algebra presented here does not depend on these considerations but only on the existence of a function @xmath325 . since we can commute the order of the derivatives , the elastic constants verify @xmath117 , and since @xmath118 is symmetric , @xmath119 . the second order expansion of the energy with respect to the components of @xmath96 thus reads : @xmath326 where @xmath327 is the stress in the reference configuration . using ( [ eqn : tension ] ) and ( [ eqn : second ] ) , we find for the thermodynamic tension : @xmath328 the elastic constant should not be confounded with the quantities which enter the stress - strain relations hence , the wave equation . these are the elastic stiffnesses , @xmath329 , and appear in the expansion of the energy with respect to tensor @xmath330 : @xcite @xmath331 the relation of the elastic stiffnesses to the elastic constants is obtained by replacing the definition ( [ eqn : epsilon ] ) of @xmath96 in equation ( [ eqn : second ] ) and expanding in terms of @xmath332 : @xcite @xmath333 note that this expression indicates that elastic stiffnesses do not enjoy the full symmetry of the elastic constants . instead , they verify : @xcite @xmath334 the cauchy stress can now be obtained from ( [ eqn : tension : exp ] ) by use of equations ( [ eqn : cauchy ] ) and ( [ eqn : stiffness ] ) : @xcite @xmath335 before proceeding to the derivation of an expression for the born term , let us derive a general formula for second derivatives with respect to the components of @xmath96 . we first write equation ( [ eqn : deriv : eta ] ) with indices explicitly expressed : @xmath336\ ] ] the derivative of this expression with respect to @xmath267 reads : @xmath337\ ] ] we then use the property that in the limit @xmath100 , @xmath338 by definition , whence we see that @xmath339 after application of equation ( [ eqn : deriv : eta ] ) . in the limit @xmath100 , we thus have for the second derivatives , after a further application of equation ( [ eqn : deriv : eta ] ) : @xmath340\end{aligned}\ ] ] we can now provide microscopic expressions for the born approximation to elastic constants . we take the energy functional @xmath277 to be the energy @xmath53 for some fixed positions of the particles in a reference frame . using equation ( [ eqn : deriv : eta : micro ] ) and ( [ eqn : deriv : f : micro ] ) it is an easy task to write the born term as : @xmath341 in order to obtain a similar expression for @xmath110 , it is convenient to write : @xmath342 with @xmath343 the pair contributions @xmath344 are then easily expressed in terms of pair contributions to the hessian : @xmath345 then , either using equation ( [ eqn : deriv : rij ] ) or applying equation ( [ eqn : deriv : eta : micro ] ) on @xmath111 , we find for @xmath344 an expression which is analogous to kirkwood s formula for the stress : @xmath346 this general formula permits relating @xmath110 to elementary contributions @xmath347 of pairs to the hessian . we furthermore notice the similarity between equation ( [ eqn : xi : micro ] ) and equation ( [ eqn : born : micro ] ) . it allows writing the following expression for the born term : @xmath348 we provide here specific expressions for the case when the interaction potential can be written as : @xmath349 in this case , the force on individual bonds reads : @xmath350 with , @xmath351 introducing the bond tensions and stiffnesses , @xmath352 the components of the hessian @xmath353 can be expressed in terms of : @xmath354 the elements of the hessian are then , @xmath355 for the off - diagonal terms and @xmath356 for the diagonal terms . to obtain an expression for the field @xmath107 , we write : @xmath357 with ( no implicit sum on @xmath40 and @xmath358 ) : @xmath359 whence , @xmath360 inserting this expression in the sum ( [ eqn : xii : xiij ] ) , the second term disappears because the system is at mechanical equilibrium . this yields : @xmath361
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we study exact results concerning the non - affine displacement fields observed by tanguy _ et al _ [ europhys .
lett . * 57 * , 423 ( 2002 ) , phys .
rev .
b * 66 * , 174205 ( 2002 ) ] and their contributions to elasticity . a normal mode analysis permits us to estimate the dominant contributions to the non - affine corrections to elasticity and relate these corrections to the correlator of a fluctuating force field .
we extend this analysis to the visco - elastic dynamical response of the system .
+ keywords : amorphous solids , born - huang approximation , visco - elasticity , non - affine a straightforward estimate of the elastic constants of simple crystals can be performed in the ( classical ) zero temperature limit : the relative initial positions of atoms are known ; elementary deformations are homogeneous even at the microscopic level .
it is thus a simple task to add up all contributing interactions .
these assumptions zero temperature and homogeneous displacement of the particles constitute the basis of the born - huang theory .
@xcite these assumptions can also be used to estimate the elastic constants of a disordered structure : they provide approximate expressions involving integrals over the pair correlation ; in liquid theory , these expressions correspond to the infinite frequency moduli .
@xcite of course , the two assumptions of zero temperature and homogeneous displacement are not valid in general and corrections to the born - huang approximation are expected to arise from the failure of either .
early studies by squire , holt and hoover , @xcite focused on thermal contributions to elasticity in crystals .
more recently , a surge of interest for athermal materials , like granular materials or foams , attracted some attention to corrections to the born - huang approximation which arise solely from the non - trivial structure of the potential energy landscape .
@xcite namely , in disordered solids at zero temperature , the assumption that particles follow homogeneous ( affine ) displacement fields is incorrect : when a material is strained even by vanishingly small amounts of deformation particles minimize the potential energy of the system by following non - affine displacements .
this idea was recently recognized in numerical simulations of compressed emulsions @xcite and lennard - jones glasses .
@xcite in particular , tanguy _ et al _
@xcite have clearly shown that non - affine corrections to the born - huang approximation hold in the continuum limit and amount to an important fraction of the born - huang term itself .
it is thus essential to understand these corrections well if we ever want to be able to construct approximations to the elastic constants of amorphous materials .
formal expressions for the non - affine ( or `` inhomogeneous '' , in the language of wallace @xcite ) corrections to the born - huang approximation were written early on @xcite .
these formal expressions have been used almost exclusively as a tool to calculate elastic constants in computer simulations , but were given little attention in light of their basic importance .
we believe that this arose from two limitations : ( i ) prior works remained at an essentially technical level , aiming merely to provide tools for numerical simulations ( ii ) the derivations of formal analytical expressions for elastic constants have always relied on simplifying assumptions either dealing with an infinite system or valid at zero stress .
these simplifications make it difficult to ascertain the domain of validity of various formulae or symmetries . in response to these issues , we wish to attack this problem from two opposite angles : provide an even more systematic and general treatment than before , yet relate this formalism to numerical observations similar to tanguy _
et al_. @xcite we thus hope that our treatment could serve , at least , as an introduction to the subject . from the formalism , we wish to extract information about the contribution of various scales to the non - affine corrections to elasticity : the question we have at heart is whether these corrections originate from small or large scales , or involve a broad distribution of scales as suggested by the observation of vortex - like patterns .
@xcite this question is directly related to the existence of a continuous limit for elastic properties of amorphous structures .
@xcite to address these questions , we perform a normal mode decomposition of the non - affine displacement field : it permits quantifying the contribution of every frequency shell to the non - affine corrections to elastic constants . in the large size limit
, these contributions seem to be self - averaging quantities ( in the sense that an ensemble average over subsystems will produce results which are equivalent to a single large system ) . the existence of this self - averaging property leads to an expression for the elastic constants in the continuum limit which resembles the sum rules of liquid theory .
finally , our attention was attracted by several related issues in the recent literature .
studies of sound propagation and attenuation in granular materials @xcite or of the visco - elastic response of dense emulsions and foams , @xcite emphasize the need for a deep theoretical understanding of the visco - elastic response of disordered solids .
related experiments by mckenna and coworkers indicated that features of the visco - elastic response of amorphous materials are related to measurable changes in their elastic constants .
@xcite we thus complement the study of static response by a study of dynamical response , and derive a formal expression for the visco - elastic moduli .
we shall establish that the relaxation spectrum is directly and simply related to a correlator emerging from the normal mode analysis .
the present work is meant to present the general framework of our analysis , which will be the basis of future numerical studies .
although , here , we will use numerical simulations to illustrate analytical developments , the main core of our numerical study will be presented in a dedicated article .
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a frequently used method for edge - preserving image denoising is the variational approach which minimizes the rudin - osher - fatemi ( rof ) functional @xcite . in a discrete ( penalized ) form the rof functional can be written as @xmath2 where @xmath3 is the given corrupted image and @xmath4 denotes the discrete gradient operator which contains usually first order forward differences in vertical and horizontal directions . the regularizing term @xmath5 can be considered as discrete version of the total variation ( tv ) functional . since the gradient does not penalize constant areas the minimizer of the rof functional tends to have such regions , an effect known as staircasing . an approach to avoid this effect consists in the employment of higher order differences / derivatives . since the pioneering work @xcite which couples the tv term with higher order terms by infimal convolution various techniques with higher order differences / derivatives were proposed in the literature , among them @xcite . in various applications in image processing and computer vision the functions of interest take values on the circle @xmath0 or another manifold . processing manifold - valued data has gained a lot of interest in recent years . examples are wavelet - type multiscale transforms for manifold data @xcite and manifold - valued partial differential equations @xcite . finally we like to mention statistical issues on riemannian manifolds @xcite and in particular the statistics of circular data @xcite . the tv notation for functions with values on a manifold has been studied in @xcite using the theory of cartesian currents . these papers were an extension of the previous work @xcite were the authors focus on @xmath0-valued functions and show in particular the existence of minimizers of certain energies in the space of functions with bounded total cyclic variation . the first work which applies a cyclic tv approach among other models for imaging tasks was recently published by cremers and strekalovskiy in @xcite . the authors unwrapped the function values to the real axis and proposed an algorithmic solution to account for the periodicity . an algorithm which solves tv regularized minimization problems on riemannian manifolds was proposed by lellmann et al . in they reformulate the problem as a multilabel optimization problem with an infinite number of labels and approximate the resulting hard optimization problem using convex relaxation techniques . the algorithm was applied for chromaticity - brightness denoising , denoising of rotation data and processing of normal fields for visualization . another approach to tv minimization for manifold - valued data via cyclic and parallel proximal point algorithms was proposed by one of the authors and his colleagues in @xcite . it does not require any labeling or relaxation techniques . the authors apply their algorithm in particular for diffusion tensor imaging and interferometric sar imaging . for cartan - hadamard manifolds convergence of the algorithm was shown based on a recent result of bak @xcite . unfortunately , one of the simplest manifolds that is not of cartan - hadamard type is the circle @xmath0 . in this paper we deal with the incorporation of higher order differences into the energy functionals to improve denoising results for @xmath0-valued data . note that the ( second - order ) total generalized variation was generalized for tensor fields in @xcite . however , to the best of our knowledge this is the first paper which defines second order differences of cyclic data and uses them in regularization terms of energy functionals for image restoration . we focus on a discrete setting . first we provide a meaningful definition of higher order differences for cyclic data which we call _ absolute cyclic differences_. in particular our absolute cyclic first order differences resemble the geodesic distance ( arc length distance ) on the circle . as the geodesics the absolute cyclic second order differences take only values in @xmath1 $ ] . this is not necessary the case for differences of order larger than two . following the idea in @xcite we suggest a cyclic proximal point algorithm to minimize the resulting functionals . this algorithm requires the evaluation of certain proximal mappings . we provide analytical expression for these mappings . further , we suggest an appropriate choice of the cycles such that the whole algorithm becomes very efficient . we apply our algorithm to artificial data as well as to real - world interferometric sar data . the paper is organized as follows : in section [ sec : diff ] we propose a definition of differences on @xmath0 . then , in section [ sec : prox ] , we provide analytical expressions for the proximal mappings required in our cyclic proximal point algorithm . the approach is based on unwrapping the circle to @xmath6 and considering the corresponding proximal mappings on the euclidean space . the cyclic proximal point algorithm is presented in section [ sec : cpp ] . in particular we describe a vectorization strategy which makes the matlab implementation efficient and provides parallelizability , and prove its convergence under certain assumptions . section [ sec : numerics ] demonstrates the advantageous performance of our algorithm by numerical examples . finally , conclusions and directions of future work are given in section [ sec : conclusions ] . let @xmath0 be the unit circle in the plane @xmath7 endowed with the _ geodesic distance _ ( arc length distance ) @xmath8 given a base point @xmath9 , the _ exponential map _ @xmath10 from the tangent space @xmath11 of @xmath0 at @xmath12 onto @xmath0 is defined by @xmath13 this map is @xmath14-periodic , i.e. , @xmath15 for any @xmath16 , where @xmath17 denotes the unique point in @xmath18 such that @xmath19 , @xmath20 . some useful properties of the mapping @xmath21 ( which can also be considered as mapping from @xmath6 onto @xmath22 ) are collected in the following remark . [ lem : mod_pi ] the following relations hold true : 1 . @xmath23 for all @xmath24 . 2 . if @xmath25 then @xmath26 for all @xmath27 . while i ) follows by straightforward computation relation ii ) can be seen as follows : for @xmath28 there exists @xmath29 such that @xmath30 hence it follows @xmath31 and since @xmath32 further @xmath33 to guarantee the injectivity of the exponential map , we restrict its domain of definition from @xmath6 to @xmath18 . thus , for @xmath34 , there is now a unique @xmath35 satisfying @xmath36 . in particular we have @xmath37 . given such representation system @xmath38 of @xmath39 , @xmath40 centered at an arbitrary point @xmath12 on @xmath0 the geodesic distance becomes @xmath41 actually we need only @xmath42 in the minimum . clearly , this definition does not depend on the chosen center point @xmath12 . we want to determine general finite differences of @xmath0-valued data . let @xmath43 with @xmath44 where @xmath45 denotes the vector with @xmath46 components one . we define the _ finite difference operator _ @xmath47 by @xmath48 by , we see that @xmath49 vanishes for constant vectors and is therefore translation invariant , i.e. , @xmath50 [ diff_n ] for the binomial coefficients with alternating signs @xmath51 we obtain the ( forward ) differences of order @xmath46 : @xmath52 note that @xmath53 does not only fulfill , but vanishes exactly for all ` discrete polynomials of order @xmath54 ' , i.e. , for all vectors from @xmath55 . here we are interested in first and second order differences @xmath56 moreover , we will apply the ` mixed second order ' difference with @xmath57 and use the notation @xmath58 we want to define differences for points @xmath59 using their representation @xmath60 with respect to an arbitrary fixed center point . as the geodesic distance these differences should be independent of the choice of the center point . this can be achieved if and only if the differences are shift invariant modulo @xmath14 . let @xmath61 . we define the _ absolute cyclic difference _ of @xmath62 ( resp . @xmath63 ) with respect to @xmath64 by @xmath65_{2 \pi } ; w \big ) \bigr\rvert = \min_{j \in { \mathbb{i}}_d } \bigl\lvert \delta \big ( [ x - ( x_j+\pi ) 1_d]_{2 \pi } ; w \big ) \bigr\rvert,\ ] ] where @xmath66_{2\pi}$ ] denotes the component - by - component application of @xmath67 if @xmath68 , @xmath69 and @xmath70_{2\pi } = \pm \pi$ ] , @xmath69 . the definition allows that points having the same value are treated separately , cf . figure [ fig : choosing_sides ] . this ensures that @xmath71 is a continuous map . for example we have @xmath72 . figures [ fig : def - d ] and [ fig : choosing_sides ] illustrate definition . for the absolute cyclic differences related to the differences in example [ diff_n ] we will use the simpler notation @xmath73 .8 , @xmath74 , on the circle ( blue ) and their inverse exponential maps at @xmath75 , @xmath74 , ( dark blue ) , where @xmath75 denotes the antipodal point of @xmath76 . in other words , we cut the circle at the point @xmath76 and unwind it with respect to the tangent line at the antipodal point @xmath75 . the absolute cyclic differences take the three pairwise different positions of the points @xmath77 , @xmath74 to each other into account . these are shown in with respect to the representation system from the arbitrary point @xmath12 in.,title="fig : " ] .8 , @xmath74 , on the circle ( blue ) and their inverse exponential maps at @xmath75 , @xmath74 , ( dark blue ) , where @xmath75 denotes the antipodal point of @xmath76 . in other words , we cut the circle at the point @xmath76 and unwind it with respect to the tangent line at the antipodal point @xmath75 . the absolute cyclic differences take the three pairwise different positions of the points @xmath77 , @xmath74 to each other into account . these are shown in with respect to the representation system from the arbitrary point @xmath12 in.,title="fig : " ] .8 , @xmath74 , on the circle ( blue ) and their inverse exponential maps at @xmath75 , @xmath74 , ( dark blue ) , where @xmath75 denotes the antipodal point of @xmath76 . in other words , we cut the circle at the point @xmath76 and unwind it with respect to the tangent line at the antipodal point @xmath75 . the absolute cyclic differences take the three pairwise different positions of the points @xmath77 , @xmath74 to each other into account . these are shown in with respect to the representation system from the arbitrary point @xmath12 in.,title="fig : " ] , @xmath74 on the circle , where @xmath78 and @xmath79 , @xmath80 . though @xmath81 denote the same point on the circle they are treated separately in the definition of the absolute cyclic differences . ] the following equivalent definition of absolute cyclic differences appears to be useful . [ cyc_diff_perm ] let @xmath62 be sorted in ascending order as @xmath82 and set @xmath83 . let @xmath84 denote the corresponding permutation matrix , i.e. , @xmath85 and @xmath86 consider the @xmath14 shifted versions of @xmath87 given by @xmath88 where @xmath89 denotes the @xmath90-th unit vector . then it holds @xmath91 the first equality in follows directly by definition . to see the second one , note that by linearity of the inner product we have @xmath92 for the geodesic distance we obtain by that @xmath93 . in general the relation @xmath94 does not hold true as the following example shows . [ conterex ] in general the @xmath46-th order absolute cyclic difference can not be written as @xmath95 consider for example the absolute cyclic third order difference for @xmath96 given by as @xmath97 we obtain @xmath98 so that @xmath99 . for @xmath100 relation holds true by the next lemma . [ diff_sec ] for @xmath100 the following relation holds true : @xmath101 note that we need only the minimum over @xmath102 in proposition [ diff_sec ] and more precisely @xmath103 , \\ \lvert\delta(x;w ) - 2\pi \sigma\rvert = 2\pi - \lvert\delta(x;w)\rvert & \mbox{if } \ ; \;\lvert\delta(x;w)\rvert \in ( \pi,2 \pi],\\ \lvert\delta(x;w)\rvert - 2\pi & \mbox{if } \lvert\delta(x;w)\rvert \in ( 2\pi,3 \pi],\\ \lvert\delta(x;w ) - 4\pi \sigma\rvert = 4\pi - \lvert\delta(x;w)\rvert&\mbox{if } \ ; \ ; \lvert\delta(x;w)\rvert \in ( 3\pi,4 \pi ) , \end{cases}\ ] ] where @xmath104 and @xmath105 since @xmath106 for @xmath107 , we see that @xmath108 and @xmath109 . first we consider @xmath110 . by lemma [ cyc_diff_perm ] we obtain @xmath111 where we can assume by the cyclic shift invariance of @xmath110 that @xmath112 . if @xmath113 , then the corresponding permutation matrix @xmath84 in lemma [ cyc_diff_perm ] is the identity matrix . further we obtain that @xmath114 and by we get @xmath115 if @xmath116 , then @xmath117 and @xmath118 $ ] . in this case we get @xmath119 this proves the first assertion . for @xmath120 we can again assume that @xmath112 . exploiting that @xmath121 we have to consider the following three cases : if @xmath122 , then @xmath84 is the identity matrix , @xmath123 and @xmath124 if @xmath125 , then @xmath126 and @xmath127 . by we have @xmath128 if @xmath129 , then @xmath130 and @xmath131 . here we obtain @xmath132 this finishes the proof . for a proper , closed , convex function @xmath133 $ ] and @xmath134 the _ proximal mapping _ @xmath135 is defined by @xmath136 see @xcite . the above minimizer exits and is uniquely determined . many algorithms which were recently used in variational image processing reduce to the iterative computation of values of proximal mappings . an overview of applications of proximal mappings is given in @xcite . in this section , we are interested in proximal mappings of absolute cyclic differences @xmath137 , i.e. , @xmath138 for @xmath139 . more precisely , we will determine for @xmath140-valued vectors represented by @xmath141 the values @xmath142 for @xmath143 and first and second order absolute cyclic differences @xmath144 , @xmath145 . here @xmath146 means that we are looking for the representative of @xmath147 in @xmath148 . in particular , we will see that these proximal mapping are single - valued for @xmath141 with @xmath149 and have two values for @xmath150 . we start by considering the proximal mappings of the appropriate differences in @xmath151 . then we use the results to find the proximal functions of the absolute cyclic differences . first we give analytical expressions for @xmath152 , where @xmath153 and @xmath154 , @xmath155 . since we could not find a corresponding reference in the literature , the computation of the minimizer of @xmath156 is described in the following lemmas . we start with @xmath157 . [ lem : proxy_r ] for given @xmath158 and @xmath159 , @xmath155 set @xmath160 then the minimizer @xmath161 of @xmath162 is given by @xmath163 and the minimum by @xmath164 since @xmath165 , there exists a component @xmath166 and we rewrite @xmath167 substituting @xmath168 , @xmath169 and @xmath170 , @xmath171 we see that @xmath172 , where @xmath173 is the minimizer of @xmath174 the ( fenchel ) dual problem of @xmath175 reads @xmath176 and the relation between the minimizers of the primal and dual problems is given by @xmath177 rewriting we see that @xmath178 is the minimizer of @xmath179 where @xmath180 . hence we obtain @xmath181 and by further @xmath182 substituting back results in and plugging @xmath161 into @xmath183 we get . [ ex1 ] let @xmath157 , @xmath184 , and @xmath185 . 1 . for @xmath186 and @xmath187 we get @xmath188 and @xmath189 so that the minimizer of @xmath190 follows by _ soft shrinkage _ of @xmath191 with threshold @xmath192 : @xmath193 2 . for @xmath194 and @xmath195 we obtain @xmath196 and @xmath197 . consequently , the minimizer of @xmath198 is given by @xmath199 3 . for @xmath200 and @xmath201 we obtain @xmath202 and @xmath203 , so that the minimizer of @xmath204 is given by @xmath205 we will apply the following corollary . [ diff_data ] let @xmath159 . further , let @xmath206 and @xmath207 be given such that @xmath208 . then @xmath209 set @xmath210 and @xmath211 . by assumption @xmath212 and according to we have to consider three cases . 1 . let @xmath213 . then by assumption also @xmath214 and we conclude by that @xmath215 2 . let @xmath216 and @xmath217 . by this implies @xmath218 since @xmath219 and @xmath217 we obtain @xmath220 . 3 . let @xmath216 and @xmath221 . by this implies @xmath222 and we are done . next we consider the case @xmath223 . [ lem : e_quad ] let @xmath224 . 1 . then , for @xmath225 and @xmath226 , the minimizer @xmath227 of @xmath228 is given by @xmath229 and the minimum by @xmath230 2 . if @xmath231 for some @xmath232 and @xmath233 , then @xmath234 1 . setting the gradient of to zero results in @xmath235 using the sherman - morrison formula @xcite it follows @xmath236 for the corresponding energy we obtain by straightforward computation @xmath237^{2}\\ & = \frac{\lambda } { 1+\lambda { \left\lvert w \right\rvert_{2}}^{2}}\bigl(\langle f , w \rangle -a\bigr)^{2 } \text{.}\end{aligned}\ ] ] 2 . follows directly from . now we turn to @xmath0-valued data represented by @xmath141 . we are interested in the minimizers of @xmath238 on @xmath239 for @xmath153 and @xmath240 . we start with the case @xmath157 . [ lem : proxy_b1 ] for @xmath145 set @xmath241 . let @xmath157 and @xmath141 , where @xmath242 is adapted to the respective length of @xmath64 . 1 . if @xmath149 , then the unique minimizer of @xmath243 is given by @xmath244 2 . [ proxy - casepi ] if @xmath150 , then @xmath243 has the two minimizers @xmath245 note that for @xmath246 case [ proxy - casepi ] ) appears exactly if @xmath247 and @xmath248 are antipodal points . by and lemma [ diff_sec ] we can rewrite @xmath249 in as @xmath250 where @xmath251 . let @xmath252 we are looking for @xmath253^d } e_{k,\sigma}(x),\ ] ] where the last equality can be seen by the following argument : if for some @xmath254 the minimizer @xmath255^d } e_{k,\sigma}(x)$ ] has components @xmath256 for @xmath257 , then we get using @xmath258 , that @xmath259 by lemma [ lem : proxy_r ] the minimizers over @xmath151 of @xmath260 are given by @xmath261 where @xmath262 by corollary [ diff_data ] the minimum of @xmath263 is determined by @xmath264 . note that @xmath265 for @xmath246 and @xmath266 for @xmath267 . we distinguish two cases . 1 . if @xmath268 , @xmath269 then @xmath270 attains its smallest value exactly for @xmath271 and @xmath272 by we obtain @xmath273 with @xmath274 as in . corollary [ diff_data ] implies that @xmath275^d } e_{k,\sigma } ( x)\qquad \forall \sigma \in { \mathbb{z}}\backslash \{r - \langle k , w \rangle \}.\ ] ] finally , there exists exactly one @xmath276 such that @xmath277 and by we conclude that @xmath278 is the unique minimizer of @xmath243 over @xmath239 . 2 . if @xmath279 , @xmath269 , then @xmath270 attains its smallest value exactly for @xmath280 and by corollary [ diff_data ] the minimum of the corresponding functions @xmath263 is smaller than those of the other functions in . we obtain @xmath281 and @xmath282 as in part 1 of the proof we conclude that @xmath283 are the minimizers of @xmath243 over @xmath239 . this finishes the proof . next we focus on @xmath284 . [ thm : p2 ] let @xmath223 in , @xmath145 and @xmath141 , where @xmath242 is adapted to the respective length of @xmath64 . 1 . if @xmath149 , then the unique minimizer of @xmath243 is given by @xmath285 2 . if @xmath150 , then @xmath243 has the two minimizers @xmath286 the proof follows the lines of the proof of theorem [ lem : proxy_b1 ] using lemma [ lem : e_quad ] . finally , we need the proximal mapping @xmath287 for given @xmath288 . the proximal mapping of the ( squared ) cyclic distance function was also computed ( for more general manifolds ) in @xcite . here we give an explicit expression for spherical data . [ theo : prox_quad ] for @xmath289 let @xmath290 then the minimizer(s ) of @xmath291 are given by @xmath292 where @xmath293 is defined by @xmath294 and the minimum is @xmath295 obviously , the minimization of @xmath249 can be done component wise so that we can restrict our attention to @xmath296 . 1 . first we look at the minimization problem over @xmath6 which reads @xmath297 and has the following minimizer and minimum : @xmath298 2 . for the original problem @xmath299 we consider the related energy functionals on @xmath6 , namely @xmath300 by part 1 of the proof these functions have the minimizers @xmath301 and @xmath302 we distinguish three cases : 1 . if @xmath303 , then the minimum in occurs exactly for @xmath304 and it holds @xmath305 for @xmath306 we see that @xmath307 and @xmath308 2 . if @xmath309 , then has its minimum exactly for @xmath310 and @xmath311 which is in @xmath18 for @xmath306 or @xmath312 and @xmath313 3 . in the case @xmath314 the minimum in is attained for @xmath315 so that we have both solutions from i ) and ii ) . this completes the proof . the proximal point algorithm ( ppa ) on the euclidean space goes back to @xcite . recently this algorithm was extended to riemannian manifolds of non - positive sectional curvature @xcite and also to hadamard spaces @xcite . a cyclic version of the proximal point algorithm ( cppa ) on the euclidean space was given in @xcite , see also the survey @xcite . a cppa for hadamard spaces can be found in @xcite . in the cppa the original function @xmath316 is split into a sum @xmath317 and , iteratively , the proximal mappings of the functions @xmath318 are applied in a cyclic way . the great advantage of this method is that often the proximal mappings of the summands @xmath318 are much easier to compute or can even be given in a closed form . in the following we develop a cppa for functionals of @xmath0-valued signals and images containing absolute cyclic first and second order differences . first we have a look at the one - dimensional case , i.e. , at signals . for given @xmath0-valued signals represented by @xmath319 , @xmath320 , and regularization parameters @xmath321 , @xmath322 , we are interested in @xmath323 where @xmath324 to apply a cppa we set @xmath325 , split @xmath326 into an even and an odd part @xmath327 and @xmath328 into three sums @xmath329 then the objective function decomposes as @xmath330 we compute in the @xmath331-th cycle of the cppa the signal @xmath332 the different proximal values can be obtained as follows : 1 . by proposition [ theo : prox_quad ] with @xmath333 playing the role of @xmath334 we get @xmath335 2 . for @xmath336 , we obtain the vectors @xmath337 by applying theorem [ lem : proxy_b1 ] with @xmath246 independently for the pairs @xmath338 , @xmath339 . 3 . for @xmath340 , we compute @xmath341 by applying theorem [ lem : proxy_b1 ] with @xmath342 independently for the vectors @xmath343 , @xmath344 . the parameter sequence @xmath345 of the algorithm should fulfill @xmath346 this property is also essential for proving the convergence of the cppa for real - valued data and data on a hadamard manifold , see @xcite . in our numerical experiments we choose @xmath347 with some initial parameter @xmath348 which clearly fulfill . the whole procedure is summarized in algorithm [ alg : cppa ] . * input * @xmath349 fulfilling and @xmath350 , @xmath351 or @xmath352 , @xmath353 , @xmath354 data @xmath355 or @xmath356 + initialize @xmath357 , @xmath306 initialize the cycle length as @xmath358 ( 1d ) or @xmath359 ( 2d ) @xmath360 @xmath361 a convergence criterion are reached @xmath362 next we consider two - dimensional data , i.e. , images of the form @xmath363 , @xmath364 . our functional includes _ _ h__orizontal and _ _ v__ertical cyclic first and second order differences @xmath365 and @xmath110 and _ _ d__iagonal ( mixed ) differences @xmath120 . for non - negative regularization parameters @xmath366 , @xmath367 and @xmath354 not all equal to zero we are looking for @xmath368 where @xmath369 here the objective function splits as @xmath370 with the following summands : again we set @xmath371 and compute the proximal value of @xmath372 by proposition [ theo : prox_quad ] . each of the sums in @xmath373 and @xmath374 can be split analogously as in the one - dimensional case , where we have to consider row and column vectors now . this results in @xmath375 functions @xmath376 whose proximal values can be computed by theorem [ lem : proxy_b1 ] . finally , we split @xmath377 . into the four sums @xmath378 and denote the inner sums by @xmath379 . clearly , the proximal values of the functions @xmath380 , @xmath381 can be computed separately for the vectors @xmath382 by theorem [ lem : proxy_b1 ] with @xmath383 . in summary , the computation can be done by algorithm [ alg : cppa ] . note that the presented approach immediately generalizes to arbitrary dimensions . since @xmath0 is not a hadamard space , the convergence analysis of the cppa in @xcite can not be applied . we show the convergence of the cppa for the 2d @xmath0-valued function under certain conditions . the 1d setting in can then be considered as a special case . in the following , let @xmath384 . our first condition is that the data @xmath385 is dense enough , this means that the distance between neighboring pixels @xmath386 is sufficiently small . similar conditions also appear in the convergence analysis of nonlinear subdivision schemes for manifold - valued data in @xcite . in the context of nonlinear subdivision schemes , even more severe restrictions such as ` almost equally spaced data ' are frequently required @xcite . this imposes additional conditions on the second order differences to make the data almost lie on a ` line ' . our analysis requires only bounds on the first , but not on the second order differences . our next requirement is that the regularization parameters @xmath387 in are sufficiently small . for large parameters any solution tends to become almost constant . in this case , if the data is for example equidistantly distributed on the circle , e.g. , @xmath388 in 1d , any @xmath389 shift is again a solution . in this situation the model loses its interpretation which is an inherent problem due to the cyclic structure of the data . finally , the parameter sequence @xmath345 of the cppa has to fulfill with a small @xmath390 norm . the later can be achieved by rescaling . our convergence analysis is based on a convergence result in @xcite and an unwrapping procedure . we start by reformulating the convergence result for the cppa of real - valued data , which is a special case of @xcite and can also be derived from @xcite . [ thm : bacak ] let @xmath391 , where @xmath392 , @xmath393 , are proper , closed , convex functionals on @xmath394 . let @xmath183 have a global minimizer . assume that there exists @xmath395 such that the iterates @xmath396 of the cppa ( see algorithm [ alg : cppa ] ) satisfy @xmath397 for all @xmath398 . then the sequence @xmath399 converges to a minimizer of @xmath183 . moreover the iterates fulfill @xmath400 + 2 \lambda_k^2 l^2 c(c+1 ) \quad \mbox{for all } \ ; x \in \mathbb r^{n \times m}.\label{noch_1 } \end{aligned}\ ] ] the next lemma states a discrete analogue of a well - known result on unwrapping or lifting from algebraic topology . we supply a short proof since we did not found it in the literature . [ lem : discretecovering ] let @xmath401 with @xmath402 . for @xmath9 not antipodal to @xmath403 fix an @xmath404 such that @xmath405 . then there exists a unique @xmath406 such that for all @xmath407 the following relations are fulfilled : 1 . @xmath408 2 . @xmath409 . we call @xmath410 the lifted or unwrapped image of @xmath411 ( w.r.t . a fixed @xmath412 ) . for @xmath413 , @xmath414 , it holds by assumption on @xmath415 that @xmath416 . hence we have @xmath417 , where with an abuse of notation @xmath413 stands for an arbitrary representative in @xmath418 of @xmath413 . then obviously @xmath419 , @xmath420 are the unique values satisfying i ) and ii ) . for @xmath421 consider @xmath422 by assumption on @xmath415 we see that @xmath423 so that @xmath424 . thus @xmath425 is the unique value with properties i ) and ii ) . proceeding this scheme successively , we obtain the whole unique image @xmath426 fulfilling i ) and ii ) . for @xmath427 we define @xmath428 where @xmath429 to measure how ` near ' the images @xmath191 and @xmath411 are to each other . [ lem : convlem2 ] let @xmath430 with @xmath431 and @xmath9 be not antipodal to @xmath432 . fix @xmath433 with @xmath434 and let @xmath435 be the corresponding lifting of @xmath191 . let @xmath436 $ ] . 1 . then every @xmath437 has a unique lifting @xmath410 w.r.t . to the base point @xmath12 with @xmath438 . 2 . for @xmath439 defined by , let @xmath440 denote its analog for real - valued data , i.e. , @xmath441 where the cyclic distances in @xmath442 and in the @xmath443 terms are replaced by absolute differences in @xmath444 and @xmath445 . then it holds @xmath446 by definition of @xmath447 and assumption on @xmath191 we have for any @xmath437 that @xmath448 and hence @xmath449 . further it holds @xmath450 . consequently , every @xmath437 has a unique lifting @xmath410 by lemma [ lem : discretecovering ] w.r.t . to the base point @xmath12 fulfilling @xmath438 . to see we show the equality for the involved summands in @xmath316 and @xmath440 separately . first we consider @xmath373 . by properties of the lifting in lemma [ lem : discretecovering ] we have @xmath451 and @xmath452 . by the definition of @xmath373 and @xmath453 , this implies @xmath454 . next we consider @xmath455 . the corresponding second order differences are given by the expressions @xmath456 and @xmath457 , respectively . we exemplarily consider the first term . since @xmath458 the distance between any two members of the triple is smaller than @xmath459 . due to the properties of the lifting @xmath426 this implies @xmath460 . then we conclude by proposition [ diff_sec ] that @xmath461 . similarly it follows that @xmath462 . concerning the data term @xmath463 we consider @xmath464 and @xmath465 . by definition of @xmath447 we have @xmath466 and by construction of @xmath435 and @xmath426 that @xmath467 , @xmath468 and @xmath469 . furthermore it holds @xmath470 . if @xmath471 , then there exists @xmath472 such that @xmath473 which is a contradiction . thus @xmath474 . similarly we conclude @xmath475 . in summary we obtain @xmath476 for all @xmath477 which implies @xmath478 . this finishes the proof . [ rem : conv ] the set @xmath447 is a convex subset of @xmath479 which means that for @xmath480 and @xmath481 $ ] we have @xmath482_t \in { \mathcal s}(f,\delta)$ ] . here @xmath482_t$ ] denotes the point reached after time t on the unit speed geodesic starting at @xmath411 in direction of @xmath483 . recall that a function @xmath484 is convex on @xmath447 if for all @xmath480 and all @xmath485 $ ] the relation @xmath486_t ) \le t\varphi(x ) + ( 1-t ) \varphi(y)$ ] holds true . let @xmath430 with @xmath431 and @xmath436 $ ] . then we conclude by lemma [ lem : convlem2 ] , since @xmath440 is convex , that @xmath439 is convex on @xmath487 . [ lem : convlem1 ] let @xmath488 and @xmath489 . let @xmath490 such that @xmath491 then any minimizer @xmath492 of @xmath316 in fulfills @xmath493 any minimizer @xmath494 of satisfies @xmath495 as a consequence we obtain @xmath496 [ rem : tildej ] lemma [ lem : convlem1 ] holds also true for real - valued data and @xmath440 in . now we combine lemma [ lem : convlem1 ] and [ lem : convlem2 ] to locate the minimizers of @xmath316 and @xmath440 . [ lem : convlem3 ] let @xmath497 with @xmath498 and @xmath499 be given . choose the parameters @xmath500 of @xmath316 in such that with @xmath501 holds true . then any minimizer @xmath492 of @xmath316 lies in @xmath487 . furthermore , if @xmath502 is the unique lifting of @xmath191 w.r.t . a base point @xmath12 and fixed @xmath503 with @xmath504 , then each minimizer @xmath505 of @xmath506 defines a minimizer @xmath507 of @xmath316 . conversely , the uniquely defined lifting @xmath508 of a minimizer @xmath492 of @xmath316 is a minimizer of @xmath506 . by lemma [ lem : convlem1 ] we obtain @xmath509 so that @xmath510 . in order to show the second statement note that the mapping @xmath511 is a bijection from @xmath487 to the set @xmath512 defined by @xmath513.\ ] ] if @xmath505 minimizes @xmath506 , then it lies in @xmath514 which follows by remark [ rem : tildej ] . by and the minimizing property of @xmath505 we obtain for any @xmath437 that @xmath515 as a consequence , @xmath516 is a minimizer of @xmath316 on @xmath447 . by lemma [ lem : convlem1 ] all the minimizers of @xmath316 are contained in @xmath447 so that @xmath516 is a minimizer of @xmath316 on @xmath479 . we proceed with the last statement . let @xmath492 be a minimizer of @xmath316 with lifting @xmath508 . then we get for any @xmath517 that @xmath518 this shows that @xmath508 is a minimizer of @xmath506 on @xmath514 . since by remark [ rem : tildej ] all minimizers of @xmath506 lie in @xmath514 , the last assertion follows . next we locate the iterates of the cppa for real - valued data on a ball whose radius can be controlled . [ lem : convlem4 ] for @xmath519 and @xmath520 with property , let @xmath521 be the sequence produced by algorithm [ alg : cppa ] for @xmath506 . assume that @xmath522 . let @xmath523 be the minimizer of @xmath506 . then , for @xmath524 and @xmath525 , it holds @xmath526 where @xmath359 denotes the number of inner iterations and @xmath527 . the assumption on the distances @xmath528 , @xmath407 , to be smaller than @xmath529 is automatically fulfilled for any unwrapping of @xmath0-valued data . by theorem [ thm : bacak ] we know that @xmath530 + 2 \lambda_k^2 l^2 c(c+1).\ ] ] as a constant @xmath531 we can choose the maximum of the lipschitz constants of the involved summands . for @xmath453 , @xmath532 and @xmath533 the lipschitz constants are @xmath534 , @xmath535 , and @xmath535 , respectively . for the quadratic data term we have @xmath536 therefore , we can set @xmath537 . plugging in the minimizer @xmath538 into and using @xmath357 yields @xmath539 by theorem [ thm : bacak ] it holds @xmath540 using the triangle inequality we obtain @xmath541 which implies the assertion by and . now we compare the proximal mappings acting on data with values in @xmath542 and @xmath543 [ lem : convlem5 ] for @xmath544 with @xmath498 , let @xmath316 be defined by with the splitting . let @xmath502 be the unique lifting of @xmath191 w.r.t . a base point @xmath12 not antipodal to @xmath432 and fixed @xmath503 with @xmath504 . further , denote by @xmath440 the functional corresponding to @xmath316 . then , for any @xmath545 , @xmath546 $ ] and its lifting @xmath426 w.r.t . @xmath12 , we have @xmath547 i.e. , the canonical projection @xmath548 commutes with the proximal mappings . the function @xmath549 is based on the distance to the data @xmath191 . since @xmath545 , we have @xmath550 for all @xmath407 . the components of the proximal mapping @xmath551 are given by proposition [ theo : prox_quad ] from which we conclude for @xmath552 the proximal mappings of @xmath318 , @xmath553 , are given via proximal mappings of the first and second order cyclic differences . we consider the first order difference @xmath554 . by the triangle inequality , we have @xmath555 as well as @xmath556 . by the explicit form of the proximal mapping in theorem [ lem : proxy_b1 ] we obtain for @xmath318 , @xmath557 . next we consider the horizontal and vertical second order differences @xmath456 and @xmath457 . we have that @xmath558 , @xmath559 as well as @xmath560 . hence all contributing values of @xmath411 lie on a quarter of the circle . applying the proximal mapping in theorem [ lem : proxy_b1 ] the resulting data lie on one half of the circle . an analogous statement holds true for the horizontal part . hence the proximal mappings of the ordinary second differences agree with the cyclic version ( under identification via @xmath548 ) . this implies for @xmath318 , @xmath561 . finally , we consider the mixed second order differences @xmath562 . as above , we have for neighboring data items that the distance is smaller than @xmath563 . for all four contributing values of @xmath411 we have that the pairwise distance is smaller by @xmath459 . thus again they lie on a quarter of the circle . hence , the proximal mapping for the ordinary mixed second differences agree with the cyclic version ( under identification via @xmath548 ) . this implies for @xmath318 , @xmath381 . we note that lemma [ lem : convlem5 ] does not guarantee that @xmath564 remains in @xmath565 . therefore it does not allow for an iterated application . in the following main theorem we combine the preceding lemmas to establish this property . [ thm : convergence ] let @xmath544 with @xmath566 . let @xmath520 fulfill property and @xmath567 for some @xmath490 , where @xmath568 and @xmath527 . further , assume that the parameters @xmath500 of the functional @xmath316 in and @xmath501 satisfy . then the sequence @xmath399 generated by the cppa in algorithm [ alg : cppa ] converges to a global minimizer of @xmath316 . let @xmath502 be the lifting of of @xmath191 with respect to a base point @xmath12 not antipodal to @xmath432 and fixed @xmath433 with @xmath504 . further , let @xmath440 denote the real analog of @xmath316 . by lemma [ lem : convlem2 ] we have @xmath569 and @xmath570 for @xmath571 such that is also fulfilled for the real - valued setting . then we can apply remark [ lem : convlem1 ] and conclude that the minimizer @xmath505 of @xmath506 fulfills @xmath572 . by we obtain @xmath573 by lemma [ lem : convlem4 ] the iterates @xmath574 of the real - valued cppa fulfill @xmath575 hence @xmath576 which means that all iterates @xmath574 stay within @xmath577 . next , we consider the sequence @xmath578 of the cppa for the @xmath0-valued data @xmath191 . we use induction to verify @xmath579 . by definition we have @xmath580 . assume that @xmath581 . by bijectivity of the lifting , cf . lemma [ lem : convlem2 ] , and since @xmath582 , we conclude @xmath583 . by lemma [ lem : convlem5 ] we obtain @xmath584 by the same argument as above we have again @xmath585 . finally , we know by theorem [ thm : bacak ] that @xmath586 and by lemma [ lem : convlem3 ] that @xmath587 is a global minimizer of @xmath316 . this completes the proof . for the numerical computations of the following examples , the algorithms presented in section [ sec : cpp ] were implemented in matlab . the computations were performed on a macbook pro with an intel core i5 , 2.6ghz and 8 gb of ram using matlab 2013 , version 2013a ( 8.1.0.604 ) on mac os 10.9.2 . .48 ( dashed red ) and disturbed signal by wrapped gaussian noise @xmath588 ( solid black ) . reconstructed signals @xmath589 using only the @xmath590 regularizer ( @xmath591 ) , only the @xmath592 regularizer ( @xmath593 ) , and both of them ( @xmath594 , @xmath595 ) . while suffers from the staircasing effect , shows weak results at constant areas . the combination of both regularizers in yields the best image.,title="fig : " ] .48 ( dashed red ) and disturbed signal by wrapped gaussian noise @xmath588 ( solid black ) . reconstructed signals @xmath589 using only the @xmath590 regularizer ( @xmath591 ) , only the @xmath592 regularizer ( @xmath593 ) , and both of them ( @xmath594 , @xmath595 ) . while suffers from the staircasing effect , shows weak results at constant areas . the combination of both regularizers in yields the best image.,title="fig : " ] .48 ( dashed red ) and disturbed signal by wrapped gaussian noise @xmath588 ( solid black ) . reconstructed signals @xmath589 using only the @xmath590 regularizer ( @xmath591 ) , only the @xmath592 regularizer ( @xmath593 ) , and both of them ( @xmath594 , @xmath595 ) . while suffers from the staircasing effect , shows weak results at constant areas . the combination of both regularizers in yields the best image.,title="fig : " ] .48 ( dashed red ) and disturbed signal by wrapped gaussian noise @xmath588 ( solid black ) . reconstructed signals @xmath589 using only the @xmath590 regularizer ( @xmath591 ) , only the @xmath592 regularizer ( @xmath593 ) , and both of them ( @xmath594 , @xmath595 ) . while suffers from the staircasing effect , shows weak results at constant areas . the combination of both regularizers in yields the best image.,title="fig : " ] the first example of an artificial one - dimensional signal demonstrates the effect of different models containing absolute cyclic first order differences , second order differences or both combined . the function @xmath596 \to [ -\pi,\pi)$ ] given by @xmath597 is sampled equidistantly to obtain the original signal @xmath598 at @xmath599 samples . this function is distorted by wrapped gaussian noise @xmath600 of standard deviation @xmath601 to get @xmath602 , see also remark [ lem : mod_pi ] . the functions @xmath603 and @xmath588 are depicted in figure [ subfig : noisy ] . note the following effects due to the cyclic data representation on @xmath18 : the linear increase on @xmath604 $ ] of @xmath191 is continuous and the change from @xmath529 to @xmath605 at @xmath606 is just due to the chosen representation system . similarly the two constant parts with the values @xmath605 and @xmath607 differ only by a jump size of @xmath608 . for the noise around these two areas , we have the same situation . we apply algorithm [ alg : cppa ] with different model parameters @xmath350 and @xmath351 to @xmath588 which yields the restored signals @xmath589 . the restoration error is measured by the ` cyclic ' mean squared error ( cmse ) with respect to the arc length distance @xmath609 we use @xmath610 and @xmath611 iterations as stopping criterion . for any choice of parameters @xmath612 the computation time is about @xmath613 seconds . the result @xmath589 in figure [ subfig : tv ] is obtained using only the @xmath590 regularization ( @xmath614 ) . the restoration of constant areas is favored by this regularization term , but linear , quadratic and exponential parts suffer from the well - known ` staircasing ' effect . utilizing only the @xmath592 regularization ( @xmath615 ) , cf . figure [ subfig : tv2 ] , the restored function becomes worse in flat areas , but shows a better quality in the linear parts . by combining the regularization terms ( @xmath616 , @xmath617 ) as illustrated in figure [ subfig : tv12 ] both the linear and the constant parts are reconstructed quite well and the cmse is smaller than for the other choices of parameters . note that @xmath350 and @xmath351 were chosen in @xmath618 with respect to an optimal cmse . the complex - valued synthetic aperture radar ( sar ) data is obtained emitting specific radar signals at equidistant points and measuring the amplitude and phase of their reflections by the earth s surface . the amplitude provides information about the reflectivity of the surface . the phase encodes both the change of the elevation of the surface s elements within the measured area and their reflection properties and is therefore rather arbitrary . when taking two sar images of the target area at the same time but from different angles or locations . the phase difference of these images encodes the elevation , but it is restricted to one wavelength and also includes noise . the result is the so called interferometric synthetic aperture radar ( insar ) data and consists of the ` wrapped phase ' or the ` principal phase ' , a value in @xmath18 representing the surface elevation . for more details see , e.g. , @xcite . after a suitable preregistration the same approach can be applied to two images from the same area taken at different points in time to measure surface displacements , e.g. , before and after an earthquake or the movement of glaciers . the main challenge in order to unwrap the phase is the presence of noise . ideally , if the surface would be smooth enough and no noise would be present , unwrapping is uniquely determined , i.e. , differences between two pixels larger than @xmath529 are regarded as a wrapping result and hence become unwrapped . there are several algorithms to unwrap , even combining the denoising and the unwrapping , see for example @xcite . for denoising , deledalle et al . @xcite use both sar images and apply a non - local means algorithm jointly to their reflection , the interferometric phase and the coherence . .48 using only the @xmath373 regularizer ( @xmath619 ) , only the @xmath620 regularizer ( @xmath621 , @xmath622 ) , and both of them ( @xmath623 , @xmath624 , @xmath625 ) . , title="fig : " ] .48 using only the @xmath373 regularizer ( @xmath619 ) , only the @xmath620 regularizer ( @xmath621 , @xmath622 ) , and both of them ( @xmath623 , @xmath624 , @xmath625 ) . , title="fig : " ] .48 using only the @xmath373 regularizer ( @xmath619 ) , only the @xmath620 regularizer ( @xmath621 , @xmath622 ) , and both of them ( @xmath623 , @xmath624 , @xmath625 ) . , title="fig : " ] .48 using only the @xmath373 regularizer ( @xmath619 ) , only the @xmath620 regularizer ( @xmath621 , @xmath622 ) , and both of them ( @xmath623 , @xmath624 , @xmath625 ) . , title="fig : " ] .48 using only the @xmath373 regularizer ( @xmath619 ) , only the @xmath620 regularizer ( @xmath621 , @xmath622 ) , and both of them ( @xmath623 , @xmath624 , @xmath625 ) . , title="fig : " ] .48 using only the @xmath373 regularizer ( @xmath619 ) , only the @xmath620 regularizer ( @xmath621 , @xmath622 ) , and both of them ( @xmath623 , @xmath624 , @xmath625 ) . , title="fig : " ] [ [ application - to - synthetic - data . ] ] application to synthetic data . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in order to get a better understanding in the two - dimensional case , let us first take a look at a synthetic surface given on @xmath626 ^ 2 $ ] with the profile shown in figure [ subfig:2dorig ] . this surface consists of two plates of height @xmath627 divided at the diagonal , a set of stairs in the upper left corner in direction @xmath628 , a linear increasing area connecting both plateaus having the shape of an ellipse with major axis at the angle @xmath629 , and a half ellipsoid forming a dent in the lower right of the image with circular diameter of size @xmath630 and depth @xmath631 . the initial data is given by sampling the described surface at @xmath632 sampling points . the usual insar measurement would ideally result in data as given in figure [ subfig:2dwrapped ] , i.e. , the data is wrapped with respect to @xmath14 . in the figure the resulting ideal phase is represented using the hue component of the hsv color space . again , the data is perturbed by wrapped gaussian noise , standard deviation @xmath633 , see figure [ subfig:2dnoisy ] . for an application of algorithm [ alg : cppa ] to the minimization problem , we have to fix five parameters @xmath634 which were chosen on @xmath635 such that they minimize the cmse . using only the cyclic first order differences with @xmath636 , see figure [ subfig:2dtv1 ] , the reconstructed image @xmath637 reproduces the piecewise constant parts of the stairs in the upper left part and the background , but introduces a staircasing in both linear increasing areas inside the ellipse and in the half ellipsoid . this is highlighted in the three magnifications in figure [ subfig:2dtv1 ] . applying only cyclic second order differences with @xmath638 manages to reconstruct the linear increasing part and the circular structure of the ellipsoid , but compared to the first case it even increases the cmse due to the approximation of the stairs and the background , see especially the magnification of the stairs in figure [ subfig:2dtv2 ] . combining first and second order cyclic differences by setting @xmath639 and @xmath640 , @xmath625 , these disadvantages can be reduced , cf . figure [ subfig:2dtv12 ] . note especially the three magnified regions and the cmse . .48 , @xmath641 and @xmath642).,title="fig : " ] .48 , @xmath641 and @xmath642).,title="fig : " ] [ [ application - to - real - world - data . ] ] application to real - world data . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + next we examine a real - world example . the data from @xcite is a set of insar data recorded in 1991 by the ers-1 satellite capturing topographical information from the mount vesuvius . the data is available online and a part of it was also used as an example in @xcite for tv based denoising of manifold - valued data . in figure [ fig : vesuvius ] the phase is represented by the hue component of the hsv color space . we apply algorithm [ alg : cppa ] to the image of size @xmath643 , cf . figure [ subfig : vesuv - orig ] , with @xmath644 and @xmath645 . this reduces the noise while keeping all significant plateaus , ascents and descents , cf . figure [ subfig : vesuv - denoised ] . the left zoom illustrates how the plateau in the bottom left of the data is smoothened but kept in its main elevation shown in blue . in the zoom on the right all major parts except the noise are kept . we notice just a little smoothening due to the linearization introduced by @xmath646 . in the bottom left of this detail some of the fringes are eliminated , and a small plateau is build instead , shown in cyan . the computation time for the whole image using @xmath647 iterations as stopping criterion was 86.6 sec and 11.1 sec for each of the details of size @xmath648 . in this paper we considered functionals having regularizers with second order absolute cyclic differences for @xmath0-valued data . their definition required a proper notion of higher order differences of cyclic data generalizing the corresponding concept in euclidian spaces . we derived a cppa for the minimization of our functionals and gave the explicit expressions for the appearing proximal mappings . we proved convergence of the cppa under certain conditions . to the best of our knowledge this is the first algorithm dealing with higher order tv - type minimization for @xmath0-valued data . we demonstrated the denoising capabilities of our model on synthetic as well as on real - world data . future work includes the application of our higher order methods for cyclic data to other imaging tasks such as segmentation , inpainting or deblurring . for deblurring , the usually underlying linear convolution kernel has to be replaced by a nonlinear construction based on intrinsic ( also called karcher ) means . this leads to the task of solving the new associated inverse problem . further , we intend to investigate other couplings of first and second order derivatives similar to infimal convolutions or gtv for euclidean data . finally , we want to set up higher order tv - like methods for more general manifolds , e.g. higher dimensional spheres . here , we do not believe that it is possible to derive explicit expressions for the involved proximal mappings at least not for riemannian manifolds of nonzero sectional curvature . instead , we plan to resort to iterative techniques . d. p. bertsekas . incremental gradient , subgradient , and proximal methods for convex optimization : a survey . technical report lids - p-2848 , laboratory for information and decision systems , mit , cambridge , ma , 2010 . p. grohs , h. hardering , and o. sander . optimal a priori discretization error bounds for geodesic finite elements . technical report 2013 - 16 , seminar for applied mathematics , eth zrich , switzerland , 2013 .
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in many image and signal processing applications , as interferometric synthetic aperture radar ( sar ) or color image restoration in hsv or lch spaces the data has its range on the one - dimensional sphere @xmath0 .
although the minimization of total variation ( tv ) regularized functionals is among the most popular methods for edge - preserving image restoration such methods were only very recently applied to cyclic structures . however , as for euclidean data , tv regularized variational methods suffer from the so called staircasing effect .
this effect can be avoided by involving higher order derivatives into the functional .
this is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration .
we introduce absolute higher order differences for @xmath0-valued data in a sound way which is independent of the chosen representation system on the circle .
our absolute cyclic first order difference is just the geodesic distance between points .
similar to the geodesic distances the absolute cyclic second order differences have only values in @xmath1 $ ] .
we update the cyclic variational tv approach by our new cyclic second order differences . to minimize the corresponding functional
we apply a cyclic proximal point method which was recently successfully proposed for hadamard manifolds . choosing appropriate cycles
this algorithm can be implemented in an efficient way .
the main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions . under certain conditions
we prove the convergence of our algorithm .
various numerical examples with artificial as well as real - world data demonstrate the advantageous performance of our algorithm .
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due to the growing demand in data traffic , large improvements in the spectral efficiency are required @xcite . network densification has been identified as a possible way to achieve the desired spectral efficiency gains @xcite . this approach consists of deploying a large number of low powered base stations ( bss ) known as small cells . with the addition of small cell bss , the overall system is known as a heterogeneous cellular network ( hetnet ) . co - channel deployment of small cell bss results in high intercell interference if their operation is not coordinated @xcite . interference coordination techniques such as intercell interference coordination ( icic ) has been extensively studied for multi - tier hetnet scenarios @xcite . icic relies on orthogonalizing time and frequency resources allocated to the macrocell and the small cell users . orthogonalization in time is achieved by switching off the relevant subframes belonging to the macrocell thereby reducing inter - tier interference to the small cell bss @xcite . orthogonalization in frequency can be achieved with fractional frequency reuse where the users in the inner part of the cells are scheduled on the same frequency resources in contrast to the users at the cell edge whom are scheduled on available orthogonal resources . distributed and joint power control strategies for dominant interference supression in hetnets is discussed in @xcite . the performance of multiple antenna ( i.e. , mimo ) hetnets using the above mentioned techniques is analyzed in @xcite and @xcite . the effects of random orthogonal beamforming with maximum rate scheduling for mimo hetnets is studied in @xcite . the effects of imperfect channel state information ( csi ) with limited feedback mimo is investigated in @xcite for a two - tier hetnet . in addition to orthogonalization , interference coordination can also be achieved by means of transmit beamforming at the bss . however , there seems to be limited literature on transmit beamforming techniques to coordinate interference in hetnets @xcite . transmit beamforming techniques have been well explored in the multiuser ( mu ) mimo literature to mitigate or reduce the effects of intracell interference @xcite . performance superiority at low signal - to - noise - ratio ( snr ) of the leakage based beamforming technique compared to zero - forcing beamforming ( zfbf ) is shown in @xcite . with zfbf , complete mu intracell interference cancellation takes place if perfect csi is present at the bs and the number of transmit antennas exceeds the total number of receive antennas . however , leakage based beamforming focuses on maximizing the desired signal - to - leakage - noise - ratio ( slnr ) without any restrictions on the number of transmit antennas . the focus of this paper is on the performance gains of a two - tier hetnet with active interference coordination . intracell and intercell interference is coordinated by deploying leakage based beamformers at the macrocell and microcell bss . we summarize the contributions of this paper as follows : * we evaluate the performance gains of full coordination and macro - only coordination techniques relative to no coordination for two - tier hetnets . the impact of imperfect csi on the performance of these coordination techniques is also investigated . * we demonstrate the effect of network densification with varying degrees of bs coordination on the mean per - user signal - to - interference - plus - noise - ratio ( sinr ) and compare the simulated mean per - user sinr results with the analytical approximations over a wide range of snr . the mean per - user sinr decreases with an increasing microcell count . however , we show that coordination substantially reduces the rate of sinr decrease . * we show that in the absence of coordination , network densification does not provide any gain in the sum rate , whereas with coordination , a linear increase in the sum rate is observed . _ notation : _ we use the symbols @xmath0 and @xmath1 to denote a matrix and a vector , respectively . @xmath2 , @xmath3 , @xmath4 , denote the conjugate transpose , the inverse and the trace of the matrix @xmath0 , respectively . @xmath5 and @xmath6 stand for the vector and scalar norms , respectively . @xmath7 $ ] denotes the statistical expectation . in the macrocell coverage area . ] we consider downlink transmission in a two - tier hetnet comprising of a single macrocell bs and multiple microcell bss , as shown in fig . we consider a typical scenario where the mobile users in the coverage area of a particular cell are served by the corresponding bs . we assume that lossless and delayless backhaul links are present between each bs to exchange csi , if desired . we denote the total number of cells ( including the macrocell and all microcells ) as @xmath8 . we denote the number of transmit antennas on bs @xmath9 as @xmath10 and the total number of single antenna users in cell @xmath9 as @xmath11 . the received signal at mobile user @xmath12 in cell @xmath9 is given by [ rs ] & y_n , k = + & _ + _ + _ + _ . here , @xmath13 denotes the @xmath14 complex gaussian independent and identically distributed ( i.i.d . ) channel vector from the bs @xmath9 to user @xmath12 . that is , @xmath15 , where @xmath16 denotes the received power from bs @xmath9 to user @xmath12 . @xmath17 is the @xmath18 normalized beamforming vector from bs @xmath9 to user @xmath12 . @xmath19 is the desired transmitted data symbol by bs @xmath9 to user @xmath12 . the transmitted data symbols are normalized such that @xmath20=1 $ ] . @xmath21 denotes the @xmath22 complex gaussian i.i.d . intercell interfering channel vector from bs @xmath23 to user @xmath12 located in cell @xmath9 . that is , @xmath24 . and @xmath25 are used to denote the desired and intercell interfering channels , respectively , regardless of the originating bs type ; i.e. , @xmath25 can represent the intercell interfering link from the macrocell bs for a particular user placed in a microcell.]@xmath26 from @xmath21 to simplify the notation . ] @xmath27 is the additive white gaussian noise at receiver @xmath12 having an independent complex gaussian distribution with variance @xmath28 . finally , @xmath16 is defined as @xmath29here , @xmath30 refers to the total effective radiated transmit power ( erp ) from bs @xmath9 . naturally , the erp of the macrocell bs is higher than the microcell bss . @xmath31 is a reference distance of @xmath32 meter ( m ) for far field transmit antennas , @xmath33 is the distance to mobile user @xmath12 from the bs @xmath9 , @xmath34 is the pathloss exponent for urban macro ( uma ) or urban micro ( umi ) depending on the transmitting bs and @xmath35 is the correlated shadow fading value with a standard deviation @xmath36 , obtained from the gudmundson model @xcite with a decorrelation distance of @xmath37 m. snr with respect to bs @xmath9 and user @xmath12 is defined as @xmath38 , where @xmath28 is the receiver noise variance at user @xmath12 . from ( [ rs ] ) , the sinr at user @xmath12 being served by bs @xmath9 can be expressed as @xmath39 the leakage based technique to generate beamforming vectors is as described in @xcite , where the main idea is to maximize the desired signal power relative to the noise and total interference powers caused to other users ( leakage power ) . the slnr for user @xmath12 served by the bs @xmath9 is defined as @xmath40 for single - stream transmission ( where each user is equipped with a single receive antenna ) , the leakage based beamforming vector desired for user @xmath12 being served by bs @xmath9 is given by the normalized version of the @xmath41 such that @xmath42 . the structure of ( [ slnr - bf ] ) remains unchanged regardless of the coordination strategy . however , the composition of @xmath43 depends on the coordination strategy considered , as described in section iii . for the simple case of no coordination @xmath44\ ] ] is the concatenated channel of all users being served by bs @xmath9 apart from user @xmath12 . assuming the distribution of intracell and intercell interference terms in ( [ sinr ] ) is identical to the distribution of noise , the mean sum rate for cell @xmath9 can be expressed as @xmath45=\mathbb{e}\bigg[\sum\limits_{k=1}^{k_{n}}\log_{2}(1+\gamma_{n , k})\bigg].\ ] ] the mean sum rate over @xmath8 cells can then be expressed as @xmath46=\mathbb{e}\bigg[\sum\limits_{\substack{j=1}}^{n}\sum\limits_{k=1}^{k_{j}}\log_{2}(1+\gamma_{j , k})\bigg].\ ] ] from ( [ sinr ] ) , the mean per - user sinr can be expressed as @xmath47 $ ] . exact evaluation of @xmath47 $ ] is extremely cumbersome . instead , we consider an approximation motivated by the work in @xcite , which allows us to express the mean per - user sinr as [ msinr ] & [ _ n , k ] + & . the statistical expectations in both the numerator and the denominator of ( [ msinr ] ) can be evaluated further . an approach to derive the closed - form approximation of ( [ msinr ] ) is presented in the appendix . on the other hand , ( [ msinr ] ) can be rewritten in its equivalent trace form as [ msinr2 ] & [ _ n , k ] + & , where @xmath48 and @xmath49 . the expression in ( [ msinr2 ] ) is used to approximate the mean per - cell sum rate over a wide range of snr and the mean per - user sinr over a large number of channel realizations as specified in section iv . it is idealistic to assume perfect csi at all times to generate the leakage based beamforming vectors . thus , we consider channel imperfections via channel estimation errors as mentioned in @xcite . the imperfect channel at bs @xmath9 of user @xmath12 after introducing channel estimation errors is given by @xmath50 here , @xmath51 controls the level of csi imperfection . @xmath52 results in perfect csi and @xmath53 models complete uncertainty . @xmath54 is a @xmath14 complex gaussian error vector with a statistically identical structure to @xmath13 . it is shown in @xcite that @xmath55 can be used to determine the impact of several factors on imperfect csi and can be a function of the length of the estimation pilot sequence , doppler frequency and snr . the concatenated channel and the leakage based beamforming vector for user @xmath12 in cell @xmath9 can be expressed as ( [ slnr - bf ] ) and ( [ concchannel ] ) when replacing @xmath43 with @xmath56 and @xmath13 with @xmath57 , respectively . the sinr with imperfect csi can be expressed as in ( [ sinr ] ) when replacing @xmath17 with @xmath58 , @xmath59 with @xmath60 and @xmath61 with @xmath62 , respectively . as the leakage based beamforming vectors are designed with imperfect csi , the sinr expressed in ( [ sinr ] ) will contain channel estimation errors . [ tab : tab1 ] in this section , we describe the bs coordination strategies considered . * _ no coordination _ in this case , each bs coordinates the desired and intracell interfering links locally . that is , the bss only consider maximizing the slnr of users belonging to its own coverage area . the concatenated channel used to compute the leakage based beamforming vector weights for user @xmath12 in cell @xmath9 is given in ( [ concchannel ] ) . we treat this strategy as the baseline case . * _ full coordination _ in this case , we assume that each bs has knowledge of its own users desired channels and all intracell and intercell interfering channels . the channel information may be exchanged by exploiting the intercell orthogonal reference signals via the backhaul interface @xcite . with the use of the fully acquired csi for each desired and interfering link , downlink leakage based beamformers can be designed to minimize the leakage power within the cell as well as to the other cells . the concatenated channel used to compute the leakage based beamforming vector weights for user @xmath12 in cell @xmath9 can be expressed as + [ fcconc ] _ n , k=[_n,1, ,_n , k-1,_n , k+1, ,_n , k_n , + _ n,1, ,_n , n ] . + here @xmath63 denotes the concatenated intercell interfering channels transmitted from bs @xmath9 to all users in cell @xmath64 , given by + [ iciconc ] _ n , m=[_m,1,_m,2,_m,3 ,_m , k_m ] . * _ macro - only coordination _ in this case , we assume that the macrocell bs has knowledge of the intercell interfering channels from itself to all microcell users . the macrocell bs uses this information to coordinate transmission to its own users , as well as to the users located in each microcell , respectively . the concatenated channel used to compute the leakage based beamforming weight vectors for user @xmath12 in cell @xmath9 can be expressed as ( [ fcconc ] ) and ( [ concchannel ] ) if @xmath9 is the macrocell and microcell bs , respectively . * _ no inter - tier interference _ this is an ideal case , where we assume that no cross - tier interference exists . this means that users in a particular tier only experience intra - tier interference . coordination is however present within each cell regardless of the tier . in computing the leakage based beamforming weight vector for user @xmath12 in cell @xmath9 , the concatenated channel will be given by ( [ concchannel ] ) if bs @xmath9 is the macrocell bs . otherwise , for a microcell bs it is given as + [ niticonc ] _ n , k=[_n,1, ,_n , k-1,_n , k+1, ,_n , k_n , + _ n,1, ,_n , n-1 ] , + where @xmath65 refer to microcell bs indices . table [ tab : tab1 ] summarizes the different bs coordination strategies with the respective structures for @xmath43 . we consider a two - tier hetnet system comprising of a single macrocell and two microcells ( unless otherwise stated ) . we carry out monte - carlo simulations to evaluate the system performance over @xmath66 channel realizations . the location of the macrocell bs was fixed at the origin of the circular coverage area with radius @xmath67 . the locations of the microcell bss inside the macrocell coverage area were uniformly generated subject to a spacing constraint . the minimum distance between two microcells was fixed to twice the radius of the microcell , i.e. , @xmath68 , such that there is no overlap between successive microcells . in table [ tab : tab2 ] , we specify the remainder of the simulation parameters and their corresponding values . .simulation parameters and values [ cols="^,^ " , ] [ tab : tab2 ] from ( [ sr1 ] ) vs. macrocell snr [ db ] for perfect and imperfect csi where @xmath69 . the squares denote the approximated mean per - cell sum rates computed with ( [ msinr2 ] ) . ] from ( [ sr1 ] ) vs. macrocell snr [ db ] for perfect and imperfect csi where @xmath69 . the squares denote the approximated mean per - cell sum rates computed with ( [ msinr2 ] ) . ] from ( [ sr1 ] ) vs. macrocell snr [ db ] for perfect and imperfect csi where @xmath69 . the squares denote the approximated mean per - cell sum rates computed with ( [ msinr2 ] ) . ] from ( [ sr1 ] ) vs. macrocell snr [ db ] for perfect and imperfect csi where @xmath69 . the squares denote the approximated mean per - cell sum rates computed with ( [ msinr2 ] ) . ] performance vs. number of microcells at snr=10 db with perfect csi for full , no and macro only coordination strategies . the approximated mean per - user sinrs are computed with ( [ msinr2 ] ) . ] performance from ( [ sr2 ] ) vs. number of microcells at snr=10 db with perfect csi for full , no and macro only coordination strategies . ] per macrocell from ( [ sr2 ] ) vs. number of microcells at snr=10 db with perfect csi for full , no and macro only coordination strategies . ] [ nc ] shows the mean per - cell sum rate performance given by ( [ sr1 ] ) vs. macrocell snr with no coordination in the hetnet . we consider perfect and imperfect csi at the bss . in the high snr regime , inter - tier interference causes the mean sum rates to saturate for macrocell and microcells , respectively . the dominant factor contributing to the poor mean sum rate performance of microcell users is the large inter - tier interference from the macro bs resulting from its high transmit power . this behaviour is a result of the uncoordinated nature of the hetnet . with imperfect csi , we again consider the mean sum rate performance with @xmath69 , where further degradation in the macrocell and microcell rates can be observed . the approximated mean per - cell sum rates based on ( [ msinr2 ] ) are shown to closely match the simulated responses . the variation between the simulated and analytical sinr responses can be justified from the fact that the approximation in ( [ msinr2 ] ) becomes less tight with increasing snr . the uncoordinated network performance can be compared to the case where the hetnet is fully coordinated . [ fc ] demonstrates the mean per - cell sum rate performance given by ( [ sr1 ] ) vs. macrocell snr for perfect and imperfect csi . two major trends can be observed from the result . first is the near @xmath70 increase in the microcell rates over the entire snr range relative to the baseline case ( fig . secondly , microcell to microcell interference has a marginal impact on the macrocell user rates due to their low transmit powers . this is demonstrated by comparing fig . [ fc ] to fig . [ nc ] . as the macrocell bs is the dominant source of interference to the microcell users , we consider the case where coordination takes place at the macrocell bs only . [ moc ] demonstrates the mean per - cell sum rate given by ( [ sr1 ] ) vs. macrocell snr performance of the macro only coordination strategy . both the macro and microcell rates are found to be approximately equivalent to the full coordination case , observed by comparing fig . [ fc ] and fig . this suggests that if we can coordinate the transmission to minimize the most dominant source of interference , we are able to achieve near full coordination performance . moreover , this strategy significantly reduces the backhaul overheads by eliminating the need to equip the microcell bss with out - of - cell csi . [ niti ] depicts the mean per - cell sum rate performance given by ( [ sr1 ] ) vs. macrocell snr of the no inter - tier interference coordination strategy . due to zero cross - tier interference , this strategy results in superior mean per - cell sum rate performance in comparison with the other coordination strategies . it is worth comparing fig . [ niti ] to fig . [ fc ] , and noting that the mean sum rate performance of full coordination approaches the performance of no inter - tier interference . this demonstrates the value of bs coordination in a hetnet . the effect of increasing the microcell density is shown in fig . [ mpusinr ] , where we plot the mean per - user sinr as a function of the number of microcells . we observe that the mean per - user sinr decreases linearly with increasing number of microcells . when the number of microcells is less than 5 , there is a marginal difference between macro only coordination and full coordination mean per - user sinr . this suggests that at low microcell density , it is advantageous to avoid paying the high price of backhaul overheads for full coordination performance . when there are more than 5 microcells , the gap between full coordination and macro - only coordination techniques starts to increase . approximately , a @xmath71 db difference in the mean per - user sinr is seen with @xmath37 microcells in the system . the difference in the slopes of the various strategies demonstrates the impact of bs coordination in a hetnet with network densification . thus , coordination arrests the rate of decay of the mean per - user sinr in a hetnet . in addition to the above , the result demonstrates the validity of the approximated mean per - user sinr in ( [ msinr2 ] ) . these are shown to closely match the simulated mean per - user sinr performance for all the coordination techniques . [ mrp ] shows the microcell sum rate performance as defined in ( [ sr2 ] ) at the mean , @xmath37th and @xmath72th percentiles with respect to number of microcells at a snr of @xmath37 db . with full coordination , the microcell sum rate increases linearly with the number of microcells , as majority of the interference is being suppressed by the leakage based beamformers . a similar trend can be observed for the macro only coordination case , however the microcell sum rate performance gains are lower compared to the full coordination case as the number of microcells increases . the no coordination case suffers from strong macro and other microcell interference resulting in a saturated sum rate at higher number of microcells . we now study the effect of deploying multiple macrocell bss on the microcell sum rate performance . for comparison purposes , we consider scenarios with both single and three overlapping macrocells with inter - site distances of @xmath32 km . in both cases , a maximum of @xmath37 microcell bss are randomly dropped at the edge of the macrocell at a radius of @xmath73 , such that the minimum distance between successive microcell bss is @xmath68 . [ mrp2 ] shows the mean microcell sum rate as a function of the number of microcells for both the single and three macrocell bss cases at a snr of 10db . it is seen that the sum rate of the single macrocell bs case is significantly higher than the three overlapping macrocell bs case . this is due to higher aggregate intercell interference resulting from other macrocells and microcells located within these macrocells . compared to fig . [ mrp ] where the microcells are randomly placed anywhere within the macrocell coverage area , the no coordination performance benefits the most from the microcells being deployed at the edge of the macrocell . this can be seen from the mean sum rate , as it shows a linear growth up to 7 microcells in comparison with 3 microcells . we also observe that the improvement in mean sum rate with cell edge deployment of microcells is higher for the no coordination strategy . at 10 microcells , the increase in the mean sum rate for the full coordination strategy is approximately 3.6 bps / hz , while the increase with no coordination is about 10 bps / hz . in this paper , we demonstrate the rate gains provided by bs coordination in hetnets . with bs coordination , the sum rate is seen to increase linearly and the mean per - user sinr decreases linearly with the number of microcells . however , the rate of mean per - user sinr degradation is reduced significantly with increased degrees coordination at the bss in the hetnet . at a low density of microcells , macro - only coordination performs close to full coordination . however , this is not the case with a higher density of microcells where increasing amounts of interference from the microcells is being added . in addition to the above , the impact of multiple macrocells is also investigated . here , degradation in the mean microcell sum rate is observed for all the respective coordination strategies in comparison to the case where only one macrocell is present . using an eigenvalue decomposition , ( [ app1 ] ) can be rewritten as @xmath74=\mathbb{e}\bigg[\big|\mathbf{h}_{n , k}(\mathbf{x\lambda{}x}^{*}+\sigma_{k}^{2}\mathbf{i})^{-1}\mathbf{h}_{n , k}^{*}\big|^{2}\bigg]\\ & \hspace{68pt}=\mathbb{e}\bigg[\big|\boldsymbol{\delta}_{n , k}(\mathbf{\lambda}+\sigma_{k}^{2}\mathbf{i})^{-1}\boldsymbol{\delta}_{n , k}^{*}\big|^{2}\bigg],\end{aligned}\ ] ] where @xmath75 has the same statistics as @xmath13 as @xmath76 is a unitary matrix . hence , @xmath77=\mathbb{e}\bigg[\big|\sum\limits_{i=1}^{z_{n}}(\mathbf{\lambda}_{ii}+\sigma_{k}^{2})^{-1}|\boldsymbol{\delta}_{n , k , i}|^{2}\big|^{2}\bigg],\ ] ] where @xmath78 is the @xmath79th element of @xmath80 . since @xmath78 is a zero mean complex gaussian random variable with variance @xmath16 , it follows that @xmath81 is an exponential random variable with mean @xmath16 . using the standard properties of the exponential random variable , can be expressed as @xmath82.\ ] ] a similar approach can be taken to evaluate the mean per - user intracell and intercell interference powers . further averaging over the eigenvalues , @xmath83 in ( [ app4 ] ) is possible as the density of eigenvalues is known . however , due to space limitations we leave this approach for future work . 1 cisco , cisco visual networking index : global mobile data traffic forcast update , 20112016 " , white paper , feb . a. ghosh , n. mangalvedhe , r. ratasuk , b. mondal , m. cudak , e. vistosky , t.a . thomas , j.g . andrews , p. xia , h.s . jo , h.s . dhillon and t.d . novlan , heterogeneous cellular networks : from theory to practice " , _ ieee commun . mag . 50 , no . 6 , pp . 5464 , jun . 2012 . a. alexiou , wireless world 2020 " , _ ieee veh . technol . mag . _ , vol . 9 , no . 1 , pp . 46 - 53 , mar . 2014 . c. kosta , b. hunt , a.u . quddus and r. tafazolli , on interference avoidance through inter - cell interference coordination ( icic ) based on ofdma mobile systems " , _ ieee commun . surveys tuts . 3 , pp . 973995 , aug . 2012 . d. lopez - perez , i. guvenc , g.d.l . roche , m. kountouris , t. quek and j. zhang , enhanced intercell interference coordination challenges in heterogeneous networks " , _ ieee commun . mag . 2230 , jun . 2011 . g. boudreau , j. panicker , n. guo , r. chang , n. wang and s. vrzic , interference coordination and cancellation for 4 g networks " , _ ieee commun . mag . 4 , pp . 7481 , apr . 2009 . e. hossain , m. rasti , h. tabassum and a. abdelnasser , evolution toward 5 g multi - tier cellular wireless networks : an interference management perspective " , _ ieee wireless commun . 3 , pp . 118127 , jun . 2014 . dhillon , m. kountouris and j.g . andrews , downlink mimo hetnets : modelling , ordering results and performance analysis " , _ ieee trans . wireless commun . 10 , pp . 52085222 , oct . dhillon , m. kountouris and j.g . andrews , downlink coverage probability in mimo hetnets " , _ in proc . ieee 46th conference on signals , systems and computers ( asilomar ) _ , pp . 683687 , nov . s. park , w. seo , y. kim , s. lim and d. hong , beam subset selection strategy for interference reduction in two - tier femtocell networks " , _ ieee trans . wireless commun . _ , vol . 9 , no . 11 , pp . 34403449 , oct s. akoum , m. kountouris and r.w . heath , on imperfect csi for the downlink of a two - tier network " , _ ieee intern . symp . on info . theory ( isit ) _ , pp . 553337 , july 2011 . w. liu , s. han and c. yang , hybrid cooperative transmission in heterogeneous networks " , _ in proc . ieee 23rd conference on personal , indoor and mobile radio communications ( pimrc ) _ , pp . 921925 , sep . m. hong , r - y . sun , h. baligh and z - q . luo , joint base station clustering and beamformer design for partial coordinated transmission in heterogeneous networks " , _ submitted to ieee j. on sel . areas commun . _ , nov . 2012 . available online : arxiv.org/pdf/1203.6390 . t. yoo and a. goldsmith , on the optimality of multiantenna broadcast scheduling using zero - forcing beamforming " , _ ieee j. sel . areas in commun . 3 , pp . 528541 , mar . peel , b.m hochwald and a.l . swindlehurst , a vector perturbation technique for near capacity multiantenna multiuser communication - part i : channel inversion and regularization " , _ ieee trans . 53 , no.1 , pp . 195202 , jan . m. sadek , a. tarighat and a. sayed , a leakage - based precoding scheme for downlink multi - user mimo channels " , _ ieee trans . wireless commun . _ , vol . 6 , no . 5 , pp . 17111721 , may 2007 . m. gudmundson , correlation model for shadow fading in mobile radio systems " , _ electronics letters _ 23 , pp . 21452146 , nov . l. yu , w. liu and r. langley , sinr analysis of the subtraction - based smi beamformer " , _ ieee trans . signal process . _ , vol.58 , no.11 , pp . 59265932 , nov . h. suraweera , p.j . smith and m. shafi , capacity limits and performance analysis of cognitive radio with imperfect channel knowledge " , _ ieee trans . technol . _ , 4 , pp . 18111822 , may 2010 . a. ahn , r.w . heath , performance analysis of maximum ratio combining with imperfect channel estimation in presence of cochannel interferences " , _ ieee trans . wireless commun . _ , vol . 8 , no . 3 , pp . 10801085 , mar . n. lee and w. shin , adaptive feedback scheme on k - 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in this paper we demonstrate the rate gains achieved by two - tier heterogeneous cellular networks ( hetnets ) with varying degrees of coordination between macrocell and microcell base stations ( bss ) .
we show that without the presence of coordination , network densification does not provide any gain in the sum rate and rapidly decreases the mean per - user signal - to - interference - plus - noise - ratio ( sinr ) .
our results show that coordination reduces the rate of sinr decay with increasing numbers of microcell bss in the system .
validity of the analytically approximated mean per - user sinr over a wide range of signal - to - noise - ratio ( snr ) is demonstrated via comparison with the simulated results .
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let @xmath0 be a set of @xmath1 points in @xmath2 , where @xmath7 is a small constant . let @xmath13 be a family of geometric `` regions , '' called _ ranges _ , in @xmath2 , each of which can be described algebraically by some fixed number of real parameters ( a more precise definition is given below ) . for example , @xmath13 can be the set of all axis - parallel boxes , balls , simplices , or cylinders , or the set of all intersections of pairs of ellipsoids . in the _ @xmath13-range searching _ problem , we want to preprocess @xmath0 into a data structure so that the number of points of @xmath0 lying in a query range @xmath14 can be counted efficiently . similar to many previous papers , we actually consider a more general setting , the so - called _ semigroup model _ , where we are given a weight function on the points in @xmath0 and we ask for the cumulative weight of the points in @xmath15 . the weights are assumed to belong to a semigroup , i.e. , subtractions are not allowed . we assume that the semigroup operation can be executed in constant time . in this paper we consider the case in which @xmath13 is a set of constant - complexity semialgebraic sets . we recall that a _ semialgebraic set _ is a subset of @xmath2 obtained from a finite number of sets of the form @xmath16 , where @xmath17 is a @xmath7-variate polynomial with integer coefficients , by boolean operations ( unions , intersections , and complementations ) . specifically , let @xmath18 denote the family of all semialgebraic sets in @xmath2 defined by at most @xmath19 polynomial inequalities of degree at most @xmath20 each . if @xmath21 are all regarded as constants , we refer to the sets in @xmath18 as _ constant - complexity semialgebraic sets _ ( such sets are sometimes also called _ tarski cells _ ) . by _ semialgebraic range searching _ we mean @xmath18-range searching for some parameters @xmath21 ; in most applications the actual collection @xmath13 of ranges is only a restricted subset of some @xmath18 . besides being interesting in its own right , semialgebraic range searching also arises in several geometric searching problems , such as searching for a point nearest to a query geometric object , counting the number of input objects intersecting a query object , and many others . this paper focuses on the _ low storage _ version of range searching with constant - complexity semialgebraic sets the data structure is allowed to use only linear or near - linear storage , and the goal is to make the query time as small as possible . at the other end of the spectrum we have the _ fast query _ version , where we want queries to be answered in polylogarithmic time using as little storage as possible . this variant is discussed briefly in section [ sec : concl ] . as is typical in computational geometry , we will use the _ real ram _ model of computation , where we can compute exactly with arbitrary real numbers and each arithmetic operation is executed in constant time . motivated by a wide range of applications , several variants of range searching have been studied in computational geometry and database systems at least since the 1980s . see @xcite for comprehensive surveys of this topic . the early work focused on the so - called _ orthogonal range searching _ , where ranges are axis - parallel boxes . after three decades of extensive work on this particular case , some basic questions still remain open . however , geometry plays little role in the known data structures for orthogonal range searching . the most basic and most studied truly geometric instance of range searching is with _ halfspaces _ , or more generally _ simplices _ , as ranges . studies in the early 1990s have essentially determined the optimal trade - off between the worst - case query time and the storage ( and preprocessing time ) required by any data structure for simplex range searching . is assumed to be _ fixed _ and the implicit constants in the asymptotic notation may depend on @xmath7 . this is the setting in all the previous papers , including the present one . of course , in practical applications , this assumption may be unrealistic unless the dimension is really small . however , the known lower bounds imply that if the dimension is large , no efficient solutions to simplex range searching exist , at least in the worst - case setting . ] simplex range searching . lower bounds for this trade - off have been given by chazelle @xcite under the semigroup model of computation , where subtraction of the point weights is not allowed . it is possible that , say , the counting version of the simplex range searching problem , where we ask just for the number of points in the query simplex , might admit better solutions using subtractions , but no such solutions are known . moreover , there are recent lower - bound results when subtractions are also allowed ; see @xcite and references therein . we also refer to @xcite and references therein for recent lower bounds for the case where subtractions are also allowed . the data structures proposed for simplex range searching over the last two decades @xcite match the known lower bounds within polylogarithmic factors . the state - of - the - art upper bounds are by ( i ) chan @xcite , who , building on many earlier results , provides a linear - size data structure with @xmath22 expected preprocessing time and @xmath3 query time , and ( ii ) matouek @xcite , who provides a data structure with @xmath23 storage , @xmath24 query time , and @xmath25 preprocessing time . denotes an arbitrarily small positive constant . the implicit constants in the asymptotic notation may depend on it , generally tending to infinity as @xmath26 decreases to @xmath27 . ] a trade - off between space and query time can be obtained by combining these two data structures @xcite . yao and yao @xcite were perhaps the first to consider range searching in which ranges were delimited by graphs of polynomial functions . agarwal and matouek @xcite have introduced a systematic study of semialgebraic range searching . building on the techniques developed for simplex range searching , they presented a linear - size data structure with @xmath28 query time , where @xmath29 . for @xmath30 , this almost matches the performance for the simplex range searching , but for @xmath31 there is a gap in the exponents of the corresponding bounds . also see @xcite for related recent developments . the bottleneck in the performance of the just mentioned range - searching data structure of @xcite is a combinatorial geometry problem , known as the _ decomposition of arrangements into constant - complexity cells_. here , we are given a set @xmath32 of @xmath33 algebraic surfaces in @xmath2 ( i.e. , zero sets of @xmath7-variate polynomials ) , with degrees bounded by a constant @xmath34 , and we want to decompose each cell of the arrangement @xmath35 ( see section [ sec : cross ] for details ) into subcells that are constant - complexity semialgebraic sets , i.e. , belong to @xmath18 for some constants @xmath20 ( bound on degrees ) and @xmath19 ( number of defining polynomials ) , which may depend on @xmath7 and @xmath34 , but not on @xmath33 . the crucial quantity is the total number of the resulting subcells over all cells of @xmath35 ; namely , if one can construct such a decomposition with @xmath36 subcells , with some constant @xmath37 , for every @xmath33 and @xmath32 , then the method of @xcite yields query time @xmath28 ( with linear storage ) . the only known general - purpose technique for producing such a decomposition is the so - called _ vertical decomposition _ @xcite , which decomposes @xmath35 into roughly @xmath38 constant - complexity subcells , for @xmath39 @xcite . an alternative approach , based on _ linearization _ , was also proposed in @xcite . it maps the semialgebraic ranges in @xmath2 to simplices in some higher - dimensional space and uses simplex range searching there . however , its performance depends on the specific form of the polynomials defining the ranges . in some special cases ( e.g. , when ranges are balls in @xmath2 ) , linearization yields better query time than the decomposition - based technique mentioned above , but for general constant - complexity semialgebraic ranges , linearization has worse performance . in a recent breakthrough , guth and katz @xcite have presented a new space decomposition technique , called polynomial partitioning . for a set @xmath40 of @xmath1 points and a real parameter @xmath5 , @xmath41 , an _ @xmath5-partitioning polynomial _ for @xmath0 is a nonzero @xmath7-variate polynomial @xmath8 such that each connected component of @xmath42 contains at most @xmath11 points of @xmath0 , where @xmath43 denotes the zero set of @xmath8 . the decomposition of @xmath2 into @xmath12 and the connected components of @xmath10 is called a _ polynomial partition _ ( induced by @xmath8 ) . guth and katz show that an @xmath5-partitioning polynomial of degree @xmath9 always exists , but their argument does not lead to an efficient algorithm for constructing such a polynomial , mainly because it relies on ham - sandwich cuts in high - dimensional spaces , for which no efficient construction is known . our first result is an efficient randomized algorithm for computing an @xmath5-partitioning polynomial . [ thm : partition - algo ] given a set @xmath0 of @xmath1 points in @xmath2 , for some fixed @xmath7 , and a parameter @xmath44 , an @xmath5-partitioning polynomial for @xmath0 of degree @xmath9 can be computed in randomized expected time @xmath45 . next , we use this algorithm to bypass the arrangement - decomposition problem mentioned above . namely , based on polynomial partitions , we construct _ partition trees _ @xcite that answer range queries with constant - complexity semialgebraic sets in near - optimal time , using linear storage . an essential ingredient in the performance analysis of these partition trees is a recent combinatorial result of barone and basu @xcite , originally conjectured by the second author , which deals with the complexity of certain kinds of arrangements of zero sets of polynomials ( see theorem [ t : basu ] ) . while there have already been several combinatorial applications of the guth - katz technique ( the most impressive being the original one in @xcite , which solves the famous erds s distinct distances problem , and some of the others presented in @xcite ) , ours seems to be the first _ algorithmic _ application . we establish two range - searching results , both based on polynomial partitions . for the first result , we need to introduce the notion of _ @xmath46-general position _ , for an integer @xmath47 . we say that a set @xmath40 is in @xmath46-general position if no @xmath48 points of @xmath0 are contained in the zero set of a nonzero @xmath7-variate polynomial of degree at most @xmath46 , where @xmath49 . this is the number one expects for a `` generic '' point set .- variate polynomials of degree at most @xmath46 have at most @xmath50 distinct nonconstant monomials . the veronese map ( e.g. , see @xcite ) maps @xmath2 to @xmath51 , and hyperplanes in @xmath51 correspond bijectively to @xmath48-variate polynomials of degree at most @xmath46 . it follows that any set of @xmath50 points in @xmath2 is contained in the zero set of a @xmath7-variate polynomial of degree at most @xmath46 , corresponding to the hyperplane in @xmath51 passing through the veronese images of these points . similarly , @xmath48 points in general position are not expected to have this property , because one does not expect their images to lie in a common hyperplane . see @xcite for more details . ] [ t : const - r ] let @xmath21 and @xmath52 be constants . let @xmath40 be an @xmath1-point set in @xmath53-general position , where @xmath53 is a suitable constant depending on @xmath54 , and @xmath26 . then the @xmath18-range searching problem for @xmath0 can be solved with @xmath55 storage , @xmath22 expected preprocessing time , and @xmath56 query time . we note that both here and in the next theorem , while the preprocessing algorithm is randomized , the queries are answered deterministically , and the query time bound is worst - case . of course , we would like to handle arbitrary point sets , not only those in @xmath53-general position . this can be achieved by an infinitesimal perturbation of the points of @xmath0 . a general technique known as `` simulation of simplicity '' ( in the version considered by yap @xcite ) ensures that the perturbed set @xmath57 is in @xmath53-general position . if a point @xmath58 lies in the interior of a query range @xmath59 , then so does the corresponding perturbed point @xmath60 , and similarly for @xmath61 in the interior of @xmath62 . however , for @xmath61 on the boundary of @xmath59 , we can not be sure if @xmath63 ends up inside or outside @xmath59 . let us say that a _ boundary - fuzzy _ solution to the @xmath18-range searching problem is a data structure that , given a query @xmath64 , returns an answer in which all points of @xmath0 in the interior of @xmath59 are counted and none in the interior of @xmath62 is counted , while each point @xmath58 on the boundary of @xmath59 may or may not be counted . in some applications , we can think of the points of @xmath0 being imprecise anyway ( e.g. , their coordinates come from some imprecise measurement ) , and then boundary - fuzzy range searching may be adequate . [ c : fuzzy ] let @xmath21 , and @xmath52 be constants . then for every @xmath1-point set in @xmath2 , there is a boundary - fuzzy @xmath18-range searching data structure with @xmath55 storage , @xmath22 expected preprocessing time , and @xmath56 query time . actually , previous results on range searching that use simulation of simplicity to avoid degenerate cases also solve only the boundary - fuzzy variant ( see e.g. @xcite ) . however , the previous techniques , even if presented only for point sets in general position , can usually be adapted to handle degenerate cases as well , perhaps with some effort , which is nevertheless routine . for our technique , degeneracy appears to be a more substantial problem because it is possible that a large subset of @xmath0 ( maybe even all of @xmath0 ) is contained in the zero set of the partitioning polynomial @xmath8 , and the recursive divide - and - conquer mechanism yielded by the partition of @xmath8 does not apply to this subset . partially in response to this issue , we we next present a different data structure that , at a somewhat higher preprocessing cost , not only gets rid of the boundary - fuzziness condition but also has a slightly improved query time ( in terms of @xmath1 ) . the main idea is that we build an auxiliary recursive data structure to handle the potentially large subset of points that lie in the zero set of the partitioning polynomial . [ t : large - r ] let @xmath21 , and @xmath52 be constants . then the @xmath18-range searching problem for an arbitrary @xmath1-point set in @xmath2 can be solved with @xmath55 storage , @xmath65 expected preprocessing time , and @xmath66 query time , where @xmath67 is a constant depending on @xmath21 and @xmath26 . we remark that the dependence of @xmath67 on @xmath20 , @xmath19 , and @xmath26 is reasonable , but its dependence on @xmath7 is superexponential . our algorithms work for the semigroup model described earlier . assuming that a semigroup operation can be executed in constant time , the query time remains the same as for the counting query . a reporting query report the points of @xmath0 lying in a query range also fits in the semigroup model , except one can not assume that a semigroup operation in this case takes constant time . the time taken by a reporting query is proportional to the cost of a counting query plus the number of reported points . our algorithm is based on the polynomial partitioning technique by guth and katz , and we begin by briefly reviewing it in section [ sec : gk ] . next , in section [ sec : algo ] , we describe the randomized algorithm for constructing such a partitioning polynomial . section [ sec : cross ] presents an algorithm for computing the cells of a polynomial partition that are crossed by a semialgebraic range , and discusses several related topics . section [ sec : range1 ] presents our first data structure , which is as in theorem [ t : const - r ] . section [ s : cad ] describes the method for handling points lying on the zero set of the partitioning polynomial , and section [ sec : range2 ] presents our second data structure . we conclude in section [ sec : concl ] by mentioning a few open problems . in this section we briefly review the guth - katz technique for later use . we begin by stating their result . [ t : gk ] given a set @xmath0 of @xmath1 points in @xmath2 and a parameter @xmath44 , there exists an @xmath5-partitioning polynomial for @xmath0 of degree at most @xmath9 ( for @xmath7 fixed ) . the degree in the theorem is asymptotically optimal in the worst case because the number of connected components of @xmath10 is @xmath68 for every polynomial @xmath8 ( see , e.g. , warren ( * ? ? ? * theorem 2 ) ) . the guth - katz proof uses the _ polynomial ham sandwich _ theorem of stone and tukey @xcite , which we state here in a version for finite point sets : _ if @xmath69 are finite sets in @xmath2 and @xmath46 is an integer satisfying @xmath70 , then there exists a nonzero polynomial @xmath8 of degree at most @xmath46 that simultaneously bisects all the sets @xmath71 . _ here `` @xmath8 bisects @xmath71 '' means that @xmath72 in at most @xmath73 points of @xmath71 and @xmath74 in at most @xmath73 points of @xmath71 ; @xmath8 might vanish at any number of the points of @xmath71 , possibly even at all of them . guth and katz inductively construct collections @xmath75 of subsets of @xmath0 . for @xmath76 , @xmath77 consists of at most @xmath78 pairwise - disjoint subsets of @xmath0 , each of size at most @xmath79 ; the union of these sets does not have to contain all points of @xmath0 . initially , we have @xmath80 . the algorithm stops as soon as each subset in @xmath81 has at most @xmath11 points . this implies that @xmath82 . having constructed @xmath83 , we use the polynomial ham - sandwich theorem to construct a polynomial @xmath84 that bisects each set of @xmath83 , with @xmath85 ( this is indeed an asymptotic upper bound for the smallest @xmath46 satisfying @xmath86 , assuming @xmath7 to be a constant ) . for every subset @xmath87 , let @xmath88 and @xmath89 . we set @xmath90 ; empty subsets are not included in @xmath77 . the desired @xmath5-partitioning polynomial for @xmath0 is then the product @xmath91 . we have @xmath92 by construction , the points of @xmath0 lying in a single connected component of @xmath10 belong to a single member of @xmath81 , which implies that each connected component contains at most @xmath11 points of @xmath0 . we begin by observing that @xmath93 is the number of all nonconstant monomials of degree at most @xmath46 in @xmath7 variables . thus , we fix a collection @xmath94 of @xmath95 such monomials . let @xmath96 be the corresponding _ veronese map _ , which maps a point @xmath97 to the @xmath48-tuple of the values at @xmath98 of the monomials from @xmath94 . for example , for @xmath99 , @xmath100 , and @xmath101 , we may use @xmath102 , where @xmath94 is the set of the eight monomials appearing as components of @xmath103 . let @xmath104 be the image of the given @xmath71 under this veronese map , for , @xmath105 . by the standard _ ham - sandwich theorem _ ( see , e.g. , @xcite ) , there exists a hyperplane @xmath106 in @xmath107 that simultaneously bisects all the @xmath108 s , in the sense that each open halfspace bounded by @xmath106 contains at most half of the points of each of the sets @xmath108 . in a more algebraic language , there is a nonzero @xmath48-variate linear polynomial , which we also call @xmath106 , that bisects all the @xmath108 s , in the sense of being positive on at most half of the points of each @xmath108 , and being negative on at most half of the points of each @xmath108 . then @xmath109 is the desired @xmath7-variate polynomial of degree at most @xmath46 bisecting all the @xmath71 s . in this section we present an efficient randomized algorithm that , given a point set @xmath0 and a parameter @xmath110 , constructs an @xmath5-partitioning polynomial . the main difficulty in converting the above proof of the guth - katz partitioning theorem into an efficient algorithm is the use of the ham - sandwich theorem in the possibly high - dimensional space @xmath107 . a straightforward algorithm for computing ham - sandwich cuts in @xmath107 inspects all possible ways of splitting the input point sets by a hyperplane , and has running time about @xmath111 . compared to this easy upper bound , the best known ham - sandwich algorithms can save a factor of about @xmath1 @xcite , but this is insignificant in higher dimensions . a recent result of knauer , tiwari , and werner @xcite shows that a certain incremental variant of computing a ham - sandwich cut is @xmath112$]-hard ( where the parameter is the dimension ) , and thus one perhaps should not expect much better exact algorithms . we observe that the exact bisection of each @xmath71 is not needed in the guth - katz construction it is sufficient to replace the stone tukey polynomial ham - sandwich theorem by a weaker result , as described below . we say that a polynomial @xmath8 is _ well - dissecting _ for a point set @xmath113 if @xmath72 on at most @xmath114 points of @xmath113 and @xmath74 on at most @xmath114 points of @xmath113 . given point sets @xmath69 in @xmath2 with @xmath1 points in total , we present a las - vegas algorithm for constructing a polynomial @xmath8 of degree @xmath115 that is well - dissecting for at least @xmath116 of the @xmath71 s . as in the above proof of the stone tukey polynomial ham - sandwich theorem , let @xmath46 be the smallest integer satisfying @xmath70 . we fix a collection @xmath94 of @xmath48 distinct nonconstant monomials of degree at most @xmath46 , and let @xmath103 be the corresponding veronese map . for each @xmath117 , we pick a point @xmath118 uniformly at random and compute @xmath119 . let @xmath106 be a hyperplane in @xmath107 passing through @xmath120 , which can be found by solving a system of linear equations , in @xmath121 time . if the points @xmath120 are not affinely independent , then @xmath106 is not determined uniquely ( this is a technical nuisance , which the reader may want to ignore on first reading ) . in order to handle this case , we prepare in advance , before picking the @xmath122 s , _ auxiliary _ affinely independent points @xmath123 in @xmath107 , which are in general position with respect to @xmath124 ; here we mean the `` ordinary '' general position , i.e. , no unnecessary affine dependences , that involve some of the @xmath125 s and the other points , arise . the points @xmath125 can be chosen at random , say , uniformly in the unit cube ; with high probability , they have the desired general position property . ( if we do not want to assume the capability of choosing a random real number , we can pick the @xmath125 s uniformly at random from a sufficiently large discrete set . ) if the dimension of the affine hull of @xmath120 is @xmath126 , we choose the hyperplane @xmath106 through @xmath120 and @xmath127 . if @xmath106 is not unique , i.e. , @xmath128 are not affinely independent with respect to @xmath129 , which we can detect while solving the linear system , we restart the algorithm by choosing @xmath123 anew and then picking new @xmath130 . in this way , after a constant expected number of iterations , we obtain the uniquely determined hyperplane @xmath106 through @xmath120 and @xmath127 as above , and we let @xmath131 denote the corresponding @xmath7-variate polynomial . we refer to these steps as one _ trial _ of the algorithm . for each @xmath71 , we check whether @xmath8 is well - dissecting for @xmath71 . if @xmath8 is well - dissecting for only fewer than @xmath132 sets , then we discard @xmath8 and perform another trial . we now analyze the expected running time of the algorithm . the intuition is that @xmath8 is expected to well - dissect a significant fraction , say at least half , of the sets @xmath71 . this intuition is reflected in the next lemma . let @xmath133 be the indicator variable of the event : _ @xmath71 is * not * well - dissected by @xmath8_. [ l : exi ] for every @xmath134 , @xmath135 \le 1/4 $ ] . let us fix @xmath136 and the choices of @xmath137 ( and thus of @xmath138 ) for all @xmath139 . let @xmath140 be the dimension of @xmath141 , the affine hull of @xmath142 . then the resulting hyperplane @xmath106 passes through the @xmath143-flat @xmath144 spanned by @xmath141 and @xmath145 , irrespective of which point of @xmath71 is chosen . if @xmath122 , the point chosen from @xmath71 , is such that @xmath146 lies on @xmath141 , then @xmath106 also passes through @xmath147 . put @xmath148 , and let us project the configuration orthogonally to a 2-dimensional plane @xmath149 orthogonal to @xmath144 . then @xmath144 appears as a point @xmath150 , and @xmath108 projects to a ( multi)set @xmath151 in @xmath149 . the random hyperplane @xmath106 projects to a random line @xmath152 in @xmath149 , whose choice can be interpreted as follows : pick @xmath153 uniformly at random ; if @xmath154 , then @xmath152 is the unique line through @xmath155 and @xmath156 ; otherwise , when @xmath157 , @xmath152 is the unique line through @xmath156 and @xmath158 ; by construction , @xmath159 . the indicator variable @xmath133 is @xmath160 if and only if the resulting @xmath152 has more than @xmath161 points of @xmath151 , counted with multiplicity , ( strictly ) on one side . the special role of @xmath158 can be eliminated if we first move the points of @xmath151 coinciding with @xmath156 to the point @xmath158 , and then slightly perturb the points so as to ensure that all points of @xmath151 are distinct and lie at distinct directions from @xmath156 ; it is easy to see that these transformations can not decrease the probability of @xmath162 . finally , we note that the side of @xmath152 containing a point @xmath163 only depends on the direction of the vector @xmath164 , so we can also assume the points of @xmath151 to lie on the unit circle around @xmath156 . using ( a simple instance of ) the standard planar ham - sandwich theorem , we partition @xmath151 into two subsets @xmath165 and @xmath166 of equal size by a line through the center @xmath156 . then we bisect @xmath165 by a ray from @xmath156 , and we do the same for @xmath166 . it is easily checked ( see figure [ f : semi2-quarters ] ) that there always exist two of the resulting quarters , one of @xmath165 and one of @xmath166 ( the ones whose union forms an angle @xmath167 between the two bisecting rays ) , such that every line connecting @xmath156 with a point in either quarter contains at least @xmath168 points of @xmath151 on each side . referring to these quarters as `` good '' , we now take one of the bisecting rays , say that of @xmath165 , and rotate it about @xmath156 away from the good quarter of @xmath165 . each of the first @xmath169 points that the ray encounters has the property that the line supporting the ray has at least @xmath169 points of @xmath151 on each side . this implies that , for at least half of the points in each of the two remaining quarters , the line connecting @xmath156 to such a point has at least @xmath169 points of @xmath151 on each side . hence at most @xmath170 points of @xmath108 can lead to a cut that is not well - dissecting for @xmath108 . we conclude that , still conditioned on the choices of @xmath137 , @xmath139 , the event @xmath162 has probability at most @xmath171 . since this holds for every choice of the @xmath137 , @xmath139 , the unconditional probability of @xmath162 is also at most @xmath171 , and thus @xmath135\le 1/4 $ ] as claimed . hence , the expected number of sets @xmath71 that are not well - dissected by @xmath8 is @xmath172 = \sum_{i=1}^k \ex [ x_i ] \le k/4.\ ] ] by markov s inequality , with probability at least @xmath173 , at least half of the @xmath71 s are well - dissected by @xmath8 . we thus obtain a polynomial that is well - dissecting for at least half of the @xmath71 s after an expected constant number of trials . it remains to estimate the running time of each trial . the points @xmath174 can be chosen in @xmath55 time . computing @xmath106 involves solving a @xmath175 linear system , which can be done in @xmath121 time using gaussian elimination . note that we do _ not _ actually compute the entire sets @xmath176 . no computation is needed for passing from @xmath106 to @xmath8we just re - interpret the coefficients . to check which of @xmath177 are well - dissected by @xmath8 , we evaluate @xmath8 at each point of @xmath178 . first we evaluate each of the @xmath48 monomials in @xmath94 at each point of @xmath113 . if we proceed incrementally , from lower degrees to higher ones , this can be done with @xmath179 operations per monomial and point of @xmath113 , in @xmath180 time in total . then , in additional @xmath180 time , we compute the values of @xmath181 , for all @xmath182 , from the values of the monomials . putting everything together we obtain the following lemma . [ l : dissect ] given point sets @xmath69 in @xmath2 ( for fixed @xmath7 ) with @xmath1 points in total , a polynomial @xmath8 of degree @xmath115 that is well - dissecting for at least @xmath116 of the @xmath71 s can be constructed in @xmath183 randomized expected time . we now describe the algorithm for computing an @xmath5-partitioning polynomial @xmath8 . we essentially imitate the guth katz construction , with lemma [ l : dissect ] replacing the polynomial ham - sandwich theorem , but with an additional twist . the algorithm works in phases . at the end of the @xmath184-th phase , for @xmath185 , we have a family @xmath186 of @xmath184 polynomials and a family @xmath77 of at most @xmath78 pairwise - disjoint subsets of @xmath0 , each of size at most @xmath187 . similar to the guth katz construction , @xmath77 is not necessarily a partition of @xmath0 , since the points of @xmath188 do not belong to @xmath189 . initially , @xmath190 . the algorithm stops when each set in @xmath77 has at most @xmath11 points . in the @xmath184-th phase , the algorithm constructs @xmath84 and @xmath77 from @xmath191 and @xmath83 , as follows . at the beginning of the @xmath184-th phase , let @xmath192 be the family of the `` large '' sets in @xmath193 , and set @xmath194 . we also initialize the collection @xmath195 to @xmath196 , the family of `` small '' sets in @xmath193 . then we perform at most @xmath197 dissecting steps , as follows : after @xmath19 steps , we have a family @xmath198 of polynomials , the current set @xmath195 , and a subfamily @xmath199 of size at most @xmath200 , consisting of the members of @xmath201 that were not well - dissected by any of @xmath198 . if @xmath202 we choose , using lemma [ l : dissect ] , a polynomial @xmath203 of degree at most @xmath204 ( with a suitable constant @xmath205 that depends only on @xmath7 ) that well - dissects at least half of the members of @xmath206 . for each @xmath207 , let @xmath208 and @xmath209 . if @xmath210 is well - dissected , i.e. , @xmath211 , then we add @xmath212 to @xmath195 , and otherwise , we add @xmath210 to @xmath213 . note that in the former case the points @xmath214 satisfying @xmath215 are `` lost '' and do not participate in the subsequent dissections . by lemma [ l : dissect ] , @xmath216 . the @xmath184-th phase is completed when @xmath217 , in which case we set is not necessarily well - dissecting , because it does not control the sizes of subsets with positive or with negative signs . ] @xmath218 . by construction , each point set in @xmath195 has at most @xmath187 points , and the points of @xmath0 not belonging to any set of @xmath195 lie in @xmath219 . furthermore , @xmath220 where again the constant of proportionality depends only on @xmath7 . since every set in @xmath83 is split into at most two sets before being added to @xmath77 , @xmath221 . if @xmath77 contains subsets with more than @xmath11 points , we begin the @xmath222-st phase with the current @xmath195 ; otherwise the algorithm stops and returns @xmath223 . this completes the description of the algorithm . clearly , @xmath224 , the number of phases of the algorithm , is at most @xmath225 . following the same argument as in @xcite , and as briefly sketched in section [ sec : gk ] , it can be shown that all points lying in a single connected component of @xmath10 belong to a single member of @xmath226 , and thus each connected component contains at most @xmath11 points of @xmath0 . since the degree of @xmath84 is @xmath227 , @xmath228 , and @xmath229 , we conclude that @xmath230 as for the expected running time of the algorithm , the @xmath19-th step of the @xmath184-th phase takes @xmath231 expected time , so the @xmath184-th phase takes a total of @xmath232 expected time . substituting @xmath228 in the above bound and summing over all @xmath184 , the overall expected running time of the algorithm is @xmath45 . this completes the proof of theorem [ thm : partition - algo ] . theorem [ thm : partition - algo ] is employed for the preprocessing in our range - searching algorithms in theorems [ t : const - r ] and [ t : large - r ] . in theorem [ t : const - r ] we take @xmath5 to be a large constant , and the expected running time in theorem [ thm : partition - algo ] is @xmath55 . however , in theorem [ t : large - r ] , we require @xmath5 to be a small fractional power of @xmath1 , say @xmath233 . it is a challenging open problem to improve the expected running time in theorem [ thm : partition - algo ] to @xmath234 when @xmath5 is such a small fractional power of @xmath1 . the bottleneck in the current algorithm is the subproblem of evaluating a given @xmath7-variate polynomial @xmath8 of degree @xmath235 at @xmath1 given points ; everything else can be performed in @xmath236 expected time . finding the signs of @xmath8 at those points would actually suffice , but this probably does not make the problem any simpler . this problem of _ multi - evaluation _ of multivariate real polynomials has been considered in the literature , and there is a nontrivial improvement over the straightforward @xmath237 algorithm , due to nsken and ziegler @xcite . concretely , in the bivariate case ( @xmath99 ) , their algorithm can evaluate a bivariate polynomial of degree @xmath238 at @xmath1 given points using @xmath239 arithmetic operations . it is based on fast matrix multiplication , and even under the most optimistic possible assumption on the speed of matrix multiplication , it can not get below @xmath240 . although this is significantly faster than our naive @xmath237-time algorithm , which is @xmath241 in this bivariate case , it is still a far cry from what we are aiming at . however , its running time is still a far cry from what we are aiming at . let us remark that in a different setting , for polynomials over finite fields ( and over certain more general finite rings ) , there is a remarkable method for multi - evaluation by kedlaya and umans @xcite achieving @xmath242 running time , where @xmath243 is the cardinality of the field . in this section we define the crossing number of a polynomial partition and describe an algorithm for computing the cells of a polynomial partition that are crossed by a semialgebraic range , both of which will be crucial for our range - searching data structures . we begin by recalling a few results on arrangements of algebraic surfaces . we refer the reader to @xcite for a comprehensive review of such arrangements . let @xmath32 be a finite set of algebraic surfaces in @xmath2 . the _ arrangement _ of @xmath32 , denoted by @xmath35 , is the partition of @xmath2 into maximal relatively open connected subsets , called _ cells _ , such that all points within each cell lie in the same subset of surfaces of @xmath32 ( and in no other surface ) . if @xmath244 is a set of @xmath7-variate polynomials , then with a slight abuse of notation , we use @xmath245 to denote the arrangement @xmath246 of their zero sets . we need the following result on arrangements , which follows from proposition 7.33 and theorem 16.18 in @xcite . [ t : make - arrg ] let @xmath247 be a set of @xmath19 real @xmath7-variate polynomials , each of degree at most @xmath20 . then the arrangement @xmath245 in @xmath2 has at most @xmath248 cells , and it can be computed in time at most @xmath249 . each cell is described as a semialgebraic set using at most @xmath250 polynomials of degree bounded by @xmath251 . moreover , the algorithm supplies an explicitly computed point in each cell . a key ingredient for the analysis of our range - searching data structure is the following recent result of barone and basu @xcite , which is a refinement of a series of previous studies ; e.g. , see @xcite : see @xcite : [ t : basu ] let @xmath252 be a @xmath48-dimensional algebraic variety in @xmath2 defined by a finite set @xmath253 of @xmath7-variate polynomials , each of degree at most @xmath20 , and let @xmath244 be a set of @xmath19 polynomials of degree at most @xmath254 . then the number of cells of @xmath255 ( of all dimensions ) that are contained in @xmath252 is bounded by @xmath256 . let @xmath0 be a set of @xmath1 points in @xmath2 , and let @xmath8 be an @xmath5-partitioning polynomial for @xmath0 . recall that the _ polynomial partition _ @xmath257 induced by @xmath8 is the partition of @xmath2 into the zero set @xmath12 and the connected components @xmath258 of @xmath10 . as already noted , warren s theorem @xcite implies that @xmath259 . we call @xmath260 the _ cells _ of @xmath261 ( although they need not be cells in the sense typical , e.g. , in topology ; they need not even be simply connected ) . @xmath261 also induces a partition @xmath262 of @xmath0 , where @xmath263 is the _ exceptional part _ , and @xmath264 , for @xmath265 , are the _ regular parts_. by construction , @xmath266 for every @xmath267 , but we have no control over the size of @xmath268this will be the source of most of our technical difficulties . next , let @xmath59 be a range in @xmath269 . we say that @xmath59 _ crosses _ a cell @xmath270 if neither @xmath271 nor @xmath272 . the _ crossing number _ of @xmath59 is the number of cells of @xmath261 crossed by @xmath59 , and the _ crossing number _ of @xmath261 ( with respect to @xmath269 ) is the maximum of the crossing numbers of all @xmath273 . similar to many previous range - searching algorithms @xcite , the crossing number of @xmath274 will determine the query time of our range - searching algorithms described in sections [ sec : range1 ] and [ sec : range2 ] . [ l : cr ] if @xmath261 is a polynomial partition induced by an @xmath5-partitioning polynomial of degree at most @xmath46 , then the crossing number of @xmath261 with respect to @xmath269 , with @xmath275 , is at most @xmath276 , where @xmath277 is a suitable constant depending only on @xmath7 . let @xmath273 ; then @xmath59 is a boolean combination of up to @xmath19 sets of the form @xmath278 , where @xmath279 are polynomials of degree at most @xmath20 . if @xmath59 crosses a cell @xmath270 , then at least one of the ranges @xmath280 also crosses @xmath270 , and thus it suffices to establish that the crossing number of any range @xmath59 , defined by a single @xmath7-variate polynomial inequality @xmath281 of degree at most @xmath20 , is at most @xmath282 . we apply theorem [ t : basu ] with @xmath283 , which is an algebraic variety of dimension @xmath284 , and with @xmath285 and @xmath286 , where @xmath8 is the @xmath5-partitioning polynomial . then , for each cell @xmath270 crossed by @xmath59 , @xmath287 is a nonempty union of some of the cells in @xmath288 that lie in @xmath252 . thus , the crossing number of @xmath59 is at most @xmath289 . we need to perform the following algorithmic primitives ( for @xmath7 _ fixed _ as usual ) for the range - searching algorithms that we will later present : * given an @xmath5-partitioning polynomial @xmath8 of degree @xmath235 , compute ( a suitable representation of ) the partition @xmath261 and the induced partition of @xmath0 into @xmath290 . + by computing @xmath291 , using theorem [ t : make - arrg ] , and then testing the membership of each point @xmath58 in each cell @xmath270 in time polynomial in @xmath5 , the above operation can be performed in @xmath292 time , this would be the second step , together with an improved construction of an @xmath5-partitioning polynomial @xmath8 ( concretely , an improved multi - point evaluation procedure for @xmath8 ) as discussed at the end of section [ sec : algo ] , needed to improve the preprocessing time in theorem [ t : large - r ] . ] where @xmath293 . * given ( a suitable representation of ) @xmath261 as in ( a1 ) and a query range @xmath294 , i.e. , a range defined by a single @xmath7-variate polynomial @xmath17 of degree @xmath275 , compute which of the cells of @xmath261 are crossed by @xmath59 and which are completely contained in @xmath59 . + we already have the arrangement @xmath291 , and we compute @xmath295 . for each cell of @xmath295 contained in @xmath296 , we locate its representative point in @xmath291 , and this gives us the cells crossed by @xmath59 . for the remaining cells , we want to know whether they are inside @xmath59 or outside , and for that , it suffices to determine the sign of @xmath17 at the representative points . using theorem [ t : make - arrg ] , the above task can thus be accomplished in time @xmath297 , with @xmath293 . we are now ready to describe our first data structure for @xmath269-range searching , which is a constant fan - out ( branching degree ) partition tree , and which works for points in general position . let @xmath0 be a set of @xmath1 points in @xmath2 , and let @xmath298 be constants . we choose @xmath5 as a ( large ) constant depending on @xmath21 , and the prespecified parameter @xmath26 . we assume @xmath0 to be in @xmath53-general position for some sufficiently large constant @xmath299 . we construct a partition tree @xmath300 of fan - out @xmath301 as follows . we first construct an @xmath5-partitioning polynomial @xmath8 for @xmath0 using theorem [ thm : partition - algo ] , and compute the partition @xmath261 of @xmath2 induced by @xmath8 , as well as the corresponding partition @xmath302 of @xmath0 , where @xmath259 . since @xmath5 is a constant , the ( a1 ) operation , discussed in section [ sec : cross ] , performs this computation in @xmath55 time . we choose @xmath53 so as to ensure that it is at least @xmath303 , and then our assumption that @xmath0 is in @xmath53-general position implies that the size of @xmath304 is bounded by @xmath53 . we set up the root of @xmath300 , where we store 1 . the partitioning polynomial @xmath8 , and a suitable representation of the partition @xmath261 ; 2 . a list of the points of the exceptional part @xmath268 ; and 3 . @xmath305 , the sum of the weights of the points of the regular part @xmath306 , for each @xmath267 . the partition polynomial @xmath8 , a suitable representation of the partition @xmath261 , a list of the points of the exceptional part @xmath268 , and @xmath305 , the sum of weights of all points of @xmath306 , for each @xmath267 . the regular parts @xmath306 are not stored explicitly at the root . instead , for each @xmath306 we recursively build a subtree representing it . the recursion terminates , at leaves of @xmath300 , as soon as we reach point sets of size smaller than a suitable constant @xmath307 . the points of each such set are stored explicitly at the corresponding leaf of @xmath300 . since each node of @xmath300 requires only a constant amount of storage and each point of @xmath0 is stored at only one node of @xmath300 , the total size of @xmath300 is @xmath55 . the preprocessing time is @xmath22 since @xmath300 has depth @xmath308 and each level is processed in @xmath55 time . to process a query range @xmath273 , we start at the root of @xmath300 and maintain a global counter which is initially set to @xmath27 . among the cells @xmath260 of the partition @xmath261 stored at the root , we find , using the ( a2 ) operation , those completely contained in @xmath59 , and those crossed by @xmath59 . actually , we compute a superset of the cells that @xmath59 crosses , namely , the cells crossed by the zero set of at least one of the ( at most @xmath19 ) polynomials defining @xmath59 . for each cell @xmath271 , we add the weight @xmath305 to the global counter . we also add to the global counter the weights of the points in @xmath309 , which we find by testing each point of @xmath268 separately . then we recurse in each subtree corresponding to a cell @xmath270 crossed by @xmath59 ( in the above weaker sense ) . the leaves , with point sets of size @xmath179 , are processed by inspecting their points individually . by lemma [ l : cr ] , the number of cells crossed by any of the polynomials defining @xmath59 at any interior node of @xmath300 is at most @xmath310 , where @xmath311 is a constant independent of @xmath5 . the query time @xmath312 obeys the following recurrence : @xmath313 o(n ) & \mbox{for $ n \le n_0 $ , } \end{array } \right .\ ] ] it is well known ( e.g. , see @xcite ) , and easy to check , that the recurrence solves to @xmath314 , for every fixed @xmath52 , with an appropriate sufficiently large choice of @xmath5 as a function of @xmath315 and @xmath26 , and with an appropriate choice of @xmath307 . this concludes the proof of theorem [ t : const - r ] . now we consider the case where the points of @xmath0 are not necessarily in @xmath53-general position . as was mentioned in the introduction , we apply a general perturbation scheme of yap @xcite to the previous range - searching algorithm . yap s scheme is applicable to an algorithm whose input is a sequence of real numbers ( in our case , the @xmath316 point coordinates plus the coefficients in the polynomials specifying the query range ) . it is assumed that the algorithm makes decision steps by way of evaluating polynomials with rational coefficients taken from a finite set @xmath317 , where the input parameters are substituted for the variables . the algorithm makes a 3-way branching depending on the sign of the evaluation . the set @xmath317 does not depend on the input . the input is considered degenerate if one of the signs in the tests is 0 . yap s scheme provides a black box for evaluating the polynomials from @xmath317 that , whenever the actual value is @xmath27 , also supplies a nonzero sign , @xmath318 or @xmath319 , which the algorithm may use for the branching , instead of the zero sign . thus , the algorithm never `` sees '' any degeneracy . yap s method guarantees that these signs are consistent , i.e. , for every input , the branching done in this way corresponds to some infinitesimal perturbation of the input sequence , and so does the output of the algorithm ( in our case , the answer to a range - searching query ) . for us , it is important that if the degrees of the polynomials in @xmath317 are bounded by a constant , the black box also operates in time bounded by a constant ( which is apparent from the explicit specification in @xcite ) . thus , applying the perturbation scheme influences the running time only by a multiplicative constant . it can be checked the range - searching algorithm presented above is of the required kind , with all branching steps based on the sign of suitable polynomials in the coordinates of the input points and in the coefficients of the polynomials in the query range , and the degrees of these polynomials are bounded by a constant . for producing the partitioning polynomial @xmath8 , we solve systems of linear equations , and thus the coefficients of @xmath8 are given by certain determinants obtained from cramer s rule . the computation of the polynomial partition and locating points in it is also based on the signs of suitable bounded - degree polynomials , as can be checked by inspecting the relevant algoritms , and similarly for intersecting a polynomial partition with the query range . the key fact is that all computations in the algorithm are of constant - bounded depth each of the values ever computed is obtained from the input parameters by a constant number of arithmetic operations . we also observe that when yap s scheme is applied , the algorithm never finds more than @xmath53 points in the exceptional set @xmath268 ( in any of the nodes of the partition tree ) . indeed , if @xmath320 input points lie in the zero set of a polynomial @xmath8 as in the algorithm , then a certain polynomial in the coordinates of these @xmath320 points vanishes ( see , e.g. , ( * ? ? ? * lemma 6.3 ) ) . thus , assuming that the algorithm found @xmath320 points on @xmath12 , it could test the sign of this polynomial at such points , and the black box would return a nonzero sign , which would contradict the consistency of yap s scheme . after applying yap s scheme , the preprocessing cost , storage , and query time remain asymptotically the same as in theorem [ t : const - r ] ( but with larger constants of proportionality ) . since the output of the algorithm corresponds to some infinitesimally perturbed version of the input ( point set and query range ) , we obtain a boundary - fuzzy answer for the original point set . if the points of @xmath0 are not in @xmath53-general position , we perturb them infinitesimally using the general perturbation scheme of yap @xcite , so that the perturbed set is in @xmath53-general position . then we construct the above data structure on the perturbed point set . by answering the query for this perturbed set , we obtain a boundary - fuzzy answer for the original point set . the preprocessing cost , storage , and query time remain asymptotically the same as in theorem [ t : const - r ] . this concludes the proof of corollary [ c : fuzzy ] . as mentioned in the introduction , if we construct an @xmath5-partitioning polynomial @xmath8 for an arbitrary point set @xmath0 , the exceptional set @xmath263 may be large , as is schematically indicated in fig . [ f : semi2-patches ] ( left ) . since @xmath268 is not partitioned by @xmath8 in any reasonable sense , it must be handled differently , as described below . following the terminology in @xcite , we call a direction @xmath321 _ good _ for @xmath8 if , for every @xmath322 , the polynomial @xmath323 does not vanish identically ; that is , any line in direction @xmath324 intersects @xmath12 at finitely many points . as argued in ( * ? ? ? * and pp . 314315 ) , a random direction is good for @xmath8 with probability 1 . by choosing a good direction and rotating the coordinate system , we assume that the @xmath325-direction , referred to as the _ vertical _ direction , is good for @xmath8 . in order to deal with @xmath268 , we partition @xmath12 into finitely many pieces , called _ patches _ , in such a way that each of the patches is _ monotone _ in the vertical direction , meaning that every line parallel to the @xmath325-axis intersects it at most once . this is illustrated , in the somewhat trivial 2-dimensional setting , in fig . [ f : semi2-patches ] ( right ) : ( middle ) , there are five one - dimensional patches @xmath326 , plus four 0-dimensional patches . then we treat each patch @xmath149 separately : we project the point set @xmath327 orthogonally to the coordinate hyperplane @xmath328 , and we preprocess the projected set , denoted @xmath329 , for range searching with suitable ranges . these ranges are projections of ranges of the form @xmath330 , where @xmath331 is one of the original ranges . in fig . [ f : semi2-patches ] ( right ) , ( middle ) , the patch @xmath332 is drawn thick , a range @xmath59 is depicted as a gray disk , and the projection @xmath333 of @xmath334 is shown as a thick segment in @xmath335 . the projected range @xmath336 is typically more complicated than the original range @xmath59 ( it involves more polynomials of larger degrees ) , but , crucially , it is only @xmath337-dimensional , and @xmath337-dimensional queries can be processed somewhat more efficiently than @xmath7-dimensional ones , which makes the whole scheme work . we will discuss this in more detail in section [ sec : range2 ] below , but first we recall the notion of _ cylindrical algebraic decomposition _ ( cad , or also _ collins decomposition _ ) , which is a tool that allows us to decompose @xmath12 into monotone patches , and also to compute the projected ranges @xmath336 . given a finite set @xmath247 of @xmath7-variate polynomials , a _ cylindrical algebraic decomposition adapted to @xmath244 _ is a way of decomposing @xmath2 into a finite collection of relatively open _ cells _ , which have a simple shape ( in a suitable sense ) , and which refine the arrangement @xmath245 . we refer , e.g. , to ( * ? ? ? * chap . 5.12 ) for the definition and construction of the `` standard '' cad . here we will use a simplified variant , which can be regarded as the `` first stage '' of the standard cad , and which is captured by ( * ? ? ? * theorem 5.14 , algorithm 12.1 ) . we also refer to ( * ? ? ? * appendix a ) for a concise treatment , which is perhaps more accessible at first encounter . let @xmath244 be as above . to obtain the first - stage cad for @xmath8 , one constructs a suitable collection @xmath338 of polynomials in the variables @xmath339 ( denoted by @xmath340 in @xcite ) . roughly speaking , the zero sets of the polynomials in @xmath341 , viewed as subsets of the coordinate hyperplane @xmath335 ( which is identified with @xmath342 ) , contain the projection onto @xmath335 of all intersections @xmath343 , @xmath344 , as well as the projection of the loci in @xmath345 where @xmath345 has a vertical tangent hyperplane , or a singularity of some kind . the actual construction of @xmath341 is somewhat more complicated , and we refer to the aforementioned references for more details . having constructed @xmath341 , the first - stage cad is obtained as the arrangement @xmath346 in @xmath2 , where the polynomials in @xmath341 are now considered as @xmath7-variate polynomials ( in which the variable @xmath325 is not present ) . in geometric terms , we erect a `` vertical wall '' in @xmath2 over each zero set within @xmath335 of a @xmath337-variate polynomial from @xmath341 , and the cad is the arrangement of these vertical walls plus the zero sets of @xmath347 . the first - stage cad is illustrated in fig . [ f : semi2-cad ] , for the same ( single ) polynomial as in fig . [ f : semi2-patches ] ( left ) . in our algorithm , we are interested in the cells of the cad that are contained in some of the sero sets @xmath345 ; these are going to be the monotone patches alluded to above . we note that using the first - stage cad for the purpose of decomposing @xmath12 into monotone patches seems somewhat wasteful . for example , the number of patches in fig . [ f : semi2-patches ] is considerably smaller than the number of patches in the cad in fig . [ f : semi2-cad ] . but the cad is simple and well known , and ( as will follow from the analysis in section [ sec : range2 ] ) possible improvements in the number of patches ( e.g. using the vertical - decomposition technique @xcite ) do not seem to influence our asymptotic bounds on the performance of the resulting range - searching data structure . the following lemma summarizes the properties of the first - stage cad that we will need ; we refer to ( * ? ? ? * theorem 5.14 , algorithm 12.1 ) for a proof . [ l : cad ] given a set @xmath348 $ ] of polynomials , each of degree at most @xmath46 , there is a set @xmath338 of @xmath349 polynomials in @xmath350 $ ] , each of degree @xmath351 , which can be computed in time @xmath352 , such that the first - stage cad defined by these polynomials , i.e. , the arrangement @xmath346 in @xmath2 , has the following properties : 1 . ( `` cylindrical '' cells ) for each cell @xmath353 of @xmath346 , there exists a unique cell @xmath354 of the @xmath337-dimensional arrangement @xmath355 in @xmath335 , such that one of the following possibilities occur : 1 . @xmath356 , where @xmath357 is a continuous semialgebraic function ( that is , @xmath353 is the graph of @xmath358 over @xmath354 ) . @xmath359 , where each @xmath360 , @xmath361 , is either a continuous semialgebraic real - valued function on @xmath354 , or the constant function @xmath362 , or the constant function @xmath363 , and @xmath364 for all @xmath365 ( that is , @xmath353 is a portion of the `` cylinder '' @xmath366 between two consecutive graphs ) . 2 . ( refinement property ) if @xmath367 , then @xmath368 , and thus each cell of @xmath346 is fully contained in some cell of @xmath369 . returning to the problem of decomposing the zero set of the partitioning polynomial @xmath8 into monotone patches , we construct the first - stage cad for @xmath286 , and the patches are the cells of @xmath346 contained in @xmath12 . if the @xmath325-direction is good for @xmath8 , then every cell of @xmath346 lying in @xmath12 is of type ( a ) , and so if any cell of type ( b ) lies in @xmath12 , we choose another random direction and construct the first - stage cad in that direction . putting everything together and using theorem [ t : make - arrg ] to bound the complexity of @xmath346 , we obtain the following lemma . [ l : patches ] let @xmath8 be a @xmath7-variate polynomial of degree @xmath46 , and let us assume that the @xmath325-direction is good for @xmath8 . then @xmath12 can be decomposed , in @xmath370 time , into @xmath371 monotone patches , and each patch can be represented semialgebraically by @xmath370 polynomials of degree @xmath372 . the first - stage cad can also be used to compute the projection of the intersection of a range in @xmath18 with a monotone patch of @xmath8 . essentially , this is done by forming the arrangement of @xmath8 and the polynomials defining @xmath59 , and by collecting the monotone patches in this arrangement that are contained in @xmath12 ; see the full version @xcite for more details . [ l : project ] let @xmath373 be the decomposition of the zero set of a @xmath7-variate polynomial @xmath8 of degree @xmath46 into monotone patches , as described in lemma [ l : patches ] , and let @xmath59 be a semialgebraic set in @xmath18 , with @xmath374 . for every patch @xmath375 , the projection of @xmath376 in the @xmath325-direction can be represented as a member of @xmath377 , i.e. , by a boolean combination of at most @xmath378 polynomial inequalities in @xmath337 variables , each of degree at most @xmath379 , where @xmath380 and @xmath381 . the representation can be computed in @xmath382 time . the task of computing @xmath336 , the projection of @xmath330 , is similar to the operation ( a2 ) discussed in section [ sec : cross ] . in more abstract terms , it can also be viewed as a quantifier elimination task : we can represent @xmath383 by a quantifier - free formula @xmath384 ( a boolean combination of polynomial inequalities ) ; then @xmath336 is represented by @xmath385 , and by eliminating @xmath386 we obtain a quantifier - free formula describing @xmath336 . more concretely , we use a procedure based on the first - stage cad ( lemma [ l : cad ] ) and the arrangement construction ( theorem [ t : make - arrg ] ) . by definition , @xmath59 is a boolean combination of inequalities of the form @xmath387 , where @xmath279 are @xmath7-variate polynomials , each of degree at most @xmath275 . we set @xmath388 , we compute the set @xmath389 of @xmath337-variate polynomials as in lemma [ l : cad ] , and the first - stage cad is then computed as the @xmath7-dimensional arrangement @xmath390 according to theorem [ t : make - arrg ] . since by lemma [ l : cad](ii ) , @xmath390 refines @xmath391 ( the first - stage cad from the preprocessing phase ) , each patch @xmath392 is decomposed into subpatches . since the sign of each @xmath393 is constant on each cell of @xmath394 , and thus on each cell of @xmath390 , @xmath330 is a disjoint union of subpatches . the projections of these subpatches into @xmath335 are cells of @xmath395 , and thus we obtain , in time @xmath382 , a representation of @xmath336 as a member of @xmath377 by theorem [ t : make - arrg ] , where @xmath380 and @xmath396 . [ sec : range2 ] we now describe our second data structure for @xmath269-range searching . compared to the first data structure from section [ sec : range1 ] , this one works on arbitrary point sets , without the @xmath53-general position assumption , or , alternatively , without the fuzzy boundary constraint on the output , and has slightly better performance bounds . the data structure is built recursively , and this time the recursion involves both @xmath1 and @xmath7 . let @xmath0 be a set of @xmath1 points in @xmath2 , and let @xmath20 and @xmath19 be parameters ( not assumed to be constant ) . the data structure for @xmath18-range searching on @xmath0 is obtained by constructing a partition tree @xmath300 on @xmath0 recursively , as above , except that now the fan - out of each node is larger ( and non - constant ) , and each node also stores an auxiliary data structure for handling the respective exceptional part . we need to set two parameters : @xmath397 and @xmath398 . neither of them is a constant in general ; in particular , @xmath5 is typically going to be a tiny power of @xmath1 . the specific values of these parameters will be specified later , when we analyze the query time . we also note that there is yet another parameter in theorem [ t : large - r ] , namely , the arbitrarily small constant @xmath52 entering the preprocessing time bound . however , @xmath26 enters the construction solely by the requirement that @xmath5 should be chosen smaller than @xmath399 , for a sufficiently large constant @xmath205 . it will become apparent later in the analysis that @xmath400 can be assumed , provided that some other parameters are taken sufficiently large ; we will point this out at suitable moments . when constructing the partition tree @xmath300 on an @xmath1-point set @xmath0 , we distinguish two cases . for @xmath401 , @xmath300 consists of a single leaf storing all points of @xmath0 . for @xmath402 , we construct an @xmath5-partitioning polynomial @xmath8 of degree @xmath235 , the partition @xmath261 of @xmath2 induced by @xmath8 , and the partition of @xmath0 into the exceptional part @xmath268 and regular parts @xmath403 , where @xmath259 . set @xmath404 and @xmath405 , for @xmath265 . the root of @xmath300 stores @xmath8 , @xmath261 , and the total weight @xmath305 of each regular part @xmath306 of @xmath0 , as before . still in the same way as before , we recursively preprocess each regular part @xmath306 for @xmath18-range searching ( or stop if @xmath406 ) , and attach the resulting data structure to the root as a respective subtree . a new feature of the second data structure is that we also preprocess the exceptional set @xmath268 into an auxiliary data structure , which is stored at the root . here we recurse on the dimension , exploiting the fact that @xmath268 lies on the algebraic variety @xmath12 of dimension at most @xmath407 . we choose a random direction @xmath324 and rotate the coordinate system so that @xmath324 becomes the direction of the @xmath325-axis . we construct the first - stage cad adapted to @xmath408 , according to lemma [ l : cad ] and theorem [ t : make - arrg ] . we check whether all the patches are @xmath325-monotone , i.e. , of type ( a ) in lemma [ l : cad](i ) ; if it is not the case , we discard the cad and repeat the construction , with a different random direction . this yields a decomposition of @xmath12 into a set @xmath409 of @xmath371 monotone patches , and the running time is @xmath370 with high probability . next , we distribute the points of @xmath268 among the patches : for each patch @xmath392 , let @xmath329 denote the projection of @xmath410 onto the coordinate hyperplane @xmath411 . we preprocess each set @xmath329 for @xmath377-range searching . here @xmath381 is the number of polynomials defining a range and @xmath412 is their maximum degree ; the constants hidden in the @xmath413 notation are the same as in lemma [ l : project ] . for simplicity , we treat all patches as being @xmath337-dimensional ( although some may be of lower dimension ) ; this does not influence the worst - case performance analysis . the preprocessing of the sets @xmath329 is done recursively , using an @xmath414-partitioning polynomial in @xmath342 , for a suitable value of @xmath414 . the exceptional set at each node of the resulting `` @xmath337-dimensional '' tree is handled in a similar manner , constructing an auxiliary data structure in @xmath415 dimensions , based on a first - stage cad , and storing it at the corresponding node . the recursion on @xmath7 bottoms out at dimension @xmath160 , where the structure is simply a standard binary search tree over the resulting set of points on the @xmath416-axis . we remark that the treatment of the top level of recursion on the dimension will be somewhat different from that of deeper levels , in terms of both the choice of parameters and the analysis ; see below for details . this completes the description of the data structure , except for the choice of @xmath5 and @xmath307 , which will be provided later as we analyze the performance of the algorithm . let us assume that , for a given @xmath0 , the data structure for @xmath269-range searching , as described above , has been constructed , and consider a query range @xmath417 . the query is answered in the same way as before , by visiting the nodes of the partition tree @xmath300 in a top - down manner , except that , at each node that we visit , we also query with @xmath59 the auxiliary data structure constructed on the exceptional set @xmath268 for that node . specifically , for each patch @xmath149 of the corresponding collection @xmath409 , we compute @xmath418 , the weight of @xmath419 . if @xmath420 then @xmath421 , and if @xmath422 then @xmath418 is the total weight of @xmath327 . otherwise , i.e. , if @xmath59 crosses @xmath149 , then @xmath418 is the same as the weight of @xmath423 , where @xmath336 is the @xmath325-projection of @xmath330 , because @xmath149 is @xmath325-monotone . by lemma [ l : project ] , @xmath424 and can be constructed in @xmath382 time . we can find the weight of @xmath425 by querying the auxiliary data structure for @xmath329 with @xmath336 . we then add @xmath418 to the global count maintained by the query procedure . this completes the description of the query procedure . the analysis of the storage requirement and preprocessing time is straightforward , and will be provided later . we begin with the more intricate analysis of the query time . for now we assume that @xmath307 and @xmath5 have been fixed ; the analysis will later specify their values . a straightforward analysis shows that the size of the data structure is linear and that it can be constructed in time @xmath65 , for any constant @xmath426 , by choosing @xmath5 sufficiently large , so we focus on analyzing the query time . let @xmath427 denote the maximum overall query time for @xmath269-range searching on a set of @xmath1 points in @xmath2 . for @xmath428 and @xmath429 , @xmath430 . for @xmath431 and @xmath432 , @xmath433 because any range in @xmath434 is the union of at most @xmath435 intervals . finally , for @xmath436 and @xmath432 , an analysis similar to the one in section [ sec : range1 ] gives the following recurrence for @xmath427 : @xmath437 @xmath438 where @xmath293 , @xmath277 is a constant depending on @xmath7 , @xmath439 , and both @xmath440 and @xmath441 are bounded by @xmath442 with @xmath235 and @xmath443 . ( these are rather crude estimates , but we prefer simplicity . ) the leading term of the recurrence relies on the crossing - number bound given in lemma [ l : cr ] . in order to apply that lemma , we need that @xmath444 , which will be ensured by the choice of @xmath5 given below . the second term corresponds to querying the auxiliary data structures for the exceptional set @xmath268 . the last term covers the time spent in computing the cells of the polynomial partition crossed by the query range @xmath59 and for computing the projections @xmath336 for every @xmath445 ; here we assume that the choice of @xmath5 will be such that @xmath446 . ultimately , we want to derive that if @xmath298 are constants , the recurrence ( [ eq : qtime ] ) implies , with a suitable choice of @xmath5 and @xmath307 at each stage , @xmath447 where @xmath448 is a constant depending on @xmath449 , and @xmath19 . however , as was already mentioned , even if @xmath298 are constants initially , later in the recursion they are chosen as tiny powers of @xmath1 , and this makes it hard to obtain a direct inductive proof of ( [ qt : good ] ) . instead , we proceed in two stages . first , in lemma [ l : weakerq ] below we derive , without assuming @xmath298 to be constants , a weaker bound for @xmath427 , for which the induction is easier . then we obtain the stronger bound ( [ qt : good ] ) for constant values of @xmath298 by using the weaker bound for the @xmath337-dimensional queries on the exceptional parts , i.e. , for the second term in the recurrence ( [ eq : qtime ] ) . [ l : weakerq ] for every @xmath450 there exists @xmath451 such that , with a suitable choice of @xmath5 and @xmath307 , @xmath452 for all @xmath453 ( with @xmath454 , say ) . \(i ) this lemma may look similar to our first result on @xmath18-range searching , theorem [ t : const - r ] , but there are two key differences the lemma works for arbitrary point sets , with no general position assumption , and @xmath20 and @xmath19 are not assumed to be constants . \(ii ) since query time @xmath55 is trivial to achieve , we may assume @xmath455 , for otherwise , the bound ( [ qt : bad ] ) in the lemma exceeds @xmath1 . the case @xmath431 is trivial because @xmath456 clearly implies ( [ qt : bad ] ) , assuming that @xmath457 and that @xmath1 is sufficiently large so that @xmath458 . we assume that ( [ qt : bad ] ) holds up to dimension @xmath407 ( for all @xmath450 , @xmath20 , @xmath19 , and @xmath1 ) , and we establish it for dimension @xmath7 by induction on @xmath1 . we consider @xmath451 yet unspecified but sufficiently large ; from the proof below one can obtain an explicit lower bound that @xmath451 should satisfy . we set @xmath459 this value of @xmath307 is roughly the threshold where the bound ( [ qt : bad ] ) becomes smaller than @xmath1 . since we assume @xmath460 , our choice of @xmath5 satisfies the assumptions @xmath444 and @xmath446 , as needed in ( [ eq : qtime ] ) . in the inductive step , for @xmath428 , @xmath461 so we assume that @xmath432 and that the bound ( [ qt : bad ] ) holds for all @xmath462 . using the induction hypothesis , i.e. , plugging ( [ qt : bad ] ) into the recurrence ( [ eq : qtime ] ) , we obtain @xmath463 by the choice of @xmath5 , the first term of the right - hand side of ( [ eq : induct ] ) can be bounded by @xmath464 which is half of the bound we are aiming for . next , we bound the second term . we use the estimates @xmath465 , @xmath466 , and @xmath467 . then @xmath468 we choose @xmath469 where @xmath470 ; i.e. , we choose @xmath471 . since @xmath472 and @xmath473 , the fraction in ( [ eq : q - aux ] ) can be bounded by @xmath474 because @xmath454 . finally , recall that @xmath293 , so our choice of @xmath451 ( again , choosing @xmath475 sufficiently large ) ensures that @xmath476 . hence , the right hand side in ( [ eq : induct ] ) is bounded by @xmath477 as desired . this establishes the induction step and thereby completes the proof of the lemma . now we want to obtain the improved bound ( [ qt : good ] ) , i.e. , @xmath478 , with @xmath479 , assuming that @xmath298 are constants and @xmath480 . to this end , in the top - level ( @xmath7-dimensional ) partition tree , we set @xmath481 , where @xmath482 is a suitable small constant to be specified later . then we use the result of lemma [ l : weakerq ] with @xmath483 for processing the @xmath337-dimensional queries on the sets @xmath329 . thus , in the forthcoming proof , we do induction only on @xmath1 , while @xmath7 is fixed throughout . we choose @xmath397 sufficiently large ( we will specify this more precisely later on ) , and we assume that @xmath402 and that the desired bound ( [ qt : good ] ) holds for all @xmath462 . in the inductive step we estimate , using the recurrence ( [ eq : qtime ] ) , the induction hypothesis , and the bound in ( [ qt : bad ] ) , @xmath484 the first term simplifies to @xmath485 . thus , if we choose @xmath67 depending on @xmath486 ( which is a small positive constant still to be determined ) so that @xmath487 , then the first term will be at most half of the target value @xmath488 . thus , it suffices to set @xmath486 so that the remaining two terms are negligible compared to this value . for the @xmath489 term , any @xmath490 will do . the second term can be bounded , as in the proof of lemma [ l : weakerq ] , by @xmath491 thus , with @xmath492 , the term is at most @xmath493 . again , this establishes the induction step and concludes the proof of the final bound for the query time . we remark that our choice of @xmath486 requires us to choose @xmath494 making its dependence on @xmath7 super - exponential . first we derive a weaker bound for @xmath427 without assuming @xmath298 to be constants . namely , we prove that for every constant @xmath450 there exists a constant @xmath451 such that , with a suitable choice of @xmath5 and @xmath307 , @xmath452 for all @xmath453 ( with @xmath454 , say ) . we can assume that @xmath495 because otherwise the query time is trivially @xmath55 . we choose @xmath473 , which ensures that @xmath496 . next , we derive the stronger bound ( [ qt : good ] ) for constant values of @xmath298 by using this weaker bound for the @xmath337-dimensional queries on the projected exceptional parts , i.e. , for the second term in the recurrence ( [ eq : qtime ] ) . in this stage , we choose @xmath497 for a sufficiently small constant @xmath498 . our choice of @xmath486 and @xmath451 implies that @xmath499 . additional details can be found in the full version @xcite . let @xmath500 denote the size of the data structure on @xmath1 points in @xmath2 for @xmath269-range searching , with the settings of @xmath5 and @xmath307 as described above . for @xmath501 we have @xmath502 . for larger values of @xmath1 , the space occupied by the root of the partition tree , not counting the auxiliary data structure for the exceptional part @xmath268 , is bounded by @xmath489 , where @xmath293 . furthermore , since @xmath500 is at least linear in @xmath1 , the total size of the auxiliary data structure constructed on @xmath268 is @xmath503 , where @xmath404 . we thus obtain the following recurrence for @xmath500 : @xmath504 for @xmath505 , and @xmath502 for @xmath401 . using @xmath506 , @xmath507 , and @xmath508 , for both types of choices of @xmath5 , the recurrence easily leads to @xmath509 where the constant of proportionality depends on @xmath7 . it remains to estimate the preprocessing time ; here , finally , the parameter @xmath52 in theorem [ t : large - r ] comes into play . let @xmath510 be a constant such that @xmath511 ( at all stages of the algorithm ) . as was remarked in the preceding analysis of the query time , we can make @xmath510 arbitrarily small , by adjusting various constants ( and , generally speaking and as already remarked above , the smaller @xmath510 , the worse constant @xmath448 we obtain in the query time bound ) . let @xmath512 denote the maximum preprocessing time of our data structure for @xmath18-range searching on @xmath1 points , with @xmath513 a constant as above . using the operation ( a1 ) of section [ sec : cross ] , we spend @xmath292 time to compute @xmath514 and the partition of @xmath0 into the exceptional part and the regular parts , and we spend additional @xmath292 time to compute @xmath409 and @xmath329 for every @xmath515 , where @xmath293 . the total time spent in constructing the secondary data structures for all patches of @xmath409 is bounded by @xmath516 . hence , we obtain the recurrence @xmath517 for @xmath402 , and @xmath518 for @xmath401 . using the properties @xmath506 and @xmath507 , a straightforward calculation shows that @xmath519 where the constant of proportionality depends on @xmath7 . hence , by choosing @xmath520 , the preprocessing time is @xmath65 . this concludes the proof of theorem [ t : large - r ] . this concludes the proof of theorem [ t : large - r ] . we conclude this paper by mentioning a few open problems . \(i ) a very interesting and challenging problem is , in our opinion , the fast - query case of range searching with constant - complexity semialgebraic sets , where the goal is to answer a query in @xmath521 time using roughly @xmath522 space . there are actually two , apparently distinct , issues . the standard approach to fast - query searching is to parameterize the ranges in @xmath13 by points in a space of a suitable dimension , say @xmath33 ; then the @xmath1 points of @xmath0 correspond to @xmath1 algebraic surfaces in this @xmath33-dimensional `` parameter space '' , and a query is answered by locating the point corresponding to the query range in the arrangement of these surfaces . first , the arrangement has @xmath523 combinatorial complexity , and one would expect to be able to locate points in it in polylogarithmic time with storage about @xmath524 . however , such a method is known only up to dimension @xmath525 , and in higher dimension , one again gets stuck at the arrangement decomposition problem , which was the bottleneck in the previously known solution of @xcite for the low - storage variant , as was mentioned in the introduction . it would be nice to use polynomial partitions to obtain a better point location data structure for such arrangements , but unfortunately , so far all of our attempts in this direction have failed . the second issue is , whether the point location approach just sketched is actually optimal . this question is exhibited nicely already in the simple instance of range searching with disks in the plane . the best known solution that guarantees logarithmic query time uses point location in @xmath526 and requires storage roughly @xmath527 , but it is conceivable that roughly quadratic storage might suffice . \(ii ) our range - searching data structure for arbitrary point sets the one with large fan - out is so complex and has a rather high exponent in the polylogarithmic factor , because we have difficulty with handling highly degenerate point sets , where many points lie on low - degree algebraic surfaces . this issue appears even more strongly in combinatorial applications , and in that setting it has been dealt with only in rather specific cases ( e.g. , in dimension 3 ) ; see @xcite for initial studies . it would be nice to find a construction of suitable `` multilevel polynomial partitions '' that would cater to such highly degenerate input sets , as touched upon in @xcite . \(iii ) another open problem , related to the construction of polynomial partitions , is the fast evaluation of a multivariate polynomial at many points , as briefly discussed at the end of section [ sec : algo ] . p. k. agarwal and j. erickson , geometric range searching and its relatives , in : _ advances in discrete and computational geometry _ ( b. chazelle , j. e. goodman and r. pollack , eds . ) , ams press , providence , ri , 1998 , pp . 156 . p. k. agarwal , j. matouek , m. sharir , on range searching with semialgebraic sets ii , in arxiv . s. barone and s. basu , refined bounds on the number of connected components of sign conditions on a variety , _ discrete comput . _ 47 ( 2012 ) , 577597 . s. basu , r. pollack , and m .- f . roy , on the number of cells defined by a family of polynomials on a variety , _ mathematika _ 43 ( 1996 ) , 120126 . b. chazelle , h. edelsbrunner , l. j. guibas , and m. sharir , a singly exponential stratification scheme for real semi - algebraic varieties and its applications , _ theoret . _ , 84 ( 1991 ) , 77105 . also in _ proc 16th int . colloq . on automata , languages and programming _ ( 1989 ) , pp . 179193 . elekes , h. kaplan and m. sharir , on lines , joints , and incidences in three dimensions , _ j. combinat . theory , ser . a _ 118 ( 2011 ) , 962977 . l. guth and n. h. katz , algebraic methods in discrete analogs of the kakeya problem , _ advances math . _ 225 ( 2010 ) , 28282839 . l. guth and n. h. katz , on the erds distinct distances problem in the plane , in arxiv:1011.4105 . h. kaplan , j. matouek and m. sharir , simple proofs of classical theorems in discrete geometry via the guth katz polynomial partitioning technique , _ discrete comput . geom . _ 48 ( 2012 ) , 499517 . h. kaplan , j. matouek , z. safernov and m. sharir , unit distances in three dimensions , _ combinat . _ 21 ( 2012 ) , 597610 . k. kedlaya and ch . umans , fast modular composition in any characteristic , _ proc . 49th annu . ieee sympos . sci . _ ( 2008 ) , 146155 . knauer , h. r. tiwari and d. werner , on the computational complexity of ham - sandwich cuts , helly sets , and related problems , _ proc . 28th annu . aspects comput . ( 2011 ) , 649660 . m. sharir and h. shaul , semi - algebraic range reporting and emptiness searching with applications , _ siam j. comput . _ 40 ( 2011 ) , 10451074 . j. solymosi and t. tao , an incidence theorem in higher dimensions , _ discrete comput . _ 48 ( 2012 ) , 255280 . a. h. stone and j. w. tukey , generalized sandwich theorems , _ duke math . j. _ 9 ( 1942 ) , 356359 .
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let @xmath0 be a set of @xmath1 points in @xmath2 .
we present a linear - size data structure for answering range queries on @xmath0 with constant - complexity semialgebraic sets as ranges , in time close to @xmath3 .
it essentially matches the performance of similar structures for simplex range searching , and , for @xmath4 , significantly improves earlier solutions by the first two authors obtained in 1994 .
this almost settles a long - standing open problem in range searching .
the data structure is based on the polynomial - partitioning technique of guth and katz [ arxiv:1011.4105 ] , which shows that for a parameter @xmath5 , @xmath6 , there exists a @xmath7-variate polynomial @xmath8 of degree @xmath9 such that each connected component of @xmath10 contains at most @xmath11 points of @xmath0 , where @xmath12 is the zero set of @xmath8 .
we present an efficient randomized algorithm for computing such a polynomial partition , which is of independent interest and is likely to have additional applications .
| 26,888 | 297 |
type ia supernovae ( sne ia ) , characterized by no @xmath1 but strong si lines in the spectra at the maximum brightness , are brighter than most of sne classified into the other types and exhibit uniform light curves . thus they are used as a standard candle to measure distances to remote galaxies . a plausible explosion model for sne ia is the accreting white dwarf ( wd ) model , in which a white dwarf in a binary system accretes material from the companion star , increases its mass , usually up to the chandrasekhar mass limit ( @xmath2 ) , and then explodes ( e.g. , * ? ? ? . there have been significant progresses in the accreting wd model since @xcite introduced the stellar wind from the wd while it accretes materials from the companion . their model succeeded in sustaining a stable mass transfer in the progenitor systems of sne ia . according to their model , there are two main evolutionary paths leading to sne ia , the super soft x - ray source ( sss ) channel and the symbiotic channel @xcite . accordingly their model predicts which companion stars lead to sne ia . the above evolutionary scenario for sne ia has not been confirmed by observations , which will require the identification of the companion star that should remain in the vicinity of the explosion site . @xcite argued that their group identified a g2iv star as the companion of tycho brahe s supernova remnant ( snr ) by measuring the velocities and distances of stars in the vicinity of the center of the snr . they concluded that this g2iv star was moving much faster than the other neighbor stars and that the distance to this star seemed consistent with the distance to tycho s snr . although @xcite has argued that the observed velocity of tycho g might correspond to the velocity of stars belonging to the thick disk population , the expected stellar mass in thick disk stars within the cone with 2.87 arcsec radius ( which corresponds to the angular distance from the center of tycho s snr to the tycho g star ) , at 3 kpc from earth , is only 2 @xmath3 , which makes the thick disk star alternative very unlikely . for this estimate we use the density in the vicinity of tycho s snr @xcite . although the coincidence of the kinematic characteristics of tycho g with its being at the position and distance of the snr appears significant , confirmation by other means would nonetheless be very useful . in this paper we propose a direct method to prove that the companion star is located inside the sn ejecta . a hint was dropped by observations for a star called s - m star discovered by @xcite near a type ia snr 1006 . @xcite proved that the s - m star was not the companion of this supernova by investigating features of fe ii absorption lines in the ultraviolet ( uv ) spectrum . very broad wings were observed in both blue and red sides of the absorption lines . the line width was a few thousand km / s much larger than the thermal velocity of stellar atmosphere , which is thought to be @xmath410 km / s . the broad wings are likely to be formed by fe ii in the ejecta of snr 1006 ; photons in the blue wing are absorbed by the matter ejected toward us , and those in the red wing away from us . thus it was proved that the s - m star is located behind the snr 1006 ejecta . when a star is inside the ejecta of sne ia , photons emitted from the star are absorbed only by the ejecta moving toward us . hence the broad wing must be present only in the blue side . the absorption line with only blue wing enables us to identify companion stars of sne ia . @xcite used the uv range to observe the s - m star with the faint object spectrograph on the hubble space telescope . however , in addition to difficulties in uv observations from the ground , companion stars on the evolutionary paths suggested by the above mentioned scenario @xcite may not be bright in the uv range . then we will focus on absorption lines in the visible range . furthermore the corresponding transitions need to be from the ground state because most fe ions in the ejecta are expected to be in the ground state . thus only fe i can produce such absorption lines in sne ia ejecta . in this paper , we estimate the amount of fe i in the ejecta of tycho s snr by taking account of collisional processes in non - equilibrium and ionizations by photons emitted from the shocked ejecta , and calculate spectra of a star located at the center of a snr and discuss whether we can identify the feature of fe i absorption lines in the spectrum of the companion star . [ cols="^,^,^,^,^,^ " , ] [ tbl : ew ] even when we find a star that exhibits the absorption feature discussed in this paper , there is a chance that a star other than the companion star happens to be inside the ejecta . a star existing in the vicinity of a sn ia gets a fraction of the explosion energy and is accelerated . if the companion star similar to tycho g star with the mass of @xmath5 1.2 @xmath6 and the radius of @xmath7 2 @xmath8 is located at the distance of @xmath9 5.5 @xmath8 from the progenitor , the size of the roche lobe is comparable to the size of the companion . thus the star in this situation will get the maximum velocity after the explosion . then the orbital velocity before the explosion is @xmath10 km / s . in a 3-d hydrodynamical simulation of sne ia by @xcite , they obtain a plausible kick velocity @xmath11 km / s . therefore , including the orbital velocity , the velocity of the companion star becomes @xmath12 km / s . if the companion star has been moving away from the explosion site at that speed , the companion star is now at the distance of @xmath130.08 pc from the center . the stellar mass density in the neighborhood of tycho s snr estimated from a model of the galaxy @xcite is less than 0.03 @xmath14 . thus the expected stellar mass inside this volume is only @xmath15 . therefore if we find a star showing absorption lines with broad blue wings in the spectrum , it is likely that the star is the companion star . as a consequence , we have demonstrated that there exhibit very deep absorption lines with unique shapes in the spectrum of the companion star located at the center of a young snr such as tycho s snr . there are , however , a few factors that might reduce or even erase these distinct absorption features . first , the number of fe i in the freely expanding ejecta is very sensitive to the number of ionizing photons emitted from the shocked ejecta . an increase in the number of ionizing photons by a few factors might decrease the number of fe i by a few orders of magnitude or more . since the main source of ionizing photons is o in the outer ejecta , the distributions of o and density in the outer ejecta need to be known precisely . unfortunately , these regions in w7 have some problems to reproduce the observed optical spectra ( * ? ? ? * and references therein ) . due to this uncertainty in the explosion model , it is not conclusive if the companion star of tycho s sn will exhibit unique fe i absorption features discussed in this paper . nevertheless , it is true that every sn ia has a period during which the companion star has the distinct absorption features discussed above because the ejecta become cool enough to have plenty of fe i for a time after the optical brightening . second , it is assumed in our calculations that ions in the shocked region are ionized to fe@xmath16 , si@xmath17 , o@xmath18 , c@xmath19 immediately after the shock passes following the procedure taken by @xcite . since ions in lower ionization stages are a strong source of ionizing photons , ionization may be more advanced in real young snrs . arnaud , m. rothenflug , r. 1985 , a&as , 60 , 425 branch , d. 1998 , , 36 , 17 colella , p. , woodward , p. 1984 , jcoph , 54 , 174 decourchelle , a. , et al . 2001 , a&as , 365 , l218 dehnen , w. , binney , j. , 1998 , mnras , 294 , 429 engstrom , l. , et al . 1992 , jphb , 25 , 2459 fuhrmann , k. 2005 , mnras , 359 , l35 hachisu , i. , kato , m. , nomoto , k. 1996 , apj , 470 , l97 hachisu , i. , kato , m. , nomoto , k. , umeda , h. 1999 , apj , 519 , 314 hachisu , i. , kato , m. , nomoto , k. 1999 , apj , 522 , 487 hamilton , a. , sarazin , c. 1984 , apj , 287 , 282 hamilton , a. , fesen , r. 1988 , apj , 327 , 178 itoh , h. , masai , k. , nomoto , k. 1988 , apj , 334 , 279 kato , t. 1976 , apjs , 30 , 397 reinecke , m. , hillebrandt , w. , & niemeyer , j. c. 2002 , , 391 , 1167 lotz , w. 1967 , apjs , 14 , 207l marietta , e. , burrows , a. , fryxell , b. 2000 , apjs , 128 , 615 mihalas , d. , 1978 , stellar atmospheres . nomoto , k. 1982 , , 253 , 798 nomoto , k , thielemann , f. , yokoi , k. 1984 , apj , 286 , 644 ruiz - lapuente , p. , et al . 2004 , nature , 431 , 1069 schweizer , f. middleditch , j. 1980 , apj , 241 , 1039 sorokina , e. i. , et al . 2004 , astl , 30 , 737 spitzer , l. 1978 , physical processes in the interstellar medium . tunklev , m. , et al . 1997 , phys , 55 , 707 verner , d. , et al . 1996 , apj , 465 , 487 vink , j. , et al . 2003 , apjl , 587 , l31 wang , l. , baade , d. , hflich , p. , wheeler , j. c. , kawabata , k. , & nomoto , k. 2004 , , 604 , l53 woosley , s. e. , & weaver , t. a. 1986 , lecture notes in physics , berlin springer verlag , 255 , 91 wu , c. , et al . 1993 , apj , 416 , 247
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we propose a method to identify the companion stars of type ia supernovae ( sne ia ) in young supernova remnants ( snrs ) by recognizing distinct features of absorption lines due to fe i appearing in the spectrum .
if a sufficient amount of fe i remains in the ejecta , fe i atoms moving toward us absorb photons by transitions from the ground state to imprint broad absorption lines exclusively with the blue - shifted components in the spectrum of the companion star . to investigate the time evolution of column depth of fe i in the ejecta
, we have performed hydrodynamical calculations for snrs expanding into the uniform ambient media , taking into account collisional ionizations , excitations , and photo - ionizations of heavy elements . as a result
, it is found that the companion star in tycho s snr will exhibit observable features in absorption lines due to fe i at @xmath0 nm and 385.9911 nm if a carbon deflagration sn model @xcite is taken .
however , these features may disappear by taking another model that emits a few times more intense ionizing photons from the shocked outer layers . to further explore the ionization states in the freely expanding ejecta
, we need a reliable model to describe the structure of the outer layers .
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group field theories ( gfts ) @xcite are quantum field theories on group manifolds , with the group chosen to be the local gauge group of spacetime in d dimensions , i.e. the lorentz group , or a suitable extension of it , for models aiming at a quantization of d - dimensional gravity . they are characterized by a non - local pairing of field arguments in the interaction term , designed in such a way as to produce , in perturbative expansion , feynman diagrams with a combinatorial structure that are in 1 - 1 correspondence with d - dimensional simplicial complexes . because of these basic properties , gfts can be understood as a generalization of matrix models @xcite for 2-dimensional quantum gravity , obtained in two steps : 1 ) by passing to generic tensors , instead of matrices , as fundamental variables , thus obtaining a generating functional for the sum over 3d simplicial complexes that was the essence of the dynamical triangulations approach to 3d quantum gravity@xcite ; 2 ) adding group structure defining extra geometric degrees of freedom . the last step is what turns a generic tensor model into a proper field theory . in fact , the first example of a gft was the group - theoretic generalization of 3d tensor models proposed by boulatov @xcite . it was already at this initial stage , that group field theories allowed a direct contact between simplicial quantum gravity and what we now call spin foam models @xcite . in fact the boulatov model , sharing the same combinatorics of tensor models and thus reproducing a 3d sum over triangulations , in perturbative expansion , produces weights for these triangulations given by the so - called ponzano - regge spin foam model , thanks to the additional su(2 ) group structure . we now know that this is just one example of a very general result @xcite : any group field theory produces feynman amplitudes that can be re - expressed as spin foam models , and , conversely , any spin foam model for fixed 2-complex can be understood as the feynman amplitude for a given feynman diagram of a corresponding gft . in turn , spin foam models @xcite have been a very active ( and growing ) area of quantum gravity research in the past ten years , for two main ( and related ) reasons . first , one obtains a spin foam model when considering , once more , a discretization of continuum 1st order gravity , formulated as a constrained bf theory , on a simplicial complex , and quantizes it via path integral methods . second , spin foams as 2-complexes with faces labelled by group representations arise naturally when considering the dynamics of the kinematical quantum states of geometry as identified by canonical loop quantum gravity @xcite ; indeed , from the lqg perspective , spin foams represents the histories of spin networks and are thus the crucial ingredient of any path integral or covariant formulation of the quantum gravity dynamics in lqg . from both the simplicial and canonical perspective , a sum over spin foams , weighted by appropriate amplitudes , is necessary to define in full the dynamics of the gravitational field : in simplicial quantum gravity because only such sum ( lacking a suitably defined refinement procedure ) can compensate the truncation of geometric degrees of freedom that the restriction to a given lattice imposes ; in lqg , because a complete path integral formulation of the dynamics needs , in general , i.e. except in the purely topological case or for very special choices of observables , a sum over all the histories between given spin network states . at present , group field theories are the only known tool to define uniquely such sum over spin foams , i.e. with fully specified weights , in a perturbative expansion of the gft partition function . in this property , lies the main reason of interest in gfts , from the lqg perspective . and indeed , up to now , group field theories have been mainly considered and used just as such : as a _ tool _ to define a sum over spin foams with prescribed weights , i.e. as an auxiliary formalism to define / construct spin foam models . a different perspective is however possible , even if only tentatively or as a working hypothesis , at present , given the very limited understanding and control we have of the whole group field theory formalism . this perspective has been advocated and described in more detail in @xcite , to which we refer . + group field theories can be understood as a 2nd quantized formulation of both loop quantum gravity and simplicial gravity , in which wave functions over the space of geometries are turned into classical fields first and then into operators . in fact , the quanta of the gft field correspond to d - valent spin network vertices @xcite and , at the same time , to ( d-1)-simplices . since both loop quantum gravity and simplicial quantum gravity ( in its quantum regge calculus as well as in its dynamical triangulations formulation ) are meant to be themselves a quantization of what is already a classical field theory , i.e. general relativity , a 2nd quantization of the same provides a formalism very similar in spirit to what was ( somewhat improperly ) dubbed third quantizationof gravity @xcite . this was a formal quantum field theory on superspace ( the space of all 3-geometries on , say , @xmath0 ) which would have canonical quantum gr corresponding to its quantum 1-particle sector , and would produce a sum over topologies in perturbative feynman expansion , obtained as interaction processes of quantum universes ( the particlesof this formalism ) . clearly , the mathematical difficulties as well as those regarding the physical interpretation of the whole formalism are formidable , and consequently they prevented any substantial development . + the group field theory formalism @xcite greatly improves the situation , both at the mathematical and physical level , by turning to a discrete and local picture of superspace , in which what is propagated , created and annihilated are local chunks of a ( d-1)-dimensional quantum space ( again , spin network vertices or ( d-1)-simplices ) . one feature of 3rd quantization is retained , and even generalized , however : on the one hand these fundamental building blocks of quantum space can be combined at the kinematical level to represent , in principle , any spatial geometry and topology , and on the other hand their interactions still produce , in perturbative feynman expansion , quantum spacetimes of any topology . + this step from a continuum to a discrete setting has another important consequence . if the quantum gravity feynman diagrams in the 3rd quantized formalism were smooth manifolds , and the continuum path integral for quantum gravity on the given manifold represented , by construction , the feynman amplitude for each of them , in the discrete version of 3rd quantization provided by group field theories the feynman diagrams are given by combinatorial and un - embedded 2-complexes or , dually , by simplicial complexes , and the feynman amplitudes have the interpretation of discrete path integrals ( sum over discrete geometries ) over each simplicial complex . + this feature allows not only a better mathematical control on the quantities involved , but it also makes group field theories , at least in principle , a common framework for several approaches to quantum gravity , providing the basis for potential cross - fertilization and mutual enrichment . + as mentioned , each group field theory , in fact completely defines a possible dynamics for loop quantum gravity spin network states , thus proposing an explicit solution ( even if yet to be put to test ) for one outstanding challenge of the lqg programme . at the same time , it does so by defining a sum over histories that includes the crucial ingredients of both main simplicial quantum gravity approaches . as in quantum regge calculus @xcite , it defines the dynamics of a quantum and classical simplicial geometry in terms of a sum over discrete geometric data ( edge lengths , areas , etc , according to dimension ) for a given base simplicial complex . in addition to this , it provides a possible way to encode all the continuum geometric degrees of freedom by a covariant procedure that is alternative to the infinite refinement procedure for fixed incidence matrix , that has proven to be problematic at the quantum level , in quantum regge calculus : a sum over inequivalent triangulations . in this it obviously agrees with the dynamical triangulations ( dt ) approach @xcite ; in fact , freezing the discrete geometric data attached to each complex ( this can be done in various ways ) turns the perturbative expansion of gfts into a form very similar to the sum over triangulations weighted by a purely combinatorial factor that characterizes the dt approach ; such formulation , in its _ causal _ restriction ( corresponding to summing over a certain restricted class of lorentzian triangulations ) @xcite , has recently proven to be much more successful in recovering a continuum spacetime from the quantum theory . as a bonus , with respect to the dynamical triangulations approach , the gft approach permits a better control over the classical and quantum _ simplicial _ dynamics of geometry , which is here akin to the few - particlephysics in the usual qfts . this stems from the possibility of a finer control over simplicial geometric variables , that puts gfts and their spin foam amplitudes in closer relation with lqg , regge calculus and other formulations of classical simplicial gravity . this seems also to be confirmed by recent results on the spin foam / lqg lattice graviton , which match at least partially the results of simplicial gravity , in the regime where a connection between the two approaches can be made precise @xcite . the links between group field theories and other approaches to quantum gravity are detailed in @xcite , to which we refer . + it should be clear , however , that much more remains to be understood about these links and that only further work can confirm or refute the idea that group field theories really represent in concrete terms a unifying framework for all of them , as we have been suggesting . as for the relationship with lqg , the open issues concern both the exact correspondence between gft boundary states and their hilbert space , in specific models , and that of usual su(2 ) lqg spin networks , as well as the relation between the gft transition amplitudes and the canonical lqg hamiltonian constraint , defining the quantum dynamics . these issues will not concern us here . the focus of our present work is instead on the relation between gfts and simplicial quantum gravity , and the aim is to make the correspondence between the two framework detailed and clear , with a precise matching between gft feynman amplitudes and simplicial quantum gravity sum over histories . in order to achieve this , we introduce and analyze in the present paper a new class of group field theories , characterized by feynman amplitudes which have , in any dimension and in any signature , _ exactly _ the form of a simplicial gravity path integral . its amplitudes , that is , are expressed as a ( real ) measure part times a phase factor , with phase clearly identified with a simplicial gravity action . in 3 dimensions , this will mean obtaining a path integral for 3-dimensional simplicial gravity in first order formalism , corresponding to a 1st order regge calculus action plus higher order ( @xmath1-like ) corrections . in higher dimensions , we will obtain instead a path integral , augmented by a sum over triangulations of any topology , for what can be interpreted as topological bf theory with an additional orientation dependence , and , again , higher order ( quantum ) corrections to the action . this work can be understood as the further development of the line of research on the issue of causality in spin foam quantum gravity and gft , and on the construction of a unified gft framework for loop quantum gravity , spin foam models , quantum regge calculus and dynamical triangulations , that started from an analysis of the issue of causality in spin foam models @xcite , continued with the development of a refined technique for the construction of causal spin foam models , based on the particle analogy which implicitly introduced additional variables into the usual spin foam formalism , in @xcite , with the explicit construction of causal spin foam models for pure gravity and gravity coupled to matter in 3d @xcite , and with the construction of a generalized gft formalism @xcite based on the techniques and variables introduced in @xcite . the present gft construction is indeed , in a sense , a much improved and further developed version of the one in @xcite , in a sense to be detailed in the following . + we will detail both the motivation , the basic ideas and the results of our work in the next section . in section iii we present the general definition of the new class of gfts , and the general structure of its feynman amplitudes . section iv and v report instead the detailed form of the amplitudes of these models in 3 and 4 dimensions , in both riemannian and lorentzian cases , and a discussion of their properties . we conclude with a summary of our results and an outlook on their relevance for further developments in this area . we assume now that the above perspective on group field theory as a discrete quantum field theory ( 3rd quantized ) of spin networks or of simplicial geometry , and as a potential common unifying framework for different approaches to quantum gravity , is agreed upon , if tentatively only . let us now focus on the following questions : what types of amplitudes we then expect or want the gft feynman diagrams to be assigned ? that is , what properties and main features our spin foam amplitudes should have ? how should they look like , if they are indeed feynman amplitudes for a field theory on a simplicial superspace ? from a path integral for a field theory on superspace , continuum or discrete , we expect to obtain a definition of causal transition amplitudes between quantum gravity states , i.e. the quantum gravity analogue of what , in ordinary field theory , is represented by expectation values in the vacuum state of time - ordered products of field operators . however , no time coordinate is allowed , in a fully background independent and diffeomorphism invariant theory of quantum gravity , to enter the definition of transition amplitudes , as it does through the minkowskian time when we define time - ordered 2-point functions in the usual qft . therefore , in quantum gravity , the difference between the various possible 2-point functionscan be characterized purely in terms of symmetry properties and of other formal features of them , independent of any spacetime coordinates . let us look at some of these features . for further discussion on this , see @xcite , and for a classic treatment of the issue of causality and different transition amplitudes in quantum gravity , within the covariant path integral approach , see @xcite . consider the simplest case of a 4d spacetime of topology @xmath2 , with compact @xmath3 . this spacetime has two boundaries , call them @xmath4 and @xmath5 , to which we associate a 3d spatial geometry @xmath6 and @xmath7 , respectively . assume now that we can uniquely associate ( within a canonical quantum theory ) a state @xmath8 to the geometry @xmath6 , and @xmath9 to the geometry @xmath7 . the basic idea underlying the time - lesscharacterization of the causal quantum gravity transition amplitudes @xmath10 is that , even if these can not correspond to any time - ordering , they do implement a time - less orderingor , better , a _ causal ordering_. this consists in the requirement that @xmath7 lies in the causal future of @xmath6 , which in turn can be formulated , when a canonical decomposition of the gravity variables is possible , as the requirement that the lapse function ( which in a suitable gauge is equivalent to a proper time ) between the two boundaries can only take positive values @xcite . note that the formulation of this criterion does not require any use of coordinates . notice also that the above has a direct analogue in the definition of different green functions for a relativistic particle @xcite , where it defines indeed the feynman propagator . moreover , this criterion can be generalized to the situation in which no canonical decomposition is available , for example , keeping the same boundaries and boundary states , for spacetimes of non - trivial topology . in such cases it can be formulated as the requirement that the amplitude is _ orientation dependent _ , i.e. that it turns into its complex conjugate if the spacetime orientation is reversed . if the dynamics is defined by a quantum gravity path integral , in metric variables , all these requirements are automatically encoded in the definition of the configuration space as the space of all metrics up to diffeomorphisms and of the quantum amplitude as the exponential of @xmath11 times the einstein - hilbert action ( or some higher - derivatives extension ) , times a diffeo - invariant real measure : h_2h_1=_g h_1,h_2ge^is_eh(g ) . [ pathintegral]indeed , this corresponds , in a canonical formulation to : h_2h_1= nn^i , [ pathintegralcan]with the integration range @xmath12 over the lapse function @xmath13 @xcite . the above amplitude is _ complex _ , _ causal _ ( @xmath7 is in the causal future of @xmath6 ) and _ orientation dependent _ ( it turns into its own complex conjugate under switch of spacetime orientation , as @xmath14 under this transformation , equivalent to switching positive to negative lapse ) . moreover , it defines , at least formally , a _ green function _ of the hamiltonian constraint operator , the dynamical equation of motion of pure gravity , for trivial spacetime topology , not a solution of it , i.e. it satisfies : @xmath15 . notice that the same definition for the quantum gravity path integral results in an orientation dependent transition amplitude also in the case of riemannian quantum gravity , i.e. the quantization of riemannian geometry , even though then the causality interpretation is not applicable to the same restriction on the lapse function . + a quick comparison with the usual ( say , free ) quantum field theory case , or , for that matter , with the relativistic particle case , shows that these same properties are shared by the time - ordered product of field operators and by the usual feynman propagator . actually , in these simplified contexts , the above properties _ select uniquely _ the feynman propagator or time - ordered n - point function among the various qft n - point functions or particle propagators . in the formal 3rd quantized framework , as in usual qft , the path integral of the field theory itself provides , after field insertions , a definition of the causal transition amplitudes . how this is realized in the 3rd quantized formalism is only apparent in perturbative expansion , and , once more , at a rather formal level , given the poor understanding of the formalism itself . however , also in the gft case , we are at present lacking control of the theory beyond the perturbative regime . therefore it is instructive to recall how the causal transition amplitudes are characterized in usual qft perturbative expansion . consider then some time ordered product of field operators in the vacuum state ; one inserts the appropriate combination of fields in the path integral expression for the partition function of the field theory , and expands in powers of the coupling constant , obtaining the usual sum over feynman diagrams weighted by amplitudes obtained by gluing propagators with interaction vertices . the choice of field insertion characterizes the boundary states and the original time ordering is reflected in the presence of the feynman propagator in each individual particle line of a feynman diagram . this propagator , in position variables , can be expressed as a sum over histories for the single particle it refers to , i.e. by a path integral weighted by the usual relativistic particle action @xcite . if one does the same for all the particles involved in a given feynman diagram , the whole feynman amplitude can be put in the form of a path integral for a discrete system of particles , weighted by the exponential of the classical action , and with a constraint on their histories in position space , representing the particle interactions . the causality restriction enters , as we said , in the use of the feynman propagator , which results from restricting the proper time ( or lapse in canonical variables ) along each particle history to be positive , in turn giving a complex amplitude given by the exponential of @xmath11 times the particle action @xcite . the restriction to positive proper times or lapses is also equivalent to a dependence of the amplitude on the orientation of the particle trajectories . + the same happens in the formal 3rd quantization setting for gravity @xcite : each feynman diagram , a discrete history of universe interactions , is weighted by an amplitude given by the path integral ( [ pathintegral ] ) , the exponential of the gravity action , i.e. the geometric action for the particle - universe for each line of propagation , plus appropriate joining conditions representing the interactions . as we have discussed above , such path integral for gravity is a green function of the dynamical constraint equations ( as the feynman propagator for the klein - gordon equation ) , and it is causal and orientation dependent in the sense specified @xcite . note also that the end result of combining the path integrals for gravity on trivial topology with the joining conditions for interactions is again a path integral for gravity with the same type of amplitude on a spacetime with non - trivial topology . the analogy with particle dynamics and usual qfts is even clearer if one recalls @xcite that the einstein s equations define indeed the dynamics of a free particle moving in superspace . in a gft , in light of its proposed physical interpretation , we would expect a similar structure for the feynman amplitudes . the discrete histories of possible interactions for the gft quanta are , as discussed , combinatorial 2-complexes dual to simplicial complexes , because the quanta themselves are identified with spin network vertices or ( d-1)-simplices , as said . therefore , if the gft degrees of freedom and dynamics are to represent a quantum geometry and its evolution , we would expect the amplitudes associated to its discrete ( virtual ) histories , the gft feynman amplitudes , to have the form of a path integral for discrete gravity , i.e. an exponential of some classical discrete gravity action . this way , they would have the sought for properties of causality / orientation dependence and complexity on top of making the relation with classical and quantum discrete gravity manifest . once more , it is the _ complex nature _ of the feynman amplitudes and their having the form of an _ exponential of some simplicial gravity action _ , that would permit the interpretation of gfts as 3rd quantized theories of simplicial geometry and as providing a definition of discrete quantum gravity transition amplitudes . + this expected and , we argue , _ needed _ general feature of gft models does not , by itself , select the _ specific form of the simplicial action _ that has to appear in the gft feynman amplitudes . on the contrary , because gfts are supposed to describe quantum gravity _ at all scales _ ( unless the future development of the formalism will turn out to show some unavoidable breakdown or incompleteness of the same , beyond some energy or distance scale ) , one should , a priori , expect a very general formulation of classical simplicial gravity to describe the classical dynamics of the gft quanta . this may correspond , for example , to some generalized action for regge calculus , with higher - order terms , e.g. higher powers of the curvature , appearing as the phase of our complex gft feynman amplitudes . indeed , we will see how our new proposed gfts produce very general types of simplicial actions inside their _ complex _ feynman amplitudes , the regge action being just one contribution , although the dominant one in physically interesting limits . this is _ not _ what happens in current gft models . the feynman amplitudes / spin foam models of _ all _ current gfts are instead : real , a - causal and orientation independent , so that they do not reflect the orientation of the underlying simplicial complex nor allow for the identification of any ordering between the boundary states . in this sense they define a - causal transition amplitudes . this structure is due to an underlying @xmath16 symmetry of the spin foam amplitudes , first identified and interpreted in @xcite , which erases the orientation dependence of the same amplitudes at the level of each dual 2-cell or ( d-2)-simplex . still in @xcite , and in @xcite , it was shown how breaking this symmetry and restoring the orientation dependence would lead to feynman amplitudes with a much more direct relation with 1st order discrete gravity and with the expected exponential form . + usual spin foam models correspond , then , to a sort of symmetrized discrete path integralsfor gravity in 1st order form . the continuum ( and 2nd order ) counterpart of such symmetrized discrete gravity path integral is given , formally , by the same formula ( [ pathintegralcan ] ) , but with the range of integration over the lapse function extended to the full real interval @xmath17 . this formula gives a covariant definition of the physical inner product between canonical quantum gravity states and thus of the projector operator onto solutions of the hamiltonian constraint operator . in fact the resulting quantity , morally a definition of @xmath18is a _ solution _ of the hamiltonian constraint equation in both its arguments @xcite , and it is a real quantity , as expected from a canonical inner product , as well as a - causal in that nothing constraints one boundary geometry to lie in the causal future of the other . this is not so surprising , and maybe not even un - welcome , given that spin foam models have been introduced exactly in order to define in a covariant way the canonical physical inner product @xcite . however , @xmath19 _ does not _ correspond to the lagrangian path integral ( [ pathintegral ] ) , and thus it is _ not _ what we would expect to obtain , in perturbative expansion , in a properly defined field theory on superspace . the difference lies in the requirement of an additional symmetry on top of the lagrangian 4-diffeos @xcite : positive and negative lapses correspond in fact to the _ same _ class of 4-geometries , and the difference between the two half - ranges @xmath12 and @xmath20 is only that they correspond to _ opposite spacetime orientations_. in other words , the above quantity in lagrangian formulation is given by a different _ symmetric _ choice of quantum amplitude , but for the same space of 4d geometries : h^2 h^1 _ h = g ( e^is(g ) + e^-is(g ) ) . [ pathintegralsym ] another insightful way of looking at the difference between the two ranges of lapse integration , or between ( [ pathintegral ] ) and ( [ pathintegralsym ] ) , is by recalling the difference between lagrangian and hamiltonian symmetries , the first being the 4d spacetime diffeomorphisms and the second being the transformations generated by the canonical operators @xmath21 and @xmath22 @xcite . the second set of symmetries is actually _ larger _ than the first , and the two coincide only after imposition of ( some ) equations of motion @xcite . the range @xmath23 is symmetric under transformations of the lapse corresponding to 4d spacetime diffeos , while it is not under canonical symmetries that can connect positive and negative lapses , thus requiring a symmetric range @xmath24 @xcite . therefore , the lagrangian path integral needs a further symmetrization ( [ pathintegralsym ] ) to satisfy the wheeler - dewitt equation . in a formulation of quantum gravity as a quantum field theory on superspace , then , one would expect the symmetrized amplitudes ( [ pathintegralsym ] ) to arise as feynman amplitudes only from a _ restriction _ or in a special _ subsector _ of a more general field theory producing instead ( [ pathintegral ] ) as the typical feynman amplitude . + once more , all this has a very precise analogue in the sum - over - histories formulation of the dynamics of a relativistic particle , with the symmetrized path integral corresponding to the hadamard propagator , given in momentum space by @xmath25 , thus imposing the hamiltonian constraint equation @xmath26 , instead of the feynman propagator , given in momentum space by @xmath27 , thus relaxing the same hamiltonian constraint at the quantum level . some confusion may arise from the fact that usual spin foam model _ do _ indeed come from a path integral quantization of a discrete action , at least in 3d . this is , however , bf theory which , although closely related to gravity , does not coincide with it . the difference between 3d bf theory and 3d gravity in 1st order form is that the triad field in a gravity path integral is summed over those configurations corresponding to positive volume element only , while the b field in 3d bf theory is summed over both positive and negative volumes ; this has been emphasized in @xcite . this again is the first order counterpart of the symmetrization ( [ pathintegralsym ] ) , and corresponds to a larger set of symmetries at the quantum level in bf theory with respect to 1st order gravity . a similar mismatch between constrained bf - like theories and 1st order gravity arises in higher dimensions as well @xcite . such mismatch also explains @xcite why , in the semiclassical , large representation limit , and thus after suppression of quantum interference between configurations associated to opposite orientations at the level of each dual face , the spin foam vertex amplitude of all known models still gives the cosine of the discrete gravity action for a single simplex , as opposed to the exponential of it @xcite . moreover , we note here that the large-@xmath28 semiclassical limit can be understood as suppressing , together with interference between quantum configurations , also all quantum corrections to the regge action somehow hidden in usual spin foam models , leaving only the dominant regge term , even though still within a sum over opposite orientations producing the mentioned cosine factor . this second suppression mechanism , in a slightly different form , will be shown at work also in our new gft models , where it will indeed reduce the gft feynman amplitudes to the form of a simple exponential of the regge action for simplicial gravity , with its quantum ( higher order ) corrections being negligible in the limit . this being the situation , we would then like to identify the true gft analogue of the quantum gravity causal transition amplitudes , or , more precisely , we would like to construct group field theories for which , as in ordinary qft , the feynman expansion of n - point functions produces feynman amplitudes given by the exponential of a discrete gravity action , i.e. with the causality restrictions implicitly , automatically , but also clearly implemented . this , for us , would be a clear sign that we are capturing the causal dynamics of discrete gravity correctly . + a subsequent analysis of how the usual spin foam models arise from a suitable restriction of these generalized amplitudes , or as a special limit of this new class of gfts would then shed some additional light on the exact relation between bf theories and gravity as well as on the role of the canonical physical inner product between quantum gravity states within a covariant field theory on superspace , in the more rigorous setting of both loop quantum gravity and group field theories . an additional motivation for constructing this new class of gfts is that they would bring simplicial quantum gravity approaches in much closer contact with the spin foam formalism for discrete gravity path integrals , and , via spin foams , with loop quantum gravity . actually , there is hope that this new class of gfts may represent _ the _ common unified framework in which both simplicial quantum gravity approaches , quantum regge calculus and dynamical triangulations , as well as loop quantum gravity / spin foam one can be subsumed , for mutual benefit and further development of each . the general idea of gfts as a common framework has been explained in the introduction , and it remains an interesting and intriguing perspective , in our opinion , regardless of our present results . however , one crucial step is needed in order to make such perspective a concrete reality , and provide a solid basis for understanding the _ exact _ links between gfts , loop quantum gravity and simplicial quantum gravity . this step has the same goal as the motivations we already gave above arising from a purely 3rd quantization perspective : this goal is to construct gft models with feynman amplitudes given _ exactly _ by the exponential of a simplicial gravity action , times some appropriate measure . this is the step we take with the present work . in fact , both quantum regge calculus and dynamical triangulations , although differing in the choice of variables used to encode the geometry of gravity at a simplicial level ( geometric data / edge lengths in the first case , combinatorics of simplicial complexes in the second ) , identify the quantum amplitude to be associated to each spacetime geometry with the exponential of the regge action for simplicial gravity , indeed a beautiful coordinate - free description of classical simplicial geometry @xcite . a gft _ with the same type of quantum amplitudes _ for its feynman diagrams and with a sum over both geometric data and triangulations would represent a unification and a generalization of both approaches in a very literal , transparent sense . + an interesting difference , as for the classical simplicial action used , between usual simplicial quantum gravity and gfts , can be already envisaged . gfts are based on a 1st order formulation of gravity and a group theoretic description of geometry . in other words they refer to a palatini - like or bf - like formulation of gravity in terms of a d - bein field and a connection field . we would then expect that the sought for generalized gfts would produce amplitudes given by exponentials of a simplicial gravity action in 1st order variables as well , i.e. with a double set of geometric variables : one corresponding to d - beins , and thus assigning volume information to the simplicial complex , and the other corresponding to a connection , and thus defining a group - theoretic notion of curvature , in terms of holonomies . in pure bf theory , in fact , the b field is just a lagrange multiplier and one can achieve a formulation of the quantum amplitudes using only the connection variables , and basically imposing the flatness condition everywhere . in gravity , however , one relaxes this condition and the d - bein is a true dynamical field , that we would expect to find as a dynamical variable in the path integral as well as a configuration / momentum variable in the definition of a field theory on superspace , alongside the connection field . + in a simplicial setting , therefore , we would expect to obtain a formulation of gravity in terms of some 1st order version of the regge action , with variables being a discretized d - bein @xmath29 , associated to each link of the simplicial complex , or a discretized bivector field @xmath30 , associated to each ( d-2)-face of the complex , in a bf - like formulation of gravity , plus a discretized connection variable , represented for example in terms of discrete parallel transports ( group elements ) of the same along dual links @xmath31 of the simplicial complex , as in all current spin foam models @xcite . this action would have a general form of the type : s = _ f v_f(e , b ) _ f ( g_e * ) + ( higher order ) , [ discreteaction]with @xmath32 being the volume associated to the ( d-2)-face @xmath33 , which is a function of either @xmath34 variable or @xmath35 variables , and @xmath36 being the corresponding deficit angle , i.e. the discretized curvature . similar 1st order formulations of discrete gravity have been proposed and studied , e.g. , in @xcite . + in dimension @xmath37 , one would have to add to ( [ discreteaction ] ) suitable ( non - local ) constraints on the discrete b variables , if a bf - like formulation is the one sought for , ensuring their geometric interpretation . in absence of such constraints , in fact , we would just have a discrete version of classical bf theory , as it clear from the fact that the variation of the action with respect to @xmath35 would produce the flatness condition @xmath38 for any face of the complex . in 4 spacetime dimensions , the issue of discrete bf constraints is also related to the issue of the constraints on area variables in the so - called area regge calculus @xcite . we will discuss these issues in slightly more detail later in this article . + with the aim of reproducing the above type of classical action , plus the hope that this will lead to a more straightforward way of imposing the above constraint than in the usual spin foam procedure , and the main motivation of imposing the causality / orientation dependence condition , which is in fact a restriction on the integration range over the b field , we are thus led to introduce additional variables , directly identifiable with the b field of bf theory , into the usual gft formalism , which is based on the connection variables only ( group elements ) . we will indeed obtain , from our new gfts , a 3rd quantized version of discrete bf theory in any dimension and any signature , _ with an additional restriction on orientation _ automatically imposed , as well as incorporating what can be interpreted as quantum corrections to the above classical action ( akin to higher derivative corrections to the einstein - hilbert action in effective approaches to continuum gravity ) . this means that we will obtain a 3rd quantization of discrete 1st order gravity in 3d , and of an orientation - restricted bf theory in higher dimension . in 4d the corresponding gft model would represent , we believe , the most suitable framework in which to implement the classical constraints on the b field that reduce bf theory to gravity , that have been extensively studied in the construction of spin foam models for 4d gravity ( see e.g. the recent @xcite ) , because it may make the identification and implementation of the geometric constraints at the simplicial level more straightforward , and in a context in which the needed orientation dependence / causality restriction is already implemented . before proceeding to the discussion of our results and of the new gft formalism , let us conclude by anticipating motivation , interpretation and advantages of the specific way we have chosen of introducing the additional variables corresponding to the continuum b field . + in the usual spin foam models , such as the ponzano - regge model for bf theory , as well as in the gft models that generate them , such as the boulatov model , the variable that is interpreted as the discrete counterpart of the b field of the original continuum bf theory is the representation label j associated to each ( d-2)-face of the simplicial complex . this is first of all justified by the way it enters the expression for the spin foam amplitudes , after peter - weyl decomposition of the same . in the riemannian 3d case , for example , one indeed gets , for each dual face @xcite : d^3 b_f = ( g_f ) = _ j_f d_j_f ^j_f(g_f ) . [ facebf ] @xmath30 is the original discretized b field , given by an @xmath39 lie algebra element , with which one starts from when deriving the spin foam amplitudes from a discrete lagrangian path integral , but that does not appear in the corresponding gft derivation , from which one just obtains the result of the @xmath30 integral above , i.e. the delta function over the group , forcing the flatness condition on the su(2 ) holonomy @xmath40 . starting from this delta function , by harmonic analysis on su(2 ) one gets the last expression in ( [ facebf ] ) , which indeed resembles the starting expression but with a discrete replacement for the @xmath30 variables : the representation labels @xmath41 . the same happens in the usual gfts . apart from the formal similarities , one physical rationale for the identification of the @xmath41 with a discretized b field is the fact that it is conjugate to the connection variables , i.e. to the group elements @xmath42 , in the sense of fourier transforms , just as the b field is canonically conjugate to the a field in the hamiltonian formulation of classical bf theory . this reasoning is of course sensible , and it is indeed supported by the respective role group representations and group elements play in loop quantum gravity , again following canonical analysis in the continuum , but it is also not fully conclusive . there are a few reasons for being dissatisfied with this interpretation , even if they are , admittedly , not at all conclusive either . one is that the @xmath41 corresponds more precisely to just _ one _ component of the original b field , its ( discretized ) absolute value , with the other components still missing any identification within the formalism . the situation , in this respect , has been ameliorated somewhat by the recent development of new spin foam models for bf theory and gravity @xcite based on coherent states . here , the additional parameters labelling a coherent state basis of vectors in each representation space @xmath41 are interpreted as the spin foam analogue of the missing components of the b field with modulus @xmath41 . this is justified by the fact that the _ expectation value _ of a lie algebra generator in a coherent state is given by a ( bi-)vector with modulus @xmath41 and components proportional to the coherent state parameters . however , there are still several questions unanswered about the relation between a generic b field and the lie algebra generators , the exact physical role played by coherent states , apart from their mathematical convenience , etc . moreover , since the above is a relation that concerns the expectation values of quantum operators , one may suspect that it should be understood as a semi - classical one , holding only in some approximation of the dynamics of the quantum theory . as we will discuss later on , we feel that our approach of introducing additional independent variables playing in a straightforward sense the role of the discrete b field , and whose relation with lie algebra generators for the group considered is governed by the _ dynamics _ of the theory , may help to clarify , with further work , the role that coherent states play at the level of spin foam amplitudes . + our main concern with the identification of representation labels , and , before that , of the generators of the lie algebra , acting on connection group elements , with the discrete b field comes from looking at the issue from a more general lagrangian , rather than canonical perspective ( which is available only for trivial topology ) . namely , we are looking for a group field theory discrete realization of the path integral for a _ gravity _ theory in 1st order form , which , as we have discussed , is likely to imply a restriction on the configurations of the b field summed over , that would give a different result for the face amplitudes with respect to ( [ facebf ] ) , as it happens , for example , in the model of @xcite . in such a path integral two sets of variables are present , the geometric b field and the connection , and the relation between the two is _ one of the equations of motion _ of the theory ( the one imposing metricity of the connection ) and thus should be imposed only by the dynamics of the theory , and not imposed already at the kinematical level at the level of each history being summed over in the path integral , as it appears to be done in current gft models . we feel that imposing such condition already at the kinematical level results , in the usual spin foam models , in freezing a part of the degrees of freedom of the theory . in particular it may be this restriction is what is responsible for turning what should have been causal transition amplitudes into rather awkward , from the gft and 3rd quantization perspective , a - causal transition amplitudes , which correspond , as said , to a sort of symmetrized path integral , imposing the canonical dynamical constraints even in situations , e.g. non - trivial spacetime topologies , where a canonical interpretation is problematic and certainly not expected . + this feeling , admittedly not much more than this , in absence of more rigorous arguments , is however confirmed by the way the new variables we introduce should be restricted in order to reduce our new gft models to the usual ones , at the level of their feynman amplitudes , i.e. spin foam models . + in any case , we would like to have at our disposal a more general framework , that reproduces the full bf or 1st order gravity path integral , in which the mutual relations between the discrete counterparts of the continuum variables mimic more closely the continuum classical and quantum dynamics . in this way , we may both confirm that we are indeed reproducing at the discrete level the features of the continuum dynamics and also , hopefully , shed some light on the usual spin foam models and procedures . as was explained in the previous section , the new models should be thought of as the causal analogues of the usual gfts associated with bf theory . + bf theory in d dimensions for a group g with a lie algebra @xmath43 is a topological field theory defined by the following action [ eq : bfaction ] s = _ m tr ( b f(a ) ) where m is a d - dimensional manifold , b can be thought of locally as a @xmath43-valued ( d-2)-form and f is the curvature of the g - connection a , so it can also be thought of locally as a @xmath43-valued 2-form . + let us now describe our main strategy and its rationale , illustrating it for simplicity in the d=3 case . the extension to different dimensions is straightforward and follows the same type of arguments . it will be discussed in detail in the following . * we would like to introduce additional variables , corresponding to a discrete b field associated to each 1-simplex in the simplicial complex , in the gft perturbative expansion . this means that there should be one such variable for each argument of the gft field . * we would like the new variables to be identified with the generators of the lie algebra of the relevant group . this implies that they must have the same number of components . the field should then be a complex function @xmath44 in this 3d example . the complexity of the field , together with symmetry under _ even _ permutation of the arguments , is needed to ensure orientability of the simplicial complexes arising in perturbative expansion . * the identification should follow from some equation of motion of the theory , so to be part of the dynamics ; at the same time , it should belong to the _ kinematical _ sector of the gft , because we would like boundary states to satisfy it , at least partially , so to have a similar structure to that of loop quantum gravity spin network states . this condition would then follow from some sort of _ asymptotic condition _ on boundary states in computing gft transition amplitudes ( notice however that a gft equivalent of the cluster decomposition principle of ordinary qft or of the ordinary s - matrix theory has not yet been developed in full detail ) . * such equation of motion should then be of the type @xmath45 , where the indices @xmath46 label the arguments of the field , while the indices @xmath11 are vector indices in @xmath47 , and @xmath48 are the generators of the @xmath49 or @xmath50 lie algebra . however , the above equation is not invariant under group transformations , so we turn it into a covariant form , obtaining : @xmath51 , for each argument of the field . * the field being a function on the group , the generators of the corresponding lie algebra act on the group arguments of the field as derivative operators , so that the above equation is actually implemented , in configuration space ( with respect to @xmath52 ) as : @xmath53 , where @xmath54 is the laplace - beltrami operator acting on the group manifold . * after harmonic analysis , the @xmath55 is turned into the invariant casimir of the group @xmath52 , in a given representation @xmath56 , acting as a multiplicative operator on the field now function on the same representation parameters @xmath56 . * @xmath57 act here as multiplicative operators ; however , we can independently perform fourier analysis on the @xmath35 variables as well , going to conjugate variables @xmath58 , also in @xmath47 , and turn the quantity @xmath59 into a differential operator , a new laplace - beltrami operator acting on @xmath47 . * by means of this extension of the group field theory formalism , we want also to reproduce proper simplicial gravity path integrals in perturbative expansion . considering that , in the regge formalism for discrete gravity , d - simplices are assumed to be flat and the whole dynamics of geometry comes from the boundary terms @xcite , we obtain a further motivation for restricting the modification of the gft dynamics with respect to usual models to be confined to only the kinematical term in the gft action . * the interaction term is only modified by the extension in the number of variables as well as in a peculiar orientation dependence , in the variables @xmath60 , fourier conjugate to @xmath35 , that is necessary to ensure the proper matching of @xmath35 variables across simplices , and encoded in the dependence on the complex structure of the field @xmath61 , as we will see . as for the dependence on the group @xmath52 , it maintains the same structure of the usual models describing bf theory . in this way we obtain a new kinetic term given by a differential operator acting on the field , very similar to the usual klein - gordon operator of scalar field theory , but with a product structure coming from the independent action of one operator of the above type acting on each argument of the field : @xmath62 . + notice that there is almost nothing in the above choices that can select any specific dynamics of the geometric data ( @xmath35 variables and group elements , say ) at the level of the individual feynman diagram . the only dynamical ingredient above is the choice of a certain _ relation _ between them , but nothing seems to dictate , at the level of the gft action , the _ individual dynamics _ of each set of variables . what we put in is then only a ) some _ complex structure _ resulting from the propagator representing the inverse of the chosen kinetic term , due to its singular nature as a differential operator , b ) the mentioned mutual relation between @xmath35 s and @xmath63 s , and c ) the combinatorics of the feynman diagrams ( dictated by the combinatorics of the variables in the action ) . it is only to be expected , then , that the simplicial action describing their dynamics at the level of each feynman diagram , and appearing in the exponent of the phase part of the feynman amplitudes ( simplicial gravity path integral ) will be pretty generic . the non trivial tests will be to show : 1 ) that this phase can be interpreted at all as a simplicial gravity action , because of the way the gft variables will enter in it ; 2 ) that this generalized action will reduce to the usual regge action ( in 1st order form ) in appropriate , clearly identified and physically meaningful limits . our proposed formalism passes these tests . the strongest support for the mentioned choices , and for the resulting form of the gft action for the new models , is , therefore , simply the resulting expression for the feynman amplitudes of the corresponding gft , which indeed fulfill _ all _ the expectations and goals we have stated above . some additional nice features of the resulting model can be already underlined at this point . as we mentioned , the kinetic operator above is a singular differential operator , which implies that its inverse has to be defined in the complex domain , just as it happens in the usual klein - gordon case . on the one hand , this complexification is responsible for the complexity of the resulting feynman amplitudes , and ultimately , as we shall see , for the wanted exponential form of the same amplitudes ; on the other hand , the differential form introduces the dynamical correlations between simplices that we would expect in a discrete theory of quantum geometry . also , the propagator corresponding to the kinetic operator will introduce quantum corrections , virtual degrees of freedom , not captured by the on - shell condition @xmath64 , thus relaxing it at the quantum level , again matching our expectations . finally let us mention that the presence of derivatives in the gft kinetic terms allows for the identification of a non - trivial symplectic structure on the space of fields , and makes a canonical analysis _ of the gft itself _ possible . this nice feature is shared also by the generalized models introduced in @xcite , of which the new ones represent a sort of relativistic extension(fixing some pathologies of the same arising in perturbative expansion ) , as we will discuss , and it is at the basis of the canonical analysis performed in @xcite . we now give the definition of the action for the new class of gft models , for general dimension d and general gauge group g. + let g be a semi - simple group ( we will deal with the double covers of the rotation and the lorentz groups in d dimensions ) and let x be a space isomorphic , as a metric vector space , to the lie algebra @xmath43 of g. the basic variable of the theory is a complex - valued field @xmath61 @xmath65 where d is the dimension of the model , which is the dimension of the generated simplicial complexes ( we will concentrate on the 3 and 4 dimensional cases ) . + the field is interpreted as a ( d-1)-simplex , with the group and lie algebra variables corresponding to its geometry . the group elements represent discrete parallel transports of a connection ( the discrete analogue of the @xmath66 of bf theory ) from the centre of the simplex to the boundaries , while the @xmath60 variables allow us the reconstruction of the volumes of the boundary ( d-2)-simplices , and are thus related to the b field of bf theory . + the field is assumed to be invariant under _ even _ permutation of the labelling of its ( pairs of ) arguments @xmath67 ) , and to turn into its own complex conjugate under change of this labelling by an _ odd _ permutation . in this way , the orientation of the corresponding ( d-1)-simplex is encoded in the complex structure of the field@xcite . + as in usual gft models , geometric closure of the ( d-2)-simplices which form this ( d-1)-simplex translates into invariance of the field under the global symmetry @xmath68 @xcite . we will impose this symmetry in the usual way by taking the field to be arbitrary and then projecting it onto the diagonal subspace , i.e. the field @xmath69 is given by @xmath70 , where @xmath71 is now arbitrary . below , to reduce clutter , we will write the actions in terms of the @xmath61 s instead o the @xmath72 s . + also , we will denote both the field and its complex conjugate by @xmath73 , with @xmath74 and @xmath75 and @xmath76 . + the model is defined by the following action @xmath77}_{\textrm{kinetic operator } } \phi(g_i ; x_i ) + { } \nonumber \\ & & { } + \frac{\lambda}{(d+1 ) ! } \sum_{\nu_1 , \ldots , \nu_{d+1 } } \int_{g^{d(d+1 ) } } \bigg ( \prod_{i \neq j = 1}^{d+1 } \ , dg_{ij}\bigg ) \ , \int_{x^{d(d+1)}}\bigg ( \prod_{i \neq j}^{d+1 } dx_{ij } \bigg ) \ , \ , \,\underbrace { \bigg [ \prod_{i < j } \delta(g_{ij } g_{ji}^{-1 } ) \delta(\nu_i x_{ij } + \nu_j x_{ji } ) \bigg ] } _ { \textrm{vertex } } \ , \ , \ , \times { } \nonumber \\ \label{eqnarray : actiongx } & & { } \times \phi^{\nu_1}(g_{1j } ; x_{1j } ) \ldots \phi^{\nu_{d+1}}(g_{d+1 j } ; x_{d+1 j}).\end{aligned}\ ] ] @xmath78 and @xmath79 are the laplace - beltrami operators on x and on g respectively , corresponding to the killing form , is obtained by extending , using e.g. left - invariance , the _ negative _ of the metric used to define @xmath78 . the reason for this choice of conventions comes from the fact that if one uses the same sign for the @xmath78 as the one used in the mathematical literature @xcite for the @xmath79 , then one gets a negative - definite operator in the case g is compact . so , for example , if g is su(2 ) , then the corresponding @xmath78 would have been given ( in the appropriate coordinates ) by @xmath80 . on the other hand , with our conventions it is just the usual laplacian on flat space . ] and the cartan - killing metric , and d is the dimension of g and @xmath81 . + as in usual gfts , the combinatorics of arguments in the action is designed in such a way as to mimic the combinatorics of the ( d-2)-faces of a d - simplex in the interaction term , and the gluing of two d - simplices across their common boundary in the kinetic term . the sum over @xmath82 in the second term makes the action real . interpreting the @xmath61 as representing ( d-1)-simplices which are incomingor in the past boundary , and the @xmath83 as representing ( d-1)-simplices which are outgoingor in the future boundarywith respect to the d - simplex corresponding to the gft interaction vertex , we see that there are d+2 possible vertices , corresponding to the cases in which ( d+1)-n initial(d-1)-simplices interact to give rise to n final(d-1)-simplices after the interaction has taken place . in turn these various terms correspond to the well - known ( d-1)-dimensional pachner moves . as noticed above , the orientation of the ( d-1)-simplices , inducing a _ pre - order _ @xcite also on the set of d - simplices , and turning the resulting feynman diagrams into _ directed graphs _ is encoded in the complex structure of the fields . for simplicity of presentation , we have chosen the weight the various interaction terms corresponding to different choices of @xmath82 s with the same coupling constant @xmath84 ; it is straightforward to relax this assumption defining coupling constants @xmath85 , with @xmath86 in order to ensure reality of the action , as it was done also in @xcite . + let us remark that it is possible to choose a different vertex from the one above . one in which there is no dependence on the @xmath87 , in the x variables , and this dependence is instead shifted to the p variables : @xmath88\ ] ] below , we will call the model given by ( [ eqnarray : actiongx ] ) model a , while the one with this new vertex model b. + note also that the the kinetic operator is just a product of d copies of the klein - gordon one for a massive scalar field living in @xmath89 , one for each pair of arguments of the field . + let us write the above action in momentum space with respect to the x variables . we will denote the space dual to x as p. thus ( [ eqnarray : actiongx ] ) is equal to ] where the vectors denote the coordinates in which the appropriate killing form has a canonical form ( diagonal matrix with @xmath901 along the diagonal . ] @xmath91}_{\textrm{kinetic operator } } \phi(g_i ; p_i ) + { } \nonumber \\ & & { } + \frac{\lambda}{(2\pi)^{dd(d+1 ) } ( d+1 ) ! } \sum_{\nu_1 \ldots \nu_{d+1 } } \int_{g^{d(d+1 ) } } \bigg ( \prod_{i \neq j = 1}^{d+1 } \ , dg_{ij } \bigg ) \ , \int_{p^{d(d+1 ) } } \bigg ( \prod_{i \neq j = 1}^{d+1 } dp_{ij } \bigg ) \ , \ , \,\underbrace { \bigg [ \prod_{i < j } \delta(g_{ij } g_{ji}^{-1 } ) \delta(p_{ij } - p_{ji } ) \bigg ] } _ { \textrm{vertex } } \times { } \nonumber \\ \label{eqnarray : actiongp } & & { } \times \phi^{\nu_1}(g_{1j } ; p_{1j } ) \ldots \phi^{\nu_{d+1}}(g_{d+1 j } ; p_{d+1 j}).\end{aligned}\ ] ] @xmath92 is the magnitude of p with respect to the killing form . the kinetic term can be interpreted as the product of d klein - gordon operators on the group g , and for a particle / field of ( variable ) mass square @xmath93 . + the written action is the one associated with model a ( i.e. equation ( [ eqnarray : actiongx ] ) ) . notice that the orientation dependence , i.e. the dependence of the vertex term on the @xmath82 s , is apparently lost in going to the @xmath94 variables ( of course , the vertex has this form in the p variables exactly because of the @xmath95-dependence in the x variables , thus this dependence is retained ) . in model b , instead , the vertex in the ( g , p ) variables becomes @xmath96\ ] ] so the vertex of model b depends explicitly on the @xmath95 s in the ( g , p ) variables , and not in the x variables . we will see that the feynman amplitudes , when we use the p variables , thus in both the ( g , p ) and ( j , p ) representations , are the same for both models . the difference between them is apparent only when the x variables are invoked . since , as we shall see later on , it is the p variables that have clear physical significance , and we are going to deal extensively only with the ( g , p ) and ( j , p ) representations , we shall not draw the distinction between the two versions of the model in what follows , apart when we briefly report the feynman amplitudes in the ( j , x ) variables at the end of this section . + we can also perform the fourier transform with respect to the group variables . expanding the field harmonically on the group and using its invariance under the global right shifts @xcite , we get @xmath97 the j s label the representations of the group g. the index j can go over both discrete and continuous values in general as is the case for the lorentz group . the d s are the representation functions ( the components of the representation matrices ) . @xmath98 is an appropriate normalized intertwiner ( between the representations labelled by @xmath99 ) , and @xmath100 labels the different basis elements of the space of normalized intertwiners . + a very important property of the laplace operator is that it is multiplicative on the representation functions ( see @xcite and references therein ) . more precisely , @xmath101 where @xmath102 is the appropriate casimir operator and the minus sign is used for compact groups while the plus sign for the noncompact ones . which is minus the usual one . the space of casimirs of the rotation and the lorentz groups in 3 dimensions is one dimensional , while in 4 dimensions it is 2 dimensional . in 4 dimensions , the casimir that corresponds to the laplace - beltrami operator in the riemannian case , where the representations of spin(4 ) are labelled by a pair of spins @xmath103 , is proportional to @xmath104 , while for the lorentzian case , where representations of @xmath105 are labelled by an integer @xmath106 and a real number @xmath107 , it is proportional ( in our normalizations ) to @xmath108 . ] inserting the above into ( [ eqnarray : actiongp ] ) we get @xmath109}_{\textrm{kinetic operator } } \phi_{\alpha_i}^{j_i \lambda}(p_i ) + { } \nonumber \\ & & { } + \frac{\lambda}{(2\pi)^{dd(d+1 ) } ( d+1 ) ! } \sum_{j_{ij } , \alpha_{ij } , \lambda_i , \nu_i ; i \neq j = 1 , \ldots , d+1 } \int_{p^{d(d+1 ) } } \bigg ( \prod_{i \neq j = 1}^{d+1 } dp_{ij } \bigg ) \phi_{\alpha_{1j}}^{\nu_1 j_{1j } \lambda_1}(p_{1j } ) \ldots { } \nonumber \\ \label{eqnarray : actionjp } & & { } \ldots \phi_{\alpha_{(d+1 ) j}}^{\nu_{d+1 } j_{(d+1 ) j } \lambda_{(d+1 ) j}}(p_{(d+1 ) j } ) \underbrace{\bigg [ \big \ { j - \textrm{symbol}\big \}(j_{ij } ; \lambda_i ) \big ( \prod_{i < j}^{d+1 } \delta(p_{ij } - p_{ji } ) \ , \delta_{\alpha_{ij } \alpha_{ji } } \big ) \bigg ] } _ { \textrm{vertex}}.\end{aligned}\ ] ] the interaction term is essentially the standard one , which is a product of fields whose arguments are contracted in the pattern of a d - simplex multiplied by the appropriate j - symbol , always obtained by the contraction ( along pairwise identified tensor indices , following the combinatorics of faces of a d - simplex ) of d+1 d - valent intertwiners of the group g. the only difference being that now it is not only the alphas and the j s that are contracted but also the p s as well . + for completeness we also list the action in the x , j variables , which is easily obtained by taking the fourier transform of ( [ eqnarray : actionjp ] ) with respect to the p variables . note that the kinetic term is just the product of d copies of the klein - gordon one on flat @xmath110 , with the metric whose signature is decided by the appropriate killing form , and with ( variable ) mass square @xmath111 . + it is given by : @xmath112}_{\textrm{kinetic operator } } \phi_{\alpha_i}^{j_i \lambda}(x_i ) + { } \nonumber \\ & & { } + \frac{\lambda } { ( d+1 ) ! } \sum_{j_{ij } , \alpha_{ij } , \lambda_i , \nu_i ; i \neq j = 1 , \ldots , d+1 } \int_{x^{d(d+1 ) } } \bigg ( \prod_{i \neq j = 1}^{d+1 } dx_{ij } \bigg ) \phi_{\alpha_{1j}}^{\nu_1 j_{1j } \lambda_1}(x_{1j } ) \ldots \phi_{\alpha_{(d+1 ) j}}^{\nu_{d+1 } j_{(d+1 ) j } \lambda_{(d+1 ) j}}(x_{(d+1 ) j } ) \times { } \nonumber \\ \nonumber \\ \label{eqnarray : actionjx } & & { } \times \underbrace{\bigg [ \big \ { j - \textrm{symbol}\big \}(j_{ij } ; \lambda_i ) \big ( \prod_{i < j}^{d+1 } \delta(\nu_i x_{ij } + \nu_j x_{ji } ) \ , \delta_{\alpha_{ij } \alpha_{ji } } \big ) \bigg ] } _ { \textrm{vertex}}.\end{aligned}\ ] ] it is easy to see that the new models we are presenting are essentially a sort of relativistic extensionof the generalized group field theories ( gfts ) introduced in @xcite . in fact , the new models encode the orientation of the feynman diagrams / triangulations resulting from the perturbative expansion of the partition function , that we are going to discuss in the following , in the action and in the quantum feynman amplitudes in almost the same way as the models in @xcite ( see also the discussion of these models in @xcite ) . the difference from the models outlined there is the fact that the field is now a function of more variables , passing , in momentum space , from a variable mass - energy @xmath113 valued on the real line to the set of momentum variables @xmath114 . consequently , the kinetic operator in each argument of the field turns from a schroedinger - type one into a klein - gordon one . while this could be considered a somewhat minor modification at the level of the action alone , the step from a non - relativistic type of dynamics to a relativistic one has huge consequences at the level of the feynman amplitudes and for the whole quantum dynamics of the corresponding models . we quantize the theory now via the path - integral method . the partition function is given by @xmath115}.\ ] ] in lack of a better understanding of the quantum theory and of more powerful tools , we study the quantum dynamics of the theory in perturbative expansion around the vacuum , expanding the partition function in feynman diagrams @xmath116 in the usual way . we get @xmath117 where @xmath118 is the number of vertices in the feynman diagram @xmath116 , sym(@xmath116 ) is the symmetry factor of the diagram ( order of automorphisms of the diagram / complex ) , and @xmath119 is the feynman amplitude for the graph @xmath116 obtained as is customary by taking the product of vertex functions and feynman propagators , obtained by inverting the kinetic operator in the action . + we then set out to extract vertex and propagator from our classical action . let us begin with the vertex contribution . it is clear that the interaction term in the ( g , p ) variables ( equation ( [ eqnarray : actiongp ] ) ) is exactly like the interaction terms in the usual gfts for bf theory with the sole difference being the extra variables ( which are contracted in exactly the same way as the group variables ) . the vertex amplitude is then just the usual one , which is nothing but a product of delta functions connecting the group arguments in the d - simplex pattern , with the addition of extra delta functions connecting the p variables paralleling the group ones . in other words , if we represent the vertex in the standard way @xcite we see that it consists of ( d+1 ) bundles , of d - strands each , joined together in a pattern of a d - simplex ( in the shaded area of the picture ) . each strand represents a product of a delta function on the group with a delta function on the lie algebra . the dark dots represent the arguments of the delta functions . since we never have a situation when several strands meet at a point , it is obvious that there is no real interaction enforced by this vertex , at least not in usual local qft sense , just a rerouting of the strands . it is in this sense that gfts are sometimes referred to as combinatorially non - local field theories + we now move on to the propagator . the easiest way to get it is to use the action written in terms of the ( j , p ) variables , and the expansion of a delta function on a group in terms of the characters . we wo nt really need the explicit form of the character functions nor the precise values of the coefficients , rather just the fact that such an expansion is possible . for all the groups that we will consider in this work this is indeed the case . we will use the following notation for this expansion [ eq : delta ] ( g ) = _ j _ j _ j(g ) , where @xmath120 is the character of the representation labelled by j , and as before the index j can go over both discrete and continuous values ( the sum standing for the usual sum or for the integral , respectively ) . + from this expansion and from the expression of the kinetic operator in ( [ eqnarray : actionjp ] ) we can immediately read off the expression of the feynman propagator [ eq : propagatorjp ] d_f [ g_i , h_i ; p_i , q_i ] = _ g dh _ i=1^d ( _ j_i _ j(g_i h h_i^-1 ) ( p_i - q_i ) ) . the above expression , indeed , satisfies [ eq : propagator equation ] _ i=1^d ( p_i^2 + _ g_i - m^2 ) d_f [ g_i , h_i ; p_i , q_i ] = i _ g dh _ i=1^d ( ( g_i h h_i^-1 ) ( p_i - q_i ) ) . the fact that it is the feynman propagator , as opposed to some other green s function , follows as usual from the i@xmath121 prescription used in ( [ eq : propagatorjp ] ) , as it is clear by recalling that the kinetic operator in ( [ eqnarray : actionjp ] ) , as noticed above , is essentially the klein - gordon operator in momentum variables . + note that by taking the fourier transform with respect to the p variables we can obtain the expression for the same propagator in the ( j , x ) variables . instead of computing it this way , which in fact does not easily lead to an explicit expression , we note that the kinetic term is perfectly symmetric in the way it treats the group manifold @xmath52 and the space @xmath60 . because of this symmetry , we can just reproduce the very same steps that will lead us to the propagator in the ( p , g ) variables , to obtain instead the same propagator in the ( x , j ) variables . clearly this will be just the product of klein - gordon propagators for a scalar massive ( of mass@xmath122 = @xmath123 ) field in the flat space x. + to sum the series and obtain the propagator in the ( g , p ) variables as opposed to the ( j , p ) above , take advantage of the fact that , once more , the kinetic term in the ( g , p ) variables in ( [ eqnarray : actiongp ] ) is just the product of klein - gordon ones on the group g with the mass given by @xmath124 , and use the feynman - schwinger - dewitt parametrization of the propagator@xcite . this parametrization relates the klein - gordon propagator of a massive scalar field on a space to the schroedinger evolution kernel on that space , in a fictitious proper time parameter @xmath125 . to see this connection between the propagator and the kernel recall the schroedinger equation on the group , which is given by [ eq : schrodinger ] i + _ g ( g , t ) = 0 . the general solution to this equation is given by the aforementioned schroedinger evolution kernel @xmath126 $ ] which gives the solution @xmath127 at time @xmath128 given the solution at time @xmath129 . @xmath130 \ , \psi(g_0 , t_0),\ ] ] where @xmath131 is the heaviside step function . many properties of @xmath132 immediately follow from this equation and the fact that ( [ eq : schrodinger ] ) is invariant under left and right shifts , notably symmetry , composition and green function property . + by the symmetry property of the kernel , we mean the fact that it is invariant under shifting both arguments on the group on one hand and that it is a central function on the other . this latter fact means that the kernel is expandable in characters , a feature we will use shortly . in formulae the kernel satisfies [ eq : central ] k[g , h , t ] = k[g h^-1 , t ] k[g , t ] = k[h g h^-1 , t ] . the composition property of the kernel is the standard one : [ eq : composition ] _ g dg_2 k [ g_1 , g_2 , t ] k [ g_2 , g_3 , s ] = k [ g_1 , g_3 , t+s ] . this equation will be useful when we will compute the feynman amplitudes of our model . + finally and most importantly the kernel satisfies the following two equations @xmath133}{\pp t } + \box_g \ , k [ g , h , t ] = 0 \ ; \ ; \textrm{and } \ ; \ ; i \frac{\pp \big ( \theta(t ) \ , k[g , h , t ] \big ) } { \pp t } + \box_g \ , \big ( \theta(t ) \ , k [ g , h , t ] \big ) = i \ , \delta(t ) \ , \delta(g h^{-1}).\ ] ] these coupled with the boundary condition @xmath134 = \delta(g h^{-1})$ ] mean that @xmath135 \big ) $ ] is the retarded propagator for the schroedinger equation . it is this last feature that links the schroedinger kernel with the feynman propagator . + to see this link take the fourier transform of the inhomogeneous equation with respect to t , going to the conjugate variable @xmath136 , which is the mass ( square ) of the particle in the proper time parametrization of the klein - gordon propagator @xcite , or the energy of the same in the usual schroedinger equation if we denote by @xmath137 $ ] the fourier transform of @xmath138 \big ) $ ] then it follows that @xmath139 = i \delta(g h^{-1}).\ ] ] comparing this to ( [ eq : propagator equation ] ) we see immediately that k satisfies essentially the same equation as @xmath140 . from this we can easily deduce that [ eq : propagatoriskernel ] d_f [ g_i , h_i ; p_i , q_i ] = _ g dh _ i=1^d ( k ( p_i - q_i ) ) . alternatively , we could use the known character expansion of the @xmath141 $ ] , which is given by @xcite @xmath142 = \sum_j \delta_j \ , \chi_j(g ) \ , e^{\mp i c_j t}.\ ] ] if we fourier transform this ( multiplied by the step function ) with respect to @xmath125 we get [ eq : kernelexpansion ] k [ g , ] = _ j _ j _ j(g ) . comparing this with the character expansion of the @xmath143 $ ] given in ( [ eq : propagatorjp ] ) we re - obtain ( [ eq : propagatoriskernel ] ) . + as was mentioned above , the feynman propagator for our theory in the ( g , p ) variables is just ( a product of d copies of ) the klein - gordon propagator , here written in terms of the schroedinger evolution kernel , for a free particle on the group , with the mass equal to @xmath144 as anticipated , the above procedure can be reproduced in order to obtain the propagator in the ( j , x ) variables , which is a product of propagators ( one for each argument of the field ) for a scalar field with mass@xmath122 = @xmath123 on the ( flat ) space x. which means that the propagator in these variables is given by s labelling different field components in ( [ eqnarray : actionjx ] ) , however they just contribute trivial kronecker deltas , and since it is customary in the literature to not write them explicitly we do the same here . their effect on the feynman amplitude is just to contribute a factor @xmath145 to the weight of every dual face . ] [ eq : propagatorjx ] d_f [ x_i , y_i ; j_1 , i , j_2,i ] = _ i=1^d ( _ j_1,i j_2,i k ) . the only difference with the ( g , p ) expression , that results , as we will see , in a very different form for the full feynman amplitudes in the two representations , is the absence of the analogue of the additional integration over h @xmath146 g , coming from the gauge invariance requirement on the gft field , and which breaks the symmetry between the x and g spaces in the gft action . + analogously to the vertex above we represent the propagator by a bundle of d strands as in the picture . + each strand represents a multiplicand from the right hand side of equation ( [ eq : propagatoriskernel ] ) , i.e. the i - th strand is @xmath147 \ , \delta(p_i - q_i).\ ] ] the box across all the strands represents the common integral over the h. the dark dots at the ends of the strands represent the remaining arguments of the propagator ( ( @xmath148 ) on one side and ( @xmath149 ) on the other ) . + the reason why we have drawn the strands in the propagator differently from those in the vertex is that in distinction to the situation in the usual models where the strands represent the same thing ( simple delta functions ) , this is not the case here , as a true propagation of degrees of freedom takes place between simplices , even though only a re - routing of the same occurs within each simplex . + since we have now both the necessary ingredients , the vertex and the propagator , we proceed now to construct explicitly the feynman amplitudes . a feynman diagram @xmath116 is obtained by gluing several vertices together using the propagators . if we pick one of the strands and follow it around the diagram , in absence of external legs , as in the diagrams merging from the perturbative expansion of the partition function , the strand closes back on itself . we can think of this loop as the boundary of a 2-dimensional surface which we assume has the topology of a disk . the combinatorics of the vertex is such that if we take all these disks together they form the dual 2-complex @xmath150 of a simplicial d - complex @xmath151 , the original disks being the 2-cells topologically dual to the ( d-2)-dimensional subsimplices in the simplicial complex ( for details consult @xcite ) . in the ( g , p ) variables the feynman graph amplitude factorizes per dual face ( or , equivalently , per edge of the triangulation ) , i.e. the amplitude for a graph @xmath116 is a product of dual face amplitudes : @xmath152 where , @xmath153 is the number of the dual edges in @xmath154 and @xmath155 is the number of the dual faces , and @xmath156 is the amplitude assigned to each dual face @xmath157 . this amplitude depends on the group elements @xmath158 that are assigned to the dual edges @xmath31 on the boundary of the dual face @xmath157 , and that result from the gauge symmetry of the field @xmath61 under @xmath52 ( see @xcite ) , and on a single p variable associated to the whole dual face @xmath157 left after doing all the delta functions over intermediate momenta . more precisely , this amplitude is just a product of kernels with delta functions , integrated over the common group and momentum @xmath94 variables , and for a dual face with n vertices ( and thus n links ) it is given by [ eq : ampl ] a [ h_1 , , h_n ; p ] = _ g^2n ( _ i=1^n dg_i dg_i ) ( _ i=1^n k ) ( _ i=1^n ( g_i-1 g_i^-1 ) ) , where @xmath159 . the first multiplicand is just the the propagators which are sitting on the dual edges , while the second multiplicand is the delta functions coming from the vertices which is a consequence of the translational symmetry in the p variables , leaving the detailed treatment of this symmetry for future work . ] . + ) in the amplitude for a dual face . the black boxes represent the integrals over the hs._,scaledwidth=70.0% ] we can use the delta functions coming from the vertices to do the integrals over the @xmath160 s obtaining : @xmath161 = \int_{g^n } \bigg ( \prod_{i=1}^n dg_i \bigg ) \bigg ( \prod_{i=1}^n k\bigg [ g_{i-1 } h_i g_i^{-1 } , \big ( p^2 - \frac{d}{24 } \ , m^2 \big ) \bigg ] \bigg ) .\ ] ] we would like to do the integrals over the remaining g s and obtain something which depends only on the holonomy around the dual face , computed through the @xmath162 variables only , as in usual spin foam models and gfts . however , the schroedinger kernels in the mass representation in the g - variables do not compose in any simple way . to bypass this difficulty we use again the feynman - schwinger - dewitt representation for the kernels in the previous equation @xmath161 = \int_{g^n } \bigg ( \prod_{i=1}^n dg_i \bigg ) \int_{\mathbb{r}^n } \bigg ( \prod_{i=1}^n dt_i \bigg ) \bigg ( \prod_{i=1}^n e^{i t_i ( p^2 - \frac{d}{24 } \ , m^2 ) } \ , \theta(t_i ) \ , k\bigg [ g_i h_i g_{i+1}^{-1 } , t_i \bigg ] \bigg ) .\ ] ] now , since the kernels in the proper time representation do satisfy the composition identity ( [ eq : composition ] ) we can ( after interchanging the order of integration ) perform the group integrals obtaining @xmath161 = \int_{\mathbb{r}^n } dt_1 \dots dt_n \ , e^{i ( p^2 - \frac{d}{24 } \ , m^2 ) ( t_1 + \dots + t_n ) } \ , \theta(t_1 ) \dots \theta(t_n ) \ , k \bigg [ h_1 \dots h_n \ , , \ , t_1 + \dots + t_n \bigg ] .\ ] ] the product of the group elements in the kernel is exactly the holonomy around the dual face which we will denote by h. thus @xmath163 = a_n [ h ; p ] $ ] . + to do the integrals over the proper times we change variables @xmath164 = \int_{\mathbb{r}^n } dt \ , dt_2 \dots dt_n \ , e^{i ( p^2 - \frac{d}{24 } \ , m^2 ) \ , t } \ , \theta(t - t_2 - \dots - t_n ) \ , \theta(t_2 ) \dots \theta(t_n ) \ , k \big [ h \ , , \ , t \big ] .\ ] ] the integrals over @xmath165 can now be performed as these variables appear only in the step functions giving [ eq : amplitudegp ] a_n [ h ; p ] = _ dt e^i ( p^2 - m^2 ) t ( t ) t^n-1 k . what we have shown above is that the dual face amplitude in the ( g , p ) variables is the value at @xmath166 of the fourier transform of a monomial multiplied by the retarded schroedinger kernel in the ( proper ) time @xmath151 . we will use this equation repeatedly in what follows . from the case there is a problem with this approach however due to the fact that the integral above does not converge for @xmath168 _ unless _ some of the parameters of the kernel are complexified ; and while it is possible to find the complexification needed by a careful analysis using distribution theory , it is much simpler to simply do the integral above explicitly , since then the required complexification is then easy to see . ] the explicit form of this object depends on the details of the group under consideration @xcite . we will give the explicit formulae for the rotation and lorentz groups in three and four dimensions in the next sections . + the above discussion gives the feynman amplitude @xmath169 in terms of the ( g , p ) variables . to make connection with the usual spinfoam we want to write this amplitude in terms of the ( j , p ) variables as well . this is done by returning to the general expression of the face amplitude ( [ eq : ampl ] ) , inserting the character expansion of the propagator ( [ eq : propagatorjp ] ) and using the fact that the characters satisfy is the kronecker delta in the discrete case and the dirac delta in the continuous . the easiest way to see that this equation is true comes from seeing that it follows from the fact that the delta functions on the groups compose , i.e. that @xmath170 . ] @xmath171 it is then easy to see that the dual face amplitude is given by ) as if we take the given character expansion of the kernel , plug it into the right hand side of ( [ eq : amplitudegp ] ) and evaluate the integral over t , we obtain exactly the answer given in ( [ eq : amplitudejp ] ) . ] : [ eq : amplitudejp ] a_n [ h ; p ] = _ j _ j(h ) , where n is again the number of dual edges ( vertices ) in the dual face @xmath157 . going through the standard computations @xcite of group integrals , we can obtain from this formula the spin foam picture of our model . the amplitude of the dual 2-complex ( the feynman amplitude ) obtained from our model is given by [ eq : spinfoamjp ] z_t^ * = ( _ f^*t^ * _ j_f^ * _ p dp_f^ * ) ( _ f^ * t^ * _ v^ * t^ * \ { } ) . the sum goes over all labellings of the dual 2-complex by representations of g , and j - symbol stands for the appropriate symbol coming from the representation theory of g ( it is the 6-j symbol in 3 dimensions and 15-j symbol in 4 dimensions ) . note that the feynman amplitude in these variables is now factorized differently , as it is no longer just a product of amplitudes assigned to dual faces , but , as a result of the group integrations , there are contributions coming from the dual vertices . + it is easy to see that , in the spin foam representation , i.e. in momentum space , from the gft perspective , the difference between the new models and the usual ones lies in the amplitudes assigned to the dual faces . these amplitudes are just just products of the coefficients of the character expansion of the propagators above . however , albeit limited , this difference is crucial and has many consequences : 1 ) it makes the feynman amplitudes complex ; 2 ) it produces truly dynamical propagating quantum degrees of freedom , as the usual feynman propagator of qft does ; 3 ) it selects as dominant contributions to the amplitudes the solutions of the kinematical qft equations of motion , i.e. those for which @xmath172 , which , up to a constant @xmath173 , implies the identification of the @xmath114 s with the lie algebra generators for the group @xmath52 , which , as explained in the previous section , is what we want to mimic the structure of a bf path integral , given the identification ( that we will confirm in detail in presenting the 3d and 4d models ) of the p variables with the discrete analogue of the b field of bf theory . we report here for completeness also the expression for the feynman amplitudes in the ( j , x ) representation : the propagator was already given above ( [ eq : propagatorjx ] ) . the vertex is completely analogous to the one we gave in the ( g , p ) variables ( under the substitutions g @xmath174 x and p @xmath174 j ) , with the sole difference that the whole expression is now multiplied by the appropriate @xmath175 . now , however , there is a difference between models a and b mentioned at the beginning of this section , as the delta functions on the x variables are different . + @xmath176 = \prod_{i=1}^d \bigg ( \delta_{j_{1,i } j_{2,i } } \ , k \bigg [ x_i - y_i , \pm c_j + \frac{d}{24 } m^2 \bigg ] \bigg ) , \ ] ] where k is the schroedinger kernel in the mass representation . note that this expression is basically the same as the propagator in the ( g , p ) variables ( [ eq : propagatoriskernel ] ) , showing that the formulations of the theory in the ( g , p ) variables and in the ( j , x ) ones are dual to each other . there is an important difference however , which spoils this duality which is the extra symmetry the field satisfies in the g variables ( shift invariance ) which has no analogue in the x ones . this is the reason there is no integral in the above expression over the space x analogous to the integral over h in ( [ eq : propagatoriskernel ] ) . + the vertex is completely analogous to the one we gave in the ( g , p ) variables ( under the substitutions g @xmath174 x and p @xmath174 j ) , with the sole difference that the whole expression is now multiplied by the appropriate @xmath175 . + if we use now these ingredients to calculate the feynman amplitudes @xmath177 , we get the following @xmath178 with the dual face amplitude @xmath179 being given by either arising from contracting all the representation kronecker deltas around a dual face . ] @xmath180,\ ] ] or @xmath181,\ ] ] where , as mentioned above , k is just the schroedinger kernel on x. @xmath182 is a function , with value either @xmath183 or @xmath184 , of the combinatorics of the triangulation and the assignment of the orientation data ( the @xmath95 s ) to the triangulation whose exact form is not important here , as we are not going to use or discuss this particular representation of the feynman amplitudes . + notice the marked difference with the same quantity in the ( g , p ) representation ( which is especially clear in model b where the above amplitudes are completely independent of the x variables ) . this is due to the absence of the analogue of the shifting gauge integrals . let us summarize what we have discussed so far . we have defined a new class of generalized gft models in ( [ eqnarray : actiongx ] , [ eqnarray : actiongp ] , [ eqnarray : actionjp ] , [ eqnarray : actionjx ] ) . we then analyzed the feynman rules of the theory . the vertex is easily seen to be almost the standard one . the propagator for the theory ( which in a sense encodes most of the new features of the model ) is obtained using the schwinger - dewitt parametrization . we have then constructed the feynman amplitudes of the model in both the ( g , p ) , ( j , p ) and ( j , x ) variables . some general features of the new models are already apparent at this stage , such as the complexity of the amplitudes , the presence of propagating degrees of freedom at the quantum level , the relaxation at the quantum level of the relation between ( the discrete analogue of ) the b field and the generators of the lie algebra of the group g. we will now move on , and present in detail the model one obtains from this general definition in the 3d and 4d cases , in both riemannian and lorentzian settings . in doing so , the above features will become even clearer , as in particular it will become clearer the geometric interpretation of both the p and the g variables . moreover , we will see that the feynman amplitudes of the new models , in the ( g , p ) variables , have indeed the form of path integrals for simplicial quantum gravity of the form of a bf theory restricted to positive orientation . this extra condition is what makes the feynman amplitudes we get not triangulation independent . here we discuss the relation between the new and the usual models . there are two ways in which one re - obtains the more traditional gfts and spin foam models for bf theory , as an appropriate restriction , from these generalized ones ( the same was true for the models proposed in @xcite ) . * the conventional gfts are obtained when we take the static - ultra - local limit @xcite of the action ( [ eqnarray : actiongx ] ) , and for a specific choice of the mass parameter @xmath185 ( which however does not play any role in the resulting amplitudes ) . in this limit one gets rid of the propagation in the theory by replacing the derivatives in the kinetic term with delta functions : + @xmath186\;\;\rightarrow\;\ ; \bigg [ \prod_{i=1}^d\,\delta(g_i , g_i')\delta(\nu_i x_i + \nu_i ' x_i ' ) \bigg ] .\ ] ] + if we do this in ( [ eqnarray : actiongx ] ) we will obtain essentially the usual gft model but with the sole difference of having extra arguments which the field depends on . + how are the feynman amplitudes affected by these extra variables ? since there is no coupling between the group and the x ( or p ) variables , they are just propagated in parallel around the feynman graph . the upshot of this is that the extra variables x ( or p ) contribute just an overall ( infinite ) constant and thus do not affect the amplitudes , that reduce then to the usual spin foam models . + * another way of looking at the relation between the new model and the conventional , which clarifies the fact the new model is the causal analogue of the usual ones , comes from considering the theory in the ( j , p ) variables . + take a single propagator and look at its character expansion ( [ eq : propagatorjp ] ) . as is clear from this equation that the coefficients of the characters are just the usual klein - gordon propagators on a flat space x , whose dimensions is equal to the dimension of the group g and which has a metric which is the killing form . also , it is clear that it is from here that the complexity ( thus the causal nature , as we discussed ) of the amplitudes comes . using sohozki s formula @xmath187 and the reality of characters ) ( from which the propagator is derived ) in a symmetric way , such that the coefficients of characters which are complex conjugates of each other are real and equal . ] , it follows that the real part of the propagator is given by + [ eq : onshell ] d_f [ g_i , h_i ; p_i , q_i ] = _ i=1^d ( ( p_i - q_i ) _ g dh ) . notice that taking the real part of the propagator is the same as going on - shell with respect to the corresponding equation of motion , which is the classical relation between the p variables and the lie algebra generators of the group g , or , as we will confirm in the next sections and we have discussed in the previous , between the b and the a field of bf theory ( metricity of the connection ) . if we do now the integrals over the p variables it is immediate that we just get the propagator and thus the whole spin foam amplitudes of the usual gfts , as the delta functions integrate to one . let us stress once more that the causalnature of the new models , i.e. the fact that their amplitudes are complex functions of the geometric data , interpretable , as we will confirm in the next sections , as discrete gravity path integrals , results exactly from the lifting of a classical equation of motion ( a discrete analogue , we argue , of the relation between b and a in bf theory ) to allow for off - shell propagation . one could go further and argue that it is this quantum lifting of a classical condition that allows us to go beyond usual bf theory , where it is only the classically allowed flat configurations that have non - zero amplitude in the path integral . + notice also that the situation here is entirely analogous to what happens in the case of a free relativistic quantum particle . the real part of the propagator @xmath188 is simply the on - shell condition @xmath189 . it is crucial to allow the momentum to go over the classically disallowed value in order to have genuine quantum behaviour of the system . this is also what characterizes time - ordered products of field operators with respect to other 2-point functions in scalar field theory + however , even though the above interpretation is intriguing and in line with what our initial motivations were and our results of the next sections will show , we feel that there is much more left to understand about the physics behind the above outlined relation between the new models and the traditional bf ones , as well as about the relation between the two ways , discussed here , in which one can re - obtain the traditional models from these new ones . we leave this for future work . we now specialize the class of models considered above to the case d=3 and g = su(2 ) . the usual models ( with the trivial kinetic term ) , for this choice of dimension and group , give euclidean 3-d bf theory , augmented by a sum over topologies , in perturbative expansion . + the action ( [ eqnarray : actiongp ] ) becomes @xmath190 \phi(g_i ; p_i ) + { } \\ & & { } + \frac{\lambda}{(2\pi)^{36 } 4 ! } \sum_{\nu_1 \ldots \nu_4 } \int_{g^{12 } } \bigg ( \prod_{i \neq j = 1}^{4 } \ , dg_{ij } \bigg ) \ , \int_{p^{12 } } \bigg ( \prod_{i \neq j = 1}^{4 } dp_{ij } \bigg ) \ , \ , \ , \bigg [ \prod_{i < j } \delta(g_{ij } g_{ji}^{-1 } ) \delta(p_{ij } - p_{ji } ) \bigg ] \times { } \\ & & { } \times \phi^{\nu_1}(g_{1j } ; p_{1j } ) \ldots \phi^{\nu_4}(g_{4 j } ; p_{4 j}).\end{aligned}\ ] ] su(2 ) is a compact group of rank one , hence the kernel depends on a single periodic parameter . it is convenient to choose this parameter to be the angle of rotation in the usual representation of su(2 ) . more precisely , if h @xmath146 g then @xmath191 where @xmath192 is the angle of rotation , @xmath193 is the axis of rotation and @xmath194 are the pauli matrices . the angle @xmath192 is a _ multivalued _ function of the group element . this should be clear as @xmath192 and @xmath195 for @xmath196 correspond to the same group element . in other words , any choice of @xmath106 in the expression @xmath197 provides a possible definition of the angle characterizing the holonomy @xmath198 . what this means is that from a geometrical point of view , the angle of rotation , is intrinsically an equivalence class of real numbers modulo addition of @xmath199 . we will denote this equivalence class by @xmath200 = \theta(h ) \ , \textrm{mod } \ , 4 \pi$ ] , and identify @xmath201 $ ] , i.e. @xmath197 for any choice of @xmath106 , with the holonomy angle . however , since the equivalence class is not a number , to write any formula involving the angle of rotation , one should pick a representative of the equivalence class ( i.e. choose a specific @xmath106 , for example @xmath202 thus restricting oneself to the @xmath203 $ ] range ) . this random choice , does not matter if the function is automatically periodic when @xmath204 ( e.g. the character function ) . however , when the functional expression one is dealing with is not periodic ( the evolution kernel below ) , one needs to sum over all the equivalence classes ( all possible @xmath106 to obtain a function with the correct boundary conditions , i.e. a function on the group . + the explicit form of the evolution kernel on su(2 ) in ( proper ) time @xmath151 is given @xcite by the following formula [ eq : su2kernel ] k [ h , t ] = _ n=- ^ ( exp ) . note the sum enforcing periodicity in @xmath205 . to avoid writing the sums which enforce periodicity in what follows , we adopt the following notation : whenever we have a sum which enforces periodicity of a certain function , i.e. whenever have an expression of the form @xmath206 , we will just write @xmath207)$ ] . the sum , which is required to convert an expression involving @xmath208 $ ] to a legitimate one involving just real numbers , will be kept implicit . this is perfectly reasonable from the geometric point of view as well , as it is exactly the entire equivalence class that has the meaning of an angle of rotation . this sum has also the meaning of a sum over all geodesics over the group ( i.e. @xmath0 ) connecting the same two points @xcite . + once more , we define the partition function of the model as a perturbative expansion in feynman diagrams : @xmath117 and , again , the feynman amplitudes factorize per dual face : @xmath209 according to ( [ eq : amplitudegp ] ) , to get the amplitude @xmath156 for a dual face with n vertices in the ( g , p ) variables , we should multiply the expression for the evolution kernel by @xmath210 and then take the fourier transform of the result , with respect to @xmath151 , at the value @xmath211 , where @xmath212 is the 3x3 identity matrix . as a consequence , since the metric on the dual to x is given in terms of the inverse of this killing form , @xmath213 . ] . + thus [ eq : su2amplitude ] a_n = _ n=- ^ ( _ 0^ dt t^n - exp ) the integral can be evaluated explicitly @xcite using the formula [ eq : hankel ] _ 0^ dt t^-1 exp = i^+1 q^ h_^(1)(q p ) , where @xmath214 is a hankel function of the first kind of order @xmath95 . the two coefficients p and q are complex numbers in general , but what is very important is that they should satisfy ( im@xmath215 and im@xmath216 ) . it should be obvious that this should be the case as the integrals will simply not converge otherwise . note that while the left hand side has @xmath217 in it , the right hand side has @xmath218 . the fact that we have to take a square root will be very important in the lorentzian case . + for us [ eq : pq ] = n - , p = ^2 - q^2 = . it is clear that @xmath219 and @xmath220 ) imply that both the @xmath221 and the @xmath192 should be complexified and given small positive imaginary parts . this complexification is nothing but the usual feynman @xmath222 prescription . the root of @xmath217 is defined in the usual way , by taking a cut along the negative real axis , letting @xmath223 and extending by continuity . as both the numerator and denominator have small phases ( both are positive ) , their ratio also has a small phase . thus the square root of @xmath217 is very close to the real axis and is very nearly equal to @xmath224 |}{|\vec{p}^2 - \frac{m^2 - 1}{4}|}$ ] . + plugging ( [ eq : hankel ] ) into ( [ eq : su2amplitude ] ) we get @xmath225 = \frac{i^{n-2}}{16 \sqrt{\pi } ( n-1 ) ! } \bigg [ \ , \ , \frac { [ \ , \theta(h ) \ , ] } { \textrm{sin}\big(\frac { [ \ , \theta(h ) \ , ] } { 2}\big ) } \bigg ( \frac{| \ , [ \ , \theta(h ) \ , ] \ , |}{\sqrt{\vec{p}^2 - \frac{m^2 - 1}{4 } } } \bigg ) ^{n- \frac{3}{2 } } \bigg ] \ , h^{(1)}_{n-\frac{3}{2 } } \big ( \sqrt{\vec{p}^2 - \frac{m^2 - 1}{4 } } \ , | \ , \ , \ , [ \ , \theta(h ) \ , ] \ , | \big ) .\ ] ] the hankel function of half - integer order can be given explicitly in terms of elementary functions via [ eq : polynomial ] h_n - ^(1)(z ) = i^- ( n-1 ) \ { _ k=0^n-2 ( -1)^k } e^iz . using this expression we get that the dual face amplitude has the form [ eq : ramplitudemeasure ] a_n = ( [ ( h ) ] , || , n ) e^i | [ ( h ) ] | , with @xmath136 being given by @xmath226 \ , , \ , ( n-1 ) ! } \bigg [ \frac{1}{\textrm{sin } ( [ \ , \theta(h ) \ , ] ) } \bigg ( \frac{| [ \ , \theta(h ) \ , ] |}{\sqrt{\vec{p}^2 - \frac{m^2 - 1}{4}}}\bigg ) ^{n-1 } \bigg ] \times { } \nonumber \\ \label{eqnarray : measure } & & { } \times \bigg \ { \sum_{k=0}^{n-2 } \big ( -1 \big ) ^k \frac{(n + k -2)!}{k ! ( n - k-2 ) ! } \frac{1}{\big ( 2 i \sqrt{\vec{p } - \frac{m^2 - 1}{4 } } \ , \ , | [ \ , \theta(h ) \ , ] | \big ) ^k } \bigg \}.\end{aligned}\ ] ] above , we have given the amplitude for just one dual face , or recalling that in 3d a dual face is dual to an edge of the triangulation , it is the amplitude for a single edge . however , as was mentioned in the previous section , the amplitude of the dual complex in the ( g , p ) variables is just the product of the dual - face amplitudes , or in the 3d context the product of edge amplitudes . thus , we can easily write the amplitude for the whole triangulation ( feynman graph ) @xmath227 . it is [ eq : partition ] z_t = _ g^e^ * ( _ e^ * t^ * dg_e^ * ) _ p^e ( _ e t d_e ) ( g_e^ * , ^2_e , n_e ) e^i _ e | [ _ e ] | , where the products go over all edges in the triangulation @xmath228 and all dual edges in the dual 2-complex @xmath229 , and the factor @xmath230 is a product of all the @xmath136 s coming from each dual face i.e. @xmath231 with @xmath232 given by ( [ eqnarray : measure ] ) . + now , consider the exponent in the above expression . we see immediately that it is just the regge action for euclidean 3d gravity @xmath233 in 1st order form , after identification of @xmath234 with @xmath235 . here the sum goes over all edges of the triangulation . @xmath235 stands for the length of the edge e and @xmath236 for the deficit angle , i.e. the discretized curvature , around the dual edge , which coincides with the angle of rotation @xmath237 $ ] that characterizes our holonomies @xmath238 ( again , equivalent to @xmath197 for any choice of @xmath106 ) . + this reconfirms and makes precise the interpretation for the new variables , the @xmath94 s , which was proposed in the introduction , as that they give the length of the edges to which they are associated , and thus as representing the discretized triad ( b field ) associated with these edges , while the group elements are confirmed as a discretization of the lorentz connection field @xmath66 . indeed , we obtain an expression for the simplicial gravity action of the same type as the ones in @xcite , and , as there , with the edge lengths ( hinge volumes ) restricted to have a positive orientation . note that this identification of the length with the variable p becomes especially nice if we set @xmath239 , as then it is the length of @xmath240 directly , @xmath241 , which coincides with the length of the edge @xmath235 . for this reason , as well as to simplify the formulae , we will adopt this choice for @xmath242 in the following discussion of the amplitude in the ( g , p ) variables . + it is clear that the variation of the above action with respect to the edge lengths , or the variables @xmath240 , gives the classical equation @xmath243 , i.e. imposes flatness of the discrete geometry as the only _ classically _ allowed configuration , as we expect from 3d gravity . the variation with respect to the connection variables is more involved , and we would expect it to provide a discrete analogue of the continuum conditions enforcing metricity of the connection . we leave its analysis for future work @xcite . + the amplitude for the triangulation @xmath227 is then just the partition function for discrete 3d euclidean gravity , in 1st order form , with a measure factor @xmath244 , as desired . let us now consider the measure factor @xmath245 , in more detail . this factor is a complex number in general as should be evident from ( [ eqnarray : measure ] ) . thus if we write @xmath246 with s_c ( g_e^ * , ^2_e , n_e ) = _ e [ sc ] , we see that the full feynman amplitude for the whole triangulation has the form @xmath247 } .\ ] ] the modulus of the quantum measure @xmath248 , i.e. @xmath249 is then what should be considered as a proper quantum measure factor in our path integral , while the phase @xmath250 gives what can be interpreted as quantum corrections to the regge action ( hence the subscript ) . we thus see that the amplitudes of our model , more precisely , have the form of a path integral ( with an explicitly defined measure ) of an extended 1st order regge calculus , in which the regge action is extended by ( also explicitly computable ) quantum corrections . + let us then study in more detail these quantum corrections , in order to confirm their geometric meaning and thus their proposed interpretation . we then study the explicit formula ( [ sc ] ) for @xmath251 as well as the expression ( [ eqnarray : measure ] ) . also , we focus on the dependence on the geometric data @xmath94 and @xmath252 , neglecting constant factors , which give a constant contribution to the phase at every edge ( equal to @xmath253 ) . + one of the most important properties of this part of expression ( [ eqnarray : measure ] ) is that it depends on @xmath254 $ ] and @xmath240 solely through the combination ( @xmath255 |$ ] ) , weighted by factors that will necessarily be purely combinatorial , i.e. dependent on @xmath256 only . + under the interpretation discussed above for the p variables and for the @xmath252 , a first possible interpretation of the powers of the expression ( @xmath257 ) is that they represent the discrete analogues of higher order corrections to the einstein - hilbert action , given by powers of the ricci scalar @xcite . one could then expect the correspondence @xmath258 | \big ) ^k \sim \ , \,\int r^k(g ) \ , \ , \textrm{vol } , \ ] ] where @xmath259 is the mentioned combinatorial factor , vol is the volume form and the aforementioned correspondence holds in the continuum approximation ( in the sense of measures ) @xcite . + however , the simplicial geometry of such higher powers of the regge term is subtle ( see again @xcite for an extensive and detailed analysis ) . in particular , for the square power of the above expression , another plausible interpretation is provided by the square of the riemann tensor , giving : @xmath260 | \big ) ^2 \sim \ , \,\int r_{\mu\nu\rho\sigma}(g ) r^{\mu\nu\rho\sigma}(g ) \ , \ , \textrm{vol } .\ ] ] in general , in fact , higher order curvature terms , as traditionally defined in simplicial gravity , involve an additional geometric ingredient , a normalization of the hinge volumes , that gives them the correct dimensionless character . this is taken to be the contribution of the d - simplex volume associated to the specific hinge considered , @xmath261 , giving a complete quadratic term of the form @xmath262 arguments , to be most likely irrelevant for the continuum correspondence , but of course this is not at all obvious . with this choice of normalizing factor , one can indeed show that ( the discrete analogue of ) both @xmath263 and @xmath264 agree when restricted to a single hinge . therefore the difference between the two types of higher order terms depends exclusively on how different hinges are coupled , each being weighted individually by the quadratic expression above . the simplest choice of coupling @xmath265 to the square of the riemann tensor . other constructions are however possible for both the riemann tensor itself and the quadratic terms that can be constructed from it @xcite . also , we are not aware of similar detailed analyzes for higher powers , thus for curvature invariants beyond the quadratic order . + in our model , the normalizing volume factor can be interpreted as being given by the planck length to the appropriate power and multiplied by our purely combinatorial factor @xmath259 , a function of @xmath256 . therefore a more complete interpretation scheme for the higher order corrections to the regge action provided by our gfts does involve a careful analysis of these combinatorial factors and in particular of the way they couple different hinges in the same d - simplex and beyond . this analysis will be performed and reported elsewhere @xcite . + from a more general perspective , however , these corrections to the regge action , predicted by our model(s ) share two main features : 1 ) they involve , as mentioned , both positive and negative powers of the curvature invariants , and 2 ) they depend on two independent sets of geometric variables , the ( discrete analogues of ) the d - bein and the connection fields . this implies , therefore , that the corrections to the bare regge action produced by the model are of the general f(r ) type in the metric affine formalism @xcite . + we would like to emphasize once more that these corrections are not arbitrary , rather their form , including relative coefficients weighting them , and their behaviour in the various regimes of the theory are fully determined by the our choice of the original gft action . this also means of course that one can modify the exact dependence on them of the simplicial action appearing in our feynman amplitudes , by modifying the same gft action , thus constructing different specific models within the general class of gfts we have defined . + let us analyze further the physics behind the corrections @xmath251 . we are most interested in two approximations , both of which can be given a clear physical interpretation . + the first regime is when the lengths becomes large , i.e. when @xmath266 ( remember that we are working in planck units ) . equivalently , this is the regime of large actions , in units of the planck s constant , due to the way in which the edge lengths enter the discrete regge action . this approximation can thus be considered as a semiclassical approximation as it corresponds to the case where the relative size of the quantum fluctuations of the action ( and of the edge lengths ) is small . this is the analogue , for our models , of the asymptotics usually considered in the standard spin foams ( the large j asymptotic ) . + the second regime is approached when the edge lengths and discrete curvatures become small , and the triangulation becomes finer and finer , i.e. when @xmath267 | ) \rightarrow 0 $ ] and @xmath268 . this can be thought of as the continuum approximation . + let us first look at the behaviour of the measure and thus of the quantum corrections @xmath251 at the heuristic level . consider then the explicit expression for the ( complex ) measure in ( [ eqnarray : measure ] ) , and in particular to the part of it within curly brackets . + in the first case ( large lengths @xmath241 ) it is the first term in the sum in ( [ eqnarray : measure ] ) that dominates , and since this term is real , it means the regge action remains the dominant contribution to the phase of the path integral amplitude . we expect then the phase , including corrections , to be of the general form @xmath269 , thus with inverse powers of the curvature to play the role of quantum corrections to the regge action , and the full feynman amplitude ( discrete gravity path integral ) to be approximated by @xmath270 \bigg [ \frac{[\theta_e]^{2(n_e-1)}}{\textrm{sin } ( [ \ , \theta_e \ , ] ) } \bigg ( \frac{1}{|\vec{p}_e| [ \theta_e]}\bigg ) ^{n_e-1 } \bigg ] \times { } \\ & & { } \times \bigg [ 1 + o \bigg ( \frac{1}{|\vec{p}_e| [ \theta_e ] } \bigg ) \bigg ] \bigg \ } \ , \ , e^{i \big [ s_{regge } ( g_{e^ * } , |\vec{p}_e| ) + o ( 1 / |\vec{p}_e| [ \theta_e ] ) \big ] } .\end{aligned}\ ] ] in the second case ( small edge lengths and very fine triangulation , i.e. high @xmath256 ) it is the last term in the sum in ( [ eqnarray : measure ] ) that dominates . this term also contributes just a constant to @xmath251 ( equal to @xmath271 ) . we expect then the phase , including corrections , to be of the general form @xmath272 , thus with positive powers of the curvature to play the role of quantum corrections to the regge action , and the full feynman amplitude ( discrete gravity path integral ) to be dominated by a term like : @xmath273)^{2(n_e-1)}}{\textrm{sin } ( [ \ , \theta_e \ , ] ) } \bigg ( \frac{1}{|\vec{p}_e| [ \theta_e]}\bigg ) ^{2n_e-3 } \bigg ] \bigg \ } \ , \ , e^{i \big [ s_{regge } ( g_{e^ * } , \vec{p}^2_e ) \ , + \ , o(\sum_e ( | \vec{p}_e| \theta_e)^2 ) \big ] } .\ ] ] we would like now to go beyond the naive heuristic considerations and analyze the form of the quantum measure , and of @xmath251 in particular , in more detail . + this can be done with full confidence for the semiclassical approximation . the reason for this is that ( [ eqnarray : measure ] ) , and thus the full feynman amplitude , is regular at the limiting point @xmath274 ( it goes simply to zero ) , for a generic triangulation . also , the proper analysis involves the asymptotic expansion of the hankel function for large values of the argument , but , for half - integer order , this _ coincides _ with the expression ( [ eq : polynomial ] ) that we have used . this allows us to obtain full understanding of the way the phase behaves in the large length limit . we can then use directly the expression ( [ sc ] ) and , expanding the arctangent in powers of @xmath275 , we get that @xmath276.\ ] ] of course , all the coefficients in the expansion can , in principle be computed within our model . as said , we can think of @xmath277 as the inverse of the scalar curvature . thus @xmath278 \textrm{vol}$ ] . since the corrections are inverse in the curvature , they are of the infrared type , as it is intuitively to be expected as we are doing a large scale approximation to out model . thus we see that the new model predicts long - distance effects , at the simplicial level , of the same type as those predicted by effective @xmath1-extended gravity models , and that have been found relevant in cosmological applications ( most notably for modelling dark energy effects ) @xcite . + the other case of interest ( the continuum limit ) is much more involved to analyze , and the purely heuristic argument can be trusted as a limited indication of the relevant physics ( it is intuitively obvious that in the small distance regime one gets quantum corrections of the ultraviolet type @xmath279 ) , but one that can not be easily confirmed by a detailed analysis , at this point . + the reason for this is that , as is not difficult to see from ( [ eqnarray : measure ] ) , the feynman amplitude has a badly singular point in ( @xmath280 ) : 1 ) it diverges in the limit like @xmath281 ; 2 ) the hankel function has a branch point at 0 , which poses extra problems one needs to deal with due to the stokes phenomenon , whose main consequence is , in this context , that the expression for the amplitude around this point depends heavily on how exactly the limit is taken , i.e. which path one takes in the complex domain to approach the singular point . finally , the limit @xmath267 | ) \rightarrow 0 $ ] by itself is not very physically meaningful . it acquires its importance when combined with the limit @xmath268 . however , it is not difficult to see that the way the amplitude behaves is sensitive to the way these two limits are combined . due to the above reasons we defer the detailed treatment of this regime of the model , as well as of the corresponding formulation of simplicial geometry for future work @xcite . + finally , let us note that the fact that the amplitude diverges as @xmath281 is very appealing intuitively , as it implies that for larger triangulations it is the small values of @xmath282 that are the most relevant ones and that they become more and more dominant as we take larger and larger triangulations ( this is because the higher powers are more divergent ) . + since we have interpreted the p variables as giving the lengths of the edges of the triangulation , this looks exactly like the behaviour one would want in order to recover a good continuum limit : for a triangulation consisting of a large number of tetrahedra , the dominant histories are those for which the basic simplices are small , corresponding moreover to a singularity in the quantum amplitudes . + the new model is a causal one in the sense of @xcite and it shares many features of the 3d model presented in @xcite . let us briefly recall the model proposed there . the action used in @xcite is a discretized version of ( [ eq : bfaction ] ) . the b field is replaced with a lie algebra element @xmath283 associated to every edge of the triangulation , and the connection a is substituted by its holonomy around the dual face @xmath284 . the discrete action is then given by @xmath285 the model is quantized via the path integral method in the usual way the only crucial difference being that the product @xmath286 is restricted to be nonnegative . this is because , as was argued in @xcite , this corresponds to restricting the discretzied volume to be positive , and thus it represents the wanted implementation of the causalityrestriction in quantum gravity transition amplitudes . thus the partition function is given by which is the deficit angle around the edge with the heaviside step function @xmath287 . ] [ eq : causalpartition ] z = _ g^e^ * ( _ e^ * t^ * dg_e^ * ) _ p^e ( _ e t d _ e ) ( _ e _ e ) e^i _ e _ e ( _ e ) . the new model , which generates amplitudes given by ( [ eq : partition ] ) , is causal in the same sense as ( [ eq : causalpartition ] ) due to the simple fact that the @xmath241 is always positive . thus , keeping the interpretation of the p s in mind , in our gft model the integral over the discretized field is also restricted to be such that the hinge volumes are positive . this restriction results @xcite in the causal analogues of usual spin foams in both the free and matter coupled cases . + there are several differences , however , between the model proposed here and the one proposed in @xcite , i.e. between ( [ eq : partition ] ) and ( [ eq : causalpartition ] ) . * first , the discretizations used in the two cases are somewhat different . although , both use the holonomies to represent the curvature and both average the b field over an edge ( and get a vector ) , the way these two objects enter into the discrete action is slightly different . notably , the two variables are totally independent in the new model and interact simply through multiplicative coupling , at least before one uses the equations of motion resulting from the variation of the simplicial action . in the old model however , the variables mix more substantially : a ) there is extra coupling introduced by the dot product @xmath288 ( @xmath289 is completely absent from the action in ( [ eq : partition ] ) ) ; and b ) the domains of integration of the two variables are interdependent , due to the step function . with regards to both these points the new model is _ simpler _ than the old one . it is well possible , however , that one can get a 3d model , in the same new class of gfts we are proposing , that is closer to the one in @xcite by imposing additional ( symmetry ) conditions on the variables appearing in the gft action . * the measure factor @xmath290 present in ( [ eq : partition ] ) is absent from ( [ eq : causalpartition ] ) . these are , as discussed above , corrections to the bare regge action ( and thus to the 3d bf action ) that have been here deduced from first principles and not added in an ad hoc way ( which of course could be done with ( [ eq : causalpartition ] ) ) . thus the new model is significantly richer than the old one . also with respect to this point , we notice that there is still freedom left in choosing specific gft actions within the general class of gfts we introduced , and thus obtaining models with modified ( and possibly simpler ) path integral measures in the perturbative expansion . * due to the fact that the factor @xmath290 depends on @xmath256 , it should be clear that if we perform the integrals over the p s in ( [ eq : partition ] ) we will get dual face amplitudes which depend on the number of vertices in each dual face , i.e. each dual face amplitude is a function of @xmath256 . this however is not the case in the old model where the dual face amplitudes , which were computed explicitly in @xcite were independent of this factor . the reason for this can be traced to the following fact . at the spin foam level , and in the construction of @xcite , the basic building block of the model was considered to be the dual face . at the gft level , it is necessarily the wedge ( i.e. the portion of the dual face contained within a d - simplex ) from which everything else is constructed @xcite . the causal restriction advocated for in @xcite was a dual - face one , and this is the reason for the independence of the resulting amplitudes from the number of wedges ( vertices ) making the dual face . one would expect that if the construction in @xcite is repeated but with the causality restriction being imposed at the level of each wedge , one would obtain a model which is closer to the one reproduced here . * finally , in @xcite the causal restriction , although shown plausible , was implemented essentially by hand simply by inserting the step function into the partition function ( [ eq : causalpartition ] ) ; therefore one could be left wondering about the possibility of different ways of implementing the same type of causal restriction . in the new model(s ) we are proposing such freedom is absent , at least for given choice of gft action : the amplitudes are built , in a _ unique _ manner , from the same building block , the propagator , and it is exactly the propagator that has the information about causality , orientation dependence and the propagation of quantum degrees of freedom . consider now the model in the ( j , p ) variables ( i.e. equation ( [ eq : amplitudejp ] ) specified for @xmath291 and g= su(2 ) ) [ eq : rspinfoamjp ] z_=(_f*t^*_j_f^ * _ p d^3 _ f^ * ) _ f^ * t^ * _ v^ * t^ * \ { 6-j } , where of course now the representations are labelled by half - integers @xmath292 . since it is the p s that give the lengths , the interpretation of the j variables is not as straightforward as it is in the usual models . looking at the expression above we can see that the j s label the different poles of the dual face amplitude . since the expression for the dual face amplitude is essentially a product of feynman propagators we can think of the j variables as labelling the different semi - classical , on - shell values of @xmath241 . the poles are @xmath293 if we make the same choice for @xmath242 as before , i.e. if we set @xmath294 , then we see that @xmath295 . + notice that if we plug this back into the path integral . we can heuristically interpret this restriction as imposing the connection metricity equation of motion ( i.e. the equation obtained by varying the connection ) into the path integral . ] into ( [ eq : partition ] ) we see that it becomes @xmath296 from which we see that the exponent is just the regge action with the edge length restricted to be @xmath297 . this matches nicely with the expression obtained in @xcite ( see also @xcite ) for the eigenvalues of the length operator in 3d canonical quantum gravity . + of course , this does not really mean that the lengths are quantized in our model . this is because the length information is given by @xmath241 s and these are unconstrained , in general . just like in the feynman propagator for a scalar particle the momentum is not constrained , in the quantum theory , to the mass shell . + we can now obtain a pure spin foam expression for the feynman amplitudes of our model , i.e. one involving only the representation variables . it is not difficult to perform the integrations over @xmath298 s in ( [ eq : rspinfoamjp ] ) . the easiest way to do this is by using cauchy s formula to @xmath299 and then closing the contour in the complex plane . by jordan s lemma , since @xmath300 , the integral of the expression we have along a semicircle centered at the origin of radius r , goes to zero as r @xmath301 . this allows us to add this bit to the integral closing the contour . ] . the result of these integrations is given by @xmath302 where the dual face amplitude is given by [ eq : rspinfoamj ] a_f^ * ( j_f^ * , n_f^ * ) = 4 ^2 ^n_f^*-1 . let us now try to extract some physical information on the model , and in particular how it depends on the combinatorics of the underlying triangulation , starting from this expression for the amplitudes . + consider the regime of large @xmath303 , i.e. consider the triangulations which are composed of many tetrahedra , which we have argued is one ingredient for approximating continuum physics in this setting . this should be combined with a small @xmath282 approximation ; however , having integrated out the p s , we can only expect to read out from the amplitudes what are the dominant configurations in the @xmath304 variables . using stirling s formula @xmath305 we can easily see that the second multiplicand in ( [ eq : rspinfoamj ] ) is asymptotic to @xmath306 . thus for large @xmath307 s [ eq : asymptotic ] a_f^ * ( j_f^ * , n_f^ * ) ~ ^n_f^*-1 , where f is a function of polynomial growth . + we conclude that the amplitude consisting of a large number of tetrahedra is dominated ( as this is when @xmath308 ) by the two lowest values of j s , @xmath309 , which can be thought of as the vacuum configuration , and @xmath310 , which is some sort of lowest excited state . so , if we interpret the values of @xmath304 as edge lengths , as in usual spin foam models , it is the shortest values that are the dominant ones for fine triangulations , as we would expect . in the limit of finer and finer triangulations ( which , again , we would expect to lead to a continuum approximation of the discrete path integral , then , the partition function can be reasonably well approximated by a purely combinatorial sum , with amplitudes given by the above quantities evaluated at @xmath311 , i.e. for purely equilateral triangulation with edge lengths @xmath312 . in other word , in this regime , the model would effectively , and _ dynamically _ , reduce to a pure dynamical triangulations model @xcite . + consider the regime of large @xmath292 s . again , having integrated out the trueedge length variables @xmath94 , we can heuristically interpret this regime as a large distance approximation . looking again at the same expression ( [ eq : rspinfoamj ] ) , it is clear that it is the lowest values of @xmath307 s that are most relevant in the limit . what this means is that if we look at the large length limit the most important feynman diagrams are represented by the simplest triangulations , more precisely those with least number of vertices for each dual face . in other words , _ if one is interested only in large distance and semi - classical physics _ , then considering simple triangulations would suffice , as the gft partition function , in perturbative expansion , is anyway dominated by such configurations . this leads further support to the nice results obtained in the calculation of the lattice graviton propagator in @xcite , working indeed in the context of the large - j limit of spin foam models , and using semi - classical boundary states based on simple boundary triangulations , as well as very simple bulk triangulations ( low order in the gft coupling constant @xmath84 ) . + these considerations should however be taken with care , since the @xmath313 are not , strictly speaking edge lengths , as we have stressed above , this role being in fact played by the @xmath94 variables . we hope that the above results underlie the fertility and potential usefulness of the proposed model in understanding quantum geometry . we now move on to the case when d=3 and g = sl(2,@xmath314 ) @xmath315 su(1,1 ) , i.e. g is the double cover of the lorentz group in three dimensions . thus , this model corresponds to the lorentzian gravity in 3d . + su(1,1 ) has two nonconjugate cartan subgroups . this is easy to see as @xmath316 can be obtained by complexifying two of the generators of @xmath39 . thus we would obtain a generator of rotation and two generators of boosts . the cartan subgroups are thus the two subgroups generated by these different elements . one cartan subgroup is just u(1 ) generated by the uncomplexified element , we will denote its conjugacy class by r ( for rotation ) . the other cartan subgroup is generated by one of the complexified elements and it is a noncompact group ( isomorphic to @xmath314 ) whose conjugacy class we will denote by b ( for boost ) . + the fact that they are cartan subgroups means that any element of su(1,1 ) is conjugate to either an element of r or of b apart from a set of elements of measure zero in the haar measure . su(1,1 ) . if @xmath317 then g @xmath146 b. if @xmath318 then g @xmath146 r. the set of elements which satisfy @xmath319 is a set of measure zero . ] the conjugacy classes of the elements of r will be parametrized by a periodic parameter @xmath252 ( angle ) for which we choose a normalization such that its period is @xmath320 . while the conjugacy classes of the elements of b will be parametrized by a real number @xmath127 ( the boost parameter , rapidity ) . + the explicit formula for the evolution kernel in proper time is given by the following formula @xcite @xmath321 & = & \frac{1}{(4 \pi i t ) ^{\frac{3}{2 } } } \frac { [ \ , \theta(h ) \ , ] } { 2 sin\big(\frac { [ \ , \theta(h ) \ , ] } { 2}\big)}\ , \ , \ , \ , exp \bigg [ \frac{i}{2 t } [ \ , \theta(h ) \,]^2 + \frac{i t}{8 } \bigg ] \qquad \qquad \qquad \textrm{when } h \in r \nonumber \\ \label{eqnarray : su11kernel } & & \\ & = & \frac{1}{(4 \pi i t ) ^{\frac{3}{2 } } } \frac{\psi(h ) } { 2 sinh\big(\frac{\psi(h)}{2 } \big)}\ , \ , \ , \ , exp \bigg [ - \frac{i}{2 t } \psi^2 + \frac{i t}{8 } \bigg ] \qquad \qquad \qquad \quad \ , \textrm{when } h \in b \nonumber , \end{aligned}\ ] ] where we have used the same notation for the periodic parameter @xmath252 as in the previous subsection . + note that when the holonomy group element is a rotation then the su(1,1 ) evolution kernel has exactly the same form as the su(2 ) one ( [ eq : su2kernel ] ) . the crucial difference between the rotation and the boost cases is the different sign sitting in front of the parameters in the two cases ; plus in the rotation case and minus in the boost case . + once more we are interested in the partition function of the theory , expanded perturbatively in feynman diagrams @xmath117 and , again , the feynman amplitudes factorize per dual face @xmath322 according to the general formula ( [ eq : amplitudegp ] ) we get the dual face amplitude @xmath156 by multiplying the above expression for the kernel ( [ eqnarray : su11kernel ] ) by @xmath323 and evaluating its fourier transform at @xmath211 . note that the killing form ( which enters into the definition of @xmath92 ) has now signature @xmath324 . thus there is one timelike direction ( the generator of the compact subgroup ) and two spacelike ones . using the same normalizations as in the case of su(2 ) , we get @xmath325 . so , the amplitude for a dual face with n vertices is given by @xmath326 = \frac{1}{(n-1 ) ! } \int_0^{\infty } dt \ , t^{n-1 } \ , e^{i t \ , ( p^2 - \frac{m^2}{8 } ) } \ , k[h \ , , \ , t].\ ] ] now , consider the case when h is a rotation ( h @xmath146 r ) . then since the formula for the kernel ( [ eqnarray : su11kernel ] ) is exactly the same as the one we used in the su(2 ) calculation ( [ eq : su2kernel ] ) we can just write down the answer . thus @xmath327 & = & \mu_r \big ( [ \ , \theta(h ) \ , ] \ , , \ , \sqrt{\vec{p}^2 } \ n \big ) \ , e^{i \sqrt{\vec{p}^2 - \frac{m^2 - 1}{4 } } \ , | \ , [ \ , \theta(h ) \ , ] \ , |}\\ \mu_r \big ( [ \ , \theta(h ) \ , ] \ , , \ , \sqrt{\vec{p}^2 } \ , , \ , n \big ) & = & \frac{- i \sqrt{2}}{16 \pi ( n-1 ) ! } \bigg [ \frac{1}{\textrm{sin } ( [ \ , \theta(h ) \ , ] ) } \bigg ( \frac{| [ \ , \theta(h ) \ , ] |}{\sqrt{\vec{p}^2 - \frac{m^2 - 1}{4}}}\bigg ) ^{n-1 } \bigg ] \times { } \nonumber \\ \label{eqnarray : rsu11amplitudegp } & & { } \times \bigg \ { \sum_{k=0}^{n-2 } \big ( -1 \big ) ^k \frac{(n + k -2)!}{k ! ( n - k-2 ) ! } \frac{1}{\big ( 2 i \sqrt{\vec{p}^2 - \frac{m^2 - 1}{4 } } \ , | [ \ , \theta(h ) \ , ] | \big ) ^k } \bigg \}.\end{aligned}\ ] ] these are exactly the same formulae as before ( [ eq : ramplitudemeasure ] ) , ( [ eqnarray : measure ] ) , with the difference being that @xmath221 is calculated with the minkowski metric and not the euclidean one , and that this formula is not valid for arbitrary su(1,1 ) element h , rather only when h is a rotation ( h @xmath146 r ) . note the exponential factor in the amplitude . restricting for the moment to the case when @xmath328 , it should be clear that if we interpret , analogously to the riemannian case , @xmath329 to be the square of the minkowski length of the edge dual to the dual face under consideration , then the above exponent gives exactly the expected contribution to the regge action coming from the edge under consideration . in order to simplify the following formulae and discussion we will set @xmath239 . this of course also has the effect of making the length of @xmath221 to be directly the square of the edge length . + now , the crucial difference between the riemannian and the lorentzian cases lies in the fact that @xmath221 can now go over both positive and negative values . as mentioned above the case when @xmath330 one just gets the exponent in the amplitude above becomes @xmath331 | } $ ] . a simple oscillating phase . + when @xmath221 goes negative , clearly latexmath:[$\sqrt{\vec{p}^2 } = \pm i previous section we know that @xmath221 should have a small positive imaginary part . this means that when @xmath221 goes from positive to negative values it does so above the origin in the complex . this means that @xmath221 has values above the cut we used to define the square root in the previous section . thus we have to choose the positive square root , i.e. @xmath333 ( see the picture ) . plugging this into our exponent we see that it is equal to latexmath:[$e^{- keeping in mind the interpretation of the p variables as that @xmath221 is the length of the corresponding edge of the triangulation , we see a very interesting phenomenon happening . the exponent as we said earlier coming from an edge contributes a summand towards the regge action of the triangulation , with @xmath335 $ ] being the deficit angle and @xmath282 being the length of the relevant edge . now , as long as the length is positive , i.e. @xmath240 is timelike we get an oscillating phase in the partition function . on the other hand when the @xmath240 goes spacelike making the length negative , we get an exponential suppression of the amplitude . + classically , in the regge action when the edge of the triangulation is timelike , the curvature defect around it has to be a rotation ( think of a massive point particle ) . we see that quantum mechanically this is not true . the edge corresponding to a rotational defect can be both timelike and spacelike , however the spacelike case is suppressed exponentially in the path integral . this is similar to the behaviour exhibited by the feynman propagator of the relativistic point particle ( which is not surprising as we have essentially the same mathematics here ) . the probability for the particle to propagate inside the lightcone is given by an oscillating phase . the particle can also leak outside the light cone ( despite being relativistic ) . but , the probability of doing so is exponentially suppressed . + this of course is an intuitively satisfying feature of the model . however , the discussion above was limited to the case when the holonomy h around the dual face is a rotation ( lies in r ) . + when h is a boost we can of course repeat the same calculation as before ) gives a hankel function of the second kind . ] . however , there is no real need to do this . look at the two expressions in ( [ eqnarray : su11kernel ] ) . note that apart from the factor in front and a phase factor ( @xmath336 ) , the case when g @xmath146 b is just the complex conjugate of the case when g @xmath146 r , due to the difference in the sign in front of the @xmath252 and @xmath127 . as , from the mathematical point of view , in order to get the dual face amplitude we are taking a fourier transform , we can apply the general theorem that relates the fourier transform of a function to the fourier transform of its conjugate . namely , if we denote the fourier transform of a function f by @xmath337 $ ] , then @xmath338 = \big ( \mathcal{f}(f)[-k ] \big ) ^*$ ] . the fourier transform of a complex conjugate of a function is the complex conjugate of the fourier transform evaluated at the negative of the argument . using this we can immediately write down the dual face amplitude in the case when g @xmath146 b. it is given by @xmath327 & = & \mu_b \big ( \psi(h ) \ , , \ , \sqrt { \vec{p}^2 } \ , , \ , n \big ) \ , e^{- i \sqrt { - \vec{p}^2 } \ , | \ , \psi(h ) } \\ \mu_b \big ( \psi(h ) \ , , \ , \sqrt{\vec{p}^2 } \ , , \ , n \big ) & = & \frac { \sqrt{2}}{16 \pi ( n-1 ) ! } \bigg [ \frac{1}{\textrm{sinh } ( \psi(h ) ) } \bigg ( \frac { \psi(h ) } { \sqrt{- \vec{p}^2}}\bigg ) ^{n-1 } \bigg ] \times { } \\ & & { } \times \bigg \ { \sum_{k=0}^{n-2 } \big ( -1 \big ) ^k \frac{(n + k -2)!}{k ! ( n - k-2 ) ! } \frac{1}{\big ( - 2 i \sqrt{- \vec{p}^2 } \ , | \psi(h ) | \big ) ^k } \bigg \}.\end{aligned}\ ] ] the formula for @xmath339 is obtained from @xmath340 by letting @xmath341 , replacing @xmath342 $ ] with @xmath343 , switching the trigonometric sine for the hyperbolic one and finally taking the complex conjugate . of course , by doing the whole calculation from scratch along the same lines as in the riemannian case , one gets the same result . + now we can easily see that the behaviour of the amplitude when h is a boost with respect to the different two possibilities of the sign of the @xmath221 is opposite of that when h is a rotation , due to the minus sign in front of @xmath221 in the formula above . in other words , when @xmath221 is positive , i.e. @xmath344 is a spacelike vector , then we just have an oscillating phase . on the other hand when @xmath221 goes negative , or equivalently , when @xmath344 becomes timelike , amplitude becomes a decaying exponent . has to have a small positive imaginary part , @xmath221 has a small negative imaginary part . thus when we go from the positive values to the negative ones , we are doing so under the cut , thus choosing the negative square root @xmath345 . ] again , this is in full agreement with expectations as classically the curvature defect around a spacelike edge is a boost . + summarizing , if we put together all the dual face amplitudes and form the amplitude for the whole triangulation then what we get is [ eq : lamplitude ] z_t = _ g^e^ * ( _ e^ * t^ * dg_e^ * ) _ p^e ( _ e t d_e ) ( g^*_e^ * , _ e^2 , n_e ) e^i s_regge where as before @xmath136 is the quantum measure factor , being a product of @xmath340 s and @xmath339 s as appropriate , and @xmath346 is given by @xmath347 here @xmath235 stands for the absolute value of the length of the edge e ( @xmath241 ) and @xmath348 stands for the deficit parameter sitting at the edge e ( an angle or a boost ) . note that they are varied independently of each other showing that we have is 1st order theory . @xmath349 is a function of both @xmath235 and @xmath348 and is given by the following table [ eq : table ] c|c|c & rotation & boost + timelike & + 1 & + i + spacelike & + i & - 1 thus as we ve said above when the variables are such that one is off - diagonal in this table ( rotation - spacelike or boost - timelike ) one gets exponential suppression of the amplitude . while when one is on the diagonal then one gets an oscillating phase . this means that the configurations that do not allow for a simultaneous classical geometric interpretation for both the discrete b field and the discrete connection , i.e. those configurations that would be classically disallowed , are not forbidden but still exponentially suppressed . we would like to stress the fact that this causal behaviour is not put into the model by hand , but rather emerges naturally from its very definition as there were no arbitrary choices made anywhere in the construction ( once the gft action has been chosen ) . + since the formulas in the lorentzian case are so close to those in the riemannian one we can easily carry over all the results from there . so , it is not difficult to see that ( [ eqnarray : measure ] ) carries over without much change . in fact , there is no change when h is a rotation apart from the definition of p. when h is a boost the angle becomes a boost parameter , the trig sine goes to a hyperbolic one as well as a few sporadic minus signs . the conclusions deduced from the measure factor carry through without any change in the case in which we have an oscillatory contribution to the partition function ( i.e. a complex exponential ) . the only difference being that when h is a boost , all the phases go to their conjugates , which of course does not affect the qualitative behaviour . + when we are off - diagonal in the table , and we have then an exponential suppression , the integrand is , apart from an overall factor ( a power of i ) , real . this is easiest to see from the fact that , as is evident from ( [ eq : polynomial ] ) , the hankel function for purely imaginary arguments is a ( multiple of ) real function . thus strictly speaking one just has the measure factor in the path integral and no complex exponential ( whose phase is to be interpreted to be the action ) . however , we find it far more clear , intuitively , and more insightful from the physical perspective to split again the integrand into a measure factor and an exponent as we did above . applying this philosophy to @xmath350 , we get corrections to the action @xmath351 of the form @xmath352 , exactly in accordance with expectations , and in complete similarity with the results obtained in the other cases . + as before , in the large minkowski length limit the quantum corrections @xmath251 coming from the phase of the factor @xmath136 are of the inverse scalar curvature type ( @xmath353 ) , indicating the infrared corrections to the bare regge action in the semiclassical limit . + moreover , in all cases , we still get an amplitude that diverges like @xmath281 as @xmath354 , which means that when the number of tetrahedra in the triangulation increases , the shorter lengths become more and more dominant ones , which is what one would expect if the model is to have a good continuum limit . the point @xmath355 is a branch point of the amplitude which diverges there , thus requiring a much more detailed treatment deferred for future work @xcite . let us now move on to the lorentzian analogue of the ( j , p ) representation ( [ eq : rspinfoamjp ] ) for the quantum amplitudes . to do this note that su(1,1 ) has two types of representations @xcite : * discrete ones labelled by a positive half integer j. the casimir @xmath102 for these representations is negative and is equal to ) . this duplicity of representations is responsible for the factor of 2 in front of the sum over the discrete representations . ] the constant @xmath145 appearing in the character expansion of the delta function is @xmath357 . * continuous ones labelled by a positive real number @xmath107 . the casimir for these representations is positive and is equal to @xmath358 . the constant @xmath359 is just @xmath360 if we plug these expressions into ( [ eq : spinfoamjp ] ) ( note that we have to pick the positive sign in front of the casimir as su(1,1 ) is noncompact ) we get [ eq : lspinfoamjp ] z_=(_f*t^ * _ p d^3 _ f^ * ) _ f^ * t^ * a_f^*(j_f^*/ _ f^ * , p_f^ * , n_f^ * ) _ v^ * t^ * \ { 6-j } , where now we get two types of the dual face amplitude in the ( j , p ) variables @xmath361 = \frac{i^{n_{f^ * } } ( 2j_{f^*}+1)}{\big ( \frac{\vec{p}_{f^*}^2}{2 } - \frac{m^2}{8 } - \frac{j_{f^*}(j_{f^*}+1)}{2 } + i \epsilon \big ) ^{n_{f^*}}},\ ] ] when the representation is of discrete type , and @xmath362 = \frac{i^{n_{f^ * } } ( 2 \rho_{f^ * } ) } { \big ( \frac{\vec{p}_{f^*}^2}{2 } - \frac{m^2}{8 } + \frac{\rho_{f^*}^2}{2 } + \frac{1}{8 } + i \epsilon \big ) ^{n_{f^*}}},\ ] ] when it is of the continuous type . + it is obvious from these two expressions that we get poles of two types , timelike and spacelike . what we mean by this is that there are two sets of poles , one when @xmath221 is positive , i.e. when @xmath344 is timelike ; and one when it is negative , i.e. @xmath344 is spacelike . the first type of poles occurs when the representation labelling the dual face is of discrete type , as then we have in the denominator of ( [ eq : lspinfoamjp ] ) the following expression @xmath363 , which vanishes when @xmath364 which , if we set @xmath239 , gives @xmath365 as in the riemannian case . + the other type of poles occurs when the relevant representation is of a continuous type as then we have in the denominator of the same equation the expression @xmath366 . which vanishes when @xmath367 which , on setting @xmath185 , gives @xmath368 . + if we interpret these formulae as giving the semiclassical values of the length , we arrive at the intriguing fact that there are no preferred spacelike lengths as @xmath107 is continuous and thus the poles at the @xmath359 s fill the line . in contrast , there are preferred timelike lengths , which are the discretely spaced @xmath145 s . + finally , by doing a wick rotation@xmath369 , we can perform the integrals over the @xmath298 s along the lines this was done in the riemannian case . the asymptotic formula ( [ eq : asymptotic ] ) goes through essentially unchanged , and we get @xmath370^{n_{f^*}-1 } \quad \textrm{or } \quad a_{f^ * } ( j_{f^ * } , n_{f^ * } ) \sim \frac{1}{f(n_{f^ * } ) } \ , \ , \ , \bigg [ \frac{8 } { \delta_{\rho_{f^*}}^2 } \bigg ] ^{n_{f^*}-1}\ ] ] the only relevant difference being that the factor second factor ( which dictates the behaviour of the asymptotic ) is now either @xmath371^{n-1}$ ] ( as before ) or alternatively equal to @xmath372^{n-1}$ ] . the conclusion is the same as before : for large @xmath373 s it is only the lowest @xmath304 s and @xmath107 s that contribute ( @xmath374 and @xmath375 ) . ) . however , the exact form of f is of no importance for us here as long as it is still a function of polynomial growth . we will continue to use this notation for this prefactor in 4d as well . ] + note that we could have performed the mentioned wick rotationanywhere in the above discussion . most importantly , we could have done it in the triangulation amplitude ( [ eq : lamplitude ] ) . since , this amplitude is just a partition function for gravity , we thus see that there is a straightforward way of performing the wick rotationin the gravity partition function coming from the new model which does not rely on the existence of any particular time slicing . we would like to point out however , that this wick rotation(although very similar to the rotation in the squares of the edge lengths performed in causal dynamical triangulations @xcite ) is not the complete story , in the sense that it does not turn the action for lorentzian gravity into one for riemannian gravity , nor it turns complex exponentials into real ones ( thus quantum mechanical amplitudes into statistical weights ) . this is due to the fact that we have a first order theory with the b and a fields being totally independent . thus , while we wick rotate the b field to a euclidean one , we do not touch the connection . in this sense , the label wick rotation is a slight abuse of language , as it really corresponds to some sort of partial or half - performed wick rotation , from a geometric perspective , hence the quotation marks . however , we find it very intriguing that even this partial transformation can be performed in such a natural way , and believe it can be a good starting point for a similarly natural , but this time complete definition of a geometric wick rotation in simplicial quantum gravity . + summarizing , we see that the lorentzian case is not particularly different from the riemannian one . there is essentially only one major , qualitative difference , which stems from the fact that the lorentzian geometry is richer than the euclidean one . due to the first order nature of the theory , in the lorentzian setting one gets additional , classically forbidden , histories , which have mismatching b and a fields . these histories are , as is customary in quantum mechanics , exponentially suppressed . as for the rest the same simplicial gravity path integral interpretation for the feynman amplitudes of our gft applies , and similar types of quantum corrections to the 1st order regge action are identified . we now come to deal with the four dimensional case ( d=4 ) . our discussion in this subsection and the next will be rather brief as , if we stick to ( causally restricted ) bf theory ( as opposed to gravity , in higher dimensions ) , there is little difference between the 3d and 4d cases . our main aim in the present section is indeed to show explicitly that there are no qualitative new features added to the model by going to the fourth dimension , in neither the riemannian nor the lorentzian signatures , which shows how our proposed new class of gfts behaves similarly in any dimension . as we shall see below , the four dimensional models are essentially carbon copies of the three dimensional ones . + the only _ absolutely crucial _ difference between 3 and 4 dimensions appears , of course , when one tries to convert bf theory into a gravitational one . at the continuum level , this is done by imposing the so - called simplicity constraints on the b field , in a plebanski - like formulation of gravity . since we re interpreting the p variables as a discretized b field , the difference between having bf and gravity lies in these variables , and indeed we expect the discrete analogue of the plebanski constraints to be imposed on them , when passing to gravity @xcite , as they indeed have the needed component structure ( see the discussion in the third subsection below ) . in this work , however , we will treat the p s as being just elements of a metric vector space isomorphic to the lie algebra of g , neglecting any further constraint . thus our discussion will be rather and will amount to little more than a presentation of the results . + the action becomes @xmath376 \phi(g_i ; p_i ) + { } \\ & & { } + \frac{\lambda}{(2\pi)^{120 } 5 ! } \sum_{\nu_1 \ldots \nu_5 } \int_{g^{20 } } \bigg ( \prod_{i \neq j = 1}^5 \ , dg_{ij } \bigg ) \ , \int_{p^{20 } } \bigg ( \prod_{i \neq j = 1}^{5 } dp_{ij } \bigg ) \ , \ , \bigg [ \prod_{i < j } \delta(g_{ij } g_{ji}^{-1 } ) \delta(p_{ij } - p_{ji } ) \bigg ] \times { } \\ & & { } \times \phi^{\nu_1}(g_{1j } ; p_{1j } ) \ldots \phi^{\nu_5}(g_{5 j } ; p_{5 j}).\end{aligned}\ ] ] the group that we are using for the riemannian version of the 4d theory is the double cover of the rotation group in 4 dimensions so(4 ) , which is just spin(4 ) @xmath377 su(2)xsu(2 ) . the fact that the group is a direct product of two copies of the group we used for the 3d riemannian case allows us to carry over easily essentially all the results we discussed in that case to the 4d setting . the reason for this is the fact that the schroedinger kernel on @xmath378 is just the product of the kernels on @xmath379 and @xmath380 . this in turn follows from the fact that the laplacian on the direct product of two groups is just the sum of the two laplacians @xmath381 . this allows us to write down the kernel on su(2)xsu(2 ) right away , essentially by squaring the expression given in ( [ eq : su2kernel ] ) [ eq : so4kernel ] k [ h , t ] = . we are of course using the same notation as before with respect to the periodic parameters @xmath382 and @xmath383 . as before , we want to calculate the feynman graph / triangulation amplitude @xmath177 . since this amplitude when written in terms of the ( g , p ) variables factorizes per dual face , we concentrate on the amplitude for a single dual face . to get the dual face amplitude for a face with n vertices @xmath384 $ ] , we multiply the expression of the kernel by @xmath323 and take the fourier transform evaluated at @xmath385 , i.e. @xmath386 = \frac{1}{(n-1 ) ! } \int_0^{\infty } dt \ , e^{i ( p^2 - \frac{m^2}{4 } ) t } \ , t^{n-1 } \ , k [ h \ , , \ , t].\ ] ] since the group is compact , its killing form , in our conventions , is positive definite . also , since the space p is isometric to the ( dual of ) @xmath387 we have @xmath388 , with @xmath389 . thus , ( with our normalizations ) @xmath390 . also , below we will denote the combination @xmath391 ^ 2 + [ \ , \theta_2(h ) \ , ] ^2}$ ] as @xmath392 $ ] . as in the 3d case , this ( equivalence class of ) parameter(s ) has the geometric interpretation as the square distance between the origin and the point on the group manifold corresponding to the holonomy @xmath198 , measured along a geodesic . + using the formula ( [ eq : hankel ] ) , we get @xmath225 = \frac{i^{n-1}}{(16 \pi)^2 ( n-1 ) ! } \ , \ , \frac { [ \ , \theta_1(h ) \ , ] [ \ , \theta_2(h ) \ , ] } { \textrm{sin}\big(\frac { [ \ , \theta_1(h ) \ , ] } { 2}\big ) \textrm{sin}\big(\frac { [ \ , \theta_2(h ) \ , ] } { 2}\big ) } \ , \ , \bigg [ \frac{[\theta(h)]}{\sqrt{\vec{p}^2 - \frac{m^2}{4 } } } \bigg ] ^{n-3 } \ , \ , h^{(1)}_{n-3 } \big ( \sqrt{\vec{p}^2 - \frac{m^2 -1 } { 2 } } \ , [ \theta(h ) ] \big ) , \ ] ] with the same analytic continuation in the variables as in the 3d case . the hankel function of integer order does not have an expression in terms of elementary functions analogous to ( [ eq : polynomial ] ) . instead it is given in terms of the following non - elementary integral . however , there is a very simple relation between a hankel function of a negative order with the one of a positive one , which is @xmath393 . this means that all we need to do when n=2 ( this is the only allowed value for n which is less than 3 , since any dual face has at least two vertices ) is multiply the given formula by a sign . ] @xmath394 as we see , the formula above still furnishes a natural split of the amplitude into an exponential piece and a measure piece . thus the dual face amplitude is equal to @xmath225 = \mu ( [ \theta(h ) ] , \vec{p } , n ) \ , e^{i \sqrt{\vec{p}^2 - \frac{m^2 -1}{2 } } | [ \theta(h ) ] |},\ ] ] with @xmath136 given by @xmath395 , \vec{p } , n ) & = & - \frac{i^n 2^{n-10}}{\pi^{\frac{5}{2 } } ( n-1 ) ! \gamma(n - \frac{5}{2 } ) } \ , \frac { [ \ , \theta_1(h ) \ , ] [ \ , \theta_2(h ) \ , ] } { \textrm{sin}\big(\frac { [ \ , \theta_1(h ) \ , ] } { 2}\big ) \textrm{sin}\big(\frac { [ \ , \theta_2(h ) \ , ] } { 2}\big ) } \ , \big [ \theta(h ) \big ] ^{2(n-3 ) } \ , \times { } \nonumber \\ \label{eqnarray : rmeasure } & & { } \times \bigg \ { \int_0^{\frac{\pi}{2}}ds \frac{\textrm{cos}^{n-\frac{5}{2}}(s ) \ , \ , e^{- i ( n - \frac{7}{2 } ) \ , s } } { \textrm{sin}^{2n - 5}(s ) } \ , \textrm{exp } \big [ -2 \ , \textrm{cot}(s ) \ , \big ( \theta(h ) \sqrt{\vec{p}^2 - \frac{m^2 -1 } { 2 } } \big ] \bigg \}.\end{aligned}\ ] ] as before , we multiply together all the dual face amplitudes and obtain the amplitude for the feynman diagram / triangulation @xmath396 the products go over all the dual edges @xmath31 of the dual complex @xmath154 and over all the triangles @xmath125 in the triangulation @xmath151 , the @xmath136 is the product of all the @xmath136 s coming from all the edges . the expression @xmath397 in the exponent is @xmath398 |\ ] ] thus the feynman amplitudes of the model are partition functions for an action of discretized bf theory type . classically , the theory given by the action @xmath397 coincides with the one given by the usual bf action , as the equations of motion that they produce are the same ( zero curvature ) . however , quantum mechanically , there is a significant difference between the two theories . the difference being that for the usual bf theory the integral over the b field is unrestricted , with the integration producing the usual a - causal , real partition function . for the model given by @xmath397 the fact that the variable p , which represents the discretized b field , enters only through its length ( which is always positive ) , means that what we have is the causal analogue of the usual bf theory ( hence the subscript ) . + it is tempting to call @xmath399 the area of the triangle t of the triangulation . however , this is untenable as the variable p , being generic and non - simple ( i.e. not itself a wedge product of 4-vectors ) , does not have an interpretation of defining the geometry of the triangle to which is associated , as one needs a simple bivector to do this . as before , the identification is cleanest if we set @xmath239 which we do in what follows to simplify the discussion and formulae . it is clear , however , that we are setting the stage for obtaining a proper causal spin foam model for 4d gravity , to be defined from the above by imposition of suitable simplicity constraints on the p variables . as before we write the @xmath136 in terms of magnitude and phase @xmath400 again , we interpret the @xmath401 as a quantum measure factor , while the phase @xmath402 gives quantum corrections to the pure bf action @xmath397 . + it is straightforward to extract the explicit expression for the phase from ( [ eqnarray : rmeasure ] ) . it is given by , @xmath403.\ ] ] although this expression looks totally different from the one we had in 3d ( [ eqnarray : measure ] ) many of the features of the three dimensional model carry through without any change . most importantly , it still depends on the @xmath404 and @xmath405 solely through the combination @xmath406 , which , at least when @xmath407 is simple , can be interpreted to be the discrete analogue of the ricci scalar r. which means that the quantum corrections arising from the measure @xmath136 are of the general form f(r ) , just like in 3d . + it is possible to analyze asymptotically the expression for the phase above , along the lines done in 3d , and compute the _ exact _ coefficients and combinatorial factors weighting the corrections to the regge action . ) , which , in the half - integer - order case , terminates and provides an explicit expression . ] in the large area asymptotic ( @xmath408 ) . the result is the same as before . one gets inverse scalar curvature corrections ( @xmath409 \ , \textrm{vol}$ ] ) to the bare bf action , i.e. one gets infrared terms arising from the factor @xmath136 in the large distance and semiclassical regime . + also , just as is the case in three dimensions , it is possible to see that the dual face amplitude goes like @xmath410 when @xmath354 . when z is close to zero . ] as before , we would like to draw the reader s attention to the fact that this type of behaviour is at least consistent with , if not suggestive of , the existence of the continuum limit . + let us move on to the j variables . since our group is a product of two copies of su(2 ) , its representation theory follows from that of the su(2 ) . more precisely , each irrep of su(2 ) @xmath411 su(2 ) , is characterized by a pair of half - integers ( @xmath412 ) . the dimension of such an irrep is @xmath413 , finally the casimir that concerns us is just the sum of the two casimirs coming from the two su(2 ) factors @xmath414 . + the 4d case corresponding to equation ( [ eq : spinfoamjp ] ) is : @xmath415 \prod_{v^ * \in t^ * } \left\ { 15-j \right\ } \bigg ) , \ ] ] from which we immediately the semiclassical values of @xmath282 . they are @xmath416 which as before have the nicest form when @xmath417 . + finally , it is not difficult to perform the integrals over the p variables , and obtain the analogue of equation ( [ eq : asymptotic ] ) . the result is @xmath418^{n_{f^*}-1}.\ ] ] as we see it is entirely analogous to the one before , with the crucial factor @xmath419^{n-1}$ ] , which determines the asymptotic being replaced by @xmath420^{n-1}$ ] . the dominant contributions come from the representations for which @xmath421 . this is satisfied only when , neither j exceeds @xmath422 . in other words , the two allowed values are those corresponding to the vacuumand to the lowest excited state . also , note that if we impose , by hand , the simplicity constraint at this level , in the way it is imposed in usual spin foam models , i.e. if we set @xmath423 then there are exactly two dominant contributions : the vacuum @xmath424 , once more , and the configuration with @xmath425 . once again , one can think of this as an indication of a dynamical reduction of the model to a purely combinatorial one of the dynamical triangulations - type . finally , let us consider the case when d=4 and g = sl(2,@xmath426 ) . the technical difference between the ( double cover of the ) lorentz group in 4 dimension and the one in 3 is that in 4d the group sl(2,@xmath426 ) has just one cartan subalgebra . thus , apart from a set of measure zero , all elements in the group are conjugate to the elements of the cartan subgroup , which is the image of the cartan subalgebra under the exponential map . as @xmath427 is spanned by three rotations and three boosts one can take the cartan subalgebra spanned by a rotation and a boost along the same direction , i.e. one compact and one noncompact element . thus the schroedinger kernel will be parametrized by one periodic parameter @xmath61 ( with period @xmath320 ) and one non - periodic one @xmath127 . one can think of them as giving the angle of rotation and the boost parameter of the given group element . intuitively , since @xmath428 can be thought of to be a complexification of @xmath429 @xcite , we will see that what happens in the lorentzian domain can be guessed by complexifying the results obtained in the riemannian one . for example , the formula for the kernel on sl(2,@xmath426 ) is effectively a complexification of that on su(2 ) @xmath411 su(2 ) given in ( [ eq : so4kernel ] ) @xmath430 = \frac{1}{(4 \pi i t ) ^3 } \frac { [ \ , \theta(h ) \ , ] \psi(h)}{4 \textrm{sin}\big(\frac { [ \ , \theta(h ) \ , ] } { 2}\big ) \textrm{sinh}\big(\frac{\psi(h)}{2}\big)}\ , \ , \ , \ , \textrm{exp } \bigg [ \frac{i}{2t}\bigg ( [ \ , \theta(h ) \ , ] ^2 -\psi^2(h ) \bigg ) + \frac{i t}{4 } \bigg ] .\ ] ] as should be easy to see , the above expression is obtained by picking one of the @xmath252 s in ( [ eq : so4kernel ] ) and analytically continuing it to @xmath431 . + again , we want to compute the feynman graph amplitude @xmath432 in the ( g , p ) variables . since in these variables the total amplitude is just a product of dual face amplitudes it is sufficient to calculate a generic amplitude @xmath433 $ ] of a dual face with n vertices . according to what should be the standard procedure by now , to get @xmath433 $ ] we multiply the kernel by @xmath434 and take the fourier transform at @xmath435 , where we have set @xmath436 as this simplifies the formulae on one hand , and gives the cleanest interpretation of the variable p on the other . @xmath386 = \frac{1}{(n-1 ) ! } \int_0^{\infty } dt \ , e^{i ( p^2 - \frac{1}{4 } ) t } \ , t^{n-1 } \ , k [ h \ , , \ , t].\ ] ] the killing form on @xmath437 has signature @xmath438 , thus in our normalization @xmath439 now , there are two ways to do the needed integral . either we use the hankel function ( [ eq : hankel ] ) formula and plow ahead with the algebra , paying attention to how we approach the cut when we take the square root . or we use similar arguments to what we used when we discussed the lorentzian case in 3 dimensions , using the fact that mathematically we are just performing a 1-d fourier transform , which allows us to rely on the relation between the fourier transforms of the function and its complex conjugate . either way , the dual face amplitude is given by @xmath225 = \mu ( [ \theta(h ) ] , \vec{p } , n ) \ , e^{i \ , \alpha ( [ \theta(h ) ] , \vec{p}^2 ) \ , \ , |\vec{p}^2| \ , \ , | [ \theta(h ) ] |},\ ] ] where @xmath440 ^ 2 $ ] is equal to @xmath441 ^ 2 - \psi^2(h)$ ] which is just the ( square of the ) length of the cartan subalgebra element parametrizing the conjugacy classes , or equivalently it is the length of a geodesic on the group manifold joining the point given by the element h to the identity . the @xmath442 is given by the following table , which is a carbon copy of the one in 3d ( [ eq : table ] ) @xmath443 the columns are labelled by the two possible cases of the @xmath444 . the rotation is when @xmath444 is positive , as it is easy to see that the h is then conjugate to a rotation ; while the boost is when @xmath444 is negative as this is when h is conjugate to a boost . + the rows , on the other hand , are labelled by the two possible cases of @xmath221 . timelike is when this vector has positive length and spacelike when this vector has negative length . + finally , the @xmath136 is , apart from sporadic signs and factors of i , just the the analytic continuation ( @xmath445 ) of the measure in the riemannian case ( [ eqnarray : rmeasure ] ) . for completenss we give the exact formula here + @xmath395 , \vec{p } , n ) & = & \frac { \mp i^{\pm n } 2^{n-10 } \alpha^{n-3 } ( [ \theta(h ) ] , \vec{p}^2)}{\pi^{\frac{5}{2 } } ( n-1 ) ! \gamma(n - \frac{5}{2 } ) } \ , \frac { [ \ , \theta_1(h ) \ , ] [ \ , \theta_2(h ) \ , ] } { \textrm{sin}\big(\frac { [ \ , \theta_1(h ) \ , ] } { 2}\big ) \textrm{sinh}\big(\frac { [ \ , \theta_2(h ) \ , ] } { 2}\big ) } \ , \big [ \theta(h ) \big ] ^{2(n-3 ) } \ , \times { } \nonumber \\ & & { } \times \bigg \ { \int_0^{\frac{\pi}{2}}ds \frac{\textrm{cos}^{n-\frac{5}{2}}(s ) \ , \ , e^ { \mp i ( n - \frac{7}{2 } ) \ , s } } { \textrm{sin}^{2n - 5}(s ) } \ , \textrm{exp } \big [ -2 \ , \textrm{cot}(s ) \ , \alpha ( [ \theta(h ) ] , \vec{p}^2 ) \ , | \vec{p } | \theta(h ) \big ] \bigg \}.\end{aligned}\ ] ] it is straightforward to compute the feynman graph amplitude @xmath446 now . it is @xmath447 where as before @xmath136 is a product of all the @xmath136 s coming from each dual face and @xmath397 is given by @xmath448 where @xmath442 is given in the table above . once again , we get a causal bf action in our partition function , i.e. we get a theory whose classical equations of motion are just like those of the standard bf , while there are profound differences at the quantum level . + as was the case in 3d we get exponential suppression of the wrong type of correlation between the variables . more precisely , had it not been for the fact that the variables p are in general not simple , we could have said that in the situation when the triangle corresponding to a holonomy given by a rotation is timelike or alternatively when it is spacelike when the holonomy is a boost , then this triangle contributes a phase to the partition function . on the other hand , if there is a mismatch between the @xmath36 and @xmath344 ( rotation - spacelike or boost - timelike ) , this triangle contributes an exponential suppression factor to the partition function . the behaviour of the model in the lorentzian case is unaffected by dimension . + also , since the measure factor is effectively the same as in the riemannian case , its phase depends on the deficit parameter @xmath36 and on the area @xmath282 in the same way as the bare bf action does , i.e. the phase of @xmath136 is a function of @xmath449 ( as well as the @xmath450 s characterizing the triangulation ) . this fact is interpreted as before to mean that there are quantum corrections arising from the factor @xmath136 of the general f(r ) type . + the semiclassical analysis is exactly the same as in the riemannian case and one sees that in the limit of large areas , we get inverse scalar curvature corrections to the bare bf action as before . finally , the amplitude is as divergent as before in the neighborhood of the @xmath355 . as in the previous section , we consider this fact to be a necessary condition for the existence of the continuum limit . + it is not difficult to write down the full feynman amplitude in the ( j , p ) variables . the relevant representations of @xmath451 are labelled by two parameters . a half integer j and a real positive parameter @xmath107 . the relevant casimir and normalizations is equal to @xmath452 . the analogue of ( [ eq : spinfoamjp ] ) is now [ eqnarray : lspinfoamjp ] z_=(_f*t^ * _ p d^3 _ f^ * ) _ f^ * t^ * _ v^ * t^ * \ { 15-j } , the poles in the expression ( [ eq : spinfoamjp ] ) are obviously located at @xmath453 so these particular values are the preferred semiclassical areas. + finally , as in 3d there exists a simple way of performing a ( partial , as explained ) wick rotation in this model , by analytically continuing ( some components of ) the p variables . this shows that also the existence of a good wick rotation in our model is independent of the dimension . we can use this wick rotation to perform the integrals over the p s in the amplitude in the ( j , p ) variables ( equation ( [ eq : spinfoamjp ] ) ) and obtain the 4d lorentzian analogue of ( [ eq : asymptotic ] ) , which is given by @xmath454^{n_{f^*}-1}.\ ] ] the relevant difference from the riemannian case is that the factor @xmath455^{n-1}$ ] , which controls the way the asymptotic behaves , gets replaced with @xmath456^{n-1}$ ] . this means that the most dominant contributions are those which satisfy @xmath457 . this does not of course force the j and the @xmath107 to each be small ( as was the case in 3d ) . however , it does force the minkowski length ( or more appropriately area ) to be small . note however , that if we restrict -by hand- to representations which are simple ( i.e. those for which either j or @xmath107 is zero ) , the expression above _ does _ force each of the parameters to be small , hinting again at a dynamical reduction to a dynamical triangulations - like sector . we have presented above a new gft model for a bf - type formulation of quantum simplicial gravity , in 4 dimensions , in the spin foam formalism . the spin foam amplitudes ( gft feynman amplitudes ) have the form , modulo a quantum measure , of the exponential of a classical 1st order action based on two types of variables : a set of bivectors associated to 2-simplices of the simplicial complex and a set of lorentz group elements representing parallel transports of a lorentz connection . the action has a regge calculus expression , augmented by higher order terms that can be interpreted as quantum corrections , that become negligible in the semi - classical limit . in this generalized simplicial gravity action , the areas as area of the corresponding triangle is , strictly speaking , not applicable until the ( discrete ) bivector is constrained to be a quadratic function of a ( discrete ) tetrad vector , as noted earlier . ] of the triangles are functions of the bivectors and the deficit angle associated , again , to each triangle obtained from the holonomy of the same lorentz connection , and thus a function of the corresponding group elements . the equations of motion following variation of the dominant contribution to the action , in the semiclassical ( large distance ) limit , restrict the holonomies to be flat , just as in ordinary bf theory , but the integration over the bivectors in the path integral does not treat on equal footing positive and negative orientations for the triangles , as bf theories do . the result is a complex amplitude , as said , and not a delta function over flat connections as in bf theory . + the above model seems to us to be a very promising starting point for the construction of a gft for quantum gravity in 4 dimensions , both riemannian and lorentzian , based on the idea of gravity as a constrained bf theory @xcite . this type of formulation has been central to the construction of all spin foam models in 4d , and lots is known already about the constraints that the bivectors have to satisfy in order to admit a true geometric interpretation , i.e. to be interpretable as coming from a discretized tetrad field , as it should be in a palatini - like formulation of gravity . + most of the model building on gfts and spin foam models from constrained bf theory have used a well - motivated but rather indirect procedure , we feel , based on the kinematical identification of bivectors with lie algebra generators , and thus translating them in constraints on the lorentz group representations and on the intertwiners appearing in the spin foam representation of the feynman amplitudes @xcite . this however , as we discussed , resulted in amplitudes with a less than straightforward relation with discrete gravity actions , and with a symmetrization over opposite orientations that is not what we should expect , we have argued , from a 3rd quantized gravity perspective . moreover , from the point of view of a path integral quantization of bf theory , the identification of bivectors with lie algebra generators acting on representation spaces remains a bit obscure , given that this holds at a quantum level while in a path integral one integrates over classical variables only ( although not solutions to the classical equations of motion ) and does not refer to quantum states if not at the boundary . a similar doubt concerns the more recent construction of spin foam models @xcite and related gfts @xcite , based on coherent states . here the geometric picture behind the chosen implementation of the constraints is much clearer , but again is justified at the quantum level in terms of the coherent states basis in each representation space . on the one hand this suggests a semiclassical validity only of the identification ; on the other hand , it results in quantum amplitudes with a less than straightforward relation with classical gravity actions , and with the same symmetrization over opposite orientations as in usual models . in the end , the procedure adopted may well result to be correct and our doubts unfounded or settled , but we feel that further work is needed to clarify the situation , and we see our new models as a useful framework in which to do so . + in fact , the new 4d bf - like model , thanks to the explicit presence of bivector variables and to their role in the discrete path integral clearly analogous to that played by the b field in continuum formulations , suggests that a much more straightforward way of implementing the simplicity constraints is possible . this is simply to constrain directly the integration over the bivector variables of our 4d model . the simplest way of doing so is to insert appropriate delta functions imposing the simplicity conditions on bivectors , and one has just to make sure that this is done consistently and as geometrically expected at the level of each feynman diagram . alternatively , and preferably , one should implement the constraints directly at the level of the new gft action , and for doing so one has to identify clearly which constraints are needed in each 4-simplex ( interaction term ) and which refer to the gluing of 4-simplices ( kinetic term ) , or whether one should instead constrain directly the field in both terms , as it is done for the other spin foam models @xcite . work on this is in progress @xcite . the expected result of this new way of implementing the simplicity constraints , starting from our 4d model , is to obtain a constrained gft whose feynman amplitude are given by a path integral for an action that could be directly interpreted as the discretization of the plebanski action for classical gravity , i.e. of the form : @xmath458 , where @xmath459 are lagrange multipliers imposing the constraints @xmath460 on the bivector variables @xmath461 associated to each dual face ( triangle ) @xmath33 incident to each dual vertex ( 4-simplex ) @xmath462 ( we have neglected here the quantum corrections to the 1st order regge action coming from the measure ) . + let us stress that a result of this type would be of interest , we believe , also from a purely simplicial gravity perspective . in fact , recent progress in spin foam models has motivated work on so - called area regge calculus@xcite , i.e. a formulation of classical and quantum simplicial gravity in which the fundamental geometric variables were the areas of the triangles of the simplicial complex as opposed to the edge lengths as in traditional regge calculus . in fact , spin foam models based on constrained bf theory ended up associating , as basic geometric variables , representation labels to triangles , with the interpretation of areas of the same . it was noted @xcite , however , that , while the correspondence areas - edges works ( almost ) fine for a single 4-simplex , constraints on the areas variables are needed in order to capture correctly the simplicial geometry as soon as more than one 4-simplex is considered . the identification of these area constraints have proven to be very difficult . our model would suggest that a better formulation of classical and quantum simplicial gravity , directly following the continuum picture of gravity as a constrained bf theory , would use our bivector variables to determine the areas of triangles , and that the needed constraints needed in order to encode the geometry are constraints on these bivector variables , and not directly on the areas . more precisely , the simplicity constraints will restrict the components of the bivectors other than their modulus ( area ) , and their re - phasing in terms of constraints on areas is , if possible , certainly not straightforward . in any case , the description of simplicial geometry implicit in our models in both 3 and 4 dimensions deserves to be studied in more detail . we leave this for future work . + the above 4d model , as well as its 3d version , in both lorentzian and riemannian versions , would also be the natural starting point for understanding in a more clear way the role that coherent states for the lorentz group play in spin foam models , and , in 4d , for understanding better the justification for the procedure used in the recently proposed spin foam models in order to impose the simplicity constraints . what we would expect is that the parameters labelling coherent states for the lorentz group in both 3d and 4d models will be directly related , if not identified , with the new ( bi-)vector variables we introduce _ in the approximation in which our free field classical equations of motions are satisfied _ , i.e. in the approximation in which one substitutes , in the amplitudes for our 3d and 4d models , the generators of the lorentz lie algebra for the ( bi-)vector variables , which indeed represents a dominant ( semiclassical ) contribution to our amplitudes , as it is clear from the structure of our propagators . however , further work is needed to confirm or refute this expectation . this work is currently in progress @xcite . we have presented a new class of gft models for the dynamics of quantum geometry , in any spacetime dimension and signature . the construction was based on the extension of the gft formalism to include additional variables with the interpretation of a discrete counterpart of the continuum b field in bf - like formulations of gravity . the feynman amplitudes for the new gfts , i.e. the corresponding spin foam models , have exactly the form of true simplicial gravity path integrals , with a clear - cut relation with discrete gravity actions , as opposed to other known models in which the connection arises only in some asymptotic limit . in 3d the new models are seen to provide a quantization of discrete quantum gravity in 1st order ( palatini ) form , in a local and discrete 3rd quantized framework in which topology is allowed to fluctuate . in 4d and higher , the new models have the form of a 3rd quantized framework for bf theory , but with an additional dependence of the amplitudes on the orientation of the simplicial complex , of the type on would expect in a path integral quantization of 1st order gravity . + the lorentzian models also present a very nice interplay between the two sets of discrete variables ( b field and connection ) which leads automatically to a suppression of all the configurations which do not match the simultaneous geometric interpretation of both of them . + the gft provides also a precise prescription for the quantum correction to the classical regge - like action ( in 1st order form ) that have to be included in the corresponding path integral , in absence of further restrictions to the models . these additional terms in the action become negligible in both the continuum limit ( large number of simplices of small size ) and in the semi - classical limit ( arbitrary number of simplices but large size of simplices , thus large associated action ) , leaving only the regge action to contribute to the path integral , as one would expect . in the general case , and as soon as one goes beyond these limiting regimes , the simplicial action provided by our gft models turns into a generic @xmath1 extended action for gravity . we feel that this has several interesting implications at the simplicial gravity level as well as from a more phenomenological perspective , that deserve to be studied in more detail . + the way the large - p limit affects the amplitudes of the new models sheds new light , we feel , on the usual large - j limit that brings usual spin foam models in closer relation with simplicial gravity path integrals , by allowing an approximation of the vertex amplitudes with the cosine of the regge action . indeed , our models suggest that this limit is a large distance limit which is equivalent , at the discrete level , to a large action and thus semi - classical limit ( because of the way the regge action , and its higher order corrections , depends on the hinge volumes ) . as such , it has two effects : it kills any quantum interference between opposite orientations for the hinges , in the usual models only , in which such opposite orientations are treated on equal footing ; it kills any short - distance effect such as @xmath463 corrections to the action , leaving only the regge term as the leading contribution to the quantum amplitudes , with next to leading order contributions being represented by inverse curvature terms @xmath464 which indeed modify the ir physics of the corresponding classical discrete gravity theory . + let us also mention that the explicit presence of a discrete analogue of the b field in our amplitudes allows a rather transparent definition of a wick rotation in simplicial quantum gravity our results , as we have discussed , support the view of gfts as local and discrete 3rd quantizations of gravity , providing a nice field theoretic description of the quantum dynamics of the fundamental building blocks of quantum space @xcite . also , the new models seem to implement nicely the ideas , discussed at length as a motivation for the work we have presented in this paper , as well as in @xcite , on the notion of causality and orientation dependence in quantum gravity , and to provide a definition of _ causal transition amplitudes _ for quantum gravity states , with all the expected properties . + at a more practical level , the new 4d models represent , in our opinion , a very convenient starting point for the construction of a gft ( and spin foam , and simplicial quantum gravity ) formulation of quantum gravity as a constrained bf theory , based on a more straightforward and geometrically clean procedure of implementing the so - called simplicity constraints that reduce bf theory to gravity , than in other spin foam formulations , as we have discussed above . also , they offer a new context in which to study the low energy physics of gfts and loop quantum gravity , e.g. graviton propagator calculations @xcite . + even more importantly , maybe , the new models , and possible modifications of the same , seem to provide the long sought for _ explicit _ unifying framework for spin foam / loop quantum gravity and simplicial quantum gravity approaches ( quantum regge calculus and dynamical triangulations ) . looking at these different approaches from the proposed common gft framework can offer , we hope , new possibilities for mutual enrichment and cross - fertilization between the various lines of research that are currently pursued as separate avenues toward a common goal , in particular regarding the outstanding issue of the continuum and semiclassical approximation of the discrete picture of quantum geometry they all seem to be based on @xcite . here we collect a few basic facts about groups @xcite used in the discussion in the main text . + a cartan subalgebra is a maximal abelian subalgebra of the lie algebra @xmath43 . the image of the cartan subalgebra under the exponential map is called a cartan subgroup . it is possible to show that all cartan subalgebras ( subgroups ) have the same dimension r , which is called the rank of the group . + in a compact group all its cartan subgroups are conjugate to each other and any element of the group is conjugate to an element of some fixed cartan subgroup . this means that we can parametrize the conjugacy classes of the group by the elements of the cartan subalgebra . + in a noncompact group the situation can be more complicated as on one hand not all cartan subgroups are conjugate . and the other , even if we do select a representative from each conjugacy class of cartan subgroups it is not true in general that each element of the group is conjugate to an element of one of the selected subgroups . fortunately , however , these exceptional elements form a set of measure zero in the haar measure in the cases that we are interested in . at the practical level , what follows from the above is that on a noncompact group there is no global spherical coordinate system @xcite , i.e. it is impossible to parametrize all the elements of the group by r parameters ( elements of a fixed cartan subalgebra ) like in the compact case . rather , there will be several different domains in general with all elements in each domain being conjugate to a cartan subgroup of a specific topology , with the elements which do nt lie in any domain forming a set of measure zero . the elements in each domain are parametrizable by a set of r parameters , which are just the elements of the corresponding cartan subalgebras . + intuitively , if one uses a cartan decomposition of @xmath43 , i.e. if one writes @xmath465 , with the killing form being negative definite on the elements of @xmath466 ( the compact part as the 1-parameter subgroup generated by any element in @xmath466 is just a circle @xmath467 ) and positive definite on the elements of @xmath468 ( the noncompact part , the 1-parameter subgroup generated by any element is diffeomorphic to the real line ) ; and if one obtains a cartan subalgebra by taking e.g. k generators from @xmath466 and p ones from @xmath468 ( with k + p = r of course ) . then the cartan subgroup corresponding to this subalgebra is not conjugate to the cartan subgroup corresponding to a subalgebra formed from a different relative proportion of compact and noncompact elements . if we call the elements generated by @xmath466 rotations and the ones generated by @xmath468 boosts , then the different cartan subgroups correspond to different relative number of rotations to boosts . the r parameters which parametrize the elements in each domain are split into two classes , periodic and aperiodic . the number of periodic parameters is equal to the number of rotations while the number of the aperiodic ones is equal to the number of boosts . j. ambjorn , b. durhuus , t. jonsson , three - dimensional simplicial quantum gravity and generalized matrix models , mod.phys.lett.*a6 * , 1133 - 1146 , ( 1991 ) ; n. sasakura , tensor model for gravity and orientability of manifold , mod.phys.lett.*a6 * , 2613 - 2624 , ( 1991 ) ; m. gross , tensor models and simplicial quantum gravity in @xmath469 2-d , nucl.phys.proc.suppl.*25a * , 144 - 149 , ( 1992 ) j.c . baez an introduction to spin foam models of quantum gravity and bf theory , lect.notes phys . * 543 * , ( 2000 ) , 25 - 94 , [ arxiv : gr - qc/9905087 ] ; d. oriti , space - time geometry from algebra : spin foam models for nonperturbative quantum gravity , rept . . phys . * 64 * , 1489 , ( 2001 ) , [ arxiv : gr - qc/0106091 ] ; a. perez , spin foam models for quantum gravity , class . . grav . * 20 * , r43 , ( 2003 ) , [ arxiv : gr - qc/0301113 ] d. oriti , _ a new perspective on the continuum in quantum gravity : group field theories as the microscopic description of the quantum spacetime fluid _ , in the proceedings of the workshop from quantum to emergent gravity : theory and experiments , trieste , pos , to appear s. b. giddings , a. strominger , baby universes , third quantization and the cosmological constant , nucl . b * 321 * , 481 , ( 1989 ) ; s. coleman , why there is nothing rather than something : a theory of the cosmological constant , nucl . b * 310 * , 643 , ( 1988 ) ; m. mcguigan , third quantization and the wheeler - dewitt equation , phys . rev . d * 38 * , 3031 , ( 1988 ) ; t. banks , prolegomena to a theory of bifurcating universes : a nonlocal solution to the cosmological constant problem or little lambda goes back to the future , nucl . b * 309 * , 493 , ( 1988 ) j. ambjorn , j. jurkiewicz , r. loll , reconstructing the universe , phys.rev.d * 72 * , 064014 , ( 2005 ) , [ arxiv : hep - th/0505154 ] ; j. ambjorn , j. jurkiewicz , r. loll the universe from scratch , contemp.phys . * 47 * , 103 - 117 , ( 2006 ) , [ arxiv : hep - th/0509010 ] c. rovelli , graviton propagator from background - independent quantum gravity , phys.rev.lett . * 97 * , 151301 , ( 2006 ) , [ arxiv : gr - qc/0508124 ] ; e. bianchi , l. modesto , c. rovelli , s. speziale , graviton propagator in loop quantum gravity , class . quant . grav . * 23 * , 6989 - 7028 , ( 2006 ) , [ arxiv : gr - qc/ 0604044 ] ; e. bianchi , l. modesto , the perturbative regge - calculus regime of loop quantum gravity , [ arxiv : 0709.2051 ] a. carlini , j. greensite , the mass shell of the universe , phys.rev.d * 55 * , 3514 - 3524 ( 1997 ) , [ arxiv : gr - qc/9610020 ] ; j. greensite , field theory as free fall , class.quant.grav . * 13 * , 1339 - 1352 ( 1996 ) , [ arxiv : gr - qc/9508033 ] l. freidel , d. louapre , asymptotics of 6j and 10j symbols , class.quant.grav . * 20 * , 1267 - 1294 , ( 2003 ) , [ arxiv : hep - th/0209134 ] ; j.w . barrett , c. m. steele , asymptotics of relativistic spin networks , class.quant.grav . * 20 * , 1341 ( 2003 ) , [ arxiv : gr - qc/0209023 ] m. caselle , a. dadda , l. magnea , regge calculus as a local theory of the poincare group , phys.lett.b * 232 * , 4 , 457 ( 1989 ) ; j. w. barrett , first order regge calculus , class.quant.grav . * 11 * , 2723 - 2730 ( 1994 ) , [ arxiv : hep - th/9404124 ] h.w . hamber , r.m . williams , higher derivative quantum gravity on a simplicial lattice , nucl.phys.b * 248 * , 392 , ( 1984 ) ; erratum : ibid.b * 260 * , 747 , ( 1985 ) ; h.w . hamber , r.m . williams , simplicial quantum gravity with higher derivative terms : formalism and numerical results in four - 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group field theories , a generalization of matrix models for 2d gravity , represent a 2nd quantization of both loop quantum gravity and simplicial quantum gravity . in this paper , we construct a new class of group field theory models , for any choice of spacetime dimension and signature , whose feynman amplitudes are given by path integrals for clearly identified discrete gravity actions , in 1st order variables .
in the 3-dimensional case , the corresponding discrete action is that of 1st order regge calculus for gravity ( generalized to include higher order corrections ) , while in higher dimensions , they correspond to a discrete bf theory ( again , generalized to higher order ) with an imposed orientation restriction on hinge volumes , similar to that characterizing discrete gravity .
the new models shed also light on the large distance or semi - classical approximation of spin foam models .
this new class of group field theories may represent a concrete unifying framework for loop quantum gravity and simplicial quantum gravity approaches .
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throughout this paper , we consider a unique server addressing two parallel queues numbered @xmath51 and @xmath52 , respectively . incoming jobs enter either queue and require random service times ; the server then processes jobs according to the so - called shortest queue first ( sqf ) policy . specifically , let @xmath6 ( resp . @xmath7 ) denote the workload in queue @xmath51 ( resp . queue @xmath52 ) at a given time , including the remaining amount of work of the job possibly in service ; the server then proceeds as follows : * queue @xmath51 ( resp . queue @xmath52 ) is served if @xmath8 , @xmath9 and @xmath10 ( resp . if @xmath11 and @xmath12 ) ; * if only one of the queues is empty , the non empty queue is served ; * if both queues are empty , the server remains idle until the next job arrival . in contrast to fixed priority disciplines where the server favors queues in some predefined order remaining unchanged in time ( e.g. , classical preemptive or non - preemptive head - of - line priority schemes ) , the sqf policy enables the server to dynamically serve queues according to their current state . the performance analysis of such a queueing discipline is motivated by the so - called sqf packet scheduling policy recently proposed to improve the quality of internet access on high speed communication links . as discussed in @xcite , sqf policy is designed to serve the shortest queue , i.e. , the queue with the least number of waiting packets ; in case of buffer overflow , packets are dropped from the longest queue . thanks to this simple policy , the scheduler consequently prioritizes constant bit rate flows associated with delay - sensitive applications such as voice and audio / video streaming with intrinsic rate constraints ; priority is thus implicitly given to smooth flows over data traffic associated with bulk transfers that sense network bandwidth by filling buffers and sending packets in bursts . in this paper , we consider the fluid version of the sqf discipline . instead of packets ( i.e. , individual jobs ) , we deal with the workload ( i.e. , the amount of fluid in each queue ) . since the fluid sqf policy considers the shortest queue in volume , that is , in terms of workload , its performance is quantitatively described by the variations of variables @xmath6 and @xmath7 . to simplify the analysis , we here suppose that the buffer capacity for both queues @xmath51 and @xmath52 is infinite . moreover , we assume that incoming jobs enter either queue according to a poisson process ; in view of the above application context , one can argue that such poisson arrivals can model traffic where sources have peak rates significantly higher than that of the output link ; such processes can hardly represent , however , the traffic variations of locally constant bit rate flows . this poisson assumption , although limited in this respect , is nevertheless envisaged here in view of its mathematical tractability and as a first step towards the consideration of more complicated arrival patterns . the above framework enables us to define the pair @xmath13 representing the workloads in the stationary regime in each queue as a continuous - state markov process in @xmath14 . in the following , we determine the probability distribution of the couple @xmath13 by studying its laplace transform . the problem can then essentially be formulated as follows . [ prob1 ] given the domain @xmath15 and analytic functions @xmath16 , @xmath17 , @xmath18 , @xmath19 , and @xmath20 in @xmath21 , determine two bivariate laplace transforms @xmath22 , @xmath23 and two univariate laplace transforms @xmath24 , @xmath25 , analytic in @xmath21 and such that equations @xmath26 for some analytic function @xmath27 , together hold for in @xmath28 . note that each condition @xmath29 or @xmath30 with @xmath31 brings the latter equations respectively to @xmath32 to the best knowledge of the authors , the mathematical analysis of the sqf policy has not been addressed in the queueing literature . some comparable queueing disciplines have nevertheless been studied : * the _ longest queue first _ ( lqf ) symmetric policy is considered in @xcite , where the author studies the stationary distribution of the number of waiting jobs @xmath33 , @xmath34 in each queue ; reducing the analysis to a boundary value problem on the unit circle , an integral formula is provided for the generating function of the pair @xmath35 ; * the _ join the shortest 2-server queue _ ( jsq ) , where an arriving customer joins the shortest queue if the number of waiting jobs in queues are unequal , is analyzed in @xcite . the bivariate generating function for the number of waiting jobs is then determined as a meromorphic function in the whole complex plane , whose associated poles and residues are calculated recursively . while the above quoted studies address the stationary distribution of the number of jobs in each queue , we here consider the real - valued process @xmath13 of workload components whose stationary analysis requires the definition of its infinitesimal generator on the relevant functional space . besides , the laplace transform of the distribution of @xmath13 proves to be meromorphic not on the entire complex plane , but on the plane cut along some algebraic singularities ( while the solution for jsq exhibits polar singularities only ) ; as a both quantitative and qualitative consequence , the decay rate of the stationary distribution at infinity for sqf may change according to the system load from that defined by the smallest polar singularity to that defined by the smallest algebraic singularity . the organization of the paper is as follows . in section [ ma ] , a markovian analysis provides the basic equations for the stationary distribution of the coupled queues ; the functional equations verified by the relevant laplace transforms are further derived in section [ ltd ] . in section [ expi ] , we specialize the discussion to the so - called symmetric exponential case where arrival rates are identical , and where service distribution are both exponential with identical mean ; the functional equations are then specified and shown to involve a key cubic equation . specifically , * problem [ prob1 ] * for the symmetric case is shown to reduce to the following . [ prob2 ] solve the functional equation @xmath36 for function @xmath1 , where given functions @xmath2 , @xmath3 and @xmath4 are related to one branch of a key cubic polynomial equation @xmath37 . for real @xmath38 , the solution @xmath39 is written in terms of a series involving all iterates @xmath40 for @xmath41 . the analytic extension of solution @xmath42 to some domain of the complex plane is further studied in section [ sqfqueue ] ; this enables us to derive the empty queue probability along with the tail behavior of the workload distribution in each queue for the symmetric case . the latter is then compared to that of the associated preemptive head of line ( hol ) policy . concluding remarks are finally presented in section [ cl ] . the proofs for basic functional equations as well as some technical results are deferred to the appendix for better readability . as described in the introduction , we assume that incoming jobs consecutively enter queue @xmath43 ( resp . queue @xmath44 ) according to a poisson process with mean arrival rate @xmath45 ( resp . @xmath46 ) . their respective service times are independent and identically distributed ( i.i.d . ) with probability distribution @xmath47 , @xmath48 ( resp . @xmath49 , @xmath50 ) and mean @xmath51 ( resp . mean @xmath52 ) . let @xmath53 ( resp . @xmath54 ) denote the mean load of queue @xmath43 ( resp . queue @xmath44 ) and @xmath55 denote the total load of the system . since the system is work conserving , its stability condition is @xmath56 and we assume it to hold in the rest of this paper . in this section , we first specify the evolution equations for the system and further derive its infinitesimal generator . first consider the total workload @xmath57 of the union of queues @xmath43 and @xmath44 . for any work - conserving service discipline ( such as sqf ) , the distribution of @xmath58 is independent of that discipline and equals that of the global single @xmath59 queue . the aggregate arrival process is poisson with rate @xmath60 and the i.i.d . service times have the averaged distribution @xmath61 with mean @xmath62 . the stationary probability for the server to be in idle state , in particular , equals @xmath63 let @xmath64 ( resp . @xmath65 ) be the number of job arrivals within time interval @xmath66 at queue @xmath51 ( resp . queue @xmath52 ) ; if @xmath67 ( resp . @xmath68 ) is the service time of the @xmath69-th job arriving at queue @xmath51 ( resp . @xmath52 ) , the total work brought within @xmath66 into queue @xmath51 ( resp . @xmath44 ) equals @xmath70 ( resp . @xmath71 . denoting by @xmath72 ( resp . @xmath73 ) the workload in queue @xmath43 ( resp . @xmath44 ) at time @xmath74 , define indicator functions @xmath75 and @xmath76 by @xmath77 respectively . with the above notation , the sqf policy governs workloads @xmath72 and @xmath73 according to the evolution equations @xmath78 for @xmath79 and some initial conditions @xmath80 , @xmath81 . this defines the pair @xmath82 , @xmath83 , as a markov process with state space @xmath84 ( see figure [ fig1 ] for sample paths of process @xmath82 ) . as a first result , integrating each equation ( [ path ] ) over interval @xmath85 $ ] , dividing each side by @xmath74 and letting @xmath86 implies @xmath87 and @xmath88 almost surely ( along with similar limits for integrals related to @xmath7 , @xmath89 and @xmath90 ) , and equating these limits readily provides identites @xmath91 for the stationary probability that the server treats queue @xmath51 and @xmath52 , respectively ; equivalently , the latter identities read @xmath92 in the symmetric case when arrival rates are equal and service times have identical distribution , i.e. , @xmath93 and @xmath94 , the above relations give @xmath95 the discrepancy in inequalities @xmath96 and @xmath97 defining the service policy ( when both queues are non empty ) does not favor queue @xmath43 with respect to queue @xmath44 , since event @xmath98 has probability 0 ; in fact , assuming for instance @xmath99 at some time @xmath74 , we have @xmath100 if a job arrival occurs with service time of amount exactly @xmath101 , which has probability 0 for any service time distribution . the distribution of process @xmath102 does not give a positive probability to the diagonal @xmath103 in state space @xmath104 . we now address the determination of the stationary distribution function @xmath105 , @xmath106 , of the bivariate workload process @xmath102 . in order to define the class of stationary distribution @xmath107 , we further assume that * distribution @xmath108 has a regular density @xmath109 ( resp . @xmath110 ) at any point @xmath111 such that @xmath112 ( resp . @xmath113 ) ; * distribution @xmath107 has a regular density @xmath114 ( resp . @xmath115 ) at any point @xmath116 ( resp . @xmath117 ) on the boundary @xmath118 ( resp . on the boundary @xmath119 ) . ( a real - valued function is here said to be regular if it is continuous and bounded over its definition domain . ) in the rest of this paper , assumptions * a.1 * -*a.2 * for the existence of regular densities will be confirmed by exhibiting their laplace transforms ; the uniqueness of the stationary distribution then _ a posteriori _ justifies such assumptions . an _ a priori _ justification for the existence of densities would otherwise imply the use of malliavin calculus @xcite on the poisson space . using , we have @xmath120 in the stationary regime ; following assumptions * a.1 * -*a.2 * above , we can then write @xmath121 for all @xmath122 , where @xmath123 ( resp . @xmath124 ) is the dirac distribution at point @xmath125 ( resp . at point @xmath126 ) . + let us now characterize the stationary distribution of process @xmath127 by means of its infinitesimal generator @xmath128 defined by @xmath129\ ] ] where the limit is uniform with respect to @xmath130 ( see @xcite or ( * ? ? ? * , p. 377 ) ) ; the symbol @xmath131 denotes any function for which the latter limit exists . in the following , we denote by @xmath132 the set of functions @xmath133 everywhere bounded , twice differentiable with bounded first and second derivatives in @xmath134 . further , introduce positive cones @xmath135 along with boundaries ( see figure [ fig1 ] ) @xmath136 we can then state the following . with the notation ( [ cones12])-([axes12 ] ) , the infinitesimal generator @xmath128 of process @xmath127 is given by @xmath137+\lambda_2 \mathbb{e}[\theta(\mathbf{u}+\mathcal{t}_2\mathbf{e}_2)-\theta(\mathbf{u } ) ] \label{gen1}\end{aligned}\ ] ] for all @xmath130 and any test function @xmath138 , where @xmath139 ( resp . @xmath140 ) denotes the generic service time of jobs arriving at queue @xmath51 ( resp . queue @xmath52 ) and with vectors @xmath141 , @xmath142 . [ generator ] using evolution equations ( [ path ] ) and given @xmath143 , expression ( [ gen1 ] ) is easily derived from uniform estimates ( with respect to @xmath130 ) for the distribution of the number of jumps of process @xmath144 on any interval @xmath145 $ ] ( all intervening poisson processes have rates lower than @xmath60 ) and for drift rates ( when non zero , the service rate is the constant @xmath146 ) . once generator @xmath128 is determined , the stationary distribution @xmath107 of @xmath147 is known ( see @xcite or @xcite ) to satisfy @xmath148 following the prerequisites of section [ ma ] , we now study integral equation . since the problem is linear in unknown distribution @xmath107 , it is tractable through laplace transform techniques . let @xmath15 and its closure @xmath149 . assumptions @xmath150 in section [ ig ] , for the existence of regular densities @xmath151 and @xmath152 with respective support @xmath153 and @xmath154 ( see equations ( [ cones12])-([axes12 ] ) ) enable us to define their laplace transforms @xmath22 , @xmath24 by @xmath155 for @xmath156 , where @xmath157 ; using the expectation operator , definitions ( [ ff ] ) equivalently read @xmath158 , \ ; \ ; g_1(s_1 ) = \e\big[e^{-s_1u_1}\ind_{\{0=u_2<u_1\}}\big].\ ] ] the laplace transforms @xmath23 and @xmath25 of regular densities @xmath159 and @xmath160 with respective support @xmath161 and @xmath162 ( see equations ( [ cones12])-([axes12 ] ) ) are similarly defined by @xmath163 for @xmath156 ; equivalent definitions can be similarly written in terms of the expectation operator . expression ( [ dphi ] ) for distribution @xmath164 and the above definitions then enable to define the laplace transform @xmath165 of the pair @xmath13 by @xmath166 for @xmath167 . finally , let @xmath168 ( resp . @xmath169 ) denote the laplace transform of service time @xmath139 ( resp . @xmath140 ) at queue @xmath43 ( resp . queue @xmath44 ) for @xmath170 ( resp . @xmath171 ) ; set in addition @xmath172 and @xmath173 * a ) * transforms @xmath22 , @xmath24 and @xmath23 , @xmath25 together satisfy @xmath174 for @xmath175 , where @xmath176 and @xmath177 . * b ) * transforms @xmath22 and @xmath25 ( resp . @xmath23 , @xmath24 ) satisfy @xmath178 for @xmath175 , with @xmath179 \\ - \lambda_2\mathbb{e } \left [ e^{-s_1u_1-s_2u_2}\ind_{\{0 \leq u_2 < u_1\}}e^{-s_2\mathcal{t}_2}\ind_{\{\mathcal{t}_2 > u_1-u_2\ } } \right ] . \label{h}\end{gathered}\ ] ] * c ) * constants @xmath180 and @xmath181 satisfy relation @xmath182 . [ resol ] * a ) * fix @xmath183 . the test function @xmath184 , @xmath185 , belongs to @xmath132 and has derivatives @xmath186 , @xmath187 . besides , we have @xmath188 hence @xmath189 = ( b_1(s_1)-1)\theta(\mathbf{u})$ ] , and similarly @xmath190 = ( b_2(s_2)-1)\theta(\mathbf{u})$ ] . applying proposition [ generator ] , formula ( [ gen1 ] ) for @xmath191 then yields @xmath192 with @xmath193 defined in ( [ ker ] ) . integrating that expression of @xmath191 over closed quarter plane @xmath134 with respect to distribution @xmath164 and using assumptions * a.1 * -*a.2 * , relation then gives @xmath194 with @xmath176 and @xmath177 ; using ( [ ldef ] ) finally provides ( [ fonct ] ) . * b ) * as detailed in appendix [ a2 ] , there exists a family of functions @xmath195 with @xmath196 , such that @xmath197 , @xmath198 _ and _ @xmath199 . for given @xmath200 , @xmath201 , the function @xmath202 defined by @xmath203 , @xmath130 , therefore belongs to @xmath132 and satisfies @xmath204 pointwise in @xmath134 with @xmath205 ( note that @xmath206 ) . apply then formula ( [ gen1 ] ) to regularized test function @xmath202 and integrate this expression over @xmath134 against distribution @xmath164 to define @xmath207 in view of ( [ stationary ] ) , we have @xmath208 and , provided that @xmath209 has a finite limit @xmath210 as @xmath211 , we must have @xmath212 . the detailed calculation of that limit @xmath210 ( depending on the pair @xmath213 ) is performed in appendix [ a2 ] and condition @xmath212 is shown to reduce to first equation ( [ fonctcomp ] ) . exchanging indices 1 and 2 provides second equation ( [ fonctcomp ] ) , after noting that @xmath214 changes into @xmath215 . * c ) * adding equations ( [ fonctcomp ] ) gives ( [ fonct ] ) if and only if @xmath182 holds . computing @xmath216 by letting @xmath217 in ( [ fonct ] ) readily gives @xmath218 with @xmath219 . identity ( [ pk ] ) is obviously pollaczek - khintchin formula for the transform @xmath220 of the total workload @xmath57 in the global @xmath59 queue , with i.i.d . service times having distribution @xmath221 defined by ( [ db ] ) . [ coroltech ] let @xmath27 be defined by ( [ h ] ) . transform @xmath24 satisfies @xmath222 for @xmath183 such that @xmath30 . similarly , transform @xmath25 satisfies @xmath223 for @xmath183 such that @xmath29 . [ c.2 ] function @xmath224 is finite for any given @xmath183 ; if @xmath30 , the product @xmath225 is therefore zero . as @xmath226 second equation ( [ fonctcomp ] ) then implies ( [ fonctg1 ] ) . relation ( [ fonctg2 ] ) is similarly derived . in this section , we first compare the sqf system with the hol queue , where one queue has head of line ( hol ) priority over the other ; such a comparison then enables us to extend the analyticity domain of laplace transforms @xmath22 , @xmath23 and @xmath24 , @xmath25 . + let @xmath227 , @xmath228 , denote the workload in queue @xmath229 when the other queue has hol priority ; similarly , let @xmath230 denote the workload in queue @xmath229 when this queue has hol priority over the other . finally , given two real random variables @xmath231 and @xmath232 , @xmath232 is said to dominate @xmath231 in the strong order sense ( for short , @xmath233 ) if and only if @xmath234 for any positive non - decreasing measurable function @xmath235 . workload @xmath236 verifies @xmath237 for all @xmath83 . [ domstoch ] we clearly have @xmath238 almost surely for all @xmath83 , where @xmath75 is defined by . equation ( [ path ] ) consequently entails that @xmath239 pathwise , which implies the strong stochastic domination . similarly , we have @xmath240 almost surely for all @xmath83 and ( [ path ] ) entails @xmath241 pathwise , hence the strong stochastic domination . assume that random variable @xmath242 has an analytic laplace transform @xmath243 in the domain @xmath244 for some real @xmath245 . laplace transform @xmath22 can be analytically extended to domain @xmath246 and transform @xmath25 can be analytically extended to @xmath247 . similarly , transform @xmath23 can be analytically extended to @xmath248 and @xmath24 can be analytically extended to @xmath249 . [ extensions ] assume first that @xmath250 and @xmath251 are real with @xmath252 ; given @xmath253 , we have @xmath254 ; using the domination property @xmath255 of proposition [ domstoch ] and the previous inequality , definition ( [ ff ] ) of @xmath22 on @xmath149 entails @xmath158 \leq \mathbb{e}\big[e^{-(s_1+s_2)\overline{u}_2}\big];\ ] ] we then deduce that @xmath22 can be analytically continued to any point @xmath213 verifying @xmath256 and @xmath257 . assuming now that @xmath258 and @xmath259 , domination property @xmath260 yields @xmath261 and definition ( [ ff ] ) of @xmath22 on @xmath149 entails in turn @xmath158 \leq \mathbb{e}\big[e^{- s_2 \overline{u}_2}\big];\ ] ] @xmath22 can therefore be analytically continued to any point @xmath213 verifying @xmath262 and @xmath263 . we conclude that @xmath22 can be analytically continued to domain @xmath264 , as claimed . writing definition ( [ ffb ] ) of @xmath25 as @xmath265 $ ] for @xmath171 , the same type of arguments as above enables us to analytically continue function @xmath25 to any point verifying @xmath266 . domains @xmath264 and @xmath267 are illustrated in figure [ fig3 ] ( assuming @xmath268 for instance ) . following proposition [ resol ] and corollary [ coroltech ] , the determination of laplace transforms @xmath22 , @xmath23 , @xmath24 and @xmath25 critically depends on both the determination of auxiliary bivariate function @xmath27 generally defined in ( [ h ] ) and the solutions to equations @xmath269 and @xmath270 . the latter , however , may be very intricate to compute for general service time distributions . to make the resolution more tractable , we will now introduce some specific assumptions . first , service times are assumed to be exponentially distributed ; this readily provides a more explicit expression for function @xmath27 . in the case of exponentially distributed service times , we have @xmath271 where @xmath272 are analytically defined for @xmath273 . [ hexp ] the proof of proposition [ hexp ] is deferred to appendix [ a3 ] . expression ( [ hbis ] ) consequently reduces the determination of function @xmath27 to that of two univariate functions @xmath274 and @xmath275 . in the rest of this paper , we further assume that the poisson arrival rates and service time distributions in each queue are equal , the so - called `` symmetric ( exponential ) case '' . because of its technical complexity , the asymmetric case will be treated in a forthcoming paper @xcite . as previously motivated , we assume from now on that * poisson arrival rates are equal , namely @xmath93 ; * service times in both queues are exponentially distributed with identical parameter @xmath276 , i.e. , @xmath277 ; the laplace transform of the service time distribution is then @xmath278 . by the latter symmetry assumption , queues @xmath43 and @xmath44 are now interchangeable in terms of probability distribution . definition ( [ ff ] ) of @xmath22 or @xmath23 then entails that @xmath279 for @xmath31 and we denote by @xmath280 the latter quantity ; using similar arguments , we have @xmath281 . by proposition [ resol].c , we further have @xmath282 and function @xmath283 introduced in ( [ j1j2 ] ) is simply given by @xmath284 relations then specialize to the unique equation @xmath285 where general expression for @xmath27 now simply reduces to @xmath286 where @xmath287 ( note the symmetry between transforms @xmath22 and @xmath23 mentioned above implies that @xmath288 ) . once function @xmath27 is expressed by ( [ hsym ] ) in terms of auxiliary function @xmath1 , functional equation ( [ kernelsym ] ) gives @xmath289 in terms of both @xmath290 and @xmath1 . as univariate transform @xmath290 will be later shown to depend on function @xmath1 only , our remaining task is therefore to derive the latter function . let us first assert some extension properties for analytic functions of interest . recall from ( * ? ? ? * 3.3 ) that the laplace transform of the workload @xmath291 in queue @xmath43 when queue @xmath44 has hol priority is given by @xmath292 = \frac{2(1-\rho)s\xi^+(s)}{\lambda(1-b(s))(s-\xi^+(s))}\ ] ] for @xmath293 , where @xmath294 is the unique root of equation @xmath295 which is positive for @xmath296 . specializing definition ( [ ker ] ) for @xmath297 to the present symmetric case , equation @xmath298 readily reduces to @xmath299 its roots @xmath294 and @xmath300 are therefore given by @xmath301 where discriminant @xmath302 is positive for @xmath303\zeta^-,\zeta^+[$ ] and non positive for @xmath304 $ ] , with @xmath305 functions @xmath306 are defined for real @xmath307 $ ] . with the convention @xmath308 , we can define analytic or meromorphic extensions of these functions in the complex plane as follows . function @xmath309 ( resp . @xmath310 ) can be analytically ( resp . meromorphically ) extended to the cut plane @xmath311 $ ] . [ extendxi ] function @xmath312 is well - defined for @xmath313\zeta^-,\zeta^+[$ ] , whereas function @xmath314 is well - defined for @xmath315\zeta^-,\zeta^+[$ ] and @xmath316 , with @xmath317 . it is easily checked that for @xmath318 belonging to the vertical line @xmath319 , we have @xmath320 and @xmath321 ( note this vertical line and the real line are the only subsets of the complex plane on which @xmath320 ) . the schwarz s reflection principle applied to function @xmath322 with respect to the vertical line @xmath323 then ensures that the function @xmath324 defined by @xmath325 for @xmath326 and @xmath327 for @xmath328 is globally analytic on the cut plane @xmath329 $ ] . let us then define functions @xmath330 and @xmath331 by @xmath332 respectively . by construction , function @xmath330 is a meromorphic extension of @xmath310 in @xmath333 $ ] with a pole at point @xmath334 , while function @xmath331 is an analytic extension of @xmath309 in @xmath335 $ ] . + for notation simplicity , we will still denote by @xmath310 and @xmath309 their respective analytic continuation @xmath330 and @xmath331 defined above . consider now equation @xmath336 , whose unique non - zero solution is @xmath337 . as @xmath338 , it is easily verified that solution @xmath337 is associated with branch @xmath310 if @xmath339 and with branch @xmath309 if @xmath340 . define then @xmath341 ( note that @xmath342 for all @xmath343 $ ] , as easily verified from the defining expression of polynomial @xmath344 in ( [ xi+- ] ) ) . with the above notation , laplace transform @xmath290 can be analytically extended to the half - plane @xmath345 ; function @xmath346 can be analytically extended to @xmath347 . [ extendgf0 ] by ( [ uhol ] ) and lemma [ extendxi ] , transform @xmath348 is analytic for @xmath349 . this transform may have a pole only at any point @xmath318 such that @xmath350 . by the above discussion , we actually have a pole at @xmath351 when @xmath352 ; it is not a pole when @xmath353 but the algebraic singularity at point @xmath354 instead occurs . applying then corollary [ extensions ] with @xmath355 , the extended analyticity domains for @xmath290 and @xmath346 follow . following definition ( [ msym ] ) and lemma [ extendgf0 ] , function @xmath1 is consequently analytic on the half - plane @xmath356 . as detailed in section [ sqfqueue ] , the final determination of function @xmath1 relies on the algebraic and analytic properties for the branches of a cubic polynomial equation . * a ) * for given @xmath38 and @xmath357 , relations @xmath358 can be inverted in variable @xmath318 as @xmath359 respectively , where @xmath360 and @xmath361 are the two non positive roots of cubic equation @xmath37 in variable @xmath362 , with @xmath363 for @xmath38 , @xmath300 and @xmath294 are given by @xmath364 and @xmath365 . * b ) * for @xmath366 , cubic polynomial @xmath367 has three distinct real roots @xmath360 , @xmath361 and @xmath368 such that @xmath369 and @xmath370 . [ roots - r ] * a ) * eliminating @xmath300 between first relation ( [ z - s ] ) and polynomial equation ( [ eqxi ] ) satisfied by @xmath300 , we can write @xmath371 where @xmath372 , cubic polynomial @xmath367 being defined as in ( [ cubiceq ] ) . similarly , eliminating @xmath294 between second relation ( [ z - s ] ) and equation ( [ eqxi ] ) enables us to write @xmath373 where @xmath374 with identical polynomial @xmath367 . we readily deduce , in particular , that @xmath375 , and similarly @xmath376 . * b ) * for @xmath38 , we have @xmath377 and @xmath378 since @xmath379 by the stability condition . further accounting for its values at infinity , we deduce that cubic polynomial @xmath367 has three real roots for @xmath366 ; denoting them by @xmath360 , @xmath361 and @xmath368 , the latter discussion implies the claimed inequalities . we finally verify that roots @xmath360 and @xmath361 previously characterised either in * a ) * or * b ) * actually coincide . in fact , let @xmath380 so that @xmath381 ; given the variations of the function @xmath314 for @xmath382 , @xmath318 has to be sufficiently large for @xmath381 to be positive ; this implies that we necessarily have @xmath383 where @xmath360 is the smallest root of polynomial @xmath367 . we can similarly prove that if @xmath384 , then @xmath385 where @xmath386 is the second smallest root of @xmath367 . as solutions to a polynomial equation , algebraic functions @xmath387 , @xmath388 and @xmath389 can be analytically defined in @xmath390 cut along some slits . specifically , writing @xmath391 as @xmath392 with coefficients @xmath393 , @xmath394 and @xmath395 defined by ( [ cubiceq ] ) and introducing @xmath396 any solution @xmath397 to @xmath37 can be expressed by cardano s formula @xcite as @xmath398{\frac{1}{2 } \left ( - \widetilde{q}+ \sqrt{- \frac{\delta}{27 } } \right ) } + j^n \sqrt[3]{\frac{1}{2 } \left ( - \widetilde{q } - \sqrt{- \frac{\delta}{27 } } \right ) } , \ ] ] where @xmath399 , the pair @xmath400 can take either value @xmath125 , @xmath401 or @xmath402 , and with discriminant @xmath403 defined by @xmath404 . some algebra shows that discriminant @xmath405 factorizes as @xmath406 with @xmath407 the respective analyticity domains of functions @xmath408 , @xmath409 and @xmath410 are related to the roots of discriminant @xmath405 , these roots defining the so - called ramification points for such algebraic functions . * a ) * discriminant @xmath405 has four distinct roots , namely two real roots @xmath411-\mu,0[$ ] and @xmath412 and two complex conjugate roots @xmath413 and @xmath414 . + * b ) * algebraic functions @xmath408 , @xmath409 and @xmath410 are analytic on the cut plane @xmath415 $ ] , @xmath416\cup[\eta_3,\eta_4])$ ] and @xmath417 $ ] , respectively . [ discriminant ] * a ) * the point @xmath418 is clearly a root of @xmath419 and it is simple since @xmath420 in view of expression ( [ smalldelta ] ) . moreover , as the coefficient of the leading term of the cubic polynomial @xmath421 is positive , as @xmath422 and @xmath423 , discriminant @xmath405 has at least another negative real root @xmath424 between @xmath334 and 0 . besides , the discriminant of @xmath421 is easily calculated as @xmath425 with @xmath426 ; as @xmath427 for @xmath428 , we have @xmath429 . it then follows from ( * ? ? ? * theorem 1.3.1 ) that cubic polynomial @xmath421 with real coefficients has only one real root , namely @xmath424 , the two others @xmath413 and @xmath414 being complex conjugates . * b ) * by considering the analytic continuation of function @xmath430 such that @xmath431 in @xmath416\cup[\eta_3,\eta_4])$ ] , formulas ( [ cardan ] ) enable us to analytically continue function @xmath408 to the cut plane @xmath415 $ ] , function @xmath409 to the cut plane @xmath416\cup[\eta_3,\eta_4])$ ] and function @xmath410 to the cut plane @xmath417 $ ] , respectively . the graphs of functions @xmath432 and @xmath433 are illustrated in fig . [ courbexi ] on interval @xmath434 $ ] . function @xmath435 is increasing while function @xmath436 reaches its minimum at some point @xmath437 ; @xmath436 is decreasing on interval @xmath438\zeta^+,s^*[$ ] and increasing on interval @xmath438s^*,0[$ ] . recall from proposition [ roots - r ] that @xmath383 entails @xmath439 ; conversely , we have @xmath439 for @xmath440 . function @xmath387 is thus defined and regular for @xmath441\eta_1,+\infty[$ ] where @xmath442 . using similar arguments , @xmath388 is shown to be regular for @xmath443 . on the basis of the preliminary results obtained in section [ expi ] , we are now ready to provide a final solution for auxiliary function @xmath1 ( section [ rse ] ) and determine an extended analyticity domain ( section [ ss ] ) , from which all relevant probabilistic properties for the symmetric queue can be derived ( sections [ eqp ] and [ lqa ] ) . we first provide a series expansion for laplace transform @xmath290 on some real interval of its definition domain . the proposition below states the core functional equation verified by function @xmath1 . function @xmath1 defined by ( [ msym ] ) verifies the functional equation @xmath444 for @xmath38 , with @xmath445 where @xmath371 is the unique solution to equation @xmath446 . [ c1 ] applying equation successively to points @xmath447 and @xmath448 with identical ordinate @xmath318 , we obtain @xmath449 after using expression ( [ jj ] ) for @xmath450 and formula ( [ hsym ] ) for @xmath214 ; equations ( [ msymequ ] ) hold for sufficiently large @xmath318 so that @xmath451 is positive . using the fact that @xmath371 . equating the common value of @xmath452 from ( [ msymequ ] ) and using the fact that @xmath453 gives functional equation ( [ fonctmsym ] ) . by proposition [ roots - r].a , @xmath454 depends on the branch @xmath360 only . as @xmath455 in view of defining equation ( [ eqxi ] ) , definition ( [ defqlh ] ) for @xmath456 further gives @xmath457 , \label{hz}\ ] ] and a similar rational expression is derived from ( [ defqlh ] ) for @xmath458 in terms of @xmath360 . as a consequence , given functions @xmath2 , @xmath3 , and @xmath4 depend only on the branch @xmath408 of cubic equation @xmath37 . note also that by the notation introduced in inversion relations ( [ z - s])-([z - sinvert ] ) , @xmath456 just coincides with @xmath459 ; the mapping @xmath460 is now introduced in view of its iterated composition , as will be shown in the central result below . the laplace transform @xmath290 can be expressed as @xmath461 \label{gsym}\ ] ] for sufficiently large real @xmath318 so that @xmath462 and where @xmath463 is given by the series expansion @xmath464 with functions @xmath2 , @xmath3 and @xmath4 defined by , and @xmath465 denoting the @xmath466-th iterate of function @xmath4 . [ resolsym ] iterating functional equation ( [ fonctmsym ] ) for @xmath380 yields @xmath467 ( the product being equal to 1 for @xmath468 ) , with remainder @xmath469 to show that @xmath470 as @xmath471 , let us fix some @xmath380 . we first prove that the sequence @xmath472 , @xmath473 , is strictly increasing and tends to @xmath474 when @xmath475 . in fact , as @xmath476 for @xmath380 by proposition [ roots - r].b , we deduce from expression ( [ hz ] ) that @xmath477 for @xmath38 and the sequence @xmath478 , @xmath473 , is thus strictly increasing . moreover , if that sequence were upper bounded , it would tend to a finite limit @xmath479 such that @xmath480 and the number @xmath481 is positive ; but using expression ( [ hz ] ) for @xmath456 , equality @xmath480 reduces to @xmath482,\ ] ] or equivalently @xmath483 and the latter would define a simultaneously positive and negative quantity , a contradiction . we thus conclude that @xmath484 when @xmath475 . besides , we derive from definition ( [ defqlh ] ) for @xmath2 that @xmath485 , where @xmath486 by definition ( [ msym ] ) of function @xmath1 , the sequence @xmath487 , @xmath473 , is bounded since both @xmath290 and @xmath346 vanish at infinity as laplace transforms of regular densities . it follows that remainder @xmath488 is @xmath489 = o(r^k)$ ] and therefore tends to 0 as @xmath471 . the finite sum in ( [ quotientbis ] ) thus converges as @xmath471 . formula ( [ gsym ] ) for @xmath452 eventually follows from the latter expansion inserted into second equation ( [ msymequ ] ) . we now specify the smallest singularity of laplace transform @xmath290 ; to this end , we first deal with the analyticity domain of auxiliary function @xmath1 . recall by definition ( [ msym ] ) that @xmath1 is known to be analytic at least in the half - plane @xmath356 , where @xmath490 is defined by . function @xmath1 can be analytically continued to the half - plane @xmath491 ( with @xmath492 ) defined by * @xmath493 in case @xmath494 , where we set @xmath495 ; * @xmath496 in case @xmath340 , where @xmath497 is the largest real root of discriminant @xmath405 . [ extendm ] the proof of proposition [ extendm ] is detailed in appendix [ a4 ] . we now turn to transform @xmath290 and determine its singularities with smallest module . recall by corollary [ extensions ] that @xmath290 has no singularity in @xmath498 . [ domaing ] the singularity with smallest module of transform @xmath290 is * for @xmath494 , a simple pole at @xmath499 with leading term @xmath500 with @xmath501 ; * for @xmath502 , an algebraic singularity at @xmath503 with leading term @xmath504 at first order in @xmath505 , where factor @xmath506 is given by @xmath507\ ] ] with constants @xmath508 , @xmath509 and where @xmath510 , @xmath511 are given in ( [ zeta1 + - ] ) . [ gsing ] consider again the two following cases : + * a ) * if @xmath494 , write the 1st equation ( [ msymequ ] ) as @xmath512 ; \label{gsymbis}\ ] ] as @xmath513 , we have @xmath514 while @xmath515 . proposition [ extendm ] then ensures that @xmath516 is analytic at @xmath517 since @xmath518 for @xmath494 . as @xmath452 has no singularity for @xmath519 , we conclude from expression ( [ gsymbis ] ) that @xmath290 has a simple pole at @xmath520 with residue @xmath521\ ] ] where @xmath495 . differentiating formula for @xmath294 at @xmath520 , we further calculate @xmath522 ; residue @xmath523 in leading term ( [ resgsym1 ] ) then follows ; + * b ) * if @xmath502 , let @xmath513 so that @xmath524 and @xmath525 where @xmath526 . proposition [ extendm ] then ensures that @xmath1 is analytic at @xmath527 since @xmath528 . we conclude from expression ( [ gsymbis ] ) and the latter discussion that @xmath529 is not a singularity of @xmath290 . + by definition of @xmath294 , where @xmath344 is factorized as @xmath530 with @xmath531 , we obtain @xmath532 where @xmath508 and with constant @xmath533 . by expression ( [ gsymbis ] ) for @xmath452 , we then obtain @xmath534 \nonumber \\ & = g(\zeta^+ ) + r^+(s-\zeta^+)^{1/2 } + ... \nonumber\end{aligned}\ ] ] since @xmath535 with @xmath536 defined in ( [ resgsym2 ] ) . expansion ( [ resgsym2 ] ) then follows with associated factor @xmath506 ; we conclude that the singularity with smallest module of @xmath290 is @xmath510 , an algebraic singularity with order 1 . the results obtained in the previous section enable us to give a closed - form expression for the empty queue probability in terms of auxiliary function @xmath1 only . probability @xmath537 is given by @xmath538 with @xmath1 given by series expansion ( [ mseries ] ) . [ g0sym ] apply relation ( [ gsymbis ] ) for @xmath452 with @xmath539 ; as @xmath540 , we then derive that @xmath541 hence @xmath542 ; \label{g(0)bis}\ ] ] differentiating formula for @xmath294 at @xmath539 gives @xmath543 so that the first term inside brackets in ( [ g(0)bis ] ) reduces to @xmath544 . now , applying ( [ fonctmsym ] ) to value @xmath539 ( with corresponding pair @xmath545 and @xmath546 ) shows that the right - hand side of ( [ g(0)bis ] ) also equals @xmath547 , as claimed . by ( [ p0 ] ) and ( [ g(0 ) ] ) , we derive the probability @xmath548 that either queue @xmath43 or @xmath44 is empty . we depict in figure [ figg0 ] the variations of @xmath549 in terms of load @xmath550 when fixing @xmath551 ( for comparison , the black dashed line represents the empty queue probability @xmath552 for the unique queue aggregating all jobs from either class @xmath43 or @xmath44 ) . the numerical results show that @xmath549 decreases to a positive limit , approximately @xmath553 , when @xmath554 tends to 1 ; this can be interpreted by saying that , while the global system is unstable and sees excursions of either variable @xmath6 or @xmath7 to large values , one of the queues remains less than the other for a large period of time and has therefore a positive probability to be emptied by the server . furthermore , the red dashed line depicts the empty queue probability @xmath555 if the server were to apply a preemptive hol policy with highest priority given to queue @xmath43 ; following lower bound ( [ stochlowup ] ) , we have @xmath556 . we further notice that for @xmath557 , the positive limit of @xmath549 derived above for sqf is close enough to the maximal limit @xmath558 of @xmath559 . the above observations consequently show that the sqf policy compares favorably to the optimal hol policy by guaranteeing a non vanishing empty queue probability for each traffic class at high load . we finally derive asymptotics for the distribution of workload @xmath6 or @xmath7 in either queue , i.e. , the estimates of tail probabilities @xmath560 for large queue content @xmath561 . we shall invoke the following tauberian theorem relating the singularities of a laplace transform to the asymptotic behavior of its inverse ( * ? ? ? * theorem 25.2 , p.237 ) . let @xmath165 be a laplace transform and @xmath562 be its singularity with smallest module , with @xmath563 as @xmath564 for @xmath565 and @xmath566 ( replace @xmath165 by @xmath567 if @xmath568 is finite ) . the laplace inverse @xmath235 of @xmath165 is then estimated by @xmath569 for @xmath570 , where @xmath571 denotes euler s @xmath571 function . [ tauberian ] note that the fact that @xmath568 is finite or not does not change the estimate of inverse @xmath235 at infinity . before using that theorem for the tail behavior of either @xmath6 or @xmath7 , we first state some simple bounds for their distribution tail . the global workload @xmath57 is identical to that in an @xmath572 queue with arrival rate @xmath573 and service rate @xmath276 . the complementary distribution function of @xmath58 is therefore given by @xmath574 for all @xmath575 , with @xmath495 ; the distribution tail of workload @xmath6 or @xmath7 therefore decreases at least exponentially fast at infinity . following upper bound ( [ stochlowup ] ) relating @xmath6 to variable @xmath291 corresponding to a hol service policy with highest priority given to queue @xmath44 , we further have @xmath576 for all @xmath577 . the laplace transform of @xmath291 is given by equation and is meromorphic in the cut plane @xmath311 $ ] , with a possible pole at @xmath578 . specifically , the application of theorem [ tauberian ] shows that the tail behavior of @xmath291 is given by @xmath579 for large @xmath561 . the tail behavior of @xmath291 , and therefore @xmath6 , may therefore be either exponential or subexponential according to system parameters . we precisely have the following result . the workload in queue @xmath43 is such that @xmath580 for large @xmath561 , with constants @xmath495 and @xmath581 where @xmath582 is given by ( [ zeta1 + - ] ) and @xmath506 by ( [ resgsym2 ] ) . [ asymptoticssym ] applying equation ( [ ldef ] ) to @xmath583 gives the laplace transform of @xmath6 as @xmath584 with @xmath585 by using and ( [ hsym ] ) . we now follow the results of theorem [ gsing ] on the smallest singularity of @xmath290 in order to derive the smallest singularity of transform @xmath586 expressed above . @xmath587 assume first @xmath494 . by proposition [ extendm ] , function @xmath588 is analytic for @xmath589 . it then follows from ( [ laplu1 ] ) that the singularity with smallest module of @xmath590 is at @xmath591 with leading term @xmath592 since @xmath593 and the root @xmath594 of @xmath595 is a removable singularity since @xmath596 has to be analytic for @xmath597 . by estimate ( [ resgsym1 ] ) for @xmath598 near @xmath591 , ( [ leadterm0 ] ) yields @xmath599 as @xmath600 ; smallest singularity @xmath591 is thus a simple pole for laplace transform @xmath586 . applying then theorem [ tauberian ] with @xmath601 and @xmath602 , we derive that @xmath603 for large @xmath561 with prefactor @xmath604 as claimed . @xmath587 assume now that @xmath502 . by formula ( [ laplu1 ] ) and proposition [ extendm ] , function @xmath588 is analytic for @xmath605 . it then follows from ( [ laplu1 ] ) that the singularity with smallest module of @xmath590 is at @xmath606 with leading term again specified by ( [ leadterm0 ] ) so that @xmath607 \label{smallestpolegter}\ ] ] near @xmath606 . by estimate ( [ resgsym2 ] ) , ( [ smallestpolegter ] ) yields @xmath608 as @xmath609 where @xmath610 smallest singularity @xmath606 is thus an algebraic singularity for laplace transform @xmath586 , with order @xmath611 . applying theorem [ tauberian ] with @xmath612 , @xmath613 and @xmath614 , we derive that @xmath615 for large @xmath561 with prefactor @xmath616 . @xmath587 finally , assume that @xmath617 ; the polar singularity @xmath618 and the algebraic singularity @xmath619 for @xmath290 coincide in this case . recall from proposition [ extendm].b that function @xmath620 is analytic for @xmath621 whenever @xmath340 ; @xmath424 is the only real zero @xmath622 of discriminant @xmath406 and expression ( [ smalldelta ] ) of @xmath421 gives @xmath623 , hence @xmath624 ; @xmath516 is therefore analytic at @xmath625 . near @xmath626 , formula ( [ xi+- ] ) easily gives @xmath627 expression ( [ gsymbis ] ) for @xmath598 and the discussion above then imply that @xmath628 in the neighborhood of @xmath626 . the leading term ( [ leadterm0 ] ) for @xmath590 is consequently given by @xmath629 smallest singularity @xmath626 is thus an algebraic singularity for laplace transform @xmath586 , with order @xmath630 . applying then theorem [ tauberian ] with @xmath631 , @xmath632 and @xmath633 , we derive that @xmath634 for large @xmath561 with prefactor @xmath635 . for any given load @xmath6360,1[$ ] , theorem [ asymptoticssym ] consequently provides the same exponential trend as that of upper bound ( [ asympthol ] ) for hol ; as a matter of fact , a large value of @xmath6 entails that queue @xmath43 behaves as if queue @xmath44 , with smaller workload , had a hol priority . the stationary analysis of two coupled queues addressed by a unique server running the sqf discipline has been generally considered for poisson arrival processes and general service time distributions ; required functional equations for the derivation of the stationary distribution for the coupled workload process have been derived . specializing the resolution of such equations to both exponentially distributed service times and the so - called `` symmetric case '' , all quantities of interest have been obtained by solving a single functional equation . the solution @xmath1 for that equation has been given , in particular , as a series expansion involving all consecutive iterates of an algebraic function @xmath4 related to a branch of some cubic equation @xmath37 . it must be noted that the curve represented by that cubic equation in the @xmath637 plane is singular ; in fact , whereas `` most '' cubic curves are regular ( i.e. , without multiple points ) , it can be easily checked that cubic @xmath638 has a double point at infinity . in equivalent geometric terms , cubic @xmath638 can be identified with a sphere when seen as a surface in @xmath639 , whereas most cubic curves are identified with a torus . this fact can be considered as an essential underlying feature characterizing the complexity of the present problem ; such geometric statements will be enlightened for solving the general asymmetric case in @xcite . an extended analyticity domain for solution @xmath1 has been determined as the half - plane @xmath640 , thus enabling to determine the singularity of laplace transform @xmath290 with smallest module . it could be also of interest to compare such extended domain @xmath640 to the maximal convergence domain of series expansion ( [ mseries ] ) ( recall the convergence of that series has been stated in theorem [ resolsym ] for real @xmath38 only ) ; in fact , the analyticity domain @xmath640 may not coincide with the validity domain for such a series representation . the discrete holomorphic dynamical system defined by the iterates @xmath641 , @xmath642 , definitely plays a central role for such a comparison . as an alternative approach to that of section [ sqfqueue ] , function @xmath1 may also be derived through a riemann - hilbert boundary value problem ; hints for such an approach can be summarized as follows . we successively note that * there exists @xmath643\zeta^-,\zeta^+[$ ] such that for @xmath644 , @xmath645 belongs to the analyticity domain @xmath640 determined by proposition [ extendm ] ; * denoting by @xmath646 the image by functions @xmath647 of the open interval @xmath438s_0,\zeta^+[$ ] , we note that @xmath648 for @xmath649 with @xmath650 . equations then enable us to deduce the condition @xmath651 the above riemann - hilbert problem for function @xmath1 is , however , valid on open path @xmath646 only and not on the whole closed contour @xmath652 , defined as the image by functions @xmath653 of closed segment @xmath654 $ ] . the well - posed problem , nevertheless , formulates as follows . [ rhpglob ] determine a function @xmath107 which is analytic in @xmath655 , where @xmath656 is the domain delineated by the closed contour @xmath652 , tends to 0 at infinity and such that boundary condition ( [ bv0 ] ) holds on @xmath652 ( and not only on @xmath646 ) . if the solution @xmath107 to * problem 3 * can be shown to exist and to be analytic on @xmath656 , then functions @xmath1 and @xmath107 coincide . proving the latter statement and deriving an alternative representation of solution @xmath1 ( namely , as a path integral on closed contour @xmath652 ) is an object of further study . on the application side , the performance of the sqf discipline has been characterized , both in terms of empty queue probability and distribution tail at infinity . the results show that sqf compares quite favorably with respect to the `` optimal '' priority discipline , namely hol . such performance properties will be generalized to the asymmetric case where flow patterns are allowed to be heterogeneous . before proving equations ( [ fonctcomp ] ) , we state preliminary expressions of @xmath24 and @xmath25 . given @xmath657 univariate transforms @xmath24 and @xmath25 satisfy @xmath658 [ g1g2prov ] as transforms of regular densities , we have @xmath659 , @xmath660 when @xmath661 for fixed @xmath250 with @xmath200 . besides , we have @xmath662 , @xmath663 when @xmath661 with fixed @xmath664 , where @xmath665 is the laplace transform of the restriction of density @xmath159 on the boundary @xmath154 and @xmath181 is the value at @xmath666 of density @xmath160 on boundary @xmath162 ; as a consequence , @xmath667 for fixed @xmath664 . now , letting @xmath251 tend to @xmath474 in each side of ( [ fonct ] ) , the above limit results entail @xmath668 with @xmath669 , which provides identity ( [ g1g2comp ] ) for @xmath670 . identity ( [ g1g2comp ] ) for @xmath671 is symmetrically deduced by letting @xmath250 tend to @xmath474 in ( [ fonctcomp ] ) with fixed @xmath672 . we now address the derivation of equations ( [ fonctcomp ] ) . recall that subsets @xmath153 , @xmath154 , etc . of state space @xmath134 are defined in ( [ cones12])-([axes12 ] ) . given @xmath196 , define the function @xmath673 by @xmath674 ; @xmath673 is twice continuously differentiable over @xmath675 , @xmath676 for each @xmath677 and @xmath678 ( the dirac mass at @xmath679 ) for the weak convergence of distributions . for given @xmath200 , @xmath201 , let then be the test function @xmath203 , @xmath130 , with @xmath680 function @xmath202 belongs to @xmath681 and is 0 on the outside of @xmath153 ; moreover , we have @xmath198 so that @xmath204 pointwise in @xmath134 , with limit function @xmath131 defined by @xmath682 , @xmath130 . by direct differentiation , we further calculate @xmath683 for @xmath130 , with @xmath684 after ( [ iepsilon ] ) ; note that derivative @xmath685 tends to @xmath686 for the weak convergence of distributions as @xmath211 . + let us now calculate the limit @xmath687 with @xmath209 introduced in ( [ mepsilon ] ) ; to this end , we address successive terms of @xmath688 according to definition ( [ gen1 ] ) . integrating first @xmath689 over @xmath690 against @xmath164 reduces to @xmath691 \varphi_1(\mathbf{u})\mathrm{d}\mathbf{u}\ ] ] since @xmath689 vanishes on the outside of @xmath153 ; on account of the above mentioned weak convergence properties , we then obtain @xmath692 with @xmath693 defined as in lemma [ g1g2prov ] and where @xmath694 defines the laplace transform of density @xmath151 restricted to the positive diagonal @xmath695 ( function @xmath696 is determined below ) . besides , the integral of @xmath697 over @xmath698 equals 0 as this function vanishes on the outside of @xmath153 . further , we have @xmath699 for given @xmath130 and @xmath700 , therefore @xmath701 by the dominated convergence theorem ; hence @xmath702 where random variable @xmath127 has distribution @xmath107 . for given @xmath130 , we similarly have @xmath703 and therefore @xmath704 finally , noting that @xmath705 and adding limit terms ( [ lim1 ] ) , ( [ lim3 ] ) , ( [ lim4 ] ) according to ( [ gen1 ] ) gives limit @xmath706 the final expression @xmath707 \\ & + \lambda_1 \left [ b_1(s_1)g_2(s_2 ) - \int_0^{+\infty } e^{-s_2u_2}\mathbb{e } \left ( e^{-s_1\mathcal{t}_1 } \ind_{\{\mathcal{t}_1 > u_2\ } } \right ) \psi_2(u_2)\mathrm{d}u_2 \right ] \\ & + \lambda_2 b_2(s_2 ) f_1(s_1,s_2 ) + \lambda_2 \ ; \bigg [ \int_{\gamma_2 } e^{-\mathbf{s } \cdot \mathbf{u } } \mathbb{e } \left ( e^{-s_2\mathcal{t}_2 } \ind_{\{\mathcal{t}_2 > u_1 - u_2\ } } \right ) \varphi_2(\mathbf{u } ) \mathrm{d}\mathbf{u } \\ & + \int_0^{+\infty } e^{-s_1u_1}\mathbb{e } \left ( e^{-s_2\mathcal{t}_2 } \ind_{\{\mathcal{t}_2 > u_1\ } } \right ) \psi_1(u_1)\mathrm{d}u_1 \bigg ] - \lambda f_1(s_1,s_2 ) = 0.\end{aligned}\ ] ] defining @xmath214 as in ( [ h ] ) to gather all remaining integrals , the latter identity reads @xmath708 with @xmath709 , @xmath693 being defined by ( [ defe21e12 ] ) and where @xmath696 defines the laplace transform of density @xmath151 restricted to the diagonal . changing index 1 into 2 , and noting that @xmath214 changes into @xmath710 , symmetrically yields second equation @xmath711 with @xmath712 , @xmath713 defined by and where @xmath714 defines the laplace transform of density @xmath159 restricted to the diagonal . to conclude the proof , we prove the following technical lemma . functions @xmath696 and @xmath714 are identically zero . [ w1w2 ] adding equations ( [ fonctcomp1 ] ) and ( [ fonctcomp2 ] ) ( and omitting arguments for the sake of simplicity ) yields @xmath715 . on the other hand , equation ( [ fonct ] ) gives @xmath716 ; equating right hand sides of the latter equations then provides the identity @xmath717 using expressions ( [ g1g2comp ] ) for @xmath670 and @xmath671 , the latter identity simply reduces to @xmath718 , showing that function @xmath719 is constant . as both @xmath696 and @xmath714 vanish at @xmath474 , this constant is 0 and since these functions are non negative by definition , this entails that @xmath720 . after using equations ( [ g1g2comp ] ) to express @xmath693 and @xmath713 in terms of @xmath671 and @xmath670 , respectively , lemma [ w1w2 ] finally enables us to reduce ( [ fonctcomp1 ] ) and ( [ fonctcomp2 ] ) to equations ( [ fonctcomp ] ) . this concludes the proof of proposition [ resol ] . for an exponentially distributed service time @xmath139 with parameter @xmath721 , the factor of @xmath45 in definition ( [ h ] ) of @xmath214 reads @xmath722 = \\ \int_0^{+\infty } \int_0^{+\infty } \left [ \int_{u_2-u_1}^{+\infty } e^{-s_1x_1 } \mu_1 e^{-\mu_1x_1 } \mathrm{d}x_1 \right ] e^{-s_1u_1-s_2u_2}\ind_{\{0 \leq u_1 < u_2\}}\mathrm{d}\phi(u_1,u_2),\end{gathered}\ ] ] for @xmath723 and @xmath724 . by definition ( [ dphi ] ) , the latter term is equal to @xmath725 \\ = \frac{\mu_1}{\mu_1+s_1 } \big [ g_2(s_1+s_2+\mu_1 ) + f_1(-\mu_1,s_1+s_2+\mu_1 ) \big ] \end{gathered}\ ] ] where , by corollary [ extensions ] , each term inside brackets is analytically defined for @xmath213 such that @xmath726 and @xmath727 , respectively , that is at least for @xmath728 . similarly , for an exponentially distributed service time @xmath140 with parameter @xmath729 , the factor of @xmath46 in definition ( [ h ] ) of @xmath214 reads @xmath730 = \\ - \frac{\mu_2}{\mu_2+s_2}\left [ g_1(s_1+s_2+\mu_2 ) + f_2(s_1+s_2+\mu_2,-\mu_2)\right ] \end{gathered}\ ] ] where , by corollary [ extensions ] , each term inside brackets is analytically defined for @xmath213 such that @xmath731 and @xmath732 , respectively , hence for @xmath733 . adding up the two above expressions , we obtain claimed expressions . * by lemma [ discriminant ] , function @xmath387 is analytic on the cut plane @xmath736 $ ] , where ramification points @xmath737 , @xmath424 are determined as the real negative roots of discriminant @xmath405 . as @xmath738 , function @xmath387 is , in particular , analytic in the half - plane @xmath739 ; * by definition ( [ cubiceq ] ) , we may have @xmath740 only if @xmath741 , that is , @xmath742 or @xmath743 or @xmath744 ; in the case @xmath745 , we have @xmath746 and in the case @xmath747 , @xmath748 we conclude that we can not have @xmath749 if @xmath750 ; * by corollary [ extensions ] , transform @xmath290 is analytic on @xmath751 where @xmath752 if @xmath494 and @xmath354 if @xmath502 . * a ) * assume first that @xmath352 . in the @xmath756 plane , the diagonal @xmath757 intersects the curve @xmath758 at @xmath759 ( see fig . [ courbexi1 ] ) . further , we easily verify that @xmath760 for @xmath761 and condition ( [ conditionaa ] ) is therefore fulfilled in this first case . we then conclude that function @xmath1 is analytic for @xmath762 , and thus for @xmath763 ( recall by definition ( [ msym ] ) that @xmath1 is the sum of two non - negative laplace transforms ) . * b ) * assume now that @xmath340 ( see fig . [ courbexi2 ] ) . we have shown above that we can not have @xmath764 , which would otherwise imply @xmath765 . we thus necessarily have @xmath766 , which entails that @xmath767 for @xmath768 and condition ( [ conditionaa ] ) is therefore fulfilled in this second case . we then conclude that function @xmath1 is analytic for @xmath768 , hence for @xmath621 .
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we analyze the so - called shortest queue first ( sqf ) queueing discipline whereby a unique server addresses queues in parallel by serving at any time that queue with the smallest workload . considering a stationary system composed of two parallel queues and assuming poisson arrivals and general service time distributions , we first establish the functional equations satisfied by the laplace transforms of the workloads in each queue .
we further specialize these equations to the so - called `` symmetric case '' , with same arrival rates and identical exponential service time distributions at each queue ; we then obtain a functional equation @xmath0 for unknown function @xmath1 , where given functions @xmath2 , @xmath3 and @xmath4 are related to one branch of a cubic polynomial equation .
we study the analyticity domain of function @xmath1 and express it by a series expansion involving all iterates of function @xmath4 .
this allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue .
this tail appears to be identical to that of the head - of - line preemptive priority system , which is the key feature desired for the sqf discipline .
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it is well known that the current through a voltage - biased superconducting quantum point contact ( sqpc ) is carried by localized states . these states , called andreev states , are confined to the normal region of the contact . the energy of the states the andreev levels exist in pairs , ( one above and one under the fermi level ) , and lie within the energy gap of the superconductor , with positions which depend on the change @xmath0 in the phase of the superconductors across the junction . the applied bias affects this phase difference through the josephson relation , @xmath1 . with a constant applied bias @xmath2 much smaller than the gap energy @xmath3 , @xmath0 will increase linearly in time , and the andreev levels will move adiabatically within the gap . this motion is a periodic oscillation in @xmath0 , indicating that no energy is transfered to the sqpc and a pure ac current will flow through the contact . this is actually the ac josephson effect . we wish to study this system in a non - equilibrium situation , one way to accomplish this is by introducing microwave radiation with a frequency @xmath4 , which will couple the andreev levels to each other . the radiation will represent a non - adiabatic perturbation of the sqpc system . however , if the amplitude of the electromagnetic field is sufficiently small , the field will not affect the adiabatic dynamics of the system much unless the condition for resonant optical interlevel transitions is fulfilled . such resonances will only occur at certain moments determined by the time evolution of the andreev level spacing . the resonances will provide a mechanism for energy transfer to the system to be nonzero when averaged over time and hence for a finite dc current through the junction . the rate of energy transfer is in an essential way determined by the interference between different scattering events @xcite , and therefore it is not surprising that oscillatory features appear in the ( dc ) current - voltage characteristics of an irradiated sqpc . dephasing and relaxation will affect the interference pattern and may even conspire to produce a dc current flowing in the reverse direction with respect to the applied voltage bias . the mechanism behind this negative resistance is very similar to the one responsible for the `` somersault effect '' discussed by gorelik et al . @xcite . a peculiar feature of the andreev bound states , in comparison with normal bound states , is that they can carry current . this is why microwave - induced transitions between andreev levels can be detected by means of transport measurements . in fact , it is possible to do andreev energy - level spectroscopy in the sense that the andreev level spectrum at least in principle can be reconstructed from a measurement of the microwave - induced subgap current . such microwave spectroscopy of andreev states in mesoscopic superconductors is the topic addressed in this work . for an unbiased , mesoscopic sqpc which by construction has a normal region @xmath5 which is much shorter than the coherence length @xmath6 the andreev spectrum of each transport mode has the form @xmath7^{1/2}.\ ] ] here @xmath8 is the transparency of the mode and the energy is measured from the fermi energy @xcite . with a small bias voltage applied , the levels move along the adiabatic trajectories @xmath9 in energy - time space , oscillating with a period of @xmath10 , as shown in fig . [ fig : junction ] . in equilibrium the lower andreev level will obviously be occupied , while the upper level will be empty . in the discussion below we consider a single - mode sqpc , although we note that the theory also applies for the case of a single dominant mode in a multi - mode junction . the transparency @xmath8 of the mode is taken to be arbitrary but energy - independent , @xmath11 . a high frequency electromagnetic field is applied to the gate situated near the contact , see inset in fig . [ fig : junction ] . the time - dependent electric field induced by the gate is concentrated within the non - superconducting region of the junction , and hence the charge carriers will couple to the electromagnetic field only there . when the criterion @xmath12 for adiabaticity is obeyed , the rate of interlevel transitions is exponentially small keeping the level populations constant in time @xcite . the presence of a weak electromagnetic field [ on the scale of @xmath13 does not affect the adiabatic level trajectories except for short times close to the resonances at @xmath14 , when @xmath15 . here the dynamics of the system is strongly non - adiabatic with a resonant coupling which effectively mixes the adiabatic levels . this is an analog of the well known landau - zener transition , which describes interlevel scattering as a resonance point is passed . in our case these transitions give rise to a splitting of the quasiparticle trajectory at the points @xmath16 into two paths ; @xmath17 and @xmath18 , forming a loop in @xmath19 space ( see fig . [ fig : junction ] ) . the resonant scattering opens a channel for energy absorption by the system ; a populated upper level when approaching the edge of the energy gap ( at point @xmath20 in fig . [ fig : junction ] ) creates real excitations in the continuum spectrum , which carry away the accumulated energy from the contact . as a result , the net rate of energy transfer to the system is finite ; it consists of energy absorbed both from the electromagnetic field and from the voltage source . the confluence of the two adiabatic trajectories at @xmath21 gives rise to a strong interference pattern in the probability for real excitations at the band edge , point @xmath20 . the interference effect is controlled by the difference of the phases acquired by the system during propagation along the paths @xmath17 and @xmath18 . in order to describe the time evolution of the andreev states in more detail we use the time - dependent bogoliubov - de gennes , ( bdg ) , equation @xcite for the quasiclassical envelopes @xmath22 of the two - component wave function @xmath23 , @xmath24 { \bf u } \ , . \label{bdg}\ ] ] in this equation @xmath25 is a four - component vector , while @xmath26 is the hamiltonian of the electrons in the electrodes of the point contact , @xmath27 , \ ] ] where @xmath28 and @xmath29 denote pauli matrices in electron - hole space and in @xmath30 space respectively . for a mesoscopic junction the gap function @xmath31 need not be calculated self - consistently , which is why in eq . ( [ h0 ] ) it is assumed to have the constant value @xmath32 in the superconducting reservoir on one side of the sqpc and the different constant value @xmath33 in the reservoir on the other side . the gate potential @xmath34 in eq . ( [ bdg ] ) oscillates rapidly in time and the amplitude is assumed to be small compared to the andreev level spacing , @xmath35 . the function @xmath36 , which is smooth on the scale of the fermi wavelength , has a discontinuity at @xmath37 , i.e. at the point contact , whose spatial extension can be neglected . the discontinuity translates to a boundary condition for @xmath36 , @xmath38 which can be found @xcite by matching at the point @xmath37 two scattering state solutions to the bdg equation approaching from left and right . under the assumption that @xmath39 the system of andreev levels experiences an adiabatic evolution in time this is true at all times except close to the resonances ( points @xmath40 and @xmath41 in fig . [ fig : junction ] ) and at point c , which will be discussed later . for a resonant transition to occur , the deviation of the interlevel spacing from the resonance value @xmath42 , where @xmath43 is the duration of the resonance , has to be less than the quantum mechanical resolution of the energy levels @xmath44 . using this criterion we can estimate the duration @xmath43 of the non - adiabatic evolution as @xmath45^{1/2}$ ] , which implies that @xmath43 is much shorter than the period @xmath10 of josephson oscillations if @xmath46 . hence , if this inequality is obeyed we may consider the non - adiabatic dynamics as temporally localized scattering events . we shall use this later to derive an analytical result for the induced dc current . for the time being , however , we will keep the discussion a little more general . we therefore introduce an ansatz for the wave function * u * in the normal region of the junction in terms of a linear combination of the eigenstates @xmath47 corresponding to the adiabatic andreev levels @xmath48 . the rapidly oscillating terms are explicitly introduced in this so called resonance approximation , @xmath49 following ref @xcite we insert the ansatz ( [ b ] ) into the bdg equation ( [ bdg ] ) to find , using the notation @xmath50 , an equation for the coefficients @xmath51 , @xmath52 \vec{b}(t ) . \label{eq.coupled}\ ] ] here @xmath53/2\hbar$ ] is a measure of the deviation from resonance . this equation describes the time evolution of the coefficients @xmath54 and @xmath55 , which embody the dynamics of the population of the andreev states under irradiation . we recall that the coupling to the time - dependent field is finite only in the normal region , which explains why @xmath56 in ( [ eq.coupled ] ) obtains as the matrix element of @xmath57 between the andreev states @xmath47 ; @xmath58 , where @xmath59 denotes a scalar product in 4-dimensional space . for the case of a double barrier sqpc structure @xmath56 was calculated in ref . @xcite . for a single barrier junction an analogous calculation gives us , @xmath60 , where the constant @xmath61 is determined by the position of the barrier . we note , that this matrix element is proportional to the _ reflectivity _ of the junction ; reflection mixes electron states with @xmath62 and @xmath63 allowing optical transitions between the andreev levels . in a perfectly transparent sqpc ( @xmath64 ) , the upper and lower andreev levels correspond to opposite electron momenta the two levels can not be coupled by radiation since momentum can not be conserved and the effect under consideration does not exist , cf . refs . @xcite . before we proceed to calculate the current through the irradiated sqpc we need to discuss the boundary condition at @xmath65 ( @xmath66 is an integer ) . in the vicinity of these points , the andreev levels approach the continuum and the adiabatic approximation is unsatisfactory , even at small applied voltages and weak electromagnetic fields . the duration @xmath43 of the non - adiabatic interaction between the andreev level and the continuum states can be estimated using the same argument as for the microwave - induced landau - zener scattering . one finds that , @xmath67 . if we evaluate the condition , @xmath68 , we find that we would require that @xmath69 . this means that we can safely treat the non - adiabatic region as short . to derive the boundary condition , for example , at point @xmath20 in fig . [ fig : junction ] , one needs to calculate the transition amplitude connecting the states @xmath70 at time @xmath71 and @xmath72 at time @xmath73 : @xmath74 . here @xmath75 is the exact propagator corresponding to the hamiltonian in eq . ( [ bdg ] ) . we will now proceed with symmetry arguments to show that this amplitude is exactly zero . it can be shown that both the hamiltonian ( [ h0 ] ) and the boundary condition for @xmath36 at @xmath37 ( [ bc ] ) are invariant under the simultaneous charge- and parity inversion described by the unitary operator @xmath76 , where @xmath77 is the parity operator in @xmath78-space . this implies that at any time any non - degenerate eigenstate of the hamiltonian is an eigenstate of the symmetry operator @xmath79 with eigenvalue @xmath80 or @xmath81 and that this property persists during the time evolution of the state . specifically , if we take a state on each side of @xmath82 , @xmath83 next we insert @xmath84 , where the operator @xmath85 changes the sign of the phase @xmath0 , into eq . ( [ sym2 ] ) and apply @xmath86 from the left and arrive at , @xmath87 which shows that @xmath88 . this means that the two states , @xmath89 and @xmath90 are orthogonal . this is consistent with the results of shumeiko et.al . @xcite , who have shown that the andreev state wave functions are @xmath91-periodic whereas the energy levels and the current are @xmath92-periodic . since the state evolving from the adiabatic state @xmath70 is orthogonal to the adiabatic state @xmath93 , the probability for an adiabatic andreev state to be scattered " into a localized state after passing the non - adiabatic region is identically zero . in reality , the andreev state as it approaches the continuum band edge decays into the states of the continuum . such a decay corresponds to a delocalization in real space and is the mechanism for transferring energy to the reservoir @xcite . the orthogonality property shown above guarantees that the coherent evolution of our system persists during only one period of the josephson oscillation and that the equilibrium population of the andreev levels is reset at each point @xmath65 @xcite . this imposes the boundary condition @xmath94 at the beginning of each period . the quasiclassical equation for the total time dependent current at the junction ( @xmath37 ) reads @xmath95 . by manipulating the bdg equation as given by ( [ bdg ] ) and ( [ h0 ] ) and omitting oscillating terms which will not contribute to the dc current the quantity of interest one may turn this expression into the following form , @xmath96 . \label{j}\ ] ] in the static limit , @xmath97 and @xmath98 , this result is clearly equivalent to the standard expression @xmath99 for the andreev level current . in the general nonstationary case @xmath36 is a linear combination of @xmath47 according to eq . ( [ b ] ) and we can calculate the current averaged over the period @xmath100 . this is done through eq . ( [ j ] ) with help of eq . ( [ eq.coupled ] ) and the normalization condition @xmath101 , leading to @xmath102 where @xmath103 is the population of the upper level at the end of the period and @xmath104 at the beginning of the period . the result simplifies further since according to eq . ( [ bcphi ] ) @xmath105 . the direct current through the contact can be viewed as resulting from photon - assisted pair tunneling or equivalently as being due to the distortion of the ac pair current due to the induced interlevel transitions . to facilitate an understanding of the current expression in another way , let us study energy conservation . we have two sources of energy , the applied field and the applied bias . if we consider a single josephson period @xmath100 , the energy absorbed by the system from the voltage source is @xmath106 , and from the applied field it is @xmath107 ( assuming for simplicity that @xmath105 ) . energy conservation for the period can be stated as , @xmath108 , with the energy leaving the system stated as @xmath109 or , inserting the value of @xmath100 , @xmath110 this is exactly the same result as obtained by the more tedious method used above . this discussion allows us to interpret the current as voltage - bias mediated , photon assisted tunneling . generally the current is given by eq . ( [ idc ] ) with the boundary condition eq . ( [ bcphi ] ) inserted . the actual calculation of the current has to be done numerically in most cases . however , in the limit of a weak applied external field and when the frequency of the applied field is such that we have two ( in time ) well separated resonances , we can treat the resonances as temporally localized scattering events . this approach can be quite rewarding as will become clear below . formally , we can describe the system s evolution through a resonance by letting a scattering matrix @xmath111 connect the coefficients @xmath51 before and after the splitting points @xmath40 and @xmath41 . approximating the time dependence of the andreev levels to be linear , a standard analysis of the landau - zener interlevel transitions ( see e.g. @xcite ) gives the scattering matrix elements at the points @xmath40 and @xmath41 as , @xmath112 , \hat{s}_b= \left [ \begin{array}{cc}r & d e^{i \theta } \\ -d e^{-i \theta } & r \end{array } \right ] , \label{sc_mat}\end{aligned}\ ] ] where @xmath113 is the probability of the landau - zener interlevel transition . here @xmath114 where @xmath56 is once again the matrix element for the interlevel transitions . by introducing the matrix @xmath115 , @xmath116 which describes the ballistic " dynamics of the system between the landau - zener scattering events , we connect the coefficients @xmath51 at the end of the period of the josephson oscillation , @xmath82 , with the coefficients @xmath117 at the beginning of the period , @xmath118 , @xmath119 the time - averaged current through the junction can be directly expressed through these coefficients . combining the boundary condition ( [ bcphi ] ) with eqs . ( [ idc ] ) and ( [ bb0 ] ) , one finds @xmath120 where @xmath121 is the phase of the probability amplitude for the landau - zener transition , which only weakly depends on @xmath2 . in fig . [ fig : compare ] the current is calculated both numerically and with the expression above , eq . ( [ i ] ) . the current is plotted as a function of inverse voltage in order to show a periodicity in @xmath122 . the close fit in fig . [ fig : compare ] ensures us that the scattering approach works well for the situation when the resonances are well separated and the applied field is weak . equation ( [ i ] ) is the basis for presenting the biased sqpc as a quantum interferometer . there is a clear analogy between the sqpc interferometer and a standard squid in that they both rely on the presence of trajectories that form a closed loop . in a squid , which is used to measure magnetic fields , the loop is determined by the device geometry ; in the sqpc the voltage ( analog of the magnetic field ) is well defined while the geometry " of the loop in @xmath19 space can be measured . this loop is determined by the andreev - level trajectories in @xmath19 space and is controlled by the frequency of the external field . this gives us an immediate possibility to reconstruct the phase dependence of the andreev levels from the frequency dependence of the period @xmath123 of oscillations of the current versus inverse voltage , see fig . , it follows from eq . ( [ phi ] ) , that @xmath124 this shows that the position of the andreev energy levels can be reconstructed from measurements of microwave - induced subgap currents through the sqpc . in order to be able to do interferometry it is necessary to keep phase coherence during at least one period of the josephson oscillation . there are three dephasing mechanisms that impose limitations in practice : ( i ) deviations from an ideal voltage bias , ( ii ) microscopic interactions , and ( iii ) radiation induced transitions to continuum states . the main source of fluctuations of the applied voltage is the ac josephson effect . according to the rsj model , a fixed voltage across the junction can only be maintained if the ratio between the intrinsic resistance @xmath125 of the voltage source and the normal junction resistance @xmath126 is small . if @xmath127 the amplitude of the voltage fluctuations @xmath128 is estimated as @xmath129 . effects of voltage fluctuations on the accumulated phase @xmath130 can be neglected if @xmath131 , i.e. if @xmath132 , which corresponds to a lower limit for the bias voltage @xcite . the dephasing time due to microscopic interactions is comparable to the corresponding relaxation time @xcite . this mechanism of dephasing can be neglected as soon as the relaxation time exceeds the josephson oscillation period , @xmath133 . taking electron - phonon interaction as the leading mechanism of inelastic relaxation , we estimate @xmath29 to be of the order of the electron - phonon mean free time at the critical temperature , @xmath134 , since the large deviations from equilibrium in our case occur in the energy interval @xmath135 . this gives @xcite another limitation on how small the applied voltage can be , @xmath136 . the third mechanism of dephasing becomes important when the andreev levels are closer than @xmath137 to the continuum band edge . one can estimate the corresponding relaxation time as @xmath138 . for small radiation amplitudes @xmath139 exceeds @xmath140 , while for optimal amplitudes they are about equal . the effect of the level - continuum transitions on the interference oscillations depends on the frequency . if @xmath141 the loop region " [ @xmath142 is optically disconnected from the continuum , and transitions can not destroy interference . possible level - continuum transitions at times outside the loop will only decrease the amplitude of the effect by a factor @xmath143 , where @xmath144 is the relative fraction of the period @xmath140 during which transitions to the continuum are possible . accordingly , this factor is of the order of unity for the voltages that correspond to the maximum amplitude of oscillations . if @xmath145 , the interference is impeded by the optical transitions into the continuum , and the current oscillations decrease . still , a nonzero average current through the junction will persist . in any real system , both relaxation and dephasing will be present as discussed above . therefore , to complete our analysis we will simulate these effects on our system . if we assume that the relaxation and dephasing times , @xmath146 are long compared to the duration of the non - adiabatic resonances , @xmath43 , we can use a technique @xcite , where dissipation is modelled by adding a term to the time evolution equation of the density matrix for the two - level andreev system . the equations are , where relaxation enters in the diagonal terms and dephasing through the off - diagonal terms , @xmath147_{nn}- \frac{\rho_{nn}-\rho_{nn}^{eq}}{\tau } , \label{eq.d1 } \\ & & \dot \rho_{nn'}(t)=-\frac{i}{\hbar}[h_0(t),\rho(t)]_{nn'}- \frac{\rho_{nn'}}{\tau_{\phi } } , \ ; \mbox{n$\neq$n'}. \label{eq.d2}\end{aligned}\ ] ] with @xmath148 as the characteristic time for relaxation of the system to the equilibrium population , @xmath149 and @xmath150 as the characteristic time for dephasing . these equations describe the `` ballistic '' dynamics of the dissipative system , replacing eq . ( [ phi ] ) . the exact form of the density matrix for the system considered here will be a @xmath151 matrix for the discrete two level space of the andreev states . ( to avoid confusion we will use @xmath152 to denote the pauli matrices in this discrete space . ) the hamiltonian for the density matrix is found in eq . ( [ eq.coupled ] ) , and the diagonal elements @xmath153 and @xmath154 will represent the population of the upper and lower andreev levels . the resonant events which are unaffected by the dissipation can still be modeled with the scattering matrices introduced earlier , eq . ( [ sc_mat ] ) , @xmath155 to calculate the effect of relaxation and dephasing on the current we will use the standard expression , @xmath156 , for the current carried by a populated andreev state and average over one josephson period . the result is , as a function of the upper levels population , @xmath157dt , \label{eq.dispcurr}\end{aligned}\ ] ] where we have used the josephson relation @xmath1 and the conservation of probability , @xmath158 . by solving the time evolution equations of the density matrix for the whole josephson period and imposing the boundary condition @xmath159 , we arrive at the following result , @xmath160 where @xmath161.\end{aligned}\ ] ] inserting these expressions into eq . ( [ eq.dispcurr ] ) we find , @xmath162.\end{aligned}\ ] ] the first important property of this expression is that for @xmath163 and @xmath164 it reduces to the previous current expression that we derived , eq . ( [ i ] ) . one obvious effect on the current is that the oscillations of the current decrease when the dephasing time , @xmath150 , becomes small . another is that when the relaxation time , @xmath148 , is decreased , the current diminishes , see fig . [ diss_plot ] . we note that the number of parameters has become quite large . it is now possible to calculate the current for different combinations of @xmath165 and the transmission coefficient @xmath8 . a curious effect occurs when the relaxation time is such that the upper level , after being populated at point a , is fully depopulated at @xmath166 , the middle of the josephson period . what we find is a negative dc current for a positive bias , a regime where our system shows a _ negative conductance _ , see fig . [ diss_plot ] , plots a and d. the relaxation time is such that the upper level is mainly populated when it s derivative is negative ( it carries a negative current ) and then the lower level is highly populated for the part of the period when it s derivative is negative . this effect will be most pronounced when the energy of the applied field , @xmath137 , approaches the energy of the gap , @xmath167 , as in fig . [ diss_plot]d . actually , the physical mechanism which is behind the appearance of this negative resistance is similar to the one responsible for the somersault effect " effect discussed in ref . @xcite . in that case the method for periodically bringing a pair of andreev levels in a superconducting quantum point contact in resonance with a microwave field was different . rather than applying a voltage bias , the time - dependence of the gate potential was assumed to have two components ; one slow component which shifts the andreev levels through its effect on the transparency @xmath168 and one fast ( microwave ) component for coupling the two levels . we have shown that irradiation of a voltage - biased superconducting quantum point contact at frequencies @xmath169 can remove the suppression of subgap dc transport through andreev levels . quantum interference among resonant scattering events can be used for microwave spectroscopy of the andreev levels . the same interference effect can also be applied for detecting weak electromagnetic signals up to the gap frequency . due to the resonant character of the phenomenon , the current response is proportional to the ratio between the amplitude of the applied field and the applied voltage , @xmath170 . at the same time , for common sis detectors a non - resonant current response is proportional to the ratio between the amplitude and the frequency of the applied radiation @xcite ) , @xmath171 , i.e. it depends entirely on the parameters of the external signal and can not be improved . finally , we note that the classic double - slit interference experiment , where two spatially separated trajectories combine to form an interference pattern , clearly demonstrates the wave - like nature of electron propagation . for a 0-dimensional system , with no spatial structure , we have shown that a completely analogous interference phenomenon may occur between two distinct trajectories in the _ temporal _ evolution of a quantum system . * acknowledgment . * support from the swedish kva , ssf , materials consortia 9 & 11 , nfr and from the national science foundation under grant no . phy94 - 07194 is gratefully acknowledged . nil and mj are grateful for the hospitality of the institute for theoretical physics , uc santa barbara , where part of this work was done . the role of photoinduced interference effects in a normal junction was discussed by l. y. gorelik , a. grincwajg , v. kleiner , r. i. shekhter , and m. jonson , phys . lett . * 73 * , 2260 ( 1994 ) ; l. y. gorelik , f. maa , r. i. shekhter , and m. jonson , phys . rev . lett . * 78 * , 3169 ( 1997 ) . this result is obvious for a pure sns contact ( @xmath64 ) since the two andreev states involved in the transition at the band edge have momenta @xmath172 and @xmath63 pointing in opposite directions . see also @xcite .
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we show that irradiation of a voltage - biased superconducting quantum point contact at frequencies of the order of the gap energy can remove the suppression of subgap dc transport through andreev levels .
quantum interference among resonant scattering events involving photon absorption is furthermore shown to make microwave spectroscopy of the andreev levels feasible .
we also discuss how the same interference effect can be applied for detecting weak electromagnetic signals up to the gap frequency , and how it is affected by dephasing and relaxation .
| 8,121 | 128 |
the generation of the mass of hadrons composed of light quarks is a central problem in quantum chromodynamics ( qcd ) , the theory of the strong interaction . unlike any other composite system , hadrons are built out of constituents with masses which are negligible compared to their total mass which is generated by dynamical effects . a central role is played by the spontaneous breaking of chiral symmetry , a fundamental symmetry of qcd . without this symmetry breaking , hadrons would appear as mass degenerate parity doublets . however , in the spectrum of free particles large mass splitting is observed between chiral partners , for baryons and for mesons . the order parameter of the symmetry breaking , the quark condensate , is expected to be density and temperature dependent so that the symmetry could be at least partially restored for hadrons embedded in nuclear matter . already in 1991 brown and rho @xcite suggested scaling laws for hadron in - medium masses ; and shortly after , lutz , klimt , and weise @xcite discussed in - medium properties of mesons in the framework of the nambu - jona - lasinio ( njl ) model . a recent overview of theory and experiment is given in @xcite ; results from photon - induced reactions are summarized in @xcite . although originally in - medium effects were mainly discussed in connection with heavy ion collisions , they should be also observable at normal nuclear density @xcite in systems that can be more easily interpreted than the strongly time - dependent density and temperature variations in heavy ion collisions . among such effects is the mass - shift of the @xmath9-meson in normal dense nuclear matter . this scalar @xmath10 , isoscalar @xmath11 state is mostly regarded as the chiral partner of the @xmath12 pion @xcite . however , due to its unconventionally large width ( mass between 400 - 550 mev , width 400 - 700 mev @xcite ) , in some models it is treated as a correlated state of two pions ( respectively four quarks ) rather than a quark - antiquark state @xcite . the mass split between the @xmath9 and the pion in vacuum is large . however , predictions from different models @xcite indicated that already at normal nuclear matter density @xmath13 the mass of the @xmath9 should drop by @xmath14 200 mev @xcite , while the pion , which is protected by its goldstone boson nature , should almost be unaffected . the strong coupling of the @xmath9 to scalar , isoscalar pion pairs should lead to a modification of the invariant - mass distribution of such pairs in nuclear matter . this effect is predicted by different model studies independent of the assumed nature of the @xmath9 @xcite . in models , in which the @xmath9 is not treated as a quark - antiquark state , the corresponding effect comes from an in - medium modification of pion - pion final state interaction for scalar , isoscalar pion pairs . this effect must not be confused with the final state interaction of individual pions with nuclear matter by re - absorption and re - emission processes . this prediction has been experimentally tested with pion- and photon - induced reactions . pion induced reactions were studied by the chaos collaboration @xcite , which observed a mass shift for isoscalar @xmath15 pairs in heavy nuclei but no effect for the like - sign @xmath16 pairs where the @xmath9 can not contribute . the experiment had only a limited detector acceptance , which complicated the interpretation of the results . a similar effect was observed for the @xmath17 reaction by the crystal ball collaboration at bnl @xcite which , however , could not measure an isovector channel for comparison . pion - induced reactions have the disadvantage that due to the large absorption cross section only the nuclear surface is probed . photons can probe the entire volume of nuclei . although also in this case final - state interaction ( fsi ) effects can not be avoided , they can be studied in a systematic way because they depend strongly on the kinetic energy of the produced pions @xcite . nuclei are almost transparent for low - energy pions , which means that invariant - mass distributions of pion pairs produced close to the production threshold are much less affected by fsi than at higher energies . photon induced production of pion pairs off nuclei has been previously measured with the taps detector at mami @xcite . these experiments found for heavy nuclei a systematic shift of the invariant mass of @xmath18 pairs towards small values while the distributions for mixed - charge pairs @xmath19 were much less effected . however , as discussed in @xcite the spectra were reproduced by the results from calculations with the boltzmann - uehling - uhlenbeck ( buu ) transport model @xcite . this model treated the pion - nucleus fsi effects in great detail , but did not include any explicit in - medium modification of @xmath18 pairs . however , the model results had also non - negligible systematic uncertainties , mainly from the input for the elementary reaction cross sections . a more systematic study was limited by the statistical quality of the data and the range of nuclear masses investigated . as discussed above , fsi effects are minimized close to the production threshold ; but the production cross sections are small . the lightest nucleus included in the previous study was @xmath4c . invariant - mass distributions for the elementary processes off the ( quasi)-free nucleon ( see @xcite for an overview ) have been studied previously for the @xmath20 @xcite , the @xmath21 @xcite , and @xmath22 @xcite final states . data for @xmath23 had low statistical quality in the threshold region @xcite and no data were available for @xmath24 . the present experiment therefore aimed at a precise measurement of pion - pion invariant - mass distributions for the @xmath18 and @xmath19 final states at low incident photon energies over a large range of nuclear masses , including @xmath25h , @xmath3li , @xmath4c , @xmath5ca , and @xmath6pb targets . such data are needed to separate the fsi effects from non - trivial in - medium modifications of the double pion channels . the experimental setup is described in @xcite where the data from the @xmath26li target have been analyzed for other meson production reactions . the measurement was performed at the tagged photon beam of the mainz mami accelerator @xcite . the primary electron beam of 883 mev produced bremsstrahlung in a copper radiator of 10@xmath27 m thickness . the photons were tagged with the glasgow magnetic spectrometer @xcite . four solid - state targets ( thicknesses 5.4 cm ( li ) , 1.5 cm ( c ) , 1 cm ( ca ) , 0.05 cm ( pb ) ) and a liquid - deuterium target ( thickness 4.8 cm , 0.6% of radiation length ) were used . the thickness of the solid - state targets was chosen such that they were comparable in units of the radiation lengths of the materials ( from 3.5% for lithium to 9% for lead ) . for the measurement with the lead target a lower beam energy was used ( 645 mev ) in order to enhance the statistical quality of the data in the threshold region . in this letter , we therefore summarize only the results for incident photon energies below 600 mev . the reaction products were detected with an almost @xmath28-covering electromagnetic calorimeter assembled from the crystal ball ( cb ) @xcite and taps @xcite detectors . the targets were mounted in the center of the crystal ball . a particle identification detector ( pid ) @xcite was mounted around the targets in cylindrical geometry for identification of charged particles . for the same purpose the taps detector was equipped with individual plastic detectors in front of each baf@xmath29 module . the trigger conditions were chosen relatively open in order to minimize systematic uncertainties . the crystal ball and taps were subdivided into logical sectors , taps into 8@xmath3064 modules arranged in a pizza - like geometry and the crystal ball into 45 rectangles . events were accepted when at least two sectors of the calorimeter were hit and the total analog sum energy in the crystal ball was above 50 mev . the charged pion was not used in the trigger . events where the decay photons of the neutral pions had not satisfied the trigger condition were removed in the off - line analysis because of the systematic uncertainties in the simulation of the trigger efficiency for charged pions . the analysis of data measured with the above setup is discussed for different final states in @xcite . the specific analysis steps for @xmath18 and @xmath19 production off nuclear targets are described in detail for the previous measurements with the taps detector @xcite . the main improvements of the present experiment compared to the previous measurements @xcite are the almost @xmath28 coverage of the solid angle and the use of the pid for the identification of charged pions via the @xmath31 method . the first step of the analysis was the assignment of hits to photons , charged pions , protons , and neutrons . the identification of hits in taps ( see @xcite for details ) used the information from the plastic scintillators in front of the taps crystals and a pulse - shape analysis ( psa ) , using the narrow- and wide - gate energy integration of the baf@xmath29 signals for the separation of photons from nucleons . a time - of - flight ( tof ) versus energy analysis showed the characteristic bands for photons , charged pions , and nucleons . as discussed in @xcite , at forward angles the charged - pion band in taps is contaminated with protons . therefore , charged pions were not accepted in taps ( this excluded charged pions with polar angles @xmath32 from the analysis ) . in the crystal ball @xcite , charged hadrons were identified via their @xmath31 spectra constructed from the energy loss @xmath33 in the pid and the energy deposition in the crystal ball . this analysis also used the correlation between the azimuthal angles of the hits in the pid and the cb . a typical @xmath31 spectrum shown in fig . [ fig : banana ] , left hand side , demonstrates the separation of charged pions from protons . in the cb , neutrons can not be distinguished from photons , as opposed to in taps . therefore , neutral hits in the cb were accepted as candidates for photons and for neutrons , while neutral hits in taps were separated at this stage into disjunct samples of photons and neutrons through psa and tof - versus - energy . events were then assigned to the @xmath18 and @xmath19 final states . accepted were events with four photons ( for the @xmath34 final state ) and events with two photons and one charged pion ( for the @xmath19 final state ) . detection of recoil nucleons was allowed but not required and after identification , recoil nucleons were ignored in the further analysis . consequently , three different sub - classes of events were accepted for both reactions . for the double neutral channel , these were : events with exactly four photon candidates ( 50% ) ; events with four photons and one proton ( 32% ) ; and events with four photons and one neutron candidate ( 18% ) . similarly , for @xmath19 pairs , events with : exactly two photon candidates and exactly one candidate for a charged pion in the crystal ball ( 59% ) ; events with an additional proton ( 28% ) ; and events with an additional candidate for a neutron ( 22% ) were accepted . in the next step the invariant mass of the pairs of photon candidates was constructed . for events with a neutron candidate in the crystal ball ( which is indistinguishable from a photon ) , all possible pion combinations of neutral hits were tested and the ` best ' combination was selected using a @xmath35-test minimizing @xmath36 where @xmath37 is the nominal pion mass , and @xmath38 , @xmath39 are invariant mass and uncertainty of all possible combinations of ` photon ' pairs . the unpaired neutral hit , not assigned to be a pion decay photon , was then taken as neutron and thereafter disregarded . the resulting two - dimensional spectrum of pion - invariant masses for double @xmath8 production is shown in fig . [ fig : banana ] , right hand side . events with both invariant masses in the range 110 - 160 mev were selected for further analysis . the small combinatorial background underneath the peak , was extrapolated in a side - bin analysis and subtracted . subsequently , the nominal mass of the pion @xmath37 was used to improve the resolution as in @xcite by replacing the measured photon energies @xmath40 by @xmath41 where @xmath42 is the invariant mass corresponding to the measured photon four vectors . this data sample was tested for residual background with missing mass spectra ( the recoil nucleon was treated as a missing particle even if it was detected ) : @xmath43 where @xmath44 and @xmath45 are the four momenta of the incident photon and nucleon ( the latter assumed at rest ) and @xmath46 , @xmath47 the four momenta of the two pions . the corresponding spectra are shown in fig . [ fig : mismas ] , bottom row . they demonstrate the cleanness of the signal and the excellent agreement between measured data and monte carlo simulations . for the mixed - charge channel , the invariant - mass spectra were fitted with the simulated line - shape of the pion peak and a polynomial background ( second degree polynomial ) . some background arises from the quasi - free @xmath48 reaction from protons which contaminate the pion band in the @xmath31 spectra . the fraction of protons that intrude is small , but the cross section for single @xmath8 production can be larger than for @xmath19 production by more than two orders of magnitude . the most efficient way to remove this background uses the missing mass calculated for the hypothesis of the @xmath48 reaction ( see @xcite for details ) , i.e. it is assumed that the ` charged pion ' is a misidentified proton : @xmath49 with the same notation as in eq . ( [ eq : misma_1 ] ) . typical spectra are shown in fig . [ fig : mismas ] . for the deuteron target background peak and double pion production are separated , for the heavier nuclei , due to the larger momenta of the bound nucleons and fsi , the background is only visible as a shoulder on the tail of the signal peak . only events with @xmath50 140 mev have been accepted . the center row of fig . [ fig : mismas ] shows the missing mass @xmath51 ( see eq . ( [ eq : misma_1 ] ) ) calculated under the assumption of quasi - free @xmath19 production . the ( blue ) triangles represent the raw spectra and the ( black ) dots the events that passed the above @xmath52 cut . the latter agree well with the simulated line - shape ( ( red ) lines ) . residual background was removed with the condition @xmath53 mev . charged pions may deposit more than their kinetic energy in the detector . when they are stopped , they may interact with nuclei or decay and deposit some fraction of their rest mass as energy ( see @xcite for details ) . the additional energy deposition was taken into account by a correction derived from the comparison of kinetic and deposited energies of charged pions in the monte carlo simulations . the charged pion reconstruction was tested as in @xcite with an analysis of the position and shape of the invariant mass peak from the @xmath54 decay . the results for data and monte carlo simulation were in good agreement , and the peak position agreed with the nominal mass of the @xmath55-meson . since charged pion detection enters twice in this analysis , this test is very sensitive . total cross sections were extracted from the measured yields , the target surface densities , the photon flux , and the simulated detection efficiencies . systematic uncertainties from the targets were in the range from 1% to 3% . the photon flux was determined from the counting of the deflected electrons in the tagger focal plane and the tagging efficiency , i.e. the fraction of correlated photons passing the collimator ( typically @xmath14 50% ) . the systematic uncertainty was estimated at the 5% level . systematic uncertainties from the cuts applied for reaction identification and possible residual backgrounds were estimated at @xmath14 3% for @xmath18 and @xmath14 5% ( for @xmath56 500 mev ) to 10% ( for @xmath56 400 mev ) for @xmath19 . the detection efficiency was simulated with the geant4 program package @xcite . the simulations were based on the event generator used for the previous experiments @xcite , which incorporates fermi - smearing and fsi processes in a glauber - type approximation . the efficiency corrections were applied to the data as a function of incident photon energy and invariant mass of the pion pairs . in this way , the influence of the properties of the event generator on the results is minimized . depending on incident photon energy , pion - pion invariant mass , and target , typical values of the detection efficiency were in the range 30% - 70% for @xmath18 pairs and between 5% - 40% for the @xmath19 channel . total cross sections were extracted by integration of the invariant mass distributions . we estimate a systematic uncertainty of the detection efficiency of @xmath14 10% for neutral pairs and @xmath14 20% for the mixed charge pairs . it is larger for the latter because their detection efficiency depends strongly on the kinetic energy of the charged pions ( see @xcite for details ) . the total systematic uncertainties are indicated in fig . [ fig : invdiff ] as shaded bands ( error bars in all figures represent statistical uncertainties ) . one should , however , keep in mind that for the comparisons of reaction channels and the results for different nuclei , systematic uncertainties cancel to a large extent . invariant - mass distributions of the pion pairs for incident photon energies between 400 mev - 460 mev are summarized in fig . [ fig : invdiff ] . the distributions measured for pion pairs produced off quasi - free nucleons from the deuteron differ in shape and magnitude from those measured off free protons ( fig . [ fig : invdiff ] , top row ) . this may be partly attributed to fermi - smearing , but includes also differences in the elementary cross sections for protons and neutrons . this is important for comparisons of the nuclear data to model results , which have to rely on experimental input for the elementary cross sections . the data for carbon and lead from messchendorp et al . @xcite are in reasonable agreement with the present results for the neutral pairs ; but not for the mixed - charge channel . agreement between the @xmath19 data for calcium from bloch et al . @xcite and the present results is much better ( see fig . [ fig : invdiff2 ] ) . it had already been noticed that the @xmath19 data from references @xcite are probably in conflict because the mass dependence for the chain carbon - calcium - lead would have been very unnatural ( the magnitude of the calcium cross section normalized by mass number had been larger than both the carbon and lead data , but one would have expected it in between of them ) . the present measurement favors the previous results from bloch et al . @xcite with which they agree within systematic uncertainties . comparison of the invariant - mass distributions in fig . [ fig : invdiff ] shows , that for the @xmath18 and the @xmath19 pairs , strength shifts to small invariant masses for increasing mass of the nucleus . this general trend is certainly not related to properties of the @xmath9 meson , which does not couple to the mixed - charge channel . the effect can be studied in more detail with the help of the composite ratios defined by @xmath57 where @xmath58 are the invariant mass distributions and @xmath59 the total production cross sections for two nuclei with mass numbers @xmath60 and @xmath61 . results for incident - photon energies from 400 mev - 460 mev are summarized in fig . [ fig : ratio ] . they demonstrate that , at least in this energy range , the behavior of the invariant - mass distributions is almost identical for both isospin states , so that the dominant effect is most likely due to fsi of the pions with the nucleus . previous results @xcite for the composite ratio of carbon and lead indicated a stronger enhancement at low invariant masses for neutral pion pairs than for the mixed - charge pairs . however , comparison to the present invariant - mass distributions ( see fig . [ fig : invdiff ] ) suggests that this was due to statistical fluctuations , in particular in the lead data . using the lighter nuclei ( deuteron and @xmath26li ) as reference , the low - mass enhancement becomes larger , but in the same way for neutral and mixed - charge pairs . an estimate of how important fsi effects are for the pions can be deduced from the scaling of the cross sections with nuclear mass number @xmath62 . total cross sections as a function of incident photon energy , scaled by the mass number @xmath62 , are shown for both isospin states in the inserts of fig . [ fig : alpha ] . the scaling changes from threshold to higher incident photon energies . this behavior can be parameterized with the scaling coefficient @xmath63 from @xmath64 which is shown in fig . [ fig : alpha ] . their interpretation is straightforward . at incident photon energies close to the production threshold @xmath63 is close to unity , which means that the cross sections scale with the number of nucleons ( or the nuclear volume ) , indicating negligible losses due to pion absorption . the coefficient then drops as a function of @xmath65 and approaches 2/3 for beam energies between 500 mev and 600 mev , indicating a scaling proportional to the surface of the nuclei , which means strong absorption . this is expected from the absorption properties of nuclear matter for pions @xcite . nuclei are transparent for pions with kinetic energies below @xmath14 40 mev and ` black ' for pions with energies above @xmath14 100 mev , which may excite the @xmath66-resonance . in quasi - free kinematics , pions from pion pairs produced with incident photon energies around 500 mev ( 600 mev ) have kinetic energies around 50 mev ( 75 mev ) when the energy is symmetrically shared by the two pions ( up to 138 ( 150 ) mev for pions from extremely asymmetric energy distribution ) . the scaling behavior is similar for both isospin channels , so that comparable fsi effects are to be expected . the shapes of pion - pion invariant - mass distributions for different ranges of incident photon energies are compared in fig . [ fig : compinv ] . for both isospin channels , at the lowest incident photon energies , the distributions are similar for all nuclei . they basically agree with a fermi - smeared version of the elementary cross section average of proton and neutron cross sections , represented by the deuteron data . at higher incident photon energies they start to differ , and for fixed photon energy , the heaviest nuclei have the softest distributions . this general trend can be related to the increase of fsi with rising beam energy and rising mass . it can be easily understood that fsi tends to soften the distributions . two effects are important . inelastic scattering tends to decrease the kinetic energy of the pions and their mean - free path increases with decreasing energy . any effects due to in - medium modification of the @xmath9-meson can only be studied on top of the fsi effects , which requires detailed model descriptions of fsi . an efficient way to treat these ` trivial ' nuclear effects are transport - theoretical approaches @xcite . we compare typical invariant - mass distributions to the results of the giessen boltzmann - uehling - uhlenbeck ( gibuu ) model @xcite in figs . [ fig : invdiff ] and [ fig : invdiff2 ] . on an absolute scale , the production cross section for the neutral pairs is underestimated by the model at very low incident photon energies and agrees better at higher energies , while for the mixed - charge pairs agreement is good at low energies and becomes worse for the highest energies . these discrepancies are at least partly related to the uncertainty in the elementary input to the model ( the production cross sections off neutrons are not well known for most isospin channels ) . in order to facilitate the comparison of the shapes , rescaled versions of the model results are shown . the shapes of the invariant - mass distributions of the mixed - charge pairs agree for all discussed ranges of incident photon energy almost perfectly with the data . this demonstrates that for @xmath19 the observed evolution of the distributions with photon energy and atomic mass can be indeed explained by fsi . agreement is not as good for the neutral pairs where , at least for low incident photon energies , the measured distributions tend to be softer than the predicted ones , exhibiting additional strength at low invariant masses . however , this effect is not large and part of it is probably also related to the model input . the shape of the distribution measured for the deuteron target is softer than for the proton ( see fig . [ fig : invdiff ] ) and this effect is not included in the model . precise results have been obtained for the invariant - mass distributions of @xmath18 and @xmath19 pairs produced from deuterium , li , c , ca , and pb targets from production threshold up to incident photon energies of 600 mev . a pronounced shift of strength towards small invariant masses as a function of the nuclear mass number is observed for _ both _ final states . this effect increases with increasing beam energy and can be related to re - absorption and re - emission processes of the pions in the nuclei . an analysis of the scaling behavior of the total cross sections demonstrates that fsi are negligible close to production threshold and saturate at energies above 500 mev . the general trend of the shape change of the invariant - mass distributions correlates with the energy dependence of the fsi effects . consequently , the dominant effect on the invariant - mass distributions comes from fsi . as a consequence of these results , possible in - medium modifications of the @xmath9 meson ( respectively pion - pion interaction in the scalar , isoscalar state ) can not be easily tested with an analysis of the shape change of the pion - pion invariant - mass distributions as a function of mass number and/or beam energy as attempted in many recent works . predictions of the invariant - mass distributions in the framework of the gibuu transport model reproduce their shapes for the mixed - charge pairs excellently , demonstrating that the nuclear effects for this channel are understood . small shape discrepancies for the neutral pairs require further investigation , in particular more precise input for the elementary reactions into models . in view of these uncertainties , no clear evidence for a contribution of @xmath9 in - medium modification can be deduced , although the observed shapes of the low - energy invariant - mass distributions of @xmath18 pairs suggest that a small effect might exists . we wish to acknowledge the outstanding support of the accelerator group and operators of mami . we thank o. buss and u. mosel for interesting discussions and the provision of the gibuu model predictions . this work was supported by schweizerischer nationalfonds , deutsche forschungsgemeinschaft ( sfb 443 ) , uk science and technology facilities council , stfc , european community - research infrastructure activity ( fp6 ) , the us doe , us nsf , and nserc ( canada ) . brown and manque rho , phys . 66 ( 1991 ) 2720 . m. lutz , s. klimt , and w. weise , nucl . phys . a 542 ( 1992 ) 521 . s. leupold , v. metag , u. mosel , int . j. of mod . e 19 ( 2010 ) 147 . b. krusche et al . , eur . j. special topics 198 ( 2011 ) 199 v. bernard , u .- g . meissner , i. zahed , phys . 59 ( 1987 ) 966 . t. hatsuda , t. kunihiro , h. shimizu , phys . rev . 82 ( 1999 ) 2840 . z. aouissat et al . , phys . c 61 ( 1999 ) 012202 . j. beringer et al . d 86 ( 2012 ) 010001 . h.c . chiang , e. oset , m.j . vicente vacas , nucl . a 644 ( 1998 ) 77 . l. roca , e. oset , m.j . vicente vacas , phys . b 541 ( 2002 ) 77 . g. chanfray et al . , eur . j. a 27 ( 2006 ) 191 . m. abaladejo and j.a . oller phys . d 86 ( 2012 ) 034003 . r. rapp et al . c 59 ( 1999 ) r1237 . f. bonutti et al . , phys . 77 ( 1996 ) 603 . f. bonutti et al . , phys . c 60 ( 1999 ) 018201 . f. bonutti et al . , nucl . phys . a 677 ( 2000 ) 213 . p. camerini et al . , nucl . phys . a 735 ( 2004 ) 89 . n. grion et al . phys . a 763 ( 2005 ) 80 . a. starostin et al , phys . rev . lett . 85 ( 2000 ) 5539 . b. krusche et al . j. a 22 ( 2004 ) 277 . messchendorp et al , phys . ( 2002 ) 222302 . f. bloch et al , eur . j. a 32 ( 2007 ) 219 . o. buss et al . , eur . j. a 29 ( 2006 ) 189 . b. krusche and s. schadmand , prog . 51 ( 2003 ) 399 . f. hrter et al . b 401 ( 1997 ) 229 . m. wolf et al . j. a 9 ( 2000 ) 5 . sarantsev et al . , phys . b 659 ( 2008 ) 94 . w. langgrtner et al . 87 ( 2001 ) 052001 . a. zabrodin et al . , phys . c 60 ( 1999 ) 055201 . v. kleber et al . , eur . j. a 9 ( 2000 ) 1 . y. maghrbi et al . , eur . j. a 49 ( 2013 ) 38 . h. herminghaus et al . , ieee trans . on nucl . science . 30 ( 1983 ) 3274 . th . walcher , prog . ( 1990 ) 189 . mcgeorge et al . , eur . j. a 37 ( 2008 ) 129 . a. starostin et al . , phys . c 64 ( 2001 ) 055205 . gabler et al . , nucl . . meth . a 346 ( 1994 ) d. watts , in _ calorimetry in particle physics , proceedings of the 11th internatinal conference , perugia , italy 2004 _ , edited by c. cecchi , p. cenci , p. lubrano , and m. pepe ( world scientific , singapore , 2005 , p. 560 ) . s. schumann et al . , eur . j. a 43 ( 2010 ) 269 . f. zehr et al . , eur . j. a 48 ( 2012 ) 98 . s. agostinelli et al . , instr . meth . a 506 ( 2003 ) 250 . o. buss et al . , phys . ( 2012 ) 1 .
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photoproduction of @xmath0 and @xmath1 pairs from nuclei has been measured over a wide mass range ( @xmath2h , @xmath3li , @xmath4c , @xmath5ca , and @xmath6pb ) for photon energies from threshold to 600 mev .
the experiments were performed at the mami accelerator in mainz , using the glasgow photon tagging spectrometer and a 4@xmath7 electromagnetic calorimeter consisting of the crystal ball and taps detectors . a shift of the pion - pion invariant mass spectra for heavy nuclei to small invariant masses has been observed for @xmath8 pairs but also for the mixed - charge pairs .
the precise results allow for the first time a model - independent analysis of the influence of pion final - state interactions .
the corresponding effects are found to be large and must be carefully considered in the search for possible in - medium modifications of the @xmath9-meson .
results from a transport model calculation reproduce the shape of the invariant - mass distributions for the mixed - charge pairs better than for the neutral pairs , but also for the latter differences between model results and experiment are not large , leaving not much room for @xmath9-in - medium modification . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
| 8,709 | 450 |
after the formidable achievement of bose - einstein condensation in spin - polarized alkali gases @xcite , the next challenge that experimentalists have already set themselves is to realize quantum degenerate conditions also in fermionic alkali vapors . one particular motivation in this respect is the prediction that a gas of spin - polarized atomic @xmath0li becomes superfluid at densities and temperatures comparable with those at which the bose - einstein experiments are performed @xcite . as a result of this experimental interest , the first theoretical studies of an ideal fermi gas trapped in a harmonic external potential have recently appeared @xcite . moreover , the effects of an interatomic interaction have also been considered @xcite . it is interesting to note that oliva s calculations for atomic deuterium were already performed a decade ago , even though magnetically trapped deuterium had not been observed at that time . in fact , it has still not been observed , because the loading of the trap can not be accomplished in the same way as for atomic hydrogen @xcite . this is presumably caused by the fact that deuterium binds more strongly to a superfluid helium film , that the surface recombination rate is much larger , and that the sample is contaminated with atomic hydrogen @xcite . fortunately , such problems do not arise for experiments with alkali gases and both @xmath0li and @xmath1k have indeed been trapped already @xcite . the most important qualitative feature of a trapped fermi gas is that its density profile ` freezes ' at low temperatures . this is a result of the pauli exclusion principle and can be easily understood by considering an ideal gas in the trapping potential @xmath2 . in general the extent of the gas cloud is determined by the classical turning point of the most energetic particles . if the gas is fully classical , i.e. , it obeys maxwell - boltzmann statistics , these particles have an energy of a few @xmath3 and the size of the cloud @xmath4 follows from @xmath5 , implying that @xmath6 . we see that , as we lower the temperature , the size of the cloud shrinks . moreover , if we keep the number of particles @xmath7 fixed , the density increases . this process gradually continues until we reach zero temperature and the density profile becomes equal to @xmath8 . if the gas obeys fermi - dirac statistics , however , the most energetic particles have in the degenerate ( nonclassical ) regime an energy that is equal to the fermi energy @xmath9 and the size of the gas cloud is always given by @xmath10 for temperatures @xmath11 . comparing this also with the density profile for an ideal bose gas , which in the degenerate regime consists of a large and narrow condensate peak with a width of about @xmath12 on top of a broad thermal background of size @xmath13 , we conclude that the density profile of a degenerate fermi gas is indeed ` frozen ' . in contrast to the ideal bose gas , the ideal fermi gas also does not have a phase transition . from a point of view of condensed matter physics , an atomic fermi gas thus appears much less interesting than a bose gas . the main objective of this contribution is to argue that this is no longer true if there are interactions between the atoms . for atomic alkali gases the most important interatomic interaction is the so - called central interaction @xmath14 , which consists of a sum of the usual singlet and triplet interactions . more precisely , we have that @xmath15 where @xmath16 and @xmath17 denote the projection operators on the subspace of singlet and triplet states , respectively . besides this interaction that is the net result of the coulomb attractions and repulsions between the electrons and nuclei of the atoms , we also have to consider the weak magnetic dipole - dipole interactions . of these , the electron - electron magnetic dipole interaction is most important and obeys @xcite @xmath18 here @xmath19 is the electron magnetic moment and the tensor operator @xmath20 is obtained from coupling the pauli spin matrices @xmath21@xmath22 and @xmath21@xmath23 of the two valence electrons to a tensor of rank 2 . it should be noted that both these interactions do not commute with the electron spin operators @xmath24@xmath21@xmath25 and therefore also do not commute with the atomic hamiltonian , which in a magnetic field contains both a hyperfine and a zeeman term . as a consequence the central and dipole - dipole interactions are not fully diagonal in the basis in which the atoms are in definite hyperfine states . this is important in principle , because it implies that two atoms can also collide inelastically , i.e. , their hyperfine states can change during the collision . together with three - body recombination events , these inelastic collisions in fact always seriously limit the lifetime of a trapped alkali gas . nevertheless , we will in the following mostly neglect the nondiagonal parts of the interatomic interaction by restricting ourselves to ( doubly ) spin - polarized gases for which the ` good ' elastic collisions dominate the ` bad ' inelastic ones . clearly such a restriction is a minimum requirement for our discussion to be also of some experimental interest . as we will see in detail below , the most common phase transition in a weakly - interacting fermi gas is due to the formation of so - called cooper pairs . the main idea behind the famous bardeen - cooper - schrieffer theory for this phenomenon is in fact a bose - einstein condensation of these pairs @xcite . to see how we can arrive at a mean - field theory for this phase transition , let us first recapitulate the mean - field ( hartree ) theory for bose - einstein condensation in an atomic bose gas . at zero temperature we can then use a variational many - body wave function in which all the atoms are in the same one - particle state @xmath26 , so @xmath27 in the language of second quantization this essentially reads @xmath28 where @xmath29 is the creation operator for an atom at postion @xmath30 and @xmath31 is the ` vacuum ' state in which there are no atoms present in the trap . calculating the average of the hamiltonian @xmath32 in this variational wave function and minimizing with respect to @xmath26 leads of course to the gross - pitaevskii equation @xcite if @xmath33 . the above variational wave function has a definite number of particles . for practical calculations , in particular if we want to consider also nonzero temperatures or corrections to the mean - field theory , it is much more convenient to consider a variational wave function in which not the number of particles @xmath7 , but instead the phase @xmath34 is fixed . this state is given by @xmath35 as can be seen from the fact that now the matrix element @xmath36 has a definite phase in contrast to the matrix element @xmath37 . it is not difficult to show that by minimizing the average energy in the state @xmath38 we again recover the gross - pitaevskii equation . more important for our purposes is that we can easily show by fourier analysis that @xmath39 which shows that the number of particles @xmath7 and the phase @xmath34 are conjugate variables and obey the commutation relation @xmath40 = i$ ] . the heisenberg equation of motion for the average phase thus becomes @xmath41 \right\rangle = -i \left\langle \frac{\partial h}{\partial n } \right\rangle \equiv -i \mu~,\ ] ] with @xmath42 the chemical potential of the gas . we therefore recover the important josephson relation @xmath43 which is also well known from the hydrodynamic formulation of the gross - pitaevskii equation @xcite . indeed , taking the gradient of this equation and using the definition of the superfluid velocity , i.e. , @xmath44 , we obtain the desired result that @xmath45 let us now return to cooper - pair formation in degenerate fermi gases . in analogy with bose - einstein condensation we can now use at zero temperature the variational wave function @xmath46 a few remarks are in order . first , we have denoted the various hyperfine states of the atoms by @xmath47 . in our previous discussion of bose - einstein condensation we should in principle also have indicated the hyperfine state of the atoms and used @xmath48 instead of @xmath29 in the variational wave function . however , as long as all the atoms are in the same hyperfine state , the precise atomic state which is trapped is unimportant from a theoretical point of view and only influences the interatomic interaction , i.e. , the particular value of the scattering length , that should be used in the gross - pitaevskii equation . for clarity we therefore suppressed the spin degrees of freedom in that case . second , the fermionic creation operators anticommute . as a result the cooper - pair wave function must obey @xmath49 , reflecting the pauli exclusion principle . there are essentially two ways to fulfill this antisymmetrization requirement . if all the atoms are in the same hyperfine state we have @xmath50 and the orbital part of the cooper - pair wave function is antisymmetric with respect to an exchange of the atoms . the relative angular momentum of the pairs must therefore be odd . in particular , we can have @xmath51-wave pairing . this situation occurs in doubly spin - polarized fermi gases and also in liquid @xmath52he @xcite . if we have an equal number of atoms in two hyperfine states , which is implicitly assumed in the above variational wave function , the spin part of the cooper - pair wave function can already be antisymmetric and the relative angular momentum of the pairs must then be even . we now can have @xmath53-wave pairing as in ordinary superconductors . in principle , we can of course also have @xmath54-wave pairing as in the high - temperature superconductors , but this turns out to be extremely unlikely for dilute gases . third , in actual applications it is again more convenient to use a variational wave function with a well defined phase . in this case it is obtained by multiplying the cooper - pair wave function in the right - hand side of eq . ( [ fixedn ] ) with @xmath55 and summing over all even values of @xmath7 . we then find @xmath56 and exactly the same relation between the states @xmath57 and @xmath38 as for a condensate of single atoms . the josephson relation given in eq . ( [ jr ] ) is therefore also valid in this case . moreover , in the state @xmath38 we have a nonvanishing expectation value @xmath58 which suggest that @xmath59 is the appropriate order parameter for the bardeen - cooper - schrieffer transition , just like @xmath60 is the order parameter for bose - einstein condensation . although this identification of the order parameter is correct , it turns out that it is more convenient in practice to work with the so - called bcs gap parameter @xmath61 where @xmath62 is a shorthand notation for the elastic part of the interatomic interaction , and @xmath63 and @xmath64 are the relative and center - of - mass coordinates of the cooper pair , respectively . knowing the order parameter of the phase transition of interest , it is straightforward to obtain the corresponding mean - field theory . the main idea is first to write in the interaction part of the hamiltonian @xmath65 the operators @xmath66 and @xmath67 as a sum of a mean value and fluctuations , and then to neglect terms that are quadratic in the fluctuations . in this manner we arrive at the mean - field hamiltonian @xmath68 where the renormalized chemical potentials are in a good approximation ( but see below ) given by @xmath69 and the density profile of atoms in spin state @xmath47 obeys @xmath70 . it is important to note that in trapped alkali gases it is indeed appropriate to have a chemical potential for each spin state , because the time scale for relaxation towards equilibrium in spin space is always much larger than the equilibration time for the spatial degrees of freedom . this is for example quite dramatically demonstrated by the two overlapping condensate experiments by myatt @xcite . to complete the mean - field theory we should now calculate the mean values of the operators @xmath66 and @xmath67 in a thermal ensemble with the hamiltonian @xmath71 . this clearly makes the theory selfconsistent . to perform the calculation we write the annihilation operators at the positions @xmath30 and @xmath72 as @xmath73 respectively . substituting this in the mean - field hamiltonian and neglecting gradients in the center - of - mass coordinate @xmath74 , we find that @xmath75 diagonalizing the above hamiltonian by means of a bogoliubov transformation , the spin - density profiles can then be calculated from @xmath76 and , most importantly , the bcs gap parameter from @xmath77 notice that the averages in the right - hand side of eqs . ( [ spin ] ) and ( [ gap ] ) depend on the position @xmath74 , since the mean - field hamiltonian @xmath71 depends parametrically on this position . to arrive at such a simplyfied ( thomas - fermi ) description of the inhomogeneity of the gas we have to be able to neglect gradients of the densities @xmath78 and the bcs gap parameter @xmath79 . this is indeed true for present experiments with fermionic alkali gases , because the number of trapped atoms is so large that both the correlation length and the size of the cooper pairs are small compared the typical length scale on which the trapping potential varies @xcite . we refer for the details of the diagonalization to our previous work @xcite . for our present purposes it is however important to mention that in the optimal case of equal spin densities , the final result of eq . ( [ gap ] ) is the famous bcs gap equation @xcite @xmath80 where the so - called bogliubov dispersion @xmath81 obeys @xmath82 in principle , one can solve the bcs equation for any interatomic potential @xmath83 . however , for alkali gases the complete central interaction is usually not very well known and we have only information on the two - body scattering length @xmath84 . we therefore would like to reformulate the gap equation in such a way that only this scattering length enters . this is achieved by noting that the gap equation is very similar to the lippmann - schwinger equation for the two - body t(ransition ) matrix . indeed the latter reads @xcite @xmath85 using the lippman - schwinger equation with @xmath86 , we can after some algebraic manipulation show that the gap equation is equivalent to @xmath87 this result serves our purposes since the two - body t matrix is directly related to the two - body scattering length as we will see next . in doubly spin - polarized fermi gases all the atoms are in the same hyperfine state . although it is straightforward to generalize the following to an arbitrary state @xmath47 , we will in first instance restrict ourselves to the fully stretched state @xmath88 in which both the electron as well as the nuclear spin have a maximal projection on the magnetic field axis . the relevant interaction matrix element is then equal to @xmath89 with @xmath90 the angle between the interatomic separation @xmath91 and the magnetic field @xmath92 . for this potential we have to calculate the two - body t matrix . treating the weak electron - electron magnetic dipole interaction in born approximation @xcite and making use of the fact that for the central interaction only @xmath51-waves contribute at low momenta , we find @xmath93 here @xmath84 is the triplet @xmath51-wave scattering length and @xmath94 denotes the angle between the momentum transfer @xmath95 and the magnetic field axis . we now have two cases to consider . generically we expect the @xmath51-wave scattering length to be of the order of the range of the triplet potential and therefore @xmath96 to be much smaller than one . in that case the contribution of the triplet potential to the two - body t matrix is negligible and the effective interaction between the atoms is dominated by the long - range dipole - dipole interaction . due to the complicated angular dependence of this interaction it is not possible to solve the bcs gap equation analytically . however , we can nevertheless make progress by noting that the magnetic dipole - dipole interaction is most attractive when @xmath91 is directed along the magnetic field . we thus expect that if cooper pairs are formed their wavefunction @xmath97 is most likely proportional to @xmath98 , which implies that @xmath99 . since this gap is anisotropic in the relative wave vector @xmath100 , the gas is below the critical temperature an anisotropic superfluid , just like @xmath52he in the so - called a phases @xcite . to obtain an estimate for the critical temperature we explicitly consider only the @xmath51-wave part of the dipole - dipole interaction , which results in the approximation @xmath101 this explicitly confirms that the dipole - dipole interaction is only attractive in the channel @xmath102 . substituting the above two - body t matrix into the bcs gap equation and linearizing with respect to @xmath103 to obtain an equation for the critical temperature , we find @xmath104 here we used the notations @xmath105 for the cauchy principle value part of the integral and @xmath106 for the fermi distribution function @xmath107 evaluated at @xmath108 . note that we have also used that the bcs gap equation will have a nontrivial solution in the center of the trap first , since the density of the gas is highest there . introducing a ` scattering length ' @xmath109 for the electron - electron magnetic dipole interaction by means of @xmath110 , i.e. , @xmath111 a@xmath112 with @xmath113 the mass of a hydrogen atom and a@xmath112 its bohr radius , the equation for the critical temperature becomes identical to the linearized bcs gap equation that has been studied previously in the context of @xmath53-wave superconductors by s de melo @xcite . using their result , we have for the critical temperature @xmath114 with @xmath115 euler s constant . unfortunately , the bcs transition to an anisotropic superfluid thus occurs at extremely low temperatures in this case and appears to be out of reach experimentally . for example for @xmath0li at a density of @xmath116 @xmath117 , we have @xmath118 nk and @xmath119 . the second case to consider appears to be more promising . as mentioned before , the triplet @xmath51-wave scattering length is in general too small to be able to dominate over the dipole - diple interaction . for @xmath0li it is only @xmath120 a@xmath112 , for instance . however , we can imagine that it is possible , in the same way as in the recent experiment by inouye @xcite , to ( optically ) trap a hyperfine state that has a @xmath51-wave feshbach resonance @xcite and tune the external bias magnetic field such that the @xmath51-wave scattering length becomes large and negative . the two - body t matrix is then well approximated by @xmath121 substituting this into the bcs gap equation , it is not difficult to show that it is solved by the _ ansatz _ @xmath122 where @xmath123 are the spherical components of a vector @xmath124 that is normalized as @xmath125 . furthermore , linearizing the bcs gap equation with respect to @xmath103 , we find that the critical temperature is now determined by @xmath126 the solution can be obtained by the same methods as before and reads @xmath127 it is important to realize that up to this point the direction of the vector @xmath124 is arbitrary . this reflects the rotational symmetry of the problem . however , as we have seen , the magnetic dipole - dipole interaction breaks this symmetry and will cause * d * to lie either parallel or perpendicular to the magnetic field , depending on precisely which hyperfine state @xmath47 is trapped . because we are dealing with an attractive interaction we also have to make sure that the gas is mechanically stable . this leads to the restriction that @xmath128 , or equivalently @xmath129 . unfortunately , the latter again severely limits the feasibility of experimentally observing the in principle interesting possibility of a transition to an anisotropic superfluid . we now turn our attention to the case of fermionic gases that are a mixture of two hyperfine states . in such a gas we can have @xmath53-wave collisions between atoms in different hyperfine states , which has as an advantage that interacting effects are expected to be much more important . moreover , it has an additional advantage that it is now in principle possible to use evaporative cooling to cool the gas to low temperatures . a fermionic gas consisting of three hyperfine states has recently been considered as well @xcite , but since it does not lead to any qualitative different physics we restrict ourselves here to mixtures of only two spin states . we denote the two hypefine states involved from now on by @xmath130 and @xmath131 . experimentally , there are two ways to realize such a system . we can either magnetically trap two low - field seeking states , or use an optical trap . in the latter case it is presumably necessary to load the trap by precooling the gas in a magnetic trap using sympathetic cooling . if the @xmath53-wave scattering length @xmath84 between two atoms in different spin states is positive , a first many - body effect that we have to consider is the phase separation of the gas into two phases with opposite ` magnetization ' . roughly speaking , this implies that instead of having overlapping spin densities @xmath132 and @xmath133 , the gas prefers to separate into two phases in which ( almost ) all the atoms are either in the state @xmath130 or in the state @xmath131 . the driving force behind this instability is that although the phase separation increases the kinetic energy of the gas , this increase is more than compensated by the decrease in interaction energy . more precisely , the gas is stable if the free - energy surface @xmath134 $ ] has a positive curvature in all directions . using that @xmath135 the free energy in the degenerate regime is given by @xmath136 = \int d{\bf r}~ \left\ { ( 6\pi^2)^{2/3 } \frac{3\hbar^2}{10 m } \left ( n_{\uparrow}^{5/3}({\bf r } ) + n_{\downarrow}^{5/3}({\bf r } ) \right ) + \frac{4\pi a \hbar^2}{m } n_{\uparrow}({\bf r } ) n_{\downarrow}({\bf r } ) \right\}\ ] ] and stability of the gas requires that the spin densities obey @xmath137 . in the particular case of equal spin densities this reduces to the condition that @xmath138 . note that above we have used the t - matrix approximation to evaluate the average interaction energy . because the chemical potential @xmath139 is equal to the derivative @xmath140 , we must for consistency use the same approximation to determine the renormalized chemical potentials @xmath141 . therefore , we use in the following always that @xmath142 instead of the less accurate ( only born approximation ) relation given in eq . ( [ chem ] ) . even though the interaction between the atoms is effectively repulsive , there can nevertheless occur a bcs pairing transition to a superfluid state due to the so - called luttinger - kohn instability @xcite . physically , the instability is a result of the fact that two atoms in the same hyperfine state can exchange a fluctuation ( phonon ) in the density of the other hyperfine state , leading to an effectively attractive interaction between the atoms involved . the @xmath51-wave transition associated with the luttinger - kohn effect has been studied by baranov @xcite . these authors obtain for the critical temperature the estimate @xmath143 where notably @xmath84 is the @xmath53-wave scattering length . it should , however , be kept in mind that to be able to observe the transition we must require that the gas is mechanically stable and does not phase separate . as we have already seen , this implies for equal spin densities that @xmath138 . moreover , for generic values of the scattering length we will even have that @xmath144 at the low densities of interest . it therefore seem once again practically impossible to observe the above transition in a real atomic gas . our last chance of achieving a superfluid phase in a fermionic gas thus appears to be a spin - polarized gas with a large and negative @xmath53-wave scattering length . besides the possibility of using a feshbach resonance to tailor the scattering length , we can also directly make use of the anomalously large @xmath0li triplet scattering length of @xmath145 a@xmath112 by trapping a spin - polarized @xmath0li gas in a bias magnetic field of at least @xmath146 t @xcite . substituting eq . ( [ t2b ] ) into the bcs gap equation we see that the solution @xmath79 is now independent of the wave vector @xmath100 and therefore equal to @xmath147 . furthermore , a linearization in @xmath147 gives @xmath148 which results in the critical temperature @xmath149 since the mechanical stability of the gas requires also for a negative scattering length only that @xmath150 , this expression shows that the prospects of observing a bcs transition in this case are indeed most favorable . in view of this encouraging situation we have studied in more detail the behavior of the spin density profile @xmath78 and the bcs gap parameter @xmath147 for a spin - polarized @xmath0li gas in the same magnetic trap that has been used for the bose - einstein condensation experiments with @xmath151li @xcite . the results for equal spin densities are shown in fig . [ profiles ] and lead to three important conclusions . first , we see explicitly that the density profile of a degenerate fermi gas is indeed completely ` frozen ' . clearly , even the bcs transition to a superfluid has essentially no effect on the density profile . from an experimental point of view this is somewhat unfortunate , because it implies that the appearance of a condensate of cooper pairs can not be observed by the same methods that have been so successful in the bose - einstein condensation experiments . we come back to this problem shortly . second , for a total density of @xmath152 @xmath117 the critical temperature is about @xmath153 nk . in view of the achievements with bosonic alkali gases , this appears to be a density - temperature combination which is certainly within reach experimentally . finally , we have also compared the density profile of @xmath0li with that of a noninteracting fermi gas with the same number of particles . the strong attractive interaction between the @xmath0li atoms evidently results in a substantial increase of the density in the center of the trap . experimentally , this is a favorable effect because for a fixed number of atoms in the trap it enhances the critical temperature as is shown quantitatively in fig . [ tc ] . for future convenience we mention that in a good approximation the critical temperature obeys @xmath154 with @xmath155 the thermal de broglie wavelength and @xmath156 the size of the harmonic oscillator ground state . it must be kept in mind that the latter formula can only be used for values of @xmath157 that are less then about @xmath158 , since for larger values the gas is mechanically unstable and undergoes a spinodal decomposition first . we have argued that for condensed matter physics in trapped atomic fermi gases , a spin - polarized gas with a large and negative @xmath53-wave scattering length between the atoms in the two different hyperfine states appears to be most promising . in view of the large uncertainties in the interatomic interaction potential of @xmath1k , the most likely candidate for the achievement of a gaseous bcs superfluid is at present @xmath0li . however , before successful experiments with atomic @xmath0li can be performed , some experimental problems need to be resolved . one problem is that in spin - polarized atomic @xmath0li not only the @xmath53-wave scattering length but also the exchange and dipolar decay rates are anomalously large . as a result the gas has usually a very short lifetime . to enhance the lifetime to about @xmath159 s at the densities of interest , we can either apply a bias magnetic field of about @xmath160 t or use optical methods to trap two high - field seeking states . the latter solution seems to be most practical and is actively being persued at the moment . assuming that we are able to achieve the necessary conditions for the bcs transition , the next problem that arises is the detection of the cooper pair condensate . as we have seen , the density profile shows essentially no sign of the phase transition . time - of - flight measurements , that were the ` smoking gun ' for the bose - einstein condensation experiments , are therefore not appropriate here . another possible signature that comes to mind are the frequencies of the collective modes . because of the large @xmath53-wave scattering length required for relatively high critical temperatures , the collective modes are always in the hydrodynamical regime . they are therefore described by the ( local ) conservation laws and the josephson relation . in the case of equal spin densities we thus obtain the following set of equations . the continuity equation for a superfluid is @xmath161 with the total density @xmath162 and the total current density @xmath163 consisting of a normal and superfluid contribution . in addition , newton s law gives @xmath164 where @xmath51 denotes the pressure in the gas . finally , we have also the josephson relation , which leads to ( cf . ( [ vs ] ) ) @xmath165 in principle , we must also take into account the conservation of energy . however , for a degenerate fermi gas the specific heats at constant pressure and volume are almost equal and the continuity equation for the total energy density essentially decouples from the previous ones . this has important consequences , because for density fluctuations the josephson relation just copies newton s law and we must conclude that the first sound modes are not affected by the bcs transition . of course , a measurement of second sound modes would be a clear signature of the transition , but this is presumably quite hard experimentally because second sound is primarily a temperature wave due to the fact that the energy fluctuations are almost decoupled . in analogy with sound attenuation in superconductors , it has been suggested by fetter @xcite that the damping of the first sound modes might be strongly influenced by the appearance of a cooper pair condensate , but more work is needed to make sure whether this interesting suggestion would work . a property of the gas that is certainly influenced by the bcs transition is the decay of the gas . qualitatively this can be easily understood from the correlator method devised by kagan @xcite . in this approach we can relate the decay rate constant @xmath166 for two - body decay to the rate constant in the normal phase @xmath167 by @xmath168 hence , the cooper pair condensate enhances the decay of the gas . a quantitative estimate of the effect is somewhat complicated by the fact that we can not use a pseudopotential to calculate the cooper pair wavefunction @xmath169 from our knowledge of the gap parameter @xmath147 . in any case , the increase in the two - body decay rate of the gas can only be used as a detection method if the gas is trapped in a magnetic trap , since in an optical trap the two - body decay will essentially be eliminated and the lifetime of the gas is determined by the rate at which photons scatter off the atoms in the gas . because this is a one - atom problem , it is also not affected by the bcs transition . a final detection method that we would like to mention is the scattering of a beam of @xmath0li atoms from the gas cloud . since such an experiment is quite similar to a tunneling experiment , it is directly sensitive to the existence of the gap parameter @xmath147 . therefore , a measurement of the angular distribution of the scattered atoms appears to be a promising way to get detailed information about the condensate of cooper pairs . of course , to be most sensitive we need a very cold beam . however , this is clearly not an impossible requirement , because the first results of such experiments with a condensate of @xmath170rb atoms have recently been reported @xcite . from a theoretical point of view , we are at present performing a study of the above scattering process to see how large the effects of the bcs transition are . it is interesting to note that it is also possible to detect vortices in this manner , because the atomic beam ( in contrast to a laser beam ) does not only see the core of the vortex but its complete velocity profile . this is in fact also true for a bose condensate . an experiment of this sort may therefore also be of interest in the context of ongoing research on the properties of trapped atomic bose gases . most of the work presented here has been performed in close collaboration with ian mcalexander , cass sackett , and randy hulet . we are very grateful for their continued encouragement and crucial contributions . we also thank yvan castin , eric cornell , jean dalibard , allan griffin , massimo inguscio , tony leggett , andrei ruckenstein , guglielmo tino and peter zoller for helpful discussions . we acknowledge support from the stichting fundamenteel onderzoek der materie ( fom ) , which is financially supported by the nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) . ( to be published , cond - mat/9807027 ) . . private communication . . . . . and . . ( to be published , cond - mat/9712262 ) . . private communication . . private communication .
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we present an overview of the various phase transitions that we anticipate to occur in trapped fermionic alkali gases .
we also discuss the prospects of observing these transitions in ( doubly ) spin - polarized @xmath0li and @xmath1k gases , which are now actively being studied by various experimental groups around the world .
| 9,042 | 82 |
the inner region of disk accretion onto neutron stars may be characterized by two unique radii : ( i ) the marginally stable orbit due to general gravity ( gr ) . for nonrotating neutron stars this is located at r_gr=6gmc^2=12.4m_1.4 km , where @xmath11 is the neutron star mass , and @xmath12 . for finite rotation rates , @xmath13 is somewhat smaller . the flow behavior near @xmath13 has been subjected to numerous studies , especially in the context of black hole accretion disks ( e.g. , muchotrzeb & paczyski 1982 ; matsumoto et al . 1984 ; abramowicz et al . 1988 ; narayan et al . 1997 ; chen et al . 1997 ) : close to @xmath13 the inward radial velocity of the accreting gas increases steeply with decreasing radius and becomes supersonic . the existence of such marginally stable orbit for neutron star is predicated on the fact that neutron star models constructed using different nuclear equations of state generally give a stellar radius less than @xmath13 ( arnett & bowers 1977 ; kluniak & wagoner 1985 ) . ( ii ) the magnetospheric radius , @xmath14 , below which magnetic stress dominates disk plasma stress . while the precise value of @xmath14 depends on the ( rather uncertain ) details of the magnetic field disk interactions , it is estimated to close to or slightly less than ( by a factor of a few ) the spherical alfven radius , i.e. , r_mr_ar(b_0 ^ 2r^3m)^2/7 = 18 r_10 ^ 12/7m_1.4 ^ -1/7b_8 ^ 4/7m_17 ^ -2/7 ( km ) , [ alfven]with @xmath15 ( e.g. , pringle & rees 1972 ; lamb , pethick & pines 1973 ; ghosh & lamb 1979 ; arons 1987 ) , where we have scaled various quantities to values appropriate for neutron stars in low - mass x - ray binaries ( lmxbs ) : @xmath16 km is the neutron star radius , @xmath17 g is the dipolar surface field strength , and @xmath18 is the mass accretion rate ( the eddington accretion rate is about @xmath19 g s@xmath20 ) . for highly magnetized neutron stars ( such as x - ray pulsars , typically having @xmath21 g ) , @xmath14 is much greater than @xmath13 and the stellar radius , the disk is therefore truncated near @xmath14 , within which the disk plasma becomes tied to the closed field lines and is funneled onto the magnetic poles of the star , although some plasma may continue to fall in the equatorial plane as a result of interchange instabilities ( spruit & taam 1990 ; see also arons & lea 1980 ) . for weakly magnetized neutron stars , such as those expected in lmxbs , @xmath14 and @xmath13 are comparable , and the plasma may not climb onto the field lines before reaching the stellar surface . a question therefore arises as to how the magnetic field affects the the dynamics of the inner disk and changes the sonic point . in this paper , we present an unified ( albeit phenomenological ) treatment of neutron star accretion disks under the combined influences of magnetic fields and strong gravity . our study is motivated by the recent observations using the rossi x - ray timing explorer ( rxte ) ( bradt , rothschild & swank 1993 ) which revealed kilo - hertz quasi - periodic oscillations ( qpos ) in the x - ray fluxes of at least thirteen lmxbs ( see van der klis 1997 for a review ) . these khz qpos are characterized by their high levels of coherence ( with @xmath22 up to @xmath23 ) , large rms amplitudes ( up to @xmath24 ) , and wide span of frequencies ( @xmath25 hz ) which , in most cases , are strongly correlated with the x - ray fluxes . in several sources , the x - ray power spectra show twin khz peaks moving up and down in frequency together , with the separation frequency roughly constant . moreover , in five atoll sources single qpos ( with a much higher level of coherence ) have been seen during one or more x - ray bursts , with frequencies equal to the frequency differences between the two peaks or twice that . this is a strong indication of beat phenomena ( strohmayer et al . while the origin of these qpos is uncertain , it is clear that the action must take place close to the neutron star , either in the accreting atmosphere ( klein et al . 1996 ) or in the inner disk ( strohmayer et al . 1996 ; miller , lamb and psaltis 1996 ) . a generic beat - frequency model assumes that the qpo with the higher frequency is associated with the kepler motion at some preferred orbital radius around the neutron star , while the lower - frequency qpo results from the beat between the kepler frequency and the neutron star spin frequency . it has been suggested that this preferred radius is the magnetosphere radius ( strohmayer et al . 1996 ) or the sonic radius of the disk accretion flow ( miller et al . 1996 ) . in this paper , we are not concerned with the actual mechanisms by which khz qpos in the x - ray fluxes of lmxbs may be produced ( see miller et al . 1996 and van der klis 1997 for extensive discussion on various possibilities ) . rather , our main purpose is to understand what physical effects determine the characteristics of the inner accretion disks in lmxbs . in the sonic - point model , miller et al . ( 1996 ) suggest that some accreting gas can penetrate inside the magnetosphere , whose boundary is located at a larger radius than the sonic radius . for unknown reasons , they assume that these gases are unaffected by the magnetic field once they are inside the magnetosphere and remains in a keplerian disk . they further suggest that the variation of qpo frequency results from the change in radiative forces on the accretion disk . we note , however , that the effect of radiative forces on the disk fluid ( rather than test particle orbiting the central star ) , is far from clear . calculating particle trajectories without solving for the global disk structure ( m. c. miller 1997 , private communication ) is inadequate for determining the magnitude of the radiative forces . while the radiative forces may be important for high - luminosity z - sources , their effects on the disk dynamics are expected to be be small for low - luminosity systems ( @xmath26 less than @xmath27 of @xmath28 ) . on the other hand , it is well known that millisecond pulsars have magnetic fields in the range of @xmath29 g , and one expect that neutron stars in lmxbs to have the similar range of field strengths . while the magnetic field may not be strong enough to induce a corotating magnetosphere outside the neutron star , it can nevertheless influence the dynamics of the inner disk flow by transporting away angular momentum from the disk . ideally , to properly assess the dynamical effect of magnetic fields on the accretion disks , one needs to solve for both the fluids and the fields self - consistently . this is a difficult task if not impossible . despite many decades of theoretical studies ( e.g. , pringle & rees 1972 ; lamb et al . 1973 ; ghosh & lamb 1979 ; aly 1985 , 1991 ; arons 1987 ; spruit & taam 1990 ; sturrock 1991 ; shu et al . 1994 ; lovelace et al . 1987 , 1995 ; stone & norman 1994 ; miller & stone 1997 ) , there remain considerable uncertainties on the nature of the stellar magnetic field disk interactions . particularly outstanding are the issues related to the efficiency of magnetic field dissipation in and outside the disk and whether the stellar field threads the disk in a closed configuration or it becomes open due to differential shearing between the star and the disk . it seems unlikely that some of these issues can be resolved on purely theoretical grounds . in this paper , we shall not attempt a self - consistent magnetohydrodynamics ( mhd ) calculations . rather , _ we shall adopt a phenomenological approach _ and consider rather general field configurations . we believe that such an approach is useful in bridging the gap between full mhd theories and observations . indeed , as we show in this paper , if the observed khz qpos are associated with the sonic point kepler frequency , then various systematics of khz qpos should provide useful constraints on the nature of magnetic field disk interactions as well as on the magnetic field structure in lmxbs . in 2 we introduce a model of magnetic slim accretion disk . numerical solutions are presented in 3 . because of various uncertainties in disk parameters , we shall focus on the simplest models , leaving more complete exploration to future studies . however , in 4 we introduce the notion of `` generalized marginally stable orbit '' including both the gr and magnetic effects . we derive an analytical expression for the sonic radius , which shows that the sonic point depends mainly on two parameters characterizing the disk magnetic field ( in addition to the neutron star mass ) . we show how different modes of magnetic field disk interactions can lead to different sonic - point orbital frequencies and their scalings with the field strength and mass accretion rate . section 5 concerns the equilibrium spin periods of neutron stars in our slim magnetic disk model . some applications to khz qpos in lmxbs are discussed in 6 , where we show that our phenomenological approach can be used to learn about the physics of magnetic field disk interactions . unless otherwise noted , we use geometrized units in which the speed of light and newton s gravitation constant are unity . we now consider geometrically thin axisymmetric accretion disk in steady state , taking into account of the transonic nature of the flow , and the deviation from keplerian motion in the inner region of the disk . our models generalize the usual `` slim disks '' around black holes ( e.g. , muchotrzeb & paczyski 1982 ; matsumoto et al . 1984 ; abramowicz et al . 1988 ; narayan et al . 1997 ; chen et al . 1997 ) by including the effect of magnetic fields . gr effects are included in our purely newtonian treatment by using the pseudo - newtonian potential introduced by paczyski & wiita ( 1980 ) = -mr-2 m . this potential correctly reproduces the marginally stable orbit ( where @xmath30 is the spin angular momentum ) , we have @xmath31 , and @xmath32 . for a spin frequency of @xmath33 hz ( strohmayer et al . 1996 ) , this amounts to a correction of @xmath34 to @xmath13 and @xmath27 to @xmath35 . these corrections are neglected in this paper . ] located at @xmath36 , and is adequate for this initial exploration , considering the much greater uncertainties in the magnetic field disk interactions . the self - gravity of the disk is neglected . we assume that the accreting material is confined to a thin disk , and we do not formally introduce a magnetosphere in our model . as discussed in 1 , when the field strength is sufficiently high ( as in x - ray pulsars , which typically have @xmath21 g ) , there is no question that a corotating magnetosphere exists outside the neutron star surface , located near @xmath37 ( see eq . [ [ alfven ] ] ; note that , theoretically , the magnetosphere radius is not known to within a factor of two , nor is it clear what the width of the transition zone is ) . in such a high - field regime , we shall find that the sonic point as obtained from our model is approximately equal to the usual alfven radius . although in our model the flow continues to be confined in the disk plane even inside the magnetosphere , in reality it may well behave differently ( e.g. , the plasma may `` jump '' onto the field lines and get funneled onto the magnetic poles ) . thus for high magnetic systems , the supersonic portion of our flow ( inside the sonic point ) may not be realistic . however , for low magnetic systems ( such as lmxbs ) , which is the main focus of this paper , there needs not be a genuine magnetosphere to truncate the disk flow , but the magnetic forces can still shift the sonic point to a radius larger than @xmath38 . in such low - field regimes , we expect our global flow solutions to have a wider validity . the mass continuity equation takes the form m=-2ru , [ mdot]where @xmath39 is the radial velocity of the flow ( @xmath40 for accretion ) , and @xmath41 is the surface density of the disk . the disk half - thickness since only height - integrated equations are used . ] is given by @xmath42 , where @xmath43 is the isothermal sound speed , and @xmath35 is the keplerian angular velocity ( for the pseudo - newtonian potential ) : _ k=(mr^3)^1/2rr-2 m . [ omegak]the radial momentum equation reads ududr=-1dpdr+(^2-_k^2)r + b_zb_r2|_z = h , [ uup ] where @xmath44 is the integrated disk pressure , @xmath45 is the angular velocity . the last term in eq . ( [ uup ] ) represents the dominant radial magnetic force , obtained by integrating over height the force per unit volume @xmath46 and dividing by @xmath47 . note that in eq . ( [ uup ] ) , @xmath48 is evaluated at the upper disk plane , and @xmath49 . in deriving the magnetic force , we have also neglected the @xmath50-component of the magnetic stress . the angular momentum equation reads u dldr = r , [ dl]where the second term on the right - hand - side is the magnetic torque per unit mass , obtained by integrating over height the torque per unit volume @xmath51 and dividing by @xmath47 . equation ( [ dl ] ) can be integrated in @xmath52 to give the conservation equation for angular momentum : m l_0=m l+2r^3 + m n_b(r ) , [ l0 ] where @xmath53 is the integration constant , and m n_b(r)=-_r^drr^2 b_zb_|_z = h . [ n_b]the three terms on the right - hand - side of eq . ( [ l0 ] ) correspond to advective angular momentum transport , viscous torque , and magnetic torque due the threading field lines from @xmath52 to @xmath54 , respectively . the constant @xmath53 is the eigenvalue of the problem ; it should be determined by requiring the flow to be regular at the sonic point . we shall adopt the standard @xmath55prescription for the disk kinematic viscosity , i.e. , @xmath56 ( shakura & sunyaev 1973 ) . the energy equation describing the thermal state of the flow can be written in the form : t u d sdr = e_visc+e_joule -2f_z . here @xmath57 is the specific entropy ( per unit mass ) , @xmath58 is the viscous heating rate per unit area . with the @xmath59-prescription for the disk viscosity , we have e_visc=2h1(rddr)^2 = ( rddr)^2 . the vertical ( optically thick ) radiative transport flux is f_z =- c3ddz(at^4 ) , where @xmath60 is the opacity and @xmath61 is the radiation energy density . the joule heating rate @xmath62 depends on the field dissipation in the disk , and its specific form depends on our ansatz for the magnetic field ( 2.2 ) . finally we need equations of state . for the inner disk region of interest in this paper , radiation pressure dominates over gas pressure . thus we have @xmath63 and @xmath64 . also , the opacity is dominated by thomson scattering , @xmath65 @xmath66 g . the equations above can be applied to general axisymmetric magnetic field disk configurations , as long as mass loss from possible disk wind is negligible , and the accreting material is confined to a thin disk plane . we now specify our ansatz for the magnetic fields . the vertical field component is assumed to take the form b_z = b_0(rr)^n . [ bz]we shall mostly focus on the @xmath67 case , corresponding to a central stellar dipole field threading the disk ( e.g. , ghosh & lamb 1979 ; knigl 1991 ; yi 1995 ; wang 1995 ) , although we will also consider more general values of @xmath68 , as in the cases when high - order multipoles are important ( arons 1993 ) or when field lines become open due differential shearing between the disk and the star ( aly 1985 , 1991 ; sturrock 1991 ; newman et al . 1992 ; lynden - bell & boily 1994 ; lovelace et al . in reality , the power - law relation in eq . ( [ bz ] ) is most likely to be valid only for a small range of @xmath52 , but as we shall see in 4 , the sonic point is mainly determined by the local behavior of the magnetic field . for the azimuthal component of the magnetic field , we consider two possibilities : \(i ) if the stellar magnetic field threads the accretion disk in a closed configurations ( e.g. , ghosh & lamb 1979 ) , then @xmath69 is governed by @xmath70 ( where @xmath71 is the field dissipation time ) . in steady - state , this gives @xmath72 , where @xmath73 is the rotation frequency of the star . we define a dimensionless parameter @xmath74 such that b_|_z = h=(_s-_k)b_z , [ bphi]where @xmath35 is the kepler frequency and @xmath75 is the disk orbital frequency . various ( uncertain ) estimates for the field dissipation timescale in the magnetically threaded disk configurations have been summarized in wang ( 1995 ) . \(ii ) if the magnetic field becomes open ( e.g. , lovelace et al . 1995 ) , we assume b_|_z = h =- b_z , where @xmath76 specifies the maximum twist angle of any field line connecting the star and the disk . clearly , eq . ( [ bphi ] ) encompasses the second possibility if we set @xmath77 . however , we note that the physical meaning of @xmath53 in these two cases are rather different : for closed field configurations ( i ) , @xmath78 measures the total torque on the star , while for the ( partially ) open field configurations ( ii ) , @xmath78 also include the angular momentum carried away from the disk by the magnetic fields of disk outflow . in both cases , @xmath78 is the total angular momentum transported away from the disk per unit time . similar to eq . ( [ bphi ] ) , our ansatz for the radial component @xmath79 of the disk magnetic field is b_r|_z = h=_r(-u_k r)b_z . we expect @xmath80 to be of the same order of magnitude as @xmath74 . with the particular ansatz for the magnetic fields given by eq . ( [ bphi ] ) , the joule heating rate @xmath62 can be calculated as e_joule = dz14b_zb_(r ) = 12r b_z^2(_s-)^2_k . the dissipation due to the current associated with @xmath79 is much smaller and has been neglected . it is useful to define dimensionless field strengths @xmath81 and @xmath82 via @xmath83 where @xmath84 . roughly speaking , @xmath6 is the ratio of the total magnetic torque and the characteristic accretion torque on the neutron star . comparing with eq . ( [ alfven ] ) we see that for dipole magnetic fields , @xmath85 . in all our calculations , we choose @xmath86 ; the flow structure and the sonic radius are rather insensitive to the value of @xmath82 . the radial force equation and the continuity equation can be cast in the form which reveals the existence of a sonic point : ( dudr)=a_s^2r+l^2r^3-_k^2 r + b_r^2u^2l_rr^3_k(rr)^2n-3 , [ up]where @xmath87 . the sonic point ( where @xmath88 ) is a critical point of the differential equation . the other equations can also be rewritten in the forms convenient for numerical integration . the angular momentum equation is l_0=l - r^2 c_s^2u_kddr + n_b , [ l0_2]with = -b^2l_rr(rr)^2n-3 -_s_k . [ dnb]the energy equation is = -c_s^2r 2u_k(ddr)^2 + 12b^2l_rr^2_k ( rr)^3(_s-)^2 -3c c_s_km . [ energy_2 ] the eigenvalue @xmath53 is adjusted so that the solution is regular at the sonic point . equations ( [ up])-([energy_2 ] ) are integrated inward from an outer radius ( far from the sonic point ) where the disk is approximately keplerian . note that in this outer keplerian region , eq . ( [ dnb ] ) can be integrated to give n_b(r)=b^2l_r2n-3(rr)^2n-3\ { 1-(4n-6)_s(4n-9)(m / r^3)^1/2 } , [ nbb]and the angular momentum equation yields u(r)=-3r c_s^22(l_k - l_0+n_b ) ( 1 - 2m/3r1 - 2m / r ) , where @xmath89 . the sound speed @xmath90 can be obtained from the energy equation ( [ energy_2 ] ) . since the radial velocity is small at large radius , the entropy advection term can be neglected . substituting @xmath91 from eq . ( [ l0_2 ] ) into eq . ( [ energy_2 ] ) , we find c_s(r)=m_k4c . [ cs ] equations ( [ up])-([energy_2 ] ) turn out to be a rather stiff set of equations . we have opted to adopt a further simplification by assuming the disk is isothermal . we estimate the range of @xmath90 using the thin disk expression ( [ cs ] ) evaluated near the sonic radius . the physical rationale behind this approximation is that the transition from keplerian disk to supersonic flow is very sharp , and we do not expect the sound speed to change significantly in this transition region ( near the sonic point ) . as we shall see in 4 , the sonic radius is insensitive to the thermal state of the disk when the sound speed is small . we have not studied the general dependence of the sonic radius on the thermal state of the disk . however , considering the very large uncertainties in the disk magnetic fields , the isothermal approximation should be adequate for use in the first step in our investigation . figure 1 depicts two examples of transonic accretion flows , with closed dipolar stellar fields threading the disks ( @xmath67 ) . we choose a standard set of parameters : @xmath92 , @xmath93 , @xmath94 ( a typical value for a @xmath95 , @xmath96 km neutron star ) , and @xmath97 ( corresponding to stellar spin frequency of @xmath98 hz ) . as expected , the magnetic fields slow down the tangential flow velocity , and , together with the strong relativistic gravity , make the radial velocity supersonic at small radii . for small @xmath6 , the deviation of @xmath45 from @xmath35 is small ; for larger @xmath6 , the sonic radius @xmath99 larger , and @xmath45 gradually approaches @xmath73 inside the sonic point . in these examples , the sonic points are located at @xmath100 ( for @xmath101 ) and @xmath102 ( for @xmath103 ) , the corresponding eigenvalues ( @xmath53 ) are @xmath104 and @xmath105 , respectively . in figure 2 we show the sonic radius @xmath99 and the corresponding specific angular momentum @xmath53 as a function of @xmath6 for several different values of @xmath59 and @xmath90 . we assume @xmath94 and @xmath106 for these models . the following trends have been found : for a given @xmath6 , a larger @xmath59 tends to make @xmath99 larger ( i.e. , viscosity tends to `` destablize '' the disk ) , while a larger @xmath90 tends to make @xmath99 smaller ( i.e. , pressure `` stablizes '' the disk ) . however , we emphasize that dependences of @xmath99 on these disk parameters ( @xmath107 ) are rather weak . moreover , as @xmath90 decreases , the sonic @xmath99 converges to a value independent of @xmath59 and @xmath90 the reason for this convergence will become clear in 4 . as our numerical results in 3 indicate , in the limit of small disk pressure and viscosity , the sonic radius approaches a value independent of the disk parameters ( @xmath108 and the equation of state ) . asymptotic sonic radius _ can be derived analytically using a simple mechanical model : consider the equation of motion of a test mass around a neutron star = l^2r^3-_k^2 r , l_0=l+n_b(r ) . [ testeq]imagine that the test mass is `` attached '' to a magnetic field line so that its orbital angular momentum @xmath109 is not conserved by itself . the conserved angular momentum @xmath53 includes the contributions from both the orbital motion and the angular momentum @xmath110 [ cf . ( [ n_b ] ) ] carried by the threading magnetic field . equation ( [ testeq ] ) can be obtained by setting the pressure and viscosity to zero in our general slim disk equations ( 2.1 ) . the radial component of the magnetic force has been neglected . an equilibrium orbit is determined by the condition l_0-n_b=_k^2 r = l_k(r ) . deviation @xmath111 from the equilibrium is governed by the perturbation equation of the form + _ eff^2r=0 , where the the effective epicyclic frequency @xmath112 is given by _ eff^2=2_krddr(l_k+n_b)= m(r-6m)r(r-2m)^3 - 2b^2l_r_kr^2 ( rr)^2n-3(1-_s_k ) , [ kappa2](recall that @xmath113 and @xmath114 is the neutron star radius ) . setting @xmath115 , we obtain a critical orbit , which we dub the _ `` generalized marginal stable orbit '' _ , located at @xmath116 . clearly , @xmath117 depends only on the gravitational potential and the local magnetic torque @xmath118 . the magnetic field enters only through the dimensionless ratio @xmath81 . the corresponding constant eigenvalue @xmath119 , however , depends on the global field structure . in figure 2 we plot @xmath117 and @xmath53 against @xmath6 for @xmath106 ( corresponding to spin frequency of @xmath98 hz ) and @xmath94 . we see that @xmath117 is the upper limit to the numerically determined @xmath99 , i.e. , _ @xmath117 is the asymptotic sonic radius as the disk viscosity and pressure diminish_. it is of interest to consider two limiting cases : ( i ) in the newtonian limit ( neglecting the gr effect ) , or equivalently when @xmath6 is large ( so that @xmath120 ) , we have = ^2/(4n-5 ) . [ largeb]for @xmath67 , this is the standard result for the inner radius of the keplerian disk , as determined by @xmath121 ( e.g. , arons 1993 ; wang 1995 ) ; ( ii ) in the limit of small @xmath6 ( so that @xmath117 is close to @xmath38 ) , we find = 6 + 16b^23(rm)^(4n-5)/2 . [ smallb]this gives the correction to the standard general - relativity - induced mso located at @xmath36 . the consideration of the limiting cases clearly indicates the sonic point ( or the generalized mso ) constructed in our slim disk model includes the essential physics embodied in the determination of the usual magnetosphere radius . as we argued at the beginning of 2 , for high magnetic systems , a genuine magnetosphere should certainly exist outside the neutron star . for low magnetic systems , however , the accretion flow may well be confined in the disk , and the distinction between the sonic point and the magnetosphere boundary may not exist . we emphasize that our analysis given here is phenomenological . it only takes account of the dynamics of disk under a fixed magnetic field configuration , while a full mhd treatment should include perturbations of both disk fluid and magnetic fields . the usefulness of our analytical result lies in the fact that @xmath117 provides a good approximation to the sonic radius ; and the sonic point is induced by both the gr effect and the magnetic effect . we note that in the presence of disk viscosity and magnetic fields , a fluid element continuously falls inward by the viscous stress and the magnetic stress , and therefore the concept of `` marginally stable orbit '' does not strictly apply . nevertheless , we use the term `` generalized mso '' to refer to the asymptotic sonic radius as determined by our analytical expressions . figure 3 depicts the orbital frequency , rather than the approximate pseudo - newtonian eq . ( [ omegak ] ) . ] at the generalized mso ( approximately the sonic radius ) as a function of the mass accretion rate @xmath3 for several different combinations of model parameters ( @xmath68 and @xmath73 ) . note that once we specify @xmath68 and @xmath73 in units of @xmath122 , the numerical value of @xmath123 depends on the other parameters only through the combination @xmath124 ( see eqs . [ [ kappa2]]-[[smallb ] ] ) . we have therefore used x = m_1.4 ^ 2n-2r_10 ^ -2nm_17b_7 ^ 2 = 0.07317b^2(m_1.4r_10)^(4n-5)/2 [ xlabel]as the @xmath125-variable in fig . 3 ( where @xmath126 is the surface field @xmath2 in units of @xmath127 g ) . for large @xmath128 ( or small @xmath6 ) , @xmath117 approaches @xmath36 and @xmath129 approaches @xmath130 khz . the scaling of @xmath131 with @xmath3 depends mainly on @xmath68 , the index which specifies the shape of the magnetic field lines ( see eq . [ [ bz ] ] ) . note that the sonic point converges to the mso only in the limit of small @xmath90 . thus in reality , the dependence of the sonic - point kepler frequency @xmath132 on @xmath3 may be different if @xmath90 is not small . for example , @xmath133 for radiation - dominated optically - thick disk , and therefore the dependence of @xmath132 on @xmath3 is slightly steeper than what is shown in fig . 3 . we shall discuss some of the applications of fig . 3 to qpos in lmxbs in 6 . as discussed before ( 2.2 ) , for closed field configurations , the quantity @xmath78 measures the total torque on the neutron star due to the accreting matter and the threading magnetic fields . it is of interest to consider how the equilibrium stellar rotation rate @xmath134 , at which @xmath135 , is determined in the slim disk model . we shall restrict to the dipole fields ( @xmath67 ) in this section . first consider the result when the gr effect is neglected . in this case the keplerian disk boundary @xmath37 is determined by the condition @xmath136 , or @xmath137 , where @xmath138 . we find = ^2/7 , [ ghosh](cf . [ [ largeb ] ] ) . the total torque on the star is given by n_tot = m l_0=m l_k(r_m)76 . [ torquen]equilibrium is obtained for @xmath139 . using eq . ( [ ghosh ] ) we then find @xmath140 and @xmath141 , which gives _ s , eq=3.44m_1.4 ^ 1/2r_10 ^ -3/2b^-6/7(khz ) = 1.467m_1.4 ^ 5/7r_10 ^ -18/7m_17 ^ 3/7 ( b_8 ^ 2)^-3/7(khz ) . [ omegeq]this is the standard result that the equilibrium spin frequency is equal to the keplerian frequency at the alfven radius . this result is plotted in fig . 4 . in the slim magnetic disk model , the total torque on the neutron star @xmath78 is obtained from the eigenvalue @xmath53 . in the asymptotic regime discussed in 4 , @xmath53 can be determined from the analytical expressions : @xmath99 is obtained from the condition @xmath142 in eq . ( [ kappa2 ] ) , and then @xmath143 with @xmath110 given by eq . ( [ nbb ] ) ( specialized to @xmath67 ) . it is straightforward to show that in the limit of @xmath144 , we recover the results given in eqs . ( [ ghosh]-[omegeq ] ) . the equilibrium @xmath134 is obtained by requiring @xmath135 . in fig . 4 we plot the equilibrium spin frequency @xmath145 as a function of @xmath6 . clearly , for large @xmath6 , our calculation agrees with the usual nonrelativistic result . relativistic corrections are significant only when @xmath6 is small , for which the sonic point lies close to the neutron star . we note that the discussion in this section is valid only if the magnetic fields are closed throughout the disk . only in these cases does @xmath53 measure the net torque on the neutron star . thus our results in this section are much more restrictive compared to the location of the sonic point ( which depends only on the local field structure ) discussed in 3 - 4 . we have presented in 2 - 4 an unified treatment of the inner region of accretion flow under the combined influences of general relativistic gravity and stellar magnetic fields in lmxbs . we have shown that even relatively weak magnetic fields ( @xmath0 g ) can slow down the orbital motion in the inner disk by taking away angular momentum from the disk , thereby changing the position of the sonic point significantly ( cf . figs . 2 - 3 ) . while the mechanisms responsible for the khz qpos in lmxbs are still uncertain , it is tempting to associate them with orbital motions at a certain preferred radius . if this is the case , then the keplerian frequency at inner radius of the disk , or more precisely the sonic radius is certainly a natural choice ( see paczyski 1987 for an earlier suggestion on the importance of the disk sonic point ) . one can envisage a number of different mechanisms that will lead to qpos at @xmath146 and the beat frequency with the stellar spin ( miller et al . 1996 ; van der klis 1997 ) . the following discussion is based on the hypothesis that the khz qpo frequency corresponds to @xmath131 or the sonic - point kepler frequency . we note that although our treatment of the magnetic field effects in 2 - 4 is rather general ( albeit based on a phenomenological prescription ) , other physical effects might be at work ( such as radiation forces for high - luminosity systems ; see 1 ) . as a result , some of our conclusions below should be considered tentative . \(a ) _ range of qpo frequencies and constraint on the magnetic field strength _ : as emphasized by van der klis ( 1997 ) , similar qpo frequencies ( @xmath25 hz ) have been observed in sources with widely different average luminosities @xmath147 ( from a few times @xmath148 to near @xmath28 , corresponding to @xmath149 between a few times @xmath150 g s@xmath20 to @xmath19 g s@xmath20 ) , while for an individual source @xmath151 often correlates strongly with the x - ray flux . this peculiar lack of correlation between the qpo frequency and @xmath147 can be explained in our model , as long as the star s magnetic field strength correlates with its @xmath149 in such a way as to leave @xmath152 in a certain range ( see fig . 3 ) . for example , if we use the model with @xmath67 ( dipole field ) and @xmath153 hz , then to produce @xmath154 in the range of @xmath25 hz would require @xmath155 ; on the other hand , if we use the model with @xmath156 and @xmath157 ( open field configuration ) , we would require @xmath158 ( we have adopted @xmath159 in these examples ) . indeed , such correlation between the magnetic field strength and the mean mass accretion rate among lmxbs has been suggested independently on the basis of z and atoll source phenomenology ( hasinger & van der klis 1989 ) , although the origin for this correlation is still unclear . we note , however , that the correlation needs not be very strong , considering the wide range of other controlling parameters such as @xmath68 ( the shape of the magnetic field ) and @xmath160 ( stellar rotation ) ( see fig . 3 ) . while there is a natural upper limit to @xmath131 ( corresponding to @xmath161 ; but see ( c ) below ) , the existence of a ( source - dependent ) lower limit to the observed qpo frequency needs an explanation . if we rely on inner disk accretion to explain these qpos ( e.g. , van der klis 1997 ) , then one possibility is that for large @xmath117 ( small @xmath131 ) , the accreting gas can be channeled out of the disk plane by the magnetic field this must happen for sufficiently small @xmath3 ( or sufficiently large @xmath162 , as in the case of accreting x - ray pulsars ) . the precise location where the plasma leaves the disk depends on the near - zone field structure , and is clearly source - dependent . the observed qpo frequencies may already be used to probe the magnetic fields in lmxbs and the nature of the magnetic field disk interactions . as an example , consider the atoll source 4u 0614 + 091 which has extremely small luminosity ( see ford et al . 1997 and references therein ) : to obtain @xmath163 hz requires @xmath164 ( this constraint depends somewhat on the field structure and @xmath160 ; see fig . 3 ) . at @xmath165 , this translates to @xmath166 a stronger magnetic field would push the sonic point ( or the generalized mso ) to a larger radius . we are left with two possibilities : ( i ) if @xmath167 , we would require @xmath168 , i.e. , the dissipation of toroidal fields near the disk must be very efficient ; ( ii ) if @xmath76 ( as expected for open field configurations ) , then we would require @xmath169 . indeed , if the khz qpo sources represent a fair sample of lmxbs , then we might conclude that the magnetic fields in lmxbs are systematically weaker than those in millisecond pulsars . this may indicate that the magnetic field of a neutron star is `` buried '' during the lmxb phase ( e.g. , romani 1990 ; urpin & geppert 1995 ; konar & bhattacharya 1997 ; brown & bildsten 1998 ) , and later regenerates or re - emerges as accretion stops . \(b ) _ scaling of @xmath151 with @xmath3 _ : for most sources , it was found that the khz qpo frequency strongly correlates with the xte count rate ( @xmath170 kev ) , with power - law index greater than unity . however , the scaling relation between the count rate and @xmath3 is not well established , and it has been suggested the flux of the black - body component is a better indicator of qpo frequency ( ford et al . as discussed in 3 , @xmath131 depends primarily on the magnetic field structure near the sonic point , particularly on the `` field shape '' index @xmath68 . if the scaling of @xmath151 with @xmath3 can be established observationally , it may be possible to distinguish a closed field configuration from an open one . for example , if we believe the scaling @xmath171 with @xmath172 , then we may conclude that @xmath173 ( see eq . [ [ largeb ] ] and the discussion following eq . [ [ xlabel ] ] ) , which indicates that magnetic fields in lmxbs do not have dipolar shape , but rather have complex topology ( see arons 1993 for discussion on related issues ) . \(c ) _ the maximum value of @xmath151 _ : it has been suggested ( zhang et al . 1997 ) based on the narrow range of the maximal qpo frequencies ( @xmath174 hz ) in at least six sources that these maximum frequencies correspond to the kepler frequency at @xmath36 , which then implies that the neutron star masses are near @xmath175 ( see also kaaret et al . while we agree that this conclusion seems most natural , we nevertheless add the following cautionary notes : ( i ) the inferred large stellar masses may be problematic : all neutron stars with well - determined masses ( including a few that certainly had accreted mass , although not necessarily in the same accretion mode as in lmxbs ) have masses consistent with being @xmath176 ( e.g. , van kerkwijk et al . in particular , the @xmath177 ms recycled pulsar b1855 + 09 , which is thought to have gone through a lmxb phase ( phinney & kulkarni 1994 ) , has a mass @xmath178 ( kaspi et al . . moreover , accretion of @xmath179 might have spun up the neutron stars to near break - up ( see cook et al . 1994 for calculations of spin - up tracks in the nonmagnetic case ) , in contrary to the observed spin rates ( @xmath180 hz ) . ( ii ) if the maximum @xmath151 is indeed @xmath181 , then the correlation between @xmath151 and @xmath3 should weaken as @xmath3 increases , and eventually @xmath151 should approaches a constant independent of @xmath3 ( see fig . 3 ) . this has not been observed . therefore in our opinion it is premature to identify the maximal @xmath151 with the kepler frequency at @xmath38 . an alternative is that as @xmath151 approaches @xmath174 hz , the rms qpo amplitude decreases as the observations have indicated , making it difficult to detect higher qpo frequencies . \(d ) _ horizontal - branch oscillations ( hbos ) in z - sources _ : in several z - sources ( e.g. , sco x-1 , gx 5 - 1 and gx 17 + 2 ) , hbos with frequencies @xmath182 hz have been detected _ simultaneously _ with the khz qpos ( see van der klis 1997 ) . one standard interpretation of hbos is that they are associated with the beat between the kepler frequency at the magnetosphere boundary and the neutron star spin ( alpar & shaham 1985 ) . since the spin frequencies @xmath160 of these sources ( as determined from the difference in the twin khz qpo frequencies ) lie around @xmath33 hz ( see white & zhang 1997 ) , the putative magnetosphere boundary must be located at a large radius where @xmath183 hz . as we have shown in this paper , such a strong magnetic field must necessarily push the ( generalized ) disk sonic point to a large radius where the kepler frequency drops below the kilo - hertz range . ( recall that for low field systems such as lmxbs , the distinction between the sonic point and the magnetosphere boundary probably does not exist , and the two separate radii are replaced by a single generalized sonic point [ see 3 - 4 ] . ) therefore if the khz qpos are associated with the sonic - point kepler frequency , then the magnetospheric beat frequency model for hbos can not work , and the origin of hbos must lie elsewhere . alternative models for hbos have been discussed by biehle & blandford ( 1993 ) and stella & vietri ( 1997 ) . the author thanks zhiyun li , rob nelson and brian vaughan for valuable discussions and lars bildsten for comment . he also thanks the referee for constructive comments which improved the presentation of the paper . this research is supported by a richard chace tolman fellowship at caltech , nasa grant nag 5 - 2756 , and nsf grant ast-9417371 .
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the inner regions of accretion disks of weakly magnetized neutron stars are affected by general relativistic gravity and stellar magnetic fields . even for field strengths
sufficiently small so that there is no well - defined magnetosphere surrounding the neutron star , there is still a region in the disk where magnetic field stress plays an important dynamical role .
we construct magnetic slim disk models appropriate for neutron stars in low - mass x - ray binaries ( lmxbs ) which incorporate the effects of both magnetic fields and general relativity ( gr ) . the magnetic field
disk interaction is treated in a phenomenological manner , allowing for both closed and open field configurations .
we show that even for surface magnetic fields as weak as @xmath0 g , the sonic point of the accretion flow can be significantly modified from the pure gr value ( near @xmath1 for slowly - rotating neutron stars ) .
we derive an analytical expression for the sonic radius in the limit of small disk viscosity and pressure .
we show that the sonic radius mainly depends on the stellar surface field strength @xmath2 and mass accretion rate @xmath3 through the ratio @xmath4 , where @xmath5 measures the azimuthal pitch angle of the magnetic field threading the disk .
the sonic radius thus obtained approaches the usual alfven radius for high @xmath6 ( for which a genuine magnetosphere is expected to form ) , and asymptotes to @xmath7 as @xmath8 .
we therefore suggest that for neutron stars in lmxbs , the distinction between the disk sonic radius and the magnetosphere radius may not exist ; there is only one `` generalized '' sonic radius which is determined by both the gr effect and the magnetic effect .
we apply our theoretical results to the khz quasi - periodic oscillations ( qpos ) observed in the x - ray fluxes of lmxbs .
if these qpos are associated with the orbital frequency at the inner radius of the disk , then the qpo frequencies and their correlation with mass accretion rate can provide useful diagnostics on the ( highly uncertain ) nature of the magnetic field
disk interactions .
in particular , a tight upper limit to the surface magnetic field @xmath2 can be obtained , i.e. , @xmath9 g , where @xmath10 , in order to produce khz orbital frequency at the sonic radius .
current observational data may suggest that the magnetic fields in lmxbs have complex topology .
| 12,679 | 631 |
for a long time , the phenomenological approach dominated in description of multiplicity distributions in multiparticle production . the very first attempts to apply qcd formalism to the problem failed because in the simplest double - logarithmic approximation it predicts an extremely wide shape of the distribution that contradicts to experimental data . only recently it became possible to get exact solutions of qcd equations which revealed much narrower shapes and such a novel feature of cumulant moments as their oscillations at higher ranks . these moments are extremely sensitive to the tiny details of the distribution . surprisingly enough , those qcd predictions for parton distributions have been supported by experimental data for hadrons . qcd is also successful in qualitative description of evolution of these distributions with decreasing phase space bins which gives rise to notions of intermittency and fractality . however , there are some new problems with locations of the minimum of cumulants at small bins . the experimentally defined truncated generating functions possess an intriguing pattern of zeros in the complex plane of an auxiliary variable . it recalls the pattern of lee - yang zeros of the grand canonical partition function in the complex fugacity plane related to phase transition . before demonstrating all these peculiarities let us define the multiplicity distribution @xmath1 where @xmath2 is the cross section of @xmath3-particle production processes , and the generating function @xmath4 the ( normalized ) factorial and cumulant moments of the @xmath5 distribution are @xmath6 @xmath7 where @xmath8 is the average multiplicity . they describe full and genuine @xmath9-particle correlations , correspondingly . let us point out here that the moments are defined by the derivatives at the origin and are very sensitive to any nearby singularity of the generating function . in practice , one deals with distribution truncated due to finiteness of the available phase space and the summation in all formulae above is cut off at some finite value of @xmath10 which depends on the phase space region chosen , and increases with its increase . it is a polynomial of the power @xmath11 and has @xmath11 zeros in the complex @xmath12-plane . to shorten the presentation , i omit here all the details of calculations and references to original papers . the reader can find them in my review paper in physics - uspekhi * 37 * ( 1994 ) 715 . main qualitative results are described and demonstrated in figures in the subsequent three sections . their physics implications are discussed in the last section . first , let us consider qcd without quarks , i.e. gluodynamics . the generating function of the gluon multiplicity distribution in the full phase - space volume satisfies the equation @xmath13 . \label{8}\ ] ] here @xmath14 is the initial momentum , @xmath15 is the angular width of the gluon jet considered , @xmath16 where @xmath17 is the jet virtuality , @xmath18const , @xmath19 @xmath20 is the running coupling constant , and the kernel of the equation is @xmath21 . \label{10}\ ] ] it is the non - linear integro - differential equation with shifted arguments in the non - linear part which take into account the conservation laws , and with the initial condition @xmath22 and the normalization @xmath23 the condition ( [ 12 ] ) normalizes the total probability to 1 , and the condition ( [ 11 ] ) declares that there is a single particle at the very initial stage . after taylor series expansion at large enough @xmath24 and differentiation in eq . ( [ 8 ] ) , one gets the differential equation @xmath25 , \label{14}\ ] ] where @xmath26 , and higher order terms have been omitted . leaving two terms on the right - hand side , one gets the well - known equation of the double - logarithmic approximation which takes into account the most singular components . the next term , with @xmath27 , corresponds to the modified leading - logarithm approximation , and the term with @xmath28 deals with next - to - leading corrections . the straightforward solution of this equation looks very problematic . however , it is very simple for the moments of the distribution because @xmath29 and @xmath30 are the generating functions of @xmath31 and @xmath32 , correspondingly , according to ( [ 3 ] ) , ( [ 4 ] ) . using this fact , one gets the solution which looks like @xmath33}{q^2 \gamma ^2 + q\gamma ^{\prime } } , \label{13}\ ] ] where the anomalous dimension @xmath34 is related to @xmath35 by @xmath36 the formula ( [ 13 ] ) shows how the ratio @xmath37 behaves in different approximations . in double - log approximation when @xmath38 , it monotonously decreases as @xmath39 that corresponds to the negative binomial law with its parameter @xmath40 i.e. to very wide distribution . in modified - log approximation ( @xmath41 ) it acquires a negative minimum at @xmath42 and approaches asymptotically at large ranks @xmath9 the abscissa axis from below . in the next approximation given by ( [ 13 ] ) it preserves the minimum location but approaches a positive constant crossing the abscissa axis . in ever higher orders it reveals the quasi - oscillatory behavior about this axis . this prediction of the minimum at @xmath43 and subsequent specific oscillations is the main theoretical outcome . it is interesting to note that the equation ( [ 8 ] ) can be solved exactly in the case of fixed coupling constant . all the above qualitative features are noticeable here as well . while the above results are valid for gluon distributions in gluon jets ( and pertain to qcd with quarks taken into account ) , the similar qualitative features characterize the multiplicity distributions of hadrons in high energy reactions initiated by various particles . as an example , i show in fig.1 the ratio @xmath37 as a function of @xmath9 in the @xmath44 data of delphi collaboration at 91 gev , where the oscillations and the location of minima are of a special interest . the multiplicity distributions can be measured not only in the total phase space ( as has been discussed above for very large phase - space volumes ) but in any part of it . for the homogeneous distribution of particles within the volume , the average multiplicity is proportional to the volume and decreases for small volumes but the fluctuations increase . the most interesting problem here is the law governing the growth of fluctuations and its possible departure from a purely statistical behavior related to the decrease of the average multiplicity . such a variation has to be connected with the dynamics of the interactions . in particular , it has been proposed to look for the power - law behavior of the factorial moments for small rapidity intervals @xmath45 @xmath46 inspired by the idea of intermittency in turbulence . in the case of statistical fluctuations with purely poissonian behavior , the intermittency indices @xmath47 are identically equal to zero . experimental data on various processes in a wide energy range support this idea ( e.g. , see fig.2 ) , and qcd provides a good basis for its explanation as a result of parton showers . the generating function technique is not applicable here , and one should consider the feynman graphs of evolution of a jet with its subjet hitting the phase - space window under consideration ( see the abovementioned review ) . at moderately small rapidity windows , one can get in the double - log approximation the power - law behavior with @xmath48 the running property of qcd coupling constant is not important in that region . this property becomes noticeable at ever smaller windows when ( e.g. , at @xmath9=2 ) @xmath49 , and leads to smaller numerical values of @xmath50 compared to ( [ 16 ] ) . the general trends in this region decline somewhat from the simple power law ( [ 15 ] ) due to logarithmic corrections . qualitatively , these predictions correspond to experimental findings at relatively small ranks @xmath9 where the steep increase in the region of @xmath51 on the log - log plot of the dependence ( [ 15 ] ) is replaced by slower one at smaller intervals @xmath45 ( see fig.2 ) . the transition point between the two regimes depends on the rank in qualitative agreement with qcd predictions also . namely , the transition happens at smaller bins for higher ranks . these findings can be interpreted as an indication on fractal structure of particle distributions within the available phase space . when interpreted in terms of fluctuations , they show that the fluctuations become stronger in small phase - space regions in a definite power - like manner and , surely , exceed trivial statistical fluctuations . let us turn now to the @xmath9-behavior of moments at small bins . the phenomenon of the oscillations of cumulants discussed above reveals itself here as well if one goes beyond the double - log approximation of ( [ 16 ] ) . in terms of factorial moments , it means the non - monotonous behavior of the intermittency indices as functions of @xmath9 . ( compare it to the steady increase with @xmath9 at @xmath52 given by ( [ 16 ] ) . ) it gives rise to the negative values of @xmath32 and @xmath37 . the fate of the first minimum can be easily guessed from the formula ( [ q ] ) . for large enough virtualities ( i.e. small @xmath53 ) , the minimum location moves to higher values of rank @xmath9 for jets with larger virtuality @xmath17 since the qcd coupling constant is running as @xmath54 . therefore , the predicted shift of the minimum is @xmath55 it follows that @xmath56 moves to higher ranks at higher energies because more massive jets become available . another corollary is that it should shift to smaller values of @xmath9 for smaller bins at fixed energy . while former statement finds some support in experiment , the second one does not look to be true as shown in fig.3 . on the contrary , the minimum appears at higher ranks for smaller bins . there is no solution of this problem yet but it should be ascribed to the higher - order effects . actually , one can guess that the higher order terms shown as @xmath57 in ( [ q ] ) become so important at small bins that they overpower the weak @xmath17-dependence of @xmath58 in the first term of ( [ q ] ) . it is important to stress here that at large rapidity intervals the modified leading - log term with @xmath28 does not influence the value of @xmath56 , and only increases the value of @xmath37 by @xmath59 . thus , the next- to - leading corrections should be in charge of the additional shift of @xmath56 , and , therefore , small bins help us look into higher orders of qcd . there is another fascinating feature of multiplicity distributions it happens that zeros of the truncated generating function form a spectacular pattern in the complex plane of the variable @xmath12 . namely , they seem to lie close to a single circle . at enlarged values of @xmath11 they move closer to the real axis pinching it at some positive value of @xmath12 . it is demonstrated in fig.4 for ua5 data on @xmath60 interactions at 200 and 900 gev for various rapidity windows . no qcd interpretation of the fact exists because it is hard to exploit the finite cut - off in analytic calculations . the interest to it stems from the analogy to the locations of zeros of the grand canonical partition function as described by lee and yang who related them to possible phase transitions in statistical mechanics . in that case , @xmath12 variable plays the role of fugacity , and pinching of the real axis implies existence of two phases in the system considered . in particle physics , it shows up the location of the singularity of the generating function i.e. the number of zeros of truncated generating functions increases and they tend to move to the singularity point when @xmath61 . since it happens to lie close to the origin , it drastically influences the behavior of moments ( see ( [ 3 ] ) , ( [ 4 ] ) ) , and , therefore , determines the shape of the distribution . the study of the singularities is at the very early stage now , and one can only say that the singularity is positioned closer to the origin in nucleus - nucleus collisions and it is farthest in @xmath44 that appeals to our intuitive guess . let us discuss the implications of the above findings . the qcd prediction of quasioscillating behavior of cumulant moments of the multiplicity distributions reveals the tiny features which were overlooked by simpleminded fits of the negative binomial distribution . the truncated negative binomial distribution has the oscillating cumulants as well but the oscillations should die out asymptotically while they persist in qcd . it demonstrates that the qcd distribution belongs to the class of non - infinitely - divisible ones and shows that the poissonian cluster models are ruled out by qcd since they lead to positive values of cumulants and can not reproduce their oscillations at asymptotically high energies . moreover , this prediction is closely related to the existence of a new expansion parameter in describing multiparticle production in qcd . in terms of moments , this parameter is equal to the product of the moment s rank to the anomalous dimension of qcd @xmath62 . let us recall that the similar parameter appears when calculating the feynman tree graphs and looks like a product of the number of final particles to qcd coupling constant @xmath63 . both parameters become large if large number of particles is involved . the invalidity of simple perturbative approach in multiple production processes would ask for more convenient basis ( compared to particle number representation ) to be used . as one of possible examples of effectiveness of such an approach , we would mention lasers where coherent state basis is more suitable . that is why studies of coherent and squeezed states as well as statistical approach and search for collective effects , in general , are welcome . the recent findings of zeros of the truncated generating functions point out in that direction , and provoke speculations to their possible relation to the problem of phase transition in hadronic matter . the success of qcd in predicting the qualitative features of moments of multiplicity distributions both for large phase - space regions ( oscillation of cumulants ) and for small bins ( intermittency indices , fractality ) seems even more surprising if one recognizes that they are derived for parton distributions while hadrons are observed in experiment . nevertheless , these qualitative features reveal themselves in hadron distributions as well . why such a local parton - hadron duality persists even at higher moments is an open question . another problem appears in such a tiny property as the evolution of the minimum location at smaller bins . its solution can show a way to take into account higher order effects of qcd more properly . to conclude , we would like to stress that recent theoretical studies of qcd multiplicity distributions predict rather exotic features of the moments of the distributions which find some support in experiment and , at the same time , provoke new problems to be solved . fig.1 the ratio of cumulant to factorial moments @xmath37 as the function of the rank @xmath9 . the delphi - data on @xmath44 at 91 gev in the total phase space are shown by dots . the dashed line shows the fit by the negative binomial distribution with the parameters given by delphi . the solid line is drawn just to guide the eye . fig.3 behavior of the ratio of cumulant to factorial moments @xmath37 as a function of the rank @xmath9 for some single hemisphere multiplicity distributions from the delphi collaboration : ( a ) full hemisphere ; ( b ) rapidity window @xmath66 ; ( c ) rapidity window @xmath67 . the minimum shifts to higher @xmath9 for smaller bins . fig.4 the locations of zeros of the truncated generating function for ua5-data on @xmath60 interactions at 200 and 900 gev ( c.m.s . ) in various rapidity windows . the upper halfplane of @xmath68 is shown only because of the up - down symmetry . ( for further experimental information see the talk of g. gianini at multiparticle dynamics-95 ) .
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the solution of qcd equations for generating functions of multiplicity distributions reveals new peculiar features of cumulant moments oscillating as functions of their rank .
this prediction is supported by experimental data on @xmath0 collisions .
evolution of the moments at smaller phase space bins leads to intermittency and fractality .
the experimentally defined truncated generating functions possess zeros in the complex plane of an auxiliary variable recalling lee - yang zeros in statistical mechanics . * novel features of multiplicity distributions in qcd and experiment * i.m.dremin lebedev physical institute , moscow 117924 , russia contents + \1 .
introduction + 2 .
_ oscillations _ of cumulants of multiplicity distributions in qcd + 3 .
_ evolution _ of distributions with decreasing phase space volume + intermittency and fractality + 4 .
_ zeros _ of truncated generating functions + 5 .
discussion and conclusions +
| 4,261 | 248 |
the emergence of non trivial collective behaviour in multidimensional systems has been analized in the last years by many authors @xcite @xcite @xcite . those important class of systems are the ones that present global interactions . a basic model extensively analized by kaneko is an unidimensional array of @xmath0 elements : @xmath1 where @xmath2 , is an index identifying the elements of the array , @xmath3 a temporal discret variable , @xmath4 is the coupling parameter and @xmath5 describes the local dynamic and taken as the logistic map . in this work , we consider @xmath5 as a cubic map given by : @xmath6 where @xmath7 $ ] is a control parameter and @xmath8 $ ] . the map dynamic has been extensively studied by testa et.al.@xcite , and many applications come up from artificial neural networks where the cubic map , as local dynamic , is taken into account for modelizing an associative memory system . @xcite proposed a gcm model to modelize this system optimazing the hopfield s model . the subarmonic cascade , showed on fig-[fig:2 ] prove the coexistence of two equal volume stable attractors . the later is verified even as the gcm given by eq.[eq : sist ] has @xmath9 . janosi et . @xcite studied a globally coupled multiattractor quartic map with different volume basin attractors , which is as simple second iterate of the map proposed by kaneko , emphazasing their analysis on the control parameter of the local dynamic . they showed that for these systems the mean field dynamic is controlled by the number of elements in the initial partition of each basin of attraction . this behaviour is also present in the map used in this work . in order to study the coherent - ordered phase transition of the kaneko s gcm model , cerdeira et . @xcite analized the mechanism of the on - off intermitency appearing in the onset of this transition . since the cubic map is characterized by a dynamic with multiple attractors , the first step to determine the differences with the well known cuadratic map given by kaneko is to obtain the phase diagram of eq.[eq : sist ] and to study the the coherent - ordered dynamical transition for a fixed value of the control parameter @xmath10 . the later is done near an internal crisis of the cubic map , as a function of the number of elements @xmath11 with initial conditions in one basin and the values of the coupling parameter @xmath4 , setting @xmath0 equal to 256 . after that , the existence of an inverse period doubling bifurcation as function of @xmath4 and @xmath11 is analized . the dynamical analysis process breaks the phase space in sets formed by synchronized elements which are called clusters . this is so , even when , there are identical interactions between identical elements . the system is labeled as _ 1-cluster _ , _ 2-cluster _ , etc . state if the @xmath12 values fall into one , two or more sets of synchronized elements of the phase space . two different elements @xmath13 and @xmath14 belong to the same cluster within a precision @xmath15 ( we consider @xmath16 ) only if @xmath17 thus the system of eq.[eq : sist ] , shows the existence of different phases with clustering ( coherent , ordered , partially ordered , turbulent ) . this phenomena appearing in gcm was studied by kaneko for logistic coupled maps when the control and coupling parameters vary . a rough phase diagram for an array of 256 elements is determined for the number of clusters calculated from 500 randomly sets of initial conditions within the precision specified above . this diagram displayed in fig-[fig:1 ] , was obtained following the criteria established by this author . therefore , the @xmath18 number of clusters and the number of elements that build them are relevant magnitudes to characterize the system behaviour . in order to study phase transition , the two greatest lyapunov exponents are shown in fig-[fig:4 ] and fig-[fig:5 ] . they are depicted for a=3.34 as a function of @xmath4 and for three different values of initial elements @xmath11 . in the coherent phase , as soon as @xmath4 decrease , the maximum lyapunov exponent changes steeply from a positive to a negative value when the two cluster state is reached . a sudden change in the attractor phase space occurs for a critical value of the coupling parameter @xmath19 in the analysis of the transition from two to one cluster state . besides that , in the same transition for the same @xmath19 , a metastable transient state of two cluster to one cluster chaotic state is observed , due to the existence of an unstable orbit inside of the chaotic basin of attraction , as is shown in fig-[fig:3 ] the characteristic time @xmath20 in which the system is entertained in the metastable transient is depicted in fig-[fig:6 ] , for values of @xmath4 near and above @xmath19 . for a given set of initial conditions , it is possible to fit this transient as : @xmath21 this fitting exponent @xmath22 , depends upon the number of elements with initial conditions in each basin as is shown in the next table for three @xmath11 values and setting @xmath23 . [ cols="<,<,<",options="header " , ] it is worth noting from the table that @xmath22 increases with @xmath11 up to @xmath24 , and for @xmath11 due to the basins symmetry . in order to analize the existence of period doubling bifurcations , the maxima lyapunov exponent @xmath25 is calculated as function of @xmath11 and @xmath4 . for each @xmath11 , critical values of the coupling parameter , called @xmath26 , are observed when a negative @xmath25 reaches a zero value without changing sign . this behaviour is related to inverse period doubling bifurcations of the gcm . fitting all these critical pair of values @xmath27 , a rough @xmath11 vs @xmath26 graph is shown in fig-[fig:7 ] , and different curves appears as boundary regions of the parameter space where the system displays @xmath28 ( @xmath29 ) periods states . this is obtained without taking into accout the number of final clusters . it is clear that greater values of @xmath11 , correspond to smaller @xmath26 for the occurrence of the bifurcation . evidence of period 16 appears for values of @xmath30 smaller than 30 . in fig-[fig:7 ] t=2(symmetric ) means period two orbit , with clusters oscillating with equal amplitud around zero , t=2(asymmetric ) means period two orbit , with clusters oscillating with different amplitud . the study of systems with coexistence of multiple attractors gives a much richer dynamics and a new control parameter must necessarily be added . although the dimensionality in the parameter space is increased by one , the dynamics is rather simple to characterize . some of the relevant aspects of this kind of systems are shown in this work . the phase diagram that was obtained shows the existence of similar phases to those using the cuadratic and quartic map , this behaviour suggests some kind of universality in the dynamics of the gcm . another interesting issue found , concerns the metastable transition between two to one cluster state , along with a sudden jump in the maximum lyapunov exponent , as it was displayed in fig.[fig:7 ] . the characteristic time given by eq.[eq:1 ] also correspond to the above transition where the critical exponent @xmath31 and the critical coupling parameter @xmath32 shows a strong dependence on the number of initial elements in each basin . an inverse bifurcation cascade appears when the system is in two or more clusters state where @xmath32 and @xmath30 are the critical parameters of the bifurcation , which means the maximum lyapunov exponent is equal to zero . this work is partially supported by conicet ( grant pip 4210 ) . mfc and lr also express their acknowledgment to the ictp where the initial discussion of the work was performed .
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a system of n unidimensional global coupled maps ( gcm ) , which support multiattractors is studied .
we analize the phase diagram and some special features of the transitions ( volumen ratios and characteristic exponents ) , by controlling the number of elements of the initial partition that are in each basin of attraction .
it was found important differences with widely known coupled systems with a single attractor .
| 2,117 | 95 |
stars during their early stage of evolution experience a phase of mass loss driven by strong stellar winds @xcite . the stellar winds can entrain and accelerate ambient gas and inject momentum and energy into the surrounding environment , thereby significantly affect the dynamics and structure of their parent molecular clouds @xcite . both outflows and bubbles are manifestations of strong stellar winds dispersing the surrounding gas . in general , collimated jet - like winds from young embedded protostars usually drive powerful collimated outflows , while wide - angle or spherical winds from the pre - main - sequence stars are more likely to drive less - collimated outflows or bubbles @xcite . a bubble is a partially or fully enclosed three - dimensional structure whose projection is a partial or full ring @xcite . the kinetic energy of an outflow is very large ( @xmath10-@xmath11 erg ; * ? ? ? * ; * ? ? ? * ) , implying a substantial input of mechanical energy into its parent molecular cloud @xcite . feedback from young stars has been proposed as a significant aspect of self - regulation of star formation @xcite . feedback may maintain the observed turbulence in molecular clouds and it may also be responsible for stabilizing the clouds against gravitational collapse @xcite . the impact of outflows on surrounding gas has been studied primarily in small regions such as orion kl @xcite , l1551 @xcite and gl 490 @xcite on scales less than 10@xmath12 . recently there have been a few studies related to outflow feedback in nearby clouds . @xcite undertook a complete survey of outflows in perseus and found that outflows have an important impact on the environment immediately surrounding localized regions of active star formation , but that outflows have insufficient energy to feed the observed turbulence in the entire perseus complex . @xcite and @xcite studied the outflows in the @xmath13 ophiuchi main cloud and serpens south , respectively . both studies concluded that outflows can power the supersonic turbulence in their parent molecular cloud but do not have enough momentum to support the entire cloud against the global gravitational contraction . @xcite identified 20 outflows in the taurus region and concluded that outflows can not sustain the observed turbulence seen in the entire cloud . in this paper , we report a systematic and detailed search for outflows around sources from the spitzer space telescope ( hereinafter referred to as spitzer ) young stellar object ( yso ) catalog and then estimated their impact on the entire taurus molecular cloud . similar to outflows , bubbles are important morphological features in star formation process , which can give information about spherical stellar winds and physical properties of their surrounding environments @xcite . parsec - scale bubbles are usually found in massive star - forming regions @xcite . the conventional thought has been that high - mass stars can drive spherical winds and easily create the observed bubbles , while the spherical winds from low- and intermediate - mass stars are too weak to produce bubbles . however , @xcite studied shells ( bubbles ) in perseus , a nearby low - mass star - forming molecular cloud , and concluded that the total energy input from outflows and shells is sufficient to maintain the turbulence . the taurus molecular cloud is at a distance of 140 pc @xcite . it covers an area of more than 100 deg@xmath1 @xcite . using the j=2 - 1 line of @xmath2co , 13 outflows have been found around low - mass embedded ysos in taurus @xcite . there are 13 high velocity molecular outflows in taurus included in the catalog of @xcite . using jcmt - harp @xmath2co j=3 - 2 observations , 16 outflows have been found in l1495 , a ` bowl - shaped ' region in the nw corner of taurus @xcite . recently , 20 outflows have been identified , 8 of which were new detections with the five college radio astronomy observatory ( fcrao ) @xmath2co j=1 - 0 and @xmath3co j=1 - 0 data cubes covering the entire taurus molecular cloud @xcite . the up - to - date catalog of ysos @xcite from the spitzer provides an opportunity to search for outflows and bubbles in a more comprehensive manner . here we present a systematic and detailed search for outflows and bubbles in the vicinity of young stellar objects and estimate their impact on the overall taurus molecular cloud . the paper is organized as follows . in [ data ] we describe the data used in the study . the details including searching methods , morphology and physical parameters of outflows and bubbles are presented in [ outflows ] and [ bubbles ] , respectively . the driving sources of outflows and bubbles , their energy feedback to the parent cloud and the related comparison between taurus and perseus are discussed [ discussion ] . in [ conclusions ] we summarize the main results . in our study we used the @xmath2co(1 - 0 ) and @xmath3co(1 - 0 ) data observed with 13.7 m fcrao telescope @xcite . we also adopted the up - to - date catalog of spitzer ysos , where 215 ysos and 140 new yso candidates in taurus are reported @xcite . the fcrao co survey was taken between 2003 and 2005 . the @xmath2co and @xmath3co maps are centered at @xmath14=@xmath15 , @xmath16=@xmath17 ( j2000 ) covering an area of approximately 100 @xmath18 . the full width at half maximum ( fwhm ) beam width is @xmath19 for @xmath2co and is @xmath20 for @xmath3co . the pixel size of the resampled data is @xmath21 , which corresponds to 0.014 pc at a distance of 140 pc . there are 80 channels for @xmath2co and 76 channels for @xmath3co , covering approximately @xmath225 to @xmath2314.9 km s@xmath8 . the width of a velocity channel is 0.254 km s@xmath8 for @xmath2co and 0.266 km s@xmath8 for @xmath3co @xcite . the mips ( multi - band imaging photometer for spitzer ; * ? ? ? * ) maps were created as part of the final products from the spitzer legacy taurus i and ii surveys @xcite . the data were obtained in fast scan mode in three bands : 24 , 70 , and 160 @xmath24 m , over an area of 44 @xmath18 . the observations were performed in three epochs between 2005 and 2007 , with an integration time of 30 s ( 24 @xmath24 m ) and 15 s ( 70 & 160 @xmath24 m ) . the maps were created using the basic calibrated data ( bcds ) and coadded using the spitzer software package mopex ( mosaicking and point source extractor ; * ? ? ? * ) . despite the fact that the data were taken with interleaved scan legs to provide optimal coverage at 70 and 160 @xmath24 m , some small gaps remained , in particular at 160 @xmath24 m . to mitigate this effect , the 160 @xmath24 m final mosaic was created using 32 arcsec pixels , instead of the native 16 arcsec / pixel scale . this pixel scale matches quite well with the @xmath2540 arcsec beam at 160 @xmath24 m wavelength . the 24 and 70 @xmath24 m maps adopted the standand 2.5 and 4 arcsec / pixel scale , respectively , to properly sample their respective 6 and 18 arcsec beams . the maps were used successfully for photometric purposes to identify new sources in the taurus molecular cloud @xcite . we identified 55 outflows around the spitzer ysos in the 44 deg@xmath1 area of taurus . in total 31 of the detected outflows were previously unknown . in the following subsections , we describe the searching procedure of outflows , the morphology , physical properties , and the comparison between our findings and the known ones . instead of a blind search , we focused on seeking outflows around ysos . the search procedure was performed with an interactive data language ( idl ) pipeline . we plotted spectra , position velocity diagrams ( hereafter p - v diagrams ) and integrated intensity maps to identify the outflows around the 355 ysos which spitzer identified in taurus . detailed steps of the search are the following . \(1 ) we plotted @xmath2co contours ( hereafter contour map ) overlaid on a @xmath3co grey - scale image around a yso . according to the scale and velocity range of previously - detected outflows in taurus , we chose two sizes ( @xmath26 and @xmath27 ) and three sets of velocity intervals ( -1 to 3.5 km s@xmath8 , -1 to 4.5 km s@xmath8 and -1 to 5.5 km s@xmath8 for blue ; 7.5 to 13 km s@xmath8 , 8.5 to 13 km s@xmath8 and 9.5 to 13 km s@xmath8 for red ) to plot the contour maps . we plotted the maps with 3 sets of velocity intervals and two scales around the 355 ysos automatically . in total , 2130 maps were obtained . we inspected these maps to identify outflow candidates according to the morphology of the blue and red lobes . in the end , 74 candidates were selected . \(2 ) we plotted @xmath2co p - v diagrams along four directions ( at position angles of @xmath28 , @xmath29 , @xmath30 and @xmath31 ) on three scales ( 20@xmath12 , 40@xmath12 and 60@xmath12 ) around the 74 candidates . the size and high - velocity range of outflow candidates were determined roughly by checking the p - v diagrams . when the velocity bulge appears along the direction away from the central velocity , we marked it as the start of the high - velocity wing . and along the above direction , the maximum velocity corresponding to the outermost contour is the end of this high - velocity wing . we further confirmed each of the diagrams individually by visual inspection . the position range of the entire high - velocity bulge along the position axis was considered to be the rough size of the outflow . if more than one central velocity is found in the p - v diagram , it likely has multiple velocity components @xcite and thus will be excluded from the list of outflow candidates . therefore , 19 candidates with multiple velocity components were eliminated and the remaining 55 outflow candidates were considered to be possible outflows . \(3 ) using the rough sizes and velocity ranges obtained in step ( 2 ) , we plotted contour maps for the remaining 55 outflows . p - v diagrams were plotted through the midpoint of the blue and red peaks ( bipolar outflow ) or through the peak of the lobe in the case of monopolar outflow , at position angles spaced by @xmath32 . we chose the angle with the most prominent bulge along the velocity axis to determine the velocity interval of the outflows . then we plotted the contour map again with this velocity interval . \(4 ) finally we plotted the average spectra of the blue and red lobes . according to the morphology of p - v diagrams and contour maps , we divided the outflows into five classes . the higher is the ranking , the more likely it is that we have identified an outflow . we define a typical p - v diagram ( tpv ) and a representative contour map ( rcm ) as follows . if there is obvious high velocity gas which can be seen by the protuberance along the velocity axis on the p - v diagram and the high velocity range is not less than 1 km s@xmath8 , then we regard the p - v diagram as a tpv . if the outermost contour of the lobe is closed and we can see a clear and unbroken lobe on the contour map , then we regard the contour map as a rcm . table [ outflowclass ] shows our criteria for outflow classification with an `` @xmath33 '' meaning that it satisfies a certain condition . having both @xmath2co tpv and @xmath2co rcm is required for high ranking ( class a@xmath34 ) , but having @xmath3co tpv or @xmath3co rcm gives a lower ranking ( class a@xmath35 and class b@xmath35 ) because @xmath3co is generally optically thin in outflows and we usually found lobes of outflows with @xmath2co not @xmath3co . having only @xmath2co rcm was divided into the lowest ranking ( class c@xmath34 ) because the high velocity gas in p - v diagram is not obvious . the primary condition to identify an outflow is having high velocity gas which can be seen from the protuberance along the velocity axis on the p - v diagram . lccccrr class & @xmath2co & @xmath2co & @xmath3co & @xmath3co & outflow & percentage + & tpv & rcm & tpv & rcm & numbers & + a@xmath34 & @xmath33 & @xmath33 & & & 24 & 43.6% + a@xmath35 & @xmath33 & @xmath33 & @xmath33 & @xmath33 & 18 & 32.7% + b@xmath34 & @xmath33 & & & & 4 & 7.3% + b@xmath35 & @xmath33 & & @xmath33 & & 1 & 1.8% + c@xmath34 & & @xmath33 & & & 8 & 14.5% + following the steps in [ outflowstep ] we found 55 outflows in the 44 deg@xmath1 area of taurus molecular cloud . all the outflows we detected were listed in table [ outflowlist ] . each outflow is referred as a taurus molecular outflow " ( tmo ) . we present the locations , polarities and scales of the outflows overlaid on the spitzer mips image in fig . [ bigmap ] . there are 31 new ones among all the detected outflows . we have thus increased the total number of known outflows by a factor of 1.3 . lccllclcl tmo_01 & 04 11 59.7 & 29 42 36 & - & iii & a@xmath34 & mr & n & 1 + tmo_02 & 04 14 12.2 & 28 08 37 & iras 04113 + 2758 ( l1495 ) & i & a@xmath34 & bi & n & 1 , 2 , 3 + tmo_03 & 04 14 14.5 & 28 27 58 & - & ii & a@xmath34 & bi & y & 4 + tmo_04 & 04 18 32.0 & 28 31 15 & - & flat & a@xmath34 & bi & y & 4 + tmo_05 & 04 19 41.4 & 27 16 07 & iras 04166 + 2706 & i & a@xmath34 & mb & n & 1 , 2 , 5 , 6 , 7 + tmo_06 & 04 19 58.4 & 27 09 57 & iras 04169 + 2702 & i & a@xmath34 & bi & n & 1 , 2 , 3 , 7 + tmo_07 & 04 21 07.9 & 27 02 20 & iras 04181 + 2655 & i & a@xmath34 & bi & n & 2 , 3 , 6 , 7 + tmo_08 & 04 22 15.6 & 26 57 06 & fs tau b & i & a@xmath34 & bi & n & 1 , 2 + tmo_09 & 04 23 25.9 & 25 03 54 & - & ii & a@xmath34 & mr & y & 4 + tmo_10 & 04 24 20.9 & 26 30 51 & - & ii & a@xmath34 & bi & y & 4 + tmo_11 & 04 24 45.0 & 27 01 44 & - & iii & a@xmath34 & mr & y & 4 + tmo_12 & 04 29 30.0 & 24 39 55 & haro 6 - 10 & i & a@xmath34 & mr & n & 1 , 6 , 8 + tmo_13 & 04 31 10.4 & 25 41 29 & - & - & a@xmath34 & mr & y & 4 + tmo_14 & 04 31 58.4 & 25 43 29 & - & iii & a@xmath34 & bi & y & 4 + tmo_15 & 04 32 14.6 & 22 37 42 & - & flat & a@xmath34 & bi & y & 4 + tmo_16 & 04 32 31.7 & 24 20 02 & l1529 & ii & a@xmath34 & bi & n & 1 , 6 , 9 , 10 + tmo_17 & 04 32 32.0 & 22 57 26 & iras 04295 + 2251 ( l1536 ) & i & a@xmath34 & mr & n & 3 , 7 + tmo_18 & 04 32 43.0 & 25 52 31 & - & ii & a@xmath34 & bi & y & 4 + tmo_19 & 04 34 15.2 & 22 50 30 & - & ii & a@xmath34 & mr & y & 4 + tmo_20 & 04 37 24.8 & 27 09 19 & - & iii & a@xmath34 & mr & y & 4 + tmo_21 & 04 39 53.9 & 26 03 09 & l1527 & i & a@xmath34 & bi & n & 1 , 3 , 6 , 7 , 11 , 12 , 13 + tmo_22 & 04 41 08.2 & 25 56 07 & iras 04381 + 2540 ( tmc-1 ) & flat & a@xmath34 & mb & n & 1 , 6 , 7 , 14 + tmo_23 & 04 41 12.6 & 25 46 35 & iras 04381 + 2540 ( tmc-1 ) & i & a@xmath34 & mr & n & 1 , 6 , 7 , 14 + tmo_24 & 04 42 07.7 & 25 23 11 & iras 04390 + 2517 ( lkh@xmath14 332 ) & ii & a@xmath34 & mr & n & 3 + tmo_25 & 04 18 58.1 & 28 12 23 & iras 04158 + 2805 ( l1495 ) & flat & a@xmath35 & mb & y & 4 , 7 + tmo_26 & 04 23 18.2 & 26 41 15 & - & ii & a@xmath35 & mr & y & 4 + tmo_27 & 04 26 56.2 & 24 43 35 & iras 04239 + 2436 ( hh 300 ) & i & a@xmath35 & mr & n & 1 , 3 , 6 , 15 + tmo_28 & 04 27 02.6 & 26 05 30 & iras 04240 + 2559 ( dg tau ) & i & a@xmath35 & bi & n & 1 , 16 + tmo_29 & 04 27 02.8 & 25 42 22 & - & ii & a@xmath35 & bi & y & 4 + tmo_30 & 04 27 57.3 & 26 19 18 & iras 04248 + 2612 & flat & a@xmath35 & mr & n & 1 , 3 + tmo_31 & 04 28 10.4 & 24 35 53 & - & flat & a@xmath35 & mr & y & 4 + tmo_32 & 04 30 51.7 & 24 41 47 & iras 04278 + 2435 ( zz tau irs ) & flat & a@xmath35 & mr & n & 1 , 6 , 17 + tmo_33 & 04 32 15.4 & 24 28 59 & iras 04292 + 2422 ( haro 6 - 13 ) & flat & a@xmath35 & bi & n & 1 , 3 + tmo_34 & 04 33 07.8 & 26 16 06 & - & iii & a@xmath35 & mb & y & 4 + tmo_35 & 04 33 10.0 & 24 33 43 & - & iii & a@xmath35 & mr & y & 4 + tmo_36 & 04 33 16.5 & 22 53 20 & iras 04302 + 2247 & i & a@xmath35 & bi & n & 1 , 3 , 7 + tmo_37 & 04 33 34.0 & 24 21 17 & - & ii & a@xmath35 & mr & y & 4 + tmo_38 & 04 33 36.7 & 26 09 49 & - & ii & a@xmath35 & mb & y & 4 + tmo_39 & 04 35 57.6 & 22 53 57 & iras 04328 + 2248 ( hp tau ) & ii & a@xmath35 & bi & n & 3 + tmo_40 & 04 39 11.2 & 25 27 10 & hh706 & - & a@xmath35 & mr & n & 1 + tmo_41 & 04 39 13.8 & 25 53 20 & iras 04361 + 2547 ( tmr-1 ) & i & a@xmath35 & bi & n & 1 , 3 , 6 , 7 , 18 + tmo_42 & 04 48 02.3 & 25 33 59 & tau a 8 & iii & a@xmath35 & bi & y & 4 + tmo_43 & 04 18 51.4 & 28 20 26 & hh156 & i & b@xmath34 & mb & y & 2 + tmo_44 & 04 20 21.4 & 28 13 49 & - & flat & b@xmath34 & bi & y & 4 + tmo_45 & 04 26 53.3 & 25 58 58 & - & i & b@xmath34 & bi & y & 4 + tmo_46 & 04 39 56.1 & 26 28 02 & - & - & b@xmath34 & bi & y & 4 + tmo_47 & 04 35 35.3 & 24 08 19 & iras 04325 + 2402 ( l1535 ) & i & b@xmath35 & bi & n & 1 , 3 , 6 , 17 , 19 , 20 + tmo_48 & 04 15 35.6 & 28 47 41 & - & i & c@xmath34 & bi & y & 4 + tmo_49 & 04 17 33.7 & 28 20 46 & - & ii & c@xmath34 & mr & y & 4 + tmo_50 & 04 18 10.5 & 28 44 47 & - & i & c@xmath34 & mr & y & 4 + tmo_51 & 04 18 31.1 & 28 16 29 & - & ii & c@xmath34 & mr & y & 4 + tmo_52 & 04 18 31.2 & 28 26 17 & - & i & c@xmath34 & mr & y & 4 + tmo_53 & 04 18 41.3 & 28 27 25 & - & flat & c@xmath34 & mr & y & 4 + tmo_54 & 04 21 54.5 & 26 52 31 & - & ii & c@xmath34 & bi & y & 4 + tmo_55 & 04 29 04.9 & 26 49 07 & iras 04260 + 2642 & i & c@xmath34 & mr & n & 4 + table [ outflowclass ] lists the numbers and percentages in the five classes of outflows . we can see class a@xmath34 and class a@xmath35 account for 76.3% of all the detected outflows . these two types can be considered as the most probable " outflows in our study . table [ outflowclassnum ] lists the numbers of previously known and newly detected outflows in different classes . we found more new outflows of class a@xmath34 and class a@xmath35 , which account for 64.5% of all the newly detected outflows . that is , most of the new outflows we found are likely true outflows . lcc type & previously & newly + & known & detected + a@xmath34 & 13 & 11 + a@xmath35 & 9 & 9 + b@xmath34 & 0 & 4 + b@xmath35 & 1 & 0 + c@xmath34 & 1 & 7 + table [ outflownumyso ] lists the outflow numbers and percentages according to the types of their driving sources . class i accounts for 36.4% , which is the largest proportion of all the ysos driving outflow . the outflows driven by the class i ysos are closer to the ysos and have more collimated bi - polar morphology . compared with the class i , class iii ysos drive a small proportion ( 12.7% ) of outflows , which tend to be farther from the ysos . this indicates that the outflows from class iii ysos are more evolved than those from class i ysos . we also found three outflows ( tmo_13 , tmo_40 , tmo_46 ) without ysos , indicating that they are possibly class 0 objects . among the three outflows , tmo_13 and tmo_40 are newly found in our study , while tmo_46 has been reported in @xcite . lrr yso & number of & percentage + type & outflows & + i & 20 & 36.4% + flat & 10 & 18.2% + ii & 15 & 27.3% + iii & 7 & 12.7% + no yso & 3 & 5.4% + we found 25 bipolar , 22 monopolar redshifted and 6 monopolar blueshifted outflows . bipolar and redshifted outflows account for the vast majority of outflows in the taurus molecular cloud . this is consistent with the results of @xcite . [ sample - figure01 ] - fig . [ sample - figure55 ] show the @xmath36 integrated intensity map , @xmath36 p - v diagram and average spectrum for each outflow . for class a@xmath35 and b@xmath35 outflows , we also plotted the @xmath37 integrated intensity maps and @xmath37 p - v diagrams . using the fcrao large - scale survey data @xcite and the latest yso catalog from @xcite we were able to identify the previously known outflows , obtain more complete morphology , and find additional new outflows . the yso catalog is also convenient for identifying the driving sources of the outflows . comparing with the previous works , we confirmed more driving sources of outflows . l1527 ( tmo_21 ) is a typical outflow in taurus @xcite . the p - v diagram and contour map in our work are very similar to those in @xcite and @xcite . tmo_08 ( sst 042215.6 + 265706 ) and fs tau b in @xcite are the same outflow with the same location . they have the same structure , which can be seen in our fig . [ sample - figure08 ] and figure 15 in @xcite . in addition , tmo_30 ( sst 042757.3 + 261918 ) , tmo_32 ( sst 043051.7 + 244147 ) , tmo_33 ( sst 043215.4 + 242859 ) and tmo_41 ( sst 043913.8 + 255320 ) also have the same morphology as iras 04248 + 2612 , zz tau irs , iras 04292 + 2422 and iras 04361 + 2547 in @xcite , respectively . these confirm the general consistency between the two works in terms of strong and extended outflows . for tmo_02 ( sst 041412.2 + 280837 ) , we obtained a good bipolar structure shown in the upper left panel of fig . [ sample - figure01 ] , while @xcite considered this outflow ( iras 04113 + 2758 ) to be redshifted only . @xcite did not identify the driving source of this outflow ( named by w - co - flow1 ) , while we determined that the yso sst 041412.2 + 280837 is driving the outflow . @xcite only presented the central spectrum of iras 04390 + 2517 and iras 04328 + 2248 , while we illustrated the two outflows ( tmo_24 and tmo_39 ) more clearly through contour maps and p - v diagrams . the morphology of tmo_07 ( sst 042107.9 + 270220 ) shown in fig . [ sample - figure07 ] is similar to that of j04210795 + 2702204 in @xcite . this outflow was also reported by @xcite , @xcite and @xcite . however , @xcite did not find it with the same fcrao survey data . @xcite reported that iras 04240 + 2559 was a monopolar redshifted outflow with @xmath2co ( 3 - 2 ) line . but at this location we found the well - defined bipolar outflow as shown in fig . [ sample - figure28 ] . as for l1529 , @xcite presented high - velocity @xmath2co wings observed by antenna no . 2 of the caltech 10.4 m array , but @xcite did not find any high - velocity gas in observations at fcrao . we identified a bipolar outflow named tmo_16 ( sst 043231.7 + 242002 ) and demonstrated the result of @xcite with the fcrao data . at the position of iras 04295 + 2251 @xcite showed line wings while @xcite found no outflow . we found a red monopolar outflow as shown in fig . [ sample - figure17 ] . we have found 31 new outflows which are labeled y " in the eighth column of table [ outflowlist ] . two of these new outflows were not identified as outflows in the literature . @xcite considered iras 04158 + 2805 ( l1495 ) but did not find any sign of outflow activity in the @xmath2co ( 2 - 1 ) transition at the location of tmo_025 ( sst 041858.1 + 281223 ) . @xcite had some doubt about co flow of coku tau-1 when analyzing the @xmath2co ( 3 - 2 ) emission , while we found tmo_043 ( sst 041851.4 + 282026 ) at this site . the rest of the new outflows have not been reported in the literature and are identified as outflows for the first time . all of the new outflows are of small angular extent , less than 10@xmath12 . they were missed in previous searches perhaps because of their small sizes . to study the effects of outflows on their environment we calculated their masses , momenta , kinetic energy and energy deposition rates . the total column density of the outflowing gas is @xmath38 where @xmath39 , @xmath40 erg s , @xmath41 , @xmath42 and @xmath43 is the observed source antenna temperature with proper correction for antenna efficiency . we assumed an excitation temperature of 25 k. the excitation temperature assumed in the literature @xcite ranges from 11 k to 50 k. the lowest temperature will decrease the mass estimate by a factor of 3 and the highest temperature will increase the mass estimate by a factor of 2.2 . the detailed derivations regarding the physical parameters of the outflows are given in appendix a. @xcite described three major issues that can cause uncertainties in the calculation of an outflow s parameters , namely , the inclination , opacity , and blending . our prescriptions are the following . ( 1 ) we defined the inclination angle of the outflow as the angle between the long axis of the outflow and the line of sight . since the outflows with * small * inclination angle ( especially when * the outflow is perpendicular to the plane of the sky * ) are hard to detect , our outflow searching is biased to those with * large * inclination angle . if the inclination angle @xmath44 is randomly distributed , the average value is given by @xmath45 from the above formula we got the average inclination angle of * 57.3@xmath46 * , which differs from the usually used median value of 45@xmath46 . then the velocity and the dynamic age , @xmath47 , should be scaled up by a factor of * 1.9 * and * 0.64 * , respectively . ( 2 ) using the @xmath36 and @xmath37 data we can correct for the opacity in the @xmath2co line when @xmath36 emission of an outflow is optically thick . the algorithm for the opacity correction is described in appendix a. ( 3 ) we probably missed some low - velocity outflowing gas , which blended into the ambient gas , when we conservatively determined the emission only from outflows . previous studies @xcite showed that neglecting this gas results in the underestimate of the outflow mass almost by a factor of 2 . table [ outflowparameters ] gives the length , mass , momentum , kinetic energy , dynamical timescale and the luminosity of the outflows in taurus . llcccccccc tmo_01 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.5 & 11 @xmath33 25 & 1.11 & 0.083 & 0.205 & 0.51 & 4.4 & 0.37 + tmo_02 & blueshifted & 3.4 & 8 @xmath33 12 & 0.58 & 0.168 & 0.570 & 1.92 & 1.7 & 3.63 + & redshifted & 3.6 & 6 @xmath33 5 & 0.31 & 0.061 & 0.220 & 0.79 & 0.8 & 2.97 + tmo_03 & blueshifted & 3.1 & 6 @xmath33 9 & 0.41 & 0.050 & 0.156 & 0.49 & 1.3 & 1.19 + & redshifted & 2.4 & 11 @xmath33 2 & 0.46 & 0.013 & 0.030 & 0.07 & 1.9 & 0.12 + tmo_04 & blueshifted & 2.6 & 6 @xmath33 8 & 0.41 & 0.026 & 0.069 & 0.18 & 1.5 & 0.36 + & redshifted & 2.5 & 3 @xmath33 8 & 0.33 & 0.005 & 0.013 & 0.03 & 1.3 & 0.08 + tmo_05 & blueshifted & 4.6 & 9 @xmath33 18 & 0.83 & 0.078 & 0.355 & 1.61 & 1.8 & 2.89 + & redshifted & - & - & - & - & - & - & - & - + tmo_06 & blueshifted & 2.8 & 5 @xmath33 5 & 0.29 & 0.038 & 0.109 & 0.31 & 1.0 & 0.96 + & redshifted & 3.9 & 3 @xmath33 5 & 0.21 & 0.010 & 0.038 & 0.15 & 0.5 & 0.88 + tmo_07 & blueshifted & 2.1 & 7 @xmath33 7 & 0.38 & 0.012 & 0.025 & 0.05 & 1.8 & 0.09 + & redshifted & 2.7 & 2 @xmath33 2 & 0.13 & 0.002 & 0.005 & 0.01 & 0.5 & 0.08 + tmo_08 & blueshifted & 3.1 & 5 @xmath33 4 & 0.26 & 0.013 & 0.039 & 0.12 & 0.8 & 0.48 + & redshifted & 2.8 & 2 @xmath33 2 & 0.12 & 0.001 & 0.003 & 0.01 & 0.4 & 0.07 + tmo_09 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 1.8 & 17 @xmath33 18 & 1.01 & 0.311 & 0.565 & 1.02 & 5.5 & 0.59 + tmo_10 & blueshifted & 2.3 & 3 @xmath33 7 & 0.29 & 0.010 & 0.023 & 0.05 & 1.3 & 0.13 + & redshifted & 2.9 & 7 @xmath33 3 & 0.30 & 0.016 & 0.047 & 0.14 & 1.0 & 0.43 + tmo_11 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.6 & 7 @xmath33 7 & 0.40 & 0.020 & 0.052 & 0.14 & 1.5 & 0.29 + tmo_12 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.5 & 4 @xmath33 3 & 0.22 & 0.014 & 0.036 & 0.09 & 0.8 & 0.33 + tmo_13 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.2 & 7 @xmath33 4 & 0.34 & 0.053 & 0.115 & 0.25 & 1.5 & 0.51 + tmo_14 & blueshifted & 2.6 & 5 @xmath33 5 & 0.29 & 0.019 & 0.050 & 0.13 & 1.1 & 0.38 + & redshifted & 3.2 & 7 @xmath33 4 & 0.31 & 0.031 & 0.099 & 0.32 & 0.9 & 1.05 + tmo_15 & blueshifted & 1.9 & 3 @xmath33 3 & 0.18 & 0.013 & 0.024 & 0.05 & 0.9 & 0.16 + & redshifted & 1.5 & 3 @xmath33 4 & 0.20 & 0.010 & 0.016 & 0.02 & 1.3 & 0.06 + tmo_16 & blueshifted & 2.3 & 2 @xmath33 2 & 0.11 & 0.005 & 0.012 & 0.03 & 0.4 & 0.21 + & redshifted & 2.0 & 4 @xmath33 4 & 0.23 & 0.023 & 0.046 & 0.09 & 1.1 & 0.25 + tmo_17 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 1.7 & 5 @xmath33 14 & 0.60 & 0.042 & 0.072 & 0.12 & 3.5 & 0.11 + tmo_18 & blueshifted & 2.0 & 10 @xmath33 3 & 0.42 & 0.012 & 0.024 & 0.05 & 2.1 & 0.07 + & redshifted & 1.9 & 3 @xmath33 4 & 0.20 & 0.007 & 0.013 & 0.02 & 1.0 & 0.08 + tmo_19 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.2 & 4 @xmath33 4 & 0.23 & 0.008 & 0.017 & 0.04 & 1.0 & 0.11 + tmo_20 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 4.0 & 7 @xmath33 3 & 0.31 & 0.017 & 0.067 & 0.26 & 0.8 & 1.08 + tmo_21 & blueshifted & 3.1 & 4 @xmath33 2 & 0.17 & 0.004 & 0.012 & 0.04 & 0.5 & 0.22 + & redshifted & 2.9 & 5 @xmath33 3 & 0.22 & 0.011 & 0.032 & 0.09 & 0.7 & 0.40 + tmo_22 & blueshifted & 1.6 & 10 @xmath33 22 & 0.97 & 0.226 & 0.351 & 0.54 & 6.1 & 0.28 + & redshifted & - & - & - & - & - & - & - & - + tmo_23 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 3.7 & 3 @xmath33 6 & 0.27 & 0.019 & 0.072 & 0.26 & 0.7 & 1.18 + tmo_24 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 3.2 & 4 @xmath33 6 & 0.29 & 0.015 & 0.047 & 0.15 & 0.9 & 0.52 + tmo_25 & blueshifted & 2.4 & 5 @xmath33 5 & 0.31 & 0.267 & 0.640 & 1.52 & 1.2 & 3.87 + & redshifted & - & - & - & - & - & - & - & - + tmo_26 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.4 & 13 @xmath33 8 & 0.60 & 0.369 & 0.880 & 2.09 & 2.5 & 2.68 + tmo_27 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.4 & 6 @xmath33 7 & 0.37 & 0.074 & 0.174 & 0.41 & 1.5 & 0.86 + tmo_28 & blueshifted & 2.2 & 4 @xmath33 6 & 0.30 & 0.170 & 0.371 & 0.80 & 1.3 & 1.91 + & redshifted & 3.9 & 3 @xmath33 2 & 0.15 & 0.031 & 0.122 & 0.47 & 0.4 & 3.80 + tmo_29 & blueshifted & 1.9 & 9 @xmath33 4 & 0.40 & 0.264 & 0.492 & 0.91 & 2.1 & 1.37 + & redshifted & 2.1 & 6 @xmath33 5 & 0.32 & 0.076 & 0.161 & 0.34 & 1.5 & 0.74 + tmo_30 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.4 & 9 @xmath33 9 & 0.49 & 0.286 & 0.673 & 1.57 & 2.1 & 2.43 + tmo_31 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.1 & 5 @xmath33 14 & 0.62 & 0.277 & 0.587 & 1.24 & 2.9 & 1.37 + tmo_32 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 3.3 & 9 @xmath33 6 & 0.42 & 0.131 & 0.429 & 1.39 & 1.2 & 3.53 + tmo_33 & blueshifted & 2.1 & 8 @xmath33 5 & 0.38 & 0.120 & 0.249 & 0.51 & 1.8 & 0.90 + & redshifted & 3.7 & 11 @xmath33 13 & 0.68 & 0.896 & 3.299 & 12.08 & 1.8 & 1.174 + tmo_34 & blueshifted & 3.2 & 5 @xmath33 3 & 0.23 & 0.061 & 0.198 & 0.63 & 0.7 & 2.86 + & redshifted & - & - & - & - & - & - & - & - + tmo_35 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 3.2 & 6 @xmath33 4 & 0.28 & 0.049 & 0.156 & 0.49 & 0.9 & 1.79 + tmo_36 & blueshifted & 2.6 & 9 @xmath33 14 & 0.65 & 0.484 & 1.241 & 3.17 & 2.5 & 4.03 + & redshifted & 1.9 & 8 @xmath33 6 & 0.42 & 0.092 & 0.177 & 0.34 & 2.1 & 0.51 + tmo_37 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.1 & 15 @xmath33 19 & 0.99 & 0.348 & 0.737 & 1.55 & 4.6 & 1.08 + tmo_38 & blueshifted & 2.3 & 13 @xmath33 8 & 0.61 & 0.377 & 0.860 & 1.95 & 2.6 & 2.35 + & redshifted & - & - & - & - & - & - & - & - + tmo_39 & blueshifted & 2.3 & 7 @xmath33 5 & 0.35 & 0.136 & 0.313 & 0.72 & 1.5 & 1.51 + & redshifted & 2.2 & 6 @xmath33 5 & 0.32 & 0.076 & 0.167 & 0.37 & 1.4 & 0.81 + tmo_40 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 4.7 & 4 @xmath33 4 & 0.25 & 0.129 & 0.607 & 2.84 & 0.5 & 7.683 + tmo_41 & blueshifted & 3.8 & 3 @xmath33 4 & 0.23 & 0.028 & 0.104 & 0.39 & 0.6 & 2.09 + & redshifted & 3.9 & 5 @xmath33 15 & 0.63 & 0.317 & 1.226 & 4.72 & 1.6 & 9.401 + tmo_42 & blueshifted & 3.2 & 5 @xmath33 15 & 0.65 & 0.105 & 0.338 & 1.08 & 2.0 & 1.72 + & redshifted & 2.3 & 3 @xmath33 3 & 0.17 & 0.004 & 0.010 & 0.02 & 0.7 & 0.10 + tmo_43 & blueshifted & 2.7 & 9 @xmath33 9 & 0.51 & 0.034 & 0.092 & 0.24 & 1.9 & 0.41 + & redshifted & - & - & - & - & - & - & - & - + tmo_44 & blueshifted & 2.4 & 15 @xmath33 14 & 0.84 & 0.148 & 0.359 & 0.87 & 3.4 & 0.81 + & redshifted & 1.4 & 9 @xmath33 13 & 0.63 & 0.070 & 0.097 & 0.13 & 4.4 & 0.10 + tmo_45 & blueshifted & 2.9 & 2 @xmath33 4 & 0.19 & 0.013 & 0.037 & 0.11 & 0.6 & 0.53 + & redshifted & 2.1 & 3 @xmath33 2 & 0.14 & 0.006 & 0.013 & 0.03 & 0.6 & 0.14 + tmo_46 & blueshifted & 2.8 & 6 @xmath33 13 & 0.58 & 0.050 & 0.141 & 0.39 & 2.0 & 0.61 + & redshifted & 2.9 & 5 @xmath33 18 & 0.76 & 0.123 & 0.362 & 1.06 & 2.5 & 1.34 + tmo_47 & blueshifted & 1.5 & 8 @xmath33 3 & 0.36 & 0.193 & 0.282 & 0.41 & 2.4 & 0.53 + & redshifted & 1.9 & 5 @xmath33 10 & 0.45 & 0.304 & 0.583 & 1.11 & 2.3 & 1.53 + tmo_48 & blueshifted & 5.2 & 2 @xmath33 3 & 0.15 & 0.002 & 0.012 & 0.06 & 0.3 & 0.71 + & redshifted & 3.4 & 2 @xmath33 2 & 0.11 & 0.001 & 0.004 & 0.02 & 0.3 & 0.16 + tmo_49 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 4.4 & 7 @xmath33 17 & 0.76 & 0.042 & 0.182 & 0.79 & 1.7 & 1.47 + tmo_50 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.2 & 4 @xmath33 13 & 0.55 & 0.020 & 0.044 & 0.10 & 2.4 & 0.12 + tmo_51 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.9 & 2 @xmath33 2 & 0.10 & 0.001 & 0.003 & 0.01 & 0.4 & 0.08 + tmo_52 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.3 & 2 @xmath33 3 & 0.16 & 0.002 & 0.004 & 0.01 & 0.7 & 0.04 + tmo_53 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.4 & 2 @xmath33 10 & 0.41 & 0.005 & 0.011 & 0.03 & 1.7 & 0.05 + tmo_54 & blueshifted & 1.9 & 5 @xmath33 5 & 0.30 & 0.013 & 0.025 & 0.05 & 1.5 & 0.10 + & redshifted & 1.8 & 8 @xmath33 5 & 0.38 & 0.034 & 0.061 & 0.11 & 2.1 & 0.17 + tmo_55 & blueshifted & - & - & - & - & - & - & - & - + & redshifted & 2.0 & 4 @xmath33 11 & 0.49 & 0.011 & 0.023 & 0.05 & 2.4 & 0.06 + the distributions of length , mass , energy and dynamical timescale of outflows are shown in fig . [ outflow - histogram ] . the extents of outflows are in the range of 0.1 - 1.11 pc . 79% of outflows are smaller than 0.6 pc . the mass of 54% of outflows is between @xmath48 and @xmath49 . the outflows with mass lower than @xmath48 and higher than @xmath49 account for 17% and 29% of the total , respectively . the energy of 48% of outflows is in the range @xmath50 - @xmath51 erg . the outflows with energy lower than @xmath50 erg and higher than @xmath51 erg account for 31% and 21% , respectively . the dynamical timescales of outflows are between @xmath52 yr and @xmath53 yr . 85% of outflows have dynamical timescale shorter than @xmath54 yr . the mass , momentum , energy and luminosity in table [ outflowparameters ] are only lower limits because we did not take into account the inclination and blending correction in the calculation . the mass should be multiplied by a factor of 2 due to blending . assuming the average inclination angle of outflows is * 57.3@xmath46 * , the velocity and the dynamic age should be scaled up by a factor of * 1.9 * and * 0.64 * , respectively . combining the correction factors due to blending and inclination , the momentum , the kinetic energy and luminosity of outflows should be multiplied by a factor of * 3.8 , 6.8 and 11 * , respectively . after correction , the total mass , momentum , energy , and luminosity of all outflows found in taurus are approximately @xmath55 , @xmath56 , @xmath57 * erg * , and @xmath58 , respectively . the totals of the previously known outflows are about @xmath59 , @xmath60 , @xmath61 * erg * , and @xmath62 , respectively . we found 1.8 times more outflowing mass , * 1.6 * times more momentum and 1.5 times more energy from outflows injecting into the taurus molecular cloud than previously study . a high spatial dynamic range and systematic spectral line survey with good angular resolution is clearly necessary for obtaining a more complete picture of the influence of outflows on their parent cloud . following the method of identifying bubbles in @xcite we have identified 37 bubbles in the @xmath0 deg@xmath1 region of taurus . the procedures for bubble searching , the morphology and physical parameters of bubbles are described in the following sections . we undertook a blind search for bubbles using the fcrao @xmath63co data cube . the integrated intensity map , p - v diagram and channel maps of each bubble were examined . the detailed steps of the search were as follows . \(1 ) we first searched for circular or arc - like ( hereafter bubble - like ) structures in @xmath63co data cube channel by channel through visual inspection . if there is a bubble - like structure in at least three contiguous channels , we considered it as a bubble candidate . the approximate central position and radius of each candidate were recorded for further analysis . we also marked the channels where the bubble - like structure appears . with the marked channels we obtained the expanding velocity interval of a bubble . \(2 ) we plotted @xmath63co contour maps around the central position of the bubble candidates with the expanding velocity intervals . \(3 ) we plotted p - v diagrams in @xmath63co through the central position of each candidate of every @xmath32 in position angle . we chose the one with the most obvious circular or v " structure to show in the figures . the circular or v " structure in the p - v diagram is described in the expanding bubble model @xcite . \(4 ) we plotted the @xmath63co channel maps of each candidate to look over the variation of radius with velocity . \(5 ) finally , we fitted a gaussian profile to the azimuthally averaged profile of @xmath63co intensity of each candidate in the channel where the bubble morphology is most like a ring or arc . the radius of a bubble was obtained from the peak position of the fitted profile . the contour map , p - v diagram , channel maps and gaussian fitted curves helped us not only to analyze the morphology but also to determine the confidence level of a bubble . the bubble candidates were classified into 6 categories according to the characteristic of the above four types of plots . the criteria for bubble classification , as well as the numbers and ratios of bubbles in different classes are illustrated in table [ bubbleclass ] . in this table @xmath33 " means that it meets a certain condition . for each type of plot the condition is as follows . \(a ) there is an obvious bubble - like structure in the contour map . \(b ) the p - v diagram has an obvious circular or v " structure . \(c ) there is an obvious bubble - like structure in the channel map and the radius of bubble is increasing or decreasing with channel . \(d ) the average intensity distribution can be fitted with a gaussian profile . meeting all the above four items is required for a high ranking ( class a ) . if the plots only meet ( b ) and ( c ) , we then divided the bubbles into the lower ranking ( class b1 ) because only the expanding velocity is detected but there is no obvious bubble - like structure and good gaussian fitted profile . then b2 , b3 and b4 are in descending order of ranking . a candidate bubble only meeting ( a ) is assigned the lowest ranking ( class c ) because the gas with expanding velocity is not obvious . lccccrr a & @xmath33 & @xmath33 & @xmath33 & @xmath33 & 13 & 35.2% + b1 & & @xmath33 & @xmath33 & & 6 & 16.2% + b2 & @xmath33 & & @xmath33 & & 4 & 10.8% + b3 & @xmath33 & @xmath33 & & & 4 & 10.8% + b4 & & & @xmath33 & & 4 & 10.8% + c & @xmath33 & & & & 6 & 16.2% + following the above procedures we found 37 bubbles in the entire 100 deg@xmath1 area of taurus molecular cloud . each bubble is referred as a `` taurus molecular bubble '' ( tmb ) . the positions and classifications are listed in table [ bubble ] . the numbers and percentage of each class are illustrated in table [ bubbleclass ] . the @xmath63co integrated intensity maps , p - v diagrams , gaussian fitting profiles and channel maps for the bubbles are presented in fig . [ bubble_fig1 . ] - fig . [ bubble_fig37 . ] . if the morphology of a contour map is a closed ring , we then considered it to be an expanding bubble . if the ring on the contour map is incomplete , we then called it a broken bubble . there are 3 expanding bubbles ( tmb_07 , tmb_10 and tmb_24 ) and 34 broken bubbles among all the bubbles in taurus . to examine the impact of bubbles on the host cloud we calculated the mass , momentum , kinetic energy , dynamical timescale and energy deposition rate of the bubbles . assuming the @xmath3co(1 - 0 ) emission of the bubble is optically thin , the total column density is derived as follows : @xmath64 where @xmath39 , @xmath40 erg s , @xmath41 and @xmath65 . the excitation temperature , @xmath66 is assumed to be 25 k. @xmath43 is the observed source antenna temperature with proper correction for antenna efficiency . the optical depth correction factor , @xmath67 , is estimated from the following formulae @xcite . @xmath68 dv } , \label{equ : ftau}\ ] ] where @xmath69 is the opacity of the @xmath3co transition . assuming equal excitation temperatures for @xmath3co and @xmath2co , we can get @xmath70 where @xmath71 and @xmath72 are the brightness temperature of @xmath2co and @xmath3co , respectively . @xmath73 is the opacity of the @xmath2co transition . assuming the @xmath2co emission from the bubbles is optically thick ( @xmath74 ) , the opacity of @xmath3co can be obtained from @xmath75 with the column density and area we can obtain the bubble mass . using the bubble mass and expansion velocity we can then get the momentum and kinetic energy of the bubble using @xmath76 and @xmath77 , respectively . the kinetic timescale of bubble can be calculated as @xmath78 , where @xmath79 is the radius and @xmath80 is the expansion velocity of the bubble . the bubble energy injection rate , @xmath81 , can be estimated as @xmath82 . the physical parameters of all bubbles are listed in table [ bubble ] . the momentum and kinetic energy are lower limits mainly because of the underestimate of the minimum expansion velocity . the total mass , momentum , energy and energy injection rate of all detected bubbles in taurus molecular cloud are about @xmath83 , @xmath84 , @xmath85 erg and @xmath86 , respectively . cccccccrrrccr tmb_01 & 04 12 08 & 24 53 33 & a & n & 0.98 & 1.3 & 25 & 31 & 0.39 & 0.8 & 0.17 & 15.5 + tmb_02 & 04 14 28 & 27 45 53 & b1 & y & 0.60 & 1.3 & 6 & 7 & 0.09 & 0.5 & 0.06 & 3.5 + tmb_03 & 04 16 20 & 28 28 53 & a & y & 0.76 & 1.8 & 10 & 18 & 0.33 & 0.4 & 0.25 & 9.0 + tmb_04 & 04 19 05 & 27 33 33 & a & y & 0.62 & 1.3 & 7 & 9 & 0.12 & 0.5 & 0.08 & 4.5 + tmb_05 & 04 21 12 & 26 55 33 & b3 & y & 0.56 & 1.8 & 4 & 8 & 0.14 & 0.3 & 0.14 & 4.0 + tmb_06 & 04 25 17 & 25 32 13 & b1 & n & 0.77 & 1.3 & 10 & 12 & 0.16 & 0.6 & 0.08 & 6.0 + tmb_07 & 04 25 29 & 26 10 13 & b1 & n & 1.12 & 2.3 & 102 & 234 & 5.31 & 0.5 & 3.51 & 117.0 + tmb_08 & 04 27 07 & 24 20 13 & a & n & 0.70 & 2.3 & 18 & 42 & 0.95 & 0.3 & 1.00 & 21.0 + tmb_09 & 04 27 31 & 26 16 53 & b4 & n & 0.49 & 1.0 & 4 & 4 & 0.04 & 0.5 & 0.03 & 2.0 + tmb_10 & 04 28 52 & 24 14 33 & a & y & 1.58 & 1.5 & 213 & 325 & 4.92 & 1.0 & 1.53 & 162.5 + tmb_11 & 04 29 32 & 26 32 33 & a & y & 0.84 & 2.0 & 41 & 83 & 1.68 & 0.4 & 1.31 & 41.5 + tmb_12 & 04 29 44 & 26 32 53 & b4 & y & 0.70 & 2.5 & 23 & 58 & 1.47 & 0.3 & 1.72 & 29.0 + tmb_13 & 04 30 31 & 24 26 13 & a & y & 0.73 & 2.3 & 18 & 41 & 0.93 & 0.3 & 0.94 & 20.5 + tmb_14 & 04 31 14 & 29 25 53 & b2 & y & 0.84 & 2.0 & 10 & 20 & 0.40 & 0.4 & 0.31 & 10.0 + tmb_15 & 04 31 30 & 24 14 33 & a & y & 0.84 & 1.3 & 34 & 43 & 0.54 & 0.6 & 0.27 & 21.5 + tmb_16 & 04 31 32 & 24 09 53 & b3 & n & 0.70 & 1.8 & 22 & 39 & 0.69 & 0.4 & 0.57 & 19.5 + tmb_17 & 04 31 35 & 23 35 13 & b4 & n & 0.42 & 1.0 & 2 & 2 & 0.02 & 0.4 & 0.02 & 1.0 + tmb_18 & 04 31 50 & 24 22 13 & c & n & 0.56 & 2.0 & 14 & 29 & 0.59 & 0.3 & 0.69 & 14.5 + tmb_19 & 04 31 59 & 25 43 13 & b1 & y & 0.70 & 2.0 & 16 & 32 & 0.64 & 0.3 & 0.60 & 16.0 + tmb_20 & 04 32 03 & 25 36 53 & b2 & n & 0.42 & 1.3 & 4 & 5 & 0.07 & 0.3 & 0.07 & 2.5 + tmb_21 & 04 32 37 & 29 29 13 & a & n & 0.62 & 2.3 & 6 & 14 & 0.31 & 0.3 & 0.37 & 7.0 + tmb_22 & 04 32 39 & 24 46 13 & a & y & 1.06 & 1.3 & 28 & 36 & 0.45 & 0.8 & 0.17 & 18.0 + tmb_23 & 04 33 10 & 26 08 53 & b4 & n & 0.28 & 1.3 & 2 & 2 & 0.03 & 0.2 & 0.04 & 1.0 + tmb_24 & 04 33 13 & 25 24 53 & c & n & 0.49 & 1.3 & 5 & 7 & 0.09 & 0.4 & 0.07 & 3.5 + tmb_25 & 04 33 34 & 24 20 53 & a & y & 0.28 & 3.0 & 5 & 15 & 0.46 & 0.1 & 1.61 & 7.5 + tmb_26 & 04 34 47 & 29 37 13 & b3 & n & 0.46 & 2.3 & 3 & 6 & 0.14 & 0.2 & 0.23 & 3.0 + tmb_27 & 04 36 02 & 28 23 13 & c & y & 1.40 & 2.5 & 64 & 161 & 4.08 & 0.5 & 2.39 & 80.5 + tmb_28 & 04 36 23 & 25 36 33 & b2 & y & 1.40 & 3.3 & 386 & 1275 & 41.84 & 0.4 & 31.93 & 637.5 + tmb_29 & 04 37 04 & 25 46 33 & c & y & 0.84 & 1.5 & 20 & 30 & 0.46 & 0.5 & 0.27 & 15.0 + tmb_30 & 04 38 11 & 26 05 53 & b3 & y & 1.90 & 1.8 & 340 & 604 & 10.68 & 1.0 & 3.23 & 302.0 + tmb_31 & 04 39 11 & 29 05 13 & b2 & n & 1.26 & 1.5 & 74 & 113 & 1.71 & 0.8 & 0.67 & 56.5 + tmb_32 & 04 39 48 & 28 35 33 & c & n & 0.70 & 2.0 & 7 & 14 & 0.27 & 0.3 & 0.26 & 7.0 + tmb_33 & 04 41 10 & 25 31 13 & c & n & 0.70 & 1.0 & 10 & 10 & 0.10 & 0.7 & 0.05 & 5.0 + tmb_34 & 04 44 20 & 28 36 53 & b1 & n & 1.26 & 2.8 & 143 & 399 & 11.08 & 0.4 & 7.95 & 199.5 + tmb_35 & 04 46 12 & 25 07 33 & a & n & 0.62 & 1.5 & 8 & 13 & 0.19 & 0.4 & 0.15 & 6.5 + tmb_36 & 04 46 43 & 24 59 13 & b1 & y & 0.56 & 1.8 & 12 & 21 & 0.38 & 0.3 & 0.39 & 10.5 + tmb_37 & 04 48 12 & 24 50 33 & a & n & 0.63 & 2.3 & 8 & 18 & 0.41 & 0.3 & 0.48 & 9.0 + the distribution of radius , mass , energy and dynamical timescale of bubbles are shown in fig . [ bubble - histogram ] . the radius of bubbles is in the range 0.28 - 1.9 pc . 78% of the bubbles are smaller than 1 pc . the mass of 65% of the bubbles is between @xmath87 and @xmath88 . the bubbles with mass lower than @xmath87 and higher than @xmath88 account for 16% and 19% , respectively . the highest bubble mass is @xmath89 . the energy of 60% of bubbles is in the range @xmath90 - @xmath91 erg . the bubbles with energy lower than @xmath90 erg and higher than @xmath91 erg account for 16% and 24% , respectively . the dynamical timescales of bubbles are between @xmath92 yr and @xmath93 yr . almost 95% of bubbles are younger than @xmath93 yr . compared to the outflows , the bubbles have about * 110 * times larger mass and * 24 * times higher energy . the extents of bubbles are larger than outflows , which can be seen from fig . [ yso - distribute ] . the dynamical timescales of bubbles are longer than that of outflows . among the 55 outflows we found that bipolar , monopolar redshifted and monopolar blueshifted outflows account for 45% , 44% and 11% , respectively . there are more red lobes than blue ones , which can be seen from the histograms in fig . [ outflow - histogram ] . the occurrence of more red lobes may result from the fact that taurus is thin @xcite . red lobes tend to be smaller and younger . the total mass and energy of red lobes is similar to blue lobes on average , which can be seen from the upper right panel and lower left panel of fig . [ outflow - histogram ] . the outflows are driven by four types of ysos . from table [ outflownumyso ] we can see class i , flat , class ii and class iii account for 36.4% , 18.2% , 27.3% and 12.7% of all the driving sources , respectively . [ yso - distribute ] shows the distribution of different classes of ysos driving outflows ( hereafter outflow - driving yso ) and ysos inside the bubbles ( hereafter bubble - driving yso ) . the rough dividing line shows that there are more outflow - driving ysos in class i , flat and class ii while few outflow - driving ysos in class iii , which indicates that outflows are more likely appear in the earlier stage ( class i ) than in the later phase ( class iii ) of star formation . there are more bubble - driving ysos of class ii and class iii while there are few bubble - driving ysos of class i and flat , implying that the bubble structures are more likely to occur in the later stage of star formation . from the size of the symbols we can see that the larger outflows and bubbles are , the higher energy they have . with the complete sample of outflows and bubbles we can estimate the overall impact of dynamical structures on the taurus molecular cloud . we investigated whether the outflows and bubbles have enough energy to potentially unbind the entire taurus molecular cloud or drive the turbulence in the cloud . using a total mass of @xmath94 @xcite and an effective radius of 13.8 pc @xcite for the 100 deg@xmath1 region of taurus , we calculated the magnitude of the gravitational binding energy @xmath95 to be @xmath9 * erg*. the total kinetic energy of outflows from the 44 deg@xmath1 region of taurus is @xmath57 erg , much less than the gravitational binding energy . given that we searched for outflows around ysos only in the spitzer 44 deg@xmath1 survey region not the overall area of taurus , we may well have missed some outflows . most of the gas of taurus is centered on the spitzer 44 deg@xmath1 survey region , which can be seen from fig . 2 of @xcite . there are not many ysos outside the spitzer coverage in taurus and those ysos are generally clustered , which can be seen from figure 1 of @xcite . so there should be few outflows outside the spitzer coverage in taurus and the outflows we found around ysos in the 44 deg@xmath1 area account for the majority of outflows in taurus . similarly , the total kinetic energy of the detected bubbles in the 100 deg@xmath1 taurus region is @xmath96 erg , which also can not balance the gravitational potential energy of the entire cloud . * the turbulent energy of the taurus molecular cloud is given approximately by * @xmath97 * where @xmath98 is the three dimensional turbulent velocity dispersion , which can be calculated by * @xmath99 * here @xmath100 = 2 km s@xmath101 is the one dimensional fwhm velocity dispersion based on typical @xmath102co spectra in taurus @xcite . then we get @xmath103 km s@xmath101 . the total mass of the 100 deg@xmath104 region of taurus is @xmath105 @xcite . using eq . ( [ equ : energyturb ] ) we obtain the turbulent energy of the taurus to be @xmath106 erg . * the energy of all detected outflows ( * @xmath107 erg * ) is about two orders of magnitude less than the turbulent energy of the cloud . the lower limit of the total energy of the bubbles ( @xmath96 erg ) is 29% of the turbulent energy . * we conclude that the total energy of outflows and bubbles can not balance the turbulence in taurus . * we also estimated the total outflow energy rate ( outflow luminosity , @xmath108 ) and the total bubble energy rate ( bubble luminosity , @xmath109 ) with the energy rate needed to maintain the turbulence ( turbulent energy dissipation , @xmath110 ) . * the luminosity of outflows is @xmath111 erg s@xmath101 after inclination and blending correction . summing up the luminosity in table [ outflowparameters ] we get the energy injection rate of bubbles to be @xmath112 erg s@xmath101 . * * the turbulent dissipation rate can be calculated as * @xmath113 * where @xmath114 is the turbulent dissipation time . we estimate the turbulent dissipation time through two methods based on numerical simulations . * * first , the turbulent dissipation time of the cloud is given by @xcite * @xmath115 * where @xmath116 pc is the cloud diameter and @xmath117 is the one dimensional turbulent velocity dispersion along the line of sight , * @xmath118 * here @xmath100 = 2 km s@xmath101 is the same as that in [ turbulentenergy ] . then we get @xmath119 km s@xmath8 . combining eq . ( [ equ : t_diss_mckee ] ) and eq . ( [ equ : sigma_1d ] ) we obtain the turbulent dissipation time , @xmath120 yr , which is about 6 times larger than the result ( @xmath121 yr ) in @xcite . then using eq . ( [ equ : l_turb ] ) we get the turbulent dissipation rate to be @xmath122 erg s@xmath101 , which is 51% of the luminosity of outflows and only 10% of the energy injection rate of bubbles . * * second , we follow @xcite to calculate the dissipation time of the cloud by * @xmath123 * where @xmath124 is the free - fall timescale , @xmath125 is the mach number of the turbulence , and @xmath126 is the ratio of the driving length to the jean s length of the cloud , * @xmath127 * using our extensive data sets we get @xmath128 0.53 pc , which is the average size of outflows and bubbles we found in taurus . the jean s length is defined as * @xmath129 * where @xmath130 is the sound speed @xcite . for an ideal gas * @xmath131 * where @xmath132 is the boltzmann s constant , @xmath133 = 10 k is the temperature of the taurus molecular cloud , and @xmath134 = 2.72 is the mean molecular weight @xcite . then we get the sound speed , @xmath135 = 0.3 km @xmath136 . @xmath137 is the representative volume density of the region where dissipation takes place . we estimate the volume density to be @xmath138 1500 @xmath139 . using eq . ( [ equ : lambda_j ] ) we obtain the jean s length of the region , @xmath140 pc , which is about 4 times larger than that ( @xmath141 pc ) in perseus @xcite . and then using eq . ( [ equ : kappa ] ) we obtain @xmath142 , which is different from the assumption ( @xmath143 1 ) by @xcite and @xcite . * * the mach number of the turbulence can be calculated by @xcite * @xmath144 * using eq . ( [ equ : sigma3d ] ) , eq . ( [ equ : c_s ] ) and eq . ( [ equ : m_rms ] ) we get the mach number , @xmath125 = 5 , which is different from the assumption ( @xmath145 10 ) by @xcite and @xcite . * * the free - fall timescale of the cloud , * @xmath146 * where * @xmath147 * is the average volume density of the cloud . then we get the free - fall timescale , @xmath148 yr . * * using the formulas from eq . ( [ equ : t_diss ] ) to eq . ( [ equ : rho_cloud ] ) we obtain the turbulent dissipation time to be @xmath149 yr . then we get the turbulent dissipation rate to be @xmath150 erg s@xmath101 , which is about 2.4 times larger than the luminosity of outflows but 48% of the energy injection rate of bubbles . the turbulent dissipation rate we obtained is close to that ( @xmath151 erg s@xmath101 ) of @xcite . in table [ dissipationrate ] we list the parameters related to the dissipation rate we get from the above two methods and compare them with the results of @xcite . * lcccc method & @xmath152 & @xmath153 & @xmath154 & @xmath110 + & & & ( @xmath93 yr ) & ( @xmath155 erg s@xmath8 ) + mo07 & - & - & @xmath156 & @xmath157 + ml99 & 0.64 & 5 & @xmath158 & @xmath159 + na12 & 1 & 10 & @xmath160 & @xmath161 + * both methods invoke numerical simulations to calibrate the numerical factors in addition to essentially dimensional arguments . the main difference of the two methods is the scale of the region where dissipation takes place . @xcite adopted the dimension of the entire cloud , while when we use the method given by @xcite the scale is the average size of outflows and bubbles . none of the simulations so far implements the physics ( excitation , radiative transfer , etc . ) necessary for actually modeling dissipation . thus we should treat the calculations above with caution and take them as dimensional and order - of magnitude estimates . * * comparing the energy injection rate of outflows and bubble with the turbulent dissipation rate , we conclude that in the current episode of star formation in taurus , both outflows and bubbles can sustain the currently observed turbulence in taurus . * protostellar winds will inject energy into the cloud and may help sustain turbulence @xcite . the winds can clear the gas surrounding the young star and form a bubble structure @xcite . to assess whether the winds can drive bubbles in taurus we compared the wind energy injection rate into the cloud ( @xmath162 ) with the total energy injection rate from bubbles of the cloud . following @xcite , we estimated the wind energy injection rate using equation 3.7 from @xcite : @xmath163 where @xmath164 is the wind velocity , which is generally assumed to be close to the star escape velocity . for the low- and intermediate - mass stars the escape velocity is about @xmath165 km s@xmath8 @xcite . similar to @xcite we assumed @xmath166 km s@xmath8 . the total mass loss rate from the protostellar winds is given by @xmath167 , which can be estimated by the sum of the wind mass loss rate for each bubble ( @xmath168 ) . the wind mass loss rate required to produce the bubbles is roughly estimated by equation ( 2 ) from @xcite : @xmath169 where @xmath170 is the total momentum of bubbles . the wind velocity , @xmath164 , is the same as that in eq . ( [ equ : windenergyinjection ] ) . @xmath171 is the wind timescale , which is assumed @xmath172 myr @xcite . ( [ equ : windmassloss ] ) we obtained the wind mass loss rates of each bubble ( @xmath168 ) , which are listed in table [ bubble ] . summing @xmath168 of all bubbles we find @xmath167 to be @xmath173 m@xmath174 yr@xmath8 . using eq . ( [ equ : windenergyinjection ] ) we obtained the wind energy injection rate ( @xmath162 ) to be @xmath175 * * erg s**@xmath101 , 31% of the total energy injection rate from bubbles in taurus , which is comparable to the turbulent dissipation rate in taurus . therefore , the protostellar winds can drive bubbles to sustain turbulence in taurus . * the origin of turbulence in the molecular cloud has been intensely debated over the past three decades ( e.g. * ? ? ? * ; * ? ? ? @xcite suggests that for a large fraction of clouds the turbulent driving is external . numerical simulations shows that the external sources of turbulence are likely to be large - scale hi streams @xcite , shocks @xcite , alfv@xmath176n waves @xcite , supernovae explosion and galactic differential rotation @xcite . it is unclear which one is the source of turbulence in taurus . * we have studied the dynamic structures including outflows and bubbles within the taurus molecular cloud using the 100 deg@xmath1 fcrao large - scale @xmath2co(1 - 0 ) and @xmath3co(1 - 0 ) maps and the spitzer protostellar catalog . the high sensitivity and large spatial dynamic range of the maps provide us an excellent opportunity to undertake an unbiased search for outflows and bubbles in this region . we also analyzed the energy injection of these dynamic structures into the entire cloud . our conclusions regarding the dynamic structures in taurus and their properties are as follows . we identified 55 outflows around the spitzer ysos in the main 44 deg@xmath1 area of taurus . in total , 31 of the detected outflows were previously unknown , increasing the number of outflows by a factor of 1.3 . we classified the outflows into 5 categories according to the morphology of contour maps and p - v diagrams . the classifications indicate the confidence level of the outflows . 76.3% of the outflows are in the `` most probable '' category in our study . most of the outflows are driven by class i , flat and class ii ysos while few outflows were found around class iii ysos , which indicates that the outflow activity likely occurred in the earlier stage rather than the late phase of the star formation . more bipolar and monopolar redshifted outflows were identified while few monopolar blueshifted ones were detected in our study . we detected 37 bubbles in the 100 deg@xmath1 region of taurus . all the bubbles were previously unknown . the bubbles were identified by the integrated intensity maps , p - v diagrams , gaussian fitting profiles and channel maps . the gravitational binding energy of the taurus molecular cloud is @xmath9 * erg*. the total kinetic energy of outflows and bubbles in taurus are @xmath4 erg and @xmath5 erg , respectively . neither outflows nor bubbles can balance the overall gravitational binding energy of taurus . \7 . the turbulent energy of the taurus molecular cloud is @xmath177 erg . the energy of all detected outflows and bubbles can not have generated the observed turbulence in taurus . the rate of turbulent dissipation in taurus is * between @xmath178 to @xmath179 erg s@xmath8*. the energy injection rates of outflows and bubbles are @xmath180 erg s@xmath8 and @xmath7 erg s@xmath8 , respectively . * both outflows and bubbles can sustain the turbulence in taurus at the current epoch . * the stellar winds can drive bubbles to sustain turbulence in the taurus molecular cloud . [ summary ] we are grateful to dr . yue , dr . z.y . zhang , dr . t. liu , dr . x.y . gao and dr . z.y . ren for their kind and valuable advice and support . we would like to thank the anonymous referee for the careful inspection of the manuscript and constructive comments particularly the important suggestions to examine the turbulent dissipation issue to improve the quality of this study . we also thank prof . w. butler burton for help in the review process . this work is partly supported by the china ministry of science and technology under state key development program for basic research ( 2012cb821802 ) , and the national natural science foundation of china ( 11373038 , 11373045 ) , the hundred talents program of the chinese academy of sciences and the young researcher grant of national astronomical observatories , chinese academy of sciences . to calculate molecular outflow parameters we need first to obtain the column density first . a simple solution of the equation of radiative transfer is @xmath181(1-e^{-\tau_\nu } ) , \label{equ : radiativetransfer}\ ] ] where @xmath43 is the source temperature , @xmath66 is the excitation temperature and @xmath182 is the background temperature , and @xmath183 is the frequency of the transition . the modified planck function @xmath184 is defined as @xmath185 where @xmath186 is boltzmann s constant and @xmath187 is planck s constant . the definition of optical depth in terms of upper level column density is expressed as @xcite @xmath188 assuming @xmath189 @xmath190 1 , @xmath191 @xmath190 @xmath66 and @xmath182 @xmath190 @xmath66 , we get the column density of the rotational upper level of the transition in the outflow by combining eq . ( [ equ : radiativetransfer ] ) with eq . ( [ equ : opticaldepth ] ) . @xmath192 where @xmath43 is the observed source antenna temperature with proper correction for antenna efficiency , @xmath193 is the speed of light , and @xmath194 is the spontaneous transition rate from the upper level(@xmath195 ) to the lower level(@xmath184 ) , which can be expressed as @xmath196 where @xmath197 is the permanent electric dipole moment of a molecule and @xmath184 = 0 . + the total column density of the outflow is @xmath198 where @xmath199 is the fraction of the @xmath36 in the upper level of the transition . under local thermal equilibrium(lte ) conditions , @xmath200 is given by @xmath201 where the statistical weight of the upper level @xmath202 , the lte partition function ( for @xmath203 @xmath204 @xmath205 ) @xmath206 and the rotational constant @xmath207 $ ] for the @xmath208 transition @xcite . then we can derive the total column density of outflow from eq . ( [ equ : columndensityupperlevel ] ) , eq . ( [ equ : aul ] ) , eq . ( [ equ : columndensitytotal ] ) and eq . ( [ equ : correctionfactor ] ) , @xmath209 if there is a high velocity wing in @xmath36 but not in @xmath37 profile , we assume @xmath36 is optically thin . then we can calculate the column density of outflow from eq . ( [ equ : derivedtotalcolumndensity ] ) . if there is high velocity wing both in @xmath36 and @xmath37 profile , we can correct the optical depth of @xmath36 using the following equation @xmath210 here @xmath211 and @xmath212 are the antenna temperatures of @xmath213 and @xmath214 ( with proper correction for antenna efficiency ) , respectively . @xmath73 and @xmath69 are the optical depths of @xmath36 and @xmath37 , respectively . we assume the abundance ratio of @xmath36 to @xmath37 is 65 @xcite . the correction factor for opacity is defined as @xmath215 then we get the corrected total column density of outflow as @xmath216 after obtaining the column density we can calculate other parameters of outflow . the mass of outflow can be calculated from @xmath217\mu_{\rm g}m({\rm h})s , \label{equ : outflowmass}\ ] ] where @xmath218=2.72 is the mean molecular weight @xcite , @xmath219=@xmath220 g is the mass of a hydrogen atom , and @xmath221/[\rm{co}]$ ] is assumed to be @xmath222 , and @xmath223 is the area of the outflow . + the momentum ( @xmath224 ) and energy ( @xmath225 ) of the outflow can be calculated from @xmath226 @xmath227 where @xmath228 is the average velocity of the outflow relative to the cloud systemic velocity and @xmath229 is obtained from eq.([equ : outflowmass ] ) . + the dynamical timescale @xmath47 can be estimated from @xmath230 where @xmath231 is the typical linear scale of the outflow lobe . the outflow luminosity , @xmath108 , can be estimated by dividing the kinetic energy by the dynamical timescale . it can be expressed as @xmath232
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we have identified outflows and bubbles in the taurus molecular cloud based on the @xmath0 deg@xmath1 five college radio astronomy observatory @xmath2co(1 - 0 ) and @xmath3co(1 - 0 ) maps and the spitzer young stellar object catalogs . in the main 44 deg@xmath1 area of taurus we found 55 outflows , of which 31 were previously unknown .
we also found 37 bubbles in the entire 100 deg@xmath1 area of taurus , all of which had not been found before .
the total kinetic energy of the identified outflows is estimated to be @xmath4 erg , which is * 1% * of the cloud turbulent energy .
the total kinetic energy of the detected bubbles is estimated to be @xmath5 erg , which is 29% of the turbulent energy of taurus .
the energy injection rate from outflows is @xmath6 , * 0.4 - 2 times * the dissipation rate of the cloud turbulence .
the energy injection rate from bubbles is @xmath7 erg s@xmath8 , * 2 - 10 times * the turbulent dissipation rate of the cloud .
the gravitational binding energy of the cloud is @xmath9 * erg * , * 385 * and 16 times the energy of outflows and bubbles , respectively .
we conclude that neither outflows nor bubbles can * provide enough energy to balance the overall gravitational binding energy and the turbulent energy of taurus .
however , * in the current epoch , stellar feedback is sufficient to maintain the observed turbulence in taurus .
= 5000 = 1000
| 25,253 | 419 |
superconducting mgb@xmath1 exhibits a number of rather peculiar properties , originating from the involvement of two sets of bands of different anisotropy and different coupling to the most relevant phonon mode @xcite . among them are pronounced deviations of the upper critical field , @xmath0 , from predictions of the widely used anisotropic ginzburg - landau theory ( aglt ) . apart from two - band superconductivity , mgb@xmath1 provides a link between low and high @xmath8 superconductors on a phenomenological level , particularly concerning vortex physics . in both high and low @xmath8 superconductors , for example , a phase transition of vortex matter out of a quasi - ordered `` bragg glass '' have been identified , with rather different positions in the @xmath5-@xmath6 plane . studying the intermediate mgb@xmath1 may help establishing a `` universal vortex matter phase diagram '' . here , we present a torque magnetometry study of the anisotropic upper critical field , equilibrium magnetization , and the vortex matter phase diagram of single crystalline mgb@xmath1 @xcite . we will show direct evidence of a temperature dependence of the @xmath0 anisotropy , discuss strong indications of a difference between the anisotropies of the penetration depth and @xmath0 , and present the @xmath5-@xmath6 phase diagram for @xmath7 . single crystals were grown with a cubic anvil high pressure technique , described in this issue @xcite . three crystals were used in this study , labeled a , b , and c. sharp transitions to the superconducting state indicate a high quality of the crystals . an @xmath9 curve of crystal b with @xmath10 can be found in ref.@xcite . the torque @xmath11 , where @xmath12 is the magnetic moment of the sample , was recorded as a function of the angle @xmath13 between the applied field @xmath14 and the @xmath15axis of the crystal in various fixed fields @xcite . for measurements close to @xmath8 , in fields up to @xmath16 , a non - commercial magnetometer with very high sensitivity was used @xcite . for part of these measurements , a vortex - shaking process was employed to speed up the relaxation of the vortex lattice @xcite . crystal a was measured in this system . crystals b and c were measured in a wider range of temperatures down to @xmath17 , in a quantum design ppms with torque option and a maximum field of @xmath18 . for crystals b and c , @xmath19 measurements at fixed angles were performed in addition to @xmath20 measurements in fixed @xmath5 . early measurements on polycrystalline or thin film mgb@xmath1 samples with various methods and single crystals by electrical transport yielded values of the anisotropy parameter of the upper critical field @xmath21 in a wide range of values of @xmath22 @xcite . more recently , several papers reported a temperature dependence of the @xmath0 anisotropy , ranging between about @xmath23 at @xmath24 and @xmath25 close to @xmath8 @xcite . in this section , we present direct evidence of a temperature dependence of the @xmath0 anisotropy @xmath2 and discuss details of it s behaviour , comparing the torque data with numerical calculations @xcite . four angular torque dependences are shown in fig.[mgb2hc2raw ] . panels a ) and b ) correspond to measurements at @xmath26 . for fields nearly parallel to the @xmath27-axis , both curves are flat , apart from a small background visible in panel b ) . only when @xmath5 is nearly parallel to the @xmath28plane there is an appreciable torque signal . the curve can be interpreted in a straight - forward way : for @xmath5 parallel to the @xmath15axis the sample is in the normal state , while for @xmath5 parallel to the @xmath28plane it is in the superconducting state . the crossover angle @xmath29 between the normal and the superconducting state is the angle for which the fixed applied field is the upper critical field . from the existence of both superconducting and normal angular regions follows immediately that @xmath30 and @xmath31 . in panel c ) , on the other hand , the crystal is seen to be in the superconducting state for all values of the angle @xmath13 , and therefore @xmath32 . finally , the data in panel d ) show only a small background contribution @xmath33 form and angular regime of the deviation from a straight line are incompatible with a superconducting signal . therefore , the crystal is here in the normal state for any @xmath13 , and we have @xmath34 . from figure [ mgb2hc2raw ] we therefore have two limitations for the upper critical field anisotropy , hereafter called @xmath2 , without any detailed @xmath0 criterion , and without any model fits : @xmath35 these relations show that _ the upper critical field anisotropy @xmath2 of mgb@xmath1 can not be temperature independent_. as an immediate implication , the _ anisotropic ginzburg - landau theory _ ( aglt ) in it s standard form _ does not hold for mgb@xmath1_. the deviation is strong , within a change of temperature of about @xmath36 , @xmath2 changes , _ at least _ , by a fifth of it s value . although it is clear that aglt with it s effective mass anisotropy model can not describe the data measured at _ different _ temperatures consistently , the detail analysis of the @xmath13 dependence of @xmath0 we used is based on aglt . we will show that as long as we stay at a _ fixed _ temperature , aglt is able to describe @xmath37 remarkably well @xcite . although the location of @xmath29 , for example in fig . [ mgb2hc2raw]a ) , seems clear at first sight , this clarity disappears , when examining the transition region in a scale necessary for the precise determination of @xmath29 ( see fig . 1 in ref . @xcite ) . for an strict analysis , it is necessary to take into account that the transtion at @xmath0 is rounded off by fluctuations . in sufficiently high fields , @xmath38 , the so - called `` lowest landau level '' ( lll ) approximation was used successfully to describe the effects of fluctuations around @xmath0 @xcite . in the case of the cuprates , the value of @xmath39 , and thus of the regime of applicability of the lll approximation , is a controversial issue ( see , e.g. , ref.@xcite for a theoretical discussion of the limits of the lll approximation ) . however , in the case of mgb@xmath1 , even using the theoretical criterion of ref . @xcite , which led to the high estimation @xmath40 in the case of yba@xmath1cu@xmath41o@xmath42 , we obtain an upper limit of @xmath43 . we therefore used a lll scaling analysis for the determination of @xmath44 or @xmath37 @xcite . from the resulting @xmath37 curve , the anisotropy parameter @xmath2 is then extracted by an analysis with aglt , which predicts the angular dependence of the upper critical field to be @xcite @xmath45 we note that in the rescaling of the torque according to the lll fluctuations theory , the target parameter @xmath2 is used , which is obtained only later with eq . ( [ hc2_theta ] ) . therefore , scaling analysis and determination of @xmath2 with eq . ( [ hc2_theta ] ) had to be performed iteratively in order to self consistently find @xmath2 . however , the @xmath44 and @xmath37 points obtained with the scaling analysis depend not very strongly on the value of @xmath2 used in the scaling and the procedure converges rather fast . figure [ hc2crystc ] shows the angular dependence of @xmath0 of crystal b and c. the curves shown in the figure are fits of eq . ( [ hc2_theta ] ) to the data , showing that the angular dependence of @xmath0 is well described by aglt at both temperatures . on the other hand , the anisotropy parameter @xmath2 needed to describe the data with eq.([hc2_theta ] ) is temperature dependent , as is best seen in the inset . dependence of crystals b and c , at @xmath46 and @xmath47 . shown are results obtained both from @xmath20 measurements in fixed @xmath5 [ b at @xmath26 ( @xmath48 ) and at @xmath47 ( @xmath49 ) , c at @xmath26 ( @xmath50 ) and at @xmath47 ( @xmath51 ) ] and from @xmath19 measurements at fixed @xmath13 [ b at @xmath26 ( @xmath52 ) ] . inset : calculated curves from the main panel , scaled to the fitted value of @xmath53 . ] vs.temperature @xmath6 . open symbols correspond to @xmath54 , full symbols to @xmath55 , from fits of eq . ( [ hc2_theta ] ) to the @xmath37 data . up triangles are from measurements on sample a ( with @xmath29 determined with a simple `` straight line crossing '' criterion ) and squares are from measurements on sample b , using a fluctuation analysis ( see text ) . full lines are theoretical calculated curves @xcite . b ) temperature dependence of the upper critical field anisotropy @xmath21 , determined from fits of eq . ( [ hc2_theta ] ) to @xmath37 . the full line is again from the theoretical calculation of ref . @xcite . ] the irreversible properties of the two crystals ( b and c ) are different in a pronounced way ( see secs . [ lockin ] and [ pesec ] ) , showing that they have a rather different defect structure . the good agreement both in value and angular dependence of @xmath0 of crystals b and c that is observable in fig . [ hc2crystc ] indicates that such differences in the defect structure do not influence the upper critical field much , at least in the region between @xmath46 and @xmath47 , and therefore can not influence our conclusion of a @xmath6 dependent @xmath0 anisotropy . small , but systematic , deviations from the angular dependence of @xmath0 according to eq . ( [ hc2_theta ] ) were observed only at temperatures close to @xmath8 . it may indicate that we are approaching @xmath39 in this region and the values of @xmath44 and @xmath37 obtained from the lll scaling analysis ( see above ) start to deviate from the mean field values . the good general approximation of @xmath37 by eq.([hc2_theta ] ) is in agreement with recent calculations @xcite . however , the calculations predict small deviations at low temperatures @xcite , rather than close to @xmath8 . our experimental limitation of fields up to @xmath56 may prevent the observation of deviations from eq . ( [ hc2_theta ] ) at low temperatures . the upper critical fields parallel and perpendicular to the layers obtained with the scaling analysis and eq . ( [ hc2_theta ] ) are shown in fig . [ hc2mgb2sumup]a ) . results obtained for two crystals measured in two magnetometers are depicted as different symbols . the @xmath57dependence of @xmath53 is in agreement with ( isotropic ) calculations by helfand _ @xcite , with @xmath58 . on the other hand , @xmath59 exhibits a slight positive curvature near @xmath8 . these features are common to highly anisotropic ( layered ) superconductors . although mgb@xmath1 as a whole is rather isotropic , superconductivity is dominant on the quasi-2d bands , which may well account for the different @xmath6 dependence of @xmath53 and @xmath60 . this may also be the origin of the positive curvature of @xmath0 observed in other measurements of bulk , thin film and single crystal mgb@xmath1 @xcite . due to the lack of low @xmath6 data and the @xmath61 dependence , only an estimation @xmath62 can be given . the anisotropy data [ fig . [ hc2mgb2sumup]b ) ] show that @xmath2 _ systematically _ _ decreases _ with increasing temperature , from @xmath63 at @xmath64 to @xmath65 at @xmath66 . from the experimental data shown in fig . [ hc2mgb2sumup ] we estimate @xmath67 , while at zero temperature , @xmath2 may become as large as @xmath68 . comparing our data with the data reported by other authors @xcite , we note that electrical transport measurements @xcite yield too high values of @xmath53 @xcite . all bulk measurements ( torque @xcite , magnetization @xcite , thermal conductivity @xcite , and specific heat @xcite ) agree well on the @xmath69 dependence and value . concerning @xmath59 , and consequently @xmath61 , however , reported values differ from each other . exchanging the samples between different groups could help clarifying , whether the discrepancies of @xmath59 values are mainly due to sample differences or due to differences in the experimental methods employed . very recently , @xmath59 , @xmath69 , and @xmath61 , have been calculated for mgb@xmath1 @xcite . the fermi surface was modeled as consisting of two separate sheets , approximated as simple spheroids , but with average characteristics taken from first principles calculations . the result of these calculations are compared with our experimental data in fig.[hc2mgb2sumup ] . very good agreement is seen for the upper critical field perpendicular to the layers ( @xmath70 ) . qualitatively , calculations and experiment also agree well for the upper critical field parallel to the layers ( @xmath71 ) . this shows that the essential source of the deviations of the upper critical field from aglt predictions is captured with a simple effective two band model , while further details of the fermi surface and superconducting gap are negligible . in fig . [ hc2mgb2sumup ] , we see good quantitative agreement between experimental data and the theoretical curve between @xmath72 and @xmath73 . the deviations at lower @xmath6 may , on the one hand , be due to a decreased the accuracy of our analysis because the field limitation of @xmath56 restricts the angular range where @xmath0 data could be obtained . this can lead to deviations larger than the estimated error bars , especially since the theoretical calculations indicate deviations of the @xmath74 dependence from the prediction of eq.([hc2_theta ] ) at low @xmath6 @xcite . on the other hand , @xmath0 at low temperatures depends on the shape of the fermi surface in rather subtle manner , and the model fermi surface used for the calculations @xcite may be too simple for a quantitatively correct description at low @xmath6 . the deviations at higher @xmath6 may be due to the limitations of the lll scaling approach in low fields , and or due to the influence of disorder , which is not accounted for in the calculations . close to @xmath8 , non - locality is not important , and consequently , aglt is expected to hold even if this is not the case at lower @xmath6 . despite of this , fig . [ hc2mgb2sumup ] clearly indicates that the variation of @xmath2 with temperature is the strongest close to @xmath8 . therefore , in mgb@xmath1 , aglt seems to have a very limited range of applicability indeed . an alternative method to obtain the anisotropy parameter @xmath75 of a superconductor , used often and with success in the case of cuprates @xcite , consists of measuring the torque , as a function of angle , well below @xmath0 , and analyzing the data with a formula developed by kogan @xcite : @xmath76 where @xmath77 , @xmath78 is the effective mass anisotropy , @xmath79 is the in - plane penetration depth , @xmath80 is the volume of the crystal , @xmath81 is the flux quantum , and @xmath82 is a constant of the order of unity depending on the vortex lattice structure . equation ( [ tau_rev ] ) is valid in the limits of fields @xmath83 and not too close to @xmath8 . a further restriction is that eq . ( [ tau_rev ] ) describes the reversible torque only . to obtain the true reversible torque , we employed a vortex - shaking process @xcite . in the investigated field and temperature region , the shaked torque was found to be well reversible . in @xmath84 [ a ) ] and @xmath85 [ b ) ] . shown together with the measured values ( open circles ) are best fits of eq . ( [ tau_rev ] ) ( full curves ) . ] in fig . [ mgb2taurev ] , normalized torque @xmath86 vs angle @xmath13 curves , measured in different fields at @xmath47 , are compared . increasing @xmath5 from @xmath87 [ panel a ) ] to @xmath85 [ panel b ) ] leads to an unexpectedly large shift of the maximum torque towards @xmath88 , which may indicate an increase of the anisotropy @xmath3 with increasing @xmath5 . this is confirmed by the analysis of the data with eq . ( [ tau_rev ] ) . the best agreements of the equation with the data are obtained for @xmath89 in @xmath84 and @xmath90 in @xmath85 ( full curves in fig . [ mgb2taurev ] ) . although descriptions with @xmath90 in @xmath84 or @xmath89 in @xmath85 are also possible without obvious discrepancies to the data , the corresponding qualities of the fit as expressed by the parameter @xmath91 are worse by more than an order of magnitude in both cases . from the analysis of reversible torque data for crystal a measured in the range of fields and temperatures from @xmath92 to @xmath93 and from @xmath94 to @xmath95 @xcite , a few points are worth to be emphasized : 1 . ) @xmath3 is field dependent , increasing nearly linearly from @xmath87 in zero field to @xmath96 in @xmath93 . no clear @xmath6 dependence is visible between @xmath94 and @xmath95 . 3 . ) the effective anisotropy @xmath3 , as obtained from the analysis with eq.([tau_rev ] ) is different from the @xmath0 anisotropy @xmath2 . especially concerning point 3 . ) , it is important to recognize that , also theoretically , the anisotropy @xmath3 is not necessarily the same as the @xmath0 anisotropy @xmath2 . when aglt is not applicable , the anisotropies of the penetration depth , @xmath97 , and of the upper critical field , @xmath2 , differ in general . calculations of @xmath97 of mgb@xmath1 @xcite indeed found values much lower than the upper critical field anisotropy values . there is also experimental support for a low @xmath97 @xcite . in eq . ( [ tau_rev ] ) , @xmath3 appears twice , and in a first approximation @xcite , the appearance outside of the logarithm can be thought of as due to the @xmath98 anisotropy , while the appearance in the logarithm is linked to the @xmath0 anisotropy . a corresponding calculation with different ( fixed ) @xmath97 and @xmath2 yields @xcite a field dependent ( common ) effective anisotropy @xmath3 similar to the experimental observations . field dependent point - contact spectroscopy @xcite and specific heat @xcite measurements indicate that the small gap disappears in fields of the order of @xmath99 , i.e. , superconductivity in the ( 3d ) @xmath100 fermi sheets is rapidly suppressed by even low fields , whereas it persists in the ( 2d ) @xmath101 sheets up to much higher fields . this should result in an increase of the effective ( bulk ) anisotropy with increasing @xmath5 . further studies are needed for a complete understanding of the detailed interplay of the effects described above . in the `` unshaked '' torque data of crystals a and b , a pronounced peak in the irreversible torque for field alignments close to @xmath102 , was observed ( see , e.g. , upper inset of fig . 5 of ref.@xcite ) . it is tempting to ascribe this feature , also observed by other authors @xcite , to `` intrinsic pinning '' , in analogy to observations on strongly anisotropic cuprate superconductors . however , the observation of such `` intrinsic pinning '' in mgb@xmath1 is rather counter - intuitive , since the `` intrinsic pinning '' is mostly determined by the ratio of the @xmath27-axis coherence length to the separation of the superconducting layers , which is much larger in mgb@xmath1 than in the case of cuprates in the region where `` intrinsic pinning '' is commonly observed by torque magnetometry ( see , e.g. , @xcite ) . the apparent paradox is resolved by further measurements : torque measurements on crystal c with the same conditions show no sign of `` intrinsic '' pinning for @xmath102 @xcite , indicating the _ extrinsic _ origin of the feature . the most likely cause of the peak in the irreversible torque for @xmath102 is a small amount of stacking faults . it may indicate the presence of some stacking faults in crystals a and b , while they would seem to be absent in crystal c @xcite . in low @xmath8 superconductors , such as nbse@xmath1 , the order - disorder transition is signified experimentally by a peak effect ( pe ) in the critical current density @xcite . we observed such a pe by torque measurements on mgb@xmath1 single crystals b and c , both in @xmath20 and @xmath19 measurements , as can be seen in fig.[peakmgb2 ] . in sec . [ lockin ] , we have noticed that crystals b and c behave quite differently for @xmath102 , which may be due to the presence of a small number of stacking faults in the former one . the presence of the pe in two crystals with such pronounced differences strongly indicates that the pe , or rather it s underlying mechanism , is an intrinsic feature of mgb@xmath1 . a study with a `` minor hysteresis loop '' technique on crystal b @xcite revealed a history dependent critical current density in the pe region , compatible with and expected for the behaviour at the order - disorder transition . , vs field , at @xmath103 . shown are representative curves , measured at angles ( from left to right ) @xmath104 , @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 , @xmath110 , and @xmath111 . horizontal lines indicate zero and the criterion of @xmath112 chosen for the determination of the irreversibility line . for the curve measured at @xmath113 , peak maximum and onset are indicated ( see text ) . ] figure [ perawtheta ] shows the irreversible part @xmath114 of the torque , scaled by @xmath115 , vs field , at @xmath103 for various angles . the scaling was chosen to minimize the angle and field dependence intrinsic to the torque . since the peak is not visible at all temperatures and angles as well as in fig.[peakmgb2 ] , onsets and maxima were determined from irreversible torque curves as those shown in fig.[perawtheta ] . @xmath116 was defined as the field , where the irreversible torque starts to deviate from a straight line behaviour , as indicated in the figure for the curve measured at @xmath117 . @xmath116 , defined in this way , is close to @xmath118 as indicated in fig.[peakmgb2]b ) . however , we note that with the determination of onsets and maxima from the irreversible torque , the fine details of the differences in the field increasing and decreasing branch of the hysteresis loops are lost . . a ) torque @xmath119 vs @xmath5 , measured at an angle of @xmath120 between the @xmath27-axis of the sample and the applied field . due to the anisotropy , the location in fields of the pe is much higher than for @xmath7 , but the angular scaling is straight forward @xcite . b ) magnetization @xmath121 vs @xmath5 , for @xmath7 , determined by squid magnetometry . c ) absolute magnetic susceptibility @xmath122 vs @xmath123 , measured with an ac amplitude @xmath124 and a frequency of @xmath125 . the pe visible in panels a ) and c ) is indicated by arrows . ] it can be seen in fig . [ perawtheta ] that the height of the peaks varies in a pronounced way with the angle @xmath13 . one possible explanation for this behaviour is an interaction of the peak effect with stacking faults @xcite . although the presence and the location of the peak effect are not affected by stacking faults , the extent of hysteresis may be . the difference of how pronounced the peaks of crystals b and c are [ see fig . [ peakmgb2]b ) ] supports such a scenario . the location in higher fields of the peak effect in crystal c indicates that there is less point - like disorder present in this crystal than in crystal b. however , the smaller ratio @xmath126 in crystal c , compared to crystal b , is difficult to explain with only one sort of disorder . individual strong pinning , e.g. , by sparse stacking faults , should be much more efficient in the disordered phase than in the bragg glass with it s nearly perfect ordered lattice @xcite . if the peak height observed in crystal b is affected by stacking faults , a more pronounced pe close to @xmath102 is natural , since the pinning efficiency of stacking faults , similar to twin boundary pinning , is strongly direction - dependent @xcite . on the other hand , the peak height can be influenced by the natural angle dependence of the torque , despite the scaling made . this is because the @xmath127 is only an approximation , which is not appropriate for all angles @xmath13 , in a superconductor with pronounced anisotropy ( see also ref . @xcite ) . the angular dependence of the _ onsets and maxima _ of the pe tracks the one of @xmath0 , i.e. , it follows eq.([hc2_theta ] ) @xcite . this indicates that the pe ( or rather it s underlying mechanism ) is a feature for all directions of the applied field , and not just of the angular region where it is readily discernible . to directly check the situation for @xmath7 and @xmath102 , where torque measurements are not possible , squid and ac susceptibility @xcite measurements were performed . ( crystal b ) for @xmath7 . data points shown are from the projection of the torque vs field data presented in ref . @xcite and the projection of additional @xmath20 measurements in fixed @xmath5 . the dashed line is a calculation @xcite of the @xmath128 dependence ( cf . [ hc2mgb2sumup ] ) , the dotted lines are guides for the eye . the different phases of vortex matter are labeled ( see text ) . ] in fig . [ pedifftech ] , we compare measurement curves obtained on crystal b at @xmath129 , using different experimental techniques . torque measurements performed at an angle of @xmath120 show [ fig . [ pedifftech]a ) ] a clearly discernible pe located in a field of about @xmath130 . scaled with eq . ( [ hc2_theta ] ) to @xmath7 , this corresponds to @xmath131 to @xmath132 . as can be seen in fig.[pedifftech]b ) , there is no sign of a pe observable in squid data in this field region . generally , no sign of a peak effect was observed by squid magnetometry at any temperature , for both field directions . this is likely due to insufficient sensitivity of the squid . in ac susceptibility data [ fig . [ pedifftech]c ) ] , on the other hand , a pe _ is _ visible for @xmath7 in the appropriate field region . a report of the ac susceptibility results will be published elsewhere @xcite . a pe in mgb@xmath1 was also reported recently by other authors , in the case of @xmath7 from transport data @xcite and ac susceptibility with a local hall probe @xcite , in the case of @xmath102 from transport data @xcite . the phase diagram for @xmath7 obtained from torque magnetometry , based on both @xmath20 and @xmath19 measurements and the angular scaling of eq . ( [ hc2_theta ] ) is presented in fig.[mgb2phdhc ] . the magnitude of the peaks is reduced quickly by increasing the temperature , and above @xmath133 , the pe is no longer discernible in the torque data . this is due to the decreased sensitivity of the torque magnetometer in lower fields and due to thermal smearing of the effective pinning potential . in a recent report of low frequency ac susceptibility measurements @xcite , the peak effect was observed for @xmath7 at temperatures up to about @xmath73 , and interpreted in terms of the order - disorder transition as well . in ref.@xcite , the transformation of the pe into a `` step - like '' ac susceptibility is reported for the temperature interval between @xmath134 and @xmath135 , and interpreted as a signature of thermal melting . in our case , no step - like feature in the reversible torque was observed in the continuation of the pe . it should be emphasized , that thermal melting so far below @xmath0 would be at odds @xcite with theoretical expectations @xcite . the equilibrium order - disorder transition , which corresponds to @xmath136 @xcite , is located in fields of about @xmath4 in crystal b and in about @xmath137 in crystal c. the peak effect observed in other crystals by transport was reported to be located even closer to @xmath0 @xcite . these differences are natural for a disorder - induced phase transition in crystals with varying degrees of disorder . form and location of the pe observed in mgb@xmath1 resembles results obtained on nbse@xmath1 single crystals with varying degrees of disorder @xcite , but are rather different from the order - disorder transition in cuprate superconductors @xcite . in summary , studying the anisotropic superconducting state properties of mgb@xmath1 revealed a strong temperature dependence of the upper critical field anisotropy @xmath2 , and indicated a difference of the anisotropies of the penetration depth and the upper critical field . these findings , which imply a breakdown of the standard form of the widely used anisotropic ginzburg - landau theory in mgb@xmath1 , can be explained by superconductivity in this compound involving two band systems of different dimensionality , in accordance with microscopic studies . a pronounced peak effect in the magnetic hysteresis is a signature of an `` order - disorder '' transition of vortex matter , similar to transitions in both high @xmath8 cuprate and low @xmath8 superconductors . despite the intermediate importance of thermal fluctuations in mgb@xmath1 , the phase diagram resembles quite closely the one of the low @xmath8 superconductor nbse@xmath1 . on the other hand , chances of a proper identification of mainly thermally induced melting at higher temperatures are better in mgb@xmath1 , due to the increased thermal fluctuations . we thank v. g. kogan and b. batlogg for enlightening and stimulating discussions . this work was supported by the swiss national science foundation , by the european community ( contract ica1-ct-2000 - 70018 ) , by the polish state committee for scientific research ( 5 p03b 12421 ) , and by the swiss federal office bbw ( 02.0362 ) . note that there are two torque conventions used in the literature , leading to opposite signs of the torque shown , but to the same physical consequences . the convention used here ( and in refs.@xcite ) is that the diamagnetism is expressed by a negative value of the magnetic moment . the other convention , used in refs.@xcite , is that the diamagnetism is expressed solely by the direction of the ( positive ) magnetic moment vector . l. lyard , p. samuely , p. szabo , c. marcenat , t. klein , k. h. p. kim , c. u. jung , h .- s . lee , b. kang , s. choi , s .- lee , l. paulius , j. marcus , s. blanchard , a. g. m. jansen , u. welp , and w. k. kwok , cond - mat/0206231 . this justifies our aglt based analysis _ a posteriori_. an only small deviation of the @xmath138dependence of @xmath139 from aglt predictions ( with a @xmath6 dependent anisotropy ) was also found in numerical calculations @xcite . in a very recent study , kogan [ v. g. kogan , cond - mat/0207688 ] performed detailed calculations of the torque density from the free energy of the london model with different anisotropies @xmath2 and @xmath97 . the obtained result is more complicated than the first approximation used in ref . @xcite . x ray investigations of crystals b and c are in progress . stacking faults have been observed in other samples of mgb@xmath1 , see , e.g. , y. zhu , l. wu , v. volkov , q. li , g. gu , a. r. moodenbaugh , m. malac , m. suenaga , and j. tranquada , physica c * 356 * , 239 ( 2001 ) . see , e.g. , s. s. banerjee , n. g. patil , s. ramakrishnan , a. k. grover , s. bhattacharya , p. k. mishra , g. ravikumar , t. v. c. rao , v. c. sahni , m. j. higgins , c. v. tomy , g. balakrishnan , and d. m. paul , phys . b * 59 * , 6043 ( 1999 ) . in the bragg glass , the strong elastic forces disfavour non - collective pinning . similar ideas were discussed in the context of cuprate superconductors with twin boundaries . see a. i. larkin , m. c. marchetti , and v. m. vinokur , phys . . lett . * 75 * , 2992 ( 1995 ) . similar observations were made on a yba@xmath1cu@xmath41o@xmath42 crystal containing two twin boundaries . see w. k. kwok , j. a. fendrich , c. j. v. der beek , and g. w. crabtree , phys . lett . * 73 * , 2614 ( 1994 ) .
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the angular and temperature dependence of the upper critical field @xmath0 in mgb@xmath1 was determined from torque magnetometry measurements on single crystals .
the @xmath0 anisotropy @xmath2 was found to decrease with increasing temperature , in disagreement with the anisotropic ginzburg - landau theory , which predicts that the @xmath2 is temperature independent .
this behaviour can be explained by the two band nature of superconductivity in mgb@xmath1 .
an analysis of measurements of the reversible torque in the mixed state yields a field dependent effective anisotropy @xmath3 , which can be at least partially explained by different anisotropies of the penetration depth and the upper critical field .
it is shown that a peak effect in fields of about @xmath4 is a manifestation of an _ order - disorder _ phase transition of vortex matter .
the @xmath5-@xmath6 phase diagram of mgb@xmath1 for @xmath7 correlates with the intermediate strength of thermal fluctuations in mgb@xmath1 , as compared to those in high and low @xmath8 superconductors .
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it is believed that a neutron star begins its life as a proto - neutron star ( pns ) in the aftermath of a supernova explosion . the evolution of the pns depends upon the star s mass , composition , and equation of state ( eos ) , as well as the opacity of neutrinos in dense matter . previous studies @xcite have shown that the pns may become unstable as it emits neutrinos and deleptonizes , so that it collapses into a black hole . the instability occurs if the maximum mass that the equation of state ( eos ) of lepton - rich , hot matter can support is greater than that of cold , deleptonized matter , and if the pns mass lies in between these two values . the condition for metastability is satisfied if `` exotic '' matter , manifested in the form of a bose condensate ( of negatively charged pions or kaons ) or negatively charged particles with strangeness content ( hyperons or quarks ) , appears during the evolution of the pns . even if collapse to a black hole does not occur , the appearance of exotic matter might lead to a distinguishable feature in the pns s neutrino signature ( _ i.e. _ , its neutrino light curve and neutrino energy spectrum ) that is observable from current and planned terrestrial detectors . this was investigated recently by pons _ @xcite who studied the evolution of a pns in the case where hyperons appeared in the star during the latter stages of deleptonization . although the possibility of black hole formation was first discovered in the context of kaon condensation in neutron star matter @xcite , a full dynamical calculation of a pns evolution with consistent eos and neutrino opacities in kaon condensed matter has not been performed so far . one of the objectives of this paper is to investigate @xmath1 condensation in finite temperature matter , including the situation of trapped neutrinos in more detail . an impetus for this study is the recent suggestion that a mixed phase of kaon - condensed and normal matter might exist which could greatly affect the structure @xcite and its neutrino opacity @xcite . another objective of our study is to identify differences in thermodynamic quantities such as the pressure , entropy or specific heat that might produce discriminating features in the star s neutrino emission . in separate works , we will examine neutrino interactions in kaon - condensed matter and neutrino signals from pns evolution calculations in a consistent fashion . since we wish to isolate the aforementioned effects due to kaons in this paper , we deliberately exclude consideration of hyperons . hyperons and kaons were considered together in refs . @xcite and @xcite . hyperons tend to delay the appearence of kaons in matter , especially if the @xmath2 appears first . however , the @xmath2 couplings are not as well determined as those of the @xmath3 and even in this case the data are restricted to nuclear or subnuclear densities . relatively small variations in the coupling constants can lead to a situation where the threshold density for the appearance of @xmath2 particles is larger than that for kaons . these uncertainties remain unresolved ; further hyper - nuclear experiments are needed to pin down their couplings . the original investigations of kaon condensation in neutron star matter ( _ e.g. _ refs . @xcite and its astrophysical conseqences @xcite ) employed a chiral @xmath4 model in which the kaon - nucleon interaction occurs directly via four point vertices . however , one can also employ an indirect , finite - range interaction which arises from the exchange of mesons . several studies have been performed along these lines @xcite . @xcite found that the chiral and meson exchange approaches give similar results provided that the kaon - nucleon couplings are chosen to yield similar optical potentials in nuclear matter . allowing kaons to interact via the exchange of mesons has the advantage that it is more consistent with the walecka - type effective field - theoretical models usually used to describe nuclear matter @xcite . in most studies of kaon condensation it has been found that the transition to a phase in which kaons condense is second order for modest values of the kaon optical potential , @xmath5 , of order -100 mev . for magnitudes of @xmath5 well in excess of 100 mev , however , the phase transition becomes first order in character . even when the transition is first order , it is not always possible to satisfy gibbs criteria for thermal , chemical and mechanical equilibrium , so a maxwell construction , which satisfies only thermal and mechanical equilibrium , was sometimes employed to construct the pressure - density relation . recently , glendenning and schaffner - bielich ( gs ) @xcite modified the meson exchange lagrangian in such a way that the gibbs criteria for thermal , chemical and mechanical equilibrium in a first order phase transition was possible . the extended mixed phase of kaon - condensed and normal matter which results produces a qualitative difference for the structure of a neutron star , since the eos is softened over a wider region than in the case in which there is no mixed phase . this has implications for the mass - radius relation and the maximum mass , among other properties of the star . in this paper , we investigate the phase transition involving kaon - condensed matter and its influence upon the equation of state . we find that the precise form assumed for the scalar interactions ( particularly , their density dependence ) , both for baryon - baryon and kaon - baryon interactions , determines whether or not the transition is first or second order , and , in the case of a first order phase transition , establishes whether or not a gibbs construction is possible . since the form of the scalar interactions is not experimentally well constrained at present , we have explored several different models in this study of the effects of kaon condensation on the eos and the structure of a pns . for each model , we have performed a detailed study of the thermal properties which are summarized in terms of phase diagrams in the density - lepton content and density - temperature planes . in sec . ii we present the various lagrangians and derive exressions for the thermodynamic properties of each . we also develop the theoretical formalism necessary to describe baryons and kaon condensed matter in both the pure and mixed phases . this is followed by a discussion of the determination of the various coupling constants . section iii contains a comparison of the results for the eos and for the structure of neutron stars for typical values of entropy and lepton content in a proto - neutron star as it evolves . our conclusions and outlook for evolution of a proto - neutron simulation are presented in sec . iv . in appendix a , the extent of the correspondence between a meson exchange formalism and a chiral model to describe kaon condensation in matter is examined . the role of higher order kaon self - interactions in determining the order of the phase transition to a kaon condensed state is studied in appendix b. we begin with the well - known relativistic field theory model of walecka @xcite supplemented by nonlinear scalar self - interactions @xcite . here nucleons ( @xmath6 ) interact via the exchange of @xmath7- , @xmath8- , and @xmath9-mesons . explicitly , the lagrangian is @xmath10 where @xmath11 is the nucleon field , the @xmath9-meson field is denoted by @xmath12 and the quantity @xmath13 is the isospin operator which acts on the nucleons . scalar self - couplings @xcite , which improve the descripton of nuclear matter at the equilibrium density , are included in the potential @xmath14 , with @xmath15 denoting the vacuum nucleon mass . the field strength tensors for the vector mesons are given by the expressions @xmath16 and @xmath17 . in the standard walecka model the nucleon effective mass m^*_gm = m - g_(the label gm refers to the glendenning - moszkowski parameters @xcite that we will use with this expression ) . we shall also study an alternative form due to zimanyi and moszkowski ( labelled by zm ) @xcite : @xmath18 by redefining the nucleon field , @xmath19 , the lagrangian can be written exactly in the form eq . ( [ hyp1 ] ) , but the nucleon effective mass becomes m^*_zm = m(1+g_/m)^-1 . for small values of @xmath7 this is equivalent to the walecka form . however the zm effective mass has the property that , in the limit of large @xmath7 , @xmath20 remains positive whereas @xmath21 can become negative @xcite , which is unphysical . in the mean field approximation the thermodynamic potential per unit volume for both lagrangians is @xmath22 here the inverse temperature is denoted by @xmath23 , @xmath24 and the subscripts @xmath25 or @xmath26 have been suppressed . the chemical potentials are given by @xmath27 note that in a rotationally invariant system only the time components of the vector fields contribute to eq . ( [ hyp2 ] ) and for the isovector field only the @xmath28 component contributes . the contribution of antinucleons is not significant for the thermodynamics of interest for a pns and is ignored . using @xmath29 , the thermodynamic quantities can be obtained in the standard way . the nucleon pressure is @xmath30 , and the number density @xmath31 and the energy density @xmath32 are given by @xmath33 where the fermi distribution function @xmath34 . the entropy density is then given by @xmath35 . the contribution from the leptons and antileptons is adequately given by its non - interacting form , since their interactions give negligible contributions @xcite . thus the thermodynamic potential per unit volume of the leptons and antileptons is : @xmath36\ , , \label{zlept}\end{aligned}\ ] ] where @xmath37 and @xmath38 denote the degeneracy and the chemical potential , respectively , of leptons of species @xmath39 . the degeneracy @xmath37 is 2 for electrons and muons and it is 1 for neutrinos of a given species . since the star is in chemical equilibrium with respect to the weak processes @xmath40 , where the lepton @xmath39 is either an electron or a muon , the chemical potentials obey @xmath41 . if there are no neutrinos trapped in the star the neutrino chemical potentials @xmath42 are zero or , equivalently , the total neutrino concentration @xmath43 , where we define the concentration for particle @xmath44 to be @xmath45 . the pressure , density and energy density of the leptons are obtained from eq . ( [ zlept ] ) in standard fashion . the two kaon lagrangians of the meson - exchange type which have been previously suggested ( in refs . @xcite and @xcite , respectively ) , can both be written in the form @xmath46 where @xmath47 denote the charged kaon fields and we have defined the combined vector field @xmath48 with @xmath49 and @xmath50 denoting @xmath8 and the @xmath28 fields , respectively ; @xmath51 and @xmath52 are coupling constants . since only the time components of the vector fields survive , in practice only @xmath53 is non - zero . the two lagrangians differ in the forms chosen for the quantity @xmath54 . both have the standard vacuum mass term , but the interaction terms differ . specifically , knorren , prakash and ellis ( kpe ) @xcite take @xmath55 with @xmath56 denoting the vacuum kaon mass , while glendenning and schaffner - bielich ( gs ) @xcite choose @xmath57 note that the coupling constant @xmath58 is defined here to be twice that defined in gs . a similar remark applies to the @xmath59 coupling constant @xmath60 . it is remarkable , as pointed out in appendix a , that to leading order in the kaon condensate intensity , the equations obtained with the chiral kaplan - nelson @xcite lagrangian at zero temperature agree precisely with those from the kpe lagrangian . to see the significance of the term involving the vector fields in eq . ( [ algs ] ) , consider the invariance of the lagrangian under the transformation @xmath61 . this allows the conserved kaon current density to be identified as @xmath62 now for the combined gs lagrangian , @xmath63 , the equation of motion for the omega field is @xmath64 since the nucleon current @xmath65 is conserved , as is the kaon current @xmath66 , taking the divergence of eq . ( [ veceom ] ) immediately yields @xmath67 ( and similarly for the @xmath9 field ) . this is the required condition for a vector field @xcite so as to reduce the number of components from four to three . on the other hand , @xmath68 does not contain an @xmath69 term , so that the kaon current does not appear on the right hand side of eq . ( [ veceom ] ) . at the mean field level , however , where the vector fields are constants , any derivative is necessarily zero so that the divergence condition is automatically satisfied . for the coupling of the kaon fields to the scalar @xmath7 field , kpe use a linear coupling , whereas gs have an additional quadratic term there is little guidance on the form that should be used to generate the kaon effective mass so the choice is somewhat arbitrary , although , as we shall see , it can significantly affect the thermodynamics . both the kpe and gs choices lead to problems for sufficiently large values of the @xmath7 field ; in one case the effective mass becomes imaginary , in the other it becomes negative . we are therefore led to consider a third form in the spirit of the zm model for nucleons . for specificity we start with the gs lagrangian which can be written @xmath70 in terms of a covariant derivative @xmath71 , and replace it by @xmath72 we observe that the form of @xmath73 above is one of many possibilities . making the transformation @xmath74 and noting that @xmath7 is a constant mean field , the kaon lagrangian can be put in the form of eq . ( [ kaonlag ] ) with @xmath75 the label @xmath76 denotes this work " . while eqs . ( [ alkpe ] ) , ( [ algs ] ) and ( [ altw ] ) all give @xmath77 for small @xmath7 , they differ at order @xmath78 and beyond , _ i.e. _ , for large values of @xmath7 . the kaon partition function at finite temperature can be obtained for a lagrangian of the form ( [ kaonlag ] ) by generalizing the procedure outlined in kapusta @xcite . first , we transform to real fields @xmath79 and @xmath80 , @xmath81 and determine the conjugate momenta @xmath82 the hamiltonian density is @xmath83 , and the partition function of the grand canonical ensemble can then be written as the functional integral @xmath84[d\pi_2]\int_{periodic}[d\phi_1][d\phi_2 ] \exp\left\{\int\limits_0^{\beta}d\tau\int d^3x\left ( i\pi_1\frac{\partial\phi_1}{\partial\tau}+ i\pi_2\frac{\partial\phi_2}{\partial\tau}-{\cal h}_k(\pi_i,\phi_i ) + \mu j_0(\pi_i,\phi_i)\right)\right\}\;.\label{zint}\ ] ] here the fields obey periodic boundary conditions in the imaginary time @xmath85 , namely @xmath86 , where @xmath87 . the chemical potential associated with the conserved kaon charge density is denoted by @xmath88 and chemical equilibrium in the reaction @xmath89 requires that @xmath90 . the gaussian integral over momenta in eq . ( [ zint ] ) is easily performed . next the fields are fourier decomposed according to @xmath91 where the first term describes the condensate , so that in the second term @xmath92 . the pion decay constant @xmath93 has been inserted so that the condensate angle @xmath94 is dimensionless . the matsubara frequency @xmath95 . the partition function can then be written @xmath96[d\phi_{2,n}({\mbox{\bfp } } ) ] e^s\;,\qquad{\rm where}\nonumber\\ s&=&\thalf \beta v(f\theta)^2(\mu^2 + 2\mu x_0-\alpha ) -\thalf\sum_{n,{\mbox{\scriptsize{\bfp}}}}\bigl(\phi_{1,-n}(-{\mbox{\bfp}}),\phi_{2,-n}(-{\mbox{\bfp}})\bigr ) { \mbox{\bfd}}\left(\matrix{\phi_{1,n}({\mbox{\bfp}})\cr\phi_{2,n}({\mbox{\bfp}})\cr}\right ) \;,\nonumber\\ { \mbox{\bfd}}&=&\beta^2\left(\matrix{\omega_n^2+p^2+\alpha-2\mu x_0-\mu^2 & 2(\mu+x_0)\omega_n\cr-2(\mu+x_0)\omega_n & \omega_n^2+p^2+\alpha-2\mu x_0-\mu^2\cr}\right).\end{aligned}\ ] ] @xmath11 is a normalization constant . we define the @xmath47 energies according to @xmath97 so that the three approaches give @xmath98 using the definition ( [ defom ] ) and suppressing the explicit dependence of @xmath99 on @xmath100 , the determinant of @xmath101 is @xmath102 \left[\omega_n^2+(\omega^++\mu)^2\right]\;,\ ] ] giving @xmath103 where the normalization constant @xmath11 has been dropped since it is irrelevant to the thermodynamics . performing the sum over @xmath104 and neglecting the zero - point contribution , which contributes only beyond the mean field approach and in any case is small @xcite , we obtain the grand potential for the kaon sector : @xmath105\;.\label{zkexch}\ ] ] the kaon pressure , @xmath106 , and the kaon number density is easily found to be @xmath107\;,\ ] ] and the bose occupation probability @xmath108 . the kaon energy density is @xmath109\;,\ ] ] and the kaon entropy density is @xmath110 . it is useful first to define the quantity @xmath111\;.\ ] ] then the mean @xmath8 , @xmath9 and @xmath7 fields , as well as the condensate amplitude @xmath94 , determined by extremizing the total grand potential @xmath112 , can be written @xmath113 \frac{\partial\alpha}{\partial x_0}\right\}\nonumber\\ & & m_{\rho}^2b_0 = \thalf g_{\rho}(n_p - n_n)-g_{\rho k } \left\{\mu(f\theta)^2+n_k^{th}-x_0a_k^{th}-\thalf[(f\theta)^2+a_k^{th } ] \frac{\partial\alpha}{\partial x_0}\right\}\nonumber\\ & & m_{\sigma}^2\sigma = -\frac{du(\sigma)}{d\sigma } -2\frac{\partial m^*}{\partial\sigma}\sum_{n , p } \int\frac{d^3k}{(2\pi)^3 } \frac{m^*}{e^*}f_f(e^*-\nu_{n , p } ) -\thalf\left[(f\theta)^2+a_k^{th}\right]\frac{\partial\alpha } { \partial\sigma}\nonumber\\ & & \theta(\mu^2 + 2\mu x_0-\alpha)= \theta[\mu-\omega^-(0)][\mu+\omega^+(0 ) ] = 0\;. \label{hhyp5}\end{aligned}\ ] ] the derivative @xmath114 is zero for the kpe case and @xmath115 for the gs and tw cases . the derivatives with respect to the @xmath7 field are @xmath116 note that the last of eqs . ( [ hhyp5 ] ) yields either @xmath117 ( no condensate ) or the condition for a condensate to exist . since @xmath88 is positive here , we only have the possibility of a @xmath1 condensate with @xmath118 . note also that the contribution of the condensate to the kaon pressure @xmath119 vanishes , as it should . the remaining condition to be imposed is that neutron star matter must be charge neutral . for a single phase this implies @xmath120 where @xmath121 and @xmath122 are the net negative lepton number densities . the mixed phase thermodynamics is discussed below . the sum of the nucleon and kaon energy densities can be simplified somewhat by using the equations of motion . this gives @xmath123\;.\end{aligned}\ ] ] in the theory discussed above there are two independent chemical potentials , which we can take to be @xmath124 and @xmath88 , each connected with a conserved charge , the baryon number and charge of the system , respectively . glendenning @xcite pointed out that in the presence of a first order phase transition , conservation laws must be globally , not locally , imposed , if possible , in the mixed phase region . a maxwell construction would have been appropriate had there been just a single conserved charge . however , relaxing the condition of local charge neutrality does not guarantee that the model lagrangian , solved in the mean field approximation , will provide a description of the mixed phase , which is only possible if the gibbs criteria can be satisifed . a simple , yet general , procedure to check if the gibbs criteria can be fulfilled by a specific model is discussed in appendix b. denoting the phase containing a condensate with a subscript @xmath94 , and the phase without a condensate with @xmath117 , the total pressures in the two phases must be equal @xmath125 each of the chemical potentials is the same in the two phases . if the volume fraction of the non - condensed phase is @xmath126 , then global conservation of charge requires @xmath127_{\theta=0}+(1-\chi)[n_p - n_k - n_e - n_\mu]_{\theta}=0 \ , . \label{g2}\ ] ] the densities of the individual species in the mixed phase are evident here . the total energy density is the weighted sum of the two phases @xmath128 the total entropy density is obtained through a similar equation . in the effective lagrangian approach adopted here , knowledge of two distinct sets of coupling constants , one parametrizing the nucleon - nucleon interactions , and one parametrizing the kaon - nucleon interactions , are required for numerical computations . these are associated with the exchange of @xmath129 and @xmath9 mesons . we consider each of these in turn . the nucleon - meson coupling constants are determined by adjusting them to reproduce the properties of equilibrium nuclear matter at @xmath130 . the properties used are the saturation density , @xmath131 , the binding energy / particle , @xmath132 , the symmetry energy coefficient , @xmath133 , the compression modulus , @xmath134 , and the dirac effective mass at saturation , @xmath135 . not all of these quantities are precisely known and the values we choose are listed in table i. for completeness , we list the equations needed to obtain the coupling constants , assuming that the scalar self coupling has the form @xmath136 , where @xmath137 . from the equation of motion for the @xmath138 field and the fact that the pressure is zero at saturation density in nuclear matter , the value of @xmath139 is given by @xmath140 where @xmath141 and @xmath142 . the @xmath9 meson coupling constant can be determined for a given symmetry energy through the relation @xmath143 an expression involving the compression modulus can be deduced by differentiating the @xmath7 equation of motion : @xmath144 ^ 2 \left\ { \frac{9n_0m^{*2 } } { e_f^{*2}[k+9(e_a+e_f^*-m)]-3k_f^2e_f^ * } -3 \left ( \frac{n_0}{e_f^*}-\frac{n_s}{m^ * } \right ) \right\ } + n_s \frac{df}{d\phi_0 } - \frac{d^2u(\phi_0)}{d\phi_0 ^ 2 } \;. \label{comp}\ ] ] here @xmath145 , the value of @xmath146 at saturation density , is obtained directly from the dirac effective mass . the function @xmath147 depends on the particular expression used for the effective mass . the scalar density @xmath148 \}$ ] . the @xmath7 equation of motion at saturation can be written in the form @xmath149\;,\ ] ] which together with eq . ( [ comp ] ) allows the @xmath7 coupling to be obtained . finally the constants appearing in the scalar self - coupling @xmath150 are determined from : @xmath151 \nonumber \\ b & = & \frac{1}{2 m \phi_0 } \left[\frac{d^2u(\phi_0)}{d\phi_0 ^ 2 } - 3 c \phi_0 ^ 2 \right ] \,.\end{aligned}\ ] ] the constants determined in this way are given in table i. note that in principle the potential should be bounded from below for large values of the @xmath7 field requiring @xmath152 to be positive ; this is the case for the zm model . in order to investigate the effect of a kaon condensate on the eos in high - density matter , the kaon - meson coupling constants have to be specified . empirically known quantities can be used to determine these constants , but it should be borne in mind that laboratory experiments give information only about kaon - nucleon interaction in free space or in nearly isospin symmetric nuclear matter . on the other hand , the physical setting in this work is matter in the dense interiors of neutron stars which has a different composition and spans a wide range of densities ( up to @xmath153 ) . therefore , kaon - meson couplings as determined from experiments might not be appropriate to describe the kaon - nucleon interaction in neutron star matter , and the particular choices of coupling constants should be regarded as parameters that have a range of uncertainty . with the above caveats in mind , we now examine the relationship between the optical potential of a single kaon in infinite nuclear matter and the kaon - meson couplings in our lagrangian . lagrange s equation for an @xmath154-wave @xmath1 with a time dependence @xmath155 , where @xmath156 is the asymptotic energy , defines the optical potential @xcite for our lagrangian ( [ kaonlag ] ) according to @xmath157~k^-({\bf x } ) & = & [ -2x_0e + \alpha - m_k^2]~k^-({\bf x } ) \nonumber\\ & \equiv & 2~m_k~u_k~k^-({\bf x})\;. \end{aligned}\ ] ] in nuclear matter , @xmath158 , so for a kaon with zero momentum ( @xmath159 ) the optical potential is @xmath160 utilizing the functional forms for @xmath54 in eqs . ( [ alkpe ] ) , ( [ algs ] ) , and ( [ altw ] ) , the optical potentials for the kpe , gs and tw models are easily obtained . for the kpe case this may be written exactly as @xmath161 whereas for the gs and tw cases there are higher order corrections in addition to the terms linear in the fields . we choose @xmath51 to be @xmath162 and @xmath52 to be @xmath163 on the basis of simple quark and isospin counting arguments . note that this value for @xmath51 is also suggested by comparison to the chiral approach ( see ref . @xcite and appendix a ) and it leads to a @xmath164 mev contribution to the optical potential . the total optical potential is shown in table ii for various choices of the @xmath7 coupling . the linear form eq . ( [ linear ] ) , exact for kpe , is an accurate fit to the the gs and tw cases for moderate values of the optical potential . for orientation , chiral models suggest that the magnitude of the optical potential is at most 120 mev @xcite , while fits to kaonic atom data have been reported with values in the range 50200 mev @xcite . we note that glendenning and schaffner - bielich @xcite label their results according to values of the optical potential obtained in the linear approximation ( henceforth , @xmath165 ) . in order to make an apposite comparison with their results , we will parametrize the kaon coupling for each model simply by specifying the value of @xmath166 . the effects of kaon condensation on the eos are more pronounced at zero temperature than at finite temperature , since the fraction of thermally excited kaons increases with temperature relative to the fraction of kaons residing in the condensate . we therefore begin by examining results for the zero temperature case . we have considered two different nucleon lagrangians , gm and zm , and three different kaon lagrangians , kpe , gs and tw . below densities of about @xmath167 , matter is composed of neutron - rich nuclei immersed in a neutron sea . for this regime , we use the potential model results of negele and vautherin @xcite in the range @xmath168 and those of baym , bethe , and pethick @xcite for @xmath169 . for cold stars , the eos in this regime has little effect on maximum masses or stellar radii . furthermore , since the entropy in the stellar mantle @xmath170 is quickly radiated away in neutrinos , the eos in this regime does not substantially affect the results of this paper . in fig . [ fig1 ] , we compare the pressures for the different nucleon and kaon lagrangians as a function of baryon density , @xmath171 . the solid lines show results for both the pure nucleon and kaon condensed phases with no attempt to enforce the gibbs conditions of chemical and mechanical equilibrium . in all cases , a first order phase transition is found to occur , as long as the magnitude of the optical potential @xmath172 is in excess of 100 mev . where possible , the pressure in the mixed phase obtained by imposing gibbs criteria for mechanical and chemical equilibrium is shown as a dashed line . for the gm+kpe , zm+kpe and zm+gs models it was not possible to satisfy gibb s criteria , despite the occurrence of a first order phase transition for large enough @xmath173 . the reason for this is connected with the form of the kaon lagrangian , as discussed below . we also point out in appendix b that non - linear kaon self - interactions lead to a second order , rather than a first order , transition . the qualitative similarity of the results shown in fig . [ fig1 ] for the different nuclear lagrangians enables us to simplify our analysis by allowing us to focus on three , rather than six , possible lagrangian combinations . for a given kaon lagrangian , fairly similar results can be obtained with different nuclear lagrangians by making relatively small shifts in the kaon optical potential @xmath165 . the following discussion will therefore focus on the three cases gm+kpe , gm+gs and zm+tw . the case gm+kpe is chosen to compare with the results of kpe , the case gm+gs is chosen to compare with the results of gs , and the case zm+tw demonstrates the usefulness of lagrangians in which anomalous values of effective masses are implicitly eliminated . the results for the model gm+gs shown here and elsewehere in this paper are identical to those found by gs for the same interactions . note that in all models considered , the phase transition is second order in nature for moderately low values of the optical potential . in figs . [ fig2 ] and [ fig3 ] , the density dependence of the scalar , vector , and iso - vector fields , the electron chemical potential @xmath174 , the condensate amplitude @xmath94 , and the nucleon and kaon effective masses are displayed in the pure nucleon and kaon condensed phases , ignoring any possible mixed phase for the present . for the optical potential chosen , @xmath175 mev , a first order phase transition occurs in all three cases . after the onset of condensation a rapid change in the behavior of the electron chemical potential and some of the field strengths is seen to occur . the differences in the variation of the scalar ( @xmath176 ) and isovector ( @xmath177 fields between the models are particularly illuminating . for gm+kpe , the scalar field exhibits a relatively rapid increase with density after the onset of condensation . this in turn causes both the nucleon and kaon effective masses to drop rapidly with density . in fact , for sufficiently large density , the gm+kpe kaon effective mass vanishes ( see fig . [ fig3 ] ) . the variations of the effective masses in the models gm+gs and zm+tw are more moderate . the variation of the isovector field with density , which in large part controls the variation of the electron chemical potential @xmath88 and hence the electron concentration , is also more dramatic in the case of gm+kpe than in the gm+gs and zm+tw models . notice that in the kpe model it goes to zero for asymptotic densities ( this follows from eq . ( [ hhyp5 ] ) ) , so that the proton and neutron abundances become equal . this does not occur for the other two cases considered here . finally , it is worth noting that in all three models the condensate amplitude rises rapidly once the threshold density is reached . we turn now to a discussion of the results obtained by imposing gibbs criteria for mechanical and chemical equilibrium at zero temperature . in fig . [ fig3a ] , we show the chemical potentials associated with the two conserved charges , charge and baryon number , as functions of each other , for the model gm+gs for a kaon optical potential of @xmath178 mev . quantities associated with the pure nucleon phase , phase i , are shown as solid lines here and in subsequent figures . phase ii refers to the high - density phase in which nucleons and the kaon condensate are in equilibrium , and quantities associated with it are shown as dashed lines . both phases , i and ii , coexist in the mixed - phase region which is displayed as a dotted line . this figure illustrates the way a mixed phase is built from the two pure phases . for electron chemical potentials below the solid curve , matter is positively charged in phase i. a similar interpretation of positive or negative charge for @xmath88 below or above the dashed curve is not possible , since two different types of particles , kaons and leptons , can furnish charge . in other words , a decrease in @xmath88 , or , equivalently , the number of electrons , does not necessarily lead to a positive net charge in phase ii . for @xmath179 mev , only phase i with nucleons and leptons are present . for @xmath180 mev , a mixed phase of positively charged phase i and negatively charged phase ii obeying the gibbs conditions ( [ g1 ] ) is favored . qualitatively , a similar situation is encountered in the construction of the mixed phase for the zm+tw model , but the mixed phase region is quite small . as noted earlier , however , it was not possible to satisfy gibbs criteria for models with the kaon lagrangian kpe . in fig . [ fig3b ] we show the individual charge densities of phase i and ii in the mixed phase , as a function of baryon density . the dotted curve in this figure shows the volume fraction of phase i. the results are for the gm+gs model with @xmath181 mev ( upper panel ) and for the zm+tw model with @xmath182 mev ( lower panel ) . near the lower threshold , matter in phase i is very slightly positively charged and occupies most of the volume . as the density increases , the volume fraction of phase i , @xmath126 , decreases and its charge density increases . note that the negative charge density of matter in phase ii at the lower transition point , @xmath183 @xmath184 , and the positive charge density of matter in phase i at the higher transition point , @xmath185 @xmath184 , are rather large in the case of gm+gs compared to the case zm+tw . this is due to the stronger density dependence of the scalar and isovector densities in the former case . note also that a first order transition allows for the existence of a very dense and nearly isospin symmetric matter in the mixed phase . in figs . [ fig4 ] and [ fig5 ] , we show the magnitudes of the various fields , the electron chemical potential , the nucleon and kaon effective masses , and the condensate amplitude for the gm+gs and zm+tw models , respectively . both models show the same qualitative behavior . at the lower phase boundary , in which phase ii just begins to appear , the scalar field in phase ii is much larger than in phase i and the condensate amplitude @xmath94 in phase ii takes a large value which decreases with increasing @xmath186 through the mixed - phase region . thus , the effective masses of both kaons and nucleons in phase ii are much smaller than in phase i. the densities demarking the mixed phase region and its overall extent are dependent upon the interaction models , and upon the assumed values of the kaon optical potentials , here taken to be @xmath187 mev in the case of gm+gs and @xmath188 mev in the case of zm+tw . the region in density over which the mixed phase extends is much smaller in the latter case , chiefly due to the more moderate behavior of the scalar interaction with density variations in this case . it is instructive to compare the behavior of the two models at the threshold of the mixed phase region . phase i will have a net small positive charge and a volume proportion @xmath126 close to 1 ( see fig . [ fig3b ] ) . this has to be counterbalanced by a large net negative charge in phase ii since it is weighted by the small proportion @xmath189 . focusing on phase ii , the condensate condition for the models gm+gs and zm+tw from the last of eqs . ( [ hhyp5 ] ) is @xmath190 and the kaon number density , which has to be large , is @xmath191 in order to ensure that @xmath192 , the quantity @xmath193 , and hence @xmath194 , has to be positive definite . in the zm+tw model the kaon effective mass is relatively large so that @xmath53 is positive and therefore @xmath94 is relatively small . on the other hand in the gm+gs model @xmath194 is quite small so that @xmath53 is negative and @xmath94 has to be large . the negative value of @xmath195 implies a large negative value of @xmath196 which is clearly sensitive to the value of @xmath52 . in fact , if this coupling is reduced by more than about 15% from our chosen value it is no longer possible to satisfy the gibbs criteria . by comparing the pure phase results in figs . [ fig2 ] and [ fig3 ] with the mixed phase results of figs . [ fig4 ] and [ fig5 ] , it is clear that substantial modifications of the various fields are required to satisfy gibbs criteria . we examine now the kpe model for which it was not possible to satifsfy the gibbs criteria . in this case , eq . ( [ alkpe ] ) and the last of eqs . ( [ hhyp5 ] ) leads to the condensate condition @xmath197 whereas the functional form of the number density of kaons is identical to that in eq . ( [ kden ] ) . ( [ culprit ] ) differs in important ways from eq . ( [ lucky ] ) . for the kpe model , even if @xmath198 is positive , @xmath88 has the proclivity to turn negative for large @xmath124 ( or equivalently , for large baryon densities ) , leading to @xmath199 or imaginary values of the kaon effective mass @xmath194 . this may be seen in fig . [ fig5a ] where we show the electron chemical potential @xmath88 as a function of the ( negative ) charge density in pure phase ii for a typical value of the neutron chemical potential @xmath200 mev . it is now possible to understand qualitatively why a mixed phase can not occur in the case of the kaon lagrangian kpe . in comparison with the gm+gs and zm+tw models , a distinctive feature of the kpe model is that @xmath88 decreases rapidly with the ( negative ) charge density . in constructing a mixed phase , we are attempting to balance the positive charge in phase i with the negative charge in the dense phase ii in which the electron chemical potential , and hence the charge content in leptons , is rapidly decreasing towards zero . the balance never occurs , hence the failure to meet the gibbs criteria . in terms of compositions , the gs or tw lagrangians introduce negative charges in matter by increasing the number density of kaons , while keeping the electron density nearly constant or even slightly increasing with the charge density . the kpe lagrangian , however , rapidly substitutes electrons by kaons , which is detrimental to meeting the gibbs criteria . for these reasons , we will concentrate on results with the other two kaon lagrangians in the remainder of this paper . the influence of the condensate on neutron star structure ( at zero temperature ) is shown in fig . [ fig6 ] in which the gravitational mass is displayed as a function of the star s central baryon number density ( left panel ) and its radius ( right panel ) . for the models shown , the transition is first order and gibbs equations for mechanical and chemical equilibrium are utilized . for all cases shown the central densities of the maximum mass stars lie in the mixed phase . the effects of the condensate are more evident in the case of the gm+gs model in which the mixed phase occurs over a wider region of density than in the zm+tw model . when the effects of the softening induced by the occurrence of the condensate are large , the limiting mass and the radius at the limiting mass are reduced significantly from their values when the condensate is absent . note , however , that the softening effects are limited by the constraint that the maximum mass must exceed that of the binary pulsar psr 1913 + 16 , 1.442 m@xmath201 . in the case of gm+gs , this constraint limits @xmath202 to be smaller than about 125 mev . in such a case , the minimum radius achieved is not as small as in the case @xmath203 mev , as shown in fig . the radii of stars with masses less than 1.2 m@xmath201 are not affected by the choice of the kaon lagrangian or the kaon optical potential , since the condensation threshold is not reached in these cases . the density dependence of @xmath204 , @xmath205 and @xmath206 have been investigated in other works @xcite , but for the most part either in isospin symmetric nuclear matter or pure neutron matter . in general , our results for @xmath204 with @xmath207 mev are consistent with those of refs . @xcite ( for an appropriate comparison , our results are to be compared with results obtained without in - medium pion contributions in ref . @xcite ) and those of ref . @xcite for nuclear matter at both @xmath208 and 3 . there is a relatively small change produced in going from nuclear matter to beta - equilibrated neutron star matter to pure neutron matter for the quantities @xmath204 and @xmath205 . note that a direct comparison of the real parts of the optical potentials between different calculations must also account for the fact that in obtaining fits to data , the imaginary parts are often found to be as large as the real parts , which indicates fragmentation of strength in the quasi - particle spectral function . relatively larger variations are found in the kaon energies in matter with varying amounts of isospin as can be seen from fig . [ newfig ] . in this figure , the top panel provides a comparison of results for beta - equilibrated neutron - star matter for the gm+kpe , gm+gs , and zm+tw models , respectively , for values of @xmath209 at the extreme ends considered here , namely , 80 and 120 mev . the bottom panel shows results for the zm+tw model for @xmath210 mev , for pure neutron matter , neutron - star matter , and isospin symmetric nuclear matter , respectively . at nuclear density where the models are calibrated , @xmath8 decreases by about a few mev in going from pure neutron matter to neutron star matter and by about a few tens of mev in going from neutron star matter to nuclear matter . with increasing density , these differences become progressively larger . this trend is chiefly due to the behavior of the vector fields in matter with different amounts of isospin . at this time , our results for the density dependence of @xmath8 can be compared with those of the potential models in refs . @xcite . for values of @xmath209 near the lower end of the range we explored , in the neighborhood of 80 mev , the behavior of @xmath8 , for example , is quite similar to the potential model results . as the authors in refs . @xcite indicated , kaon condensation may be unlikely in this case . however , the relevant comparision must also include the electron chemical potential @xmath211 , since the density where @xmath212 determines the onset of kaon condensation . as demonstrated in ref . @xcite , the behavior of @xmath211 for neutron star matter at high densities is determined by the density dependence of the nuclear symmetry energy ( see also a similar discussion in ref . potential model calculations ( see , for example ref . @xcite ) tend to have a relatively weak density dependence of the symmetry energy , which generally results in an onset of kaon condensation that is at a rather large density . in field - theoretical and dirac - brueckner - hartree - fock @xcite models , however , the symmetry energy varies relatively rapidly with density . these lead to smaller densities where kaon condensation occurs , for a given behavior of the kaon energy @xmath8 . furthermore , the calculations of ref . @xcite have been performed only for pure neutron matter which further enhaces the values of @xmath8 and discourages kaon condensation . in addition , as @xmath213 is increased in magnitude in field - theoretical models , the role of kaons increases and @xmath8 becomes progressively smaller as a function of density . nevertheless , the lack of effective constraints at high density preclude choosing any model over another at this time . in summary , choosing values of @xmath209 near the lower end of the range we explored either lead to a second order phase transition or no transition at all in a neutron star , in which case the gross properties of the star are relatively unaffected from the case without kaons . on the other hand , values near the higher end of this range lead to a first order phase transition at a relatively low density , depending on the form of the interaction chosen , and a more pronounced effect on the star . our aim has been to provide benchmark calculations in which both possibilities are entertained in order to consider their impact on thermodynamics and their astrophysical implications . we now investigate results at finite temperature and values of the lepton content characteristic of those likely to be encountered in the evolution of a pns . we choose three representative sets of pns conditions which correspond to : the initial conditions within a pns ( entropy / baryon @xmath214 , trapped neutrinos with a lepton fraction @xmath215 ) , a time after several seconds when the interior is maximally heated ( @xmath216 , no trapped neutrinos so @xmath43 ) , and a very late time when the pns has cooled ( @xmath217 identical to the zero temperature case discussed above ) . for a detailed explanation of the evolution of a cooling pns see pons _ _ @xcite . the contribution of the nucleons to the entropy per baryon @xmath218 , with @xmath171 denoting the total nucleon density , in degenerate situations ( @xmath219 ) can be written @xmath220 where @xmath135 and @xmath221 are the effective mass and the fermi momentum of species @xmath44 , respectively . for the temperatures of interest here , and particularly with increasing density , the above relation provides an accurate representation of the exact results for entropies per baryon even up to @xmath222 . the behavior with density of both the fermi momenta and the effective mass controls the temperatures for a fixed @xmath223 . for kaons it is straightforward to show that the contribution to the entropy from @xmath224 mesons can be ignored since it is exponentially suppressed in comparison to the @xmath1 contribution . for the latter , keeping the leading temperature dependence of the simplest approximation scheme for bosons given in ref . @xcite , the kaon entropy per baryon is @xmath225 \frac{n_k^{th}}{n_b}\quad{\rm where } \quad \psi t=\mu+x_0-\sqrt{\alpha+x_0 ^ 2}\;,\ ] ] and @xmath226 is determined from @xmath227 by solving the equation @xmath228 below the kaon condensation threshold as the temperature becomes very small @xmath229 so @xmath230 . above the kaon condensation threshold the last of eqs . ( [ hhyp5 ] ) implies that @xmath231 in which case @xmath232 . this simple approximation provides quite an accurate account of the kaon entropy per baryon which is fairly small for the scenarios examined here since it involves just the thermal contribution and the condensate plays no role . the total entropy per baryon @xmath233 also includes the lepton contributions ; @xmath234 is dominated , however , by @xmath223 . in figs . [ fig7 ] and [ fig8 ] , the relative concentrations of various particles are displayed versus baryon number density for our three pns conditions in the cases gm+gs and zm+tw , respectively . the cases shown allow the gibbs equations to be solved , and the boundaries of the mixed phase regions are indicated by vertical lines . the effect of finite temperature is to allow the existence of @xmath235 and @xmath1 particles at all densities , although kaons become relatively abundant only within the mixed phase region . in the third set of diagrams , trapped neutrinos are present at all densities and the appearance and abundances of the negatively charged particles @xmath235 and @xmath1 are suppressed . furthermore , the critical density for kaon condensation is shifted to higher density . in fig . [ fig9 ] the pressure is displayed as a function of baryon number density for these two lagrangians and the three pns conditions . two choices for the kaon optical potential are shown to highlight differences between cases in which kaons condense in second or first order phase transitions . the reduction of the pressure when kaons condense is obvious . for conditions in which the phase transition is first order , the result of applying the gibbs conditions and the result of assuming pure phases ( thin line ) are both shown . the application of the gibbs conditions leads to further softening of the pressure over a wider density range . in the case of model zm+tw , a first order phase transition occurs only for very low temperatures and low neutrino concentrations . in fig . [ fig10 ] we show the matter temperature as a function of the baryon density for these two lagrangians for the two pns conditions with @xmath236 ( the kaon optical potentials are as in the previous figure ) . the appearance of the kaon condensate generally leads to a reduction in specific heat which is indicated by the abrupt temperature increase which persists to high densities . in the case of first order transitions , applying the gibbs conditions leads to a further enhancement of the temperature in the mixed phase regime . this behavior is in marked contrast to the case in which additional fermionic degrees of freedom , such as hyperons or quarks , are excited @xcite causing the temperature to drop and the specific heat of the matter to be increased . the latter follows from eq . ( [ nucentropy ] ) where , in the absence of any variation of @xmath135 , a system with more components at a given baryon density has a smaller temperature than a system with fewer components ( recall that @xmath237 ) . in the present case the dropping of the effective mass is the dominant effect and this leads to larger temperatures . figure [ fig11 ] shows the phase diagram of kaon condensed matter , for the case gm+gs with @xmath178 mev . the left panel displays results for zero temperature in the density lepton concentration plane . the dashed lines show the minimum lepton concentration allowed at zero temperature ( with @xmath43 ) for each density . note that the minimum lepton concentration increases with density until the phase transition begins ; above this density , the minimum lepton concentration decreases with increasing density . also note that the phase transition to a kaon - condensed phase is pushed to higher densities when trapped neutrinos are present . this implies that in the initial pns core material , in which @xmath238 and the central density is less than 3.5 times the nuclear saturation density , a kaon condensate phase likely does not exist . however , as neutrinos leak from the star the transition density decreases and a kaon condensate eventually forms . the right panel displays results in the density versus temperature plane , assuming no trapped neutrinos ( @xmath43 ) . the phase diagram for kaon condensed matter for the case zm+tw with @xmath182 mev is shown in fig . [ fig12 ] ; the results are qualitatively similar to the gm+gs case in which @xmath239 mev in fig . [ fig11 ] . this is understandable from the perspective that the actual optical potential for these two models are nearly the same . the boundary between phases i and the mixed - phase region are nearly the same for the two cases . the major difference is the much smaller extent of the mixed - phase region for the case zm+tw . note that for both cases the density at which the phase transition begins is relatively independent of temperature , so that the heating which initially occurs in the pns has little effect on the eventual appearance of a kaon condensate . also note that the density range of the mixed phase decreases with increasing temperature , and the mixed phase persists to high temperatures . it appears that the mixed phase exists up to temperatures exceeding 60 mev , for the case gm+gs and @xmath178 mev , or 30 mev for the case zm+tw with @xmath182 mev . it becomes increasingly difficult to determine the properties of a mixed phase near the temperature at which it disappears . in fig . [ fig13 ] the gravitational mass is plotted as a function of central baryon number density for these models . results are shown for our three pns conditions which correspond to snapshots of the pns evolution . the initial configuration ( dotted curves ) has the largest maximum mass . the progression to the dashed and solid curves indicates the evolution with time and we see that the maximum masses decrease . the effect of temperature upon the structure of the pns is significant . thermal kaons play a significant role here , since the net negative charge they contribute to the system partially inhibits the appearance of the condensate which allows hot neutrino free stars to reach higher masses than cold stars . the net decrease in maximum mass during the evolution for either case is seen to be of order 0.20.3 m@xmath201 . thus there is an appreciable range of masses for the pns which will result in metastability with the star ultimately collapsing to a black hole . the central density of the maximum mass , zero temperature star is smaller for the gm+gs case than for the zm+tw case . this is in spite of the apparently softer " gm+gs equation of state in which the kaon condensed mixed - phase region extends over a wider density range . ultimately , the smaller maximum mass of the gm+gs eos leads to a smaller central density at the maximum mass . in this work , we have studied the equation of state of matter , incorporating the possible presence of a kaon condensate , and including the effects of trapped neutrinos and finite temperatures . the calculation of the neutrino spectra of different flavors emitted from a proto - neutron star as it evolves from a hot , lepton - rich state to a cold , neutrino - poor state requires the knowledge of the equation of state of matter at temperatures up to about 5060 mev and lepton fraction up to about 0.4 . since the nucleon - nucleon and kaon - nucleon interactions at high density are relatively poorly understood , we explored several possible field - theoretical models in both sectors . these models are distinguished by the form of the assumed scalar ( and in some cases vector ) interactions which chiefly determine the density dependences of the nucleon and kaon effective masses . these models produce significantly different high density behavior of the eos , even though the kaon - meson couplings in these models are calibrated to give the same the kaon - nucleus optical potential in nuclear matter . the principal findings of our studies at zero temperature were : 1 . the order of the phase transition between pure nucleonic matter and a phase containing a kaon condensate depends sensitively on the choice of the kaon - nucleon interaction . 2 . in one case we studied ( kpe ) , although a first - order phase transition resulted , it was not possible to satisfy gibbs rules for phase equilibrium which would have produced a mixed phase . we performed a detailed analysis of this situation and found that scalar , and to a lesser extent the isovector , interactions that vary rapidly with density were chiefly responsible for this failure . this was confirmed by developing a new kaon - nucleon interaction ( tw ) with more moderate variations in the scalar density and the kaon effective mass in which the gibbs criteria in a first order phase transition would be satisfied . the extent of the mixed phase region was thereby reduced . the significance of the new kaon - nucleon interaction ( tw ) we developed is that it avoids the anomalous behavior for the kaon effective mass that occurs in previous models ( kpe , gs ) at very high density . near the low - density boundary of a mixed phase region , the kaon condensed phase appears with large density , too large for the kpe interaction to produce physically acceptable effective masses . we also made detailed comparisons with earlier work which used the gs form for the scalar interactions . 3 . in all models considered ( kpe , gs and tw ) , a first - order phase transition occurs only for large values of the kaon - nucleus optical potential ; moderate values generally produce a second order phase transition . in the meson exchange models studied here , only linear kaon self - interactions were considered . in the case of a first order phase transition , the condensate amplitude was found to be rather large at the low - density boundary of the mixed phase . we therefore explored the effect of non - linear kaon self - interactions guided by the chiral model in appendix b. we found that introducing higher order interactions , using the lowest order chiral lagrangian , results in a second order , rather than a first order , phase transition . whether this behavior persists when more general higher order operators in the chiral expansion are considered remains an open question . at finite temperatures , we find the effects of condensation , in general , are less pronounced than at zero temperature . for moderate values of the optical potential , when the phase transition is first order at zero temperature , kaon condensation eventually becomes a second order phase transition at high enough temperatures , whether or not neutrino trapping is considered . the temperature at which this occurs is in the range of 3060 mev , depending upon interactions . for the cases at finite temperatures in which the transition is first order , its thermodynamics ( such as the pressure - density relation ) becomes effectively similar to that of a second order phase transition this is because of the existence of thermal kaons and because of nucleonic thermal effects . the condensate is suppressed , and moved to higher densities , both by the existence of trapped neutrinos and by finite temperatures . compared to earlier works , the new aspects of our work are : 1 . the delineation of the phase boundaries in the baryon density versus lepton number and baryon density versus temperature planes . this is helpful to anticipating the possible outcome in a full pns simulation . in particular , the critical temperatures above which the mixed phase disappears are above 30 and 60 mev , depending upon the interaction . this has implications for the temperature dependence of the surface energies , and for the melting temperatures , of the droplets in the mixed phase . the finding that thermal effects on the maximum gravitational mass of neutron stars are comparable to the effects induced by the trapped neutrino content . this is in stark contrast to previously studied cases in which nucleons - only matter , or matter containing hyperons , were considered . furthermore , compared to equations of state previously studied that allow metastable protoneutron stars , those containing hyperons or quark matter , the maximum mass does not significantly decrease during the deleptonization of the protoneutron star because of these thermal effects . only after the temperature in the protoneutron star significantly decreases does the maximum mass appreciably fall . this implies that the possible collapse of a metastable protoneutron star to a black hole occurs during the late stages of cooling , after several tens of seconds , rather than during the late stages of deleptonization , which is somewhat earlier . the support of the u.s . department of energy under contract numbers doe / de - fg02 - 87er-40317 ( jap and jml ) , doe / de - fg06 - 90er40561 ( sr ) , doe / de - fg02 - 88er-40328 ( pje ) , and doe / de - fg02 - 88er-40388 ( mp ) is acknowledged . j. pons also gratefully acknowledges research support from the spanish dgcyt grant pb97 - 1432 , and thanks j.a . miralles for useful discussions . in this appendix , we examine the conditions under which there exists a close correspondence between a meson exchange model and the chiral @xmath240 approach of kaplan and nelson @xcite . such a correspondence is most easily established for the zero temperature case by setting the scalar self - coupling terms , _ i.e. , _ @xmath241 . specializing to the case where the only baryons are nucleons and using the walecka lagrangian for the nucleons , it was shown in ref . @xcite that the chiral thermodynamic potential per unit volume can be written @xmath242 where the primes on the meson fields distinguish them from those used previously and @xmath243 is the heaviside step function . the nucleon effective masses are @xmath244 we employ the values suggested by politzer and weise @xcite , namely @xmath245 mev ( @xmath246 is the strange quark mass ) and @xmath247 mev . @xmath248 is usually taken to lie in the range @xmath249 to @xmath250 mev . if we ignore the fairly small effect of @xmath251 here and in the kaon - nucleon sigma term , @xmath252 , we can write @xmath253 as well as redefining the scalar field , we can redefine the chiral vector fields entering the chemical potentials : @xmath254 substituting in eq . ( [ omkap ] ) we find @xmath255 if we expand this in powers of @xmath94 and retain only the lowest order @xmath256 term , the last term in eq . ( [ omchmes ] ) does not contribute and our thermodynamic potential is exactly of the form given by eqs . ( [ hyp2 ] ) and ( [ zkexch ] ) for the meson exchange model provided that the @xmath68 expression is used . in order for the correspondence to be exact , the parameters for the @xmath7 and @xmath8 meson need to obey @xmath257 these are precisely the conditions found in ref . @xcite for the optical potentials of the chiral and meson exchange models to be the same in nuclear matter . the relation involving the @xmath8 meson couplings is quite well obeyed with our parameters . in addition , for the @xmath9 meson , @xmath258 this indicates that @xmath259 , a condition which is not well obeyed by the parameters used here or in other works . given that the chiral and meson exchange thermodynamic potentials can be put into precise correspondence to lowest order in @xmath256 , it follows that the equations of motion and the thermodynamics will be identical to this order . if scalar self - coupling terms are included , @xmath260 , then the transition from the chiral to the meson exchange approach will couple higher powers of the @xmath7 field to the kaon condensate ( in the braces in eq . ( [ omchmes ] ) ) . it will also introduce higher order terms . these additional contributions may not be negligible so the correspondence between the two approaches becomes less precise . our findings in appendix a naturally raise the question of whether it is sufficient to work at order @xmath256 , involving only linear kaon self - interactions , in the meson exchange models . it clearly will be sufficient at the low - density onset of a second order phase transition where @xmath94 is small . on the other hand , for a first order phase transition , the value of @xmath94 is large at the low - density onset of the mixed phase , particularly for the gs model . it is therefore interesting to explore the effect of non - linear kaon self - interactions guided by the chiral model . the order of the phase transition ( in the mean field approximation ) is determined by the behavior of the thermodynamic potential , @xmath261 , at fixed chemical potentials . a first order transition , with a mixed phase , is possible only if there exists some value of @xmath124 for which @xmath261 exhibits two degenerate minima . at the critical density corresponding to the low - density onset of the mixed phase , the @xmath117 phase should be a local minimum which is degenerate with a minimum at some finite @xmath262 . in the vicinity of the critical density , the @xmath117 phase is nearly charge neutral ( with an infinitesimal excess of positive charge and a volume fraction close to 1 which balances the negative charge in the kaon phase which has an infinitesimal volume fraction ) . this requirement enables us to determine the electron chemical potential at the critical density by charge neutrality . we focus on the gm+gs model for which the thermodynamic potential of nucleons and kaons was given in eqs . ( [ hyp2 ] ) and ( [ zkexch ] ) ; the contribution due to leptons is ignored since it does not contain any @xmath94 dependence at fixed @xmath88 . at zero temperature with a kaon optical potential @xmath175 mev , this model predicts a first order phase transition in the vicinity of @xmath263 mev , as can be deduced from fig . [ fig14 ] where @xmath264 is plotted as a function of @xmath94 . the thermodynamic potential for the model gm+gs is shown as the solid curve labelled @xmath265 . it clearly shows two minima , one at @xmath117 and the other at @xmath266 . the latter corresponds to a kaon number density @xmath267 @xmath184 which is larger than the baryon density . for such a dense condensate one would suspect that non - linear kaon self - interactions might be important . the order @xmath268 corrections to the thermodynamic potential are easily found from eq . ( [ omchmes ] ) : @xmath269 the result of adding this correction to @xmath265 is shown as the dashed curve in fig . [ fig14 ] . it greatly alters the behavior of @xmath261 for @xmath270 . the exsistence of a second minimum suggests that a first order phase transition is still possible , but at larger @xmath124 . however , we find that this is not the case and a second - order transition occurs at @xmath271 mev . it is possible to incorporate all powers of @xmath94 arising from self - interactions in the chiral model . in this case the correction to the grand potential is @xmath272 the result of including this correction is shown as the dot - dashed curve in fig . [ fig14 ] . in this case no first order phase transition is possible in the vicinity of @xmath273 . instead a second order phase transition occurs once again at @xmath271 mev ; this is because kaon self interactions play no role when @xmath94 is small . despite our findings here , it is not clear if kaon self - interactions will generically disfavor a first order transition . this is because we have ignored higher order operators in the chiral expansion which will become important with increasing @xmath94 . the indication from phenomenological chiral perturbation theory @xcite is that such effects can be significant when @xmath274 . the robust finding here is that the higher order kaon self - interactions predicted by the lowest order chiral lagrangian lead to a second order , rather than a first order , phase transition . 99 bigus m. prakash , i. bombaci , m. prakash , p.j . ellis , j.m . lattimer and r. knorren , phys . rep . * 280 * , 1 ( 1997 ) . tpl v. thorsson , m. prakash and j.m . lattimer , nucl . phys . a * 572 * , 693 ( 1994 ) . kj w. keil and h.t . janka , astron . and astrophys . * 296 * , 145 ( 1994 ) . pons , s. reddy , m. prakash , j.m . lattimer and j.a . miralles , astrophys . j. * 513 * , 780 ( 1999 ) . glendenning and j. schaffner - bielich , phys . c * 60 * , 025803 ( 1999 ) . rbp s. reddy , g. bertsch and m. prakash , phys . b 475 , 1 ( 2000 ) . kpe r. knorren , m. prakash and p.j . ellis , phys . c * 52 * , 3470 ( 1995 ) . schaffner j. schaffner and i.n . mishustin , phys . c * 53 * , 1416 ( 1996 ) . kapnel d. b. kaplan and a. e. nelson , phys . b * 175 * , 57 ( 1986 ) ; * 179 * , 409 ( 1986 ) ( e ) . politzer and m.b . wise , phys . b * 273 * , 156 ( 1991 ) . brown , k. kubodera , m. rho and v. thorsson , phys . b * 291 * , 355 ( 1992 ) . mfmt t. maruyama , h. fujii , t. muto and t. tatsumi , phys . b * 337 * , 19 ( 1994 ) . mti t. muto , t. tatsumi and n. iwamoto , phys . d * 61 * , 083002 ( 2000 ) ; _ ibid_. d * 61 * , 063001 ( 2000 ) . ty t. tatsumi and m. yasuhira , nucl . a * 670 * , 218 ( 2000 ) . sew b. d. serot and j. d. walecka , advances in nuclear physics * 19 * ed . negele and e. vogt ( plenum , ny , 1986 ) ; b. d. serot , rep . phys . * 55 * , 1855 ( 1992 ) . bb j. boguta and a. bodmer , nucl * a292 * , 413 ( 1977 ) . gm n.k . glendenning and s.a moszkowski , phys . lett . * 67 * , 2414 ( 1991 ) . zm j. zimanyi and s.a . moszkowski , phys . c * 42 * , 1416 ( 1990 ) . lrp c .- h . lee , s. reddy and m. prakash , proc . of int . workshop xxvi on gross properties of nuclei and nuclear excitations , ed . m. buballa , w. nrenberg , j. wambach and a. wirzba ( hirschegg , austria , 1998 ) p. 86 . ogievetskij and i.v . polubarinov , ann . 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[ cols="^,^,^,^,^,^ " , ] fig . [ fig1 ] : pressure versus baryon number density for the six choices of the nucleon and kaon lagrangians considered in this paper . the temperature @xmath130 and there are no trapped neutrinos ( @xmath43 ) . selected values for the kaon optical potential @xmath278 are indicated . the solid lines show the pressure in the pure phases i ( nucleons only ) and ii ( the high - density nucleon - kaon condensed phase ) . the dashed lines show the pressures obtained by imposing gibbs criteria for phase equilibrium in a mixed - phase region for the case of first order transitions . for the kpe choice of the kaon lagrangian , gibbs criteria could not be satisfied despite the occurence of first order phase transitions in some cases . fig . [ fig2 ] : the density dependences of the scalar , vector , and iso - vector fields for different choices of the nucleon and kaon lagrangians ( @xmath279 ) . the solid curves show the chemical potential @xmath280 . in this figure , results are shown only for the pure phases i and ii ; the mixed phase produced by satisfying gibbs criteria is ignored . [ fig3a ] : the electron chemical potential @xmath88 versus the neutron chemical potential @xmath124 in pure phases i and ii , and in the mixed phase . the pure phase i ( solid curve ) consists of nucleons and leptons . the pure phase ii ( dashed curve ) is comprised of a kaon condensate coexisting with nucleons and leptons . the mixed phase ( dots ) is constructed by satisfying gibbs rules for phase equilibrium . [ fig3b ] : individual charge densities of pure phases i and ii and the volume fraction @xmath126 of phase i in the mixed phase as a function of baryon density . results are for the gm+gs model with @xmath181 mev and for zm+tw model with @xmath281 mev . fig . [ fig4 ] : the density dependences of the scalar , vector , and iso - vector fields for two choices of the nucleon and kaon lagrangians ( @xmath279 ) . phase i is the pure nucleon phase and phase ii is the high - density nucleon - kaon condensed phase . the vertical lines demark the mixed phase region . [ fig5a ] : the electron chemical potential @xmath88 in phase ii matter versus charge density for different models at a fixed neutron chemical potential of @xmath283 mev . in all cases , the optical potential @xmath178 mev . [ fig6 ] : left panel : the gravitational mass as a function of the central baryon number density for the cases gm+gs and zm+tw ( @xmath279 ) . curves are labelled by the values of @xmath278 and the eos includes a mixed phase region . right panel : the gravitational mass as a function of the stellar radius . fig [ newfig ] : the density dependences of the kaon energy @xmath8 in matter with different isospin content . the top panel compares results of gm+kpe , gm+gs and zm+tw models for beta stable neutron star matter for @xmath284 and -120 mev , respectively . the bottom panel shows results for the zm+tw model with @xmath284 mev in pure neutron matter , beta stable neutron star matter and nuclear matter . fig [ fig7 ] : the relative concentrations of hadrons and leptons as functions of baryon number density for three representative snapshots during the evolution of a pns . the results shown are for the model gm+gs . to the left of the vertical line there is no kaon condensate , to the right a mixed phase is present . fig [ fig9 ] : the pressure versus baryon number density for three representative snapshots during the evolution of a pns . the cases shown in the upper panels produce only second order phase transitions . for the cases in the lower panels the transitions are first order , except for zm+tw with @xmath236 . in the lower panels , heavy curves include a mixed phase region and light curves ignore a mixed phase region . fig [ fig11 ] : the phase diagram of kaon condensed matter for the case gm+gs and @xmath175 mev . the left panel shows results at zero temperature in the density versus lepton concentration plane . the dashed curve shows the minimum lepton concentration for each density , which occurs for trapped neutrino concentration @xmath43 . the right panel shows results in the density versus temperature plane for neutrino free matter ( @xmath43 ) . fig [ fig14 ] . the thermodynamic potential as a function of the condensate order parameter @xmath94 . results are shown for the gm+gs model near the critical density ( @xmath286 mev and @xmath287 mev ) with optical potential @xmath175 mev .
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we study the equation of state of kaon - condensed matter including the effects of temperature and trapped neutrinos .
several different field - theoretical models for the nucleon - nucleon and kaon - nucleon interactions are considered .
it is found that the order of the phase transition to a kaon - condensed phase , and whether or not gibbs rules for phase equilibrium can be satisfied in the case of a first order transition , depend sensitively on the choice of the kaon - nucleon interaction . to avoid the anomalous high - density behavior of previous models for the kaon - nucleon interaction , a new functional form is developed .
for all interactions considered , a first order phase transition is possible only for magnitudes of the kaon - nucleus optical potential @xmath0 mev .
the main effect of finite temperature , for any value of the lepton fraction , is to mute the effects of a first order transition , so that the thermodynamics becomes similar to that of a second order transition . above a critical temperature , found to be at least 3060 mev depending upon the interaction , the first order transition disappears .
the phase boundaries in baryon density versus lepton number and baryon density versus temperature planes are delineated , which are useful in understanding the outcomes of protoneutron star simulations .
we find that the thermal effects on the maximum gravitational mass of neutron stars are as important as the effects of trapped neutrinos , in contrast to previously studied cases in which the matter contained only nucleons or in which hyperons and/or quark matter were considered .
kaon - condensed equations of state permit the existence of metastable neutron stars , because the maximum mass of an initially hot , lepton - rich protoneutron star is greater than that of a cold , deleptonized neutron star .
the large thermal effects imply that a metastable protoneutron star s collapse to a black hole could occur much later than in previously studied cases that allow metastable configurations . #
10= -.025em0 - 0 .05em0 - 0 -.025em.0433em0
| 22,748 | 533 |
most people have heard of the black hole information paradox @xcite . but the full strength of this paradox is not always appreciated . if we make two reasonable sounding assumptions \(a ) all quantum gravity effects die off rapidly at distances beyond some fixed length scale ( e.g. planck length @xmath0 or string length @xmath1 ) \(b ) the vacuum of the theory is unique then we _ will _ have ` information loss ' when a black hole forms and evaporates , and quantum unitarity will be violated . ( the hawking ` theorem ' can be exhibited in this form @xcite , and it can be seen from the derivation how conditions ( a),(b ) above can be made more precise and the ` theorem ' made as rigorous as we wish . ) in this article we will see that string theory gives us a way out of the information paradox , by violating assumption ( a ) . how can this happen ? one usually thinks that the natural length scale for quantum gravity effects is @xmath0 , since this is the only length scale that we can make from the fundamental constants @xmath2 . but a black hole is a large object , made by putting together some large number @xmath3 of fundamental quanta . thus we need to ask whether non - classical effects extend over distances @xmath0 or over distances @xmath4 for some constant @xmath5 . one finds that the latter is true , and that the emerging length scale for quantum corrections is order horizon radius . the information of the hole is distributed throughout a horizon sized ` fuzzball ' . hawking radiation is thus not emitted from a region which is an ` information free vacuum ' , and the information paradox is resolved . to see how the information paradox arises , we must first see how hawking radiation is produced in the traditional picture of the black hole . consider the semiclassical approximation , where we have a quantum field living on a classical spacetime geometry . if the metric of this spacetime is time dependent , then the quantum field will not in general sit in a given vacuum state , and pairs of particles will be produced . the schwarzschild black hole has a metric ds^2=-(1 - 2mr ) dt^2+dr^21 - 2mr+r^2d_2 ^ 2 this metric looks time independent , but that is an illusion ; these schwarzschild coordinates cover only the exterior of the hole , and if we look at the full geometry of the spacetime then we can not obtain a time independent slicing of the geometry . we schematically sketch some spacelike slices for the schwarzschild geometry in fig.[matfthree ] . ( this figure is not a penrose diagram ; it is just a formal depiction of the exterior and interior regions of the hole , and if we try to put any time independent coordinates on this space they will degenerate at the horizon @xmath6 . ) outside the horizon ( @xmath7 ) we can take the spacelike slice to be @xmath8 ; this part is called @xmath9 in the figure . inside the horizon @xmath10 the constant @xmath11 surface is _ timelike_. we get a spacelike surface by taking @xmath12 instead ; this part is termed @xmath13 . we can join these two parts of the spacelike surface by a ` connector region ' @xmath14 , so that we construct a spacelike surface covering regions both outside and inside the horizon . the details of such a construction can be found in the reference listed above @xcite , and we will summarize the discussion given there . we have the part @xmath9 as a @xmath15 slice . the ` connector ' part @xmath14 is almost the same on all slices , and has a smooth intrinsic metric as the surface crosses the horizon . the inner part of the slice @xmath13 is a @xmath16 surface , with the value of @xmath17 kept away from the singularity at @xmath18 . the coordinate @xmath19 is only schematic ; it will degenerate at the horizon . ] how do we make a ` later ' spacelike slice ? outside the horizon we can take the surface @xmath20 . inside the horizon we must now continue our constant @xmath17 surface for a little longer before joining it to the constant @xmath11 part . thus the later surface is not identical in its intrinsic geometry to the earlier one . we have a time dependent slicing , and there will be particle production in the region where the surface is being ` stretched ' . to see this particle production consider the evolution of wavemodes in the geometry . to leading order we can evolve the wavemode by letting the surfaces of constant phase lie along the null geodesics of the geometry . [ matffourtp ] shows a wavemode being stretched and deformed , so that even though the wavemode was not populated by particles at the start of the evolution , we have some amplitude to get particles @xmath21 and @xmath22 at the end of the stretching . the crucial point here is the state of these created quanta . this state has the form @xmath23 , where @xmath24 creates quanta on the part of the slice outside the horizon and @xmath25 creates quanta on the part of the slice inside the horizon . this state can thus be expanded in a series of terms that have @xmath26 particle pairs . to understand the essentials of the paradox we can replace the state by a simpler one with just two terms quanta outside the horizon ( the @xmath27 quanta ) is ` entangled ' with the state of the quanta inside the horizon ( the @xmath28 quanta ) . . the mode with shorter wavelength evolves for some more time before suffering the same distortion , and then it creates entangled pairs @xmath29 . ] it is important to see how the next pair of quanta are created ( fig.[matftthree ] ) . the spacelike slice stretches , moving the locations of the @xmath30 quanta further apart . in the new region that is created , an entangled pair @xmath29 is created out of the vacuum . thus the overall state can be written schematically in the form |=_k 1 [ |0_b_k|0_c_k+ |1_b_k|1_c_k ] [ one ] to see how the above state leads to the information paradox , let us make some basic observations . \(i ) the state @xmath31 is ` highly entangled ' between the @xmath32 pairs . we can compute the entropy of this entanglement by tracing over the @xmath28 quanta , obtaining the density matrix @xmath33 describing the @xmath27 quanta , and computing the entropy @xmath34 $ ] of this density matrix . this entropy is of order the bekenstein entropy @xcite of the hole . if the hole evaporates away completely then we are left with the @xmath27 quanta in their highly entangled state but we can not see anything that they are entangled _ thus an initial pure state which formed the hole has evolved to a mixed state , and we have lost unitarity . \(ii ) a common misconception is that ` subtle quantum gravity effects ' can change the state of the emitted radiation and resolve this problem . this is incorrect . suppose we change the state of each entangled pair in ( [ one ] ) a little , |=_k 1[(1+_k ) |0_b_k|0_c_k+ ( 1-_k)|1_b_k|1_c_k ] [ onep ] where @xmath35 . then the state is still highly entangled ; the entropy of entanglement has changed by a very small fraction . a pure state for the @xmath27 quanta would be a state like |=_k [ 1 ( |0_b_k+|1_b_k ) ] [ oneq ] but such a state is nowhere ` close ' to the state ( [ one ] ) ; we need an _ order unity _ change in the state of each pair @xmath36 . and the entangled pairs @xmath37 shown on the spacelike slices in the penrose diagram . ] \(iii ) if we somehow obtained a state like ( [ oneq ] ) then the emitted radiation would be in a pure state , but this would still not help ; the state of the radiation would have no dependence on the initial matter making the hole . fig.[matfseventq ] shows a penrose diagram of the hole . on any spatial slice there are three kinds of matter that we must consider . on the extreme left we have the infalling matter @xmath38 that made the hole . next we have the ` negative energy quanta ' @xmath39 and finally near spatial infinity we have the quanta @xmath40 . what we need is for the @xmath40 to form a pure state ( entangled with nothing else ) , but carrying the information of the initial matter @xmath38 . \(iv ) so what prevents the information of @xmath38 from reaching the quanta @xmath40 ? when we burn a piece of coal , the emitted radiation does manage to carry all the information of the coal . the first quantum emitted from the coal may well be in a mixed state with the part of the coal left behind ; for example the emitted quantum may be a photon , and its spin may be entangled with the spin of the emitting atom which stays behind in the coal |=1 [ |_photon|_atom+|_photon|_atom ] the next quantum emitted from the coal may also be in a mixed state with the coal , but note that the emission process will be influenced in principle by the spin of the atom left behind after the first emission . in this way the spin of later emitted quanta get related to the spins of earlier emitted quanta , and if the coal finally burns away to nothing then the emitted radiation survives in a pure state , with all the information of the initial piece of coal . we can now see the difference between this process and the evaporation of the hole . the radiation quanta @xmath36 are pairs created from the _ vacuum_. the matter @xmath38 is far away ( several miles for a typical astrophysical hole ) from the place where the spacelike slice is stretching and producing quanta , so its information does not influence the state of the created pairs . further , later pairs @xmath36 are produced in a way that does not depend on the state of earlier produced pairs . as we had seen from fig.[matftthree ] , after the quanta @xmath30 are created , the part of the spacelike slice carrying these quanta stretches in such a way that these quanta are moved away from the region near the horizon where the production of the next pair @xmath29 will occur . thus unlike the case of the coal , here the the state of later pairs does not depend on the state of earlier pairs . \(v ) we can now summarize the essential strength of the information paradox . the region near the horizon has a curvature length scale @xmath41 , which we can take to be of order several miles . consider the evolution of modes of a quantum field in this region . follow the evolution of a field mode from the time its wavelength is say @xmath42 to the time it stretches to a wavelength @xmath41 ; this evolution takes a ` time ' @xmath41 . with all length and time scales being classical , and the evolution taking place far away from the matter @xmath38 and any region of high curvature , the evolution of the mode will lead to a state like ( [ two ] ) . but to solve the information problem _ we need the actual evolution of the field mode in this situation to differ by order unity from the expected evolution_. the fuzzball program solves the paradox by showing that assumption ( a ) in the above section is incorrect ; quantum effects change the black hole interior in a way that distributes the information of the hole throughout a horizon sized region . schematically , the picture of the hole is changed from that in fig.[inter](a ) to that in fig.[inter](b ) , where the latter picture shows a ` quantum fuzz ' filling a horizon sized region . the modification of the black hole interior allows the emitted quanta to carry the information of the state of the hole . = 2.5 in while astrophysical holes are typically charge neutral , in string theory it is easier to start with supersymmetric holes which have a charge equal to their mass . thus they are ` extremal black holes ' in general relativity , and give supersymmetric solutions in string theory . the traditional picture of the extremal hole is shown schematically in fig.[firstp](a ) . we have flat space at infinity , then a ` neck ' leading to an infinite ` throat ' . there is a horizon at the end of the throat , through which a quantum can fall in finite _ proper _ time . there is a region inside the horizon , which contains a timelike singularity . the region around the horizon is a low curvature region . the important point is that if we draw a ball shaped region around the horizon then the state in this region is the _ vacuum _ state @xmath43 . thus there is no information about the hole in the vicinity of this horizon . we will term a horizon like this with no information in its vicinity an ` information free horizon ' . fig.[firstp](b ) depicts the extremal hole given by the fuzzball proposal . we have flat space at infinity , the neck and the throat . but while the throat is long , it is not infinite . the throat ends in a quantum fuzzy ` cap ' , where the precise details of the cap contain the information of the state of the hole . the fuzzball program is primarily a _ construction . _ we take a specific black hole in string theory , with some mass and charges . this hole should have @xmath44 microstates , where @xmath45 is the bekenstein entropy of the hole . we try to construct these microstates and see what they look like . all cases worked out so far have given microstates that are ` fuzzballs ' ; there is no horizon , and the details of the microstates are explicitly manifested by the gravity solution . in particular , all extremal black hole states that have been constructed have the form fig.[firstp](b ) , and not the form fig.[firstp](a ) . note that if fig.[inter](b ) or fig.[firstp](b ) was the true picture of all black hole microstates then there would be no information paradox . an infalling quantum would not encounter a vacuum all the way to a singularity , but instead would interact with the degrees of freedom of the hole , just like what happens when a photon falls on a piece of coal . so far we have a good understanding of all states for the 2-charge extremal hole ( the so called ` small black hole ' ) , and we also understand large sets of microstates for the 3-charge and 4-charge extremal holes . one family of states for the non - extremal hole has also been constructed ; moreover , these nonextremal states are found to emit radiation at exactly the rate that would be expected for the ` hawking emission ' from these special microstates . ( for some reviews on the fuzzball program , see @xcite . ) the fuzzball ` conjecture ' says that all microstates of all black holes will behave like the ones that have been constructed . let us see in more detail what this means . the essential property of the microstates found in the fuzzball program is that there is no ` information free horizon ' . consider first the extremal hole . in the traditional picture fig.[firstp](a ) we can mark a ball shaped region around the horizon where all quantum fields are in the _ vacuum _ state @xmath43 ; i.e. , we just have the expected vacuum of quantum fields on gently curved spacetime . with the fuzzball conjecture _ it is not possible to find such a ball shaped region around a horizon_. while the redshift may be large near the fuzzy region drawn in fig.[firstp](b ) , there is no region that we can mark out that will look like a piece of the traditional extremal penrose diagram straddling the horizon . any ball shaped region we draw near the fuzzball boundary will have a state @xmath31 that is _ not _ near the vacuum state @xmath43 . rather , we will have 0 | 0 for mm_plso that the state @xmath31 would be nearly orthogonal to the vacuum @xmath43 for holes with large mass @xmath46 . this absence of a traditional horizon distinguishes the fuzzball proposal from many other attempts to understand the information problem . let us list some of these alternative proposals . first , we have hawking s original proposal which says that information is indeed lost , and we should build our quantum theory without requiring a unitary s - matrix . another proposal is that the information moves into baby universes forming inside the horizon region . another recent proposal is that we should impose a ` final state boundary condition ' at the black hole singularity @xcite , so that information is forced to emerge in the hawking radiation . by contrast , the fuzzball proposal does not require ` new physics ' . instead the proposal says that when we actually construct the microstates of a black hole in the full theory of quantum gravity then we find the state to be a ` puffed up fuzzball ' , and so radiation from the microstate is no different from radiation from a piece of coal . before proceeding to see in more detail what kind of microstates we find for black holes , let us note some common misconceptions about the information puzzle and the fuzzball proposal . \(a ) ads / cft duality is one of the most remarkable results to emerge from string theory @xcite . it is sometimes believed that we can resolve the information paradox by using this duality . this is incorrect , since such an argument would be circular . as we discussed in the last section , if we are given assumptions ( a),(b ) about quantum gravity then we _ will _ have a breakdown of quantum unitarity . in this situation we will also lose the ads / cft correspondence , since this duality assumes that both sides of the duality are good unitary quantum theories . thus to save quantum theory ( and ads / cft in particular ) we have to show that at least one of the traditional assumptions ( a),(b ) breaks down in our full theory of quantum gravity . we have to resolve the problem in the _ gravity _ description of the state ; it is a circular argument to say that information will come out because there is a dual field theory that is unitary . this said , it will turn out that the ads / cft correspondence will be a very important tool in helping us understand the general set of microstates . it is easier to count and classify states in the cft , so while we must construct our microstates in the gravity picture to resolve the information paradox , we can use the cft analysis to know when we have constructed all the states ( or enough that the general state can be understood as an extrapolation of those that have been made ) . \(b ) a common question about fuzzballs is : does an infalling observer feel something very different when falling into a fuzzball than into a traditional black hole ? this is a dynamical question , and we will try to use our knowledge of the time independent fuzzball states to conjecture an answer in section [ dynamical ] . the key point will be that there are different energy and time scales for different processes . for _ heavy _ observers ( mass much larger than the hawking temperature ) and over short times ( order the infall time ) the behavior of the typical fuzzball may be no different from the behavior of the traditional black hole geometry . but over _ long _ times ( order the hawking evaporation time ) the fuzzball behaves differently from the traditional black hole , and returns information to infinity in the hawking radiation while the traditional black hole geometry leads to information loss . there are a couple of things that we need to be careful about when addressing such issues . first , it is sometimes believed that if the fuzzball state is ` too complicated ' then it is ` essentially ' the vacuum , and should be replaced by @xmath43 . this is incorrect . the generic fuzzball state is indeed very ` complicated ' , but it is important that it is close to being _ orthogonal _ to the vacuum . all we can say is that for some _ particular _ process the fuzzball state behaves almost like the vacuum state . secondly , it is sometimes believed that the fuzzball state will have a ` fine structure ' that will affect only motion over planck distances ; evolution of ordinary quanta will be just the same as in the traditional black hole geometry . this is incorrect ; in fact as noted in section [ introduction ] ( and shown in detail in the reference mentioned above @xcite ) we need the evolution of hawking wavelength quanta to change _ by order unity _ at the horizon . we will note below that for the one family of non - extremal microstates that are known , the low energy emitted quanta indeed see the detailed structure of the ` ergoregion ' of the geometry , while high energy quanta are not sensitive to the location of the ergoregion . before moving to a detailed study of fuzzballs , let us ask what would constitute a ` disproof ' of the fuzzball conjecture . to disprove the conjecture we would have to show that generic states of the hole _ do _ have an ` information free horizon ' . for extremal holes , this would need us to argue that there are two kinds of microstates : the ones that are like the ` fuzzballs ' that have so far been found , and the remainder that are _ not _ like fuzzballs . with all we know now this looks hard to do , since in the dual cft description there seems to be no sharp boundary between different classes of microstates , and for the simple case of the 2-charge extremal hole _ all _ states have been found to be fuzzballs . the remarkable thing about string theory is that it admits no free parameters it is a unique theory with all brane tensions and couplings fixed . there is of course a large freedom in which solution of the theory we choose to look at ; this freedom allows us for example to choose any value for the dilaton field which sets the local value of the string coupling @xmath47 . since we can not add anything to the theory , we must make our black hole from objects _ in _ the theory . the theory contains gravitons , as any theory of quantum gravity would , and a collection of extended objects - strings and branes - of different dimensionalities . one knows that all different versions of string theory are related by exact dualities , so we can use any one ; we will take type iib string theory for concreteness . one makes black holes by taking branes in the theory and wrapping them on compact directions ; from the viewpoint of the noncompact directions this places a given mass at a point in space , and with a suitable choice of wrapped objects we can create a black hole . among the objects in iib string theory we have 5-dimensional branes , which we will use . thus we compactify five directions as follows m_9,1m_4,1s^1t^4 [ sfour ] where we have singled out one @xmath48 for later use . the @xmath48 has length @xmath49 and the @xmath50 has volume @xmath51 . we can wrap a large number @xmath52 of strings on the @xmath48 , and this does give a large mass at one point in the noncompact space . but the strings carry charge as well , and also create distortions of the moduli the sizes of the compact directions . when all these effects are taken into account one finds that one does not get a horizon , and there is no black hole . from a statistical entropy perspective this is good , since the degeneracy of the string bound state does not grow with @xmath52 ; the strings bind together by just making one ` multiwound string ' which loops @xmath52 times around the @xmath48 before closing on itself . thus the statistical entropy vanishes , in agreement with the vanishing of the bekenstein entropy . we can do better by adding @xmath53 units of momentum along @xmath48 to the string . the strings are called the ` ns1-brane ' in string theory , and momentum is usually denoted ` p ' , so this system would be called the ns1-p system . from the viewpoint of the noncompact directions , the momentum carries kaluza - klein charge under the gauge field arising from reduction along @xmath48 , so we have two kinds of charges in the state and this is called the 2-charge system . the extremal states of this system are those that have the minimum charge for given mass , and these turn out to be supersymmetric . what is remarkable is that these lowest mass states are very numerous . as we had seen , the strings join up into one long string , and the momentum will bind to this string by creating travelling waves along the string . the total momentum can be partitioned among different harmonics in many ways , and each such state has the same energy . the number of states is @xcite ~e^2 so that the microscopic entropy is s_micro=2 [ entropy ] terms from string theory ; there is an infinite throat , ending in a horizon with a singularity inside ( c ) the actual geometries of the ns1-p system ; the throat ends in ` caps ' , with different caps for different microstates . the boundary of the ` cap ' region shown by the dotted circle has area satisfying @xmath54 . ] what about the metric that this ns1-p system will generate ? let us discuss this in three steps : \(1 ) at first it may seem reasonable to assume that all the strings and momentum charges sit at one location @xmath18 in the noncompact space ; after all we had made a bound state of these objects and so all charges should be concentrated at a given location . this gives what we will call the ` naive ' geometry of the 2-charge extremal system ds^2&=&11+q_1r^2[-dt^2+dy^2+q_pr^2(dt+dy)^2 ] & + & _ i=1 ^ 4 dx_idx_i+_a=1 ^ 4 dz_adz_a [ three ] here @xmath55 is along @xmath48 , @xmath56 are coordinates for @xmath50 and in the 4-d noncompact space @xmath57 we write @xmath58 . this metric has a singularity at @xmath18 but no horizon . this metric is sketched in fig.[geometries](a ) . \(2 ) we note that near @xmath18 the curvature of ( [ three ] ) diverges , so if there are higher derivative @xmath59 terms in the gravity lagrangian then they can be important . in string theory there is indeed a whole series of such higher derivative terms in the effective action @xcite , and it is unclear how to compute the net effect from all these terms . dabholkar @xcite considered the 2-charge system was taken with a slightly different compactification ( the @xmath50 in ( [ sfour ] ) was replaced by another 4-manifold called k3 ) . only the first of the higher derivative corrections was considered , and it was found that the naive geometry changed to one of the kind expected for an extremal hole : there is an infinite throat ending in a horizon ( fig.[geometries](b ) ) . in the presence of higher derivative terms the bekenstein entropy gets replaced @xcite by its generalization , the bekenstein - wald entropy @xmath60 , and it was found that s_bw = s_micro an order of magnitude agreement between these entropies had been earlier conjectured in @xcite . thus we see that while there are still open questions about the gravity solution of the 2-charge hole , it does seem plausible that this is a simple example of an extremal black hole . the strongest argument for thinking of the 2-charge system as a good black hole comes from the form of the microscopic entropy . we have @xmath61 for two charges , @xmath62 for three charges , and @xmath63 for four charges ; further the entropy arises in each case as a partition of momentum of a string or ` effective string ' . thus let us investigate further this 2-charge system and see how fuzzballs arise . \(3 ) let us now ask if ( [ three ] ) is indeed the correct metric for the system . we have seen that different microstates arise from different ways of carrying the momentum on the string . the first point to note is that the fundamental string has no longitudinal vibrations , so to describe the momentum carrying wave we have to specify both the harmonic order along the string as well as the transverse direction chosen for vibration . to picture the vibrations of the string let us open it up to its full length @xmath64 ; i.e. go to the @xmath52 fold cover of the @xmath48 . let us start by putting all the momentum in the lowest allowed harmonic , and choose the polarization of the vibration such that the string in the covering space executes one turn of a uniform helix ; the helix will project to a circle @xmath65 in the noncompact space . thus the string looks like a ` slinky ' , winding around the @xmath48 direction as it wanders around in the @xmath66 plane . the important part here is that the string is not sitting at @xmath18 in the noncompact space ; instead it is spread out over a sizable region ( the size of this region scales with the charges as @xmath67 ) . as a result the metric produced by this vibrating string will differ from the naive expectation ( [ three ] ) . this metric can be written down in a straightforward way @xcite ( it was earlier found in a dual form in related contexts @xcite ) . the metric has no horizon , and we have pictured the string and its metric in fig.[strings](a ) . what do we make of this metric ? this way of choosing the momentum harmonics is certainly one of the microstates that we were counting in the entropy ( [ entropy ] ) . but the geometry does not agree with ( [ three ] ) , and even if we apply the higher derivative corrections , we do not get the infinite throat of fig.[geometries](b ) . one might think that the departure from ( [ three ] ) arises because this particular state of the string has a large rotation ; by choosing the string to swing in a helical fashion we gave the state its maximal possible angular momentum . to address this issue , let us look at a microstate that has _ no _ angular momentum . since we know how to make all microstates , this is easy to do . again consider our vibrating string , but let the first half of the string describe a clockwise helix , and the other half an anticlockwise helix ( fig.[strings](b ) ) . the net angular momentum will be zero . naively , one might have thought that now we should get back the solution ( [ three ] ) ; after all the state we are making has the same mass , charges and angular momentum as the metric ( [ three ] ) . but we see immediately that we will _ not _ get the metric ( [ three ] ) ; the string has again spread over a region whose radius scales as @xmath68 . so we see that the actual microstates of our system do not give the ` naive ' metric ( [ three ] ) . further the region over which the metric departs from the naive metric is so large for these states that the higher derivative corrections turn out to have no significant effect on the geometry ; in particular it does not change the microstate geometry to one with an infinite throat . let us now consider the general state of this system . each harmonic of vibration of the string behaves like a harmonic oscillator , and strictly speaking we should specify the state of the string by giving the excitation number for each oscillator . thus an energy eigenstate would be written like |=(a^ i_1_k_1)^m_1 ( a^i_2_k_2)^m_2 (a^i_s_k_s)^m_s|0[sstate ] where @xmath43 is the state of the string with no vibrations , and the creation operator @xmath69 creates an excitation in the harmonic @xmath70 with vibration direction @xmath71 . a generic state will have @xmath72 , and so we should really write down the quantum wavefunction of the appropriate eigenstate for each harmonic oscillator . but it is easier to start with the case where the energy of the state is placed in relatively few harmonics , so that @xmath73 . in this case we have large occupation numbers for the excited oscillators , and we can replace the energy eigenstates by coherent states without losing the essential physics of the state . now we can describe the string by a classical vibration profile f(t - y ) where the vector @xmath74 is transverse to the direction @xmath55 , and is a function of only @xmath75 because the momentum moves purely upwards along the string in the extremal state ( if we had vibrations going in both directions on the string then we would have more energy than needed to give the net momentum charge of the state ) . let the string vibrations be in the noncompact directions . the metric of the string carrying such a vibration profile is given by @xcite ds^2_string&=&h[-dudv+kdv^2 + 2a_i dx_i dv ] & & + _ i=1 ^ 4 dx_idx_i+_a=1 ^ 4 dz_adz_ab_uv&=&-12[h-1 ] , b_vi = ha_ie^2&=&h [ ttsix ] where h^-1&=&1+q_1l_t_0^l_t + k&=&q_1l_t_0^l_t + a_i&=&-q_1l_t_0^l_t [ functionsq ] where we have written @xmath76 to denote the fact that this metric is in the ` string frame ' of string theory , and we have given the metric , gauge field @xmath77 and dilaton field @xmath78 which are the nonzero fields in this solution . in fig.[geometries](c ) we depict these solutions schematically . there is no horizon ; instead the throat ends in a cap whose structure depends on the choice of profile function @xmath74 . ( the same geometries can also be obtained in the language of ` supertubes ' , where the charges are dualized to ns1-d0 @xcite . ) for string vibrations in the @xmath50 directions the metrics can be found in a similar way @xcite , and fermionic excitations can be added @xcite to make general extremal solutions . the extremal and near - extremal behavior of 2-charge solutions have been studied in many different ways @xcite . it is now interesting to look at a generic state from the set of allowed states , and note at what radius @xmath79 this departure from the naive geometry becomes significant . suppose we compute the area @xmath80 of this surface @xmath12 in the naive geometry ( [ three ] ) ; since the naive geometry and the actual geometries pretty much agree at this location what we are computing is the area of the boundary of the ` fuzzball region ' in a typical microstate . interestingly , one finds that @xcite ~~s_micro [ ssize ] so we see that even though there is no horizon for any microstate , the boundary area of the typical microstate satisfies a bekenstein like relation with the entropy of the system . in the above section we looked at extremal holes made with two charges ns1 and p. in string theory we have s and t dualities , which can change one set of charges into another . these are exact symmetries of the theory , so the physics in the two descriptions will be equivalent . but it can be more convenient to describe the physics in one duality frame than in another . to see the structure of the 2-charge system it is useful to start with the ns1-p frame , as we did . this is because the bound states of this system are just states of a fundamental string carrying momentum , and it is possible to construct the metric produced by such a string . it is not obvious how to construct the metrics of the 2-charge extremal states if we start in any other duality frame . but once we have the metrics in the ns1-p frame , we can of course apply the s , t dualities to get the metrics in any other duality frame . why should we be interested in other duality frames ? we have two goals : \(1 ) first , we would like to study small excitations around the extremal states that we have constructed . the extremal states themselves are supersymmetric ground states of system for its given charge , and they have no dynamics . if we excite the system with extra energy , then the state will become non - supersymmetric and time - dependent , and we will be able to observe the dynamical behavior of excitations around our microstate . \(2 ) second , we would like to construct extremal states of the extremal black hole with _ three _ charges . the 3-charge hole has a larger entropy , and therefore a larger horizon , than the 2-charge hole . the higher derivative corrections are thus small at the horizon of the 3-charge hole , so this hole looks closer to the black holes that we are familiar with . we will find it easier to do both these things if we first take our 2-charge system to another duality frame . under the dualities the following will happen : \(1 ) the @xmath52 ns1 strings will be transformed to @xmath81 d5 branes . in the compactification ( [ sfour ] ) these d5 branes are wrapped on @xmath82 . \(2 ) the @xmath53 units of momentum will be transformed into @xmath83 d1 branes wrapped on @xmath48 . the system obtained after these dualities will be called the d1d5 system . as in the case of the ns1-p system , what we want here is the bound state of the d1 branes and the d5 branes . naively , we might think that when we bind all these branes together then we will get a pointlike mass which we can take to be sitting at the origin @xmath18 of the noncompact space @xmath84 . but our experience with the ns1-p system shows that this might not be right . in the ns1-p case the system had acquired a nontrivial transverse size due to the vibration of the ns1 in the process of carrying the momentum p. an s duality will not change the transverse size of any system , when we measure this size in the einstein metric ; this is because the einstein metric is left unchanged by an s duality . t dualities are carried out only in the compact directions , and do not change the transverse size of the system when this size is measured in the _ string _ metric . thus when we are done with our dualities from ns1-p to d1d5 , we will find that the d1-d5 microstate will also have a nontrivial transverse size . the metric ( [ ttsix ] ) for the ns1-p system gives , after duality transformations , the following d1-d5 metric @xcite ( the subscript ` string ' means that the metric is written in the string frame ) ds^2_string&=&[-(dt - a_i dx^i)^2+(dy+b_i dx^i)^2]&+&dx_idx_i+dz_adz_a [ qsix ] where the harmonic functions are h^-1&=&1+^2q_1l_t_0^l_t dv|x - f(v)|^2k&=&^2q_1l_t_0^l_t dv ( ^2 f(v))^2|x - f(v)|^2 , + a_i&=&-^2q_1l_t_0^l_t dv f_i(v)|x - f(v)|^2 [ functionsqq ] here @xmath85 is given by db=-*_4da [ vone ] and @xmath86 is the duality operation in the 4-d transverse space @xmath87 using the flat metric @xmath88 . by contrast the ` naive ' geometry which one would write for d1-d5 is & & ds^2_naive=1[-dt^2+dy^2 ] & & dx_idx_i+dz_adz_a[d1d5naive ] suppose that in ( [ qsix ] ) we look at a region |f|r [ slimit ] then we see that the metric simplifies to the form ds^2_string&=&r^2[-dt^2+dy^2]+dr^2&+&d_3 ^ 2 + dz_adz_a this metric has the form ads_3s^3t^4 thus we have an asymptotically @xmath89 space if we restrict to the region @xmath90 , and we can apply the ideas of ads / cft duality . to be able to take the limit ( [ slimit ] ) we need that @xmath91 , and it turns out that this is possible if we take the radius @xmath92 of the @xmath48 to be large @xcite 1 taking this limit , we expect by maldacena s ads / cft correspondence that there will be a cft description that is dual to the gravity description . let us see what this cft is . recall that we have wrapped @xmath93 d5 branes on @xmath82 and @xmath94 d1 branes on @xmath48 . the d1 branes are bound to the d5 branes , so as a first approximation we can say that the d1 branes vibrate inside the plane of the d5 branes . but now note how the corresponding charges behaved in the ns1-p duality frame . suppose we had a ns1 that was wound @xmath52 times around the @xmath48 , which has length @xmath95 . let us add one unit of momentum p. then we have p=2l=2n_1n_1 l=2n_1l_t thus even though we added only one unit of momentum p to the ns1 , this unit of momentum looks like @xmath52 units of the basic vibration mode allowed on the full length of the ` multiwound ns1 ' . let us call this phenomenon ` _ fractionation _ ' @xcite . after duality to the d1d5 frame , we get the following picture . suppose we have @xmath93 d5 branes in a bound state . we bind one d1 brane to these d5 branes . then this d1 brane will appear as a ` fractional d1 brane ' in the bound state ; it will behave as if there were @xmath93 ` fractional d1 branes inside the d5 branes ' , with the tension of each fractional d1 brane being @xmath96 times the tension of an isolated d1 brane @xcite . if we had @xmath94 units of d1 charge , then there will be nn_5n_1 units of ` fractional d1 charge ' inside the d5 branes . this corresponds to the @xmath97 units of ` fractional momentum ' that we would find in the ns1-p duality frame . the d1 and d5 branes each stretch like a ` string ' along the direction @xmath48 . now note that these two kinds of branes can be interchanged by a set of t dualities . indeed , if we perform a t duality in each of the four directions of the @xmath50 , the d5 branes become d1 branes and the d1 branes become d5 branes . thus our final model for the d1d5 bound state should be symmetric under the interchange of these two kinds of branes . thus rather than think of having @xmath98 units of fractional d1 branes inside the d5 branes , we should think of just having @xmath99 units of an ` effective string ' @xcite that winds around the @xmath48 . one can advance more rigorous arguments for such a model , but the above crude picture should suffice for our present discussion . let us now see what we can do with this effective string : \(1 ) _ count of ground states:_first , let us look at all the ground states of this d1d5 cft . the effective string has total winding number nn_1n_5 around the @xmath48 . we can have many different configurations with this same total winding . all strands of this effective string could be separate closed loops , as in fig.[eff](a ) . or we could join them all into one long string , as in fig.[eff](b ) . more generally , we would get @xmath100 strands with winding number @xmath101 , with _ k k m_k = n [ ssum ] counting all these different possibilities gives @xmath102 states , with s=2 this agrees with ( [ entropy ] ) , as it should , since the d1d5 system is the same as the ns1-p system under s , t dualities . identifying cft states with gravity solutions:_we have made d1d5 gravity solutions in ( [ qsix ] ) , and sketched the cft states in fig.[eff ] . but which cft state corresponds to which gravity solution ? the link is made by going through the solutions in the ns1-p language . start with the d1d5 cft state , and look at a loop with winding number @xmath101 . each separate such loop is called a ` component string ' . in the ns1-p picture the string state ( [ sstate ] ) was described by oscillators acting on the vacuum state . the oscillator @xmath103 maps to a component string with winding number @xmath101 . the polarization @xmath104 of the vibration gives a ` spin ' for the component string , which we have drawn with arrows in fig.[eff ] . thus start with a cft state , find the corresponding ns1-p state from ( [ sstate ] ) and find the profile function @xmath74 for these vibrations of the string . putting this @xmath74 in ( [ ttsix ] ) gives the metric of this ns1-p state , and performing s , t dualities gives the metric ( [ qsix ] ) in the d1d5 duality frame . this then is the metric dual to the cft state that we started with . as mentioned above , to get a well defined profile function @xmath74 we need large occupation numbers @xmath100 for each @xmath101 in ( [ ssum ] ) . if this is not the case , we get quantum fluctuations and the system is not well described by a classical geometry ; this gives the general ` fuzzball ' configuration . ( the reader can consult the references @xcite for details on the approximations needed to get a classical geometry and for more details of the cft - gravity map @xcite . ) the essential point property of fuzzballs is their size and not how ` quantum ' the solution is . as noted in ( [ ssize ] ) the _ size _ of the generic state is order horizon size ; how ` quantum ' this state is depends on whether the excitations of the ns1 are concentrated into a few harmonics or spread over many harmonics . \(3 ) _ energy gaps:_so far we have looked at ground states of the d1d5 cft . let us now add some extra energy to one of these ground states , making a non - extremal state . the dynamics of the effective string is very simple if we are at ` weak coupling ' : we just get massless bosonic and fermionic modes travelling up and down the effective string ( these are called left and right moving modes respectively ) . while the cft should actually be at strong coupling to reflect the gravity solution , we will use it at weak coupling where we can actually compute , and hope that the corrections are not large since we are ` close ' to supersymmetric configurations . in fig.[effex](a ) we take the state where all component strings are singly wound , and add an excitation on one component string ; let this excitation be in the lowest harmonic allowed on the component string . this is the lowest energy excitation of this cft state , and has an energy @xmath105 . in the gravity dual , we see that we can place a quantum in a wavefunction at the bottom of the throat . let the lowest allowed energy for such a quantum be @xmath106 . one finds ( e)_cft=(e)_gravity [ sagree ] in fig.[effex](b ) we take the cft state with winding number @xmath107 for each component string . the lowest allowed excitation energy is now _ half _ the value in fig.[effex](a ) . but the corresponding gravity dual has a deeper throat ; this makes the quantum in the geometry suffer a larger redshift , and we again get ( [ sagree ] ) . we see from this analysis that we _ must _ have ` caps ' for the geometries dual to the d1d5 cft states . if we had the naive geometry of fig.[geometries ] ( a ) or ( b ) , then we would not get agreement of energy gaps between the cft and gravity pictures . 3-charge geometries:_if we add excitations carrying momentum p _ up _ the component strings , but not _ down _ , then the state get a net momentum charge p which equals the energy added . we then get states of the 3-charge extremal hole @xcite . the generic cft state of this hole is pictured in fig.[threec](a ) . we do not yet know how to make the dual of a generic 3-charge cft state . but let us look at the simple 3-charge state depicted in fig.[threec](b ) ; because all the component strings have equal length and spins , the geometry has axial symmetry , and we _ are _ able to construct the gravity dual . this dual is given by the metric @xcite & & ds^2 = - ( dt^2-dy^2 ) + ( dt - dy)^2 & & + hf ( + d^2 ) + & & + h ( r_n^2 - n a^2 + ) ^2d^2 + & & + h ( r_n^2 + ( n+1 ) a^2 - ) & & ^2d^2 + & & + ( ^2d+ ^2d)^2 + & & + ( dt - dy ) + & & - dy & & + _ i=1 ^ 4 dz_i^2 [ em ] where & & q_1 q_5q_1 q_5 + q_1 q_p + q_5 q_pf & = & r_n^2 - a^2n ^2 + a^2 ( n+1 ) ^2 + h & = & , h_1 = 1 + , h_5 = 1 + [ deffh ] again one finds that there is no horizon , and the geometry ends in a smooth ` cap ' ( fig.[threec](c ) ) . the energy gaps for the 3-charge cft states agree exactly with the energy of quanta placed in the geometry fig.[threec](c ) . these facts suggest very strongly that all we have learnt for 2-charge extremal holes ( where we can understand all states ) will also hold for 3-charge extremal holes . \(5 ) _ non - extremal holes : _ we have seen in ( 2 ) above that we get non - extremal states if we have excitations running both up and down the string . in the case ( 2 ) we added only one excitation to one component string , so in the gravity dual we had just one quantum sitting in the geometry . we could ignore the backreaction of this single quantum , and so solved the free wave - equation on the extremal background . let us now consider the general non - extremal state , where we have an arbitrary number of left and right excitations on the component strings . we depict the general state of the non - extremal system in fig.[nonex](a ) . if we could understand the gravity dual of such cft states , we would have understood the non - extremal black hole . we can not construct the gravity duals of the generic states fig.[nonex](a ) , but we _ do _ know how to make duals of special states like the one in fig.[nonex](b ) . all the component strings have been chosen to have the same length and spins ; further , the left and right excitations are all fermionic and chosen so that they occupy the lowest allowed levels for these fermions . in this case the gravity dual is found to have the structure @xcite @xmath108 where c_i = _ i , s_i=_i @xmath109 the geometry again has no horizon , and is sketched schematically in fig.[nonex](c ) . it is exciting that we have been able to make non - extremal states and found them to also be ` fuzzballs ' rather than ` metrics with horizon ' . but more is true . we can also study hawking radiation from these non - extremal states . first consider the generic cft state in fig.[nonex](a ) . the left and right moving excitations can collide and leave the cft bound state as radiation . the rate of this process is given by an emission vertex @xmath110 times the occupation probabilities for the left and right colliding modes . symbolically , = v_l_r if we put thermal distributions for @xmath111 , then @xmath112 agrees exactly with the hawking emission from the near - extremal black hole @xcite = _ hawking of course here we have agreement only of the radiation _ rate _ , not the details of emission . the cft emission @xmath112 is a unitary process in a normal thermodynamic system , while @xmath113 is the semiclassical computation in the black hole geometry which leads to information loss . let us see if we can do better with our understanding of fuzzballs . we can not yet make the gravity dual of the general state fig.[nonex](a ) , but let us see if we can understand emission from the special states fig.[nonex](b ) that we _ can _ make . in the cft description we get the emission by replacing the occupation numbers @xmath114 with the ones appropriate to this special microstate _ cft = v|_l|_r on the gravity side , we find that the geometry ( [ 2charge ] ) is _ unstable _ , and radiates energy out to infinity @xcite . the rate of this radiation is found to exactly agree with the rate of emission from the cft @xcite _ gravity=_cft with such an explicit description of the emission from the gravity state , we can ask how and where the radiation arises . the geometry of the microstate has no horizon , but it does have an ergoregion . thus we get the process of ergoregion emission , whereby particle pairs are produced near the ergoregion ; one member of the pair falls into the ergoregion while the other escapes to infinity as radiation . but the member that falls in is not ` lost ' as would be the case for traditional hawking radiation ; instead it influences the production of further quanta from the ergoregion . this happens because of a ` bose enhancement ' process ; after @xmath115 quanta have collected in the ergoregion the probability to create the next quantum is proportional to @xmath116 . the emission thus increases exponentially , and is characterized by a set of complex frequencies ^(i)gravity=^(i)gravity_r+i^(i)gravity_i in the dual cft state fig.[nonex](b ) we also find emission peaked at certain discrete frequencies since we have taken all component strings to be excited in the same way . we again find an exponential growth of emission , with complex frequencies in exact agreement with the gravity emission @xcite ^(i)cft=^(i)gravity the emission from our special microstates is peaked at special frequencies like a laser instead of being the planckian emission spectrum expected from warm bodies . but this is of course expected ; each microstate emits somewhat differently , and if we start with a very special microstate where all excitations are at a given energy then we will get a peculiar emission behavior . the important point is that we get exact agreement between the cft computation and a gravity calculation which this time _ gives the same emission by a unitary process with no information loss_. in particular , we see that the quanta that fall into the ergoregion influence the production of the next quantum through bose enhancement . this should be compared to the discussion of section [ introduction ] where we noted that radiation from a piece of coal can carry out information because radiated quanta can ` see ' the effects of earlier radiated quanta , while in the traditional computation of hawking radiation the newly produced pairs do not see the state of earlier produced pairs . the 2-charge extremal hole requires @xmath117 corrections at its horizon to get the exact bekenstein - wald entropy . thus while we can understand all states of the 2-charge hole , we would like to study 3-charge and 4-charge extremal holes , which have a larger horizon and do not require such corrections . for 3-charge and 4-charge extremal holes we do not yet have a systematic way of constructing all states in the gravity description . but for all those states which have been constructed , we find that we get ` fuzzballs ' : the throats are finite and capped , not infinite and ending in a horizon . the simplest 3-charge extremal states are those with @xmath118 axial symmetry ; these states were constructed some years ago @xcite . how do we make more general 3-charge solutions ? it can be shown that any supersymmetric solution for n=1 supergravity in 6-d can be written as a 2-d fiber over a hyperkahler base @xcite . the @xmath118 extremal solutions @xcite can be dimensionally reduced on the @xmath50 to give solutions in 6-d , and we can then ask what this base - fiber split looks like . interestingly , the base turns out to be ` pseudo - hyperkahler ' : the signature of the base jumps from being @xmath119 to @xmath120 across a hypersurface in the base @xcite . the fiber degenerates at this hypersurface too , in such a way so that the overall 6-d metric remains smooth . thus the lesson is that while local supergravity equations tell us that the solution will have a hyperkahler base and a 2-d fiber , in the actual solutions corresponding to d1-d5-p extremal states this split can not be performed globally ; it degenerates along certain surfaces . in a very interesting series of papers @xcite , bena and warner took this story to a new level . they started from the equations of 11-d m - theory , and obtained a more detailed version of this base - fiber split . specializing the hyperkahler base to gibbons - hawking spaces ( which have an extra u(1 ) symmetry ) , they managed to get a complete solution of the supergravity field equations . the fact that the space was pseudo - hyperkahler ( rather than hyperkahler ) could be easily built into their formalism : the solutions were written in terms of harmonic functions on the base , and the sign of the sources in these harmonic functions determined the local signature of the base . with this formalism , it became possible to write down explicitly large families of supersymmetric solutions to string theory , all having the mass and charges of the 3-charge black hole . none of the solutions had a horizon or ` black hole singularity ' . the sources of the harmonic functions are held apart at fixed distances by fluxed running on spheres joining them ; these constraints are given by ` bubble equations ' , which contain the essence of the supergravity equations in the present ansatz . the above mentioned solutions had one @xmath121 symmetry the one needed to make the pseudo - hyperkahler base a gibbons - hawking space . the solutions have 4 + 1 noncompact dimensions . we can do a dimensional reduction along the circle corresponding to the remaining @xmath121 symmetry , thus getting solutions in 3 + 1 noncompact dimensions . the way to do this compactification is to make the circle the fiber of a kaluza - klein monopole . the solutions acquire a fourth charge , that of the kk monopole , and we get 4-charge solutions in 3 + 1 noncompact dimensions . ( note that if we want to make an extremal black hole with classical horizon size in 3 + 1 dimensions , then we have to use four charges . ) such solutions have recently been constructed @xcite . the bubble equation in this 3 + 1 dimensional setting become similar to equations studied earlier by denef @xcite . in fact denef had developed an elegant general formalism for making supersymmetric solutions out of more fundamental constituents . these fundamental constituents could be individual branes ( having no entropy ) or extremal black holes ( having a nonzero entropy ) . the fuzzball proposal would say that all states of the system can be written in terms of constituents _ without _ entropy . it is not clear if the elementary constituents used in the references above @xcite are ` complete ' ; it is likely that there are more complicated constructions that need to be taken into account before we have all 4-charge extremal states . a general philosophy that emerges from all these constructions is the importance of ` dipole charges ' . the supersymmetric solutions have some charges that we measure from infinity ; let us call these the ` true charges ' of the solutions . when we look at the actual microstate solutions , we find that we have flat space at infinity , then for some distance we have the uniform throat expected of the traditional black hole , and then a ` cap ' region . in this cap we find , besides the ` true ' charges , a set of charges that are not measured as charges at infinity . these are ` dipole charges ' and their net value adds up to zero . but their locations can be varied , and this gives us different solutions corresponding to the same total mass and ` true ' charges . exploring the space of such allowed solutions is therefore relevant to exploring the structure of general black hole microstates . ( one can think of these dipole systems as supertubes with more than two charges ; for some generalization of supertubes to three or more charges , see @xcite . ) with this wealth of available tools , a large variety of supersymmetric solutions have been made for the 3-charge and 4-charge cases . one can make structures that look like microstates of holes , or rings , or a collection of holes and rings . some choices of fluxes lead to ` deep throat ' solutions , which may account for a large fraction of the microstates of the hole . solutions depending on a continuous parameter were recently found @xcite by putting a supertube inside a deep throat . with such a construction it may be possible to get enough solutions that their number will go like @xmath122 $ ] for charges @xmath123 ; in that case one would have an entropy from these solutions that would account for the black hole entropy , and we would be in a situation similar to the one that we had for the 2-charge case . several other studies have been done with extremal solutions . steps have been taken to quantize the moduli space of these solutions @xcite , to study the mathematical properties of the family of such solutions @xcite , and to coarse grain over the solutions to get an ` entropy ' @xcite . while we could make the gravity states of the 2-charge extremal system with comparative ease , we have seen that it is hard to make the gravity duals for general states of the three and four charge extremal holes . one approach in this situation has been to treat some of the charges ` exactly ' , finding their exact gravity description , while letting the other charges be placed as a small perturbation in the background produced by the first set of charges . with such an approach we may be better able to think of the complete ensemble of all states , though we will lose some understanding of the full gravity description of the state since some of the charges have not been handled with full backreaction . let us see how some of these approaches proceed . since the entropy of a black hole is given by its surface area , it has always been tempting to find some degrees of freedom that live at the horizon and whose count gives the entropy of the hole . the problem with this of course is that we can not place something at the horizon and expect it to stay there ; any excitation at the horizon either falls into the hole or escapes to infinity , leaving no degrees of freedom at the horizon . this is just the standard ` no hair ' phenomenon found for traditional black hole geometries , and has been a long standing problem in understanding the entropy of black holes . but now we have learnt that at least for simple cases of extremal black hole states , we do not have a horizon but instead a geometry that ` caps off ' before a horizon is reached . in the simplest case of the 2-charge extremal d1d5 solution , the profile function @xmath74 is the helix sketched in fig.[strings](a ) . in this case the cap region has the geometry of global @xmath124 . thus let two of the charges making the hole be d1 and d5 , and let these charges be in a state which generates this particular 2-charge geometry . now let us add other excitations as perturbations , creating new excitations that we can count but for which we will not take the gravitational backreaction into account . what excitations should we take ? it has been noted @xcite that we can put ` giant gravitons ' @xcite in @xmath89 type geometries . these giant gravitons are branes which wrap spheres in the @xmath89 space or the sphere , and are preventing from collapsing to a point because they move through the gauge field flux which exists in the background geometry . counting these giant gravitons one finds enough states to account for the entropy of a 3-charge hole . note however that since we have not considered the gravitational backreaction of these giant gravitons we can not say that we understand the full gravity description in this picture . a counting has been suggested @xcite for ` brane states wrapping a black hole horizon ' . the count gives a number that agrees with the entropy of the corresponding hole . for the reasons mentioned above , it is completely clear where and how such brane states would be located in the presence of the horizon . tro see if this count could be put on a firmer footing , an attempt was made @xcite to understand such a counting of branes by replacing the effect of some of the charges by the capped geometry they would produce ; the other branes were then put as test charges in the capped geometry . there is no horizon now , but the branes wrapped a sphere which is analogous to the spherical horizon in the naive black hole geometry . with this construction the branes did not fall through a horizon , and thus could be localized and counted . but a different problem emerged . the branes wrapping the sphere turn out to act like ` domain walls ' , so that the value of the flux they produced jumps from one side of the wrapped brane to the other . regularity in the cap required no field on the ` inner ' side of the brane , so one gets a nonzero field on the ` outer ' side which extends all the way to infinity . thus wrapping a brane in this fashion on a sphere produces a nonzero gauge field strength over an infinite volume , making the state have infinite energy . perhaps some other method of wrapping branes may be more appropriate to counting the degrees of freedom of the system . the 4-charge hole has been studied similarly @xcite by letting some charges form a ` capped ' background , and letting the other charges be added as test branes . interactions between these test charges were also considered , and it seemed possible that d0 branes placed in the background geometry could swell up to d2 branes wrapping spheres by the myers mechanism @xcite . the construction has been extended to black rings @xcite . it should be checked though if in all these cases one can avoid the above mentioned problem of infinite flux energies . the fuzzball program constructs the states of the black hole , in the gravity description . these states can be thought of as energy eigenstates of the system . thus they do not individually describe time dependent processes like the formation of a black hole by collapse of a shell . but once we understand the energy eigenstates of a system , we can reconstruct its dynamics by superpositions of these eigenstates . while we do not have a comprehensive picture of all non - extremal microstates , in this section we will try to conjecture some aspects of the dynamics that should result if all black hole microstates were indeed fuzzballs . the main dynamical questions of interest are of the following type . what happens to an observer as he approaches the horizon ? how should we understand his evolution inside the hole ? how does his information come out in the hawking radiation ? if we start with a collapsing shell , how does it evolve into a fuzzball ? let us consider what we have learnt about black hole microstates and see if we can postulate how some of these questions might be answered . a common first question about fuzzballs is the following . in the traditional picture of the hole we have vacuum at the horizon , so an infalling observer feels nothing as he crosses the horizon . in the fuzzball the information of the hole is distributed throughout a horizon sized ball . so will the observer feel something drastically different as he approaches the place where he expected a horizon ? to understand this and similar issues , it is important to note that there are two different time scales of interest in the black hole problem . one is the ` crossing time scale ' @xmath125 over which an infalling quantum travels from the horizon to the singularity . the other is the much longer hawking evaporation timescale @xmath126 , which for a 3 + 1 dimensional schwarzschild hole is @xmath127 times @xmath125 . thus we can say that @xmath126 is larger than @xmath125 by a power of @xmath128 . now consider a quantum falling into the hole . the density of the ` fuzz ' for a generic state of the hole was computed @xcite , and found to be low at the horizon . thus there need not be a sharp interaction of the infalling quantum with the degrees of freedom of the hole ; in fact there is no contradiction in assuming that the motion of the quantum over the time @xmath125 resembles the free fall in the traditional black hole geometry . what we _ need _ to solve the information problem is that the interaction of the infalling quantum with the degrees of freedom of the hole happen in a time smaller than @xmath126 , so that the information of the quantum can indeed come out in the hawking radiation . since @xmath129 , there is no contradiction in assuming very different evolutions on these two different time scales . the existence of these two different scales makes it possible to preserve some part of our classical intuition about black holes while resolving the information puzzle . this could be part of a more general principle . in the extremal hole , we see two different _ length _ scales . in the traditional extremal geometry the throat has an infinite length . in the fuzzball picture , the length of the throat for a generic 3-charge geometry has been estimated @xcite . suppose the diameter of the throat is @xmath130 . the length of the throat is then a power of the charges @xmath131 times @xmath130 . from a macroscopic perspective , we can say that the depth of the throat is a power of @xmath128 times its diameter . thus if we look only a down the throat only upto a fixed multiple of @xmath130 then for @xmath132 we will see just the classical throat geometry and not the quantum fuzz at the end of the throat . these computations suggest a ` classical correspondence principle ' , which would say that to leading classical order the fuzzball states behave in a way expected from the traditional hole . we do not yet have a clear formulation of such a principle , but let us note some other computations which might help formulate such a principle . consider the 2-charge extremal geometries . if we take a simple geometry like the one pictured in fig.[strings](a ) , then an infalling quantum bounces off the end and returns back in a small time . now consider the geometry for a generic state ( fig.[geometries](c ) ) . this geometry is very complicated in the cap region , and an infalling quantum will be trapped in that region for a long time . was estimated the time of return from a generic 2-charge geometry was found @xcite to be a power of @xmath133 times the crossing time across the fuzzball ; this long time results from the many deflections a geodesic suffers before it can exit the cap region . again , we can think of this return time as a power of @xmath128 times the crossing time . for an observer who looks at the system only for a fixed multiple @xmath134 times the crossing time , the infalling quantum would appear to be lost for ever when we take the charges to infinity . thus for such an observer we can replace the boundary of the fuzzball by a traditional horizon , and obtain essentially the same effect : now the infalling quantum would never return . it is in this sense that we should understand the emergence of a horizon in the fuzzball picture . the ` horizon ' is only an effective concept describing the evolution over the short timescale @xmath125 , while the actual details of the quantum fuzz lead to the eventual leakage of information from the fuzzball , something that can not happen if we _ really _ had the traditional black hole horizon . just as we differentiated between two different time scales and length scales , we should also separate two different _ energy _ scales . the typical hawking radiation quantum has an energy @xmath135 of order the temperature @xmath136 of the hole . when we think of an infalling observer , we should ask if the energy of this observer is @xmath137 , or if @xmath138 . from our analysis of the information paradox we know that the evolution of hawking radiation quanta with @xmath137 _ must _ be modified by order unity by the detailed information in the fuzzball state ; otherwise information will not come out in the radiation . on the other hand , it is not necessary that the evolution of modes with @xmath138 be affected to leading order by the fuzzball structure , at least for time scales @xmath139 . as an explicit example of this , consider the nonextremal geometry that we considered in the last section . the hawking emission happens because of the negative effective potential in the ergoregion , and this emission does not happen from the part of the geometry which is not in the ergoregion . but the negative potential is quite small , and the emitted quanta have a low energy . if we send a high energy quantum into the geometry , it does not notice the ergoregion potential in any significant way , and its evolution does not depend sensitively on whether or not it passes through the ergoregion . thus here we have a simple example where the evolution of the hawking radiation quanta depends on sensitive details of the geometry while the evolution of a ` heavy ' infalling observer is not sensitive to the same details . if the energy eigenstates of the black hole are horizon sized fuzzballs , then any infalling shell should eventually be best described by a linear combination of these fuzzball geometries . but how will this happen ? a classical shell seems to feel no large quantum effects as it crosses the horizon , so one would think that the result should be the traditional black hole with the ` information free horizon ' . in this section we will make some simple observations which indicate why black holes may not be as classical as they at first appear . consider any state of matter which has mass @xmath46 , and which is localized in a region @xmath140 which is order the black hole radius for mass @xmath46 . a collapsing shell would be such a state as it crosses its horizon . now consider any other state which has the same mass and which is localized in the same region , for example a fuzzball state . let us ask if there is any significant amplitude for ` tunneling ' between such states , postponing for the moment the details of what this tunneling process is . ( we will see below that we are looking for ` spreading of a wavefunction ' rather than tunneling , but it is more helpful to think of a tunneling process on a first pass at the issues . ) normally the tunneling amplitude would be small , since the states have large mass and size . we will estimate the action for a tunneling process by writing s_tunnel~1 g r d^4x~1 g 1(gm)^2(gm)^4~gm^2 [ saction ] where we have assumed a length scale @xmath141 for the curvature and a volume @xmath142 over which the process takes place . thus the amplitude for tunneling from the shell to a fuzzball state ~e^-s_tunnel is very small . but now note that there are a very large number of fuzzball states that we can tunnel to . this number is given by ~e^s_bek~e^gm^2 we see that something curious happens for black holes . these objects have such a large entropy that the very small probability for tunneling between classical configurations can be compensated for by the very large number of states that we can tunnel to @xcite . this would make a black hole an essentially quantum object . note that if we took a star instead , then the action ( [ saction ] ) is larger ( the size of the object is bigger ) while the entropy is much lower , and there is no such quantum behavior . the above was just a crude order of magnitude estimate , but now let us see if we can say something more about the actual dynamical process of shell collapse . the crucial point will be the fact that there is a large number of possible states states of the hole the @xmath143 fuzzballs . in the classical picture of collapse we do not see these states which are supposed to give the entropy of the hole . we will see that it may not be correct to ignore the large phase space which these microstates represent , and when we do take all these solutions into account the quantum evolution of a collapsing shell can be very different from its classical approximation . let us proceed in three steps . \(1 ) first let us take a 2-charge extremal geometry , and throw into the throat a quantum of a scalar field @xmath78 with energy @xmath144 . we choose @xmath144 to be small , so the backreaction of the quantum on the geometry can be ignored . the quantum will fall down the throat , reach the cap , and eventually reflect back up the throat . how do we describe this evolution in terms of the energy eigenstates of the system ? we can find the energy eigenstates of the quantum by solving the wave - equation @xmath145 . ( for the simple geometries of fig.[effex ] the wavefunctions have been explicitly computed @xcite . ) we get a set of energy eigenfunctions . the lowest energy state is localized in the cap ( as shown in fig.[effex ] ) , the next one extends a little further out , the next one still further , etc . the infalling quantum starts high up the throat , so we must superpose these energy eigenfunctions with suitable coefficients to obtain this initial wavepacket |=_k c_k |e_k[sum ] where @xmath146 is the eigenfunction with energy @xmath147 . this is all just standard quantum mechanics , and we would do a similar computation for describing a localized quantum moving in the potential of a harmonic oscillator . the evolution of the wavepacket down the throat is obtained by evolving the energy eigenfunctions ; since these eigenfunctions have slightly different energies , the relative phases between their coefficients change with time and cause the wavepacket to move downwards towards the cap . the essential point in the above discussion is that even though the quantum is localized quite high up the throat up the start , if we want to express its wavefunction in terms of the stationary states of the system then we have to construct the detailed energy eigenfunctions @xmath146 in the entire geometry , and these will depend sensitively on the structure of the cap . \(2 ) now let us imagine that the energy of the infalling quantum is a bit higher . we would therefore like to take into account the small backreaction that the infalling quantum would create on the geometry . how should we do this ? we still have to follow the same basic scheme : we have to find the energy eigenstates of the system and superpose them with appropriate coefficients . the evolution will then be given by the changing phases of the coefficients . but what are the energy eigenstates this time ? clearly , we should find solutions to the full system of gravity plus scalar field @xmath78 , with the backreaction of the @xmath78 excitation included , and arrive at some eigenstates @xmath148 $ ] which are functionals of both the metric @xmath47 and the scalar field @xmath78 . note in particular that the energy @xmath147 of this state will reflect the energy of the background extremal 2-charge geometry as well as the energy of the quantum . so we are making energy eigenstates around an energy e_total = e_extremal+e_quantum [ stotal ] the number of states of the system increase with the energy , and we observe here that the set of eigenstates that will be involved in a sum like ( [ sum ] ) will be the number at energy @xmath149 , and not at the base energy @xmath150 . \(3 ) now let us imagine increasing the energy of the infalling quantum still further , so that a classical analysis would indicate the formation of a horizon at some point in the throat , much before the cap is reached . this is of course the case that we are really interested in understanding . the basic scheme will remain the same as in the above two cases , but now we have to find all energy eigenstates of the system with an energy @xmath149 where the contribution @xmath151 is not small . according to our postulate , these energy eigenstates are horizon sized fuzzballs , pictured in fig.[geometries](c ) . thus the initial infalling quantum has to be written in the form ( [ sum ] ) as a set of very quantum fuzzball states ; these states are very numerous and have a nontrivial structure all the way upto the horizon . now suppose we did not know that there were all these fuzzball states , and we wrote the sum ( [ sum ] ) with only the states that we see in the traditional picture of the black hole . then we would be using a much smaller number of states . for example if we took the infalling quantum to have spherical symmetry , then we might ( erroneously ) assume that the black hole background should be a classical spherically symmetric state . but from what we have seen of fuzzball states , they are in general _ not _ spherically symmetric . spherical symmetry of the overall state is obtained by superposing with equal coefficient a _ non - spherical _ geometry with all of its rotates . thus if we write the initial shell as a superposition of spherically symmetric fuzzball states , then these states will have large _ fluctuations _ @xmath152 . in short , the fuzzball picture would give a much larger sum of states in ( [ sum ] ) as compared to a traditional picture which does not explicitly recognize the degrees of freedom corresponding to the bekenstein entropy . as the phases of the coefficients @xmath39 evolve , the initial state with the quantum will change to a general linear superposition of fuzzball states , something we can not see in the traditional classical infall . it is interesting to note the phase evolution of the @xmath39 becomes important in a time that is shorter than the hawking evaporation time . suppose we have a shell of mass @xmath46 that collapses to form a black hole . let the schwarzschild radius of the hole be denoted by @xmath92 . to make the shell collapse we must localize the matter in the shell so that it fits in a radius @xmath153 . this needs a momentum spread for the shell p for a nonrelativistic shell , the energy of the shell is @xmath154 , and the uncertainty in @xmath144 will ; be e~ppm the different fuzzball states @xmath146 making up the shell wavefunction @xmath31 will go ` out of phase ' over a time @xmath155 so that the state will look like a linear combination of generic fuzzball states rather than a well defined shell . we have t_dephase~1emr^2 but the hawking evaporation time for a schwarzschild hole ( in all dimensions ) is t_evap~ mr^2 thus we find that the time over which the the wavefunction ` dephases to fuzzballs ' is shorter than the hawking evaporation time t_dephase t_evap this is important , since this ` dephasing ' would not be of interest if it took _ longer _ than the hawking evaporation time . ( note that if we take a relativistic shell with @xmath156 instead of @xmath154 then we get an even shorter time @xmath155 . now we would have e~p this gives t_dephaser mr^2 where we recall that we are measuring all quantities in planck units , and @xmath157 . ) having obtained a rough picture of how black hole infall may be studied using fuzzball states , let us consider a toy model which illustrates in more detail how wavefunctions ` spread ' during evolution . in fig.[potential ] we sketch a system where a quantum can move along the @xmath17 direction , from @xmath158 to @xmath18 . if we have only this direction @xmath17 to move in , the motion of a quantum would be straightforward . but now let us assume that there is another direction @xmath55 in our space . let there be a potential v = k(r ) y^2 let @xmath159 vanish at large and small @xmath17 and be high in - between , with the peak at @xmath12 . towards @xmath18 . the lines of constant potential are sketched ; they allow the wavepacket to spread as it reaches @xmath160 . ] now let us see what this toy model represents . if @xmath159 vanishes near @xmath18 , then the wavefunction can easily spread over a large range of values of @xmath55 once the quantum gets close to @xmath18 . this represents the fact that there is a large phase space of fuzzball states ( given by the bekenstein entropy ) which can be accessed once an infalling shell comes close enough to the origin . for larger @xmath17 there are much fewer states for the given energy , while at infinity there are again many states possible because of the large volume of space available . first consider a classical particle moving in this @xmath161 space . we can assume @xmath162 consistently , and the particle just reaches the point @xmath163 at the end of its motion . now consider the quantum problem , and start with a wavepacket @xmath164 at large @xmath17 . if @xmath5 is large enough , the wavepacket will manage to pass through the location of steep potential at @xmath12 , and emerge into the region at small @xmath17 . but in this region there is no potential limiting the wavefunction in the @xmath55 direction , so it can spread over the region @xmath165 . thus while the classical solution suggested that the endpoint of the motion is at @xmath163 , the actual wavefunction can spread over all @xmath55 on reaching @xmath18 . this effect becomes more pronounced if we have a large number of transverse directions like @xmath55 . in our actual problem the wavefunction of a collapsing shell can spread over the very large of @xmath44 fuzzball states after the shell becomes smaller than a certain size . it is possible that the consequent spreading of the wavefunction invalidates a classical analysis of the motion of the shell . let us summarize the above discussion on the possible dynamics of fuzzballs . a principal feature characterizing black holes is their large entropy . the traditional picture of the hole does not exhibit the microstates required to explain this entropy . if we take the presence of the large number of microstates into account , then the wavefunction of a collapsing shell might spread to a nontrivial extent over this vast phase space of allowed solutions . the resulting dynamics would not correspond to a given quantum moving on a given black hole geometry , but rather lead to a wavefunctional @xmath166 $ ] that is spread over all possible geometries . if this happens then we can not argue that the light cones of the traditional black hole geometry trap the information of the shell forever and lead to information loss . so what is the fuzzball proposal and what does it say about the information problem ? suppose we go to a condensed matter physicist , and tell him about the information paradox . we show him the principles ( a ) , ( b ) listed in section [ introduction ] , and tell him that they are reasonable conditions to assume for quantum gravity . we then prove to him that given these conditions , there will _ have _ to be a violation of quantum unitarity . since the condensed matter person uses quantum theory , he would be very concerned that quantum theory needs a fundamental modification , even though the violation may not be significant in his systems of interest . indeed , he would probably agree that resolving this paradox should be an important goal of theoretical physics . now let us see how the results of the fuzzball program change the situation . the fuzzball proposal does not require new physics , or try to develop abstract principles about what happens in black holes . it simply takes a fully consistent theory of quantum gravity string theory and explicitly makes examples of microstates of black holes . all states made so far turn out to be different from the traditional geometry of a black hole : the microstates do not have an ` information free horizon ' . thus condition ( a ) of the hawking ` theorem ' breaks down . the fuzzball ` conjecture ' just says that the microstates not yet constructed would continue to have this feature ; thus there should not be two sharply different classes of microstates , one with ` information free horizons ' and one without . given the results of the fuzzball program , what would the condensed matter physicist say ? he can not agree that there is any information _ paradox_. a paradox is a sharp contradiction that we can not find a way around . if we can find a way around the paradox for some black hole states , then we can not argue that there is any sharp contradiction with black holes , even though we have not yet constructed all possible states for all holes . thus the condensed matter person will simply tell us to go and make other fuzzball states , and come back only if we can show that there are states of black holes that are _ not _ fuzzballs . to summarize , now the ` boot is on the other leg ' ; with the results from the fuzzball program we do not have an information ` paradox ' unless we can show that the behavior of microstates found so far does not continue in natural way to the class of all microstates . but it is important to note that this does not mean that we understand all there is to know about black holes . for one thing , we still have a lot to learn about the dynamics of black holes . we have conjectured some aspects of dynamics above , and it would be good to check these ideas in concrete detail and to understand what role is played by the large phase space of fuzzball solutions . in the early days of of the fuzzball program there were some concerns that quantum corrections may destroy the fuzzball nature of 2-charge solutions , and that 3-charge microstates may not be fuzzballs like the 2-charge ones . possible quantum corrections were investigated @xcite and no evidence was found that they would be a problem ; the magnitude of these corrections was shown to be bounded because of the geometric structure of the fuzzball solution . large numbers of 3-charge and 4-charge solutions have been made , and now there are also families of nonextremal solutions . for these reasons , perhaps at this point we should accept the hypothesis that the @xmath44 states of the hole are fuzzballs , and see what this hypothesis tells us about the physics of black holes . one thing we can do with the fuzzball picture is ask if we can find evidence for various ideas that have been suggested in the past : \(a ) it has been suggested @xcite that the observations made by an infalling observer are given by a description that is ` complementary ' to the observations made by an observer at infinity . let us see if we can say anything about this suggestion from our microstate constructions . in @xcite the infall of a test quantum into the extremal 2-charge system was studied in the cft picture . it was found that there were _ three _ different logical ways to define time evolution for the quantum : one suited to an infalling quantum , one to an emerging quantum , and one symmetrical between these two , which may be appropriate for an observer at infinity . simple states in one description look very complicated in the other , with the ` complication ' determined by the entropy of the state . note that we do not have different hilbert spaces for different observers . nevertheless , it would be good to see if there is a relation between such effects and notion of complementarity . \(b ) recently it has been suggested @xcite that there is a ` future boundary condition ' that must be imposed at the black hole singularity . this makes the state at the singularity unique , and forces information to come out in the hawking radiation . with fuzzballs , we find that states of the hole ` swell up ' and become big because we need an adequate phase space to hold @xmath44 states @xcite . thus with fuzzballs there is a sense in which data can not be ` focused ' to a singularity . perhaps this effect can be interpreted as some kind of a boundary condition at a singularity , and thus a relation found with the idea of a boundary condition at the singularity @xcite . \(c ) in one of the earliest attempts at resolving the information paradox @xcite it was argued that when considering virtual quanta , we should take into account their gravitational backreaction ; thus the creation operator for a scalar quantum should be ` dressed ' with gravity excitations . for black holes , it was argued that this would lead to large gravitational backreaction from the hawking radiation quanta , destroying the traditional picture of semiclassical particle production at a low curvature horizon . this proposal has a standard counter - argument : the gravitational effects of the _ pair _ of produced quanta should cancel out at the horizon , so that the hawking derivation is not really invalidated . let us now recall our discussion of section [ spread ] , where we have seen that to follow the effect of an infalling shell we must expand its wavefunction in eigenstates of the _ total _ system ( matter+gravity ) , with energy ( [ stotal ] ) . so while the argument of @xcite may not work for the hawking quanta of a scalar field on a spherically symmetric background , with the full set of nonperturbative black hole microstates we do find support for the idea that matter states should be studied only with their full gravitational backreaction included . \(d ) there have been studies @xcite of geodesics in the traditional black hole geometry , where it was found that complex geodesics gave a dominant saddle point describing the correlation of operators in the dual cft ; these correlations were then used as a way of characterization of the singularity . fuzzballs states are not expected to have such a singularity individually ( though the quantum fuzz does get more dense towards the center for a typical state ) . but when we take an average over fuzzball states , the traditional black hole geometry can appear as a saddle point of the entire sum @xcite , and it would be interesting to see if the complex geodesics emerge naturally to describe expectation values of correlation functions in the ensemble of fuzzball states . a crucial question now is to extract the essential lessons of the fuzzball program , and see what it tells us about the structure of quantum gravity when we have large amounts of matter crushed at high densities . clearly , one feature that we have seen is that quantum gravity effects do not extend over a fixed distance like @xmath0 ; instead this distance increases with the number of quanta involved in the black hole bound state . what does this tell us about cosmology , where we also have large amounts of matter at high densities ? in @xcite the state of the early universe was modeled after the states that give the entropy of black holes , and the resulting evolution was studied . the universe did not inflate . but the nonlocal correlations in the quantum bound state extended right across the universe . so 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the black hole information paradox is one of the most important issues in theoretical physics .
we review some recent progress using string theory in understanding the nature of black hole microstates . for all cases where
these microstates have been constructed , one finds that they are horizon sized ` fuzzballs ' .
most computations are for extremal states , but recently one has been able to study a special family of non - extremal microstates , and see ` information carrying radiation ' emerge from these gravity solutions .
we discuss how the fuzzball picture can resolve the information paradox .
we use the nature of fuzzball states to make some conjectures on the dynamical aspects of black holes , observing that the large phase space of fuzzball solutions can make the black hole more ` quantum ' than assumed in traditional treatments .
_ black holes , string theory _ :
| 32,172 | 203 |
quantum entanglement is a feature of quantum mechanics that has captured much recent interest due to its essential role in quantum information processing @xcite . it may be characterized and manipulated independently of its physical realization , and it obeys a set of conservation laws ; as such , it is regarded and treated much like a physical resource . it proves useful in making quantitative predictions to quantify entanglement.when one has complete information about a bipartite system subsystems @xmath0 and @xmath1the state of the system is pure and there exists a well established measure of entanglement the _ entropy of entanglement _ , evaluated as the von neumann entropy of the reduced density matrix , @xmath2 with @xmath3 . this measure is unity for the bell states and is conserved under local operations and classical communication . unfortunately , however , quantum systems in nature interact with their environment ; states of practical concern are therefore mixed , in which case the quantification of entanglement becomes less clear . given an ensemble of pure states , @xmath4 with probabilities @xmath5 , a natural generalization of @xmath6 is its weighted average @xmath7 . a difficulty arises , though , when one considers that a given density operator may be decomposed in infinitely many ways , leading to infinitely many values for this average entanglement . the density operator for an equal mixture of bell states @xmath8 , for example , is identical to that for a mixture of @xmath9 and @xmath10 , yet by the above measure the two decompositions have entanglement one and zero , respectively . various measures have been proposed to circumvent this problem , most of which evaluate a lower bound . one such measure , the _ entanglement of formation _ , @xmath11 @xcite , is defined as the minimal amount of entanglement required to form the density operator @xmath12 , while the _ entanglement of distillation _ , @xmath13 @xcite , is the guaranteed amount of entanglement that can be extracted from @xmath12 . these measures satisfy the requirements for a physical entanglement measure set out by horodecki _ et al_. @xcite . they give the value zero for @xmath14 , which might be thought somewhat counterintuitive , since this state can be viewed as representing a sequence of random `` choices '' between two bell states , both of which are maximally entangled . this is unavoidable , however , because assigning @xmath15 a non - zero value of entanglement would imply that entanglement can be generated by local operations . the problem is fundamental , steming from the inherent uncertainty surrounding a mixed state : the state provides an incomplete description of the physical system , and in view of the lack of knowledge a definitive measure of entanglement can not be given . an interacting system and environment inevitably become entangled . the problem of bipartite entanglement for an open system is therefore one of tripartite entanglement for the system and environment . complicating the situation , the state of the environment is complex and unknown . conventionally , the partial trace with respect to the environment is taken , yielding a mixed state for the bipartite system . if one wishes for a more complete characterization of the entanglement than provided by the above measures , somehow the inherent uncertainty of the mixed state description must be removed . to this end , nha and carmichael @xcite recently introduced a measure of entanglement for open systems based upon quantum trajectory unravelings of the open system dynamics @xcite . central to their approach is a consideration of the way in which information about the system is read , by making measurements , from the environment . the evolution of the system conditioned on the measurement record is followed , and the entanglement measure is then contextual dependent upon the kind of measurements made . suppose , for example , that at some time @xmath16 the system and environment are in the entangled state @xmath17 a partial trace with respect to @xmath18 yields a mixed state for @xmath19 . if , on the other hand , an observer makes a measurement on the environment with respect to the basis @xmath20 , obtaining the `` result '' @xmath21 , the reduced state of the system and environment is @xmath22 with conditional system state @xmath23 where @xmath24 is the probability of the particular measurement result . thus , the system and environment are disentangled , so the system state is pure and its bipartite entanglement is defined by the von neumann entropy , eq . ( [ eq : von - neumann ] ) . nha and carmichael @xcite apply this idea to the continuous measurement limit , where @xmath25 executes a conditional evolution over time . in this paper we follow the lead of nha and carmichael , also carvalho _ et al . _ @xcite , not to compute their entanglement measure _ per se _ , but to examine the entanglement _ dynamics _ of a cascaded qubit system coupled through the oneway exchange of photons . the system considered has been shown to produce unconditional entangled states generally a superposition of bell states as the steady - state solution to a master equation @xcite . for a special choice of parameters ( resonance ) , a maximally entangled bell state is achieved @xmath26 except that the approach to the steady state takes place over an infinite amount of time . here we analyze the conditional evolution of the qubit system to illuminate the dynamical creation of entanglement in the general case , and to explain , in particular , the infinitely slow approach to steady - state in the special case . we demonstrate that in the special case the conditional dynamics exhibit a distinct bimodality , where the approach to the bell state is only one of two possibilities for the asymptotic evolution : the second we call an _ entangled - state cycle _ , where the qubits execute a sustained stochastic switching between two bell states . though involving just two qubits and elementary quantum transitions , the situation is similar to that of a bimodal system in classical statistical physics in the limit of a vanishing transition rate between attractors . the physical model of the cascaded qubit system is presented in sec . [ subsec2_1 ] and the quantum trajectory unraveling of its conditional dynamics in sec . [ subsec3_1 ] . in sec . [ subsec4_1 ] we analyze the quantum trajectory equations to demonstrate bimodality and the existence of entangled - state cycles . finally , a discussion and conclusions are presented in sec . [ subsec5_1 ] . in this section we briefly outline the physical model for the cascaded qubit system to be analyzed . a more detailed description , together with the techniques and assumptions used to derive the model master equation presented here , is available in @xcite . the system considered consists of two high - finesse optical cavities , each containing a single tightly - confined atom , the cavities arranged in a cascaded configuration with unidirectional coupling from cavity 1 to cavity 2 ( fig . [ fig : fig1 ] ) . for simplicity , we consider the cavity modes to be identical , with resonance frequency @xmath27 and field decay rate @xmath28 . inefficiencies and losses in the coupling between the cavities are modeled by a real parameter @xmath29 , @xmath30 , with perfect coupling corresponding to @xmath31 . the atoms are assumed to have five relevant electronic levels , of which two ground states , @xmath32 and @xmath33 , represent an effective two - state system , or qubit . and @xmath33 , to three excited states , @xmath34 , @xmath35 , and @xmath36.,scaledwidth=40.0% ] for each atom , the cavity field in combination with auxiliary laser fields ( incident from the side of the cavity ) drives two separate resonant raman transitions between states @xmath32 and @xmath33 . an additional laser field coupled to the @xmath37 transition provides a tunable light shift of the energy of state @xmath32 . all fields are assumed far detuned from the atomic excited states , so these states may be adiabatically eliminated and atomic spontaneous emission ignored . under the further assumption that the cavity field decay rate is much larger than the transition rates between @xmath32 and @xmath33 , the cavity fields may also be adiabatically eliminated to yield a master equation for the reduced two - atom density matrix @xmath12 , @xmath38}+{\left[\hat r_2,\rho\hat r_1^\dag\right]}\right)},\end{aligned}\ ] ] with @xmath39 where @xmath40 , and @xmath41 and @xmath42 are the rates of @xmath43 and @xmath44 transitions , respectively . by virtue of the cavity output , the system is an open system and solutions to master equation ( [ eq : me ] ) generally describe mixed states . under appropriate conditions , however , the system evolves to a pure and entangled steady state . if the coupling between cavities is perfect ( @xmath31 ) and the parameters of the subsystems are the same ( @xmath45 , @xmath46 ) then the steady state is the pure state @xmath47 where we use the abbreviated notation @xmath48 and @xmath49 . then when @xmath50 , which we shall refer to as the _ resonance _ condition , the steady state is a maximally - entangled bell state . this may seem to be ideal , but a problem arises when we consider the eigenvalues of the operator @xmath51 . specifically , the characteristic time for the system to reach steady state , @xmath52 , where @xmath53 denotes the eigenvalue of @xmath51 with smallest ( in magnitude ) non - zero real part , approaches infinity as the resonance condition is approached . this is shown by the plot in fig . [ fig : fig2 ] . thus the master equation itself , in particular its steady state , offers limited insight into the behavior of the system at resonance . we wish to learn more about this special case ; in particular , how does the entanglement develop dynamically . also , if additional information is factored into the description , by making measurements on the environment , can we better characterize the long term behavior , or possibly find perfect entanglement after a finite time ? we demonstrate that quantum trajectory theory can provide answers to these questions . the relaxation time @xmath54 plotted as a function of @xmath55 . note the singularity at resonance , @xmath56.,scaledwidth=40.0% ] as with any open system , the first step in unraveling the master equation is to identify the points of coupling to the environment . the first is obvious the output from cavity 2 . to measure this output , let us assume the existence of an ideal photon detector in the path of the output from cavity 2 ; we call it _ detector 1_. the second point of coupling to the environment is more subtle . our model does not assume the inter - cavity coupling to be perfect ; only a fraction @xmath29 of the output photon flux from cavity 1 makes it into cavity 2 . physically , this loss may be caused , for example , by non - ideal transmissivity of the faraday isolators or by absorption in the cavity mirrors . these imperfections cause photons to be scattered into the environment in some uncontrollable fashion . formally , though , this is equivalent to assuming that the apparatus is ideal , except that there exists a beamsplitter between the cavities , as drawn schematically in fig . [ fig : fig3 ] . we therefore further assume the existence of a second photon detector to collect photons reflected by this beamsplitter ; we call it _ detector 2_. we now proceed to develop the quantum trajectory formalism for the cascaded qubit system . in this approach the system is described by a pure state which is dependent on ( conditioned on ) the counting histories , or records , of detectors 1 and 2 . firstly , we rewrite the master equation in a form suitable for translation into the quantum trajectory language . we reexpress eq . ( [ eq : me ] ) in the form @xmath57 with @xmath58-\frac{1}{2}\sum_{i=1,2 } \left(\hat c_i^{\dag}\hat c_i \rho+\rho\hat c_i^{\dag}\hat c_i\right),\\ \mathcal{s}\rho&=\sum_{i=1,2}\hat c_i \rho\hat c_i^{\dag},\end{aligned}\ ] ] where @xmath59 then , within quantum trajectory theory , the evolution of the system is described by a pure state @xmath60 which evolves under the non - hermitian effective hamiltonian @xmath61 the continuous evolution interrupted at random times by quantum jumps , @xmath62 , where the jumps occur with probability @xmath63 in time interval @xmath64 . physically , the jump operators @xmath65 and @xmath66 account for the reduction of the state of the system , given a photon count is recorded by detector 1 or detector 2 , respectively . thus , within the quantum trajectory description of the coupled cavity system , we consider an experiment in which ideal detectors are employed , such that every scattered photon is detected and recorded . given the history of detector ` clicks ' , one has complete information about the system state , in the sense that that state is always pure ; hence , although the solution to the master equation is generally mixed , one is able to characterize the entanglement in an unambiguous ( conditional ) fashion @xcite . consider the special case where the coupling between the cavities is optimal ( @xmath31 ) . in this case there is only one output from the system , that from cavity 2 , recorded by detector 1 . standard numerical algorithms @xcite have been used to simulate typical quantum trajectories for various values of @xmath55 . specifically , we consider the evolution of the conditional expectation of the operator product @xmath67 , where @xmath68 is the pauli operator diagonal in the @xmath69representation , @xmath70 this expectation has a number of convenient properties ; for example , the steady - state value @xmath71 regardless of the value of @xmath55 , which makes it easy to compare rates of convergence to the steady state for different system parameters . -0.5 cm figure [ fig : fig4 ] contrasts the solution to the master equation and a single quantum trajectory . the solution to the master equation exhibits a completely smooth evolution that tends asymptotically towards the steady state . the quantum trajectory , on the other hand , undergoes a sequence of switches between two extreme values of @xmath72 , which occur at each photon detection . provided the parameters are chosen away from resonance , the photon detections eventually stop and the trajectory settles into the steady state ( [ eq : ss ] ) , with @xmath73 ; the steady state is clearly a dark state . at resonance , however , the photon detections may continue indefinitely . physically , this seems plausible , since it simply implies that the atoms continue to switch between states @xmath32 and @xmath33 , scattering one photon with each transition . at resonance , apparently , a unique equilibrium dark state can not be established . the cyclic behavior that replaces it is completely invisible if we consider only the ensemble average a vivid demonstration of how single quantum trajectories can provide additional insight into the evolution of an open quantum system . the oscillatory behavior featured in fig . [ fig : fig4 ] hints at a simple cyclic process . in fact , it is simple enough that we can understand why it occurs without resorting to numerics . in this section we formulate a graphical description of individual trajectories . figure [ fig : fig4 ] demonstrates that the conditional expectation @xmath72 is conserved during the periods of evolution between quantum jumps . the positively and negatively correlated subspaces @xmath74 are coupled only through quantum jumps . noting that @xmath75 are each @xmath76-dimensional ( assuming real amplitudes without loss of generality ) , we manage to break up a @xmath77-dimensional space into two @xmath76-dimensional planes , linked to one another by the quantum jumps . we refer to this representation as the _ cascaded system phase space_. trajectories within it can be viewed as lines moving continuosly within either plane and jumping discontinuously between the planes . we use phase space portraits within @xmath78 and @xmath79 to characterize the behavior of the system , where for the sake of simplicity , and without loss of generality , we are assuming @xmath80 and @xmath81 to be real . we define @xmath82 and scale time by setting @xmath83 . the master equation then takes the form ( @xmath31 ) @xmath84 where @xmath85 the resonance condition is now @xmath86 . it is useful to convert to a matrix notation , such that a pure state @xmath60 of the system is represented by a @xmath77-vector , @xmath87 and system operators are written as @xmath88 matrices , e.g. , @xmath89 and @xmath90 the evolution of @xmath60under @xmath91 is written as a linear differential equation in four variables , @xmath92\ ! { |\phi\rangle}\nonumber\\ & = \left(\begin{array}{cccc } -2&0&0&2r\\ 0&-(1+r^2)&2r^2&0\\ 0&2&-(1+r^2)&0\\ 2r&0&0&-2r^2\\\end{array}\right ) \label{eqn : basicde}\!{|\phi\rangle}.\end{aligned}\ ] ] as noted above , this evolution is constrained within either @xmath78 or @xmath79 . thus we can write @xmath60 as a vector sum of two orthogonal components @xmath93 and @xmath94 , @xmath95 , to obtain the decoupled dynamics @xmath96 eigenvectors of the two dynamical matrices correspond to states of the system that are preserved under the evolution between quantum jumps . note , however , that it does not necessarily follow that such a state is a steady state of the quantum trajectory evolution as a whole ; it must eventually experience a quantum jump if its norm decays i.e . , the corresponding eigenvalue is not zero . recall from quantum trajectory theory that the probability for a state not to jump prior to time @xmath16 is given by its norm @xcite . for the systems of equations given above we find the following ( unnormalised ) eigenstates and eigenvalues : 1 . @xmath97 , @xmath98 ; this is the steady state of the system for @xmath99 . 2 . @xmath100 , @xmath101 ; this state in @xmath78 is orthogonal to @xmath102 and must eventually jump to a state in @xmath79 . 3 . @xmath103 , @xmath104 ; this state in @xmath79 must eventually jump to a state in @xmath78 unless @xmath86 ; in the latter case it plays no role once an entangled - state cycle is initiated ( see below ) . 4 . @xmath105 , @xmath106 ; this state in @xmath79 must eventually jump to a state in @xmath78 . in the special case of resonance , @xmath86 , there are two independent steady states , @xmath102 and @xmath107 , which helps to explain the failure of the master equation evolution to approach a unique steady state . it also suggests a fundamental feature of the indefinite switching , the cyclic behavior , revealed by individual quantum trajectories : during such an _ entangled - state cycle _ , the system state must remain orthogonal to @xmath102 and @xmath107 . we verify this shortly , after examining the trajectory evolution away from resonance , where the steady state @xmath102 is always reached for perfect inter - cavity coupling . typical quantum trajectories for @xmath108 are shown in figs . [ fig : fig5 ] and [ fig : fig6 ] , where the @xmath78 and @xmath79 subspaces are drawn as circular planes . normalized states are located on the circumferences of the circles . the bell states @xmath109 lie at intersections of the circumference with the dotted lines as shown . between quantum jumps , under the influence of the non - hermitian hamiltonian @xmath91 , the norm of the state decays and the point representing it within the phase space moves to the interior of one of the circles . quantum jumps cause a switch from @xmath78 to @xmath79 or vice - versa . they are represented by the lines connecting the two planes , where for illustrative purposes , the system state is renormalized after each quantum jump ; thus jumps terminate at points on the circumference of the circles . we restrict ourselves to separable initial states located in one or other of the two subspaces ; for example , the states @xmath110 and @xmath111 , respectively , are considered in figs . [ fig : fig5 ] and [ fig : fig6 ] . the action of the jump operator @xmath65 on states located in @xmath78 ( with renormalization ) is @xmath112 while the action of @xmath65 on states in @xmath79 is @xmath113 thus , when a quantum jump occurs , any state within @xmath78 collapses onto the bell state @xmath114 in @xmath79 , while any state within @xmath79 collapses onto the state @xmath115 in @xmath78 . consider an initial normalized state in @xmath78 , @xmath116 , for some ( real ) @xmath117 . given that @xmath102 is a steady state of the evolution between quantum jumps , the probability of an eventual quantum jump to @xmath79 is @xmath118 while with probability @xmath119 the system evolves to the steady state @xmath102 without any photon emissions . if a jump from @xmath79 to @xmath120 has just occurred , then by the same argument one shows that the probability of a future quantum jump to @xmath79 is @xmath121 , or , alternatively , the probability of reaching the steady state after such a jump is @xmath122 ^ 2 $ ] . consider now an initial state in @xmath79 , @xmath123 , for some ( real ) @xmath124 . owing to the instability of both @xmath107 and @xmath125 for @xmath99 , an eventual quantum jump is guaranteed ; thus , @xmath126 armed with this information , we move to an explanation of the quantum trajectories displayed in figs . [ fig : fig5 ] and [ fig : fig6 ] . in fig . [ fig : fig5 ] we plot three typical phase - space trajectories for @xmath108 and @xmath127 . [ fig : fig5](a ) illustrates the case where the system evolves directly to the steady state @xmath102 . the probability of this event is @xmath128 , so it is the most likely occurrence for the chosen parameters . if a first quantum jump does occur , then typical trajectories are shown in figs . [ fig : fig5](b ) and ( c ) . following the jump to @xmath129 in @xmath79 , a second jump returning the state to @xmath78 is guaranteed . for @xmath108 , this leaves the system in the state @xmath130 , from which the probability of a further cycle of jumps is @xmath131 . thus , after a first quantum jump cycle , it is most likely that further cycles will follow , as seen in figs . [ fig : fig5](b ) and ( c ) , where in both cases a total of five cycles ( ten photon detections ) occur before the system finally reaches the steady state . in fig . [ fig : fig6 ] we plot three typical phase - space trajectories for @xmath108 and @xmath132 . in this case , at least one quantum jump is certain to occur , following which the probability of further jumps is @xmath131 , as above . so for this initial condition , the most likely outcome is a sequence of quantum jump cycles following a first guaranteed photon detection . in fig . [ fig : fig6 ] ( a ) only the first detection occurs , while in figs . [ fig : fig6](b ) and ( c ) this detection is followed by a sequence of cycles before the steady state is eventually achieved . the case @xmath86 is of particular interest . the normalized eigenstates of the evolution between quantum jumps are the bell states @xmath133 , @xmath134 , @xmath135 , and @xmath136 . the eigenvalues are @xmath137 and @xmath138 . the action of the jump operator @xmath65 on states within @xmath78 simplifies to @xmath139 and its action on states within @xmath79 to @xmath140 for @xmath86 , photon detections , if they occur , are associated with collapses onto one of two maximally - entangled bell states . for initial states @xmath141 and @xmath142 in @xmath78 and @xmath79 , respectively , the system evolves continuously , without the emission of any photons , to @xmath133 and @xmath143 , with probabilities @xmath144 and @xmath145 in @xmath79 or @xmath146 in @xmath78 . in this case , as both terminal states are unstable under the between - jump evolution , a second detection and quantum jump must follow . according to eqs . ( [ eq : plusjump ] ) and ( [ eq : minusjump ] ) this simply exchanges @xmath147 for @xmath146 and vice - versa . hence , a perpetual switching between bell states @xmath147 and @xmath146 occurs . we designate this behavior an _ entangled - state cycle_. thus , at resonance we find a distinctly bimodal behavior . the system either evolves into a maximally - entangled bell state without emitting photons , or an entangled - state cycle is initiated under which the system switches indefinitely between orthogonal bell states while emitting a continual stream of photons . as an aside , such behavior can be regarded as a quantum measurement that distinguishes the bell states @xmath148 from @xmath149 . the two alternative outcomes of the quantum trajectory evolution are illustrated in figs . [ fig : fig7 ] and [ fig : fig8 ] for the initial states @xmath150 in @xmath78 and @xmath132 in @xmath79 , respectively . with this choice of initial states there are equal probabilities for reaching the steady states , @xmath151 [ fig . [ fig : fig7](a ) ] and @xmath152 [ fig . [ fig : fig8](a ) ] , and for commencing an entangled - state cycle [ figs . [ fig : fig7](b ) and [ fig : fig8](b ) ] . note that once an entangled - state cycle is initiated , the trajectory remains in a plane orthogonal to the lines defining @xmath151 and @xmath152 ; the cycle continues indefinitely our original model allowed for the possibility of imperfect intercavity coupling , through the parameter @xmath29 and the jump operator @xmath66 which describe the effects of photon loss in propagation between the two cavities . focusing on the resonant case ( @xmath86 ) , we now consider the situation in which @xmath153 . typical trajectories for @xmath154 are shown in figs . [ fig : fig9](a ) and [ fig : fig10](a ) , with the two photon count records shown in frames ( b ) and ( c ) of the figures . remarkably , entangled - state cycles persist , but now the system settles into one or other of two distinct cycles , involving either the symmetric or antisymmetric bell states . to understand the behavior , consider the forms of the operators involved ; in particular , for @xmath86 , we have effective hamiltonian @xmath155 and jump operators @xmath156 and @xmath157 significantly , these operators commute with one another , @xmath158=[\hat c_1,\hat h_{\rm eff}]=[\hat c_2,\hat h_{\rm eff}]=0.\end{aligned}\ ] ] their operation upon the bell states is given by @xmath159 and @xmath160 thus , the bell states are eigenstates of @xmath91 , and the jump operators interchange bell states in @xmath78 and @xmath79 : each jump operator converts the symmetric ( antisymmetric ) bell state in @xmath78 to the symmetric ( antisymmetric ) bell state in @xmath79 and vice - versa . now , let us consider a particular quantum trajectory for which a total of @xmath161 jumps occur , separated by the time intervals @xmath162 . for an initial state @xmath163 , the ( unnormalized ) state at the conclusion of the @xmath161 jumps is written as @xmath164 where each @xmath165 is either @xmath65 or @xmath66 . since all operators in the string acting on @xmath166 commute , this expression can be rewritten in a variety of forms , two of which prove to be especially useful in explaining the distinct behaviors illustrated by figs . [ fig : fig9 ] and [ fig : fig10 ] . in the first case , we may write @xmath167 passing all @xmath168 ocurrences of @xmath65 to the right and all @xmath169 occurrences of @xmath66 to the left ( @xmath170 ) ; in the second we write @xmath171 where all jump operators are passed to the left . the arbitrary ( pure ) initial state can be expressed as a superposition of bell states , @xmath172 where @xmath173 , @xmath174 , @xmath175 , and @xmath176 are expansion coefficients , generally complex . substituting this expansion into eqs . ( [ phit1 ] ) and ( [ phit2 ] ) , and using eqs . ( [ heff1])([c2psi])assuming for simplicity that @xmath168 and @xmath169 are even the two forms for the state @xmath177 are @xmath178 where @xmath179 observe now that the ratio of the eigenvalues satisfies @xmath180 it follows that @xmath181 and @xmath182 allow us to predict quite distinct asymptotic behaviors for the system state . for sufficiently large @xmath168 , the contribution to @xmath181 from the symmetric bell states is negligible compared with the contribution from the antisymmetric bell states , in which case , using eqs . ( [ eq : cycle1 ] ) and ( [ eq : cycle1prime ] ) , @xmath183 the system is locked into a cycle between the two antisymmetric bell states , the situation illustrated in fig . [ fig : fig9 ] ( for @xmath184 , @xmath185 ) . in contrast , for sufficiently large @xmath16 , the contribution to @xmath182 from the antisymmetric bell states is negligible compared with that from the symmetric bell states , and using eqs . ( [ eq : cycle2 ] ) and ( [ eq : cycle2prime ] ) , @xmath186 the system is locked into a cycle between the two symmetric bell states , as shown in fig . [ fig : fig10 ] . which of the two cycles is chosen in a particular realization of the photon counting record is random , as is the time taken to settle into the cycle . effectively , the decision is the outcome of a competition between the periods of evolution between quantum jumps and the jumps themselves specifically , those associated with photon counts at detector 1 . considering eqs . ( [ eq : cycle1prime ] ) and ( [ eq : ratio ] ) , we see that every count at detector 1 results in an increased probability to find the system in one of the antisymmetric bell states . on the other hand , from eqs . ( [ eq : cycle2prime ] ) and ( [ eq : ratio ] ) , the periods of evolution between counts have the reverse effect they increase the probability for the system to be found in a symmetric bell state . the critical factor that decides which tendency wins is the number of photon counts occuring at detector 1 over a given ( substantial ) interval of time . if there are many , as in fig . [ fig : fig9](b ) , the entangled - state cycle between antisymmetric bell states wins out ; if there are few , fig . [ fig : fig10](b ) , the cycle between symmetric bell states occurs . the same decision mechanism is observed in other examples @xcite . note that counts at detector 2 are not involved not directly at least . they do figure indirectly as a mechanism reducing the average number of counts at detector 1 ; indeed , they are the ultimate source of the asymmetry reflected in the ratio @xmath187 . as the system approaches a particular cycle the quantum trajectory evolution tends to reinforce the establishment of the cycle . close to the antisymmetric cycle , the evolution between jumps is dominantly governed by @xmath188 and is therefore relatively fast . this leads to frequent photon counts at detector 1 [ fig . [ fig : fig9](b ) ] . close to the symmetric cycle , the between - jump evolution is dominantly governed by @xmath189 , hence is relatively slow . photon counts at detector 1 become much less frequent [ fig . [ fig : fig10](b ) ] . from the dramatic difference in count rates at detector 1 for the two cycles , it is clear that one can determine which entanglement cycle the system evolves to for a particular realization . however , without knowledge of the record of photon counts at detector 2 , which by definition we do not have , one can not know where on the cycle the system is , i.e. , whether the state is in @xmath78 or @xmath79 . thus , the ensemble average state of the system is mixed , described by one of the density operators @xmath190 consider a thought experiment where the cascaded qubit system , set to resonance , evolves freely and its entire output is collected and stored inside a black box . at some time the lasers driving the raman transitions are turned off , so the evolution ceases . the box and qubits are separated and moved to causally disconnected regions of space time . let alice and bob be standard observers of the qubits , and give eve jurisdiction over the box . we can now ask , how much entanglement exists between the qubits of alice and bob ? while this is simply a roundabout way of asking how entanglement evolves , it helps elucidate some of the key concepts behind the quantum trajectory measure of entanglement . conventional entanglement measures are based upon an analysis of the density matrix at this time . they throw away the box and look at the system of qubits alone they disregard eve and view the system from the perspective of alice and bob . yet in general every interaction between two objects entangles them , and as the qubit system and box interacted in the past , their states are intertwined . neither possess an independent reality , and neither , considered alone , can be completely described . eve s box contains information , which , if discarded , adds entropy to the qubit system of alice and bob . this entropy is the source of ambiguity in the quantification of entanglement . from this point of view , as noted in the introduction , the problem of bi - partite entanglement in an open system relates to that of tri - partite entanglement in a closed one . to completely characterize the entanglement of the present example , in addition to the entanglement between alice and bob , we must consider their entanglement with eve . a quantum description of the box is impractical , but it is feasible to extract classical information about what it contains , through measurement . quantum trajectories facilitate this , and allow us not to discard the box completely . in turn , the system state retains its purity , conditional on the classical information extracted from the box . with this extra information , we can extract more entanglement from the cascaded qubit system . working from the master equation for the cascaded system @xcite , previously it was assumed that the system evolved gradually into a pure state , whereby entanglement was generated . the behaviour at resonance , however , was unclear , since there the master equation had two zero eigenvalues and no well - defined steady state . by considering the conditional evolution we have shown that , at resonance , asymptotically the system is either in the bell state @xmath151 or oscillating ( stochastically switching ) between two bell states , @xmath146 and @xmath147 . from the density matrix point of view , the latter is an equal mixture of bell states and would yield no entanglement under any mixed state measure ; physically , alice and bob , without collaboration from eve , can not extract any entanglement from their qubits . suppose , however , that eve opens her box to count the number of photons inside . seeing whether the count is even or odd , she is able to deduce exactly which bell state alice and bob s system is in . thus , her measurement unravels the density operator , creating entanglement , despite the fact that the measurement is not causally connected to alice and bob s qubits . it is tempting to say that the entanglement was always there , as a matter of fact , until one realizes that there are many other ways in which eve could choose to measure her state , each producing a different unravelling of the qubit system and yielding a different value of entanglement . the entanglement facilitated by eve s measurements is _ contextual _ in this sense . this thought experiment demonstrates why any attempt to quantify the entanglement of an open system from the density operator alone can not be considered complete . the density operator should not be treated as a fundamental object , as it does not provide a complete description of the physical state . we have presented a simple example where oscillations between maximally entangled states are hidden within a separable density operator . the fact that the density operator contains entropy , implies that information about its entanglement with an external system was discarded at some time . in studying such a mixed state , there is benefit from considering , not only the mixed state itself , but the process through which it was generated , and the access this potentially gives to a conditional dynamics . the results of this paper could be extended by employing quantum trajectories in a broader sense . in cases where the results of environmental interactions can not be measured , such as coupling loss , wiseman and vaccaro @xcite have shown that only certain unravelings can be physically realized . a conceivable measure of entanglement would take the minimum of all physically realizable unravelings . alternatively , one might take the maximum of all physically realizable unravellings , which would measure the maximum distillable entanglement when local measurements on the environment are taken into account . h. j. carmichael , p. kochan , and l. tian , `` coherent states and open quantum systems : a comment on the stern - gerlach experiment and schrdinger cats , '' in _ coherent states : past , present , and future _ , eds . d. h. feng , j. r. klauder , and m. r. strayer ( world scientific , singapore , 1994 ) , pp . 75 - 91 .
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a system of cascaded qubits interacting via the oneway exchange of photons is studied . while for general operating conditions the system evolves to a superposition of bell states ( a dark state ) in the long - time limit , under a particular _ resonance _ condition no steady state is reached within a finite time .
we analyze the conditional quantum evolution ( quantum trajectories ) to characterize the asymptotic behavior under this resonance condition .
a distinct bimodality is observed : for perfect qubit coupling , the system either evolves to a maximally entangled bell state without emitting photons ( the dark state ) , or executes a sustained entangled - state cycle random switching between a pair of bell states while emitting a continuous photon stream ; for imperfect coupling , two entangled - state cycles coexist , between which a random selection is made from one quantum trajectory to another .
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one of the most important events in modern physics is that our universe is expanding accelerated @xcite . however , a plausible explanation for this is commonly done using the model of a very exotic fluid called dark energy , which has negative pressure . another well - known possibility is to modify einstein s general relativity ( gr ) @xcite , making the action of the theory depend on a function of the curvature scalar @xmath3 , but at a certain limit of parameters the theory falls on gr . this way to explain the accelerated expansion of our universe is known as modified gravity or generalized . considering that the gravitational interaction is described only by the curvature of space - time , we can generalize the einstein - hilbert action through analytic function of scalars of the theory , as for example the gravities @xmath0 @xcite , with @xmath4 being the ricci scalar or curvature scalar , @xmath5 @xcite , with @xmath6 being the trace of energy - momentum tensor , or yet @xmath7 @xcite , @xmath8 @xcite and @xmath9 @xcite , with @xmath10 being the energy - momentum tensor . an alternative to consistently describe the gravitational interaction is one which only considers the torsion of space - time , thus cancelling out any effect of the curvature . this approach is known as teleparallel theory ( tt ) @xcite , which is demonstrably equivalent to gr . in order to describe not only the gravitational interaction , but also the accelerated expansion of our universe , ferraro and fiorini @xcite proposed a possible generalization of the tt , which became known as @xmath1 gravity @xcite , in which up to now has provided good results in both cosmology as local phenomena of gravitation . a key problem in @xmath1 gravity is that it breaks the invariance under local lorentz transformations complicating the interpretation of the relationship between all inertial frames of the tangent space to the differentiable manifold ( space - time ) @xcite . this problem may lead to the emergence of new degrees of freedom spurious who are responsible for the breakdown of the local lorentz symmetry @xcite . a consequence of the formulated theory using a scalar which is not invariant by local lorentz transformations , the torsion scalar @xmath11 in this case , is that instead of the theory presenting differential equations of motion of fourth order , as in the case of the @xmath0 gravity , it has second - order differential equations . that seems like a benefit but is a consequence of this fact on the local lorentz symmetry . we still have which this generalization of the tt is not equivalent to generalization @xmath0 for rg . this is the main reason that will address the construction of a theory that generalize the tt , but which still keep the local lorentz symmetry on a particular case . therefore , it is clear that we must build the function of action with dependence on a scalar that at some limit is invariant under local lorentz transformations . it will be shown soon forward . the paper is organized as follows . in section [ sec2 ] we do a review of @xmath1 gravity , introducing the functional variation method used in this work , obtaining the equations of motion of this theory , noting a poorly treated point at the limit to gr . in section [ sec3 ] we propose the action of generalized teleparallel theory , we obtain the equations of motion through functional variation of the same and compared with @xmath1 gravity . we show the equivalence of our theory with @xmath0 gravity , in the case of cosmology for the line element of flat flrw metric in subsection [ subsec4.1 ] , and also in the case of a spherically symmetric line element in subsection [ subsec4.2 ] . we show still the equivalence of our theory with a particular case of @xmath12 gravity in section [ sec5 ] . in section [ sec6 ] we make four applications , one where we reconstructed the action of our theory for the universe of the model of de sitter , another where we obtain a static type - de sitter solution ; we analyse teh evolution for the state parameter to dark energy and the thermodynamics for a cosmological model . we make our final considerations in section [ sec7 ] . the geometry of a space - time can be characterized by the curvature and torsion . in the particular case in which we only consider the curvature and torsion being zero , we have defined , together with the metricity condition @xmath13 where @xmath14 are the components of the metric tensor , a riemannian geometry where the connection @xmath15 is symmetric in the last two indices . already in the particular case that we consider only torsion ( riemann tensor identically zero , case without curvature ) in the space - time , we can then work with objects that depend solely on the so - called tetrads matrices and its derivatives as dynamic fields . in the space - time having only torsion , the line element can be represented through two standard forms @xmath16 where we have the following relationships @xmath17 , @xmath18 , @xmath19 , @xmath20 e @xmath21 , with @xmath22 being the tetrads matrices and @xmath23 its inverse , and @xmath24=diag[1,-1,-1,-1]$ ] the minkowski metric . we adopt the latin indices for the tangent space and the greeks into space - time . we will first establish the equations of motion for the theory @xmath1 , thus showing that the functional variation method adopted here is consistent . we restrict the geometry to of weitzenbock where we have the following connection @xmath25 all riemann tensor components are identically zero for the connection ( [ wc ] ) . we can then define the components of the tensor of torsion and contortion as @xmath26 we can also define a new tensor , so we write a more elegant way the equations of motion , through the components of the tensor torsion and contortion , as @xmath27 we define the torsion scalar as @xmath28 some observations are important here . the first is that there is a direct analogy to a space only with torsion and another considering only curvature in that the connections are related by @xmath29 where @xmath30 is the levi - civita connection , which is symmetric in the last two indices . the second observation is that the torsion scalar @xmath11 is not a lorentz scalar ( in the tangent space ) , being only a scalar in the tensorial indices ( space - time ) @xcite . this is precisely the cause for that theory built starting this scalar breaks down the invariance by local lorentz transformations . we can in reality build the curvature scalar analog , through of the torsion scalar , to relation @xcite @xmath31 where @xmath32=\sqrt{-g}$ ] , with @xmath33 $ ] . the curvature scalar @xmath4 in ( [ r ] ) is a lorentz scalar as well as a scalar on tensorial indices . that is why the @xmath34 gravity is a theory that is invariant under local lorentz transformations and general coordinates transformations ( tensorial ) . is then possible to construct a generalization of the teleparallel theory ( tt ) using the following action of the @xmath1 gravity , @xmath35\,\label{ftaction}\end{aligned}\ ] ] where @xmath36 , @xmath1 is a function of the torsion scalar and @xmath37 is the lagrangian density of the material content . we call attention to the true sign @xmath38 in the front of the matter term . this so far has not been explicitly addressed in the literature of this theory , because we still have few models that couple content materials that need to be obtained through functional variation in principle . this signal is essential if the theory is equivalent to gr at some limit . it will soon be clear forward . making the functional variation of the action ( [ ftaction ] ) we have @xmath39\,,\nonumber\\ & = & \frac{1}{2\kappa^2}\int d^4x\left[f\frac{\partial e}{\partial e^{a}_{\;\;\sigma}}\delta e^{a}_{\;\;\sigma}+e\frac{df}{dt}\delta t\right ] -\int d^4x \delta\mathcal{l}_{matter}\,,\nonumber\\ & = & \delta s_t-\delta s_{matter}\label{dels1}\,,\end{aligned}\ ] ] with @xmath40 . now let s do first the functional variation of the matter term , @xmath41\nonumber\,,\end{aligned}\ ] ] that making the integration by part of the latter term , considering @xmath42 , we have @xmath43\nonumber\\ & = & \frac{1}{2\kappa^2}\int d^4x\;2\kappa^2 e\theta_a^{\;\;\sigma}\delta e^a_{\;\;\sigma}\,,\label{delsm1}\end{aligned}\ ] ] where @xmath44 , and we define @xmath45 as being the energy - momentum tensor . we have now the functional variation of geometric part , @xmath46\right\ } \nonumber\,.\end{aligned}\ ] ] doing integration by parts the last term , considering @xmath42 , we obtain @xmath47\right\}\delta e^{a}_{\;\;\sigma } \label{delst1}\,.\end{aligned}\ ] ] where @xmath48 . taking ( [ delsm1 ] ) and ( [ delst1 ] ) and replacing in ( [ dels1 ] ) , and imposing the principle of least action @xmath49 and multiplying by @xmath50 , we have the following equation of motion @xmath51- \kappa^2 \theta_{\nu}^{\;\;\sigma}=0\label{eqm0}\,.\end{aligned}\ ] ] substituting the derivatives @xcite @xmath52 in ( [ eqm0 ] ) we finally have the equations of motion of the @xmath1 gravity @xmath53-\kappa^2\theta_{\nu}^{\;\;\sigma}=0\label{eqmft}\,.\end{aligned}\ ] ] now we make use of identity @xcite @xmath54=-\frac{1}{2}\left[g_{\nu}^{\;\;\sigma}-\frac{1}{2}\delta^{\sigma}_{\nu}t\right]\,,\label{id}\end{aligned}\ ] ] with @xmath55 being the mixed components of the einstein tensor , for rewrite ( [ eqmft ] ) as @xmath56=\kappa^2 \theta_{\nu}^{\;\;\sigma}\label{eqmft2}\,.\end{aligned}\ ] ] this theory falls on einstein s general relativity with a cosmological constant , when we make @xmath57 . here it becomes clear that if we do not consider the sign @xmath38 in front of the matter term in action ( [ ftaction ] ) in the theory , we do not return to gr for a linear @xmath1 function , reaching a opposite signal to einstein s equation . this fact will be crucial to show later that an invariant theory by local lorentz transformations , as the @xmath0 gravity , can not fall in @xmath1 gravity , since these have opposite coupling signs to the matter term . sotiriou et al @xcite have shown that @xmath1 gravity does not preserve its equations of motion invariant by local lorentz transformations . it is in relation to this problem that we then construct a generalization of the teleparallel theory that preserves the invariant of the equations of motion for a local lorentz transformation . this will be addressed in the next section . an important identity is given by @xmath58 , where @xmath4 is the curvature scalar associated with a riemann tensor defining solely by levi - civita connection @xmath30 , where the indices @xmath59 are symmetric , and the covariant derivative @xmath60 is defined by this connection . the curvature scalar is by definition invariant through a local lorentz transformation , but it is also invariant through a general coordinate transformation . so it would be interesting to develop a theory that generalize the tt but that the functional action depends on an invariant under local lorentz transformations . this is not the case on @xmath1 gravity . we propose the following action @xmath61\,,\label{sgtt}\end{aligned}\ ] ] where we define @xmath62 this action generalizes tt and falls on a modified @xmath1 gravity as well as @xmath0 gravity . we can show this by making the limit @xmath63 , where we have @xmath64 , ergo @xmath65 , @xmath66 and the theory must be equivalent to a modified @xmath1 ( we ll see this later ) . moreover , we can regain @xmath0 gravity , making the limit @xmath67 , where we have @xmath68 , then the theory must be equivalent to @xmath0 . we show this explicitly through the equations of motion later on . by performing the functional variation of the action ( [ sgtt ] ) we obtain : @xmath69\,.\label{sgtt2}\end{aligned}\ ] ] as @xmath70 $ ] , in which @xmath71 are the matter fields , doing , @xmath72 with @xmath40 in the same manner as in @xmath1 gravity . the functional variation of the matters term ( [ sgtt3 ] ) is exactly the same as given in ( [ delsm1 ] ) . the geometric part is @xmath73\nonumber\\ & = & \frac{1}{2\kappa^2}\int d^4x \left[f\;\frac{\partial e}{\partial e^a_{\;\;\sigma}}\delta e^a_{\;\;\sigma}+e\;f_{\mathcal{t}}\delta{\mathcal{t}}\right]\,,\label{dels3}\end{aligned}\ ] ] where we use @xmath74 . the first term in ( [ dels3 ] ) is already known , we will pay attention to the second term . performing the functional variation to @xmath75 in ( [ tc ] ) we obtain @xmath76\nonumber\\ & = & -\delta t-2a_1\left[-e^{-2}\partial_\mu\left(eg^{\mu\beta}t^\alpha_{\;\;\beta\alpha}\right)\delta e+e^{-1}\delta\partial_\mu\left(eg^{\mu\beta}t^\alpha_{\;\;\beta\alpha}\right)\right]\,.\end{aligned}\ ] ] replacing in ( [ dels3 ] ) taking into account the functional variation of @xmath11 and @xmath77 we have , @xmath78\nonumber\\ & & + 2a_1\left[e^{-1}f_{\mathcal{t}}\partial_\mu(eg^{\mu\beta}t^\alpha_{\;\;\beta\alpha})\frac{\partial e}{\partial e^a_{\;\;\sigma}}\delta e^a_{\;\;\sigma}-f_{\mathcal{t}}\delta\partial_\mu(eg^{\mu\beta}t^\nu_{\;\;\beta\nu})\right]\bigg{\rbrace}\,.\label{delstc1}\end{aligned}\ ] ] now we do the integration by part in the terms containing @xmath79 and @xmath80 . the first integration by parts is given by @xmath81+\frac{1}{2\kappa^2}\int d^4x\partial_\alpha\left[ef_{\mathcal{t}}\frac{\partial t}{\partial(\partial_\alpha e^a_{\;\;\sigma})}\right]\delta e^a_{\;\;\sigma}\,,\nonumber\\\label{int1}\end{aligned}\ ] ] where the first term is zero because it is a surface term , which we consider @xmath42 . the second integration by parts is given by @xmath82+\frac{2a_1}{2\kappa^2}\int d^4x(\partial_\mu f_{\mathcal{t}})\,\delta(eg^{\mu\beta}t^\nu_{\;\;\beta\nu})\,,\label{sterm}\end{aligned}\ ] ] with the first term is null for being a surface term . then we have @xmath83\;.\label{int2}\end{aligned}\ ] ] making use of the following relationship @xmath84 and replacing ( [ int1 ] ) and ( [ int2 ] ) in ( [ delstc1 ] ) , developing the terms of @xmath85 we have @xmath86\delta e^a_{\;\;\sigma}\nonumber\\ & & + 2a_1\bigg[e^{-1}f_{\mathcal{t}}\partial_{\mu}(e\,g^{\mu\beta}t^\nu_{\beta\nu})\frac{\partial e}{\partial e^a_{\;\;\sigma}}\delta e^a_{\;\;\sigma}+(\partial_\mu f_{\mathcal{t}})\bigg[g^{\mu\beta}t^\nu_{\;\;\beta\nu}\frac{\partial e}{\partial e^a_{\;\;\sigma}}\delta e^a_{\;\;\sigma}\nonumber\\ & & -et^{\nu}_{\;\;\beta\nu}\left(g^{\beta\sigma } e_a^{\;\;\mu}\delta e^a_{\;\;\sigma}+g^{\mu\sigma}e_a^{\;\;\beta}\delta e^a_{\;\;\sigma}\right)+e\,g^{\mu\beta}\frac{\partial t^{\nu}_{\;\;\beta\nu}}{\partial e^a_{\;\;\sigma}}\delta e^a_{\;\;\sigma}+e\,g^{\mu\beta}\frac{\partial t^\nu_{\;\;\beta\nu}}{\partial(\partial_\alpha e^a_{\;\;\sigma})}\delta(\partial_\alpha e^a_{\;\;\sigma})\bigg]\bigg]\bigg{\rbrace}\,.\label{delstc2}\end{aligned}\ ] ] at this point we see that we still have to do an integration by parts in the last term , ie , @xmath87\nonumber\\ & & -\frac{2a_1}{2\kappa^2}\int d^4x\partial_\alpha\left[(\partial_\mu f_{\mathcal{t}})\,e\,g^{\mu\beta}\frac{\partial t^\nu_{\;\;\beta\nu}}{\partial(\partial_\alpha e^a_{\;\;\sigma})}\right]\delta e^a_{\;\;\sigma}\,,\nonumber\\\end{aligned}\ ] ] where once again the first term vanishes due to be a surface term . replacing this result in ( [ delstc2 ] ) we obtain @xmath88 + 2a_1\bigg[e^{-1}f_{\mathcal{t}}\partial_{\mu}(e\,g^{\mu\beta}t^\nu_{\beta\nu})\frac{\partial e}{\partial e^a_{\;\;\sigma}}\nonumber\\ & & + ( \partial_\mu f_{\mathcal{t}})\bigg[g^{\mu\beta}t^\nu_{\;\;\beta\nu}\frac{\partial e}{\partial e^a_{\;\;\sigma}}-et^{\nu}_{\;\;\beta\nu}\left(g^{\beta\sigma } e_a^{\;\;\mu}+g^{\mu\sigma}e_a^{\;\;\beta}\right)+e\,g^{\mu\beta}\frac{\partial t^{\nu}_{\;\;\beta\nu}}{\partial e^a_{\;\;\sigma}}\bigg]-\partial_\alpha\left[(\partial_\mu f_{\mathcal{t}})\,e\,g^{\mu\beta}\frac{\partial t^\nu_{\;\;\beta\nu}}{\partial(\partial_\alpha e^a_{\;\;\sigma})}\right]\bigg]\bigg{\rbrace}\delta e^a_{\;\;\sigma}\,.\label{delstc3}\end{aligned}\ ] ] now we must replace those derived from @xmath11 , @xmath77 and @xmath89 in relation tetrads and its derivatives . taking into account the results of @xmath90 , we have the following derivative , @xmath91 substituting the above derivatives in ( [ delstc3 ] ) , making @xmath92 in ( [ sgtt3 ] ) and multiplying by @xmath93 we have the following equation of motion for the generalized teleparallel theory @xmath94+a_1\bigg{\lbrace}e^{-1}f_{\mathcal{t}}\delta^\sigma_\omega \partial_\mu(eg^{\mu\beta}t^\nu_{\;\;\beta\nu})+(\partial_\mu f_{\mathcal{t}})\big[\delta^\sigma_\omega g^{\mu\beta}t^\nu_{\;\;\beta\nu}\nonumber\\ & & -\left(\delta^\mu_\omega g^{\beta\sigma}t^{\nu}_{\;\;\beta\nu}+g^{\mu\sigma}t^\nu_{\;\;\omega\nu}\right)-g^{\mu\beta}t^{\sigma}_{\;\;\beta\omega}\big]-e^{-1}e^a_{\;\;\omega}\partial_\alpha\big[e(\partial_\mu f_{\mathcal{t}})(g^{\mu\alpha}e_a^{\;\;\sigma}-g^{\mu\sigma}e_a^{\;\;\alpha})\big]\bigg{\rbrace}+\kappa^2\theta_{\omega}^{\;\;\sigma}=0\,.\label{eqmgtt}\end{aligned}\ ] ] taking the limit in which @xmath63 ( @xmath95 ) , making @xmath96 the equation of motion ( [ eqmgtt ] ) does not fall exactly on the equation of motion of the @xmath1 gravity in ( [ eqmft ] ) . this happens to the fact that the relationship between the curvature scalar and the torsion scalar are through a minus sign , which prevents a theory as @xmath0 gravity , in which the coupling signal with matter is positive , fall in a theory like @xmath1 gravity , in which the coupling signal with the matter should be negative so that it falls within the gr . in the next section we ll show the equivalence between the gtt and @xmath0 gravity . let s start this section showing the equivalence of gtt with @xmath0 gravity in the limit @xmath67 , to general tetrads . let us first establish some necessary identities , as arising from the condition of metricity @xmath97 with it the identity @xmath98 becomes @xmath99 now we can divide the equation of motion in terms such as @xmath100+a_1e^{-1}f_{\mathcal{t}}\delta^\sigma_\omega \partial_\mu(eg^{\mu\beta}t^\nu_{\;\;\beta\nu})\,,\label{t1}\\ & & t^{(2)}=a_1(\partial_\mu f_{\mathcal{t}})\big[\delta^\sigma_\omega g^{\mu\beta}t^\nu_{\;\;\beta\nu}-\delta^\mu_\omega g^{\beta\sigma}t^{\nu}_{\;\;\beta\nu}-g^{\mu\sigma}t^\nu_{\;\;\omega\nu}-g^{\mu\beta}t^{\sigma}_{\;\;\beta\omega}\big]\,,\label{t2}\\ & & t^{(3)}=-a_1e^{-1}e^a_{\;\;\omega}\partial_\alpha\big[e(\partial_\mu f_{\mathcal{t}})(g^{\mu\alpha}e_a^{\;\;\sigma}-g^{\mu\sigma}e_a^{\;\;\alpha})\big]\,.\label{t3}\end{aligned}\ ] ] developing the last term we have @xmath101\,.\label{t31}\end{aligned}\ ] ] using , , , , and in and we have the sum of terms @xmath102 and @xmath103 results in @xmath104 now we use the identity in , then we can rewrite the equation of motion , using , as follows @xmath105f_{\mathcal{t}}+\frac{1}{2}\left[-\mathcal{t}f_{\mathcal{t}}+f\right]\delta^{\mu}_{\nu}+2s_{\nu}^{\;\;\alpha\mu}\partial_{\alpha}f_{\mathcal{t}}+\kappa^2\theta^{\mu}_{\;\;\nu}=0\label{gtt1}\,.\end{aligned}\ ] ] considering @xmath106 , with @xmath107 , we have to gtt will only be equivalent to @xmath0 gravity in the limit @xmath67 , so @xmath108 and the term @xmath109 must be identically zero , as shown in section iii subsection c to @xcite . when this term vanishes , we have exactly one theory invariant by local lorentz transformations , which occurs only when @xmath67 and thus the equation becomes identical to the @xmath0 gravity , which is covariant and independent of the chosen of set of tetrads . in the next section we will specify a set of tetrads that explicitly show the equivalence between the two theories to the limit referred to above . in this section we explicitly show that the gtt equations of motion in ( [ eqmgtt ] ) , are exactly the same as @xmath0 gravity for the particular limit in which @xmath67 . we can then begin comparing the equations of motion for a easier symmetry of the metric , as the maximum symmetry for the cosmological friedmann - lemaitre - robertson - walker ( flrw ) flat metric @xmath110 considering now the case of cosmology , with line element flrw flat ( [ eleflrw ] ) , for a diagonal tetrad @xmath111=diag[1,a(t),a(t),a(t)]$ ] , we have that the equations ( [ eqmgtt ] ) become @xmath112f_{\mathcal{t}}-fa^2\bigg{\rbrace}\;,\label{eqmgttflrw00}\end{aligned}\ ] ] @xmath113f_{\mathcal{t}}-fa^2\bigg{\rbrace } \;,\label{eqmgttflrw11}\end{aligned}\ ] ] where @xmath114 and @xmath115 . we can now compare these equations with those obtained from the @xmath0 gravity , whose equations of motion are @xcite @xmath116 considering the flat flrw metric ( [ eleflrw ] ) , the equations ( [ eqmfr ] ) provide us @xmath117\ ; , \label{eqmfrflrw00}\end{aligned}\ ] ] @xmath118\;.\label{eqmfrflrw11}\end{aligned}\ ] ] subtracting ( [ eqmgttflrw00 ] ) from ( [ eqmfrflrw00 ] ) we have @xmath119\right\}\;.\label{sub1}\end{aligned}\ ] ] subtravting ( [ eqmgttflrw11 ] ) from ( [ eqmfrflrw11 ] ) we obtain @xmath120+\dot{a}^2[(4 - 6a_1)f_{\mathcal{t}}+2f_{\bar{r}}]\nonumber\\ & & + \frac{a^2}{2}\left[f(\bar{r})-f(\mathcal{t})\right]\big\}\;.\label{sub2}\end{aligned}\ ] ] now we see clearly that to the limit at which @xmath67 , we have @xmath121 , then ( [ sub1 ] ) and ( [ sub2 ] ) are identically null , showing the equivalence of equations of motion between gtt and @xmath0 for this limit . the conclusion is that the gtt is only invariant under local lorentz transformations and at the same time invariant by general coordinates transformations to the limit at which @xmath67 . we have demonstrated in general that the gtt is equivalent to gravity @xmath122 , but in addition to explain this through a metric with specific symmetry , we want to leave the equations of motion open for further analysis of this theory . let us now consider the case of a spherically symmetric and static line element @xmath123 we can choose the following diagonal tetrad @xmath111=diag[e^{a(r)/2},e^{b(r)/2},r , r\sin\theta]$ ] , which taking into account ( [ eqmgtt ] ) , provides us the following equations of motion @xmath124f_{\mathcal{t}}+2\,f\,r^2e^b\bigg{\rbrace}\ ; , \label{diag00}\end{aligned}\ ] ] @xmath125f_{\mathcal{t}}+2\,f\,r^2e^b\bigg{\rbrace}\;,\label{diag11}\end{aligned}\ ] ] @xmath126 @xmath127f_{\mathcal{t}}+2\,f\,r^2e^b\bigg{\rbrace}\;,\label{diag22}\end{aligned}\ ] ] where @xmath128 denotes derivation in relation to radial coordinate @xmath129 . taking the metric ( [ eles ] ) to the equations of the @xmath0 gravity in ( [ eqmfr ] ) , we obtain @xmath130f_{\bar{r}}+2fre^b\bigg{\rbrace}\;,\label{frs00}\\ \kappa^2\theta_1^{\;\;1}&=&\frac{e^{-b}}{4r}\bigg{\lbrace}(2ra'+8)\frac{d}{dr}f_{\bar{r}}+\left[(ra'+4)b'-2ra''-r(a')^2\right]f_{\bar{r}}+2fre^b\bigg{\rbrace}\;,\label{frs11}\\ \kappa^2\theta_2^{\;\;2}&=&\kappa^2\theta_3^{\;\;3}=\frac{e^{-b}}{2r^2}\bigg{\lbrace}2r^2\frac{d^2}{dr^2}f_{\bar{r}}-r(rb'-ra'-2)\frac{d}{dr}f_{\bar{r}}+(rb'+2e^b - ra'-2)f_{\bar{r}}+fr^2e^b\bigg{\rbrace}\;.\label{frs22}\end{aligned}\ ] ] here first we noticed that if @xmath131 , exists an equation ( [ diag21 ] ) outside the diagonal for gtt , resulting in the restriction of functional form @xmath132 , com @xmath133 . then we have the same constraint to @xmath1 gravity in this case @xcite . we also see that to the limit at which @xmath67 , @xmath121 , all equations ( [ diag00])-([diag22 ] ) for gtt are identical to @xmath0 given in ( [ frs00])-([frs22 ] ) . now choose a set of non - diagonal tetrads @xmath134\;,\end{aligned}\ ] ] the equations to gtt in ( [ eqmgtt ] ) provide us @xmath135f_{\mathcal{t}}+2\,f\,r^2e^b\bigg{\rbrace}\;,\label{ndiag00}\end{aligned}\ ] ] @xmath136e^b\nonumber\\ & & -\left(a_1r^2\left(2a''+(a')^2\right)-8(1-a_1)ra'-8(1-a_1)\right)e^{b/2}\bigg]f_{\mathcal{t}}+2\,f\,r^2e^{3b/2}\bigg{\rbrace}\;,\label{ndiag11}\end{aligned}\ ] ] @xmath137f_{\mathcal{t}}+2\,f\,r^2 e^b\bigg{\rbrace}\;.\label{ndiag22}\end{aligned}\ ] ] we can then see that in this case the equations of motion are diagonals . but equivalence of the gtt with the @xmath0 gravity only gives to the limit @xmath67 , when the equations ( [ frs00])-([frs22 ] ) and ( [ ndiag00])-([ndiag22 ] ) are identical . in this section we make an important observation . when we were finishing the calculation of the non - diagonal tetrads case of the previous subsection , we note that a group have submitted exactly the same idea of our work here . the so - call @xmath12 gravity @xcite , with @xmath138 , is a more general theory that presented here , where the algebraic function contained in action , may be any analytic function of the variables @xmath11 and @xmath139 . we noted then that the equivalence of this theory with the @xmath0 gravity is given only for the specific functional form @xmath140 . compared to our theory , we have the gtt is a particular case of @xmath12 gravity , when @xmath141 . we can show this again explicitly using equations of motion . the equation of motion for @xmath12 gravity is given by @xmath142 the first observation here is that this theory does not fall in @xmath1 gravity on general , as well as our gtt , as mentioned at the end of the section [ sec3 ] . making @xmath143 , ergo @xmath144 , the equation of motion ( [ ftb ] ) , using the identity ( [ i d ] ) , becomes @xmath145 this equation is not equal to ( [ eqmft2 ] ) for @xmath1 gravity , and can not fall on gr when @xmath146 , due to sign . this shows that the @xmath12 gravity also not returns on @xmath1 gravity on general . now we can show that the particular case @xmath147 this theory falls in our gtt . we take the flrw metric ( [ eleflrw ] ) with diagonal tetrads @xmath148=diag[1,a , a , a]$ ] , the equations of motion ( [ ftb ] ) provides us with @xmath149f_b+fa^2\bigg{\rbrace}\;,\label{ftbflrw00}\\ & & \kappa^2\theta_1^{\;\;1}=\kappa^2\theta_2^{\;\;2}=\kappa^2\theta_3^{\;\;3}=\frac{1}{2a^2}\bigg{\lbrace}4a(\dot{a})\left(\frac{d}{dt}f_t\right)+\left[4a\ddot{a}+8(\dot{a})^2\right]f_t-2a^2\left(\frac{d^2}{dt^2}f_b\right)\nonumber\\ & & + \left[6a\ddot{a}+12(\dot{a})^2\right]f_b+fa^2\bigg{\rbrace}\;.\label{ftbflrw11}\end{aligned}\ ] ] now identifing @xmath150 , recall that @xmath75 is given in ( [ tc ] ) , then @xmath151 we have that the equations ( [ ftbflrw00 ] ) and ( [ ftbflrw11 ] ) are identical from the gtt ( [ eqmgttflrw00 ] ) and ( [ eqmgttflrw11 ] ) , thus showing the equivalence between the theories . we can also confirm this by choosing the spherical symmetry for the metric ( [ eles ] ) , first for diagonal tetrads @xmath148=diag[e^{a/2},e^{b/2},r , r\sin\theta]$ ] , ergo , the equations ( [ ftb ] ) provide us @xmath152f_t-2r^2\left[2\frac{d}{dr}-b'\right]\frac{d}{dr}f_b\nonumber\\ & & + \left[-r(ra'+4)b'-4e^b+r\left(2ra''+r(a')^2 + 8a'\right)+8\right]f_b-2fr^2e^b\bigg{\rbrace}\;,\label{ftbdiag00 } \\ & & \kappa^2\theta_1^{\;\;1}=\frac{e^{-b}}{4r^2}\bigg{\lbrace}2\left[2e^b-4ra'-4\right]f_t+2r\left[ra'+4\right]\frac{d}{dr}f_b\nonumber\\ & & + \left[r\left(ra'+4\right)b'+4e^b - r\left(2ra''-r(a')^2 - 8a'\right)-8\right]f_b+2r^2fe^b\bigg{\rbrace}\;,\label{ftbdiag11 } \\ & & \kappa^2\theta_2^{\;\;1}=-\frac{1}{r^2\sin\theta}\left[\cos\theta\frac{d}{dr}f_t+\cos\theta\frac{d}{dr}f_b\right]=0\;,\label{ftbdiag21 } \\ & & \kappa^2\theta_2^{\;\;2}=\kappa^2\theta_3^{\;\;3}=\frac{e^{-b}}{4r^2}\bigg{\lbrace}2r(ra'+2)\frac{d}{dr}f_t+\left[-r\left(ra'+2\right)b'+r^2\left(2a''+(a')^2\right)+6ra'+4\right]f_t\nonumber\\ & & -2r^2\left[2\frac{d}{dr}-b'\right]\frac{d}{dr}f_b+\left[-r(ra'-4)b'-4e^b+r^2\left(2a''+(a')^2\right)+8ra'+8\right]f_b-2fr^2e^b\bigg{\rbrace}\;.\label{ftbdiag22}\end{aligned}\ ] ] again we have the equivalence of the equations of motion ( [ ftbdiag00])-([ftbdiag22 ] ) with ( [ diag00])-([diag22 ] ) , for the identifications @xmath153 and ( [ id2 ] ) . by taking the choice of non - diagonal tetrads([ndtetrad ] ) , the equations of motion from the @xmath154 gravity ( [ ftb ] ) provide us @xmath155\frac{d}{dr}f_t+\left[4re^{3b/2}b'+2(2ra'+4)e^{2b}-4(ra'+2)e^{3b/2}\right]f_t\nonumber\\ & & + \left[4r^2e^{3b/2}\frac{d}{dr}+\left(8re^{2b}-2r^2e^{3b/2}\right)\right]\frac{d}{dr}f_b\nonumber\\ & & + \left[r\left(ra'+4\right)e^{3b/2}b'+4\left(ra'+2\right)e^{2b}-\left(2r^2a''+r(a')^2 + 8ra'+8\right)e^{3b/2}\right]f_b+2fr^2e^{5b/2}\bigg{\rbrace}\;,\label{ftbndiag00}\end{aligned}\ ] ] @xmath156f_t+2r(ra'+4)e^{3b/2}\frac{d}{dr}f_b\nonumber\\ & & + \left[r(ra'+4)e^{3b/2}b'+4(ra'+2)e^{2b}-(2r^2a''+r^2(a')^2 + 8ra'+8)e^{3b/2}\right]f_b+2fr^2e^{5b/2}\bigg{\rbrace}\;,\label{ftbndiag11}\end{aligned}\ ] ] @xmath157\frac{d}{dr}f_t\nonumber\\ & & + \left[r(ra'+2)e^{3b/2}b'-4e^{5b/2}+4(ra'+2)e^{2b}-\left(2r^2a''+r^2(a')^2 + 6ra'+4\right)e^{3b/2}\right]f_t\nonumber\\ & & + \left[4r^2e^{3b/2}\frac{d}{dr}+4re^{2b}-2r^2e^{3b/2}b'\right]\frac{d}{dr}f_b\nonumber\\ & & + \left[r(ra'+4)e^{3b/2}b'+4(ra'+2)e^{2b}-\left(2r^2a''+r^2(a')^2 + 8ra'+8\right)e^{3b/2}\right]f_b+2fr^2e^{5b/2}\bigg{\rbrace}\;.\label{ftbndiag22}\\\end{aligned}\ ] ] just as before , making identifications @xmath153 and ( [ id2 ] ) , the equations ( [ ftbndiag00])-([ftbndiag22 ] ) are identical from the gtt in ( [ ndiag00])-([ndiag22 ] ) , confirming again the equivalence of these theories . a method to obtain the functional form of the algebraic function @xmath158 is the so - called reconstruction . this method is to specify a model that fix the material content of the theory in terms of scalar @xmath75 , allowing to reconstruct the functional form of @xmath158 through of the equations of motion of the theory . we will choose the particular case of flat flrw metric in which @xmath159 , a_0,h_0,t_0\in\re_{+}$ ] , it provides us with the model of de sitter universe , where @xmath160 . in this case , using ( [ tc ] ) , we have that @xmath161}$ ] , @xmath162 and @xmath163 . knowing that @xmath164 , the equation ( [ eqmgttflrw00 ] ) provide us @xmath165 ^ 2=3[h_0(\mathcal{t})]^2(2 - 3a_1)f_{\mathcal{t}}(\mathcal{t})-\frac{1}{2}f(\mathcal{t})\label{eqr1}\,,\end{aligned}\ ] ] integrating with respect to that @xmath75 results in @xmath166^{(1 - 3a_1)/(2 - 3a_1)}c_1\;,\;c_1\in\re\,.\label{ftc1}\end{aligned}\ ] ] we take here the limit at which @xmath63 in ( [ eqmgtt ] ) , that after use the identity ( [ i d ] ) and the consideration @xmath64 , results in @xmath167=-\kappa^2\theta_{\omega}^{\;\;\sigma}\label{gttft}\;.\end{aligned}\ ] ] as in @xmath0 gravity @xcite , we can consider a very specific case where @xmath168 , with @xmath169 and @xmath170 is defined by ( [ tc ] ) . in the case of a perfect fluid @xmath171 $ ] , and @xmath172 , which results in the equations @xmath173\;,\label{gttdesitter}\end{aligned}\ ] ] which taking the trace results in @xmath174 considering now the line element ( [ eles ] ) , for @xmath175 and @xmath176 ( type - dark energy ) , we can integrate the equations of motion ( [ gttdesitter ] ) , where we get the following solution @xmath177\;.\label{ssol}\end{aligned}\ ] ] this is a static type - de sitter solution where we can identify effective cosmological constant @xmath178/[6f_{t_0}(-t_0)]$ ] . a solution type - de sitter was also previously obtained in @xmath0 gravity for @xmath179 and @xmath180 @xcite . * we emphasize here that this solution it comes to different theory to @xmath1 gravity , because the gtt does not fall in @xmath1 gravity for @xmath63 , except for the special case where @xmath1 is a odd analytic function , that is @xmath181 . * a good test for our theory is the evolution of a model of the universe . this can discard or keep a theory depending on whether it is in agreement with the observational data . let s follow the procedure found in @xcite to determine the state parameter @xmath2 . for a universe permeated by a perfect fluid , of which equation of state is governed by @xmath182 , we can rewrite the equations of motion and as @xmath183\label{h2}\,,\\ & & \dot{h}=-\frac{\kappa^2}{2}g_{eff}\left(p_{m}+p_{de}+\rho_{m}+\rho_{de}\right)\,,\,p_{de}=\frac{1}{\kappa^2}\left[a_1\ddot{f}_{\mathcal{t}}+h\dot{f}_{\mathcal{t}}-\frac{1}{2}f\right]\label{hdot}\,.\end{aligned}\ ] ] now we can defining the state parameter of dark energy by @xmath184 we now assume an exponential model , as well as @xcite , defined by @xmath185\right)\label{exp}\,.\end{aligned}\ ] ] we will now test for a solution of the type power law @xmath186 * we can show that is a solution of the equations of motion and if the material part is given by the expressions @xmath187}{\mathcal{t}_st^2}}\left[\mathcal{t}_s^2t^4 + 6\alpha\mathcal{t}_st^2(a_1+\alpha(3a_1 - 2))+72a_1\alpha^2(a_1(2\alpha-1)-\alpha)\right]\beta\nonumber\\ & & + \mathcal{t}_st^2\left[-6\alpha^2 + 6\alpha a_1(2+\alpha)-\beta\mathcal{t}_st^2\right]\bigg\}\\ & & p_{mt}=\frac{1}{2\mathcal{t}_s^2\kappa^2 t^6}\bigg\{e^{\frac{6\alpha[a_1+\alpha(1 - 2a_1)]}{\mathcal{t}_st^2}}\big[-\mathcal{t}_s^3t^6 + 2\mathcal{t}_s^2\alpha t^4(-4 + 3a_1(2 - 3\alpha)+6\alpha)+288a_1\alpha^2(a_1+\alpha(1 - 2a_1))^2\nonumber\\ & & -24\mathcal{t}_st^2\alpha(3a_1-\alpha)(-\alpha+a_1(2\alpha-1))\big]\beta+\mathcal{t}_s^2t^4\left[2\alpha(-4 - 3a_1(\alpha-1)+3\alpha)+\beta\mathcal{t}_s t^2\right]\bigg\}\end{aligned}\ ] ] * the figure [ fig1 ] is the temporal evolution of the state parameter @xmath2 of the dark energy . the red curve is obtained with constant given by @xmath188 , where we can see that the fluid is always phantom @xmath189 . the blue curve is obtained with constant given by @xmath190 , where we can see that the fluid is always phantom @xmath189 , but it fluctuates approximately between the values @xmath191 and @xmath192 . the most interesting case is the green curve obtained for the constants @xmath193 . in this case we see that the fluid begins in a rather phantom phase , going through another phase type quintessence , heading toward a behavior of baryonic matter ( @xmath194 ) and finally returning the phantom phase . the result is that the current accelerated expansion of the universe and the crossing of the phantom divide from the phantom phase to the non - phantom ( quintessence ) one can be realized , as well as in @xcite . [ cols= " > , < " , ] a further application is for thermodynamics of the apparent horizon in cosmology flrw metric . we can follow the formulation given in @xcite . we can establish a similar equation of continuity , deriving over time and using @xmath195 whereas the baryonic matter is conserved ( @xmath196 ) , we can see that dark energy is not conserved , yielding the interpretation that it is a system out of equilibrium with entropy production ( non - equilibrium thermodynamics ) . following exactly the same steps in @xcite , we can establish the first law of thermodynamics @xmath197 at where @xmath198 the temperature of the apparent horizon , @xmath199 is the entropy of the apparent horizon , @xmath200 is the produced entropy , @xmath201 is the misner - sharp energy , @xmath202 the work and @xmath203 the volume element of the apparent horizon . here it is clearly seen that the first law of thermodynamics is consistent for entropy production associated with an effective newton constant @xmath204 , given in , which for the linear case of @xmath158 the entropy production vanishes and the system back to equilibrium . * if we take the same model of the previous section , i.e. and , we can explicitly show the time dependence of effective newton constant in @xmath205}{\mathcal{t}_st^2}}\right\}^{-1}\label{geff}\end{aligned}\ ] ] here are two important observations . the first is that it becomes explicit dependence of the first law of thermodynamics to the specific choice of the value on @xmath206 in . the second is that by taking the particular value @xmath207 in , clearly we have @xmath208 , which again shows the dependence of the theory in relation to the specific value of @xmath206 , and from , and we return to linear theory , where there is no entropy production . * we construct a theory that describes the gravitational interaction through effects of torsion of space - time . this theory generalizes the teleparallel theory keeping the invariance by both local lorentz transformations as general coordinates transformations for a particular case . the action of our theory is described by a general algebraic function that depends on a tensorial scalar @xmath75 which is classified by a real parameter @xmath206 . our theory falls exactly in @xmath0 gravity when we take the limit in which @xmath67 . this is shown from the equations of motion of the two theories . we make two small applications of our theory , reconstructing the action for the particular case of de sitter universe for the flat flrw metric , with a set of diagonal tetrads , and for obtain a static type - de sitter solution . we also analyse the evolution of the state parameter of the dark energy and the first thermodynamics law for the apparent horizon . our theory is a good scenery to an attempt to explain the accelerated expansion of our universe , by modifying the teleparallel usual gravitation , or analogous to einstein gravity . the real parameter @xmath206 which classifies which theory the gtt describes , it is crucial to any consideration of cosmological phenomena . we also expect new solutions of black holes arise through our theory , in which may also suggest some light on the so - called dark matter explanation on local effects of gravitation . 99 a. riess et al , _ observational evidence from supernovae for an accelerating universe and a cosmological constant _ , astron . j. * 116 * , 1009 ( 1998 ) , [ http://arxiv.org/abs/astro-ph/9805201[astro-ph/9805201 ] ] ; 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rong - jia yang , _ new types of f(t ) gravity _ , eur.phys.j . c * 71 * ( 2011 ) 179 , [ http://arxiv.org/abs/arxiv:1007.3571[arxiv:1007.3571 ] ] ; puxun wu , hong wei yu , _ the dynamical behavior of f(t ) theory _ , phys.lett . b * 692 * ( 2010 ) 176 - 179 , [ http://arxiv.org/abs/arxiv:1007.2348[arxiv:1007.2348 ] ] ; ratbay myrzakulov , _ accelerating universe from f(t ) gravity _ , eur.phys.j . c * 71 * ( 2011 ) 1752 , [ http://arxiv.org/abs/arxiv:1006.1120[arxiv:1006.1120 ] ] ; g.g.l . nashed and w. el hanafy , _ a built - in inflation in the f(t)-cosmology _ , eur.phys.j . c * 74 * ( 2014 ) 10 , 3099 , [ http://arxiv.org/abs/arxiv:1403.0913[arxiv:1403.0913 ] ] . thomas p. sotiriou , baojiu li , john d. barrow , _ generalizations of teleparallel gravity and local lorentz symmetry _ , phys.rev . d * 83 * ( 2011 ) 104030 , [ http://arxiv.org/abs/arxiv:1012.4039[arxiv:1012.4039 ] ] ; baojiu li , thomas p. sotiriou , john d. barrow , _ f(t ) gravity and local lorentz invariance _ , d * 83 * ( 2011 ) 064035 , [ http://arxiv.org/abs/arxiv:1010.1041[arxiv:1010.1041 ] ] . miao li , rong - xin miao , yan - gang miao , _ degrees of freedom of f(t ) gravity _ , jhep * 1107 * ( 2011 ) 108 , [ http://arxiv.org/abs/arxiv:1105.5934[arxiv:1105.5934 ] ] . ruben aldrovandi and jose geraldo pereira , _ teleparallel gravity , an introduction _ , springer , new york ( 2013 ) . m. hamani daouda , manuel e. rodrigues and m.j.s . houndjo , _ static anisotropic solutions in f(t ) theory _ , eur.phys.j . c * 72 * ( 2012 ) 1890 , [ http://arxiv.org/abs/arxiv:1109.0528[arxiv:1109.0528 [ physics.gen-ph ] ] ] . sebastian bahamonde , christian g. boehmer and matthew wright , _ modified teleparallel theories of gravity _ , [ http://arxiv.org/abs/arxiv:1508.05120[arxiv:1508.05120 [ gr - qc ] ] ] . s. capozziello , a. stabile and a. troisi , _ spherical symmetry in @xmath0-gravity _ , class.quant.grav . * 25 * ( 2008 ) 085004 , [ http://arxiv.org/abs/arxiv:0709.0891[arxiv:0709.0891 [ gr - qc ] ] ] . kazuharu bamba , chao - qiang geng and chung - chi lee , _ cosmological evolution in exponential gravity _ , jcap * 08 * ( 2010 ) 021 , [ http://arxiv.org/abs/1005.4574v3[arxiv:1005.4574 [ astro-ph.co ] ] ] . ednaldo l. b. junior , manuel e. rodrigues , ines g. salako and mahouton j. s. houndjo , _ reconstruction , thermodynamics and stability of cdm model in f(t , t ) gravity _ , accepted for publication in class . and quantum gravity , [ https://arxiv.org/abs/1501.00621[arxiv:1501.00621 [ gr - qc ] ] ] .
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we construct a theory in which the gravitational interaction is described only by torsion , but that generalizes the teleparallel theory still keeping the invariance of local lorentz transformations in one particular case .
we show that our theory falls , to a certain limit of a real parameter , in the @xmath0 gravity or , to another limit of the same real parameter , in a modified @xmath1 gravity , interpolating between these two theories and still can fall on several other theories .
we explicitly show the equivalence with @xmath0 gravity for cases of friedmann - lemaitre - robertson - walker flat metric for diagonal tetrads , and a metric with spherical symmetry for diagonal and non - diagonal tetrads .
we do still four applications , one in the reconstruction of the de sitter universe cosmological model , for obtaining a static spherically symmetric solution type - de sitter for a perfect fluid , for evolution of the state parameter @xmath2 and for the thermodynamics to the apparent horizon .
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it is an extrapolation of 18 orders of magnitude from the neutron radius of a heavy nucleus such as @xmath1pb with a neutron radius of @xmath2 fm to the approximately 10 km radius of a neutron star . yet both radii depend on our incomplete knowledge of the equation of state of neutron - rich matter . that strong correlations arise among objects of such disparate sizes is not difficult to understand . heavy nuclei develop a neutron - rich skin as a result of its large neutron excess ( _ e.g. , _ @xmath3 in @xmath1pb ) and because the large coulomb barrier reduces the proton density at the surface of the nucleus . thus the thickness of the neutron skin depends on the pressure that pushes neutrons out against surface tension . as a result , the greater the pressure , the thicker the neutron skin @xcite . yet it is this same pressure that supports a neutron star against gravitational collapse @xcite . thus models with thicker neutron skins often produce neutron stars with larger radii @xcite . the above discussion suggests that an accurate and model - independent measurement of the neutron skin of even a single heavy nucleus may have important implications for neutron - star properties . attempts at mapping the neutron distribution have traditionally relied on strongly - interacting probes . while highly mature and successful , it is unlikely that the hadronic program will ever attain the precision status that the electroweak program enjoys . this is due to the large and controversial uncertainties in the reaction mechanism @xcite . the mismatch in our knowledge of the proton radius in @xmath0pb relative to that of the neutron radius provides a striking example of the current situation : while the charge radius of @xmath0pb is known to better than 0.001 fm @xcite , realistic estimates place the uncertainty in the neutron radius at about 0.2 fm @xcite . the enormously successful parity - violating program at the jefferson laboratory @xcite provides an attractive electroweak alternative to the hadronic program . indeed , the parity radius experiment ( prex ) at the jefferson laboratory aims to measure the neutron radius of @xmath1pb accurately ( to within @xmath4 fm ) and model independently via parity - violating electron scattering @xcite . parity violation at low momentum transfers is particularly sensitive to the neutron density because the @xmath5 boson couples primarily to neutrons . moreover , the parity - violating asymmetry , while small , can be interpreted with as much confidence as conventional electromagnetic scattering experiments . prex will provide a unique observational constraint on the thickness of the neutron skin of a heavy nucleus . we note that since first proposed in 1999 , many of the technical difficulties intrinsic to such a challenging experiment have been met . for example , during the recent activity at the hall a proton parity experiment ( happex ) , significant progress was made in controlling helicity correlated errors @xcite . other technical problems are currently being solved such as the designed of a new septum magnet and a specific timeline has been provided to solve all remaining problems within the next two years @xcite . our aim in this contribution is to report on some of our recent results that examine the correlation between the neutron skin of @xmath0pb and various neutron - star properties @xcite . in particular , we examine the consequences of a `` softer '' equation of state that is based on a new accurately calibrated relativistic parameter set that has been constrained by both the ground state properties of finite nuclei and their linear response . further , results obtained with this new parameter set dubbed `` fsugold '' @xcite will be compared against the nl3 parameter set of lalazissis , konig , and ring @xcite that , while highly successful , predicts a significantly stiffer equation of state . the starting point for the calculation of the properties of finite nuclei and neutron stars is an effective field - theory model based on the following lagrangian density : @xmath6\psi \nonumber \\ & & - \frac{\kappa}{3 ! } ( g_{\rm s}\phi)^3 \!-\ ! \frac{\lambda}{4!}(g_{\rm s}\phi)^4 \!+\ ! \frac{\zeta}{4 ! } \big(g_{\rm v}^2 v_{\mu}v^\mu\big)^2 \!+\ ! \lambda_{\rm v } \big(g_{\rho}^{2}\,{\bf b}_{\mu}\cdot{\bf b}^{\mu}\big ) \big(g_{\rm v}^2v_{\mu}v^\mu\big ) \;. \label{lagrangian}\end{aligned}\ ] ] the lagrangian density includes an isodoublet nucleon field ( @xmath7 ) interacting via the exchange of two isoscalar mesons a scalar ( @xmath8 ) and a vector ( @xmath9 ) one isovector meson ( @xmath10 ) , and the photon ( @xmath11 ) @xcite . in addition to meson - nucleon interactions , the lagrangian density is supplemented by four nonlinear meson interactions , with coupling constants denoted by @xmath12 , @xmath13 , @xmath14 , and @xmath15 . the first three of these terms are responsible for a softening of the equation of state of symmetric nuclear matter at both normal and high densities @xcite . in particular , the cubic ( @xmath12 ) and quartic ( @xmath13 ) scalar self - energy terms are needed to reduce the compression modulus of symmetric nuclear matter , in accordance to measurements of the giant monopole resonance in medium to heavy nuclei @xcite . in turn , @xmath16-meson self - interactions ( @xmath14 ) are instrumental for the softening of the equation of state at high density thereby affecting primarily the limiting masses of neutron stars @xcite . finally , the last of the coupling constants ( @xmath15 ) induces isoscalar - isovector mixing and has been added to tune the poorly - known density dependence of the symmetry energy @xcite . as a result of the strong correlation between the neutron radius of heavy nuclei and the pressure of neutron - rich matter @xcite , the neutron skin of a heavy nucleus is highly sensitive to changes in @xmath15 . [ cols="<,^,^,^,^,^,^,^,^",options="header " , ] in conclusion , a new accurately calibrated relativistic model ( `` fsugold '' ) has been fitted to the binding energies and charge radii of a variety of magic nuclei . in this regard , the new parametrization is as successful as the nl3 set which has been used here as a useful paradigm . in particular , symmetric nuclear matter saturates at a fermi momentum of @xmath17 ( corresponding to a baryon density of @xmath18 ) with a binding energy per nucleon of @xmath19 mev . further , by constraining the fsugold parameter set by a few nuclear collective modes , we obtain a nuclear - matter incompressibility of @xmath20 mev and a neutron skin thickness in @xmath0pb of @xmath21 fm . while the description of the various collective modes imposes additional constraints on the eos at densities around saturation density , the high - density component of the eos remains largely unconstrained . thus , we made no attempts at constraining the eos at the supranuclear densities of relevance to neutron - star physics . rather , we simply explored the consequences of the new parametrization on a variety of neutron star observables and eagerly await high - quality data that will constrain the high - density component of the eos . in particular , we found a limiting neutron - star mass of @xmath22 , a radius of @xmath23 km for a @xmath24 neutron star , and no direct urca cooling in neutron stars with masses below @xmath25 . it is interesting to note that recent observations of pulsar - white dwarf binaries at the arecibo observatory suggest a pulsar mass for psrj0751 + 1807 of @xmath26 at a 95% confidence level @xcite . if this observation could be refined , not only would it redefine the high - density behavior of this ( and many other ) eos , but it could provide us with a precious boost in our quest for the equation of state .
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the nucleus of @xmath0pb a system that is 18 order of magnitudes smaller and 55 orders of magnitude lighter than a neutron star may be used as a miniature surrogate to establish important correlations between its neutron skin and several neutron - star properties .
indeed , a nearly model - independent correlation develops between the neutron skin of @xmath1pb and the liquid - to - solid transition density in a neutron star .
further , we illustrate how a measurement of the neutron skin in @xmath0pb may be used to place important constraints on the cooling mechanism operating in neutron stars and may help elucidate the existence of quarks stars .
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populations of self - sustained oscillators can exhibit various synchronization phenomena @xcite . for example , it is well known that a limit - cycle oscillator can exhibit phase locking to a periodic external forcing ; this phenomenon is called the forced synchronization @xcite . recently , it was also found that uncoupled identical limit - cycle oscillators subject to weak common noise can exhibit in - phase synchronization ; this remarkable phenomenon is called the common - noise - induced synchronization @xcite . in general , each oscillatory dynamics is described by a stable limit - cycle solution to an ordinary differential equation , and the phase description method for ordinary limit - cycle oscillators has played an essential role in the theoretical analysis of the synchronization phenomena @xcite . on the basis of the phase description , optimization methods for the dynamical properties of limit - cycle oscillators have also been developed for forced synchronization @xcite and common - noise - induced synchronization @xcite . synchronization phenomena of spatiotemporal rhythms described by partial differential equations , such as reaction - diffusion equations and fluid equations , have also attracted considerable attention @xcite ( see also refs . @xcite for the spatiotemporal pattern formation ) . examples of earlier studies include the following . in reaction - diffusion systems , synchronization between two locally coupled domains of excitable media exhibiting spiral waves has been experimentally investigated using the photosensitive belousov - zhabotinsky reaction @xcite . in fluid systems , synchronization in both periodic and chaotic regimes has been experimentally investigated using a periodically forced rotating fluid annulus @xcite and a pair of thermally coupled rotating fluid annuli @xcite . of particular interest in this paper is the experimental study on generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator @xcite ; this experimental synchronization can be considered as common - noise - induced synchronization of spatiotemporal chaos . however , detailed theoretical analysis of these synchronization phenomena has not been performed even for the case in which the spatiotemporal rhythms are described by stable limit - cycle solutions to partial differential equations , because a phase description method for partial differential equations has not been fully developed yet . in this paper , we theoretically analyze common - noise - induced phase synchronization between uncoupled identical hele - shaw cells exhibiting oscillatory convection ; the oscillatory convection is described by a stable limit - cycle solution to a partial differential equation . a hele - shaw cell is a rectangular cavity in which the gap between two vertical walls is much smaller than the other two spatial dimensions , and the fluid in the cavity exhibits oscillatory convection under appropriate parameter conditions ( see refs . @xcite and also references therein ) . in ref . @xcite , we recently formulated a theory for the phase description of oscillatory convection in the hele - shaw cell and analyzed the mutual synchronization between a pair of coupled systems of oscillatory hele - shaw convection ; the theory can be considered as an extension of our phase description method for stable limit - cycle solutions to nonlinear fokker - planck equations @xcite ( see also ref . @xcite for the phase description of spatiotemporal rhythms in reaction - diffusion equations ) . using the phase description method for oscillatory convection , we here demonstrate that uncoupled systems of oscillatory hele - shaw convection can be in - phase synchronized by applying weak common noise . furthermore , we develop a method for obtaining the optimal spatial pattern of the common noise to achieve synchronization . the theoretical results are validated by direct numerical simulations of the oscillatory hele - shaw convection . this paper is organized as follows . in sec . [ sec:2 ] , we briefly review our phase description method for oscillatory convection in the hele - shaw cell . in sec . [ sec:3 ] , we theoretically analyze common - noise - induced phase synchronization of the oscillatory convection . in sec . [ sec:4 ] , we confirm our theoretical results by numerical analysis of the oscillatory convection . concluding remarks are given in sec . [ sec:5 ] . in this section , for the sake of readability and being self - contained , we review governing equations for oscillatory convection in the hele - shaw cell and our phase description method for the oscillatory convection with consideration of its application to common - noise - induced synchronization . more details and other applications of the phase description method are given in ref . @xcite . the dynamics of the temperature field @xmath0 in the hele - shaw cell is described by the following dimensionless form ( see ref . @xcite and also references therein ) : @xmath1 the laplacian and jacobian are respectively given by @xmath2 the stream function @xmath3 is determined from the temperature field @xmath0 as @xmath4 where the rayleigh number is denoted by @xmath5 . the system is defined in the unit square : @xmath6 $ ] and @xmath7 $ ] . the boundary conditions for the temperature field @xmath0 are given by @xmath8 where the temperature at the bottom ( @xmath9 ) is higher than that at the top ( @xmath10 ) . the stream function @xmath3 satisfies the dirichlet zero boundary condition on both @xmath11 and @xmath12 , i.e. , @xmath13 to simplify the boundary conditions in eq . ( [ eq : bcty ] ) , we consider the convective component @xmath14 of the temperature field @xmath0 as follows : @xmath15 inserting eq . ( [ eq : t_x ] ) into eqs . ( [ eq : t])([eq : p_t ] ) , we derive the following equation for the convective component @xmath14 : @xmath16 where the stream function @xmath3 is determined by @xmath17 applying eq . ( [ eq : t_x ] ) to eqs . ( [ eq : bctx])([eq : bcty ] ) , we obtain the following boundary conditions for the convective component @xmath14 : @xmath18 that is , the convective component @xmath14 satisfies the neumann zero boundary condition on @xmath11 and the dirichlet zero boundary condition on @xmath12 . it should be noted that this system does not possess translational or rotational symmetry owing to the boundary conditions given by eqs . ( [ eq : bcpx])([eq : bcpy])([eq : bcxx])([eq : bcxy ] ) . the dependence of the hele - shaw convection on the rayleigh number @xmath5 is well known , and the existence of stable limit - cycle solutions to eq . ( [ eq : x ] ) is also well established ( see ref . @xcite and also references therein ) . in general , a stable limit - cycle solution to eq . ( [ eq : x ] ) , which represents oscillatory convection in the hele - shaw cell , can be described by @xmath19 the phase and natural frequency are denoted by @xmath20 and @xmath21 , respectively . the limit - cycle solution @xmath22 possesses the following @xmath23-periodicity in @xmath20 : @xmath24 . inserting eq . ( [ eq : x_x0 ] ) into eqs . ( [ eq : x])([eq : p_x ] ) , we find that the limit - cycle solution @xmath22 satisfies @xmath25 where the stream function @xmath26 is determined by @xmath27 from eq . ( [ eq : t_x ] ) , the corresponding temperature field @xmath28 is given by ( e.g. , see fig . [ fig:2 ] in sec . [ sec:4 ] ) @xmath29 let @xmath30 represent a small disturbance added to the limit - cycle solution @xmath22 , and consider a slightly perturbed solution @xmath31 equation ( [ eq : x ] ) is then linearized with respect to @xmath30 as follows : @xmath32 as in the limit - cycle solution @xmath22 , the function @xmath33 satisfies the neumann zero boundary condition on @xmath11 and the dirichlet zero boundary condition on @xmath12 . note that @xmath34 is time - periodic through @xmath20 . therefore , eq . ( [ eq : linear ] ) is a floquet - type system with a periodic linear operator . defining the inner product of two functions as @xmath35\!\!\bigr ] } } = \frac{1}{2\pi } \int_0^{2\pi } d\theta \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u^\ast(x , y , \theta ) u(x , y , \theta ) , \label{eq : inner}\end{aligned}\ ] ] we introduce the adjoint operator of the linear operator @xmath34 by @xmath36\!\!\bigr ] } } = { \ensuremath{\bigl[\!\!\bigl [ { \cal l}^\ast(x , y , \theta ) u^\ast(x , y , \theta ) , \ , u(x , y , \theta ) \bigr]\!\!\bigr]}}. \label{eq : operator}\end{aligned}\ ] ] as in @xmath33 , the function @xmath37 also satisfies the neumann zero boundary condition on @xmath11 and the dirichlet zero boundary condition on @xmath12 . details of the derivation of the adjoint operator @xmath38 are given in ref . @xcite . in the following subsection , we utilize the floquet eigenfunctions associated with the zero eigenvalue , i.e. , @xmath39 we note that the right zero eigenfunction @xmath40 can be chosen as @xmath41 which is confirmed by differentiating eq . ( [ eq : x0 ] ) with respect to @xmath20 . using the inner product of eq . ( [ eq : inner ] ) with the right zero eigenfunction of eq . ( [ eq : u0 ] ) , the left zero eigenfunction @xmath42 is normalized as @xmath43\!\!\bigr ] } } = \frac{1}{2\pi } \int_0^{2\pi } d\theta \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u_0^\ast(x , y , \theta ) u_0(x , y , \theta ) = 1.\end{aligned}\ ] ] here , we can show that the following equation holds ( see also refs . @xcite ) : @xmath44 = 0.\end{aligned}\ ] ] therefore , the following normalization condition is satisfied independently for each @xmath20 as follows : @xmath45 we now consider oscillatory hele - shaw convection with a weak perturbation applied to the temperature field @xmath0 described by the following equation : @xmath46 the weak perturbation is denoted by @xmath47 . inserting eq . ( [ eq : t_x ] ) into eq . ( [ eq : t_p ] ) , we obtain the following equation for the convective component @xmath14 : @xmath48 using the idea of the phase reduction @xcite , we can derive a phase equation from the perturbed equation ( [ eq : x_p ] ) . namely , we project the dynamics of the perturbed equation ( [ eq : x_p ] ) onto the unperturbed solution as @xmath49 \nonumber \\ & = \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u_0^\ast(x , y , \theta ) \left [ \nabla^2 x + j(\psi , x ) - \frac{\partial \psi}{\partial x } + \epsilon p(x , y , t ) \right ] \nonumber \\ & \simeq \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u_0^\ast(x , y , \theta ) \left [ \nabla^2 x_0 + j(\psi_0 , x_0 ) - \frac{\partial \psi_0}{\partial x } + \epsilon p(x , y , t ) \right ] \nonumber \\ & = \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u_0^\ast(x , y , \theta ) \left [ \omega \frac{\partial}{\partial \theta } x_0(x , y , \theta ) + \epsilon p(x , y , t ) \right ] \nonumber \\ & = \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u_0^\ast(x , y , \theta ) \ , \biggl [ \omega \ , u_0(x , y , \theta ) + \epsilon p(x , y , t ) \biggr ] \nonumber \\ & = \omega + \epsilon \int_0 ^ 1 dx \int_0 ^ 1 dy \ , u_0^\ast(x , y , \theta ) p(x , y , t),\end{aligned}\ ] ] where we approximated @xmath14 by the unperturbed limit - cycle solution @xmath22 . therefore , the phase equation describing the oscillatory hele - shaw convection with a weak perturbation is approximately obtained in the following form : @xmath50 where the _ phase sensitivity function _ is defined as ( e.g. , see fig . [ fig:2 ] in sec . [ sec:4 ] ) @xmath51 here , we note that the phase sensitivity function @xmath52 satisfies the neumann zero boundary condition on @xmath11 and the dirichlet zero boundary condition on @xmath12 , i.e. , @xmath53 as mentioned in ref . @xcite , eq . ( [ eq : theta_p ] ) is a generalization of the phase equation for a perturbed limit - cycle oscillator described by a finite - dimensional dynamical system ( see refs . however , reflecting the aspects of an infinite - dimensional dynamical system , the phase sensitivity function @xmath52 of the oscillatory hele - shaw convection possesses infinitely many components that are continuously parameterized by the two variables , @xmath11 and @xmath12 . in this paper , we further consider the case that the perturbation is described by a product of two functions as follows : @xmath54 that is , the space - dependence and time - dependence of the perturbation are separated . in this case , the phase equation ( [ eq : theta_p ] ) can be written in the following form : @xmath55 where the _ effective phase sensitivity function _ is given by ( e.g. , see fig . [ fig:5 ] in sec . [ sec:4 ] ) @xmath56 we note that the form of eq . ( [ eq : theta_q ] ) is essentially the same as that of the phase equation for a perturbed limit - cycle oscillator described by a finite - dimensional dynamical system ( see refs . we also note that the effective phase sensitivity function @xmath57 can also be considered as the collective phase sensitivity function in the context of the collective phase description of coupled individual dynamical elements exhibiting macroscopic rhythms @xcite . in this section , using the phase description method in sec . [ sec:2 ] , we analytically investigate common - noise - induced synchronization between uncoupled systems of oscillatory hele - shaw convection . in particular , we theoretically determine the optimal spatial pattern of the common noise for achieving the noise - induced synchronization . we consider @xmath58 uncoupled systems of oscillatory hele - shaw convection subject to weak common noise described by the following equation for @xmath59 : @xmath60 where the weak common noise is denoted by @xmath61 . inserting eq . ( [ eq : t_x ] ) into eq . ( [ eq : t_xi ] ) for each @xmath62 , we obtain the following equation for the convective component @xmath63 : @xmath64 as in eq . ( [ eq : p_x ] ) , the stream function of each system is determined by @xmath65 the common noise @xmath66 is assumed to be white gaussian noise @xcite , the statistics of which are given by @xmath67 here , we assume that the unperturbed oscillatory hele - shaw convection is a stable limit cycle and that the noise intensity @xmath68 is sufficiently weak . then , as in eq . ( [ eq : theta_q ] ) , we can derive a phase equation from eq . ( [ eq : x_xi ] ) as follows ) can be slightly different from the natural frequency given in eq . ( [ eq : x_x0 ] ) ; however , this point is not essential in this paper because eq . ( [ eq : lambda ] ) is independent of the value of the frequency . the theory of stochastic phase reduction for ordinary limit - cycle oscillators has been intensively investigated in refs . @xcite , but extensions to partial differential equations have not been developed yet . ] : @xmath69 where the effective phase sensitivity function @xmath57 is given by eq . ( [ eq : zeta ] ) . once the phase equation ( [ eq : theta_xi ] ) is obtained , the lyapunov exponent characterizing the common - noise - induced synchronization can be derived using the argument by teramae and tanaka @xcite . from eqs . ( [ eq : xi])([eq : theta_xi ] ) , the lyapunov exponent , which quantifies the exponential growth rate of small phase differences between the two systems , can be written in the following form : @xmath70 ^ 2 \leq 0 . \label{eq : lambda}\end{aligned}\ ] ] here , we used the following abbreviation : @xmath71 . equation ( [ eq : lambda ] ) represents that uncoupled systems of oscillatory hele - shaw convection can be in - phase synchronized when driven by the weak common noise , as long as the phase reduction approximation is valid . in the following two subsections , we develop a method for obtaining the optimal spatial pattern of the common noise to achieve the noise - induced synchronization of the oscillatory convection . considering the boundary conditions of @xmath52 , eqs . ( [ eq : bczx])([eq : bczy ] ) , we introduce the following spectral transformation : @xmath72 for @xmath73 and @xmath74 . the corresponding spectral decomposition of @xmath52 is given by @xmath75 by inserting eq . ( [ eq : z_cossin ] ) into eq . ( [ eq : zeta ] ) , the effective phase sensitivity function @xmath57 can be written in the following form : @xmath76 where the spectral transformation of @xmath77 is defined as @xmath78 the corresponding spectral decomposition of @xmath77 is given by @xmath79 for the sake of convenience in the calculation below , we rewrite the double sum in eq . ( [ eq : zeta_double ] ) by the following single series : @xmath80 in eq . ( [ eq : zeta_single ] ) , we introduced one - dimensional representations , @xmath81 and @xmath82 , where the mapping between @xmath83 and @xmath84 is bijective . accordingly , we obtain the following quantity : @xmath85 ^ 2 = \sum_{n=0}^\infty \sum_{m=0}^\infty s_n s_m q_n'(\theta ) q_m'(\theta ) , \label{eq : dzeta_squared}\end{aligned}\ ] ] where @xmath86 . from eqs . ( [ eq : lambda])([eq : dzeta_squared ] ) , the lyapunov exponent normalized by the noise intensity , @xmath87 , can be written in the following form : @xmath88 ^ 2 = \sum_{n=0}^\infty \sum_{m=0}^\infty k_{nm } s_n s_m , \label{eq : lambda_k}\end{aligned}\ ] ] where each element of the symmetric matrix @xmath89 is given by @xmath90 by defining an infinite - dimensional column vector @xmath91 , eq . ( [ eq : lambda_k ] ) can also be written as @xmath92 which is a quadratic form . using the spectral representation of the normalized lyapunov exponent , eq . ( [ eq : lyapunov ] ) , we seek the optimal spatial pattern of the common noise for the synchronization . as a constraint , we introduce the following condition : @xmath93 that is , the total power of the spatial pattern is fixed at unity . under this constraint condition , we consider the maximization of eq . ( [ eq : lyapunov ] ) . for this purpose , we define the lagrangian @xmath94 as @xmath95 where the lagrange multiplier is denoted by @xmath96 . setting the derivative of the lagrangian @xmath94 to be zero , we can obtain the following equations : @xmath97 \frac{\partial f}{\partial \lambda } & = - \left ( \sum_{n=0}^\infty s_n^2 - 1 \right ) = 0,\end{aligned}\ ] ] which are equivalent to the eigenvalue problem described by @xmath98 these eigenvectors @xmath99 and the corresponding eigenvalues @xmath100 satisfy @xmath101 because the matrix @xmath89 , which is defined in eq . ( [ eq : k ] ) , is symmetric , the eigenvalues @xmath100 are real numbers . consequently , under the constraint condition given by eq . ( [ eq : unity ] ) , the optimal vector that maximizes eq . ( [ eq : lambda ] ) coincides with the eigenvector associated with the largest eigenvalue , i.e. , @xmath102 therefore , the optimal spatial pattern @xmath103 can be written in the following form : @xmath104 where the coefficients @xmath105 in the double series correspond to the elements of the optimal vector @xmath106 associated with @xmath107 . from eq . ( [ eq : lyapunov ] ) , the lyapunov exponent is then given by @xmath108 finally , we note that this optimization method can also be considered as the principal component analysis @xcite of the phase - derivative of the phase sensitivity function , @xmath109 . in this section , to illustrate the theory developed in sec . [ sec:3 ] , we numerically investigate common - noise - induced synchronization between uncoupled hele - shaw cells exhibiting oscillatory convection . the numerical simulation method is summarized in ref . modes for the dirichlet zero boundary condition and a cosine expansion with @xmath110 modes for the neumann zero boundary condition . the fourth - order runge - kutta method with integrating factor using a time step @xmath111 ( mainly , @xmath112 ) and the heun method with integrating factor using a time step @xmath113 were applied for the deterministic and stochastic ( langevin - type ) equations , respectively . ] . considering the boundary conditions of the convective component @xmath114 , eqs . ( [ eq : bcxx])([eq : bcxy ] ) , we introduce the following spectral transformation : @xmath115 for @xmath73 and @xmath74 . the corresponding spectral decomposition of the convective component @xmath114 is given by @xmath116 in visualizing the limit - cycle orbit in the infinite - dimensional state space , we project the limit - cycle solution @xmath22 onto the @xmath117-@xmath118 plane as @xmath119 the initial values were prepared so that the system exhibits single cellular oscillatory convection . the rayleigh number was fixed at @xmath120 , which gives the natural frequency @xmath121 , i.e. , the oscillation period @xmath122 . figure [ fig:1 ] shows the limit - cycle orbit of the oscillatory convection projected onto the @xmath117-@xmath118 plane , obtained from direct numerical simulations of the dynamical equation ( [ eq : x ] ) . snapshots of the limit - cycle solution @xmath22 and other associated functions , @xmath28 and @xmath52 , are shown in fig . [ fig:2 ] , where the phase variable @xmath20 is discretized using @xmath123 grid points . we note that fig . [ fig:1 ] and fig . [ fig:2 ] are essentially reproductions of our previous results given in ref . details of the numerical method for obtaining the phase sensitivity function @xmath52 are given in refs . @xcite ( see also refs . @xcite ) . as seen in fig . [ fig:2 ] , the phase sensitivity function @xmath52 is spatially localized . namely , the absolute values of the phase sensitivity function @xmath52 in the top - right and bottom - left corner regions of the system are much larger than those in the other regions ; this fact reflects the dynamics of the spatial pattern of the convective component @xmath22 . as mentioned in ref . @xcite , the phase sensitivity function @xmath52 in this case possesses the following symmetry . for each @xmath20 , the limit - cycle solution @xmath22 and the phase sensitivity function @xmath52 , shown in fig . [ fig:2 ] , are anti - symmetric with respect to the center of the system , i.e. , @xmath124 where @xmath125 and @xmath126 . therefore , for a spatial pattern @xmath127 that is symmetric with respect to the center of the system , @xmath128 the corresponding effective phase sensitivity function @xmath57 becomes zero , i.e. , @xmath129 that is , such symmetric perturbations do not affect the phase of the oscillatory convection . the optimal spatial pattern is obtained as the best combination of single - mode spatial patterns , i.e. , eq . ( [ eq : aopt ] ) . thus , we first consider the following single - mode spatial pattern : @xmath130 then , the effective phase sensitivity function is given by the following single spectral component : @xmath131 from eq . ( [ eq : lambda ] ) , the lyapunov exponent for the single - mode spatial pattern can be written in the following form : @xmath132 ^ 2,\end{aligned}\ ] ] where @xmath133 . figure [ fig:3](a ) shows the normalized lyapunov exponent for single - mode spatial patterns , i.e. , @xmath134 . owing to the anti - symmetry of the phase sensitivity function , given in eq . ( [ eq : anti - symmetric ] ) , the normalized lyapunov exponent @xmath134 exhibits a checkerboard pattern , namely , @xmath135 when the sum of @xmath136 and @xmath137 , i.e. , @xmath138 , is an odd number . the maximum of @xmath134 is located at @xmath139 ; under the condition of @xmath140 , the maximum of @xmath134 is located at @xmath141 . the single - mode spatial patterns , @xmath142 , @xmath143 , and @xmath144 , are shown in figs . [ fig:4](b)(c)(d ) , respectively . we note that @xmath142 and @xmath143 are anti - symmetric with respect to the center of the system , whereas @xmath144 is symmetric . these spatial patterns are used in the numerical simulations performed below . we now consider the optimal spatial pattern . figure [ fig:3](b ) shows the spectral components of the optimal spatial pattern , i.e. , @xmath105 , obtained by the optimization method developed in sec . [ subsec:3c ] ; figure [ fig:4](a ) shows the corresponding optimal spatial pattern , i.e. , @xmath103 , given by eq . ( [ eq : aopt ] ) . as seen in fig . [ fig:3 ] , when the normalized lyapunov exponent for a single - mode spatial pattern , @xmath134 , is large , the absolute value of the optimal spectral components , @xmath145 , is also large . as seen in fig . [ fig:4](a ) , the optimal spatial pattern @xmath103 is similar to the snapshots of the phase sensitivity function @xmath52 shown in fig . [ fig:2 ] . in fact , as mentioned in sec . [ subsec:3c ] , the optimal spatial pattern @xmath103 corresponds to the first principal component of @xmath109 . reflecting the anti - symmetry of the phase sensitivity function , eq . ( [ eq : anti - symmetric ] ) , the optimal spatial pattern @xmath103 is also anti - symmetric with respect to the center of the system . figure [ fig:5 ] shows the effective phase sensitivity functions @xmath57 for the spatial patterns shown in fig . [ fig:4 ] . when the normalized lyapunov exponent @xmath134 is large , the amplitude of the corresponding effective phase sensitivity function @xmath57 is also large . for the spatial pattern @xmath144 , which is symmetric with respect to the center of the system , the effective phase sensitivity function becomes zero , @xmath146 , as shown in eq . ( [ eq : zeta_s ] ) . to confirm the theoretical results shown in fig . [ fig:5 ] , we obtain the effective phase sensitivity function @xmath57 by direct numerical simulations of eq . ( [ eq : x_p ] ) with eq . ( [ eq : p_aq ] ) as follows : we measure the phase response of the oscillatory convection by applying a weak impulsive perturbation with the spatial pattern @xmath77 to the limit - cycle solution @xmath22 with the phase @xmath20 ; then , normalizing the phase response curve by the weak impulse intensity @xmath147 , we obtain the effective phase sensitivity function @xmath57 . the effective phase sensitivity function @xmath57 obtained by direct numerical simulations with impulse intensity @xmath147 are compared with the theoretical curves in fig . [ fig:6 ] . the simulation results agree quantitatively with the theory . therefore , the phase response curve normalized by the impulse intensity @xmath147 converges to the effective phase sensitivity function @xmath57 as @xmath147 decreases . as shown in fig . [ fig:6](d ) , when the impulsive perturbation is not weak , the dependence of the phase response curve on the impulse intensity @xmath147 becomes nonlinear . in general , when the impulsive perturbation is not weak , the phase response curve is not equal to zero , even though the effective phase sensitivity function is equal to zero , @xmath146 . we also note that the linear dependence region of the phase response curve on the impulse is generally dependent on the spatial pattern @xmath77 of the impulse . ] . in this subsection , we demonstrate the common - noise - induced synchronization between uncoupled hele - shaw cells exhibiting oscillatory convection by direct numerical simulations of the stochastic ( langevin - type ) partial differential equation ( [ eq : x_xi ] ) . theoretical values of both the lyapunov exponents @xmath148 for several spatial patterns @xmath77 with the common noise intensity @xmath149 and the corresponding relaxation time @xmath150 toward the synchronized state are summarized in table [ table:1 ] . figure [ fig:7 ] shows the time evolution of the phase differences @xmath151 when the common noise intensity is @xmath149 . the initial phase values are @xmath152 for @xmath153 . figure [ fig:8 ] shows the time evolution of @xmath154 , which corresponds to fig . [ fig:7 ] . the relaxation times estimated from the simulation results agree reasonably well with the theory ( d ) should be constant because the effective phase sensitivity function is equal to zero , @xmath146 , for this case . as shown in fig . [ fig:6](d ) , when the perturbation is not sufficiently weak , the phase response curve is not equal to zero ; this higher order effect causes the slight variations shown in fig . [ fig:7](d ) . ] . as seen in fig . [ fig:7 ] and fig . [ fig:8 ] , the relaxation time for the optimal spatial pattern @xmath103 is actually much smaller than those for the single - mode spatial patterns . for the cases of single - mode patterns , the relaxation time for the single - mode spatial pattern @xmath142 is also smaller than those for the other single - mode spatial patterns , @xmath143 and @xmath144 . we also note that the time evolution of both @xmath151 and @xmath154 for @xmath142 is significantly different from that for @xmath144 in spite of the similarity between the two spatial patterns of the neighboring modes ; this difference results from the difference of symmetry with respect to the center , as shown in eq . ( [ eq : zeta_s ] ) . figure [ fig:9 ] shows a quantitative comparison of the lyapunov exponents between direct numerical simulations and the theory for the case of the optimal spatial pattern @xmath103 . the initial phase values are @xmath155 for @xmath156 , i.e. , the initial phase difference is @xmath157 . the results of direct numerical simulations are averaged over @xmath158 samples for different noise realizations . the simulation results quantitatively agree with the theory . figure [ fig:10 ] shows the global stability of the common - noise - induced synchronization of oscillatory convection for the case of the optimal spatial pattern @xmath103 ; namely , it shows that the synchronization is eventually achieved from arbitrary initial phase differences , i.e. , @xmath159 $ ] . although the lyapunov exponent @xmath148 based on the linearization of eq . ( [ eq : theta_xi ] ) quantifies only the local stability of a small phase difference , as long as the phase reduction approximation is valid , this global stability holds true for any spatial pattern @xmath77 with a non - zero lyapunov exponent , namely , the lyapunov exponent is negative , @xmath160 , as found from eq . ( [ eq : lambda ] ) . the global stability can be proved by the theory developed in ref . @xcite , i.e. , by analyzing the fokker - planck equation equivalent to the langevin - type phase equation ( [ eq : theta_xi ] ) ; in addition , the effect of the independent noise can also be included . our investigations in this paper are summarized as follows . in sec . [ sec:2 ] , we briefly reviewed our phase description method for oscillatory convection in the hele - shaw cell with consideration of its application to common - noise - induced synchronization . in sec . [ sec:3 ] , we analytically investigated common - noise - induced synchronization of oscillatory convection using the phase description method . in particular , we theoretically determined the optimal spatial pattern of the common noise for the oscillatory hele - shaw convection . in sec . [ sec:4 ] , we numerically investigated common - noise - induced synchronization of oscillatory convection ; the direct numerical simulation successfully confirmed the theoretical predictions . the key quantity of the theory developed in this paper is the phase sensitivity function @xmath52 . thus , we describe an experimental procedure to obtain the phase sensitivity function @xmath52 . as in eq . ( [ eq : z_cossin ] ) , the phase sensitivity function @xmath52 can be decomposed into the spectral components @xmath161 , which are the effective phase sensitivity functions for the single - mode spatial patterns @xmath162 as shown in eq . ( [ eq : zjk ] ) . in a manner similar to the direct numerical simulations yielding fig . [ fig:6 ] , the effective phase sensitivity function @xmath161 for each single - mode spatial pattern @xmath162 can also be experimentally measured . therefore , in general , the phase sensitivity function @xmath52 can be constructed from a sufficiently large set of such @xmath161 . once the phase sensitivity function @xmath52 is obtained , the optimization method for common - noise - induced synchronization can also be applied in experiments . finally , we remark that not only the phase description method for spatiotemporal rhythms but also the optimization method for common - noise - induced synchronization have broad applicability ; these methods are not restricted to the oscillatory hele - shaw convection analyzed in this paper . for example , the combination of these methods can be applied to common - noise - induced phase synchronization of spatiotemporal rhythms in reaction - diffusion systems of excitable and/or heterogeneous media . furthermore , as mentioned above , also in experimental systems , such as the photosensitive belousov - zhabotinsky reaction @xcite and the liquid crystal spatial light modulator @xcite , the optimization method for common - noise - induced synchronization could be applied . y.k . is grateful to members of both the earth evolution modeling research team and the nonlinear dynamics and its application research team at ifree / jamstec for fruitful comments . is also grateful for financial support by jsps kakenhi grant number 25800222 . h.n . is grateful for financial support by jsps kakenhi grant numbers 25540108 and 22684020 , crest kokubu project of jst , and first aihara project of jsps . e. m. izhikevich , _ dynamical systems in neuroscience : the geometry of excitability and bursting _ ( mit press , cambridge , ma , 2007 ) . g. b. ermentrout and d. h. terman , _ mathematical foundations of neuroscience _ ( springer , new york , 2010 ) . h. nakao , t. yanagita , and y. kawamura , procedia iutam * 5 * , 227 ( 2012 ) . y. kawamura , h. nakao , k. arai , h. kori , and y. kuramoto , phys . * 101 * , 024101 ( 2008 ) . [ arxiv:0807.1285 ] + y. kawamura , physica d * 270 * , 20 ( 2014 ) . [ arxiv:1312.7054 ]
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we investigate common - noise - induced phase synchronization between uncoupled identical hele - shaw cells exhibiting oscillatory convection . using the phase description method for oscillatory convection
, we demonstrate that the uncoupled systems of oscillatory hele - shaw convection can exhibit in - phase synchronization when driven by weak common noise .
we derive the lyapunov exponent determining the relaxation time for the synchronization , and develop a method for obtaining the optimal spatial pattern of the common noise to achieve synchronization .
the theoretical results are confirmed by direct numerical simulations .
| 11,153 | 149 |
road traffic prediction plays an important role in intelligent transport systems by providing the required real - time information for traffic management and congestion control , as well as the long - term traffic trend for transport infrastructure planning @xcite . road traffic predictions can be broadly classified into short - term traffic predictions and long - term traffic forecasts@xcite . short - term prediction is essential for the development of efficient traffic management and control systems , while long - term prediction is mainly useful for road design and transport infrastructure planning . there are two major categories of techniques for road traffic prediction : those based on non - parametric models and those based on parametric models . non - parametric model based techniques , such as k - nearest neighbors ( knn ) model @xcite and artificial neural networks ( ann ) @xcite , are inherently robust and valid under very weak assumptions @xcite , while parametric model based techniques , such as auto - regressive integrated moving average ( arima ) model @xcite and its variants @xcite@xcite , allows to integrate knowledge of the underlying traffic process in the form of traffic models that can then be used for traffic prediction . both categories of techniques have been widely used and in this paper , we consider parametric model based techniques , particularly starima ( space - time autoregressive integrated moving average)-based techniques . as for the estimation of parameters and coefficients in starima model , overfitting easily occurs which makes the predictive performance poor as it overreacts to minor fluctuations in the training data @xcite . furthermore , the same model and hence the same correlation structure is used for traffic prediction at different time of the day , which is counter - intuitive and may not be accurate . to elaborate , consider an artificial example of two traffic stations @xmath0 and @xmath1 on a highway , where traffic station @xmath1 is at the down stream direction of @xmath0 . intuitively , the correlation between the traffic observed at @xmath0 and the traffic observed at @xmath1 will peak at a time lag corresponding to the time required to travel from @xmath0 to @xmath1 because at that time lag , the ( approximately ) same set of vehicles that have passed @xmath0 now have reached @xmath1 . obviously , the time required to travel from @xmath0 to @xmath1 depends on the traffic speed , which varies with the time of the day , e.g. peak hours and off - peak hours . accordingly , the time lag corresponding to the peak correlation between the traffic at @xmath0 and the traffic at @xmath1 should also vary with time of the day and , to be more specific , should approximately equal to the distance between @xmath0 and @xmath1 divided by the mean speed of vehicles between @xmath0 and @xmath1 . therefore , in designing the starima model for traffic prediction , the aforementioned time - varying lags should be taken into account for accurate traffic prediction . ] ] to validate the aforementioned intuition , we analyze the cross - correlation function ( ccf ) of traffic flow data at two traffic stations ( stations 6 and 3 ) , denoted as @xmath2 , from i-80 highway ( more details of data are discussed in section [ sub : data_collection ] ) with the formulation : @xmath3}{\sigma_{uu}\sigma{}_{yy}}\label{eq : corr_u_y}\ ] ] where @xmath4 and @xmath5 are the traffic flow data collected in @xmath6 time slots from the two traffic stations , @xmath7 is the temporal order in the range of @xmath8\subset\mathbf{n}$ ] , @xmath9 and @xmath10 are respectively the standard deviation of @xmath4 and @xmath5 . a higher value of ccf indicates a stronger correlation of the traffic at both stations . as shown in fig.[fig : flow_3_6 ] , the correlation between traffic at stations 6 and 3 peaks at different time lags depending on the time of the day . during on - peak period ( approximately from 6:30am - 8:30am ) , the correlation peaks at a time lag of @xmath11 ( one time lag corresponds to @xmath12 ) while during off - peak period ( approximately from 19pm - 24pm ) , the correlation peaks at a time lag of @xmath13 , where one time lag corresponds to a time of 30@xmath14 . we observe that at peak hours , the time lag corresponding to the maximum correlation is larger than that for off - peak hours . in the latter section , we will further show that this time lag approximately equals to the distance between the two traffic stations divided by the average speed . therefore , our intuition explained in the previous paragraph is valid . the above observation motivates us to design a starima - based traffic prediction with time - varying lags which better matches the time - varying correlation structure between traffic of different stations and hence can potentially deliver more accurate traffic prediction . more specifically , the contributions of the paper are : * we analyze the ccf between the speed and traffic flow data between different detector stations and establish the relationship between the changes in the temporal lag ( corresponding to the aforementioned maximum correlation ) and the speed variations . * an unsupervised classification algorithm based on isodata algorithm is designed to classify different time periods of the day according to the variation of the speed . the classification helps to determine the appropriate time lag to use in the starima model . * a starima - based model with time - varying lags is developed for short - term traffic prediction . experimental results using real traffic data show that the developed starima - based model with time - varying lags has superior accuracy compared with its counterpart developed using the traditional cross - correlation function and without employing time - varying lags . the the rest of the paper is organized as follows . in section [ sec : related_work ] , we briefly discuss related work . section [ sec : data_method ] introduces the starima model and the isodata algorithm in section [ sec : starima_time_lag_variation ] , we present the details the proposed algorithm . the experimental results are presented in section [ sec : results ] . finally , section [ sec : conclusion ] concludes the paper . there is previous work , which predicts traffic flow using a modified arima models @xcite . in @xcite and @xcite , a multivariate arima based model , arimax , was applied for better traffic flow prediction . the difference is that the former paper considered the varibility of the speed from upstream to downstream , the other one considered different model specifications during different time periods of the day . similarly , the authors in @xcite also different configurations of temporal lags in arima model . more concretely , they firstly applied a hidden markov model ( hmm ) model along with an expectation - maximization ( em ) algorithm to evaluate the traffic state ( one of \{major accident , minor incident , instability , normal driving } ) in next time slot . after that , the arima models with different configurations of temporal lags were used to predict the state of the traffic flow . all these models have improved the accuracy of the forecasting results compared with arima model . however , the spatial information was less considered in these models . in this way , the starima based models @xcite have aroused more and more concern . ] in @xcite , the authors proposed a dynamic starima model by combining the dynamic turn ratio prediction ( dtrp ) model and the starima model . in this paper , a dynamic space weigh matrix is used to capture different impact of traffic at upstream locations on traffic at downstream locations . similarly , the research in @xcite also applied the starima model with the consideration of the dynamic space weight matrix . our work distinguish from theirs in that in our work , the space weight matrices vary with on - peak and off - peak periods to capture the time - varying correlations of road traffic at different locations . from the above related work , we can find that the " `` dynamic' of a arima or starima model in existing research is often used to indicate the dynamic of the space weight matrix , the traffic state during different time periods @xcite . however , sometimes it is not enough to only consider these aspects . for example , fig.[fig : speed_flow_data ] presents the normalized average speed and normalized flow data collected at traffic station 6 every 30 minutes in one day . theoretically , the weight matrix in time slot 3 ( or 4 , the left hollow rectangle ) and slot 15 ( the right hollow rectangle ) should be different since they are respectively in the off - peak period and peak period . however , their average speed are the same . this is caused by an inaccurate evaluation of the time range of peak or off - peak period . furthermore , few research considered the relationship between speed and the parameters ( temporal or spatial lag ) in starima model . specially , a great majority of research use pacf to evaluate temporal lag which easily causes overfitting problem . motivated by the above observations , in this paper we investigate a more efficient method to evaluate these parameters in starima model with the consideration of spatial information and the variation of average speed during different time periods . ] [ sub : data_collection ] in urban environment , the road structure is often complex . also , the sensors ( such as loop detectors or cameras ) @xcite are not deployed at every road . therefore , it is difficult to obtain comprehensive data . for simplicity , we only consider highway in this paper where there are only on - ramp / off - ramps so the traffic condition is comparatively simpler than that in the urban area . we use data collected from a segment of interstate 80 ( i-80 ) freeway located in emeryville , california @xcite . available data are collected every 30 seconds from six traffic stations numbered by 1 , 3 , 4 , 5 , 6 and 7 within 10 days . there are two traffic stations for upstream and downstream traffic respectively . the road topology is shown in fig.[fig : real_scenario_and_topology ] . note that there is no data at station 2 and there are too many interference caused by on or off - ramps at station 1 and 7 . for simplicity , we only use the data collected at stations 3 , 4 , 5 , and 6 . the travel direction is from station 3 to 6 . in this section , we first present a simple way to evaluate the temporal lag in relation to speed variation . then we propose a classification algorithm based on isodata by which we can respectively obtain a set of speed clusters and a set of time period clusters . finally , we describe a modified starima model with the varying temporal lag . from fig.[fig : ccf_3_6_different_time_period ] , the temporal lags with maximal ccf between stations 6 and 3 are different during peak and off - peak periods . this is attributable to the variation of speed . assuming the distance between two detector stations @xmath0 and @xmath1 is @xmath15 , and the vehicles keep a stable average speed @xmath16 , then approximately @xmath17 is needed for vehicles to travel from @xmath1 to @xmath0 . in other words , the traffic flow collected at station @xmath0 is strongly correlated with that at @xmath1 @xmath18 time ago . thus the temporal lag with the maximum correlation should be @xmath19 $ ] where @xmath20 is the length of one temporal lag . note that @xmath15 depend on the spatial order @xmath21 in the starima model and @xmath16 can often be measured by loop detectors @xcite . furthermore , the advance in telecommunication and electronic technology also brings a number of new techniques that allows us to estimate the travel time , e.g. via smartphones . indeed , the observation discussed in the introduction section suggests another novel way to estimate travel time : we can infer travel time from the correlation of the observed traffic . in order to validate the above discussion , we further analyze the results in fig.[fig : ccf_3_6_different_time_period ] by using the f average speed information at stations 3 and 6 , which is collected in the same day as the traffic flow data used in fig.[fig : ccf_3_6_different_time_period ] . specifically , the average speed from station 3 to station 6 between 6:30 am and 8:30 am is 44.45 feet / second . the average speed is 67.05 feet / second between 19 pm and 24 pm . the maximal temporal lag during these two time periods is respectively 3 and 2 with 30 seconds in each temporal lag . as @xmath22 , given @xmath23and @xmath24 during two time periods along with the corresponding best temporal lag @xmath25 and @xmath26 , we are able to obtain the following equation according to the theoretical analysis above : @xmath27 substituting the data into formulation , it is easy to find @xmath28 . this result agrees with our theoretical analysis and verifies our speculation that temporal @xmath29 is a function of the variation of average speed @xmath16 in section [ sec : introduction ] . an easy way to classify speed data is by dividing into peak time and off - peak periods . after that , the temporal lag can be calculated using @xmath30 , where @xmath31 is the average speed in time period @xmath32\{peak , off - peak}. however an empirical classification is often prone to error and inaccuracy . recall the analysis in section [ sec : related_work ] , the evaluation of the average speed is sensitive to the time range selected for peak or off - peak period . it is obvious that the speed is not always fast even during off - peak period from fig . [ fig : speed_flow_data ] . therefore , in this paper an isodata based speed data classification algorithm is developed to deliver an accurate classification . using this algorithm , we firstly classify the speed data collected in each time slot into different clusters . after that , the time period clusters are confirmed based on the time slots contained in different speed clusters . * input : * @xmath33 , @xmath34,@xmath35,@xmath36,@xmath37,@xmath38,@xmath39 * return : * @xmath40,@xmath41 @xmath40**@xmath42isodata**(@xmath33 , @xmath34,@xmath35,@xmath36,@xmath43,@xmath38 ) [ algo - isodata ] @xmath44 [ algo - for - start-1 ] @xmath45 , @xmath46 @xmath47 @xmath48 @xmath49 @xmath50 and @xmath51 @xmath52 @xmath49 @xmath53 @xmath54 [ algo - if-1 ] @xmath55[algo - min - start ] @xmath56[algo - min - end ] @xmath57 [ algo - endif-1 ] @xmath58 [ algo - for - end-1 ] assuming there is a set of speed data @xmath59 in which @xmath60 is the speed in time slot @xmath61 . the purpose here is to confirm a set of speed clusters , denoted as @xmath62 , where @xmath63 with cluster center @xmath64 and @xmath65 . based on @xmath66 , we can obtain another set of time period clusters , denoted as @xmath67 , in which @xmath68 . in addition , let @xmath69 be a set of continuous time slots , termed as a time range and defined as follows : @xmath70 where @xmath71 is the number of time slots contained in @xmath72 , @xmath39 is a threshold defined as the minimal number of time slots included in a time period . the speed data classification algorithm is given in algorithm [ alg : speed - data - classification ] . in line [ algo - isodata ] , the isodata algorithm is implemented to get speed clusters . the time period clusters are obtained from line [ algo - for - start-1 ] to [ algo - for - end-1 ] . it is worth mentioning that a decision is made to decide whether @xmath73 belongs to @xmath74 by comparing its capacity and threshold @xmath71 ( from line [ algo - if-1 ] to [ algo - for - end-1 ] ) . if @xmath73 does not belong to @xmath74 , line [ algo - min - start ] and [ algo - min - end ] are implemented to allocate each @xmath75 to other @xmath76 by the operation @xmath77 . @xmath78 is defined as the absolute difference between speed recorded in time slot @xmath79 and the average speed calculated during time period @xmath80 , which is presented in . @xmath81 according to the speed and time period clusters obtained from section [ sub : classification_speed ] , we propose a modified starima model , denoted as @xmath82 . the definitions of parameters @xmath83 , @xmath84 and @xmath85 in this model are the same as the original starima model , except that the temporal lag @xmath29 will vary with the spatial order @xmath21 and the average speed in different time periods . more precisely , given a time period @xmath86 , @xmath82 is defined as follows : @xmath87 in [ eq : starima_tv_model ] , @xmath88 is a @xmath89 vector in which each element @xmath90 represents the temporal lag between two station @xmath91 and @xmath92 with the spatial order @xmath21 . @xmath93 is calculated by @xmath94 where @xmath95 is the distance between these two stations and @xmath96 is the average speed in time period @xmath73 which is equal to @xmath97 . note that when @xmath98 , the " `` 0th order neighbor' of a station is itself such that the temporal lag is evaluated with the pacf used in arima . based on the data collection introduced in section [ sec : data_method ] , we utilize the speed and traffic flow data at stations 3 , 4 , 5 and 6 . at each station , there are 2880 data recorded in one day and the length of one time slot is 30 seconds . in order to eliminate noise in the data , we make a " `` smooth' operation by calculating the mean traffic flow every @xmath99 data points and regarding it as one data point . the experimental results are divided into two parts . in the first part , we provide the speed and time period clusters classified by our proposed algorithm . for the speed data @xmath100 , we choose @xmath101 . in the second part , we present the forecast results of traffic flow in different time periods and stations using @xmath82 model . we choose @xmath102 to calculate the mean traffic flow using original traffic flow data within 2 minutes . firstly , the configuration of input parameters of algorithm is given in table [ tab : the - configuration ] in which the speed data @xmath100 is the results after the smooth operation on the speed data collected from four stations . with this setting , the smallest length of time range @xmath86 is 120 minutes . the speed and time period clusters classified by algorithm [ algo - isodata ] is presented in table [ tab : speed - time - cluster ] . .the input parameters in algorithm [ algo - isodata][tab : the - configuration ] [ cols="^,^,^,^",options="header " , ] in table [ tab : mse_mape_diff_algo_one_day ] , we compare the mape / mse of one day using @xmath103 , chaos and @xmath104 at four stations . except station 3 , all the mape / mse achieved by our model are better than those of the other two models . furthermore , in table [ tab : the - mape / mse - of - two - periods - s6 ] , we present the mape / mse of the forecast results of station 6 using these three models , in which the forecast time ranges are 9 - 10am and 23 - 24pm . it can be seen that the performance of our model is almost coincident with the true data . and chaos comes to the second in the prediction during 23 - 24pm . motivated by the observation that the correlation between traffic at different traffic stations is time - varying and the time lag corresponding to the maximum correlation approximately equals to the distance between two traffic stations divided by the speed of vehicles between them , in this paper , we developed a modified starima model with time - varying lags for short - term traffic flow prediction . experimental results using real traffic data collected on a highway showed that the developed starima - based model with time - varying lags has superior accuracy compared with its counterpart developed using the traditional cross - correlation function and without employing time - varying lags . in an urban environment , the correlation between traffic tends to be much more intricate . it is part of our future work plan to develop prediction technique for urban roads that incorporates the knowledge of the underlying road topology . p. dellacqua , f. bellotti , r. berta , and a. de gloria , " time - aware multivariate nearest neighbor regression methods for traffic flow prediction , _ ieee trans . intelligent transportation systems _ , vol . 16 , no . 6 , pp . 33933402 , 2015 . kim , j .- s . et al . _ , " urban traffic flow prediction system using a multifactor pattern recognition model , _ ieee trans . intelligent transportation systems _ , vol . 16 , no . 5 , pp . 27442755 , 2015 . e. i. vlahogianni , m. g. karlaftis , and j. c. golias , " short - term traffic forecasting : where we are and where we re going , _ transportation research part c : emerging technologies _ , vol . 43 , pp . 319 , 2014 . m. lippi , m. bertini , and p. frasconi , " short - term traffic flow forecasting : an experimental comparison of time - series analysis and supervised learning , _ ieee trans . intelligent transportation systems _ , vol . 14 , no . 2 , pp . 871882 , 2013 . l. song , " improved intelligent method for traffic flow prediction based on artificial neural networks and ant colony optimization . _ journal of convergence information technology _ , vol . 7 , no . 8 , 2012 . b. l. smith , b. m. williams , and r. k. oswald , " comparison of parametric and nonparametric models for traffic flow forecasting , _ transportation research part c : emerging technologies _ , 10 , no . 4 , pp . 303321 , 2002 . x. min , j. hu , q. chen , t. zhang , and y. zhang , " short - term traffic flow forecasting of urban network based on dynamic starima model , in _ 2009 ieee int . conf . intelligent transportation systems _ , pp . 16 . a. stathopoulos and m. g. karlaftis , " a multivariate state space approach for urban traffic flow modeling and prediction , _ transportation research part c : emerging technologies _ , vol . 11 , no . 2 , pp . 121135 , 2003 . g. mao and b. d. anderson , " graph theoretic models and tools for the analysis of dynamic wireless multihop networks , in _ 2009 ieee wireless communications and networking conference_.1em plus 0.5em minus 0.4emieee , 2009 , pp . 16 . a. a. kannan , b. fidan , and g. mao , " robust distributed sensor network localization based on analysis of flip ambiguities , in _ ieee globecom 2008 - 2008 ieee global telecommunications conference_.1em plus 0.5em minus 0.4emieee , 2008 , pp . 16 .
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based on the observation that the correlation between observed traffic at two measurement points or traffic stations may be time - varying , attributable to the time - varying speed which subsequently causes variations in the time required to travel between the two points , in this paper , we develop a modified space - time autoregressive integrated moving average ( starima ) model with time - varying lags for short - term traffic flow prediction .
particularly , the temporal lags in the modified starima change with the time - varying speed at different time of the day or equivalently change with the ( time - varying ) time required to travel between two measurement points .
firstly , a technique is developed to evaluate the temporal lag in the starima model , where the temporal lag is formulated as a function of the spatial lag ( spatial distance ) and the average speed .
secondly , an unsupervised classification algorithm based on isodata algorithm is designed to classify different time periods of the day according to the variation of the speed .
the classification helps to determine the appropriate time lag to use in the starima model .
finally , a starima - based model with time - varying lags is developed for short - term traffic prediction . experimental results using real traffic data show that the developed starima - based model with time - varying lags has superior accuracy compared with its counterpart developed using the traditional cross - correlation function and without employing time - varying lags .
| 6,289 | 357 |
the hadronic particle - antiparticle correlation was already pointed out in the beginning of the nineties . however , the final formulation of these hadronic squeezed or back - to - back correlations was proposed only at the end of that decade@xcite , predicting that such correlations were expected if the masses of the mesons were modified in the hot and dense medium formed in high energy nucleus - nucleus collisions . soon after that , it was shown that analogous correlations would exist in the case of baryons as well . an interesting theoretical finding was that both the fermionic ( fbbc ) and the bosonic ( bbbc ) back - to - back correlations were very similar , both being positive and treated by analogous formalisms . in what follows , we will focus our discussion to the bosonic case , illustrating the effect by considering @xmath0 and @xmath1 pairs , considered to be produced at rhic energies@xcite . let us discuss the case of @xmath3-mesons first , which are their own antiparticles , and suppose that their masses are modified in hot and dense medium . naturally , they recover their asymptotic masses after the system freezes - out . therefore , the joint probability for observing two such particles , i.e. , the two - particle distribution , @xmath4 , can be factorized as @xmath5 $ ] , after applying a generalization of wick s theorem for locally equilibrated systems@xcite . the first term corresponds to the product of the spectra of the two @xmath3 s , @xmath6 , being @xmath7 and @xmath8 the free - particle creation and annihilation operators of scalar quanta , and @xmath9 means thermal averages . the second term contains the identical particle contribution and is represented by the square modulus of the chaotic amplitude , @xmath10 . together with the first term , it gives rise to the femtoscopic or hanbury - brown & twiss ( hbt ) effect . the third term , the square modulus of the squeezed amplitude , @xmath11 , is identically zero in the absence of in - medium mass - shift . however , if the particle s mass is modified , together with the first term it leads to the squeezing correlation function . the annihilation ( creation ) operator of the asymptotic , observed bosons with momentum @xmath12 , @xmath13 ( @xmath14 ) , is related to the in - medium annihilation ( creation ) operator @xmath15 ( @xmath16 ) , corresponding to thermalized quasi - particles , by the bogoliubov - valatin transformation , @xmath17 , where @xmath18 , @xmath19 . the argument , @xmath20 $ ] , is the _ squeezing parameter_. in terms of the above amplitudes , the complete @xmath0 correlation function can be written as c_2(k_1,k_2 ) = 1 + + , [ fullcorr ] where the first two terms correspond to the identical particle ( hbt ) correlation , whereas the first and the last terms represent the correlation function between the particle and its antiparticle , i.e. , the squeezed part . the in - medium modified mass , @xmath21 , is related to the asymptotic mass , @xmath22 , by @xmath23 , here assumed to be a constant mass - shift . the formulation for both bosons and fermions was initially derived for a static , infinite medium @xcite . more recently , it was shown@xcite in the bosonic case that , for finite - size systems expanding with moderate flow , the squeezed correlations may survive with sizable strength to be observed experimentally . similar behavior is expected in the fermionic case . in that analysis , a non - relativistic treatment with flow - independent squeezing parameter was adopted for the sake of simplicity , allowing to obtain analytical results . the detailed discussion is in ref . @xcite , where the maximum value of @xmath24 , was studied as a function of the modified mass , @xmath21 , considering pairs with exact back - to - back momentum , @xmath25 ( in the identical particle case , this procedure would be analogous to study the behavior of the intercept of the hbt correlation function ) . although illustrating many points of theoretical interest , this study in terms of the unobserved shifted mass and exactly back - to - back momenta was not helpful for motivating the experimental search of the bbc s . a more realistic analysis would involve combinations of the momenta of the individual particles , @xmath26 , into the average momentum of the pair , @xmath27 . since the maximum of the bbc effect is reached when @xmath28 , this would correspond to investigate the squeezed correlation function , @xmath29 , close to @xmath30 . for a hydrodynamical ensemble , both the chaotic and the squeezed amplitudes , @xmath31 and @xmath32 , respectively , can be written in a special form derived in @xcite and developed in @xcite . therefore , within a non - relativistic treatment with flow - independent squeezing parameter , the squeezed amplitude is written as in @xcite , i.e. , @xmath33 + 2 n^*_0 r_*^3 \exp\bigl[-\frac{(\mathbf{k}_1-\mathbf{k}_2)^2}{8 m _ * t}\bigr ] \exp \bigl[-\frac{im\langle u\rangle r(\mathbf{k_1 } + \mathbf{k_2})^2}{2 m _ * t_*}\bigr ] \exp\bigl[- \bigl ( \frac{1}{8 m _ * t _ * } + \frac{r_*^2}{2 } \bigr ) ( \mathbf{k_1 } + \mathbf{k_2})^2\bigr ] \bigl\ } $ ] , and the spectrum , as @xmath34 , where @xmath35 and @xmath36 @xcite . we adopt here @xmath37 . inserting these expressions into eq . ( [ fullcorr ] ) and considering the region where the hbt correlation is not relevant , we obtain the results shown in figure 1 . part ( a ) shows the squeezed correlation as a function of @xmath38 , for several values of @xmath39 . the top plot shows results expected in the case of a instant emission of the @xmath0 correlated pair . if , however , the emission happens in a finite interval , the second term in eq . ( [ fullcorr ] ) is multiplied by a reduction factor , in this case expressed by a lorentzian ( @xmath40 ^ -1 $ ] ) , i.e. , the fourier transform of an exponential emission . the result is shown in the plot in the middle of figure 1(a ) . we see that this represents a dramatic reduction in the signal , even though its strength is sizable for being observed experimentally . if the system expands with radial flow ( @xmath41 ) , the result is shown in the plot at the bottom of figure 1(a ) , again considering that the @xmath3 s are emitted during a finite period of time , @xmath42 fm / c . we see that , in the absence of flow , the squeezed correlation signal grows faster for higher values @xmath43 than the corresponding case in the presence of flow . however , this last one is stronger in all the investigated @xmath43 region , showing that the presence of radial flow enhances the signal . the sensitivity of the squeezed - pair correlation to the size of the region where the mass - shift occurs is shown in figure1(b ) for two values of radii , @xmath44 fm and @xmath45 fm , keeping @xmath46 gev / c fixed . the differences are reflected in the inverse width of the curves , plotted as a function of @xmath47 . in case of no in - medium mass modification , the squeezed correlation functions would be unity for all values of @xmath48 in both plots . fm ( top ) and @xmath45 fm ( bottom).,title="fig : " ] fm ( top ) and @xmath45 fm ( bottom).,title="fig : " ] in the case of the squeezed correlations of @xmath1 pairs , we show in figure 2(a ) results for the generated momenta of the pairs within the narrow interval @xmath49 mev / c , by plotting the squeezed correlation , @xmath50 versus @xmath21 and @xmath51 . for the kaons , we can fixe the value of the shifted mass to be @xmath52 mev , corresponding to one of the maxima in figure 2(a ) , and then procedure similarly to what was done in the @xmath0 case . the result is shown in figure 3 of ref.@xcite . also in this case the intensity of the squeezed correlation would be large enough to be searched for experimentally . next , we investigate how the behavior of the identical particle correlations could be affected in case of in - medium mass modification , since the femtoscopic correlation function also depends on the squeezing factor , @xmath53 . the hbt correlation function is obtained by inserting the chaotic amplitude , @xmath54 + n^*_0 r_*^3 ( |c_{_0}|^2+|s_{_0}|^2 ) \exp\bigl[-\frac{(\mathbf{k_1}+\mathbf{k}_2)^2}{8 m _ * t_*}\bigr ] \exp\bigl[-\bigl(\frac{im\langle u\rangle r}{2 m _ * t_*}\bigr)(\mathbf{k}_1 ^ 2-\mathbf{k}_2 ^ 2)\bigr ] \exp\bigl[-\bigl ( \frac{1}{8 m _ * t } + \frac{r^2_*}{2}\bigr)(\mathbf{k_1}-\mathbf{k_2})^2\bigr ] \bigr\ } , $ ] together with the expression for the spectrum , into eq.([fullcorr ] ) . we use the case of identical @xmath55 pairs as illustration , as seen in figure 2(b ) . the investigation is extended to both the cases of instant emission ( @xmath56 ) and finite emission ( @xmath57=2fm / c ) . in this figure , we can see the well - known result corresponding to the narrowing of the femtoscopic correlation function with increasing emission times , as well as the broadening the curve with flow in the absence of squeezing , as expected . however , if the squeezing originated in the mass - shift is present , its effects tend to oppose to those of flow ( for large @xmath58 , it practically cancels the broadening of the correlation function due to flow ) , another striking indication of mass - modification , even in hbt ! in the present work we suggest an effective way to search for the back - to - back squeezed correlations in heavy ion collisions at rhic , and later at lhc energies , by investigating the squeezed correlation function , @xmath29 , in terms of @xmath59 , for different values of @xmath60 . we showed that , in the presence of flow , the signal is stronger over the momentum regions analyzed in the plots , suggesting that flow may help to effectively discover the bbc signal experimentally . another important point that we find , within this simplified model and in the non - relativistic limit considered here , is that the squeezing would distort significantly the hbt correlation function as well , tending to oppose to the flow effects on those curves , practically neutralizing it for large values of @xmath58 . the analysis in terms of the variable @xmath61 would not be suited for a genuine relativistic treatment . in this case , however , a momentum variable could be constructed , as @xmath62 . in fact , it would be preferable to redefine this variable as @xmath63 , whose non - relativistic limit is @xmath64 , as discussed in ref.@xcite . finally , it is important to emphasize that all the effects and signals discussed here would exist only if the particles analyzed had their masses modified by interactions in the hot and dense medium . m. asakawa m , t. csrg and m. gyulassy _ phys . lett . _ * 83 * , 4013 ( 1999 ) . p. k. panda , t. csrg , y. hama , g. krein and sandra s. padula , _ phys b * 512 * , 49 ( 2001 ) . sandra s , padula , y. hama , g. krein , p. k. panda and t. csrg , _ phys . _ c * 73 * , 044906 ( 2006 ) . m. gyulassy , s. k. kaufmann , and l. w. wilson , phys . c * 20 * , 2267 ( 1979 ) . a. makhlin and yu . sinyukov , _ sov . phys . _ * 46 * , 354 ( 1987 ) ; yu . sinyukov , _ nucl . phys . _ a * 566 * , 589c ( 1994 ) . sandra s. padula , o. socolowski jr . , t. csrg , m. i. nagy , proc . quark matter 2008 , _ j. phys . g : nucl . part . phys . _ * 35 * , 104141 ( 2008 ) . sandra s. padula , o. socolowski jr . and danuce m. dudek , arxiv:0812.1784v1[nucl - th ] , to be published in the proc . of the xxxviii international symposium on multiparticle dynamics ( ismd 2008 )
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a novel type of correlation involving particle - antiparticle pairs was found out in the 1990 s .
currently known as squeezed or back - to - back correlations ( bbc ) , they should be present if the hadronic masses are modified in the hot and dense medium formed in high energy heavy ion collisions .
although well - established theoretically , such hadronic correlations have not yet been observed experimentally . in this phenomenological study
we suggest a promising way to search for the bbc signal , by looking into the squeezed correlation function of @xmath0 and @xmath1 pairs at rhic energies , as function of the pair average momentum , @xmath2 .
the effects of in - medium mass - shift on the identical particle correlations ( hanbury - brown & twiss effect ) are also discussed .
| 3,902 | 215 |
in the unification scheme of agn the difference between type 1 and type 2 agn is explained by angle - dependent circumnuclear obscuration of the accretion disk and broad - line region @xcite . this obscuring dusty medium commonly referred to as `` dust torus '' is optically and geometrically thick and probably extends from sub - parsec scales outward to several 10s of parsecs , or beyond for high luminosity objects . the dust in the torus absorbs the incident uv / optical radiation an re - emits the received energy in the infrared . observations have shown that type 1 agn show significantly more emission in the near - ir than type 2 agn for the same given intrinsic luminosity ( e.g. * ? ? ? this is consistent with the picture where the face - on view onto the torus in type 1 agn exposes the innermost hot dust to the observer . on the other hand in type 2 agn the torus is seen edge - on so that internal obscuration blocks the line - of - sight to the hot dust . owing to this effect , it is expected that for a given agn luminosity the infrared emission of type 1 agn is generally stronger than from type 2s . in the light of attempts at forming isotropic agn samples based on ir fluxes it seems important to know exactly how strong of a bias towards type 1s over type 2s may occur when invoking flux limits . moreover , probing the wavelength dependence of this anisotropy in the infrared has some constraining power on our understanding of how the torus obscures the agn . it may be possible to distinguish torus models where the dust is smoothly distributed from those where the dust is arranged in clouds ( e.g. * ? ? ? * ) : if the dust is smoothly distributed within the torus , a large degree of anisotropy is expected . if , however , the dust is arranged in clouds the anisotropy is expected to be smaller . a problem commonly encountered when studying agn samples in the local universe is a significant contribution of the host galaxy to the ir . this is related to the typical lower luminosity seyfert galaxies which dominate the nearby agn population . one way around this problem is the use of high - spatial resolution observations , as possible with the largest ground - based telescopes or interferometers , which are able to resolve out the host and isolate the agn emission ( for details see * ? ? ? * ) . however , it is difficult to set up representative samples owing to the observational limitations . another possibility is the use of high luminosity objects typically at higher redshift where the agn outshines the host galaxy by a large factor in the optical and near - ir . if pah features are absent , the agn most likely dominates the mid - ir wavelength region as well ( in our sample : host @xmath4 for wavelengths @xmath5 17@xmath2 ) . in this paper we aim at quantifying the wavelength dependence of the anisotropy of the agn emission in the infrared from @xmath1 . for that we use a ( nearly ) isotropically selected and complete sample of quasars and radio galaxies with hidden quasars at @xmath0 as recently presented in @xcite and described in sect . [ sec : sample ] . here we improve the analysis by using host - galaxy subtraction for the radio galaxies and use clumpy torus models for interpretation . in sect . [ sec : results ] we show the average seds of each of the subsamples which are representing obscured ( type 2 ) and unobscured ( type 1 ) agn . we further analyze the origin of the anisotropy by fitting extinction models and clumpy torus models to the observations in sect . [ sec : analysis ] . in sect . [ sec : obsaniso ] we discuss our results by comparing them to previous ir anisotropy estimates in literature . the results are summarized in sect . [ sec : summary ] . the object sample for this paper comprises all 3crr @xcite radio galaxies and quasars with @xmath6 this lobe dominated sample presents a well matched set of radio galaxies and quasars in terms of their intrinsic luminosity ( @xmath7erg / s for the quasars and @xmath8erg / s for the radio galaxies ; errors indicate standard deviation of the sample ) . the data used here have been presented previously in @xcite and @xcite and we refer the reader to these papers for further details on the source selection , data reduction , and building of the average seds . to summarize briefly , we obtained mid - ir photometry in all six filters from 3.6@xmath2 to 24@xmath2 and spectroscopy from 19@xmath2 to 38@xmath2 utilizing all three instruments onboard the _ spitzer _ space telescope @xcite . after the data reduction in a standard manner , the individual source seds were interpolated onto a common rest frame wavelength grid . the quasars and radio galaxies were then averaged into a mean sed for each class of objects . the individual seds ( including observed and interpolated photometry ) as well as the the average seds are presented in ( * ? ? ? * their figs.1 & 2 ) . for this paper , additional corrections have been applied to the radio galaxy data before the averaging process outlined above : at the shortest wavelengths considered here ( @xmath9@xmath10 m , rest frame ) the radio galaxy seds show contributions from the host galaxy . since we want to isolate the emission coming from the active nucleus , we have to correct for the stellar emission in these cases . this correction was performed by fitting the observed irac photometry with a combination of a moderately old ( 5 - 10gyr ) elliptical galaxy sed to represent the stellar emission ( taken from the grasil webpage ; @xcite ) and a hot black body for the agn dust whose temperature we allowed to vary ( see e.g. * ? ? ? * ; * ? ? ? the resulting black body temperatures in the radio galaxies range from 600 to 970k , with a median value of 860k . the fraction of host galaxy light contributing to the flux measured at the observed frame wavelengths 3.6 , 4.5 , 5.8 , and 8.0@xmath10 m was found to be 0.9 , 0.6 , 0.3 , and 0.1 , respectively . despite a negligible contribution from the host galaxy , we performed similar fits to the quasar sample as well to obtain characteristics of the hot dust emission and compare it to the radio galaxies . for the quasars we find notably hotter temperatures in the range from 8801250k ( median 1020k ) . this is consistent with the idea that the hottest dust is obscured in radio galaxies , while directly seen in quasars . in the radio galaxies we then subtracted the estimated host galaxy contribution from the observed irac photometry using the scaled template sed . for the irs and mips measurements the host galaxy contributions were considered negligible at restframe wavelength @xmath11 and no corrections have been applied . we refrain from further corrections related to possible starformation . as pointed out in @xcite neither the individual seds nor the averages showed any pah features ( see also fig . [ fig : aver_sed ] ) , which indicates that any contribution from starformation to the mid - ir is probably negligible . the radio galaxy 3c368 has a galactic m - star superimposed close to the position of the radio galaxy nucleus ( e.g. * ? ? ? * ; * ? ? ? both sources are partly blended even at the shortest irac wavelengths which makes the correction for the host galaxy emission in this source quite uncertain . consequently , we removed this source from the sample considered here . this leaves us with 11 quasars and 8 radio galaxies from which the average seds have been calculated . in fig . [ fig : aver_sed ] we show the average sed of @xmath13 quasars ( red ) and radio galaxies ( blue ) respectively . the error bars reflect the mean absolute deviation while the shaded areas show the range of the respective subsample at each wavelength point of the interpolated data ( see sect . [ sec : sample ] ) . for each of the object types we calculated a spline fit through the mean data points in order to guide the eye . it is obvious that the radio galaxies are systematically lower in infrared emission than the quasars . the discrepancy is largest in the near - ir and flattens out towards longer wavelengths . @xcite showed similar average seds . here we used additional host galaxy subtraction , which isolates the agn light much better ( see sect . [ sec : sample ] ) . this is most obvious in the near - ir part of the radio galaxies shortward of 5@xmath2 . the sed keeps on falling toward shorter wavelengths consistent with the wien tail from hot dust emission , instead of making an upward turn ( see * ? ? ? since both types have the same radio luminosities due to our selection , this difference between quasars (= type 1 agn ) and radio galaxies (= type 2 agn ) is a generic property of the sample . it either reflects a difference in line - of - sight extinction ( e.g. by cold dust in the host galaxy ) , or traces the anisotropy in re - emission of the agn - heated dust . @xcite tested the former possibility and found a surprisingly good match of the difference between radio galaxies and quasars in the mid - ir by a single extinction law . this , however , breaks down in the near - ir . in this paper we will test if , instead , a single absorber and emitter (= the dust torus ) may be responsible for the radio galaxy / quasar anisotropy ( see sect . [ sec : modcomp ] ) . to quantify the wavelength dependence of the anisotropy we plot the quasar / radio galaxy ratio in fig . [ fig : ratio ] . also shown is the mean absolute deviation of the ratio calculated by propagating the standard deviations of each subsample . from 2 to 8@xmath2 the emission ratio gradually decreases from 20 to 2 . in the silicate feature this ratio increases again up to about 3 and flattens out towards 15@xmath2 at a value of @xmath14 . our sample comprises a range of sed shapes , meaning that a range of ratios is observed . most of this sample range comes from the radio galaxies which show much less uniformity than the quasars ( see fig . [ sec : sample ] ) . we illustrated the range of ratios covered by the radio galaxies in fig . [ fig : range ] where we plot the wavelength dependence of the ratio of each radio galaxy using the average quasar sed and normalize it to 15@xmath2 . at around 10@xmath2 the radio galaxies show the silicate feature in absorption while quasars display a weak silicate emission feature . we used the spline fits described above ( see fig . [ fig : aver_sed ] ) to locate the centers of the silicate features following the method outlined by @xcite . the silicate absorption feature center is found to be at @xmath15 while the peak of the silicate emission feature was measured at @xmath16 . this difference in central wavelength is similar to the ones observed in local galaxies . as recent high spatial resolution studies of seyferts suggest , the `` shift '' in wavelength is not a pure radiative transfer effect due to the location or distribution of the dust , but implies a change of dust chemistry within the torus @xcite . as demonstrated by @xcite the silicate emission feature may be located at around 10.5@xmath2 if a fraction of the hot silicate dust consists of porous silicate grains . we note that the transition from quasars to radio galaxies is not as smooth as one may expect . in spite of some overlap in the range of seds of both types in the near - ir part in fig . [ fig : aver_sed ] , quasars generally show infrared emission characteristics expected for a type 1 agn ( blue ir color ; silicate emission feature ) and radio galaxies have ir seds with type 2 emission properties ( redder sed ; silicate absorption feature ) . in this section we analyze the scenarios that can lead to the observed anisotropy in radio galaxies and quasars . there are two possibilities : ( 1 ) extinction by a cold dust screen and ( 2 ) absorption and emission in a warm dusty medium . the former possibility may be associated with cold dust in the host galaxy while the latter one is equivalent to the dust torus in the agn unification scheme ( which we may call `` intrinsic anisotropy '' ) . we will first discuss the plausibility and consequences of a cold dust screen on the seds . if the ir anisotropy is dominated by cold host galaxy dust , then we have a situation where the ir emission originates from the torus while the absorption is coming from a different component ( i.e. dust in the host ) . this means that we require an additional component outside the agn to model the data ( e.g. as used for a significant minority of high-@xmath17 type 2 qsos in @xcite ) . however , the objects suffering extinction ( here : radio galaxies ) would be offset from quasars by only a single extinction law . this has been tested and ruled out by @xcite for the same set of objects as presented here . in fig . [ fig : ratio ] we show the anisotropy ratio between quasars and radio galaxies ( black circles with error bars ) . we overplot a standard ism extinction curve , resembling cold screen extinction , scaled to the observed 15@xmath2 anisotropy ( dark - blue dotted line ; using a mixture of 53% silicates and 47% graphite , based on updated dust opacity cuves by @xcite , and a grain size distribution according to @xcite ) . the extinction curve significantly overpredicts the anistropy in the silicate feature with better agreement in the near - ir . for reference we also plot the extinction curve based on ( * light - blue dotted line ; pixie dust ) , as used in @xcite , which results in better agreement within the silicate feature but significant offsets in the near - ir . in fact , @xcite pointed out that a good correspondence of quasars and radio galaxies can be achieved only if the quasar sed is attenuated by at least _ two _ instead of one extinction components which is reminiscent of radiative transfer (= absorption _ and _ emission ) within the torus rather than a cold screen . moreover , we found a mean anisotropy of about 1.4 at 15@xmath2 ( see sect . [ sec : results ] ) . dust opacity curves typically have opacity ratios @xmath18 . in order to obtain the observed anisotropy , the dust screen would have to have @xmath19 . while such optical depth values are in reach for galactic dust lanes ( e.g. extinction towards our own galactic center ) , it requires very edge - on views onto disk galaxies since the scale height of galactic disks is small . on statistical grounds this possibility may be viable for a minority of all radio galaxies , but it is unlikely that the whole population is dominated by host extinction . we note that the same line of argument can be made using the near - ir anisotropy leading to even higher @xmath20 and illustrating the need for more then just one cold extinction screen in this scenario . we use our clumpy torus models _ cat3d _ @xcite to test if the observed anisotropy ratio can be explained by the intrinsic anisotropy as predicted in the unification scheme . generally smooth dust torus models predict stronger anisotropy than clumpy models @xcite . @xcite argue that the observed small anisotropy in the mid - ir / x - ray correlation ( see sect . [ sec : obsaniso ] ) is qualitatively in agreement with a clumpy torus . here we aim at being more quantitative and show consistency of the observed anisotropy with torus orientation . in fig . [ fig : ratio ] we overplot the observed ratio with predicted ratios of a clumpy torus model ( red dashed line ) . as mean torus inclination in quasars we assumed 39@xmath21 while the mean radio galaxy inclination is set to 75@xmath21 . this corresponds to a mean opening angle of the torus of 60@xmath21 or a type 1/type 2 ratio of 1:1 , which is consistent with the number statistics of our complete and isotropic sample @xcite . [ fig : ratio ] shows that the model is following the continuum anisotropy curve quite well . there is , however , a slight deviation within the silicate feature where in the center of the feature the model curve predicts slightly lower anisotropy than shown by the observed curve . this may be indicative of additional , off - torus obscuration in some objects ( e.g. from the host galaxy ) , which effectively deepens silicate absorption features in radio galaxies ( the denominator of the plot in fig . [ fig : ratio ] ) and , to a lesser degree , changes the spectral slope . if this happens in individual objects , the average curve and scatter will tend to be slightly more anisotropic . some local examples of host obscuration are centaurus a , ngc 5506 , or the nucleus of the circinus galaxy where host - galactic dust lanes are projected onto the nucleus producing deep silicate features . in fact the silicate absorption feature in the average radio galaxy sed seems to be deeper than in typical local seyfert 2 galaxies without host obscuration @xcite . note that this requires a statistical alignment of cold host dust with that of the torus ( see sect . [ sec : extinct ] ) . the model used for reproducing the ratio uses a dust cloud distribution with radial power law @xmath22 and 5 clouds along the line - of - sight in equatorial direction ( for details see * ? ? ? * ) . in comparison to models for seds and mid - ir interferometry of local seyfert galaxies , these parameters suggest only a slightly more centrally condensed and transparent torus @xcite , while the half - opening angle may be wider ( 60@xmath21 instead of 45@xmath21 ) . in fact a range of torus model parameters satisfies the observed type 1/type 2 anisotropy spectrum within the error bars of the sample ( e.g. various steeper and shallower dust distributions ) . from bayesian inference analysis we found that the torus model parameters are generally poorly constrained ( broad posterior distributions for individual parameters ) . a weakness is certainly that modeling the flux ratio is not very constraining for model parameters since it does only take relative fluxes into account , while absolute fluxes ( e.g. actual silicate strength of type 1s or type 2s ) are not included . on the other hand , what the modeling shows is that the observed small ratios at long wavelengths and the change of anisotropy from the near- to the mid - ir are in agreement with expectations from clumpy torus models without fine - tuning parameters . in summary , the torus model seems to reproduce the observed anisotropy reasonably well over most of the wavelength range , while single extinction laws result in much worse fits . the model parameters used in the clumpy torus model fit are reasonable in comparison to fits to local agn . it demonstrates plausibility of the scenario that the anisotropy is related to torus orientation . this strongly suggests that the observed anisotropy is a measure for the intrinsic anisotropy of luminous type 1 and type 2 agn at @xmath12 . in sect . [ sec : analysis ] we argued that the observed ir anisotropy is probably reflecting the `` intrinsic anisotropy '' as caused by the dust torus . the isotropic and complete selection of the sample helps to minimize any biasing effects on the anisotropy . on the other hand , optically- and x - ray - selected samples often suffer from missing some of the most obscured objects . moreover , since the ratio is @xmath23 at 15@xmath2 , flux - limited mid - ir selected samples are potentially biased towards type 1 agn as well . thus , even using 12@xmath2-selected agn samples will slightly suffer from the assumption of isotropy . one popular way of comparing type 1s and type 2s is the correlation between x - ray and mid - ir luminosity . in case the x - luminosity is emitted isotropically ( and traces the dust - heating emission ) , and the mid - ir emission is radiated very anisotropically , this correlation is expected to be different for type 1 and type 2 agn . however , using high spatial resolution observations @xcite showed that local seyfert 1 and seyfert 2 galaxies , up to column densities of few @xmath24 @xmath25 , essentially follow the same correlation . a conservative estimate suggests that at 12@xmath2 the difference between both types is smaller than a factor of 3 . despite including some `` mildly '' compton - thick objects in this study , the most obscured objects are still missing due to the lack of intrinsic x - ray data . this essentially makes anisotropy estimates from the mid - ir / x - ray - correlation a lower limit on the `` true '' anisotropy . nevertheless this result is fully consistent with our finding for powerful @xmath12 agn . at 12@xmath2 we find a type 1/type 2 ratio of @xmath26 . assuming that x - ray selection misses the highest - inclination objects our result would predict that the anisotropy in the mid - ir / x - ray correlation is @xmath27 , at least for powerful agn as presented here . @xcite used a sample of local seyfert galaxies and compared the average 535@xmath2 sed of type 1s and type 2s , scaled to their respective 8.4ghz emission . however since the sources have been selected according to a flux limit at 12@xmath2 , the sample can not be considered isotropic . @xcite report generally higher fluxes for type seyfert 1 agn as compared to seyfert 2s . the anisotropy decreases from a factor of about 8 to @xmath32.5 from 5 to 8@xmath2 . this is significantly larger than what we find for our isotropically selected radio - loud sample . at longer wavelengths the seyfert 1/seyfert 2 ratio of @xcite flattens to about a factor of 23 ( with no convergence to unity as expected at long wavelengths ) which is , again , larger than what we found . the discrepancy between our results and @xcite may be either due to ( 1 ) the different selection criteria chosen , ( 2 ) a difference between low and high luminosity agn , or ( 3 ) a difference between radio - quiet and radio - loud agn . while the data in @xcite has not been corrected for host galaxy , the contribution by starformation should not be significant given the lack of pah features in the average spectrum . if a host - correction were applied , it would predominantly affect type 2 agn , thus making the anisotropy even larger . furthermore , if the mid - ir selection had any effect , then the sample would miss out agn at highest obscuration , so that the real type 1/type 2 anisotropy would again be larger . in conclusion , possible selection effects and host contamination in the @xcite study would tend to result in an underestimated anisotropy , making the difference to our findings even stronger . it is well possible that the difference in anisotropy between our high - luminosity radio - loud sample and the low - luminosity radio - quiet sample is real . this would imply that either luminosity or radio power are the drivers for the observed characteristics . radio jets are highly collimated so that any influence of the jet can be expected perpendicular to the torus plane and just in a very small solid angle . it is , therefore , more reasonable to assume that the higher agn luminosity would cause the lower anisotropy than the jet . the classical receding torus picture changes the opening angle of the torus for higher luminosity ( e.g. * ? ? ? * ; * ? ? ? this mainly affects the relative number of type 1s and type 2s in a sample . the relatively high fraction of about 50% unobscured agn in our sample would support this scenario . however , to change the anisotropy between both types , it would be necessary to also change the obscuration properties ( i.e. type 2s must on average look more like type 1s ) . such a scenario of `` radiation - limited obscuration '' has been proposed by @xcite and supported by observations of @xcite . in this case about the same effect is expected for radio - quiet and radio - loud agn . we use the sample of quasars and radio galaxies at @xmath12 recently presented in @xcite . since the sample was selected isotropically , it should cover all torus inclination angles ( weighted by solid angle ) . for these objects an infrared 117@xmath2 restframe sed has been constructed . average seds were calculated for the quasar (= type 1 agn ) and radio galaxy (= type 2 agn ) samples , respectively , in order to study the intrinsic anisotropy of the ir emission of the dust torus . it is shown that the ratio between type 1 and type 2 agn in our parameter space of very luminous radio galaxies , the value gradually decreases from 20 to 2 at wavelengths 2 to 8@xmath2 . within the 10@xmath2 silicate feature the ratio raises slightly . at longer wavelength the mid - ir emission becomes more isotropic . the intrinsic ratio between our type 1 and type 2 agn is @xmath14 at 15@xmath2 . when using ir - selected flux - limited samples this anisotropy has to be taken into account . by analyzing the silicate feature in the sample averages we find the well - established `` shift '' of the central peak of the silicate emission feature with respect to the center of the absorption feature . the resulting central wavelengths at @xmath15 for the absorption and @xmath16 for the emission feature are in agreement with previous reports ( e.g. * ? ? ? * ; * ? ? ? we discussed our results in the frame of other anisotropy estimators . our findings are consistent with upper limits derived from the x - ray / mid - ir correlation of local seyfert galaxies @xcite . some discrepancy exists with respect to a similar study of @xcite for nearby seyferts . if real it would imply that nearby , radio - quiet lower - luminosity agn show a higher degree of anisotropy in the ir than higher luminosity , radio - loud sources . this may be explained by a receding torus model with luminosity - dependent obscuration . we also show that the overall relatively small degree of anisotropy is consistent with the torus being clumpy rather than smooth . our clumpy torus model reproduces the observed type 1/type 2 ratio reasonably well . we would like to thank our referee prof . andy lawrence for helpful and constructive comments which significantly improved the paper , as well as poshak gandhi who also commented on this manuscript . the paper was made possible by deutsche forschungsgemeinschaft ( dfg ) in the framework of a research fellowship ( `` auslandsstipendium '' ) for sh . mh is supported by the nordrhein - westflische akademie der wissenschaften und der knste . this work is based on observations made with the spitzer space telescope , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . antonucci , r. 1993 , ara&a , 31 , 473 best , p. n. , longair , m. s. , & roettgering , j. h. a. 1997 , mnras , 292 , 758 buchanan , c. l. , gallimore , j. f. , odea , c. p. , baum , s. a. , axon , d. j. , et al . 2006 , aj , 132 , 401 chiar , j. e. , & tielens , a. g. g. m. 2006 , apj , 637 , 774 de breuck , c. , seymour , n. , stern , d. , willner , s. p. , eisenhardt , p. r. m. , et al . 2010 , apj , 725 , 36 draine , b. t. 2003 , apj , 598 , 1026 gandhi , p. , horst , h. , smette , a. , hnig , s. , comastri , a. , et al . 2009 , a&a , 502 , 457 haas , m. , willner , s. p. , heymann , f. , ashby , m. l. n. , fazio , g. , et al . 2008 , apj , 688 , 122 hammer , f. , proust , d. , & le fevre , o. 1991 , apj , 374 , 91 hnig , s. f. , & beckert , t. 2007 , mnras , 380 , 1172 hnig , s. f. , kishimoto , m. , gandhi , p. , smette , a. , asmus , d. , duschl , w. , polletta , m. , weigelt , g. 2010 , a&a , 515 , 23 hnig , s. f. , & kishimoto , m. 2010 , a&a , 523 , 27 laing , r. a. , riley , j. m. , & longair , m. s. 1983 , mnras , 204 , 151 lawrence , a. 1991 , mnras , 252 , 586 leipski , c. , haas , m. , meusinger , h. , siebenmorgen , r. , chini , r. , et al . 2005 , a&a , 440 , l5 leipski , c. , haas , m. , siebenmorgen , r. , meusinger , h. , albrecht , m. , et al . 2007 , 473 , 121 leipski , c. , haas , m. , willner , s. p. , ashby , m. l. n. , wilkes , b. j. , et al . 2010 , apj , 717 , 766 levenson , n. a. , radomski , j. t. , packham , c. , mason , r. e. , schaefer , j. j. , telesco , c. m. 2009 , apjl , 703 , 390 mathis , j. s. , rumpl , w. , & nordsieck , k. h. 1977 , apj , 217 , 425 polletta , m. , weedman , d. w. , hnig s. f. , lonsdale , c. j. , smith , h. e. , houck , j. r. 2008 , apj , 675 , 960 schartmann , m. , meisenheimer , k. , camenzind , m. , wolf , s. , tristram , k. r. w. , & henning , t. 2008 , a&a , 482 , 67 seymour , n. , stern , d. , de breuck , c. , vernet , j. , rettura , a. , et al . 2007 , apjs , 171 , 363 shi , y. , rieke , g. h. , hines , d. c. , gorjian , v. , werner , m. w. , et al . 2006 , apj , 653 , 127 silva , l. , granato , g.l . , bressan , a. , danese , l. , 1998 , apj , 509 , 103 simpson , c. 2005 , mnras , 360 , 565 sirocky , m. m. , levenson , n. a. , elitzur , m. , spoon , h. w. w. , armus , l. 2008 , apj , 678 , 729 smith , h. a. , li , a. , li , m. p. , khler , m. , ashby , m. l. n. , et al . 2010 , apj , 716 , 490 sturm , e. , schweitzer , m. , lutz , d. , contursi , a. , genzel , r. , et al . 2005 , apj , 629 , l21 treister , e. , virani , s. , gawiser , e. , urry , c. m. , lira , p. , et al . 2009 , apj , 693 , 1713 urry , c. m. , & padovani , p. 1995 , pasp , 107 , 803 werner , m. w. , et al . 2004 , apjs , 154 , 1
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we use restframe near- and mid - ir data of an isotropically selected sample of quasars and radio galaxies at @xmath0 , which have been published previously , to study the wavelength - dependent anisotropy of the ir emission . for
that we build average seds of the quasar subsample (= type 1 agn ) and radio galaxies (= type 2 agn ) from @xmath1 and plot the ratio of both average samples . from 2 to 8@xmath2 restframe wavelength
the ratio gradually decreases from 20 to 2 with values around 3 in the 10@xmath2 silicate feature .
longward of 12@xmath2 the ratio decreases further and shows some high degree of isotropy at 15@xmath2 ( ratio @xmath31.4 ) .
the results are consistent with upper limits derived from the x - ray / mid - ir correlation of local seyfert galaxies .
we find that the anisotropy in our high - luminosity radio - loud sample is smaller than in radio - quiet lower - luminosity agn which may be interpreted in the framework of a receding torus model with luminosity - dependent obscuration properties .
it is also shown that the relatively small degree of anisotropy is consistent with clumpy torus models .
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in this letter we address dynamical processes in highly ordered complex plasmas associated with _ spontaneous symmetry breaking_. spontaneous symmetry breaking ( ssb ) plays a crucial role in elementary particle physics but is also very common in classical physics @xcite . it happens whenever the system goes from a state which has a certain symmetry , e.g. rotational symmetry , into an ordered state , which does not have this symmetry anymore . in general , this state not necessarily has to be the ground ( vacuum ) state and the transition to the new state may or may not be associated with a phase transition . for example , in the case of magnetization the spins point all in one direction ( ordered state ) whereas above the curie temperature there is no preferred direction . another example from a mechanical system without phase transition is a vertical stick which bends under a sufficiently high force from above to one side breaking the rotational symmetry of the system without the force . different symmetries coexisting in the same phase , and symmetry transformations escorting phase transitions are widely spread in nature . for instance , the mechanisms of symmetry breaking are thought to be inherent in the molecular basis of life @xcite . ssb is also an important feature of elementary particle physics @xcite . the universe itself is believed to have experienced a cascade of symmetry - breaking phase transitions which broke the symmetry of the originally unified interaction giving rise to all known fundamental forces @xcite . symmetry effects are crucial either in 3d and 2d systems . chiral ( mirror - isomeric ) clusters @xcite , magic clusters of a new symmetry frozen - in by a solid surface @xcite , or dynamical symmetry breaking by the surface stress anisotropy of a two - phase monolayer on an elastic substrate @xcite are examples of the importance of 2d or quasi-2d systems in many applications . low pressure , low temperature plasmas are called _ complex plasmas _ if they contain microparticles as an additional thermodynamically active component . in the size domain of 1 - 10@xmath0 m ( normally used in experiments with complex plasmas ) these particles can be visualized individually , providing hence an atomistic ( kinetic ) level of investigations @xcite . the interparticle spacing can be of the order of 0.1 - 1 mm and characteristic time - scales are of the order of 0.01 - 0.1 s. these unique characteristics allow to investigate the microscopic mechanism of ssb and phase transitions at the kinetic level . common wisdom dictates that symmetry breaking is an inherent attribute of systems in an active state . hence these effects are naturally important in complex plasmas where the _ particle cloud - plasma _ feedback mechanisms underlying many dynamical processes are easy to vitalize . also in complex plasmas where different kind of phase transitions exist , e.g. in the elelectrorheological plasmas @xcite , one can find examples for classical ssb . another option , interesting in many applications , is the clustering of a new phase which is dissymmetric with regard to a background symmetry ( as an example of fluid phase separation in binary complex plasmas see @xcite ) . it is important to mention that the microparticles , collecting electrons and ions from the plasma background , become charged ( most often negatively @xcite ) and hence should be confined by external electric fields . the configuration of the confining forces might deeply affect the geometry and actual structure of the microparticle cloud . in rf discharge complex plasmas the particles are self - trapped inside the plasma because of a favorable configuration of the electric fields @xcite . one of the interesting things is the possibility to levitate a monolayer of particles under gravity conditions . in this case the particle suspension has a flat practically two dimensional structure . this is , of course , a very attractive simplification ( from a theoretical point of view ) , significantly lowering the description difficulties . below we concentrate mostly on 2d complex plasmas . depending on the discharge conditions , the monolayer can have crystalline or liquid order . 2d configurations of dust particles either in crystalline or liquid state were successfully used to study phase transitions , dynamics of waves and many transport phenomena in complex plasmas @xcite . a symmetry disordering escorting a crystalline - liquid phase transition has been investigated experimentally in @xcite . dislocation nucleation ( a shear instability ) has been reported in @xcite , albeit the importance of ssb for this phenomenon has not been explained . the results of these recent experimental observations can not be properly addressed without a deep understanding of this important issue . we would like to highlight this in the paper and report on the physics of spontaneous disordering of a cold plasma crystal , simulated melting and crystallization process , including associated defect clusters nucleation , dissociation , and symmetry alternation . these options are realizable in experimental complex plasmas , and can be mimicked in simulations , as we demonstrate below . it is well known that two broken symmetries distinguish the crystalline state from the liquid : the broken translational order and the broken orientational order . in two dimensions for ordinary crystals it is also well known that even at low temperatures the translational order is broken by spontaneous disordering mediated by thermal fluctuations @xcite . as a result , the fluctuation deflections ( disordering ) grow with distance and translational correlations decay ( algebraically , see @xcite ) . 2d plasma crystals also obey this common rule . the character of disordering may be deeply affected by the confinement forces , though . usually such an in - plane confinement is due to the bowl - shaped potential well self - maintained inside the discharge chamber , which to first order is approximately parabolic ( see , e.g. @xcite ) , that is @xmath1 , where @xmath2 is the distance , @xmath3 is the particle mass , and @xmath4 is the _ confinement parameter _ @xcite . ( the out - of - plane confining forces , controlling the position of the entire lattice , are normally much stronger ; below we consider the pure 2d - case , assuming , hence , an absolutely stiff out - of - plane confinement . ) the fluctuation spectra can be calculated in the following manner . the long - range phonon contribution to the free energy of a 2d system of particles interacting via the yukawa potential and confined by a shallow isotropic parabolic well can be conveniently represented as @xcite : @xmath5 @xmath6 where @xmath7 are the transverse ( shear wave ) and the longitudinal ( compressional wave ) sound speed , and @xmath8 are the fourier components of the vorticity @xmath9 and the divergency @xmath10 of the particle displacements @xmath11 , and @xmath12 is the wave vector ( @xmath13 ) . the unperturbed crystal is supposed to be hexagonal . the relationship ( [ eq.a ] ) provides ( see , e.g. @xcite ) the probability of the fluctuation @xmath14 . next , using it , we can calculate the averaged fluctuation spectral intensity per unit mass as @xmath15 where @xmath16 is the particle thermal velocity . it is known ( see @xcite ) that a lattice layer of a finite size @xmath17 is stably confined if roughly : @xmath18 here the parameter @xmath19 stands for an effective number of the nearest neighbors of any edge particle . note that according to ( [ eq.4 ] ) formally @xmath20 at @xmath21 . without confinement ( @xmath22 ) the fluctuation spectrum ( [ eq.3 ] ) apparently diverges @xmath23 at @xmath24 , and , as a consequence , in agreement with @xcite , the crystal ordering decays algebraically with the distance @xmath2 , i.e. the density - density correlation behaves as @xmath25 here @xmath26 , @xmath27 is the vector of the reciprocal lattice , and @xmath28 . it is assumed that @xmath29 is large compared to the interparticle separation @xmath30 . in the experiments @xmath31 is always finite ( though noticeably small , one or two orders of magnitude less than the frequency of the local caged oscillations of the individual particles @xcite ) . from ( [ eq.3 ] ) at non - vanishing @xmath4 it immediately follows that the fluctuations remain finite even at @xmath32 . this absence of a singularity alters the character of disordering from algebraic ( [ eq.5 ] ) to exponential at a scale depending on the confinement parameter : @xmath33 it is essential that both asymptotes algebraic and exponential must be treated as near - field ( @xmath34 ) and far - field ( @xmath35 ) approximations . hence it would be logical to assume that the ordering decay alternates with distance from algebraic to exponential . this is indeed in qualitative agreement with observations @xcite . remarkably ( [ eq.a])-([eq.3 ] ) are formally similar to the equations describing director fluctuations in nematic crystals in the presence of a magnetic field @xcite . the action of the magnetic field is known as suppressing the large - scale director fluctuations in liquid crystals . the length scale @xmath36 seems to be of a fundamental importance . the particles , experiencing a horizontal confinement , are distributed non - uniformly . the steady - state displacements of the particles @xmath37 in the plasma crystal from their ideal locations in a uniform 2d crystal represent a growing function with distance @xmath38 @xcite . the lattice breaks up when @xmath39 where @xmath40 is the lindemann parameter . since @xmath40=0.16 - 0.18 ( see , e.g. @xcite ) , it follows that the first row of defects most probably appears at @xmath41 . making use of ( [ eq.4 ] ) , ( [ eq.7 ] ) one can estimate the size of the domains ( or equivalent correlation length ) as : @xmath42 the correlation length ( [ eq.9 ] ) does not depend explicitly on the temperature . in other words , for purely topological reasons the big crystal spontaneously splits , assembling an array of sub - domains , even at zero temperature . the estimated values of @xmath43 agree well with those obtained in the simulation see fig . [ fig : cluster ] , and in experiments . for instance , it has been observed in @xcite that the crystal orientational order had a power law decay at distances @xmath44 in fairly good agreement with @xmath45 following from ( [ eq.9 ] ) . the one - plus correlation length ( [ eq.9 ] ) , unavoidably introducing a network of sub - domains to a lattice layer , is of crucial importance , e.g. , for observations of the so called _ hexatic state _ in the plasma crystals that is still an outstanding and controversial issue in complex plasma studies @xcite . one of the possible scenarios for melting ( recrystallization ) in a 2d complex plasma is a precipitous increase ( decrease ) in the density of the dislocations and the dislocation aggregates ( such as defect clusters , grain boundaries etc . ) @xcite . to realize this scenario in simulations , it is desirable to avoid any aforementioned complications associated with the lattice layer sectioning from the very beginning. a promising tool in that sense , allowing to create a defect - free initial lattice layer , is a hexagonal confinement cell proposed in @xcite . we performed a series of simulations that revealed several peculiarities in symmetry that are worth to mention . first , the order parameter of the paired defects dislocations , was systematically lower for 7-fold cells . this is not surprising actually from a purely geometric point of view because the 5-fold cell in a pair is more compact . second , simulations manifested that not only isolated pairs dislocations ( @xmath46@xmath47 ) , but also compact triplets like ( @xmath46@xmath47@xmath46 ) , quadruplets ( @xmath46@xmath47@xmath48@xmath49 ) etc . , or even elongated defect chains were quite frequent . actually they dominantly defined the symmetry of the entire particle suspension . it would certainly be promising to connect the cluster formation in ordered complex plasmas @xcite with the general percolation process known in many similar applications ( see , e.g. , @xcite ) . third , in such melted clusters , in agreement with recent experimental observations @xcite , the defect density permanently decreased upon cooling . at higher temperatures in the beginning of the recrystallization process , while the mutual interparticle collisions were still frequent , the defect density dropped exponentially . then , at lower temperatures , the decay rate significantly slowed down ( see fig.[fig : defect ] ) . a sharp drop in the defect numbers followed by a quasi - saturation resembles the well - known situation @xcite in which both thermal activation and tunneling events occur . hence , by analogy , the fact that in our case the system of defects behaves in a similar way could be naturally explained by an annihilation scenario which is presumably of the _ dissipative tunneling _ type @xcite at lower mean kinetic energies . nucleation of dislocations is another important example of spontaneous symmetry breaking on a scale of elementary cells . whatever the melting scenario would be true , still there would remain a question what mechanism explains nucleation of the primary dislocation clusters . recently this issue has been studied experimentally : spontaneous nucleation of the edge - dislocation pairs ( followed by their dissociation ) has been successfully observed at the kinetic level in the experiments with plasma crystals @xcite . since the burgers vector of the entire lattice is kept constant ( e.g. zero ) spontaneously created dislocations must be paired forming defect quadruplets of the type ( @xmath46@xmath47@xmath48@xmath49 ) . ( the burgers vector characterizes the magnitude and direction of the crystalline lattice distortion by a dislocation @xcite . ) these dislocation clusters were created in the lattice locations where the internal shear stress exceeded a threshold . it has also been shown that even an elementary act of nucleation is in fact a multi - scale process consisting of the latent pre - phase , prompt nucleation of a defect cluster , and dissociation of the cluster followed by the escape of free dislocations @xcite . in the experiments @xcite it was suggested that the stress that finally caused nucleation was affected by the differential crystal rotation . the exact reason of nucleation , however , was difficult to determine consistently . in simulations the nucleation conditions are certainly easier to identify . to demonstrate nucleation in simulations a deformable hexagonal cell is used ( fig.[fig : nucleation ] ) . it confines a 2d cloud of equally charged particles interacting pairwise via the yukawa ( the screened coulomb ) force : @xmath50 where @xmath51 is the relative coordinate and @xmath52 is the distance between the particles @xmath53 with the coordinates @xmath54 ; @xmath55 is the particle charge and @xmath56 is the screening length . the cell design is similar to that applied in @xcite to simulate melting and recrystallization process of the plasma crystal . the hexagonal simulation cell has the evident advantage of flexible shape , compared to , e.g. , a parabolic cell confinement . deforming the boundary of the cell , it is simple to manipulate the particles in a tractable way . an additional option of variable geometry enables an opportunity to separate or consolidate pure shear and simple shear deformation @xcite if desirable . the strain rate is controllable during deformation as well . [ fig : nucleation ] shows a simple - sheared particle lattice layer . at a properly chosen loading rate deformation affects the _ shear instability _ that ends up with nucleation of defect clusters in the bulk of the lattice layer . after a while , when deformation becomes stronger , the components of the clusters decoupled and the newly born free dislocations glided away in a similar manner as the dislocations observed in experiments . symmetry alternation is of primary importance for understanding nucleation of dislocation clusters . the compact cluster design is _ magic _ in the sense that the _ hexagonal _ symmetry of the particle system neatly turns into a nearly _ tetratic _ symmetry of the cluster core ( like lead turns into gold when touched by the philosophers stone ) , see fig . [ fig : topology ] . despite an apparent simplicity of the cluster interior only four nearest neighbor particles ( marked abcd in fig . [ fig : topology ] ) , the centers of the 5- and 7-fold cells , are in the core , to discover the cluster topology was certainly a challenge @xcite . in our case the interparticle interaction potential is of the screened yukawa type , hence more compact in contrast to the @xmath57 interaction potential in case of magnetically interacting super - paramagnetic colloid particles considered in @xcite . thus there is a unique opportunity to verify whether the core topology is universal . let us start with a simple model treating the cluster as constituted of two point - like dislocations , which are set apart at a distance @xmath2 and allowed to glide only along two fixed crystallographic planes separated by one lattice period @xmath30 , so that @xmath58 , where @xmath59 is the angle of mutual orientation of the cluster components with respect to the gliding plane . the interaction energy of the point dislocations having the counter - directed burgers vectors is @xcite : @xmath60+const.\ ] ] it has a minimum , a _ stable ground state _ , at @xmath61 . it corresponds to @xmath62 , hence tetragonal symmetry of the cluster core might be considered as preferred . this prediction agrees noticeably well with the results of simulations of finite clusters : on average in fig . [ fig : topology ] the edge - to - diagonal angle in the cluster core is @xmath63 . it is worth noting that the cluster core is nearly cyclic . a measure of it immediately follows from the famous ptolemy s inequality valid for any quadrilateral : @xmath64 where @xmath65 denote the ( ordered ) sides , and @xmath66 are the diagonals of the quadrilateral . over 80 % of the recognized clusters have @xmath67 for the simulation results shown in fig . [ fig : topology ] . for comparison a hexagonal four - side cell corresponds to @xmath68 . note also that a stable defect cluster could not be obtained only by shifting positions of four central particles from a hexagonal configuration to tetragonal one . such deformation would be reversible , hence unstable . a weakly deformed environment , impeding relaxation of the core particles back to the stable hexagonal configuration , is indeed a necessary lock making the deformation plastic , i.e. irreversible . at sufficiently strong external stress even a stable cluster dissociates . whatever is the orientation of the cluster as a whole , escaping dislocations can glide only along two crystallographic directions ( along burgers vectors , see fig . [ fig : topology ] ( a , b ) ) . this naturally explains the asymmetry of the escape directions and chirality of the defect configurations revealed by the newly nucleated dislocations in experiments @xcite . spontaneous symmetry breaking is a common and inherent feature of many systems in physics as well as other fields , it plays an important role , for example , from classical one - component plasmas to modern string representations @xcite , from the evolution of the early universe @xcite to the dynamics of a wide variety of small - scale systems @xcite . therefore it is not surprising that ssb is present also in the physics of plasma crystals . as example we have considered dislocations in plasma crystals which exhibit a spontaneous disordering , involved in the process of melting , and form clusters that are caused by a shear instability and show an interesting topological symmetry . the authors appreciate valuable discussions with dr . ivlev and dr . nosenko . k. r. stterlin , a. wysocki , a. v. ivlev , c. rth , h. m. thomas , m. rubin - zuzic , w. j. goedheer , v. e. fortov , a. m. lipaev , v. i. molotkov , o. f. petrov , g. e. morfill , and h. lwen , phys . lett . * 102 * , 085003 ( 2009 ) .
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spontaneous symmetry breaking is an essential feature of modern science .
we demonstrate that it also plays an important role in the physics of complex plasmas .
complex plasmas can serve as a powerful tool for observing and studying discrete types of symmetry and disordering at the kinetic level that numerous many - body systems exhibit .
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in the present almost frenetic rate of advance of cosmology it is useful to be reminded that the big news this year is the establishment of evidence , by two groups ( @xcite , @xcite ) , of detection of the relativistic curvature of the redshift - magnitude relation . the measurement was proposed in the early 1930s . compare this to the change in the issues in particle physics since 1930 . the slow evolution of cosmology has allowed ample time for us to lose sight of which elements are reasonably well established and which have been adopted by default , for lack of more reasonable - looking alternatives . thus i think it is appropriate to devote a good part of my assigned space to a discussion of what might be included in the standard model for cosmology . i then comment on additions that may come out of work in progress . main elements of the model are easily listed : in the large - scale average the universe is close to homogeneous , and has expanded in a near homogeneous way from a denser hotter state when the 3 k cosmic background radiation was thermalized . the standard cosmology assumes conventional physics , including general relativity theory . this yields a successful account of the origin of the light elements , at expansion factor @xmath0 . light element formation tests the relativistic relation between expansion rate and mass density , but this is not a very searching probe . the cosmological tests discussed in 3 could considerably improve the tests of general relativity . the model for the light elements seems to require that the mass density in baryons is less than that needed to account for the peculiar motions of the galaxies . it is usually assumed that the remainder is nonbaryonic ( or acts that way ) . our reliance on hypothetical dark matter is an embarrassment ; a laboratory detection would be exceedingly welcome . in the past decade many discussions assumed the einstein - de sitter case , in which there are negligibly small values for the curvature of sections of constant world time and einstein s cosmological constant @xmath1 ( or a term in the stress - energy tensor that acts like one ) . this is what most of would have chosen if we were ordering . but the evidence from the relative velocities of the galaxies has long been that the mass density is less than the einstein - de sitter value @xcite , and other more recent observations , notably the curvature of the redshift - magnitude relation ( @xcite , @xcite ) , point in the same direction . now there is increasing interest in the idea that we live in a universe in which the dominant term in the stress - energy tensor acts like a decaying cosmological constant ( @xcite - @xcite ) . this is not part of the standard model , of course , but as discussed in 3 the observations seem to be getting close to useful constraints on space curvature and @xmath1 . we have good reason to think structure formation on the scale of galaxies and larger was a result of the gravitational growth of small primeval departures from homogeneity , as described by general relativity in linear perturbation theory . the adiabatic cold dark matter ( acdm ) model gives a fairly definite and strikingly successful prescription for the initial conditions for this gravitational instability picture , and the acdm model accordingly is widely used in analyses of structure formation . but we can not count it as part of the standard model because there is at least one viable alternative , the isocurvature model mentioned in 3.3 . observations in progress likely will eliminate at least one , perhaps establish the other as a good approximation to how the galaxies formed , or perhaps lead us to something better . the observational basis for this stripped - down standard model is reviewed in references @xcite and @xcite . here i comment on some issues now under discussion . pietronero @xcite argues that the evidence from redshift catalogs and deep galaxy counts is that the galaxy distribution is best described as a scale - invariant fractal with dimension @xmath2 . others disagree ( @xcite , @xcite ) . i am heavily influenced by another line of argument : it is difficult to reconcile a fractal universe with the isotropy observed in deep surveys ( examples of which are illustrated in figs . 3.7 to 3.11 in @xcite and are discussed in connection with the fractal universe in pp . 209 - 224 in @xcite ) . -1.0truecm fig . 1 shows angular positions of particles in three ranges of distance from a particle in a fractal realization with dimension @xmath3 in three dimensions . at @xmath3 the expected number of neighbors scales with distance @xmath4 as @xmath5 , and i have scaled the fraction of particles plotted as @xmath6 to get about the same number in each plot . the fractal is constructed by placing a stick of length @xmath7 , placing on either end the centers of sticks of length @xmath8 , where @xmath9 , with random orientation , and iterating to smaller and larger scales . the particles are placed on the ends of the shortest sticks in the clustering hierarchy . this construction with @xmath10 ( and some adjustments to fit the galaxy three- and four - point correlation functions ) gives a good description of the small - scale galaxy clustering @xcite . the fractal in fig . 1 , with @xmath3 , the dimension pietronero proposes , does not look at all like deep sky maps of galaxy distributions , which show an approach to isotropy with increasing depth . this can not happen in a scale - invariant fractal : it has no characteristic length . a characteristic clustering length for galaxies may be expressed in terms of the dimensionless two - point correlation function defined by the joint probability of finding galaxies centered in the volume elements @xmath11 and @xmath12 at separation @xmath13 , dp = n^2[1+_gg(r)]dv_1dv_2 . [ eq : xigg ] the galaxy two - point function is quite close to a power law , = ( r_o / r)^,= 1.77 , 10hr10 , [ eq : xiggparameters ] where the clustering length is r_o=4.50.5 , and the hubble parameter is h_o=100h^-1 ^ -1 . the rms fluctuation in galaxy counts in a randomly placed sphere is @xmath14 at sphere radius @xmath15 mpc , to be compared to the hubble distance ( at which the recession velocity approaches the velocity of light ) , @xmath16 mpc . the isotropy observed in deep sky maps is consistent with a universe that is inhomogeneous but spherically symmetric about our position . there are tests , as discussed by paczyski and piran @xcite . for example , we have a successful theory for the origin of the light elements as remnants of the expansion and cooling of the universe through @xmath17 mev @xcite . if there were a strong radial matter density gradient out to the hubble length we could be using the wrong local entropy per baryon , based on conditions at the hubble length where the cbr came from , yet the theory seems to be successful . but to most people the compelling argument is that distant galaxies look like equally good homes for observers like us : it would be startling if we lived in one of the very few close to the center of symmetry . mandelbrot @xcite points out that other fractal constructions could do better than the one in fig . 1 . his example does have more particles in the voids defined by the strongest concentrations in the sky , but it seems to me to share the distinctly clumpy character of fig . 1 . it would be interesting to see a statistical test . a common one expands the angular distribution in a given range of distances in spherical harmonics , a_l^m = d()y_l^m ( ) , [ eq : alm ] where @xmath18 is the surface mass density as a function of direction @xmath19 in the sky . the integral becomes a sum if the fractal is represented as a set of particles . a measure of the angular fluctuations is e_l = l_-l < m < l |a_l^m|^2/(a_0 ^ 0)^2 , where ^2/^2 - 1 = _ l 1 e_l / l . [ eq : variance ] in the approximation of the sum as an integral @xmath20 is the contribution to the variance of the angular distribution per logarithmic interval of @xmath21 . it will be recalled that the zeros of the real and imaginary parts of @xmath22 are at separation @xmath23 in the shorter direction , except where the zeros crowd together near the poles and @xmath22 is close to zero . thus @xmath20 is the variance of the fractional fluctuation in density across the sky on the angular scale @xmath24 and in the chosen range of distances from the observer . i can think of two ways to define the dimension of a fractal that produces a close to isotropic sky . first , each octant of a full sky sample has half the diameter of the full sample , so one might define @xmath25 by the fractional departure of the mean density within each octant from the mean in the full sample , ( e_2)^1/2~2 ^ 3-d - 1.[eq : e_2 ] thus in fig . 1 , with @xmath3 , the quadrupole anisotropy @xmath26 is on the order of unity . second , one can use the idea that the mean particle density varies with distance @xmath13 from a particle as @xmath27 . then the small angle ( large @xmath21 ) limber approximation to the angular correlation function @xmath28 is @xcite 1+w(=/l)~^l e_l dl / l _ 0 ^ 1du [ u^2 + ( /l)^2]^-(3-d)/2 . [ eq : e_l ] to find @xmath20 differentiate with respect to @xmath21 . at @xmath3 this gives @xmath29 : the surface density fluctuations are independent of scale . at @xmath30 , @xmath31 . the x - ray background fluctuates by about @xmath32 at @xmath33 , or @xmath34 . this is equivalent to @xmath35 in the fractal model in eq . ( [ eq : e_l ] ) . the universe is not exactly homogeneous , but it seems to be remarkably close to it on the scale of the hubble length . it would be interesting to know whether there is a fractal construction that allows a significantly larger value of @xmath36 for given @xmath20 than in this calculation . expansion that preserves homogeneity requires that the mean rate of change of separation of pairs of galaxies with separation @xmath4 varies as the hubble law , v = hr.[eq : hl ] the redshift - distance relation for type ia supernovae gives an elegant demonstration of this relation ( @xcite , @xcite ) . arp ( @xcite , @xcite ) points out that such precision tests do not directly apply to the quasars , and he finds fascinating evidence in sky maps for associations of quasars with galaxies at distinctly lower redshifts . but there is a counterargument , along lines pioneered by bergeron @xcite , as follows . a quasar spectrum may contain absorption lines characteristic of a cloud of neutral atomic hydrogen at surface density @xmath37 atoms @xmath38 . if this absorption system is at redshift @xmath39 a galaxy at the same redshift is close enough that there is a reasonable chance observing it , and with high probability an optical image does show a galaxy close to the quasar and at the redshift of the absorption lines ( @xcite , @xcite ) . also , when a galaxy image appears in the sky close to a quasar at higher redshift then with high probability the quasar spectrum has absorption lines at the redshift of the galaxy . we have good evidence the galaxy is at the distance indicated by its redshift . we can be sure the quasar is behind the galaxy : the quasar light had to have passed through the galaxy to have produced the absorption lines . if quasars were not at their cosmological distances we ought to have examples of a quasar appearing close to the line of sight to a lower redshift galaxy and without the characteristic absorption lines produced by the gas in and around the galaxy . arp s approach to this issue is important , but i am influenced by what seems to be this direct and clear interpretation of the bergeron effect , that indicates redshift is a good measure of distance for quasars as well as galaxies . in the relativistic friedmann - lematre cosmological model the wavelength of a freely propagating photon is stretched in proportion to the expansion factor from the epoch of emission to detection : 1+z=_obs_em = a_obsa_em . [ eq : redshift ] the first expression defines the redshift @xmath40 in terms of the ratio of observed wavelength to wavelength at emission . the cosmological expansion parameter @xmath41 is proportional to the mean distance between conserved particles . the most direct evidence that the redshift is a result of expansion is the thermal spectrum of the cbr @xcite . in a tired light model in a static universe the photons suffer a redshift that is proportional to the distance travelled , but in the absence of absorption or emission the photon number density remains constant . in this case a significant redshift makes an initially thermal spectrum distinctly not thermal and inconsistent with the measured cbr spectrum . one could avoid this by assuming the mean free path for absorption and emission of cbr photons is much shorter than the hubble length , so relaxation to thermal equilibrium is much faster than the rate of distortion of the spectrum by the redshift . but this opaque universe is quite inconsistent with the observation of radio galaxies at redshifts @xmath42 at cbr wavelengths . that is , the universe can not have an optical depth large enough to preserve a thermal cbr spectrum in a tired light model . in the standard world model the expansion has two effects : it redshifts the photons , as @xmath43 , and it dilutes the photon number density , as @xmath44 . the result is to cool the cbr while keeping its spectrum thermal . thus the expanding universe allows a self - consistent picture : the cbr was thermalized in the past , at a time when when the universe was denser , hotter , and optically thick . i have not encountered any serious objection to this argument ; the issue is the expansion factor . in the relativistic friedmann - lematre model the expansion of the universe traces back at least as far as redshift @xmath0 , when the light elements formed in observationally reasonable amounts @xcite . in the model of arp _ et al . _ @xcite the expansion and cooling traces back to a redshift only moderately greater than the largest observed values , @xmath45 , when there would have been a burst of creation of matter and radiation followed by rapid clearing of the dust that thermalized the radiation . the arp _ _ picture for the origin of the light elements has not been widely debated . if it were agreed that it is viable then a choice between this and the friedmann - lematre model would depend on other tests , such as the angular fluctuations in the cbr , as discussed next . -1.0truecm the tests in table 1 are organized in four categories : spacetime geometry , galaxy peculiar velocities , structure formation , and early universe physics . i offer grades for three sets of parameter choices . as the tests improve we may learn that one narrowly constrained set of values of the cosmological parameters receives consistent passing grades , or else that we have to cast our theoretical net more broadly . in the relativistic friedmann - lematre cosmological model the mean spacetime geometry ( ignoring curvature fluctuations produced by local mass concentrations in galaxies and systems of galaxies ) may be represented by the line element ds^2 = dt^2 - a(t)^2 , [ eq : lineelement ] where the expansion rate satisfies the equation h^2 = ( aa ) ^2 = 83 g + 3 , which might be approximated as h^2= h_o^2[(1+z)^3 + ( 1+z)^2 + ] . [ eq : cos_pars ] the last equation defines the fractional contributions to the square of the present hubble parameter @xmath46 by matter , space curvature , and the cosmological constant ( or a term in the stress - energy tensor that acts like one ) . the time - dependence assumes pressureless matter and constant @xmath1 . other notations are in the literature ; a common practice in the particle physics community to add the matter and @xmath1 terms in a new density parameter , @xmath47 . i prefer keeping them separate , because the observational signatures of @xmath19 and @xmath48 can be quite different . by 1930 people understood how one would test the space - time geometry in these equations , and as i mentioned there is at last direct evidence for the detection of one of the effects , the curvature of the relation between redshift and apparent magnitude ( @xcite , @xcite ) . as indicated in line 1b , the measured curvature is inconsistent with the einstein - de sitter model in which @xmath49 and @xmath50 . the measurements also disagree with a low density model with @xmath51 , though the size of the discrepancy approaches the size of the error flags , so i assign a weaker failing grade for this case . the measurements are magnificent . the issue yet to be thoroughly debated is whether the type ia supernovae observed at redshifts @xmath52 are drawn from essentially the same population as the nearer ones . in a previous volume in this series krauss @xcite discusses the time - scale issue . stellar evolution ages and radioactive decay ages do not rule out the einstein - de sitter model , within the still considerable uncertainties in the measurements , but the longer expansion time scales of the low @xmath19 models certainly relieve the problem of interpretation of the measurements . thus i enter a tentative negative grade for the einstein - de sitter model in line 1a . in the analysis by falco _ _ @xcite of the rate of lensing of quasars by foreground galaxies ( line 1d ) for a combined sample of lensing events detected in the optical and radio , the @xmath53 bound on the density parameter in a cosmologically flat ( @xmath54 ) universe is @xmath55 . the sneia redshift - magnitude relation seems best fit by @xmath56 , @xmath57 , a possibly significant discrepancy . a serious uncertainty in the analysis of the lensing rate is the number density of early - type galaxies in the high surface density branch of the fundamental plane at luminosities @xmath58 , the luminosity of the milky way . if further tests confirm an inconsistency of the lensing rate and the redshift - magnitude relation the lesson may be that @xmath48 is dynamical , rolling to zero , as ratra & quillen @xcite point out . the relation between the mass density parameter @xmath19 and the gravitational motions of the galaxies is an issue rich enough for a separate category in table 1 . it has been known for the past decade that if galaxies were fair tracer of mass then the small - scale relative velocities of the galaxies would imply that @xmath19 is well below unity @xcite . if the mass distribution were smoother than that of the galaxies , the smaller mass fluctuations would require a larger mean mass density to gravitationally produce the observed galaxy velocities . davis , efstathiou , frenk & white @xcite were the first to show that such a biased distribution of galaxies relative to mass readily follows in numerical n - body simulations of the growth of structure , and the demonstration has been repeated in considerable detail ( @xcite , @xcite , and references therein ) . this is a serious argument for the biasing effect . but here are three arguments for the proposition that galaxies are fair tracers of mass for the purpose of estimating @xmath19 . first , in many numerical simulations dwarf galaxies are less strongly clustered than giants . this is reasonable , for if much of the mass were in the voids defined by the giant galaxies , as required if @xmath59 , then surely there would be remnants of the suppressed galaxy formation in the voids , irregular galaxies that bear the stigmata of a hostile early environment . the first systematic redshift survey showed that the distributions of low and high luminosity galaxies are strikingly similar @xcite . no survey since , in 21-cm , infrared , ultraviolet , or low surface brightness optical , has revealed a void population . there is a straightforward interpretation : the voids are nearly empty because they contain little mass . second , one can use the galaxy two - point correlation function in eq . ( [ eq : xigg ] ) and the mass autocorrelation function @xmath60 from a numerical simulation of structure formation to define the bias function b(r , t ) = ^1/2 . [ eq : bias ] in numerical simulations @xmath61 typically varies quite significantly with separation and redshift @xcite . that is , the galaxies give a biased representation of the statistical character of the mass distribution in a typical numerical simulation . the issue is whether the galaxies , or the models , or both , are biased representations of the statistical character of the real mass distribution . what particularly strikes me is the observation that the low order galaxy correlation functions have some simple properties . the galaxy two - point function is close to a power law over some three orders of magnitude in separation ( eq . [ eq : xiggparameters ] ) . the value of the power law index @xmath62 changes little back to redshift @xmath63 . within the clustering length @xmath64 the higher order correlation functions are consistent with a power law fractal . a reasonable presumption is that the regularity exhibited by the galaxies reflects a like regularity in the mass , because galaxies trace mass . i am impressed by the power of the numerical simulations , and believe they reflect important aspects of reality , but do not think we should be surprised if they do not fully represent other aspects , such as relatively fine details of the mass distribution . -1.0truecm the third argument deals with the idea that blast waves or radiation from the formation of a galaxy may have affected the formation of nearby galaxies , producing scale - dependent bias . in this case the apparent value of the density parameter derived from gravitational motions within systems of galaxies on the assumption galaxies trace mass would be expected to vary with increasing scale , approaching the true value when derived from relative motions on scales larger than the range of influence of a forming galaxy . 2 shows a test . the abscissa at the entry for clusters of galaxies is the comoving radius of a sphere that contains the mass within the abell radius . the estimates at larger scales are plotted at approximate values of the radius of the sample . if it were not for the last two points at the right - hand side of fig . 2 , one might conclude that the apparent density parameter is increasing to the true value @xmath65 at @xmath66 mpc . but considering the last two points , and the sizes of the error flags , it is difficult to see any evidence for scale - dependent bias . i assign a strongly negative grade for the einstein - de sitter model in line 2a in table 1 , based on galaxy motions on relatively small scales , because biasing certainly is required if @xmath49 and i have argued there is no evidence for it . the more tentative grade in line 2b is based on fig . 2 : the apparent value of the density parameter does not seem to scale with depth . the friedmann - lematre model is unstable to the gravitational growth of departures from a homogeneous mass distribution . the present large - scale homogeneity could have grown out of primeval chaos , but the initial conditions would be absurdly special . that is , the friedmann - lematre model requires that the present structure the clustering of mass in galaxies and systems of galaxies grew out of small primeval departures from homogeneity . the consistency test for an acceptable set of cosmological parameters is that one has to be able to assign a physically sensible initial condition that evolves into the present structure of the universe . the constraint from this consideration in line 3c is discussed by white _ @xcite , and in line 3b by bahcall _ ( @xcite , @xcite ) . here i explain the cautious ratings in line 3a . as has been widely discussed , it may be possible to read the values of @xmath19 and other cosmological parameters from the spectrum of angular fluctuations of the cbr ( @xcite and references therein ) . this assumes nature has kept the evolution of the early universe simple , however , and we have hit on the right picture for its evolution . we may know in the next few years . if the precision measurements of the cbr anisotropy from the map and planck satellites match in all detail the prediction of one of the structure formation models now under discussion it will compel acceptance . but meanwhile we should bear in mind the possibility that nature was not kind enough to have presented us with a simple problem . -1.0truecm an example of the possible ambiguity in the interpretation of the present anisotropy measurements is shown in fig . the two models assume the same dynamical actors cold dark matter ( cdm ) , baryons , three families of massless neutrinos , and the cbr but different initial conditions . in the adiabatic model the primeval entropy per conserved particle number is homogeneous , the space distribution of the primeval mass density fluctuations is a stationary random process with the scale - invariant spectrum @xmath67 , and the cosmological parameters are @xmath68 , @xmath69 , and @xmath70 ( following @xcite ) . the isocurvature initial condition in the other model is that the primeval mass distribution is homogeneous there are no curvature fluctuations and structure formation is seeded by an inhomogeneous composition . in the model shown here the primeval entropy per baryon is homogeneous , to agree with the standard model for light element production , and the primeval distribution of the cdm has fluctuation spectrum p(k)k^m , m = -1.8 . [ eq : piso ] the cosmological parameters are @xmath71 , @xmath72 , and @xmath73 . the lower density parameter produces a more reasonable - looking cluster mass function for the isocurvature initial condition @xcite . in both models the density parameter in baryons is @xmath74 , the rest of @xmath19 is in cdm , and space sections are flat ( @xmath75 ) . both models are normalized to the large - scale galaxy distribution . the adiabatic initial condition follows naturally from inflation , as a remnant of the squeezed field that drove the rapid expansion . a model for the isocurvature condition assumes the cdm is ( or is the remnant of ) a massive scalar field that was in the ground level during inflation and became squeezed to a classical realization . in the simplest models for inflation this produces @xmath76 in eq . ( [ eq : piso ] ) . the tilt to @xmath77 requires only modest theoretical ingenuity @xcite . that is , both models have pedigrees from commonly discussed early universe physics . the lesson from fig . 3 is that at least two families of models , with different relations between @xmath19 and the value of @xmath21 at the peak , come close to the measurements of the cbr fluctuation spectrum , within the still substantial uncertainties . an estimate of @xmath19 from the cbr anisotropy measurements thus may depend on the choice of the model for structure formation . programs of measurement of @xmath78 in progress should be capable of distinguishing between the adiabatic and isocurvature models , even given the freedom to adjust the shape of @xmath79 . the interesting possibility is that some other model for structure formation with a very different value of @xmath19 may give an even better fit to the improved measurements . i assign a failing grade to the einstein - de sitter model in line 3a because the adiabatic and isocurvature models both prefer low @xmath19 ( @xcite , @xcite ) . i add question marks to indicate this still is a model - dependent result . in their version of table 1 dekel , burstein , & white @xcite give the einstein - de sitter model the highest grade on theoretical grounds , and a cosmologically flat model with @xmath1 the next highest grade . the point is well taken : this is the order most of us would choose . the issue is whether nature agrees with our ideas of elegance , or maybe prefers physics that produces an open universe ( @xcite - @xcite ) . full closure of cosmology may come with the discovery of physics that predicts the values of @xmath48 and space curvature ( eq . [ [ eq : cos_pars ] ] ) in terms of the expansion age of the universe , consistent with all the other constraints in table 1 . but since we seem to be far from that goal i am inclined to omit entries in line 4 . we have a secure if still schematic standard model for cosmology , and the prospect for considerable enlargement from the application of the cosmological tests . the theoretical basis for the tests was discovered seven decades ago . a significant application likely will take a lot less than seven more decades : the constraints in table 1 already are serious , if debatable , and people know how to do better . application of the tests could yield a set of tightly constrained values of the cosmological parameters and a clear characterization of the primeval departure from homogeneity . if so cosmology could divide at a fixed point , the situation at @xmath80 , say , when the universe is well described by a slightly perturbed friedmann - lematre model . one branch of research would analyze evolution from these initial conditions to the present complex structure of the universe . the other would search for the physics of the very early universe that produced these initial conditions . but before making any long - term plans based on this scenario i would wait to see whether the evidence really is that the early universe is simple enough to allow such a division of labor .
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we have a well - established standard model for cosmology and prospects for considerable additions from work in progress .
i offer a list of elements of the standard model , comments on controversies in the interpretation of the evidence in support of this model , and assessments of the directions extensions of the standard model seem to be taking .
psfig * the standard cosmological model * + + +
| 7,945 | 91 |
block copolymers ( bcps ) , comprising chemically distinct polymers permanently linked together , are interesting because of the diverse array of ordered phases to which both polymer theory and experiment have been directed.@xcite the phase behavior of diblock copolymer melts is a competition between the entropic tendency to mix the two species into an isotropic melt , and an energetic penalty for having unlike species adjacent , which induces transitions into ordered phases of many symmetries , depending on the topology and composition of the polymers . near the order - disorder transition ( weak incompatibility ) entropy effects dominate , and the individual polymers retain ( within mean field ) their gaussian coil conformation through the transition,@xcite , while at much higher incompatibilities the chains are strongly stretched . it is this strongly stretched regime which we address here . leibler developed the first complete theory of ordered phases in bcp melts@xcite , and predicted the by - now classical phases of lamellar ( l ) , cylindrical ( c ) and spherical ( s ) symmetry using the random phase approximation to derive an effective landau free energy in terms of composition modulations in fourier space . the strong segregation regime was studied by helfand and co - workers @xcite and semenov @xcite , who predicted the same series of phases with increasing asymmetry , denoted by the fraction @xmath1 of polymer a in an @xmath2 diblock . ( in this work we always use a to denote the minority block ) . this treatment balances the stretching energy of a polymer chain with the interfacial energy between a and b regions . by assuming an incompressible melt , minimization of the free energy gives a preferred domain size which scales as @xmath3 , where @xmath4 is the degree of polymerization . in the strong segregation limit the free energies of all microphases scale the same way with chain length and interfacial tension , so the phase boundaries become independent of the strength of the repulsion @xmath5 between a and b monomers and depend only on the composition @xmath6 . semenov s calculation in effect gave a lower bound to the free energy of the l , c , and s phases because the phases he constructed did not fill space , but were micelles of the corresponding topology @xcite . this approximation treats the @xmath2 interface and outer block surface as having the same circular or spherical shape , and is sufficient for understanding the qualitative aspects of the transitions between the phases . experiments followed the theories of leibler and semenov and quickly discovered a new phase,@xcite , originally thought to be ordered bicontinuous double diamond ( here denoted d ) , of @xmath7 symmetry , but recently shown to be of @xmath8 symmetry @xcite and related to the minimal surface known as the gyroid ( g).@xcite the g phase occurs for compositions between those of the l and c phases , can occur directly from the disordered phase upon increasing the incompatibility @xmath9 , and is found to be unstable to the l or c phases at high enough @xmath9.@xcite although several groups attempted to describe this transition theoretically,@xcite using variations on leibler s theory , the first successful theory is due to matsen and schick @xcite , who developed a method for computing the free energy of any crystalline structure by expanding the partition function in the basis functions for the symmetry of the desired mesophase , rather than the fourier mode expansion of leibler . they found a stable gyroid phase for @xmath10 , where the upper limit was determined by extrapolation from the phase boundaries at lower @xmath9.@xcite this was followed by careful application of leibler s method,@xcite to include higher harmonics and calculate the stability of the g phase in weak segregation analytically . roughly concurrent to the calculations of matsen and schick , methods were developed to calculate the free energy of essentially arbitrary structures in the strong segregation regime ( @xmath11 ) . these methods use the results for polymer brushes,@xcite , supplemented by an ansatz about the geometry of the relevant phase and an assumption about the chain paths . olmsted and milner assumed straight paths through the @xmath2 interface and locally specified the volume fraction per molecule,@xcite , while likhtman and semenov relaxed the assumption of straight paths @xcite but enforced the constraint of constant @xmath1 per molecule only globally . the former approach corresponds to an upper bound on the free energy ( see below ) , while it is not clear that the likhtman - semenov calculations corresponds to any bound , or indeed to any systematic approximation , because the local constraint of constant composition is relaxed . by comparing upper bounds between bicontinuous , c , and l phases ( obtained for the cylindrical phase by assuming hexagonal symmetry and imposing straight paths ) , we showed that the bicontinuous phases are unstable , when comparing upper bounds , to the l and c phases . later , xi and milner extended this work to calculations with kinked polymer paths , and found an upper bound to the hexagonal phase which lies very close to the lower bound using round unit cells.@xcite experiments have found an additional phase at @xmath5 values between the g and l phases @xcite , a hexagonally - perforated lamellae ( hpl ) phase , which consists of majority lamellae connected through a minority matrix by hexagonal arrays of tubes.@xcite the stacking has been suggested to be @xmath12 @xcite or @xmath13 @xcite . theoretical attempts to justify this phase have failed in both the strong segregation limit , where fredrickson chose a catenoid as a candidate base surface;@xcite and in the weak - segregation limit by self - consistent field calculations @xcite . recent experiments @xcite have shown that the hpl phase is not an equilibrium phase in diblock melts , but may be metastable . here we present the calculations of ref . @xcite in more detail . we show that the g geometry is the most stable of the candidate bicontinuous phases , followed by the d and p geometries , and that the g phase can be stable for block - copolymers with sufficient conformational asymmetry . the outline of this paper is as follows . in section [ sec : general ] we present the formalism for calculating the free energy in general geometries . in section [ sec : classical ] we present the results for the classical diblock topologies ( lamellae , cylinders , and spheres ) , extended to include non - round unit cells and , in the case of the cylindrical topology , kinked paths . in section [ sec : bicontinuous ] we present the free energy for a generic `` saddle '' wedge , which is representative of a generic bicontinuous structure as a pie - shaped wedge is representative of cylindrical phases regardless of packing . we then introduce the geometry necessary for calculating the free energy of the p , d , and g topologies . in section [ sec : results ] we present our results for both symmetric and non - symmetric stars , and we conclude in section [ sec : summary ] . we first recall some results for polymer brushes in strong segregation under melt conditions , and then show how to apply this to a general geometry . we consider a melt of star @xmath14 copolymers , comprising @xmath15 arms of _ a_-blocks of mean square end - to - end distance @xmath16 and similarly for the _ b_-arms . the volume fraction @xmath1 of _ a _ material is @xcite @xmath17 where @xmath18 is the total chain volume , and @xmath19 and @xmath20 are the volumes of single _ a _ and _ b _ arms . our calculations are appropriate for strongly - segregated chains , for which interfaces are sharp on the scale of microphase lattice constants . in strong segregation the free energies of all microphases scale the same way with chain length and interfacial tension , so the phase boundaries become independent of the strength of the repulsion @xmath5 between _ a _ and _ b _ monomers . = 2.5truein consider an elementary wedge , as in figure [ fig : wedge ] , from which we will construct all of the strong - segregation phases . our calculations are performed in terms of the ratio @xmath21 of the cross - sectional area @xmath22 at a height @xmath23 relative to that of the outer surface , in an infinitesimal wedge of height @xmath24 . this function may be easily calculated for wedges of particular shapes by elementary geometry , and is collected in table [ table1 ] for various geometries . since @xmath25 is the projected surface area along the normal vector extending from the wedge point to the flat wedge top , it will be a quadratic function of @xmath26 . the boundary condition @xmath27 implies that @xmath25 is a sum of @xmath26 and @xmath28 terms , and the boundary condition @xmath29 fixes the sum of the corresponding coefficients to be @xmath30 , leaving a single parameter . hence we may generally write @xmath31 the location @xmath32 of the `` dividing surface '' separating the two species is determined by equating the relative volume below @xmath23 , denoted @xmath33 , to the volume fraction @xmath1 : @xmath34 where @xmath35 is the ( partial ) volume of the wedge below height @xmath23 ( see figure [ fig : wedge ] ) . .area function @xmath25 ( see eq [ eq : area ] ) for various wedge geometries , where @xmath36 and @xmath24 is the wedge height . the expression for @xmath37 for the d , p , and g wedges refers to the geometry in figure [ fig : pgwedges ] . [ cols="<,^",options="header " , ] [ table2 ] upon minimizing eq [ eq : ftot ] over the scale of the structure , _ i.e. _ the radius @xmath24 , it is convenient to normalize all energies by a characteristic energy @xmath38 to obtain a compact form for the free energy : @xmath39 where @xcite @xmath40^{1/6 } \left(\ell_{\scriptscriptstyle a } \ell_{\scriptscriptstyle b}\right)^{1/6 } \label{eq : f0}\ ] ] is the free energy of a symmetric @xmath41 lamellar phase . the procedure above applies to a single wedge . for the classical cylinder and sphere geometries with circular ( spherical ) unit cells , each wedge is identical . for non - circular unit cells and for the complex geometries of bicontinuous phases , we must assemble the structure from many different wedges and minimize over the scale factor for the entire structure . the average free energy per chain @xmath42 of a structure with many distinct wedges is @xmath43 where @xmath44 is the free energy per molecule in wedge @xmath45 of volume @xmath46 , and @xmath47 is the volume of the structure . we choose a single scale factor @xmath48 to determine the size of the whole structure . each wedge @xmath45 has its own area and volume functions @xmath49 and @xmath50 , where the position of the dividing surface of each wedge , @xmath51 , is determined by @xmath52 the dimensionless functions @xmath53 are generalizations of eq [ eq : area ] for each wedge @xmath45 with wedge height @xmath54 . these functions encode the geometry of the particular structure , and the cross - sectional area at the top of each wedge , @xmath55 , scales as @xmath56 , for a @xmath57-dimensional structure ( _ e.g. _ @xmath58 for lamellae , @xmath59 for cylinders , @xmath60 for spheres ) . expressing eq [ eq : interf],[eq : fstr ] in terms of @xmath61 and @xmath48 , we minimize over @xmath48 to find the following free energy per chain of a particular structure : @xmath62 where @xmath63 where @xmath64 is obtained from eqs [ eq : ia]-[eq : ib ] by substituting @xmath49 in place of @xmath65 , and a similar relation defines @xmath66 . by specifying the volume fraction in each wedge according to eq [ eq : zdloc2 ] , we locally satisfy the constraint arising from the fixed composition of the copolymers . in contrast , likhtman and semenov @xcite satisfied this constraint only globally within a particular structure , which would be relevant for mixtures of different diblock copolymers with overall composition @xmath1 @xcite in the strong segregation limit , in which the entropy of mixing of such different copolymers would be negligible using the results of section [ sec:2a ] we can find the energies of the classical phases of diblock - copolymers : lamellae ( l ) , cylinders ( c ) , and spheres ( s ) in the round unit cell approximation , in which the unit cells are taken to consist of identical wedges . the corresponding free energies are : @xmath67^{1/3 } \label{eq : felam}\\ { f_{cyl } \over f_0 } & = & \left [ { 2 \varepsilon\phi(1-\phi^{1/2})^3(3+\phi^{1/2 } ) \over ( 1-\phi)^2 } + { 2 \phi \over \varepsilon}\right]^{1/3 } \label{eq : fecyl } \\ { f_{sph } \over f_0 } & = & 3\left [ { \varepsilon\phi^{4/3}(1-\phi^{1/3})^3 ( \phi^{2/3}+3\phi^{1/3}+6 ) \over 10(1-\phi)^2 } + { \phi \over 10\varepsilon } \right]^{1/3}\!\!\!\!\!\!\!. \label{eq : fesph}\end{aligned}\ ] ] calculations based on round unit cells @xcite provide lower bounds for the free energy , because they in fact describe the free energy per molecule of micelles.@xcite we may imagine a volume packed with such micelles , the interstitial regions filled with compatible long homopolymer with negligible surface tension against the outsides of the micelles , and negligible entropy of mixing . then we could do work to deform the micelles into a space - filling array , expelling the homopolymer at no free energy cost . to distinguish between crystal structures within a particular topology , such as between hexagonal and square for the cylindrical topologies , we must examine the energy for packing the molecules into the particular geometry , which is performed below . = 3.0truein we can produce an _ upper _ bound for different structures by assembling small pieces of the cylindrical or spherical micelles to fill the appropriate unit cell . each wedge has a parabolic monomer chemical potential given by eq [ eq : mu ] . however , each wedge @xmath45 has a slightly different shape and geometry , and thus has a distinct potential @xmath68 . adjacent wedges are not in equilibrium with each other and will relax if allowed to do so . hence the calculation yields an upper bound . to construct the unit cell of , _ e.g. _ , hexagonal cylinders , we assume a hexagonal dividing surface scaled down by @xmath69 and assemble the unit cell from tiny pie - shaped wedges extending from the center of the hexagon to the cell boundary . we make an analogous construction for square arrays of cylinders , or for fcc and bcc packings of spheres . we calculate the volume - averaged stretching free energy per molecule using eq [ eq : avg ] . to calculate this in practice we use the following procedure . for cylindrical micelles we divide a cell of a given symmetry ( say , hexagonal ) into tiny wedges . each wedge is adjusted slightly by making the segment of the wedge on the dividing surface normal to the bisector of the wedge , which is the path of the polymer . such an adjustment introduces a negligible volume in the continuum limit of many small wedges . the surface area used for calculating the surface energy ( eq [ eq : sa ] ) is , of course , the area of the segment in the original hexagonal dividing surface ( before adjusting the wedge to account for straight paths ) . = 3.5truein for the classical phases ( see figure [ fig : classicala ] ) the ratios of the upper and lower bounds for the free energies are independent of @xmath1 , given by : @xmath70 evidently , the most favorable structures have the `` roundest '' unit cells . the hexagonal phase is favored over the square phase , and bcc is slightly favored over fcc . = 2.0truein other upper bounds can be obtained by using different prescriptions for the @xmath2 surface . for example , one could choose a circular @xmath2 surface of radius @xmath71 for the cylindrical phase . the inner ( @xmath72 ) volume may be divided into wedges , and the outer ( @xmath73 ) volume divided into wedges which each satisfy the volume constraint @xmath1 with a partner @xmath72 wedge @xcite . for a right triangular wedge which subtends an angle @xmath74 , points at angle @xmath75 on the @xmath2 surface map to points @xmath76 on the boundary of the wigner - seitz cell , @xmath77 where @xmath78 , and @xmath79 . the composition specifies the radius , according to @xmath80 the mapping which obeys the local composition constraint is @xmath81 where @xmath82 and @xmath83 for hexagons and squares ( see figure [ fig : wedgecut ] ) , respectively . minimizing over the scale of the structure , eq [ eq : fsum ] yields , after some calculation , the following free energy : @xmath84\right\}^{1/3},\end{aligned}\ ] ] where @xmath85\end{aligned}\ ] ] and similarly for @xmath86 . remarkably , the upper bound for the kinked - path - hexagonal ansatz is typically less than @xmath87 above the lower bound of cylindrical micelles , and the transition is shifted to only a slightly smaller @xmath72 fraction @xmath1 ( see figure [ fig : classicalb ] ) . apparently the extra stretching energy to maintain a hexagonal @xmath2 interface with straight paths is relaxed considerably by allowing the inner block to adopt a more nearly circular dividing surface , which is preferred . recent accurate numerical self - consistent field calculations of diblock melts have shown that in fact the @xmath2 interface is nearly circular , with a slight hexagonal modulation ( angular modulation with 6-fold symmetry ) of relative amplitude @xmath88 at @xmath89 and @xmath90 @xcite . before addressing particular symmetries ( p , d , or g ) of bicontinuous phases , we discuss the closest analogue to a round unit cell . we would like to produce a simple estimate of the free energy , analogous to the cylindrical and spherical micelle calculation , which captures the physics of bicontinuous topologies . we thus represent a generic bicontinuous phase as a wedge , shown in figure [ fig : chip ] : an infinitesimal patch of `` saddle '' surface , with edges given by the normals , terminating in a small line segment lying along the bond - lattice . we envision the surface as a minimal surface , which has zero mean curvature.@xcite the stretching free energy per molecule of the symmetric wedge can be calculated as before , given the relative area as a function of relative height along the center normal ( table [ table1 ] ) : @xmath91 , where @xmath92 , @xmath93 is the thin end of the wedge and @xmath94 is the patch of minimal surface . as before , the dividing surface location @xmath32 is determined by eq [ eq : zdloc ] . the resulting free energy is given by applying eqs [ eq : zdloc]-[eq : interf ] , and is shown with the various cylindrical bounds in figure [ fig : classicalb ] . this estimate misses by a few tenths of one percent the intersection of the lamellar phase and the kinked - path upper bound bound for the hexagonal phase , and is stable with respect to the straight - path upper bound . = 3.5truein clearly the simple wedge construction captures some important physics . the structures formed by copolymers at different volume fractions @xmath1 arise from competition between interfacial and stretching free energies . the different structures present different functions @xmath25 , which determine both the dividing surface area and the stretching energy as a function of volume fraction . the phases occur in the order they do because the progression of functions @xmath25 from quadratic @xmath28 ( spheres ) to linear @xmath26 ( cylinders ) to @xmath95 ( bicontinuous ) to constant ( lamellae ) gives progressively less volume to the `` outer '' chain to avoid stretching , but uses progressively less area to separate the two species at higher volume fractions of the minority species . = 3.5truein however , we can not argue as before that our simple estimate for the bicontinuous phase is a lower bound for actual bicontinuous phases , because there is no way to pack together copies of one infinitesimal wedge to produce a `` micelle '' that 1 ) fills some region of space , and 2 ) is bounded by some surface(s ) , the volume outside of which could be filled by homopolymer . neither can we argue that this estimate is an upper bound , because we certainly can not pack a unit cell of the region bounded by the d or g surfaces with identical copies of one infinitesimal wedge . bicontinuous phases are assembled from different wedges @xmath45 , with different shape factors @xmath96 in table [ table1 ] . the shape factor roughly gauges the splay or gaussian curvature of the surface at the top of the wedge , with @xmath97 ( cylinders ) corresponding to zero gaussian curvature , @xmath98 to positive gaussian curvature , and @xmath99 to negative gaussian curvature . [ the gaussian curvature is the product of the two radii of curvature of a surface ] . the distribution of wedges must be chosen to pack the desired structure . while the symmetric wedge has @xmath100 , this is not an optimum shape . in fact , the optimum wedge shape depends on composition , as can be seen in figure [ fig : pnwedges ] . it is evident that there are shape factors @xmath37 which have lower free energy than the straight- and kinked - path hexagonal upper bounds ( figure [ fig : wedgeshape ] ) , so it is not unreasonable to hope that a judicious packing configuration can be a stable thermodynamic phase . the effect of the shape factor @xmath37 on the topology of the phase diagram emerges upon examining conformationally asymmetric ( @xmath101 ) copolymers . following ref . @xcite , we explore the effect of conformational asymmetry on the stability of bicontinuous phases by multiplying the wedge free energy by an additional arbitrary small prefactor @xmath102 which enhances stability . [ we will see below that for @xmath103 the bicontinuous phases that we can calculate ( g , d , p ) are of order @xmath104 higher than the cylinder - lamellar crossing , depending on which upper bound one compares . ] figure [ fig : asymwedge ] shows ` phase diagrams ' as a function of conformational asymmetry @xmath105 . recall that for @xmath106 the @xmath73-block is more flexible , while the @xmath72-block is stiffer and better able to stretch . for the symmetric wedge @xmath107 conformational asymmetry reduces the stability of the stiff - minority wedge phase @xmath108 and enhances the stability of the flexible - minority wedge phase ( @xmath109 ) , and shifts all transitions to greater @xmath1 . for @xmath110 the wedge phases lose stability , as could be guessed from figure [ fig : wedgeshape ] . for @xmath111 , for which the wedge is more cylindrical - like ( @xmath97 coresponds to cylinders ) , conformational asymmetry enhances the stiff - minority wedge phase relative to both the lamellar and cylindrical phases , and decreases the stability of the flexible - minority wedge phase . we emphasize that these are not phase diagrams , for a true phase is a mixture of wedges with different shape factors which fill space . = 3.5truein = 7.0truein 2 experimentally , the g phase has been observed between the lamellar and cylindrical phases in several strongly - segregated copolymer systems , for @xmath1 around 0.3 ( and , symmetrically , around 0.7 ) . some groups have argued for phase stability of bicontinuous phases in terms of bending rigidities,@xcite . however , there is fundamentally no bending energy in the problem ; descriptions in terms of bending energies only arise from a proper accounting of stretching free energy in curved geometries . our approach is to choose a geometry as an ansatz and compute the corresponding interfacial and stretching free energies in a manner consistent with calculations for cylindrical , spherical , and lamellar phases . the structure , revealed by scattering and electron microscopy,@xcite studies , can be described as follows,@xcite . consider first the d geometry , which is easier to visualize . a skeleton formed of the bonds of a diamond lattice is shown in figure [ fig : skeletons ] . two such lattices interpenetrate , analogous to the interpenetration of two simple cubic lattices in a bcc structure . now imagine swelling the bonds in these lattices into tubes of a finite diameter . the walls of these tubes are a rough approximation of the experimentally observed `` dividing surface '' separating the regions containing the two blocks . the volume contained within the tubes corresponds to the region inhabited by the low volume - fraction monomer . to model the d geometry , we use a self - dual minimal surface , called the schwartz d ( diamond ) surface , which partitions space into two identical interpenetrating regions , each of which contains and is topologically equivalent to a diamond bond - lattice @xcite . within each of these regions is a dividing surface , which surrounds a copy of the bond - lattice . the copolymer chains then have conformations with one ( a ) species stretching towards the bond - lattice , the junction between blocks residing on the dividing surface , and the other ( b ) species stretching towards the minimal surface . in the g phase the diamond lattice is replaced by a three - fold coordinated lattice , and the surface is replaced by the gyroid minimal surface discovered by schoen in 1970 @xcite , in which the two interpenetrating g volumes are chiral enantiomers of one another . for the p phase the bond lattice is six - fold coordinated ( figure [ fig : skeletons ] ) and the candidate partitioning surface is the schwartz p minimal surface . there is no compelling reason to choose a minimal surface for the partitioning surface . however , minimal surfaces solve the variational problem of minimizing surface area with zero pressure across the interface . if the diblock phase is in fact partitioned into two equivalent connected regions , then by symmetry there can be no net pressure exerted across the dividing surface that separates the two equivalent disjoint connected regions . so a minimal surface is reasonable , but by no means certain , since there is no obvious area energy to minimize . = 2.5truein = 2.5truein = 2.5truein the p , d , and g surfaces may be conveniently calculated using the weierstrass representation.@xcite here , the three - dimensional points @xmath112 of the two - dimensional surface are parametrized by the complex number @xmath113 . the g , p , and d surfaces are triply - periodic minimal surfaces with space groups @xmath114 and @xmath7 , respectively.@xcite to generate the full surface it is enough to calculate a single patch , to which all the symmetry operations of the space group may be applied to generate the full structure . a generic surface has two radii of curvature which are generally non - zero and different . points where the surface is flat are singular points , since both radii of curvature are zero and a direction of the surface can not be determined . these flat points define the corners of the fundamental patch . the weierstrass representation is : @xmath115 where @xmath116 and @xmath117 is the domain of integration shown in figure [ fig : weier ] . the points on the corners correspond to the flat points , and it is evident that the integrand above ( excluding the measure ) is singular at these points . the angle @xmath75 determines the surface : @xmath118 = 2.1truein = 7.0truein 1.0truecm 2 for other angles the surface intersects itself . this does not , of course , exhaust the class of triply periodic minimal surfaces either mathematically,@xcite or physically @xcite . we have chosen the d , p , and g surfaces because they are the most common observed surfactant bicontinuous surfaces , and have been claimed experimentally in block copolymers . sections of these surfaces are shown in figure [ fig : surfaces ] for fairly accurate calculations ( yielding energies _ lower _ than those for the true surface by of order a few tenths of a percent ) the d surface may be approximated by a simple hyperbolic surface , @xmath119 . this suggests that the minimal d surface may not the optimal partitioning surface . however , we have varied the shape of the partitioning surface around the d surface , and found free energy variations of only a few tenths of a percent . because of this , we have not optimized the free energy with respect to adjustments in the partitioning surfaces . to produce an upper bound on the free energy of the bicontinuous phases we follow a procedure analogous to that used for the classical cylindrical and spherical topologies . namely , we divide a unit cell of these structures into a large number of wedges ( shown in figure [ fig : pgwedges ] ) , similar to figure 1 , but of varying radii and gaussian curvature , and average the free energy per molecule using eq [ eq : avg]-[eq : sa ] . each structure has a different unit cell ( a big wedge ) from which the entire structure may be generated by applying the symmetry operations of the particular space group . figure [ fig : gchip ] shows the fundamental cell for the g structure . this procedure is straightforward . first we calculate the minimal surface , and then adopt a convenient mapping from the points on this surface to the skeleton . essentially , we construct an interpolation between those high - symmetry points on the minimal surface with normals that project onto the underlying bond - lattice . for the @xmath120 and @xmath121 phases , we optimize the mapping from the partitioning surface to the line segments of the bond lattice to minimize the free energy . we perform this by a conjugate gradient algorithm that distorts the two dimensional mesh of points on the surface , and gains of order @xmath87 in energy . = 3.5truein = 3.5truein thus , each small patch on the minimal surface is connected by straight lines to a small line segment on the skeleton , and a set of wedges results . each wedge is adjusted slightly by making both the top patch and the bottom segment orthogonal to the line segment connecting the center of the patch to the skeleton . this is analogous to making the outer surface of the wedge in the upper bound for the hexagon phase orthogonal to the line segment connecting the center of the outer surface of the wedge to the center of the hexagon . such adjustments are negligible in the limit of infinitesimal wedges . as before , the area of the @xmath2 interface is the true area , rather than the ( smaller ) area that results from adjusting the wedge to assure a chain path normal to the @xmath2 interface . in this way , the unit cell of the region bounded by a minimal surface is decomposed into many small wedges , each with a known ( and different ) radius @xmath24 , shape factor @xmath37 , and volume . the location of the dividing surface within each wedge is fixed by eq [ eq : zdloc2 ] , and the free energy calculated with eqs [ eq : avg]-[eq : sa ] . we have checked the algorithm and the dependence on the fineness of the mesh by using it to successfully compute the free energies of the hexagonal phase . = 3.5truein figure [ fig : data2 ] shows the free energy curves as a function of @xmath1 for conformationally symmetric copolymers ( @xmath103 ) . the upper bound for the @xmath122 phase ( p ) lies @xmath123 above that for the @xmath7 phase ( d ) , which in turn is less than a percent ( @xmath124 at @xmath125 ) above the @xmath8 ( g ) phase . consistent with experiments and self - consistent field theory , we do not find a stable g phase . at the lamellar kinked - path - hexagons free energy crossing ( @xmath126 ) the free energy of the @xmath121 phase is of order @xmath127 larger , while at the lamellar straight - path - hexagons free energy crossing ( @xmath128 ) the free energy of the @xmath121 phase is only a few tenths of a percent ( @xmath129 ) greater . this , we have argued , may be the fairer comparison , since both calculations use straight paths . unfortunately , we do not know how to perform a kinked - paths estimate for the bicontinuous phases . note , however , that we do not expect to gain as much energy from a kinked - path calculation for the bicontinuous phases as for the cylindrical topologies . consider the hexagonal calculation . the boundaries of the wigner - seitz cell , and hence the @xmath2 dividing surface , have sharp corners into which the chains must stretch . presumably a large part of the gain in the kinked - path calculation comes from relieving the strain associated with this stretch , and relaxing the inner block to its preferred circular structure . bicontinuous phases , on the other hand , have smooth `` wigner - seitz boundaries '' ( _ i.e. _ the minimal surface ) , and expensive stretching occurs mainly at the junctions of the skeleton lattice . hence , the anomalous stretching that may be relieved by a kinked - path calculation occurs along points in the structure , rather than along lines . so we expect that our straight - path estimate is not likely to differ greatly from a kinked - path estimate , and that the free energy of the g phase remains well above that of the kinked - path hexagons . the stretching of the chains at the junctions presumably contributes to the relative stability of the @xmath130 , @xmath120 , and @xmath121 structures , which have 6- , 4- , and 3-fold coordinated bond lattices . the more highly - coordinated lattices require more chain stretching to accommodate the space , which suggests that @xmath130 , @xmath120 , and @xmath121 occur in increasing order of stability . this picture is corroborated by recent work of matsen and bates @xcite , who quantitatively examined the packing frustration in the g , d , and hpl phases . = 7.0truein 2 our results apply to the strong - segregation limit , which is attained in the limit of large @xmath5 . because the phase boundaries shift away from @xmath131 as @xmath5 increases from weak segregation @xcite , we expect our phase boundaries to be further from @xmath131 than experimental values , which is indeed the case . figure [ fig : asym ] shows free energy crossings for various values of the conformational asymmetry parameter @xmath105 . relative to a conformationally symmetric melt , conformational asymmetry stabilizes phases with a stiff minority species and destabilizes phases with a stiff majority species , moving boundaries to larger @xmath132 for @xmath106 . recall that for @xmath133 the inner @xmath72-block is more flexible , while the @xmath73-block is stiffer and better able to stretch . we find that @xmath133 reduces the relative stability of the g phase , while the stability is enhanced for @xmath106 , and becomes stable for rather large asymmetries @xmath134 ( figure [ fig : asymg ] ) . = 3.5truein previous calculations of phase diagrams of conformationally - asymmetric diblocks have been done in the weak segregation regime,@xcite and , for the generic symmetric wedge as a model bicontinuous structure , in the strong segregation regime.@xcite matsen and bates @xcite found , as we do , that conformational asymmetry stabilizes the g phase with a stiff minority phase , widening the composition window and moving the lamellar - g and g - cylinder boundaries to higher stiff compositions ; and destabilizes the g phase with a stiff majority phase , both narrowing the composition window and shifting it to a higher stiff fraction . their calculations are limited to @xmath135 , and it is inconclusive whether the @xmath136 limit of this calculation yields a stable g phase . the same qualitative behavior was found for the generic symmetric wedge in the strong segregation regime.@xcite table [ table2 ] summarizes asymmetry parameters from recently collected data @xcite . while none of these diblocks have the large conformational asymmetry required to test our prediction of a stable phase in strong segregation , @xmath137 starblock copolymers have asymmetry factors larger , by a factor of @xmath138 , than those due to intrinsic chain stiffness effects alone . for example a @xmath139@xmath140 starpolymer has an asymmetry factor @xmath141 . we have attempted to calculate energies for the hpl @xcite phase , which is now accepted as a metastable phase @xcite . the minimal crystal phases d , p , and g have obvious candidate minimal surfaces to act as an intermaterial dividing surface towards which the majority - phase ends stretch ; and the minority - phase ends stretch towards the skeletal bond lattice . on the other hand , the majority - phase ends in the hpl phase stretch towards a combination of lines ( in the hexagonally - arranged perforating tubes ) and surfaces ( within the majority - phase layer ) . similarly , it is not obvious how to partition the minority - phase ends between lines and surfaces . the result is a non - analytic mapping which is difficult to minimize over . our attempts have thus far yielded quite high energies , of order that of the @xmath130 phase . we have outlined a general method for computing the free energy of block copolymer phases in the strong segregation regime . the procedure consists of the following steps : 1 . choose a candidate geometry and an associated partitioning surface that divides space into disjoint interpenetrating regions ( the majority blocks from the two regions stretch towards this surface ) . 2 . divide the enclosed volume into infinitesimal wedges , defined by straight paths connecting the partitioning surface to a skeleton of bonds ( the minority blocks stretch towards this bond skeleton ) . the a - b interface in each wedge is located such that the fraction of wedge volume filled by a blocks is locally equal to @xmath1 . the interfacial contribution to the free energy is the area of the a - b interface times the a - b surface tension . 4 . compute the stretching free energy per chain for each wedge within the approximation of straight paths . calculations for straight paths involve slight adjustments to the wedges whose contributions vanish in the limit of small wedges . 5 . optimize the free energy per chain with respect to the overall scale of the mesophase ( _ e.g. _ , the dimension of the unit cell ) . optimize the mapping from the partitioning surface to the bond skeleton to minimize the overall free energy . in certain structures ( _ e.g. _ hpl ) the majority phase ends lie on both lines and surfaces , in which case the procedure above must be suitably generalized . we have also shown how to calculate the free energy for geometries where the shape of the @xmath2 interface is specified , for phases of cylindrical topology . this requires polymer chain paths which are kinked at the @xmath2 interface . the infinitesimal wedges are described by the relative area function @xmath142 ( eq [ eq : area ] ) which is parametrized by a single scalar @xmath37 ( eq [ eq : p ] and table [ table1 ] ) that roughly gauges the local gaussian curvature of the partitioning surface . for the classical phases all wedges are identical , while bicontinuous phases have different distributions of shape factors @xmath37 . for symmetric stars we find a metastable bicontinuous ( gyroid , or g ) phase which is most stable near the lamellae - hexagonal cylinder transition . for sufficiently asymmetric copolymers ( @xmath134 ) we predict a _ stable _ g phase . in a weakly segregated mesophase , very long polymer chains are in fact more strongly stretched than gaussian chains [ almdal , k. ; rosedale , j. h. ; bates , f. s. ; wignall , g. d. ; and fredrickson , g. h. _ phys . lett . _ * 1990 * , _ 65 _ , 1112 ] , which has been explained in terms of concentration fluctuations [ barrat , j .- l . ; and fredrickson , g. h _ j. chem . phys . _ * 1991 * , _ 95 _ , 1281 ] . hajduk , d.a . ; harper , p.e . ; gruner , s.m . ; honeker , c.c . ; kim , g. ; thomas , e.l . ; fetters , l.j . _ macromolecules _ * 1994 * , _ 27 _ , 4063 ; hajduk , d.a . ; gruner , s.m . ; rangarajan , p. ; register , r.a . ; fetters , l.j . ; honeker , c. ; albalak , r.j . ; thomas , e.l . _ macromolecules _ * 1994 * , _ 27 _ , 490 ; hajduk , d.a . ; harper , p.e . ; gruner , s.m . ; honeker , c.c . ; thomas , e.l . ; fetters , l.j . _ macromolecules _ * 1995 * , _ 28 _ , 2570 . the composition variable @xmath1 is defined in ref . @xcite as the volume fraction of the @xmath73 phase . similarly , ref . @xcite defines @xmath105 with @xmath72 and @xmath73 reversed with respect to the present article , so equations such as eqs [ eq : felam]-[eq : fesph ] are identical in the two works ( compare eqs 9 - 11 of ref . eq 8 of ref . @xcite has a misprint , and should have a factor of @xmath143 to match eq [ eq : f0 ] in section [ sec : general ] above .
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we compute phase diagrams for @xmath0 starblock copolymers in the strong - segregation regime as a function of volume fraction @xmath1 , including bicontinuous phases related to minimal surfaces ( g , d , and p surfaces ) as candidate structures .
we present the details of a general method to compute free energies in the strong segregation limit , and demonstrate that the gyroid g phase is the most nearly stable among the bicontinuous phases considered .
we explore some effects of conformational asymmetry on the topology of the phase diagram .
# 1*#1 * # 1 10truept * [ fig # 1 here ] * 2
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the nobel prize in physics awarded in 2015 to takaaki kajita and arthur mcdonald for studies of the atmospheric and solar neutrinos @xcite does not leave any doubt that neutrinos oscillate and have nonzero masses . the latter fact leads to the well - known possibility for neutrino to have non - trivial electromagnetic properties @xcite , which brought forth the research area that was investigated in details by numerous authors ( see recent review @xcite and references therein ) . in the course of these studies many phenomena that may appear in electromagnetic fields have been recognized and described thoroughly . among them the neutrino spin - flavor oscillations is the one , featuring both the above mentioned basic neutrino aspects nonzero mass and electromagnetic properties from the one side and mixing from another @xcite . owing to this , in spite of being a longstanding problem , the spin - flavor oscillations can reveal some new aspects of existence of neutrino mass and electromagnetic properties . these concern the problems of neutrino parameters relation in neutrino physical ( mass ) and flavor bases and , generally speaking , of an accurate derivation of the formulas used to describe oscillations . we below give an advanced view on the standard scheme of neutrino spin - flavor oscillations description aiming at the solid determination of parameters involved in the formalism and its rigorous derivations . we consider neutrino mixing and oscillations in presence of an arbitrary constant magnetic field with nonzero transversal @xmath0 and longitudinal @xmath1 components with respect to the direction of neutrino propagation . the electromagnetic interaction of neutrinos is determined by neutrino diagonal and transition magnetic moments that are introduced for the neutrino mass states . explicit expressions for the effective neutrino diagonal and transition magnetic moments for the flavor bases in terms of these values for the mass states are obtained . the effective evolution hamiltonian for the flavor neutrino and the corresponding oscillation probability are derived . the role the longitudinal magnetic field component is examined . in particular , it is shown that : 1 ) @xmath1 coupled to the corresponding magnetic moments shifts the neutrino energy , and 2 ) in case of nonvanishing neutrino transition magnetic moments @xmath1 produces an additional mixing between neutrino states , both in the mass and flovour neutrino bases . consider two dirac neutrino physical states , @xmath2 and @xmath3 , with masses @xmath4 and @xmath5 . also consider the neutrino electromagnetic interaction via magnetic moment matrix @xmath6 ( @xmath7 ) : @xmath8 where @xmath9 is the electromagnetic field tensor , @xmath10 and @xmath11 being the dirac matrices . in a uniform magnetic field the hamiltonian ( [ em_hamilt ] ) becomes @xmath12 where @xmath13 @xmath14 are the pauli matrices . for considering the neutrino evolution in the ultrarelativistic limit we introduce the 4-component basis @xmath15 of states with definite helicity @xmath16 . with the standard column vector notation , @xmath17 the neutrino evolution equation relevant to electromagnetic interaction has the schr@xmath18dinger - like form , @xmath19 the effective hamiltonian consists of the vacuum and interaction parts @xmath20 where the interaction part @xmath21 is composed of matrix elements of the interaction hamiltonian ( [ b_hamilt ] ) taken over the helicity neutrino states : @xmath22 . let us calculate the effective interaction hamiltonian under the assumption that neutrino moves along the @xmath23-axis . from the magnetic field interaction hamiltonian ( [ b_hamilt ] ) we have : @xmath24 for the spinors representing the free neutrino states we take @xmath25 where @xmath26 is the neutrino @xmath27 momentum . the two - component spinors @xmath28 define neutrino helicity states , and are given by @xmath29 recall that in the ultrarelativistic limit these two states correspond to the right - handed @xmath30 and left - handed @xmath31 chiral neutrinos , respectively . substituting ( [ wave func ] ) into the effective hamiltonian given by ( [ hb ] ) , we get @xmath32 decomposing the magnetic field vector into longitudinal and transversal with respect to neutrino motion components @xmath33 it is possible to show that @xmath34 in the the ultrarelativistic limit @xmath35 one gets : @xmath36 where the quantity @xmath37 , fetched out also in @xcite , we call the transition gamma - factor . similarly , it is also possible to show that @xmath38 introducing an angle @xmath39 between the @xmath40 and @xmath26 vectors and assuming that @xmath41 is aligned along the @xmath42-axis , we further obtain that @xmath43 and , similarly , @xmath44 @xmath45 @xmath46 as it was expected , in neutrino transitions without change of helicity only the @xmath47 component of the magnetic field contributes to the effective potential , whereas in transitions with change of the neutrino helicity the transversal component @xmath48 matters . performing the remaining simple algebra one can readily write out the @xmath21 matrix . thus , we obtain the evolution equation ( [ schred_eq ] ) in the following form @xmath49 this equation governs all possible oscillations of the four neutrino states with defined masses ( @xmath4 and @xmath5 ) and helicities ( @xmath50 and @xmath51 ) in the presence of a magnetic field . the obtained expression ( [ gen_evol_eq ] ) for the evolution hamiltonian explicitly reveals a particular role of the longitudinal magnetic field component @xmath52 : \1 ) the longitudinal magnetic field component @xmath52 coupled to the corresponding magnetic moment shifts the neutrino energy , \2 ) in case of nonvanishing neutrino transition magnetic moment @xmath53 , the presence of the longitudinal field @xmath52 produces mixing among neutrino species with different masses but with equal helicities . at the same time , from ( [ gen_evol_eq ] ) it is clearly seen that mixings of different helicity neutrino states are due to the corresponding magnetic moment ( or transition magnetic moment ) interactions with the transversal magnetic field @xmath0 . once having physics in the mass basis in hands , our next step is to bring it to observational terms . this means that we must elaborate a generalization of the mixing matrix for transitions between neutrino state vectors written in two four - component bases @xmath54 and @xmath55 so that @xmath56 this procedure appears to be not quite direct since we should hold the condition that polarization of the fields must be preserved under transformation of the bases elements . that is why , keeping in mind that chiral components are almost the helicity ones , we define @xmath57 then , using eqs . ( [ u_def ] ) and ( [ transformations ] ) , it is easy to obtain that @xmath58 given the transition matrix ( [ u ] ) , derivation of the evolution equation in the flavor basis is straightforward : @xmath59 so that the effective magnetic field interaction hamiltonian @xmath60 has the following structure , @xmath61 here we have introduced the following formal notations intended to manifest an analogy with the standard spin - flavor oscillation formalism ( see below ) : @xmath62 @xmath63 the hamiltonian ( [ h_b_f ] ) has been derived within much consistent procedure than is usually given in literature . it should be noted that eqs . ( [ mu_fl ] ) could be also obtained as results of general consideration @xcite of neutrino mixing and oscillations in an arbitrary constant magnetic field ( the final form like that of eqs . ( [ mu_fl ] ) was not established in @xcite ) . also , an effective neutrino oscillation hamiltonian with terms containing quantities ( [ mu_gammas_fl ] ) , however without account for possibility of spin transitions , was obtained in @xcite . let us now confront the obtained effective magnetic field interaction hamiltonian ( [ h_b_f ] ) with the one typically written straight in the neutrino flavor basis ( see , for instance , @xcite and @xcite ) : @xmath64 where @xmath65 is the common neutrino gamma - factor . first of all , we can ascertain that the structure of the obtained expression ( [ h_b_f ] ) is consistent with the standard " hamiltonian ( [ h_b_f_old ] ) . at that , the magnitudes ( [ mu_fl ] ) account for the neutrino mixings and represent the straightforward expressions for the effective magnetic moments in the flavor basis in terms of magnetic moments introduced in the mass basis . the neutrino effective magnetic moments " ( [ mu_gammas_fl ] ) determine neutrino interactions and mixings due to the longitudinal magnetic field @xmath52 . as in the mass basis , the magnetic field component @xmath52 shifts the neutrino energy also , it contributes to transitions with change of flavor . it is interesting to observe that when the transversal field component is set to zero the hamiltonian structure becomes @xmath66 so that neutrino states with different flavors and the same chirality decouple and form subsystems independently mixed by the field . for example , one would have two neutrino species @xmath67 and @xmath68 mixed in accordance with the equation ( here the standard terms for vacuum oscillations are added ) @xmath69 in this way , neutrino oscillations would be influenced by the longitudinal field @xmath52 ( see also @xcite ) and @xmath52 would produce an additional mixing with respect to the usual one described by the vacuum mixing angle @xmath70 . the oscillation probability is just strightforward , @xmath71 where @xmath72 is the denominator of the pre - sine factor and the other quantities have usual for the theory of neutrino oscillations meaning . it follows that in case the second term in the denominator is much smaller than the first one then the amplitude of the flavor oscillations @xmath73 gets its maximal value . this can be considered as an effect of neutrinos interaction with the longitudinal magnetic field due to neutrino magnetic moments . one of the authors ( a.s . ) is thankful to nicolao fornengo and carlo giunti for the kind invitation to participate in the 14th international conference on topics in astroparticle and underground physics . the work on this paper was partially supported by the russian basic research foundation grants no . 14 - 22 - 03043-ofi - m , 15 - 52 - 53112-gfen and 16 - 02 - 01023-a . 4 fukuda y _ et al _ ( super - kamiokande collaboration ) 1998 _ phys . lett . _ * 81 * 1562 ahmad q r _ et al _ ( sno collaboration ) 2001 _ phys . lett . _ * 87 * 071301 ahmad q r _ et al _ ( sno collaboration ) 2002 _ phys . lett . _ * 89 * 011301 fujikawa k and shrock r 1980 _ phys . * 45 * 963 giunti c and studenikin a 2015 _ rev . phys . _ * 87 * 531 ( _ preprint _ arxiv:1403.6344 [ hep - ph ] ) cisneros a 1971 _ astrophys . space sci . _ * 10 * 87 schechter j and valle j 1981 _ phys . * d * 24 1883 voloshin m b , vysotsky m i and okun l b 1986 _ sov . . jetp _ * 64 * 446 akhmedov e k 1988 _ sov . j. nucl . phys . _ * 48 * 382 lim c - s and marciano w j 1988 _ phys . _ d * 37 * 1368 likhachev g and studenikin a 1995 _ sov . . jetp _ * 81 * 419 akhmedov e k and khlopov m yu 1988 _ sov . phys . _ * 47 * 689 - 691 akhmedov e k and khlopov m yu 1988 _ mod . lett . _ * a3 * 451 - 457
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we consider neutrino mixing and oscillations in presence of an arbitrary constant magnetic field with nonzero transversal @xmath0 and longitudinal @xmath1 components with respect to the direction of neutrino propagation .
the electromagnetic interaction of neutrinos is determined by diagonal and transition neutrino magnetic moments that are introduced for the neutrino mass states .
explicit expressions for the effective neutrino diagonal and transition magnetic moments for the flavor basis in terms of these values for the mass states are obtained .
the effective evolution hamiltonian for the flavor neutrino and the corresponding oscillation probability are derived .
the role of the longitudinal magnetic field component is examined . in particular
, it is shown that : 1 ) @xmath1 coupled to the corresponding magnetic moments shifts the neutrino energy , and 2 ) in case of nonvanishing neutrino transition magnetic moments @xmath1 produces an additional mixing between neutrino states , both in the mass and flavor neutrino bases .
| 3,308 | 230 |
invasion of influenza virus into a host s upper respiratory tract leads to infection of healthy epithelial cells and subsequent production of progeny virions @xcite . infection also triggers a variety of immune responses . in the early stage of infection a temporary non - specific response ( innate immunity ) contributes to the rapid control of viral growth while in the late stage of infection , the adaptive immune response dominates viral clearance @xcite . the early immune response involves production of antiviral cytokines and cells , e.g. type 1 interferon ( ifn ) and natural killer cells ( nk cells ) , and is independent of virus type @xcite . in the special case of a first infection in a naive host , the adaptive immune response , mediated by the differentiation of naive t cells and b cells and subsequent production of virus - specific t cells and antibodies @xcite , leads to not only a prolonged killing of infected cells and virus but also the formation of memory cells which can generate a rapid immune response to secondary infection with the same virus @xcite . @xmath0 t cells , which form a major component of adaptive immunity , play an important role in efficient viral clearance @xcite . however , available evidence suggests they are unable to clear virus in the absence of antibodies @xcite except in hosts with a very high level of pre - existing naive or memory @xmath0 t cells @xcite . some studies indicate that depletion of @xmath0 t cells could decrease the viral clearance rate and thus prolong the duration of infection @xcite . furthermore , a recent study of human a(h7n9 ) hospitalized patients has implicated the number of effector @xmath0 t cells as an important driver of the duration of infection @xcite . this diverse experimental and clinical data , sourced from a number of host - species , indicates that timely activation and elevation of @xmath0 t cell levels may play a major role in the rapid and successful clearance of influenza virus from the host . these observations motivate our modeling study of the role of @xmath0 t cells in influenza virus clearance . viral dynamics models have been extensively applied to the investigation of the antiviral mechanisms of @xmath0 t cell immunity against a range of pathogens , with major contributions for chronic infections such as hiv / siv @xcite , htlv - i @xcite and chronic lcmv @xcite . however , for acute infections such as measles @xcite and influenza @xcite , highly dynamical interactions between the viral load and the immune response occur within a very short time window , presenting new challenges for the development of models incorporating @xmath0 t cell immunity . existing influenza viral dynamics models , introduced to study specific aspects of influenza infection , are limited in their ability to capture all major aspects of the natural history of infection , hindering their use in studying the role of @xmath0 t cells in viral clearance . some models show a severe depletion of target cells ( i.e healthy epithelial cells susceptible to viral infection ) after viral infection @xcite . depletion may be due to either infection or immune - mediated protection . either way , these models are arguably incompatible with recent evidence that the host is susceptible to re - infection with a second strain of influenza a short period following primary exposure @xcite . furthermore , as reviewed by dobrovolny _ @xcite , target cell depletion in these models strongly limits viral expansion so that virus can be effectively controlled or cleared at early stage of infection even in the absence of adaptive immunity , which contradicts the experimental finding that influenza virus remains elevated in the absence of adaptive immune response @xcite . while a few models do avoid target cell depletion @xcite , they either assume immediate replenishment of target cells @xcite or a slow rate of virus invasion into target cells resulting in a much delayed peak of virus titer at day 5 post - infection ( rather than the observed peak at day 2 ) @xcite . moreover , models with missing or unspecified major immune components , e.g. no innate immunity @xcite , no antibodies @xcite or unspecified adaptive immunity @xcite , also indicate the need for further model development . for an in - depth review of the current virus dynamics literature on influenza , we refer the reader to the excellent article by dobrovolny _ _ @xcite . t cells and b cells to produce effector @xmath0 t cells and antibodies , responsible for final clearance of virus . ] in this paper , we construct a within - host model of influenza viral dynamics in naive ( i.e. previously unexposed ) hosts that incorporates the major components of both innate and adaptive immunity and use it to investigate the role of @xmath0 t cells in influenza viral clearance . the model is calibrated against a set of published murine data from miao _ et al . _ @xcite and is then validated through demonstration of its ability to qualitatively reproduce a range of published data from immune - knockout experiments @xcite . using the model , we find that the recovery time defined to be the time when virus titer first drops below a chosen threshold in the ( deterministic ) model is negatively correlated with the level of effector @xmath0 t cells in an approximately exponential manner . to the best of our knowledge , this relationship , with support in both h3n2-infected mice and h7n9-infected humans @xcite , has not been previously identified . the exponential relationship between @xmath0 t cell level and recovery time is shown to be remarkably robust to variation in a number of key parameters , such as viral production rate , ifn production rate , delay of effector @xmath0 t cell production and the level of antibodies . moreover , using the model , we predict that people with a lower level of naive @xmath0 t cells may receive significantly more benefit from induction of additional effector @xmath0 t cells . such production , arising from immunological memory , may be established through either previous viral infection or t cell - based vaccines . the model of primary viral infection is a coupled system of ordinary and delay differential equations , consisting of three major components ( see fig . 1 for a schematic diagram ) . 13 describe the process of infection of target cells by influenza virus and are a major component in almost all models of virus dynamics in the literature . 4 and 5 model ifn - mediated innate immunity @xcite . thirdly , adaptive immunity including @xmath0 t cells and b cell - produced antibodies for killing infected cells and neutralizing influenza virus respectively are described by eqs . @xmath1 in further detail , eq . 1 indicates that the change in viral load ( @xmath2 ) is controlled by four factors : the production term ( @xmath3 ) in which virions are produced by infected cells ( @xmath4 ) at a rate @xmath5 @xcite ; the viral natural decay / clearance ( @xmath6 ) with a decay rate of @xmath7 ; the viral neutralisation terms ( @xmath8 and @xmath9 ) by antibodies ( both a short - lived antibody response @xmath10 driven by , e.g. igm , and a longer - lived antibody response @xmath11 driven by , e.g. igg and iga @xcite ) , and a consumption term ( @xmath12 ) due to binding to and infection of target cells ( @xmath13 ) . in eq . 2 , the term @xmath14 models logistic regrowth of the target cell pool @xcite . both target cells ( @xmath13 ) and resistant cells ( @xmath15 , those protected due to ifn - induced antiviral effect ) can produce new target cells , with a net growth rate proportional to the severity of infection , @xmath16 ( i.e. the fraction of dead cells ) . @xmath17 is the initial number of target cells and the maximum value for the target cell pool @xcite . target cells ( @xmath13 ) are consumed by virus ( @xmath2 ) due to binding ( @xmath18 ) , the same process as @xmath12 . note that @xmath19 and @xmath20 have different measurement units due to different units for viral load ( @xmath2 ) and infected cells ( @xmath4 ) . as already mentioned , the innate response may trigger target cells ( @xmath13 ) to become resistant ( @xmath15 ) to virus , at rate @xmath21 . resistant cells lose protection at a rate @xmath22 @xcite . this process also governs the evolution of virus - resistant cells ( @xmath15 ) in eq . 5 . eq . 3 describes the change of infected cells ( @xmath4 ) . they increase due to the infection of target cells by virus ( @xmath18 ) and die at a ( basal ) rate @xmath23 . two components of the immune response increase the rate of killing of infected cells . ifn - activated nk cells kill infected cells at a rate @xmath24 @xcite . effector @xmath0 t cells ( @xmath25 ) produced through differentiation from naive @xmath0 t cells @xmath26 in eq . 6 kill at a rate @xmath27 . of note our previous work has demonstrated that models of the innate response containing only ifn - induced resistance for target cells ( state @xmath15 ; eq . 5 ) , while able to maintain a population of healthy uninfected cells , still control viral kinetics through target cell depletion , and therefore can not reproduce viral re - exposure data @xcite . given our interest in analysing a model that prevents target cell depletion , inclusion of ifn - activated nk cells ( term @xmath24 ) is an essential part of the model construction . 4 models the innate response , as mediated by ifn ( @xmath28 ) . ifn is produced by infected cells at a rate @xmath29 and decays at a rate @xmath30 @xcite . 6 models stimulation of naive @xmath0 t cells ( @xmath26 ) into the proliferation / differentiation process by virus at a rate @xmath31 ) , where @xmath32 is the maximum stimulation rate and @xmath33 indicates the viral load ( @xmath2 ) at which half of the stimulation rate is achieved . note that this formulation does not capture the process of antigen presentation and @xmath0 t cell activation , but rather is a simple way to establish the essential coupling between the viral load and the rate of @xmath0 t cell activation in the model @xcite . in eq . 7 , the production of effector @xmath0 t cells ( @xmath25 ) is assumed to be an `` advection flux '' induced by a delayed virus - stimulation of naive @xmath0 t cells ( the first term on the righthand side of eq . the delayed variables , @xmath34 and @xmath35 , equal zero when @xmath36 . the introduction of the delay @xmath37 is to phenomenologically model the delay induced by both naive @xmath0 t cell proliferation / differentiation and effector @xmath0 t cell migration and localization to the site of infection for antiviral action @xcite . the delay also captures the experimental finding that naive @xmath0 t cells continue to differentiate into effector t cells in the absence of ongoing antigenic stimulation @xcite . the multiplication factor @xmath38 indicates the number of effector @xmath0 t cells produced from one naive @xmath0 t cell , where @xmath39 is the average effector @xmath0 t cell production rate over the delay period @xmath37 . the exponential form of the multiplication factor is derived based on the assumption that cell differentiation and proliferation follows a first - order advection reaction equation . effector @xmath0 t cells decay at a rate @xmath40 . similar to @xmath0 t cells , eqs . 8 and 9 model the proliferation / differentiation of naive b cells , stimulated by virus presentation at rate @xmath41 . stimulation subsequently leads to production of plasma b cells ( @xmath42 ) after a delay @xmath43 . the multiplication factor @xmath44 indicates the number of plasma b cells produced from one naive b cell , where @xmath45 is the production rate . plasma b cells secrete antibodies , which exhibit two types of profiles in terms of experimental observation : a short - lived profile ( e.g. igm lasting from about day 5 to day 20 post - infection ) and a longer - lived profile ( e.g. igg and iga lasting weeks to months ) @xcite . these two antibody responses are modeled by eqs . 10 and 11 wherein different rates of production ( @xmath46 and @xmath47 ) and consumption ( @xmath48 and @xmath49 ) are assumed . the model contains 11 equations and 30 parameters ( see table 1 ) . this represents a serious challenge in terms of parameter estimation , and clearly prevents a straightforward application of standard statistical techniques . to reduce uncertainty , a number of parameters were taken directly from the literature , as per the citations in table . 1 . the rest were estimated ( as indicated in table 1 ) by calibrating the model against the published data from miao _ et al . _ @xcite who measured viral titer , @xmath0 t cell counts and igm and igg antibodies in laboratory mice ( exhibiting a full immune response ) over time during primary influenza h3n2 virus infection ( see @xcite for a detailed description of the experiment ) . the approach to estimating the parameters based on miao _ et al . _ s data is provided in the _ supplementary material _ and the estimated parameter values are given in table 1 . note that the data were presented in scatter plots in the original paper @xcite , while we presented the data here in mean @xmath50 sd at each data collection time point for a direct comparison with our mean - field mathematical model . for model simulation , the initial condition is set to be @xmath51 unless otherwise specified . the initial target cell number ( @xmath17 ) was estimated by petrie _ we estimate that of order 100 cells ( resident in the spleen ) are able to respond to viral infection ( @xmath26 ) ( personal communication , n. lagruta , monash university , australia ) . note that 100 naive @xmath0 t cells might underestimate the actual number of naive precursors that could respond to all the epitopes contained within the virus but does not qualitatively alter the model dynamics and predictions ( see _ results _ where the naive @xmath0 t cell number is varied between 0 to 200 ) . in the absence of further data , we also use this value for the initial naive b cell number ( @xmath52 ) , but again this choice does not qualitatively alter the model predictions . the numerical method and code ( implemented in matlab , version r2014b , the mathworks , natick , ma ) for solving the model are provided in the _ supplementary material_. ) . note that due to the limit of detection for the viral load ( occurring after 10 days post - infection as seen in viral load data ) , the last three data points in the upper - left panel were not taken into consideration for model fitting . ] clinical influenza a(h7n9 ) patient data was used to test our model predictions on the relationship between @xmath0 t cell number and recovery time . the data was collected from 12 surviving patients infected with h7n9 virus during the first wave of infection in china in 2013 ( raw data is provided in _ dataset s1 _ ; see the paper of wang _ et al . _ @xcite for details of data collection ; this study was reviewed and approved by the shaphc ethics committee ) . note that the clinical data were scarce for some patients . for those patients , we have assumed that the available data are representative of the unobserved values in the neighboring time period . for each patient , we took the average @xmath53 @xmath0 t cell number in @xmath54 peripheral blood mononuclear cells ( pbmc ) for the period from day 8 to day 22 ( or the recovery day if it comes earlier ) post - admission as a measure of the effector @xmath0 t cell level . this period was chosen _ a priori _ as it roughly matches the duration of the @xmath0 t cell profile and clinical samples were frequently collected in this period . the average @xmath0 t cell count was given by the ratio of the total area under the data points ( using trapezoidal integration ) to the number of days from day 8 to day 22 ( or the recovery day if it comes earlier ) . for those patients for which samples at days 8 and/or 22 were missing we specified the average @xmath0 t cell level at the missing time point to be equal to the value from the nearest sampled time available . ) , effector @xmath0 t cells ( @xmath25 ) , short - lived antibody response ( @xmath10 ) and long - lived antibody response ( @xmath11 ) have been shown in fig . 2 . ] we first analyze the model behavior in order to establish a clear understanding of the model dynamics . fig . 2 shows solutions ( time - series ) for the model compartments ( viral load , @xmath0 t cells and igm and igg antibody ) calibrated against the murine data from @xcite . solutions for the remaining model compartments are shown in fig . the model ( with both innate and adaptive components active ) prevents the depletion of target cells ( see fig . 3 wherein over 50% of target cells remain during infection ) and results in a minor loss of just 1020% of healthy epithelial cells ( i.e. the sum of target cells ( @xmath13 ) and virus - resistant cells ( @xmath15 ) ; see supplementary fig . the primary driver for the maintenance of the target cell pool during acute viral infection is a timely activation of the innate immune response , and in particular the natural killer cells ( supplementary fig . since we have previously shown that the target - cell limited model ( even with the resistant cell compartment ) is unable to reproduce observations from heterologous re - exposure experiments @xcite , our model improves upon previous models where viral clearance was only achieved through depletion of target cells ( a typical solution shown in supplementary fig . importantly , our result is distinguished from that of saenz et . @xcite , wherein the healthy cell population was similarly maintained , but primarily through induction of the virus - resistant state , thereby rendering that model incapable of capturing re - infection behavior as established in animal models @xcite . the modeled viral dynamics exhibits three phases , each dominated by the involvement of different elements of the immune responses ( fig . 4 ) . immediately following infection ( 02 days post - infection ) and prior to the activation of the innate ( and adaptive ) immune responses , virus undergoes a rapid exponential growth ( fig . 4a ) . in the second phase ( 25 days post - infection ) , the innate immune response successfully limits viral growth ( fig . 4a ) . in the third phase ( 46 days post - infection ) , adaptive immunity ( antibodies and @xmath0 t cells ) is activated and viral load decreases rapidly , achieving clearance . 4b and 4c demonstrate the dominance of the different immune mechanisms at different phases . in fig . 4b models with and without immunity are indistinguishable until day 2 ( shaded region ) , before diverging dramatically when the innate and then adaptive immune responses influence the dynamics . in fig . 4c , models with and without an adaptive response only diverge at around day 4 as the adaptive response becomes active . we have further shown that this three - phase property is a robust feature of the model , emergent from its mathematical structure and not a property of fine tuning of parameters ( see supplementary fig . importantly , it clearly dissects the periods and effect of innate immunity , extending on previous studies of viral infection phases where the innate immune response was either ambiguous or ignored @xcite . and @xmath55 in the model ) . the trajectories overlap prior to the activation of the innate response , before diverging due to target cell depletion . the shaded region highlights the first phase ( exponential growth ) . in panel ( c ) , the dashed line shows viral kinetics in the absence of adaptive immunity ( by letting @xmath55 ; innate immunity remains active ) . the trajectories overlap prior to the activation of the adaptive response . the shaded region highlights the second phase ( innate response ) . note : changes in model parameters shifts where the three phases occur , but does not alter the underlying three - phase structure . i.e. existence of the three phases is robust to variation in parameters ( see _ supplementary material _ and supplementary fig . s3 in particular ) . ] as reviewed by dobrovolny _ @xcite , a number of _ in vivo _ studies have been performed to dissect the contributions of @xmath0 t cells and antibodies @xcite . we use the findings of these studies to validate our model , by testing how well it is able to reproduce the experimental findings ( without any further adjustment to parameters ) . although the determination of the role of @xmath0 t cells is often hindered by co - inhibition of both @xmath0 t cells and the long - lived antibody response ( e.g. using nude mice ) , it is consistently observed that antibodies play a dominant role in final viral clearance while @xmath0 t cells are primarily responsible for the timely killing of infected cells and so indirectly contribute to an increased rate of removal of free virus towards the end of infection @xcite . furthermore , experimental data demonstrate that a long - lived antibody response is crucial for achieving complete viral clearance , while short - lived antibodies are only capable of driving a transient decrease in viral load @xcite . we find that our model ( with parameters calibrated against miao _ s data @xcite ) is able to reproduce these observations : * virus can rebound in the absence of long - lived antibody response ( see fig . 5 and supplementary fig . both the @xmath0 t cell response and short - lived antibody response only facilitate a faster viral clearance , and are incapable of achieving clearance in the absence of long - lived antibody response ( see fig . 5 and supplementary fig . a lower level of @xmath0 t cells ( modulated by a decreased level of initial naive @xmath0 t cells , @xmath26 ) significantly prolongs the viral clearance ( see supplementary fig . in addition , the model also predicts a rapid depletion of naive @xmath0 t cells after primary infection ( see fig . 3 ) , which represents a full recruitment of naive @xmath0 t cell precursors . this result may be associated with the experimental evidence suggesting a strong correlation between the naive @xmath0 t cell precursor frequencies and effector @xmath0 t cell magnitudes for different pmhc - specific t cell populations @xcite . note that in fig . 5 no adjustments to the model ( e.g. to the vertical scale ) were made ; its behavior is completely determined by the calibration to the aforementioned murine data @xcite and so these findings represent a ( successful ) prediction of the model . t- , iga- , igg- " was modeled by letting @xmath56 and @xmath57 . external igm " ( in addition to the igm produced by plasma cells ) was modeled by adding a new term @xmath58 to eq . 1 where @xmath59 follows a piecewise function @xmath60 for @xmath61 , @xmath62 for @xmath63 , @xmath64 for @xmath65 and @xmath60 for @xmath66 . ( b ) data is from the paper of iwasaki _ et al . the data indicates that the long - lasting iga response , but not the long - lasting igg response or the short - lasting igm response , is necessary for successful viral clearance . no long - lived antibody response " was modeled by letting @xmath57 . note that miao _ only measured igm and igg , but not iga . as such , our model s long - lived antibody response was calibrated against igg kinetics ( see fig . 2 ) . therefore , we emphasise that we can only investigate the relative contributions of short - lived and long - lived antibodies . ] in summary , we have demonstrated that our model with parameters calibrated against murine data @xcite exhibits three important phases characterized by the involvement of various immune responses . advancing on previous models , our model does not rely on target cell depletion , and successfully reproduces a multitude of behavior from knockout experiments where particular components of the adaptive immune response were removed . this provides us with some confidence that each of the major components of the immune response has been captured adequately by our model , allowing us to now make predictions on the effect of the cellular adaptive response on viral clearance . having established that our model is ( from a structural point of view ) biologically plausible and that our parameterization is capable of reproducing varied experimental data under different immune conditions ( i.e. knockout experiments ) , we now study how the cellular adaptive response influences viral kinetics in detail . we focus on the key clinical outcome of recovery time , defined in the model as the time when viral titer first falls below @xmath67 , the minimum value detected in relevant experiments ( e.g. fig . 2 ) . t cells plays an important role in determining recovery time . recovery time is defined to be the time when viral load falls to 1 @xmath68 . panel ( a ) shows that the average effector @xmath0 t cell number over days 620 is linearly related to the naive @xmath0 t cell number ( i.e. @xmath69 ) . panel ( b ) shows that the recovery time is approximately exponentially related to the initial naive @xmath0 t cell number . combined , these results give panel ( c ) wherein an approximately exponential relationship is observed between the average @xmath0 t cell number and recovery time , both of which are experimentally measurable . note that the exponential / linear fits shown in the figures are not generated by the viral dynamics model but are used to indicate the trends ( evident visually ) in the model s behavior . panel ( d ) shows that varying the delay @xmath37 ( in a similar way to that shown in fig . s5 in the _ supplementary material _ ) , rather than the naive @xmath0 t cell number , does not alter the exponential relationship . in panel ( d ) , the crosses represent the results of varying @xmath37 and the empty circles are the same as those in panel ( c ) for comparison . ] time series of the viral load show that the recovery time decreases as the initial naive @xmath0 t cell number ( @xmath26 ) increases ( supplementary fig . s4 ) . with that in mind , we now examine how recovery time is associated with the clinically relevant measure of effector @xmath0 t cell level during viral infection . with an increasing initial level of naive @xmath0 t cells , the average level of effector @xmath0 t cells over days 620 increases linearly ( fig . 6a ) , while the recovery time decreases in an approximately exponential manner ( fig . combining these two effects gives rise to an approximately exponential relation between the level of effector @xmath0 t cells and recovery time ( fig . note that the exponential / linear fits shown in the figures are simply to aid in interpretation of the results . they are not generated by the viral dynamics model . if varying the delay for naive @xmath0 t cell activation and differentiation , @xmath37 , while keeping the naive @xmath0 t cell number fixed ( at the default value of 100 ) , we find that the average level of effector @xmath0 t cells is exponentially related to the delay , while the recovery time is dependent on the delay in a piecewise linear manner ( see supplementary fig . s5 ) . nevertheless , the combination still leads to an approximately exponential relationship between the level of effector @xmath0 t cells and recovery time ( supplementary fig . s6 ) , which is almost identical to that of varying naive @xmath0 t cells ( fig . we also examine the sensitivity of the exponential relationship to other model parameters generally accepted to be important in influencing the major components of the system , such as the viral production rate @xmath5 , ifn production rate @xmath29 and naive b cell number . we find that the exponential relationship is robust to significant variation in all of these parameters ( see supplementary figs . s6 and s7 and fig . 9 ) . these results suggest that a higher level of effector @xmath0 t cells is critical for early recovery , consistent with experimental findings @xcite . t cell number and recovery time . the x - axis is the average level of functional effector @xmath0 t cells ( i.e. @xmath70 cells ) over the period from day 8 to day 22 ( or the recovery day if it comes earlier ) . spearman s rank correlation test indicates a significant negative correlation between the average @xmath0 t cell numbers and recovery time ( @xmath71 ) . excluding one of the patients ( no . dataset s1 _ ; discussed in the _ discussion _ ) , all other data points ( solid dots ) are fitted by an offset exponential function @xmath72 , indicating that the best achievable recovery time for individuals with a high @xmath0 t cell response is approximately 17.5356 . ] finally , and perhaps surprisingly given our model has been calibrated purely on data from the mouse , a strikingly similar relationship as shown in fig . 6c is found in clinical data from influenza a(h7n9 ) virus - infected patients ( fig . 7 ) . excluding one patient ( no . dataset s1 _ ; the exclusion is considered further in the _ discussion _ ) , average @xmath70 cells and recovery time are negatively correlated ( spearman s @xmath73 , @xmath74 ) and well captured by an exponential fit with an estimated offset ( see fig . 7 caption for details ) . the exponential relationship ( observed in both model and data ) has features of a rapid decay for relatively low / intermediate levels of effector @xmath0 t cells and a strong saturation for relatively high @xmath0 t levels , implying that even with a very high level of naive @xmath0 t cells , recovery time can not be reduced below a certain value ( in this case , estimated to be approximately 17 days ) . of course , the exponential relationship ( i.e. the scale of @xmath0 t cell level or recovery time ) , is only a qualitative one , as we have no way to determine the scaling between different x - axis measurement units , nor adjust for particular host and/or viral factors that differ between the two experiments ( i.e. h3n2-infection in the mouse @xcite versus h7n9-infeciton in humans @xcite ) . in addition to naive @xmath0 t cells , memory @xmath0 t cells ( established through previous viral infection ) may also significantly affect recovery time due to both their rapid activation upon antigen stimulus and faster replication rate @xcite . to study the role of memory @xmath0 t cells , we must first extend our model . as we are only concerned with how the presence of memory @xmath0 t cells influences the dynamics , as opposed to the development of the memory response itself , the model is modified in a straightforward manner through addition of two additional equations which describe memory @xmath0 t cell ( @xmath75 ) proliferation / differentiation : @xmath76 accordingly the term @xmath77 in eq . 3 is modified to @xmath78 . the full model and details on the choice of the additional parameters are provided in the _ supplementary material_. note that the model component , @xmath75 , may include different populations of memory @xmath0 t cells , including those directly specific to the virus and those stimulated by a different virus but which provide cross - protection @xcite . t cells on viral clearance . recovery time is defined to be the time when the viral load falls to 1 @xmath68 . panel ( a ) demonstrates that varying the number of memory @xmath0 t cells ( @xmath75 ) reduces the recovery time for any naive @xmath0 t cell number ( i.e. @xmath69 ) . note that saturation is observed for @xmath79 where the recovery time is about 6 days , independent of the naive cell numbers . panel ( b ) demonstrates how the presence of pre - existing memory @xmath0 t cells ( solid dots ) leads to a shorter recovery time when compared to the case where no memory @xmath0 t cells are established ( open circles ) . note the time scale difference in panels ( a ) and ( b ) . this simulation is based on the assumption that the level of pre - existing memory @xmath0 t cells is assumed to be either 1% or 5% ( as indicated in the legend ) of the maximum effector @xmath0 t cell number due to primary viral infection . the memory cell number ( which is not shown in this figure ) is about 30 time as many as the naive cell number shown in the figure , i.e. 30 naive cells result in about 900 memory cells before re - infection . ] 8a shows how the pre - existing memory @xmath0 t cell number ( @xmath75 ) changes the exponential relationship between naive @xmath0 t cells and recovery time . importantly , as the number of memory @xmath0 t cells increases , the recovery time decreases for any level of naive @xmath0 t cells and the exponential relationship remains . the extent of reduction in the recovery time for a relatively low level of naive @xmath0 t cells is greater than that for a relatively high level of naive @xmath0 t cells . this suggests that people with a lower level of naive @xmath0 t cells may benefit more through induction of memory @xmath0 t cells , emphasizing the potential importance for taking prior population immunity into consideration when designing @xmath0 t cell - based vaccines @xcite . the above result is based on the assumption that the initial memory @xmath0 t cell number upon re - infection is independent of the number of naive @xmath0 t cells available during the previous infection . however , it has also been found that the stationary level of memory @xmath0 t cells is usually maintained at about 510% of the maximum antigen - specific @xmath0 t cell number during primary viral infection @xcite . this indicates that people with a low naive @xmath0 t cell number may also develop a low level of memory @xmath0 t cells following infection . in consequence , such individuals may be relatively more susceptible to viral re - infection @xcite . this alternative and arguably more realistic relationship between the numbers of naive and memory @xmath0 t cells is simulated in fig . 8b where memory @xmath0 t cell levels are set to 5% of the maximum of the effector @xmath0 t cell level . results suggest that , upon viral re - infection , pre - existing memory @xmath0 t cells are able to significantly improve recovery time except for in hosts with a very low level of naive @xmath0 t cells ( fig . this is in accordance with the assumption that a smaller naive pool leads to a smaller memory pool and in turn a weaker shortening in recovery time . although the model suggests that the failure of memory @xmath0 t cells to protect the host is unlikely to be observed ( because of the approximately 30 fold increase in the size of the memory pool relative to the naive pool ) , the failure range may be increased if the memory pool size is much smaller ( modulated by , say , changing 5% to 1% in the model ) . therefore , for people with a low naive @xmath0 t cell number , the level of memory @xmath0 t cells may be insufficient and prior infection may provide very limited benefit , further emphasizing the opportunity for novel vaccines that are able to induce a strong memory @xmath0 t cell response to improve clinical outcomes . t cell number and recovery time . recovery time is defined to be the time when viral load falls to 1 @xmath68 . different antibody levels are simulated by varying the initial number of naive b cells ( i.e. @xmath52 at @xmath80 ) . ] antibodies appear at a similar time as effector @xmath0 t cells during influenza viral infection and may enhance the reduction in the recovery time in addition to @xmath0 t cells . by varying the naive b cell number @xmath52 ( as a convenient , but by no means unique , way to influence antibody level ) , we find that increasing the antibody level shortens the recovery time regardless of the initial naive @xmath0 t cell number , leaving the exponential relation largely intact ( fig . 9 ) . a slight saturation occurs for the case in which levels of both naive b cells and @xmath0 t cells are low . moreover , variation in naive b cell number also results in a wider variation in recovery time for a lower naive @xmath0 t cell level , suggesting that people with a lower level of naive @xmath0 t cells may , once again , receive a more significant benefit ( in terms of recovery time ) through effective induction of an antibody response via vaccination . in this paper , we have studied the role of @xmath0 t cells in clearing influenza virus from the host using a viral dynamics model . the model was calibrated on a set of published murine data from miao _ et al . _ @xcite and has been further shown to be able to reproduce a range of published data from experiments with different immune components knocked out . by avoiding target - cell depletion , our model is also compatible with re - infection data @xcite , providing a strong platform on which to examine the role of @xmath0 t cells in determining recovery time from infection . our primary finding is that the time of recovery from influenza viral infection is negatively correlated with the level of effector @xmath0 t cells in an approximately exponential manner . this robust property of infection has been identified from the model when calibrated against influenza a(h3n2 ) infection data in mice @xcite , but also observed in clinical case series of influenza a(h7n9 ) infection in humans ( fig . 7 ) @xcite . our findings , in conjunction with conclusions on the potential role for a t cell vaccine that stimulates and/or boosts the memory response , suggest new directions for research in both non - human species and further studies in humans on the association between @xmath0 t cell levels and clinical outcomes . further research , including detailed statistical fitting of our model to an extensive panel of infection data ( as yet unavailable ) from human and non - human species , is required to establish the generality of these relationships and provide quantitative insights for specific viruses in relevant hosts . the non - linear relationship between effector @xmath0 t cell level and recovery time may be useful in clinical treatment . the saturated property of the relation implies that a linear increase in the effector @xmath0 t cell level may result in diminishing incremental improvements in patient recovery times . with evidence of a possible age - dependent loss of naive t cells @xcite , our model results imply that boosting the @xmath0 t cell response via t cell vaccination may be particularly useful for those with insufficient naive @xmath0 t cells . the population - level consequences of such boosting strategies , while beyond the scope of this work , have previously been considered by the authors @xcite . we also investigated the effect of memory @xmath0 t cell level on viral clearance and found unsurprisingly that a high pre - existing level of memory @xmath0 t cells was always beneficial . however , our results suggest that pre - existing memory @xmath0 t cells may be particularly beneficial for certain groups of people . for example , if the memory @xmath0 t cell number induced by viral infection or vaccination is assumed to be relatively constant for everyone , people with less naive @xmath0 t cells would benefit more upon viral re - infection ( see fig . 8a ) . on the other hand , if assuming pre - existing memory @xmath0 t cell number is positively correlated with the number of naive @xmath0 t cells ( simulated in fig . 8b ) , people with more naive @xmath0 t cells would benefit more upon viral re - infection . emerging evidence suggests that the relationship between the level of memory @xmath0 t cells and naive precursor frequencies is likely to be deeply complicated @xcite . in that context , our model predictions emphasize the importance for further research in this area , and the necessity to take prior population immunity into consideration when designing @xmath0 t cell vaccines @xcite . we modeled both short - lived and long - lived antibody responses . experimental data and model predictions consistently show that the short - lived antibody response results in a temporary reduction in virus level whereas the long - lived antibody response is responsible for complete viral clearance ( fig . we emphasize here that although the model is able to capture the observed short - lived and long - lived antibody responses ( in order to study the virus - immune response interactions ) , it is not designed to investigate the mechanisms inducing different antibody responses . the observed difference in antibody decay profile may be a result of many factors including the life times of different antibody - secreting cell types @xcite , different antibody life times @xcite and antibody consumption through neutralizing free virions . detailed study of these phenomena requires a more detailed model and associated data for parameter estimation and model validation , and is thus left for future work . similarly , @xmath81 t cells are also known to perform a variety of functions in the development of immunity , such as facilitation of the formation and optimal recall of @xmath0 t cells or even direct killing of infected cells during viral infection @xcite . their depletion due to , say , hiv infection has also been associated with more severe clinical outcomes following influenza infection @xcite . some of the major functions of @xmath81 t cells may be considered to be implicitly modeled through relevant parameters such as the rate of recall of memory @xmath0 t cells ( modeled by the delay @xmath82 ) in our extended model which includes memory @xmath0 t cells . however , a detailed viral dynamics study of the role of @xmath81 t cells in influenza infection , including in hiv infected patients with depleted @xmath81 t cells , remains an open and important challenge . in a recent theoretical study , it was found that spatial heterogeneity in the t cell distribution may influence viral clearance @xcite . resident @xmath0 t cells in the lungs have a more direct and significant effect on timely viral clearance than do naive and memory pools resident in lymph nodes . although this factor has been partially taken into consideration in our model by introducing a delay for naive / memory @xmath0 t cells , lack of explicit modeling of the spatial dynamics limits a direct application of our model to investigate these spatial effects . finally , as noted in the results , one of the influenza a(h7n9)-infected patients ( patient a79 ) was not included in our analysis of the clinical data ( fig . although our model suggests some possibilities for the source of variation due to possible variation in parameter values , large variations in recovery time are only expected to occur for relatively low levels of naive @xmath0 t cells , nominally incompatible with this patient s moderate @xmath0 t cell response but a relatively long recovery time . however , we note that @xmath83 t cell counts for this patient 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( 1988 ) . the half - lives of serum immunoglobulins in adult mice . _ eur j immunol . _ * 18(2 ) , * 313316 . riberdy , j. m. _ et al . _ ( 2000 ) . diminished primary and secondary influenza virus - specific @xmath85 t - cell responses in cd4-depleted @xmath86 mice . _ j virol . _ * 74 , * 97629765 . laidlaw , b. j. _ et al . _ @xmath87 t cell help guides formation of @xmath88 lung - resident memory @xmath84 t cells during influenza viral infection . _ immunity . _ * 41 , * 633645 . cohen , c. _ et al . _ ( 2015 ) . mortality amongst patients with influenza - associated severe acute respiratory illness , south africa , 2009 - 2013 . _ plos one . _ * 10(3 ) , * e0118884 . doi:10.1371/journal.pone.0118884 . pc is supported by an australian government national health and medical research council ( nhmrc ) funded centre for research excellence in infectious diseases modelling to inform public health policy . zw is supported by an nhmrc australia china exchange fellowship . awcy is supported by an australian postgraduate award . jm and kk are support by nhmrc career development and senior researcher fellowships respectively . jmm is supported by an australian research council future fellowship . the h7n9 clinical study was supported by national natural science foundation of china ( nfsc ) grants 81471556 , 81470094 and 81430030 . the authors would like to thank members of the modelling and simulation unit in the centre for epidemiology and biostatistics and the school of mathematics and statistics , university of melbourne , for helpful advice on the study . we have no competing interests . .model parameter values obtained by fitting the model to experimental data . @xmath89 $ ] , @xmath90 $ ] and @xmath91 $ ] represent the units of viral load , ifn and antibodies respectively . @xmath89 $ ] and @xmath91 $ ] are @xmath68 ( 50% egg infective dose ) and @xmath92 , consistent with the units of data . ifn is assumed to be a non - dimensionalized variable in the model , and therefore @xmath90 $ ] can be ignored . some parameters are obtained from the literature and the rest are obtained by fitting the model to experimental data in the paper of miao _ et al . _ @xcite , except @xmath93 which is of minor importance when considering a single infection and is thus fixed to reduce uncertainty . [ cols="<,<,<,<",options="header " , ] the delay in activation of @xmath0 t cells and b cells directly results in the delay of production of effector @xmath0 t cells and antibody - producing cells . by a time - shift transform @xmath94 , 7 in the main text becomes @xmath95 where @xmath96 and @xmath97 starts from @xmath98 . as defined in the main text , @xmath99 for any negative time , i.e. @xmath100 for @xmath101 . moreover , @xmath102 for @xmath103 . thus , the equation is trivial for @xmath104 . therefore , for @xmath104 , we replace @xmath97 back to @xmath105 and eq . s1 becomes @xmath106 where @xmath96 . similarly , eqs . 9 - 11 in the main text can be changed to @xmath107 in this way , the model presented in the main text is equivalent to the following model : @xmath108 for variables whose independent variable is not explicitly specified , they are all functions of @xmath105 , i.e. @xmath2 reads @xmath109 . this model avoids negative time and is solved by the following steps : * firstly choosing a time step size @xmath110 and using it to discretise the time domain to be @xmath111 . the results in the main text are generated using @xmath112 ( day ) , the choice of which is based on the result that further decreasing @xmath110 does not improve the solution ( results not shown ) . * given initial condition + @xmath113 , + we solve the model iteratively . for iteration from @xmath114 to @xmath115 , @xmath116 , @xmath117 and @xmath118 in eqs . s6 and s8 are already known and thus treated as parameters . the system becomes an ode system and can be easily solved by using a built - in ode solver _ ode15s _ in matlab(r2014b ) with default settings . all the variables at time @xmath115 are then updated for use in the next iteration . .... clear dt=0.1 ; % time step size time=0:dt:100 ; % parameters t0=7e+7;gt=0.8;pv=210;deltav=5.0 ; beta=5e-7;betap=3e-8 ; deltai=2;kappan=2.5;kappae=5e-5 ; phi=0.33;rho=2.6;pf=1e-5;deltaf=2 ; betacn=1;betabn=0.03 ; kappas=0.8;kappal=0.4 ; pc=1.2;pb=0.52;deltae=0.57;deltap=0.5 ; ps=12;pl=4;deltas=2;deltal=0.015 ; tauc=6;taub=4;hc=1e+4;hb=1e+4 ; % index indicating when delayed process starts indc = round(tauc / dt+1 ) ; % for tauc indb = round(taub / dt+1 ) ; % for taub % variable vectors and initial conditions v = zeros(1,length(time));v(1)=1e+4 ; t = v;t(1)=7e+7 ; i = t;i(1)=0 ; r = i ; f = i ; cn=100*ones(1,length(time ) ) ; bn=100*ones(1,length(time ) ) ; e = zeros(1,indc+length(time ) ) ; p = i ; as = zeros(1,indb+length(time ) ) ; al = as ; init=[v(1),t(1),i(1),r(1),f(1),cn(1),e(1),bn(1),p(1),as(1),al(1 ) ] ' ; options = odeset('reltol',1e-3,'abstol',1e-6 ) ; for i=2:length(time ) [ ~,y]=ode15s(@odemodel,[0 dt],init , options , e(i),al(i),as(i),tauc , taub , phi , rho , deltaf , gt , pf , pv , beta , betap , kappan , deltav , deltai , betacn , betabn , kappae , kappas , pl , ps , deltal , deltas , deltap , deltae , pc , pb , kappal , hc , hb , t0 ) ; v(i)=y(end,1);t(i)=y(end,2);i(i)=y(end,3);r(i)=y(end,4 ) ; f(i)=y(end,5);cn(i)=y(end,6);bn(i)=y(end,8);p(i)=y(end,9 ) ; e(indc+i)=y(end,7 ) ; as(indb+i)=y(end,10 ) ; al(indb+i)=y(end,11 ) ; init = y(end , : ) ' ; % initial condition for next iteration end .... .... function ynew = odemodel(~,y , e , al , as , tauc , taub , phi , rho , deltaf , gt , pf , pv , beta , betap , kappan , deltav , deltai , betacn , betabn , kappae , kappas , pl , ps , deltal , deltas , deltap , deltae , pc , pb , kappal , hc , hb , t0 ) % v : viral load % t : target cell % i : infected cell % r : resistant cell % f : ifn % cn : naive cd8 + t cells % e : effector cd8 + t cells % bn : naive b cells % p : plasma b cells % as : short - lived antibodies % al : long - lived antibodies % y=[v , t , i , r , f , cn , e , bn , p , as , al ] ynew = zeros(11,1 ) ; ynew(1)=pv*y(3)-deltav*y(1)-kappas*y(1)*as - kappal*y(1)*al - beta*y(1)*y(2 ) ; ynew(2)=gt*(y(2)+y(4))*(1-(y(2)+y(3)+y(4))/t0)-betap*y(1)*y(2)+rho*y(4)-phi*y(2)*y(5 ) ; ynew(3)=betap*y(1)*y(2)-deltai*y(3)-kappan*y(3)*y(5)-kappae*y(3)*e ; ynew(4)=phi*y(2)*y(5)-rho*y(4 ) ; ynew(5)=pf*y(3)-deltaf*y(5 ) ; ynew(6)=-betacn*y(1)/(y(1)+hc)*y(6 ) ; ynew(7)=betacn*y(1)/(y(1)+hc)*y(6)*exp(pc*tauc)-deltae*y(7 ) ; ynew(8)=-betabn*y(1)/(y(1)+hb)*y(8 ) ; ynew(9)=betabn*y(1)/(y(1)+hb)*y(8)*exp(pb*taub)-deltap*y(9 ) ; ynew(10)=ps*y(9)-deltas*y(10 ) ; ynew(11)=pl*y(9)-deltal*y(11 ) ; .... the model contains 11 equations and 30 parameters ( see table 1 in the main text ) . this represents a serious challenge in terms of parameter estimation , and clearly prevents a straightforward application of standard statistical techniques . however , based on an extensive survey of the experimental literature , we have been able to identify plausible , but by no means unique , combinations of parameters that successfully explain the available data . a number of parameters were taken directly from the literature , as per the citations in table 1 . the rest ( 18 parameters ) were estimated by calibrating the model to the published data from miao _ et al . _ @xcite who measured viral titre , @xmath0 t cell counts and igm and igg antibodies in laboratory mice ( exhibiting a full immune response ) over time during primary influenza h3n2 virus infection ( shown in fig . 2 in the main text ) . note that the data were presented in scatter plots in the original paper @xcite , while we presented the data in mean @xmath50 sd at each data collection time point ( as shown in fig . 2 in the main text ) and fit our mean - field mathematical model to the means . * we first manually determined a set of the 18 parameters which produced a model solution that reasonably matched the experimental data shown in fig . 2 in the main text . in detail , the main criteria include : 1 ) the viral load starts from about @xmath119 , reaches a peak of about @xmath120 at 12 days p.i . , then declines rapidly from about 5 days p.i . ( note that the last three data points were not considered due to the limit of detection ) ; 2 ) the @xmath0 t cell count starts to increase rapidly at about 6 days p.i . , reaches a peak of about @xmath121@xmath54 at 810 days p.i . and returns back to zero after about 20 days p.i . ; 3 ) the igm level starts to increase rapidly at 45 days p.i . , reaches a peak of about 200300 pg / ml at about 10 days p.i . and returns back to baseline after about 20 days p.i . ; 4 ) the igg level starts to increase rapidly at 45 days p.i . , reaches a peak of about 8001000 pg / ml at about 20 days p.i . and decays slowly . given the high - dimensionality of the parameter space and limited experimental data , the procedure was essential in allowing us to identify a candidate parameter set which was not far from generating a local minimum in the following optimisation process . * the candidate parameter set was then used as an initial estimate for optimization using matlab s built - in function _ fmincon _ with default settings . the target of optimization was to minimize the least - squares error ( lse ) : @xmath122 where the four lse components for viral load , effector @xmath0 t cells , igm and igg were given respectively by @xmath123 ^ 2 , \tag{s18}\ ] ] @xmath124 ^ 2 , \tag{s19}\ ] ] @xmath125 ^ 2 , \tag{s20}\ ] ] @xmath126 ^ 2 . \tag{s21}\ ] ] @xmath127 , @xmath128 , @xmath129 and @xmath130 indicate the model solution for variables @xmath2 , @xmath25 , @xmath10 and @xmath11 evaluated at time @xmath131 respectively . @xmath132 , @xmath133 , @xmath134 and @xmath135 indicate the associated data . @xmath136 is the index of the time point and the @xmath131 may differ for the four components . @xmath137 , @xmath138 , @xmath139 and @xmath140 were used to scale the errors to the same order of magnitude . due to the fact that data were not collected at a fixed frequency ( i.e. the time interval between adjacent data points is not constant ) , the errors at different time points were assigned different weights based on the length of time intervals between adjacent points . for example , if viral load was measured at time @xmath141 , the weight function @xmath142 for interior points was given by @xmath143 and for boundary points by @xmath144 this error weighting was used to weaken the domination of dense data points on model fits . it is evident that a lot of measurements were done within the first 10 days post - infection but only a few were performed after day 20 post - infection , in particular for igg data ( see fig . 2 in the main text ) . we found that using equally weighted lses led to a model fit that manifestly failed to capture those sparse data points which we believe are equally , if not more important from a more biological perspective , in determining the igg kinetics ( see fig . s8 , compared with fig . 2 in the main text ) . * the parameter constraints when using _ fmincon _ were set to @xmath145 $ ] , @xmath146 $ ] , @xmath147 $ ] , @xmath148 $ ] , @xmath149 $ ] , @xmath150 $ ] , @xmath151 $ ] , @xmath152 $ ] , @xmath153 $ ] , @xmath154 $ ] , @xmath155 $ ] , @xmath156 $ ] , @xmath157 $ ] , @xmath158 $ ] , @xmath159 $ ] , @xmath160 $ ] , @xmath161 $ ] and @xmath162 $ ] . * after obtaining a locally optimized solution for the candidate parameter set , we then checked the solution generated and evaluated its biological plausibility ( based on the criteria mentioned above ) . this step was essential as given the over - specification of the model ( in a statistical sense ) , it was possible for good fitting solutions to be identified by matlab s optimization algorithm , which were nonetheless biologically implausible . for example , oscillatory solutions for quantities such as igg , while providing a good - fit " to data , were not deemed acceptable on biological grounds ( see fig . s9 for such an example ) . if the optimised solution failed our ( qualitative ) evaluation , we returned to the first step and redetermined a new set of parameters as new initial estimates to be optimized using matlab . this entire process was repeated to arrive at the default parameter set shown in table 1 in the main text . to guarantee that the default parameter set was a good choice , we further randomly generated 10,000 sets of parameter samples near the default parameter set ( within @xmath5050% from the default values ) and used them as initial estimates with _ fmincon _ to search for locally optimized solutions . of these , 31 generated a better lse but all failed to meet the criteria mentioned above ( results not shown ) . 2 in the main text shows how the model reproduces the key dynamic behavior shown in the data . we also show in the _ results _ section in the main text , that the model behavior is robust to perturbation of model parameters and that model predictions are reasonably consistent with a range of other experimental data , demonstrating the plausibility , if not uniqueness ( of course ) , of the parameter set . we emphasize that , although the default parameter set is successful in reproducing a multitude of experimental observations ( e.g. full immune , knockout , re - infection ) as presented in the main text , we by no means claim that this parameter set is unique . it remains an open and challenging problem to reliably identify a biologically plausible and statistically identifiable solution for what is a highly complex system , where we are severely limited by available experimental data . incorporating memory @xmath0 t cells into the model in the main text , we only make two changes . the first is adding two equations to describe the memory @xmath0 t cell ( @xmath75 ) proliferation / differentiation , similar to eqs . 6 and 7 in the main text @xmath163 then we change the term @xmath77 in eq . 3 in the main text to @xmath78 . hence , similar to the approach mentioned above that moving the delayed term from viral load to effector cells , we write down the model in an equivalent form , @xmath164 , \tag{s28}\\ \frac{df}{dt } & = p_fi-\delta_ff , \tag{s29}\\ \frac{dr}{dt } & = \phi ft-\rho r , \tag{s30}\\ \frac{dc_{n}}{dt } & = -\beta_{cn}(\frac{v}{v+h_c})c_{n } , \tag{s31}\\ \frac{de(t+\tau_c)}{dt } & = \beta_{cn}(\frac{v}{v+h_c})c_{n}e^{(p_c\tau_c ) } - \delta_ee(t+\tau_c ) , \tag{s32}\\ \frac{db_n}{dt } & = -\beta_{bn}(\frac{v}{v+h_b})b_{n } , \tag{s33 } \\ \frac{dp(t+\tau_b)}{dt } & = \beta_{bn}(\frac{v}{v+h_b})b_{n}e^{(p_b\tau_b ) } - \delta_pp(t+\tau_b ) , \tag{s34}\\ \frac{da_s(t+\tau_b)}{dt } & = p_sp(t+\tau_b)-\delta_sa_s(t+\tau_b ) , \tag{s35 } \\ \frac{da_l(t+\tau_b)}{dt } & = p_lp(t+\tau_b)-\delta_la_l(t+\tau_b ) . \tag{s36}\\ \frac{dc_{m}}{dt } & = -\beta_{cm}(\frac{v}{v+h_{cm}})c_{m } , \tag{s37}\\ \frac{de_m(t+\tau_{cm})}{dt } & = \beta_{cm}(\frac{v}{v+h_{cm}})c_{m}e^{(p_{cm}\tau_{cm } ) } - \delta_ee_m(t+\tau_{cm } ) . \tag{s38}\end{aligned}\ ] ] for variables whose independent variables are not explicitly specified , they are all functions of @xmath105 , i.e. @xmath2 reads @xmath109 . memory @xmath0 t cells show a shorter delay and faster proliferation than naive @xmath0 t cells @xcite . the shortened delay may be caused by a shortened lag time to the first division and/or a reduced delay for effector cells migrating from the lymphatic compartment to the lung @xcite . the former reduction is about 15 hours @xcite and the latter is less than about 12 hours @xcite . thus , we choose @xmath165 ( days ) , correspond to a one day reduction in the delay time compared to the delay of naive cells ( @xmath166 ) . memory @xmath0 t cells show a higher division rate and a lower loss rate than naive @xmath0 t cells @xcite , based on which the net production rate of effector cells for memory @xmath0 t cells is estimated to be about 1.5 times of that for naive cells . thus , we choose @xmath167 ( @xmath168 ) . in the absence of data , we assume @xmath169 and @xmath170 . the initial number of memory @xmath0 t cells is varied as specified in the main text or figures . we assume that the effector @xmath0 t cells produced by either naive or memory @xmath0 t cells are functionally identical ( i.e. then have the same decay rate @xmath40 and killing rate @xmath171 ) . note that we do not model the process of differentiation of effector @xmath0 t cells into memory cells but use a memory cell pool as an initial condition to simulate viral re - infection . .... clear dt=0.1 ; % time step size time=0:dt:100 ; % parameters t0=7e+7;gt=0.8;pv=210;deltav=5.0 ; beta=5e-7;betap=3e-8 ; deltai=2;kappan=2.5;kappae=5e-5 ; phi=0.33;rho=2.6;pf=1e-5;deltaf=2 ; betacn=1;betabn=0.03 ; kappas=0.8;kappal=0.4 ; pc=1.2;pb=0.52;deltae=0.57;deltap=0.5 ; ps=12;pl=4;deltas=2;deltal=0.015 ; tauc=6;taub=4;hc=1e+4;hb=1e+4 ; betacm=1;pcm=1.8;taucm=5 ; % index indicating when delayed process starts indc = round(tauc / dt+1 ) ; % for tauc indcm = round(taucm / dt+1 ) ; % for taucm indb = round(taub / dt+1 ) ; % for taub % variable vectors and initial conditions v = zeros(1,length(time));v(1)=1e+1 ; t = v;t(1)=7e+7;i = t;i(1)=0;r = i;f = i ; cn=100*ones(1,length(time ) ) ; bn=100*ones(1,length(time ) ) ; e = zeros(1,indc+length(time ) ) ; p = i;as = zeros(1,indb+length(time));al = as ; cm=5000*ones(1,length(time ) ) ; em = zeros(1,indcm+length(time ) ) ; init=[v(1),t(1),i(1),r(1),f(1),cn(1),e(1),bn(1),p(1),as(1),al(1 ) , cm(1),em(1 ) ] ' ; options = odeset('reltol',1e-3,'abstol',1e-6 ) ; for i=2:length(time ) [ ~,y ] = ode15s(@odemodel_with_memory,[0 dt],init , options , e(i),al(i),as(i),em(i),tauc , taub , phi , rho , deltaf , gt , pf , pv , beta , betap , kappan , deltav , deltai , betacn , betabn , kappae , kappas , pl , ps , deltal , deltas , deltap , deltae , pc , pb , kappal , hc , hb , t0,betacm , pcm , taucm ) ; v(i)=y(end,1);t(i)=y(end,2);i(i)=y(end,3);r(i)=y(end,4 ) ; f(i)=y(end,5);cn(i)=y(end,6);bn(i)=y(end,8);p(i)=y(end,9 ) ; e(indc+i)=y(end,7 ) ; as(indb+i)=y(end,10);al(indb+i)=y(end,11 ) ; cm(i)=y(end,12);em(indcm+i)=y(end,13 ) ; init = y(end , : ) ' ; % initial condition for next iteration end .... .... function ynew = odemodel_new_with_memory(~,y , e , al , as , em , tauc , taub , phi , rho , deltaf , gt , pf , pv , beta , betap , kappan , deltav , deltai , betacn , betabn , kappae , kappas , pl , ps , deltal , deltas , deltap , deltae , pc , pb , kappal , hc , hb , t0,betacm , pcm , taucm ) % v : viral load % t : target cell % i : infected cell % r : resistant cell % f : ifn % cn : naive cd8 + t cells % e : effector cd8 + t cells % bn : naive b cells % p : plasma b cells % as : short - lived antibodies % al : long - lived antibodies % cm : memory cd8 + t cells % em : effector cd8 + t cells produced from memory cells % y=[v , t , i , r , f , cn , e , bn , p , as , al , cm , em ] ynew = zeros(13,1 ) ; ynew(1)=pv*y(3)-deltav*y(1)-kappas*y(1)*as - kappal*y(1)*al - beta*y(1)*y(2 ) ; ynew(2)=gt*(y(2)+y(4))*(1-(y(2)+y(3)+y(4))/t0)-betap*y(1)*y(2)+rho*y(4)-phi*y(2)*y(5 ) ; ynew(3)=betap*y(1)*y(2)-deltai*y(3)-kappan*y(3)*y(5)-kappae*y(3)*(e+em ) ; ynew(4)=phi*y(2)*y(5)-rho*y(4 ) ; ynew(5)=pf*y(3)-deltaf*y(5 ) ; ynew(6)=-betacn*y(1)./(y(1)+hc)*y(6 ) ; ynew(7)=betacn*y(1)./(y(1)+hc)*y(6)*exp(pc*tauc)-deltae*y(7 ) ; ynew(8)=-betabn*y(1)./(y(1)+hb)*y(8 ) ; ynew(9)=betabn*y(1)./(y(1)+hb)*y(8)*exp(pb*taub)-deltap*y(9 ) ; ynew(10)=ps*y(9)-deltas*y(10 ) ; ynew(11)=pl*y(9)-deltal*y(11 ) ; ynew(12)=-betacm*y(1)./(y(1)+hc)*y(12 ) ; ynew(13)=betacm*y(1)./(y(1)+hc)*y(12)*exp(pcm*taucm)-deltae*y(13 ) ; ....
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myriad experiments have identified an important role for @xmath0 t cell response mechanisms in determining recovery from influenza a virus infection .
animal models of influenza infection further implicate multiple elements of the immune response in defining the dynamical characteristics of viral infection . to date , influenza virus models , while capturing particular aspects of the natural infection history , have been unable to reproduce the full gamut of observed viral kinetic behaviour in a single coherent framework . here , we introduce a mathematical model of influenza viral dynamics incorporating all major immune components ( innate , humoral and cellular ) and explore its properties with a particular emphasis on the role of cellular immunity . calibrated against a range of murine data ,
our model is capable of recapitulating observed viral kinetics from a multitude of experiments .
importantly , the model predicts a robust exponential relationship between the level of effector @xmath0 t cells and recovery time , whereby recovery time rapidly decreases to a fixed minimum recovery time with an increasing level of effector @xmath0 t cells .
we find support for this relationship in recent clinical data from influenza a(h7n9 ) hospitalised patients .
the exponential relationship implies that people with a lower level of naive @xmath0 t cells may receive significantly more benefit from induction of additional effector @xmath0 t cells arising from immunological memory , itself established through either previous viral infection or t cell - based vaccines .
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continuous - variable quantum - key distribution ( cvqkd ) , as an unconditionally secure communication scheme between two legitimate parties alice and bob , has achieved advanced improvements in theoretical analysis and experimental implementation in recent years @xcite . practical implementation systems , such as fiber - based gaussian - modulated @xcite and discrete - modulated @xcite coherent - state protocol qkd systems over tens of kilometers , have been demonstrated in a few groups . the unconditional security of such systems with prepare - and - measure ( pm ) implementation has been confirmed by the security analysis of the equivalent entanglement - based ( eb ) scheme @xcite . however , the traditional security analysis of the eb scheme of cvqkd just includes the signal beam and not the local oscillator ( lo ) , which is an auxiliary light beam used as a reference to define the phase of the signal state and is necessary for balanced homodyne detection . this will leave some security loopholes for eve because lo is also unfortunately within eve s manipulating domain . the necessity of monitoring lo intensity for the security proofs in discrete qkd protocols embedded in continuous variables has been discussed @xcite . moreover , in @xcite , the excess noise caused by imperfect subtraction of balanced homodyne detector ( bhd ) in the presence of lo intensity fluctuations has been noted and quantified with a formulation . however , in the practical implementation of cvqkd , shot noise scaling with lo power measured before keys distribution is still assumed to keep constant if the fluctuations of lo intensity are small . and in this circumstance , pulses with large fluctuation are just discarded as shown in @xcite . unfortunately , this will give eve some advantages in exploiting the fluctuation of lo intensity . in this paper , we first describe bob s measurements under this fluctuation of lo intensity , and propose an attacking scheme exploiting this fluctuation . we consider the security of practical cvqkd implementation under this attack and calculate the secret key rate with and without bob monitoring the lo for reverse and direct reconciliation protocol . and then , we give a qualitative analysis about the effect of this lo intensity fluctuation on the secret key rate alice and bob hold . we find that the fluctuation of lo could compromise the secret keys severely if bob does not scale his measurements with the instantaneous lo intensity values . finally , we briefly discuss the accurate monitoring of lo intensity to confirm the security of the practical implementation of cvqkd . generally , in practical systems of cvqkd , the local oscillator intensity is always monitored by splitting a small part with a beam splitter , and pulses with large lo intensity fluctuation are discarded too . however , even with such monitoring , we do not yet clearly understand how fluctuation , in particular small fluctuation , affects the secret key rate . to confirm that the secret key rate obtained by alice and bob is unconditionally secure , in what follows , we will analyze the effects of this fluctuation on the secret key rate only , and do not consider the imperfect measurement of bhd due to incomplete subtraction of it in the presence of lo intensity fluctuations , which has been discussed in @xcite . ideally , with a strong lo , a perfect pulsed bhd measuring a weak signal whose encodings are @xmath0 will output the results@xcite , @xmath1 where _ k _ is a proportional constant of bhd , @xmath2 is the amplitude of lo , @xmath3 is the relative phase between the signal and lo except for the signal s initial modulation phase . so scaling with lo power or shot noise , the results can be recast as @xmath4 with @xmath3 in eq . ( [ eq : x0 ] ) is 0 or @xmath5 . here the quadratures @xmath6 and @xmath7 are defined as @xmath8 and @xmath9 , where @xmath10 is the quadrature of the vacuum state . however , in a practical system , the lo intensity fluctuates in time during key distribution . with a proportional coefficient @xmath11 , practical lo intensity can be described as @xmath12 , where @xmath2 is the initial amplitude of lo used by normalization and its value is calibrated before key distribution by alice and bob . if we do not monitor lo or quantify its fluctuation @xcite , especially just let the outputs of bhd scale with the initial intensity or power of lo , the outputs then read @xmath13 unfortunately , this fluctuation will open a loophole for eve , as we will see in the following sections . in conventional security analysis , like the eb scheme equivalent to the usual pm implementation depicted in fig . [ fig:1](a ) , lo is not taken into consideration and its intensity is assumed to keep unchanged . however , in practical implementation , eve could intercept not only the signal beam but also the lo , and she can replace the quantum channel between alice and bob with her own perfect quantum channel as shown in figs . [ fig:1](b ) and [ fig:1](c ) . in so doing , eve s attack can be partially hidden by reducing the intensity of lo with a variable attenuator simulating the fluctuation without changing lo s phase , and such an attack can be called a lo intensity attack ( loia ) . in the following analysis , we will see that , in the parameter - estimation procedure between alice and bob , channel excess noise introduced by eve can be reduced arbitrarily , even to its being null , just by tuning the lo transmission . consequently , alice and bob would underestimate eve s intercepted information and eve could get partial secret keys that alice and bob hold without being found under this attack . figure [ fig:1](b ) describes the loia , which consists of attacking the signal beam with a general gaussian collective attack @xcite and attacking the lo beam with an intensity attenuation by a non - changing phase attenuator * a * , such as a beam splitter whose transmission is variable . this signal - beam gaussian collective attack consists of three steps : eve interacts her ancilla modes with the signal mode by a unitary operation * u * for each pulse and stores them in her quantum memory , then she makes an optimal collective measurement after alice and bob s classical communication . figure [ fig:1](c ) is one practical loia with * u * being a beam - splitter transformation . its signal attack is also called an entangling cloner attack , which was presented first by grosshan @xcite and improved by weedbrook @xcite . in appendix [ sec : security ] , we will demonstrate that with this entangling cloner , eve can get the same amount of information as that shown in fig . [ fig:1](b ) . we analyze a practical cvqkd system with homodyne protocol to demonstrate the effect of loia on the secret key rate , and for simplicity we do not give the results of the heterodyne protocol , which is analogous to the homodyne protocol . in the usual pm implementation , alice prepares a series of coherent states centered on @xmath0 with each pulse , and then she sends them to bob through a quantum channel which might be intercepted by eve . here , @xmath14 and @xmath15 , respectively , satisfy a gaussian distribution independently with the same variance @xmath16 and zero mean . this initial mode prepared by alice can be described as @xmath17 , and @xmath18 is the quadrature variable . here @xmath9 describes the quadrature of the vacuum mode . note that we denote an operator with a hat , while without a hat the same variable corresponds to the classical variable after measurement . so the overall variance of the initial mode prepared by alice is @xmath19 . when this mode comes to bob , bob will get a mode @xmath20 , @xmath21 where @xmath22 describes eve s mode introduced through the quantum channel whose quadrature variance is @xmath23 . bob randomly selects a quadrature to measure , and if eve attenuates the lo intensity during the key distribution , but bob s outputs still scale with the initial lo intensity , just as in eq . ( [ eq : x0 ] ) , he will get the measurement @xmath24 where @xmath25 is @xmath26 ( or equivalently @xmath27 ) . however , if bob monitors lo and also scales with the instantaneous intensity value of lo with each pulse , he will get @xmath25 without any loss of course . note that , for computation simplicity , hereafter we assume the variable transmission rate @xmath28 ( or attenuation rate @xmath29 ) of each pulse of lo is the same without loss of generality . thus the variance of bob s measurements and conditional variance on alice s encodings with and without monitoring ( in what follows , without monitoring specially indicates that bob s measurement is obtained just by scaling with the initial lo intensity instead of monitoring instantaneous values , and vice versa ) can be given by @xmath30\;,\label{eq : vbm}\\ v_{b|a}&=t+(1-t)n\ ; , \\ v_{b|a}^w&=\eta\left[t+(1-t)n\right]\;,\end{aligned}\ ] ] where the superscript @xmath31 indicates without monitoring " and all variances are in shot - noise units ; the conditional variance is defined as @xcite @xmath32 hence , the covariance matrix of alice s and bob s modes can be obtained as @xmath33\mathbb{i } \end{pmatrix}\label{eq : rab},\end{split } \\ & \gamma_{ab}^w=\begin{pmatrix } v\mathbb{i } & \sqrt{\eta t(v^2 - 1)}\sigma_z\\ \sqrt{\eta t(v^2 - 1)}\sigma_z & \eta[tv+(1-t)n]\mathbb{i } \end{pmatrix},\end{aligned}\ ] ] where @xmath34 is the pauli matrix and @xmath35 is a unit matrix . from eqs . ( [ eq : vb ] ) and ( [ eq : vbm ] ) we can derive that the channel transmission and excess noise are @xmath36 , @xmath37 with monitoring , and @xmath38 , @xmath39 without monitoring . hence , by attenuating the lo intensity as fig . [ fig:1 ] shows , to make @xmath40 , eve could arbitrarily reduce @xmath41 to zero , thus she will get the largest amount of information permitted by physics . in the following numerical simulation we always make @xmath42 , namely , @xmath43 . thus , the covariance matrix @xmath44 and eve s introducing noise @xmath23 [ in an entangling cloner it is eve s epr state s variance as fig . [ fig:1](c ) shows ] should be selected to be @xmath45 to estimate the secret key rate , without loss of generality , we first analyze the reverse reconciliation then consider the direct reconciliation . from alice and bob s points of view , the secret key rate for reverse reconciliation with monitoring or not are given , respectively , by @xmath46 where the mutual information between alice and bob with and without monitoring are the same , and that is @xmath47 this is because bob s measurements in these two cases are just different with a coefficient @xmath28 , and they correspond with each other one by one , so they are equivalent according to the data - processing theorem @xcite . however , the mutual information between eve and bob given by the holevo bound @xcite in these two cases is not identical . as the previous analysis showed , in bob s point of view , channel transmission and the excess noise estimation are different . but from eve s point of view , they are identical according to the data - processing theorem because she estimates bob s measurements in these two cases just by multiplying a coefficient @xmath28 . we ll calculate the real information intercepted by eve first . it can be given by @xmath48 where @xmath49 is the von neumann entropy @xcite . for a gaussian state @xmath50 , this entropy can be calculated by the symplectic eigenvalues of the covariance matrix @xmath51 characterizing @xmath50 @xcite . to calculate eve s information , first eve s system @xmath52 can purify @xmath53 permitted by quantum physics , so that @xmath54 . second , after bob s projective measurement , the system @xmath55 is pure , so that @xmath56 . designating @xmath57 , @xmath58 , and @xmath59 , the symplectic eigenvalues of @xmath60 are given by @xmath61 where @xmath62 and @xmath63 . similarly , the entropy @xmath64 is determined by the symplectic eigenvalue @xmath65 of the covariance matrix @xmath66 @xcite , namely , @xmath67 where @xmath68 and @xmath69 stands for the moore - penrose inverse of a matrix . then @xmath70 , and the holevo bound reads @xmath71 where @xmath72 . however , eve s information estimated by bob without monitoring is given by @xmath73 by substituting eqs . ( [ eq : xbe ] ) and ( [ eq : xbem ] ) into eqs . ( [ eq : krr ] ) and ( [ eq : krrm ] ) , respectively , the secret key rate with and without bob s monitoring can be obtained . however , the secret key rate in eq . ( [ eq : krrm ] ) without monitoring is unsecured in evidence . eve s interception of partial information from @xmath74 is not detected , in other words , alice and bob underestimate eve s information without realizing it . actually , the real or unconditionally secure secret key rate @xmath74 , which we called a truly secret key rate , should be available by replacing @xmath75 in eq . ( [ eq : krrm ] ) with eq . ( [ eq : xbe ] ) . note that it is identical with the monitoring secret key rate in eq . ( [ eq : krr ] ) due to eq . ( [ eq : iab ] ) . we investigate the secret key rate @xmath74 bob measured without monitoring and the true one or equivalently monitoring one @xmath76 for reverse reconciliation under eve attacking the intensity of lo during key distribution . as fig . [ fig:2 ] shows , with various values of transmission of lo that can be controlled by eve , the truly secret key rate alice and bob actually share decreases rapidly over long distances or small channel transmissions . ( color online ) reverse reconciliation pseudosecret key rate and the truly secret one vs channel transmission @xmath36 under loia . solid lines are secret key rate estimated by bob without monitoring lo intensity and dashed lines are the truly secret ones . colored lines correspond to the lo transmissions @xmath28 as labeled . here alice s modulation variance @xmath77 . ] additionally , because the mutual information between alice and bob with and without monitoring is identical as eq.([eq : iab ] ) shows , subtracting eq . ( [ eq : krr ] ) from eq . ( [ eq : krrm ] ) we can estimate eve s intercepted information @xmath78 which is plotted in fig . [ fig:3 ] . we find that eve could get partial or full secret keys which alice and bob hold by controlling the different transmissions of lo . taking a 20-km transmission distance as an example , surprisingly , just with lo intensity fluctuation or attenuating rate 0.08 , eve is able to obtain the full secret keys for reverse reconciliation without bob s monitoring the lo . we now calculate the secret key rate for direct reconciliation , which is a little more complicated , and we investigate the effect of lo intensity attack by eve on cvqkd . the secret key rate estimated by bob with and without lo monitoring is given , respectively , by @xmath79 note that we have already calculated @xmath80 and @xmath81 in eq.([eq : iab ] ) and they are identical for direct and reverse reconciliation . for eve we have @xmath82 where @xmath54 has been already computed in the previous section , and @xmath83 using the fact that after alice s projective measurement on modes @xmath84 and @xmath85 obtaining @xmath86 in the eb scheme shown in fig . [ fig:1](a ) , the system @xmath87 is pure . to calculate @xmath88 , we have to compute the symplectic eigenvalues of covariance matrix @xmath89 , which is obtained by @xmath90 where @xmath91 and @xmath92 can be read in the decomposition of the matrix @xmath93 which is available by elementary transformation of the matrix [ see fig . [ fig:1](a ) ] @xcite @xmath94 where @xmath95 is a unit matrix . it is obtained by applying a homodyne detection on mode a after mixing @xmath84 and @xmath85 with a balanced beam - splitter transformation @xmath96 . the matrix @xmath97 and @xmath98 actually is @xmath60 in eq . ( [ eq : rab ] ) , @xmath99 is a unit matrix . so we can get @xmath100 and the symplectic eigenvalues of it @xmath101 where @xmath102 and @xmath103 . the holevo bound then reads @xmath104 substituting eqs . ( [ eq : xae ] ) and ( [ eq : xaem ] ) into eqs . ( [ eq : kdr ] ) and ( [ eq : kdrm ] ) , respectively , the secret key rates in these two cases are obtained . in fig . [ fig:4 ] , we plotted them for channel transmission @xmath36 with various values of @xmath28 and find that the difference between the pseudosecret key rate with bob not monitoring lo and the truly secret one is still increasing with the channel transmission @xmath36 becoming smaller . for eve , when bob does not monitor lo , she will get the partial or total secret key rate @xmath105 without being found by subtracting eq . ( [ eq : kdr ] ) from eq . ( [ eq : kdrm ] ) when she reduces the intensity of lo . in fig . [ fig:5 ] , we plotted the pseudosecret key rate for direct reconciliation and the mutual information overestimated by alice and bob . we find that for short distance communication ( less 15 km or 3 db limit ) , a small fluctuation of lo intensity could still hide eve s attack partially or totally . note that in the above estimation we assume each pulse s transmission rate @xmath28 ( or attenuation rate @xmath29 ) is identical . however , when @xmath28 is different for each pulse ( eve simulates the fluctuation of lo to hide her dramatic attack on lo ) , eve still could intercept as much as or even more secret key rates than above for reverse and direct reconciliation , as long as the largest value of @xmath28 among all pulses ( or approximately most pulse transmission rates ) is smaller than the above constant value . our analysis shows that reverse reconciliation is more sensitive than direct reconciliation about the fluctuation of lo intensity , and , even with a small attenuation of lo intensity , eve can get full secret keys but not be found . this is consistent with the fact that channel excess noise has a more severe impact on reverse reconciliation than on direct reconciliation . of course , when the intensity of lo fluctuates above the initial calibrated value ( i.e. , @xmath106 ) , eve could not get any secret keys , but alice and bob would overestimate eve s intercepted information due to the overestimation of channel excess noise . however , when lo fluctuates around the initial calibrated value , how to quantify eve s information is still an open question , because the distribution of the fluctuation of lo ( or @xmath28 ) is not a normal distribution and unclear for alice and bob due to eve s arbitrary manipulation . but in this circumstance , eve still could intercept partial secret keys if she increases the channel excess noise of one part of the signal pulses when she controls @xmath107 and decreases it for the other part when controlling @xmath106 , i.e. , making the overall estimated excess noise by alice and bob lower than the real one . remarkably , lo intensity fluctuation opens a loophole for eve to attack the practical system , especially in the case of communication with low channel transmission or over long distance . consequently , in the practical implementation of cvqkd , we must monitor the lo fluctuation carefully and in particular scale the measurements with instantaneous intensity values of lo . alternatively , we can also scale with the lowest intensity value of lo if the fluctuations are very small , but it will estimate the secret key rate pessimistically thus leading to the reduction of the efficiency of the key distribution . however , we can not use the average intensity value of lo to normalize the measurements as most current implementations do , because it still could overestimate the secret key rate for alice and bob . additionally , for reverse reconciliation communication over long distance , very small fluctuation of lo might compromise the secret key rate completely , which presents a big challenge for accurately monitoring lo intensity . finally , we point out that in this paper we do not consider the imperfections of bhd such as detection efficiency , electronic noise , and incomplete subtraction , which may make lo intensity fluctuation have a more severe impact on estimating the secret key rate for alice and bob . in conclusion , we have analyzed the effect of lo intensity fluctuation on the secret key rate estimation of alice and bob for reverse and direct reconciliation . incredibly , bob s estimation of the secret key rate will be compromised severely without monitoring lo or if his measurements do not scale with lo instantaneous intensity values even with monitoring but just discard large fluctuation pulses like in @xcite . furthermore , we have shown that eve could hide her attack partially by reducing the intensity of lo and even could steal the total secret keys alice and bob share without being found by a small attenuation of lo intensity , especially for reverse reconciliation . finally , we have also briefly discussed the monitoring of lo and pointed out that it would be a challenge for highly accurate monitoring . this work is supported by the national natural science foundation of china , grant no . is supported by the program for new century excellent talents . c.m . is supported by the hunan provincial innovation foundation for postgraduates . acknowledge support from nudt under grant no . kxk130201 . in this appendix , we calculate the holevo bound obtained by eve for direct and reverse reconciliation using weedbrook s entangling cloner model @xcite , and then give the secret key rate shared by alice and bob under loia . we begin the analysis by calculating the von neumann entropy of eve s intercepting state first . as fig . [ fig:1](c ) shows , the entangling cloner consists of eve replacing the gaussian quantum channel between alice and bob with a beam splitter of transmission @xmath36 and an epr pair of variance @xmath23 . half of the epr pair mode @xmath108 is mixed with alice s mode in the beam splitter and is sent to bob to match the noise of the real channel by tuning n. the other half mode @xmath109 is kept by eve to reduce the uncertainty on one output of the beam splitter , the mode @xmath110 , which can be read as @xmath111 where @xmath112 is the quadrature of mode @xmath108 . thus , the variance of mode @xmath110 is given by @xmath113 and the conditional variance @xmath114 can be calculated as , using eq . ( [ eq : cvar ] ) , @xmath115 hence , eve s covariance matrix can be obtained as @xmath116 where @xmath117 and the notation @xmath118 stands for a matrix with the arguments on the diagonal elements and zeros everywhere else . the symplectic eigenvalues of this covariance matrix are given by @xmath119 where @xmath120 , and @xmath121 . hence , the von neumann entropy of eve s state is given by @xmath122 for the direct reconciliation protocol of cvqkd , the holevo bound between eve and alice is given by eq . ( [ eq : xae0 ] ) , where @xmath123 has been calculated by eq . ( [ eq : se ] ) . @xmath124 can be obtained by the conditional covariance matrix @xmath125 and its symplectic eigenvalues are given by @xmath126 where @xmath127 , and @xmath128 . thus , the conditional entropy is @xmath129 substituting eqs . ( [ eq : se ] ) and ( [ eq : sea ] ) into eq . ( [ eq : xae0 ] ) , we can get the mutual information between alice and eve , @xmath130 under loia , bob s estimation of the holevo bound without monitoring lo intensity then reads , using eq . ( [ eq : xae1 ] ) , @xmath131 with eqs . ( [ eq : xae1 ] ) and ( [ eq : xaem1 ] ) , the secret key rates in eqs . ( [ eq : kdr ] ) and ( [ eq : kdrm ] ) then can be calculated respectively , and the calculation numerically demonstrates that they are perfectly consistent with the fig . [ fig:4 ] . the calculation of the holevo bound between eve and bob for reverse reconciliation is a bit more complicated . using eq . ( [ eq : xbe0 ] ) , we only need to calculate the conditional entropy @xmath132 , which is determined by the symplectic eigenvalues @xmath133 of the covariance matrix @xmath134 , @xmath135 where @xmath136 and @xmath137 , @xmath138 . then , @xmath134 can be recast as @xmath139 where + @xmath140 , @xmath141 , + and + @xmath142 . + hence , its symplectic eigenvalues are given by @xmath143 where @xmath144 , @xmath145 , and @xmath146 is the determinant of a matrix . so , we get the conditional entropy @xmath147 and then the holevo bound @xmath148 consequently , without monitoring lo intensity , alice and bob will give eve the holevo bound @xmath149 substituting eqs . ( [ eq : xbe1 ] ) and ( [ eq : xbem1 ] ) into eqs . ( [ eq : krr ] ) and ( [ eq : krrm ] ) , respectively , the secret key rates with and without bob s monitoring can be obtained , and for channel transmission @xmath36 with various values of @xmath28 , they are numerically demonstrated to be perfectly consistent with fig . [ fig:2 ] , too . hence , it also indirectly confirms that either for direct or reverse reconciliation , the entangling cloner could reach the holevo bound against the optimal gaussian collective attack . in this paper , lo intensity fluctuation indicates the deviation of each pulse s intensity from the initial calibrated value during the key distribution . it does not mean the quantum fluctuation of each pulse itself , because lo is a strong classical beam whose quantum fluctuation is very small relative to itself and can be neglected .
|
we consider the security of practical continuous - variable quantum key distribution implementation with the local oscillator ( lo ) fluctuating in time , which opens a loophole for eve to intercept the secret key .
we show that eve can simulate this fluctuation to hide her gaussian collective attack by reducing the intensity of the lo .
numerical simulations demonstrate that , if bob does not monitor the lo intensity and does not scale his measurements with the instantaneous intensity values of lo , the secret key rate will be compromised severely .
| 7,544 | 123 |
it is by now standard to parameterize transverse momentum distributions with functions having a power law behaviour at high momenta . this has been done by the star @xcite and phenix @xcite collaborations at rhic and by the alice @xcite , atlas @xcite and cms @xcite collaborations at the lhc . in this talk we would like to pursue the use of the tsallis distribution to describe transverse momentum distributions at the highest beam energies . + in the framework of tsallis statistics @xcite the entropy @xmath1 , the particle number , @xmath2 , the energy density @xmath3 and the pressure @xmath4 are given by corresponding integrals over the tsallis distribution : @xmath5^{-\frac{1}{q-1 } } .\label{tsallis}\ ] ] it can be shown ( see e.g. @xcite ) that the relevant thermodynamic quantities are given by : @xmath6 , \label{entropy } \\ n & = & gv\int\frac{d^3p}{(2\pi)^3 } f^q , \label{number } \\ \epsilon & = & g\int\frac{d^3p}{(2\pi)^3}e f^q , \label{epsilon}\\ p & = & g\int\frac{d^3p}{(2\pi)^3}\frac{p^2}{3e } f^q\label{pressure } .\end{aligned}\ ] ] where @xmath7 and @xmath8 are the temperature and the chemical potential , @xmath9 is the volume and @xmath10 is the degeneracy factor . we have used the short - hand notation @xmath11 often referred to as q - logarithm . it is straightforward to show that the relation @xmath12 ( where @xmath13 refer to the densities of the corresponding quantities ) is satisfied . the first law of thermodynamics gives rise to the following differential relations : @xmath14 since these are total differentials , thermodynamic consistency requires the following maxwell relations to be satisfied : @xmath15 this is indeed the case , e.g. for eq . this follows from @xmath16^{-\frac{q}{q-1 } } \nonumber \\ & = & - g\int\frac{d^3p}{(2\pi)^3}\frac{p^2}{3 } \frac{d}{pdp}\left [ 1 + ( q-1 ) \frac{e-\mu}{t}\right]^{-\frac{q}{q-1 } } \nonumber \\ & = & g\int\frac{d\cos\theta d\phi dp}{(2\pi)^3 } \left [ 1 + ( q-1 ) \frac{e-\mu}{t}\right]^{-\frac{q}{q-1 } } \frac{d}{dp}\frac{p^3}{3 } \nonumber \\ & = & n \nonumber\end{aligned}\ ] ] after an integration by parts and using @xmath17 . + following from eq . , the momentum distribution is given by : @xmath18^{-q/(q-1 ) } , \label{tsallismu}\ ] ] or , expressed in terms of transverse momentum , @xmath19 , the transverse mass , @xmath20 , and the rapidity @xmath21 @xmath22^{-q/(q-1 ) } . \label{tsallismu1}\ ] ] at mid - rapidity , @xmath23 , and for zero chemical potential , as is relevant at the lhc , this reduces to @xmath24^{-q/(q-1)}. \label{tsallisfit1}\ ] ] in the limit where the parameter @xmath0 goes to 1 it is well - known that this reduces to the standard boltzmann distribution : @xmath25 the parameterization given in eq . is close to the one used by various collaborations @xcite : @xmath26^{-n } , \label{alice}\ ] ] where @xmath27 and @xmath28 are fit parameters . this corresponds to substituting @xcite @xmath29 and @xmath30 after this substitution eq . becomes @xmath31^{-q/(q-1)}\nonumber\\ & & \left [ 1 + ( q-1)\frac{m_t}{t } \right]^{-q/(q-1 ) } . \label{alice2}\end{aligned}\ ] ] at mid - rapidity @xmath32 and zero chemical potential , this has the same dependence on the transverse momentum as eq . apart from an additional factor @xmath33 on the right - hand side of eq . . however , the inclusion of the rest mass in the substitution eq . is not in agreement with the tsallis distribution as it breaks @xmath33 scaling which is present in eq . but not in eq . . the inclusion of the factor @xmath33 leads to a more consistent interpretation of the variables @xmath0 and @xmath7 . + a very good description of transverse momenta distributions at rhic has been obtained in refs @xcite on the basis of a coalescence model where the tsallis distribution is used for quarks . tsallis fits have also been considered in ref . @xcite but with a different power law leading to smaller values of the tsallis parameter @xmath0 . + interesting results were obtained in refs . @xcite where spectra for identified particles were analyzed and the resulting values for the parameters @xmath0 and @xmath7 were considered . + the transverse momentum distributions of identified particles , as obtained by the alice collaboration at 900 gev in @xmath34 collisions , are shown in figure fig : positive . the fit for positive pions was made using @xmath35^{-q/(q-1)}. \label{tsallisfitpi}\ ] ] with @xmath0 , @xmath7 and @xmath9 as free parameters . + + in figure strange we show fits to the transverse momentum distributions of strange particles obtained by the alice collaboration @xcite in @xmath34 collisions at 900 gev . + similarly we show fits to the transverse momentum distributions obtained by the cms collaboration @xcite in figure cms and by the atlas collaboration in figure chargedatlas . + the transverse momentum distributions of charged particles were fitted using a sum of three tsallis distributions , the first one for @xmath36 , the second one for @xmath37 and the third one for protons @xmath38 . the relative weights between these were determined by the corresponding degeneracy factors , i.e. 1 for for @xmath36 and @xmath37 and 2 for protons . the fit was taken at mid - rapidity and for @xmath39 using the following expression was used @xmath40^{-\frac{q}{q-1}},\ ] ] where @xmath41 and @xmath42 , @xmath43 and @xmath44 . the factor @xmath45 in front of the right hand side of this equation takes into account the contributions of the antiparticles @xmath46 . the tsallis distribution also describes the transverse momentum distributions of charged particles in @xmath47 collisions in all pseudorapidity intervals as shown in figure ppb . + collisions obtained by the alice collaboration @xcite using the tsallis distribution.,height=377 ] collisions obtained by the alice collaboration @xcite using the tsallis distribution.,height=377 ] obtained from fits to transverse momentum spectra described in the text.,height=377 ] obtained from fits to transverse momentum spectra described in the text.,height=377 ] the tsallis distribution described here in eq . leads to excellent fits to the transverse momentum distributions in high energy @xmath34 and @xmath47 collisions . the values obtained for the tsallis parameter @xmath0 are truly remarkably consistent , a feature which does not become apparent when using the parametrization of eq . . 90 b. i. abelev et al . ( star collaboration ) , phys . c * 75 * , 064901 ( 2007 ) . a. adare et al . ( phenix collaboration ) , phys . c * 83 * , 052004 , ( 2010 ) ; phys . rev . c * 83 * , 064903 ( 2011 ) . alice collaboration , eur . j. c * 71 * 1594 ( 2011 ) ; eur . j. c * 71 * 1655 ( 2011 ) ; phys . b * 693 * ( 2010 ) 53 ; phys . ( 2013 ) 082302 . atlas collaboration , new j. phys . * 13 * ( 2011 ) 053033 . cms collaboration , phys . * 105 * ( 2010 ) 022002 ; eur . j. c * 72 * ( 2012 ) 2164 . c. tsallis , j. statist . phys . * 52 * , 479 ( 1988 ) . t.bir , g. purcsel , k. rmssy , eur . j. a * 40 * ( 2009 ) 325 . j. m. conroy , h. g. miller , a. r. plastino , phys . a * 374 * , 4581 ( 2010 ) . j. cleymans and d. worku , j. phys . g * 39 * ( 2012 ) 025006 . j. cleymans and d. worku , eur . j. a * 48 * ( 2012 ) 160 . k. rmssy , t.s . bir , phys . b * 689 * ( 2010 ) 14 . k. rmssy , t.s . bir , j. phys . g * 36 * ( 2009 ) 064044 . cheuk - yin wong , g. wilk , acta physica polonica , 43 ( 2012 ) 2047 . t. wibig , j. phys . g : nucl . part . phys . * 37 * 115009 ( 2010 ) . t. wibig , i. kurp , jhep * 0312 * 039 ( 2003 ) . l. marques , e.andrade-ii , a. deppman , arxiv:1210.1725[hep - ph ] i. sena , a. deppman , eur . j. a 49 ( 2013 ) 17 ; arxiv:1209.2367[hep - ph ] k. rmssy , arxiv:1212.0260[hep - ph ] . j. cleymans , g.i . lykasov , a.s . parvan , a.s . sorin , o.v . teryaev , d. worku phys . b * 723 * ( 2013 ) 351 . [ arxiv:1104.0620 [ hep - ph ] ] .
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an overview is presented of transverse momentum distributions of particles at the lhc using the tsallis distribution .
the use of a thermodynamically consistent form of this distribution leads to an excellent description of charged and identified particles .
the values of the tsallis parameter @xmath0 are truly remarkably consistent .
= by -1
| 3,113 | 80 |
magnetic superconductivity has attracted much research attention since it was reported in the strongly - correlated rusr@xmath13rcu@xmath13o@xmath14 ru-1212 cuprate system ( r = sm , eu , gd , y ) with the tetragonal space group p4/mbm . @xcite the ru magnetic moments order weak - ferromagnetically ( wfm ) with ordering temperature t@xmath15 130 k. high - t@xmath16 superconductivity occurs in the quasi-2d cuo@xmath13 bi - layers from doped holes with maximum superconducting transition onset t@xmath16(onset ) @xmath6 60 k for r = gd and coexists with the wfm order . a crossover from anisotropic 2d - like to less - anisotropic 3d - like structure was observed near r = sm , along with a metal - insulator transition . no superconductivity can be detected for the mott insulators r = pr and nd . since the oxygen content for all samples is close to eight after oxygen annealing , the variation of t@xmath16 with rare - earth ions indicates a self - doping of electrons from cuo@xmath13 layers to ruo@xmath17 layers . such self - doping creates hole carriers in cuo@xmath13 layers and conduction electrons in ruo@xmath17 layers . the ru l@xmath18-edge x - ray absorption near - edge spectrum ( xanes ) of rusr@xmath13gdcu@xmath13o@xmath14 indicates that ru valence is basically at ru@xmath19 ( 4d - t@xmath20 , s = 3/2 ) state with small amount ( @xmath620 @xmath21 ) of ru@xmath22 ( 4d - t@xmath23 , s = 1 in low spin state ) which corresponds to doped electrons . @xcite the strong antiferromagnetic superexchange interaction between ru@xmath19 moments is responsible for the basic g - type antiferromagnetic order observed in the neutron diffraction study . @xcite the weak ferromagnetic component observed from magnetic susceptibility and nmr spectrum is probably due to weak - ferromagnetic double - exchange interaction through doped conduction electrons in the metallic ruo@xmath17 layers . since the magnetic superexchange and double - exchange interaction is anisotropic in general , the study of anisotropic physical properties is crucial for this quasi-2d system . in this report , we align the microcrystalline rusr@xmath13rcu@xmath13o@xmath14 ( r = rare earths ) powder ( @xmath61 - 10 @xmath7 m ) in magnetic field to investigate the anisotropic properties . the stoichiometric rusr@xmath13gdcu@xmath13o@xmath14 bulk sample was synthesized by the standard solid - state reactions . high - purity ruo@xmath13 ( 99.99 @xmath21 ) , srco@xmath18 ( 99.9 @xmath21 ) , gd@xmath13o@xmath18 ( 99.99 @xmath21 ) and cuo ( 99.99 @xmath21 ) preheated powders with the nominal composition ratio of ru : sr : gd : cu = 1:2:1:2 were well mixed and calcined at 960@xmath24c in air for 16 hours . the calcined powders were then pressed into pellets and sintered in flowing n@xmath13 gas at 1015@xmath24c for 10 hours to form sr@xmath13gdruo@xmath17 and cu@xmath13o precursors . the sintered pellets were then heated at 1060 - 1065@xmath24c in flowing o@xmath13 gas for 7 days to form the ru-1212 phase and slowly furnace cooled to room temperature with a rate of 15@xmath24c per hour . for powder alignment in magnetic field , samples were ground into powders with an average microcrystalline grain size of 1 - 10 @xmath7 m and mixed with epoxy ( 4-hour curing time ) in a quartz tube ( @xmath25 = 8 mm ) with the ratio of powder : epoxy = 1:5 then immediately put into the alignment field environments ( simple field or rotation - field alignment ) . for simple powder alignment , the mixture was placed in a 14-t superconducting magnet at room temperature in flowing n@xmath13 gas and slowly hardened overnight as shown in figure 1 . the powder x - ray diffraction pattern of three typical aligned powder - in - epoxy samples rusr@xmath13rcu@xmath13o@xmath14 ( r = sm , eu , gd@xmath2dy@xmath2 ) are shown collectively in figure 2 . for r = sm ( as well as for r = pr and nd ) , no magnetic alignment can be achieved . the lack of magnetic anisotropy may closely relate to the variation of tetragonal lattice parameters where @xmath5/3 @xmath26/@xmath27 for r = sm with @xmath28 = 0.5448 nm and @xmath5 = 1.1560 nm ( space group p4/mbm ) as shown in figure 3 . for r = eu ( as well as for r = gd ) , partial ( @xmath6 90@xmath21 ) @xmath29-plane aligned along alignment magnetic field b@xmath8 is observed through the appearance of enhanced ( @xmath300 ) diffraction lines . a small amount of srruo@xmath18 impurity is presented . the @xmath29-plane alignment may be due to the fact that @xmath5/3 @xmath31/@xmath27 for r = eu ( @xmath28 = 0.5435 nm , @xmath5 = 1.1572 nm ) . for metastable compound r = gd@xmath2dy@xmath2 near the phase boundary with some unreacted precursor sr@xmath13rruo@xmath17 , partially @xmath5-axis alignment along b@xmath8 is detected with enhanced ( 00@xmath32 ) lines due to @xmath5/3 @xmath33/@xmath27 in this compound ( @xmath28 = 0.5426 nm , @xmath5 = 1.1508 nm ) . schematic diagram for the magnetic field powder alignment method in a 14 t superconducting magnet at 300 k. ] powder x - ray diffraction patterns for rusr@xmath13rcu@xmath13o@xmath14 aligned powder . ( a ) r = sm , ( b ) r = eu , ( c ) r = gd@xmath2dy@xmath2 . ] the variation of superconducting transition t@xmath16 and tetragonal lattice parameters @xmath28 , @xmath5 with rare earth ionic radius r@xmath34 for rusr@xmath13rcu@xmath13o@xmath14 system ( r = pr - dy ) . ] the phase diagram in figure 3 indicates a structural crossover from less - anisotropic 3d - like ( @xmath5/3 @xmath26 ) to anisotropic 2d - like structure ( @xmath5/3 @xmath35 @xmath28/@xmath27 ) near r = sm , along with an insulator - to - metal transition . superconductivity appears only in the quasi-2d metallic region with resistivity onset transition temperature t@xmath36 0 for r = sm , t@xmath16 = 36 k for r = eu , t@xmath16 = 56 k for gd , and t@xmath16 = 55 k for metastable r = gd@xmath2dy@xmath2 . for r = eu with @xmath29-plane aligned along b@xmath8 , @xmath5-axis can be in any direction within the plane perpendicular to b@xmath8 . to obtain the @xmath5-axis aligned powder , a field - rotation alignment method is used as shown in figure 4 . since @xmath29-plane is fixed along b@xmath8 , the rotation of quartz tube ( 10 rpm ) perpendicular to b@xmath8 forces the microcrystalline @xmath5-axis to have no choice but to be aligned along the rotation axis . the powder x - ray diffraction patterns of rusr@xmath13eucu@xmath13o@xmath14 random powder ( @xmath37 ) , partially @xmath29-plane aligned along b@xmath8 ( @xmath300 ) , and highly @xmath5-axis aligned along the rotation axis ( 00@xmath32 ) are shown collectively in figure 5 . the relative intensity of the enhanced ( 00@xmath32 ) lines and ( 113 ) line indicates a @xmath6 90@xmath21 c - axis alignment in this field - rotation aligned powder - in - epoxy sample . schematic diagram for the field - rotation powder alignment method with @xmath5-axis perpendicular to aligned magnetic field and along the rotation axis . ] powder x - ray diffraction patterns for rusr@xmath13eucu@xmath13o@xmath14 . ( a ) random powder , ( b ) @xmath29-plane aligned along b@xmath8 , and ( c ) @xmath5-axis aligned along the rotation axis . ] figure 6 shows the field dependence of paramagnetic moment of rusr@xmath13gdcu@xmath13o@xmath14 aligned powder up to 7 t at 300 k. since @xmath29-plane is aligned along the magnetic field in the alignment procedure at room temperature , magnetic anisotropy of @xmath38 at 300 k is expected . however , 300 k m - b@xmath39 data showed weak magnetic anisotropy with linear paramagnetic magnetic moment m@xmath40 0.95 m@xmath16 or susceptibility @xmath41 = m / b@xmath8 with @xmath42 . the anisotropic temperature dependence of logarithmic molar magnetic susceptibility of rusr@xmath0gdcu@xmath0o@xmath1 c - axis aligned powder in 1-t applied magnetic field is shown in figure 7 . a crossover from @xmath42 at 300 k to @xmath43 at lower temperature was observed around 185 k , followed by a weak - ferromagnetic ordering at t@xmath44(ru ) = 131 k. the field dependence of paramagnetic moment of rusr@xmath13gdcu@xmath13o@xmath14 aligned powder up to 7 t at 300 k. linear paramagnetic magnetic moment . ] temperature - dependence of logarithmic molar magnetic susceptibility @xmath45 and @xmath46 . ] the magnetic anisotropy of @xmath42 observed at 300 k is mainly due to the contribution of magnetic gd@xmath34 ions ( j = 7/2 ) . the anisotropy of @xmath45(gd ) @xmath47 @xmath46(gd ) is from the tetragonal gdo@xmath14 cage with anisotropic @xmath48-factor @xmath49 , but with little 4@xmath50 wavefunction overlap with the neighbor oxygen 2@xmath51 orbital . although there are three types of magnetic moments in this magnetic superconductor : ru@xmath19 ( s = 3/2 ) with doped electrons or ru@xmath22 ( s = 1 ) , cu@xmath52 ( s = 1/2 ) with doped holes , and gd@xmath34 moment ( j = 7/2 ) , not all moments have the same contribution in powder alignment . in the aligned magnetic field , anisotropic orbital wavefunction is tied to the spin direction , and a strong spin - orbital related short - range anisotropic exchange interaction at 300 k should dominate the magnetic alignment . in the present case , it is believed that ru moment with the strong short - range anisotropic double - exchange / superexchange interaction along the @xmath29-plane due to the jahn - teller distortion of ruo@xmath17 octahedron with @xmath45(ru ) @xmath53 @xmath46(ru ) is the dominant factor for @xmath29-plane alignment along b@xmath8 at 300 k. the shorter ru - o(1 ) bond length in the tetragonal @xmath29-basal plane provides strong 4d@xmath54(ru)-2p@xmath55(o(1))-4d@xmath54(ru ) wavefunction overlap . this exchange interaction increases with decreasing temperature , and eventually total @xmath43 was observed below 185 k as expected . the weak - ferromagnetic state below 131 k is due to the long range order of this anisotropic double - exchange / superexchange interaction . reciprocal molar magnetic susceptibility 1/@xmath45 and 1/@xmath46 for aligned rusr@xmath13gdcu@xmath13o@xmath14 powder . ] the reciprocal molar magnetic susceptibility 1/@xmath45 and 1/@xmath46 of rusr@xmath13gdcu@xmath13o@xmath14 aligned powder are shown in figure 8 . a curie - weiss behavior @xmath41= c/(t - @xmath56 ) was observed in the high temperature paramagnetic region above 200 k with a curie - weiss intercept @xmath56 = 60 k for both field orientations and the anisotropic effective magnetic moment @xmath57 = 7.44 @xmath58 per formula unit along the @xmath5-axis , and @xmath59 = 7.26 @xmath58 per formula unit along the @xmath29-plane . diamagnetic superconducting transition t@xmath16(dia ) at 39 k and gd ordering temperature t@xmath44(gd ) = 2.5 k are not clearly seen in this high applied field . the temperature - dependence of logarithmic molar magnetic susceptibility @xmath45 and @xmath46 of aligned rusr@xmath13eucu@xmath13o@xmath14 powder in 1 t applied magnetic field . ] the temperature dependence of logarithmic molar magnetic susceptibility of aligned rusr@xmath0eucu@xmath0o@xmath1 powder along @xmath5-axis and @xmath29-plane in 1-t applied magnetic field are shown in figure 9 . similar to r = gd compound , weak paramagnetic anisotropy of @xmath45 = 0.95 @xmath46 was observed at 300 k , and a crossover to @xmath60 was detected below 190 k with a weak - ferromagnetic ordering temperature t@xmath44(ru ) = 133 k. superconducting diamagnetic signal t@xmath16(dia ) at 21 k ( resistivity zero point ) is very weak in large 1-t applied field . low temperature , low field ( 1-g field - cooled ( fc ) and zero - field - cooled ( zfc ) ) anisotropic magnetic and superconducting properties of rusr@xmath13gdcu@xmath13o@xmath14 aligned powder . ] low temperature superconducting properties of aligned powder ( dispersed microcrystallines in epoxy ) , random powder and bulk rusr@xmath13gdcu@xmath13o@xmath14 sample . ] low temperature , low field ( 1-g field - cooled ( fc ) and zero - field - cooled ( zfc ) ) anisotropic magnetic and superconducting properties of rusr@xmath13gdcu@xmath13o@xmath14 aligned powder are shown in figure 10 . a clear weak - ferromagnetic ordering t@xmath44(ru ) at 131 k was observed in all data with @xmath43 in the weak - ferromagnetic state for both fc and zfc measurements . a superconducting diamagnetism setting in at the vortex melting temperature t@xmath4(dia ) of 39 k , same as the t@xmath4(dia ) in the bulk sample , can be clearly seen in the zfc measurements but with a weaker diamagnetic signal . in the fc measurements , the onset of diamagnetic signal can be recognized as a kink at t@xmath4(dia ) , the abrupt increase of magnetic signal at 30 k was attributed to the magnetic field profile inside the squid magnetometer @xcite and corresponds to the spontaneous vortex state temperature t@xmath61 , the same as the one observed in bulk samples.@xcite the antiferromagnetic gd@xmath34 order is observed at t@xmath44(gd ) = 2.5 k. the electrical resistivity data of bulk sample in figure 11 indicates a high superconducting onset temperature of 56 k , with a much lower t@xmath4(zero ) = t@xmath16(dia ) at the vortex melting temperature of 39 k. slightly larger diamagnetic signal for random powder is probably due to partially intergrain supercurrent shielding through partial grain contact . the weak diamagnetic signal observed in aligned powder - in - epoxy samples below t@xmath16(dia ) is due to pure intragain shielding with , in addition , long penetration depth @xmath62 ( @xmath11 0.55 @xmath7 m , @xmath12 0.66 @xmath7 m ) in comparison with powder grain size ( @xmath6 1 - 10 @xmath7 m ) and the two - dimensional ( 2d ) character of cuo@xmath0 layers . the anisotropic high - field ( @xmath637 t ) isothermal superconducting hysteresis loops m - b@xmath8 at 100 k ( fig . 12 ) indicate the initial m@xmath64 m@xmath16 as expected from the susceptibility . since the magnetization curves for both field orientations showed weak - ferromagnetic behavior , the weak - ferromagnetic ordered m(ru ) dominates over the paramagnetic m(gd ) in the magnetic response and should be responsible for the complex metamagnetic - like behavior around 1 t for field applied along the @xmath29-plane . anisotropic high - field hysteresis loop m@xmath16(b@xmath8 ) and m@xmath65(b@xmath8 ) at 100 k for aligned rusr@xmath13gdcu@xmath13o@xmath14 sample . ] anisotropic high - field hysteresis loop m@xmath16(b@xmath8 ) and m@xmath65(b@xmath8 ) at 10 k for aligned rusr@xmath13gdcu@xmath13o@xmath14 sample . ] the magnetization curves with applied field b@xmath8 along @xmath5-axis , m@xmath16(b@xmath8,t ) , and @xmath29-plane , m@xmath65(b@xmath8,t ) , for aligned rusr@xmath13gdcu@xmath13o@xmath14 powder at 10 k are shown in fig . 13 . a strongly anisotropic magnetization was observed for different field orientations . a superconducting hysteresis loop with a weak paramagnetic background was observed in b@xmath39 @xmath66 @xmath5-axis with a maximum diamagnetic signal at magnetization peak field @xmath67h@xmath68(peak ) of 11 g. however , no superconducting diamagnetic signal was directly observed for the initial magnetization curve for field applied along @xmath29-plane due to the much stronger paramagnetic - like background . after subtracting the paramagnetic - like background , the maximum diamagnetic signal for m@xmath65 can be obtained as @xmath67h@xmath69(peak ) = 9 g. the great difference of magnetization anisotropy between 100 k and 10 k observed can not be explained by simple superposition of magnetic and superconducting components that suggests complicated interplay between the doped electrons in weak - ferromagnetically ordered ruo@xmath70 layers and superconducting holes in cuo@xmath0-layers . anisotropic powder alignment is achieved for rusr@xmath13rcu@xmath13o@xmath14 weak - ferromagnetic superconductors ( r = eu , gd , and gd@xmath2dy@xmath2 ) . due to spin - orbital related short - range anisotropic exchange interaction , paramagnetic susceptibility @xmath45(ru / cu ) @xmath71(ru / cu ) at 300 k in rusr@xmath13gdcu@xmath13o@xmath14 and rusr@xmath13eucu@xmath13o@xmath14 , @xmath5-axis aligned powder can be achieved only using field - rotational method . total @xmath42 at room temperature is dominated by r@xmath34 . due to long - range ru anisotropic exchange interaction , total @xmath43 were observed below crossover temperature @xmath6 185 k , with weak - ferromagnetic t@xmath44(ru ) = 131 k , superconducting t@xmath16(dia ) = 39 k and t@xmath44(gd ) = 2.5 k. weak diamagnetic signal observed below t@xmath16(dia ) was due to pure intragain shielding with long penetration depth @xmath62 ( @xmath11 0.55 @xmath7 m , @xmath12 0.66 @xmath7 m ) and the two - dimensional ( 2d ) character of cuo@xmath0 layers .
|
the powder alignment method is used to investigate the anisotropic physical properties of the weak - ferromagnetic superconductor system rusr@xmath0rcu@xmath0o@xmath1 ( r = pr , nd , sm , eu , gd , gd@xmath2dy@xmath2 ) . the rusr@xmath0gdcu@xmath0o@xmath1 ru-1212 cuprate is a weak - ferromagnetic superconductor with a magnetic ordering of ru moments at t@xmath3(ru ) = 131 k , a superconducting transition in the cuo@xmath0 layers at t@xmath4 = 56 k , and a low temperature gd antiferromagnetic ordering at t@xmath3(gd ) = 2.5 k. due to weak magnetic anisotropy of this tetragonal system , highly @xmath5-axis aligned microcrystalline powder ( diameter @xmath6 1 - 10 @xmath7 m ) in epoxy can be obtained only for r = eu and gd through the field - rotation powder alignment method where @xmath5-axis is perpendicular to the aligned magnetic field b@xmath8 = 0.9 t and parallel to the rotation axis . for smaller rare
earth compound r = gd@xmath2dy@xmath2 , powder alignment can be achieved using the simple field powder alignment method where @xmath5-axis is partially aligned along the aligned magnetic field .
no powder alignment can be achieved for larger rare earths r = pr , nd or sm due to the lack of magnetic anisotropy in these compounds .
the anisotropic temperature dependence of magnetic susceptibility for the @xmath5-axis aligned powders exhibit weak anisotropy with @xmath9 at room temperature due to anisotropic rare earth , eu and gd , contribution and crossover to @xmath10 below 190 k where strong ru anisotropic short - range exchange interaction overtakes the rare earth contribution .
anisotropic diamagnetic superconducting intragrain shielding signal of aligned microcrystalline rusr@xmath0gdcu@xmath0o@xmath1 powder - in - epoxy below vortex lattice melting temperature at 39 k in 1-g field is much weaker than the intergrain polycrystalline bulk sample signal due to the small grain size ( d @xmath6 1 - 10 @xmath7 m ) , long penetration depth ( @xmath11 0.55 @xmath7 m , @xmath12 0.66 @xmath7 m ) and the two - dimensional ( 2d ) character of cuo@xmath0 layers .
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observations of nearby star - forming regions show that the vast majority of pre - main - sequence ( pms ) stars are either accreting classical t tauri stars ( cttss ) with optically thick inner disks extending inward to the dust sublimation radius and showing a narrow range of infrared ( ir ) colors or , more evolved , non - accreting weak - line t tauri stars ( wttss ) with bare stellar photospheres . the few transition objects " that are caught between the typical cttss and wttss stages present very diverse ir seds associated with a wide range of disk structures , but they usually have little or no excess at near - ir wavelength and much larger excesses at longer wavelengths ( see williams & cieza , 2011 for a recent review ) . ever since they were discovered by the _ infrared astronomical satellite _ ( strom et al . 1989 ; wolk & walter 1996 ) , the deficit of near - infrared excess in transition disks has been interpreted as a diagnostic of inner disk clearing , possibly connected to planet formation . however , in addition to planet formation , at least three other disk evolution processes can produce the inner opacity holes that are characteristic of transition objects : grain growth , photoevaporation , and dynamical interactions with ( sub)stellar companions . the four different mechanisms potentially responsible for the holes in transition disks might become distinguishable when , in addition to seds , disk masses , accretion rates and multiplicity information are available ( najita et al . 2007 ) . in our recent multi - wavelength study of ophiuchus transition disks ( cieza et al . 2010 , hereafter paper i ) , we have estimated disk masses ( from submillimeter photometry ) , accretion rates ( from optical echelle spectroscopy ) , and multiplicity information ( from near - ir adaptive optics imaging ) of 26 transition objects in order to shed some light on the origin of their inner opacity holes . of these 26 transition disks , 4 were classified as planet - forming disks candidates " based on their sed morphologies , multiplicity , accretion rates , and disk mass measurements ( objects # 11 , 21 , 31 , and 32 in the study ) . these 4 objects have seds consistent with sharp , dynamically induced inner holes ( as opposed to the smooth decrease in opacity expected from grain growth ) , yet our adaptive optics imaging showed that they lack _ stellar _ companions beyond @xmath210 au . also , their large disks masses and/or high accretion rates disfavor photoevaporation as the disk clearing mechanism . overall , the properties of these objects are those expected for protoplanetary disks with embedded giant planets . the recent discoveries of what appear to be forming planets embedded within the disks of the transition objects t cha ( huelamo et al . 2011 ) and lkca 15 ( kraus & ireland , 2012 ) give credence to this interpretation and encourages detailed studies of similar objects . while sed modeling can be a powerful tool to estimate the physical properties of circumstellar disks , it has severe limitations . it is a well known problem that different combination of parameters can reproduce the photometry data equally well , even when _ very _ well sampled seds are available ( e.g. , see cieza et al . 2011 for the t cha case ) . fortunately , resolved ( sub)millimeter images can break many of the degeneracies between model parameters and help constrain disk properties much better than it is possible with the sed alone ( andrews et al . 2009 ; brown et al . 2009 ; isella et al . 2010 ) . rxj1633.9 - 2442 is the most massive of the 4 planet forming disk " candidates identified in paper i. its sed was recently modeled by orellana et al . ( 2012 ) as a 17 m@xmath5 disk with a 7.9 au radius cavity . here we present high resolution ( 0.3@xmath1 @xmath2 35au ) sma continuum observations at 340 ghz ( 880 @xmath0 m ) of rxj1633.9 - 2442 and use the mcfost radiative transfer code ( pinte et al . 2006 ) to simultaneously model the entire optical to millimeter wavelength sed and sma visibilities in order to constrain the structure of its disk . in 2 , we present our sma data and the photometry from the literature that we use to construct to the full sed , as well as the keck aperture masking data we have obtained to search for low - mass companions and the magellan optical spectroscopy data we use to better measure the spectral type of the central star . our disk model and the degree to which each of the disk parameters can be constrained are discussed in 3 . in 4 , we analyze our results in the broader context of disk evolution and planet formation models . a summary of our main conclusions is presented in 5 . submillimeter interferometric observations of our target were conducted in service mode with the sma ( ho et al . 2004 ) , on mauna kea , hawaii , on february 9@xmath6 and february 22@xmath7 , 2010 . the receivers were tuned to a local oscillator frequency of 340 ghz ( 880 @xmath0 m ) . both the upper and lower sideband data were used , providing a total bandwidth of 4ghz . the observations were obtained with seven of the eight 6-meter antennas in the very extended configuration " , resulting on 21 baselines from 120 to 510 meter in length and an elongated synthesized beam , 0.26@xmath1@xmath80.31@xmath1 in size ( i.e. , 31 au @xmath8 37 au ) . the zenith opacities during both nights were @xmath9 0.07 . for each target , the observations cycled rapidly between the target and two gain calibrators , 1625 - 254 and 1626 - 298 , located at 2.0 and 5.3 degrees from rxj1633.9 - 2442 , respectively . in order to ensure the appropriate calibration of short - timescale phase variations , we adopted integration times of 5 minutes on target and 3 minutes on each calibrator . in each of the two nights , our target was observed between hour angles of @xmath103.2 and @xmath113.4 , amounting to a combined _ integration _ time of 7.5 hs . the raw visibility data were calibrated with the mir reduction packagecqi / mircook.html ] . the passband was flattened using @xmath260 to 90 min scans of the bright quasar 0854 + 201 and the solutions for the antenna - based complex gains were obtained using the primary calibrator 1625 - 254 . these gains , applied to our secondary calibrator , 1626 - 298 , served as a consistency check for the solutions . the absolute flux scale was determined through observations of vesta and is estimated to be accurate to 15@xmath12 . the visibilities were fourier transformed , deconvolved with the @xmath13 algorithm , and restored with the synthesized beam using the standard @xmath14 software package ( sault et al . each track was first processed independently to check for consistency , but both were later combined to increase the signal to noise of the final image . the final image is shown in figure 1 and has a rms noise of 1.1 mjy / beam . an inspection of the image gives a first - order approximation to some disk properties . first , the image shows two clear peaks , implying that the inner hole has been resolved in one direction ( east - west ) , but not in the other . also , the aspect ratio of the image suggests the disk is highly inclined ( i.e. , @xmath15 45 deg from face - on ) . fortunately , the major axes of the disk and the synthesized beam are almost perpendicular to each other , maximizing the spatial resolution along the disk . the disk is @xmath21@xmath1 ( 120 au ) across , and the diameter of the inner hole seems to be slightly larger than the 0.3@xmath1 ( 40 au ) beam . the disk diameter should be considered a lower limit as the sma observations are insensitive to the low optical depths of the outermost parts of the disk ( see [ modeling ] ) . finally , it is clear that the disk is not located at the exact center of the field . for interferometric observations , the _ uv_-plane provides more direct means than the image plane to derive quantitative constraints on disk parameters . a convenient way to encapsulate the information from all physical scales sampled by the interferometer is to deproject the visibilities to 0.0 deg inclination and 0.0 deg position angle ( pa ) for the major axis ( andrews et al . 2009 ; brown et al . 2009 ) . the deprojected _ uv_-distances are given by @xmath16 , where @xmath17 and @xmath18 , @xmath19 , and @xmath20 is the inclination ( lay et al . a first approximation to the offsets in @xmath21 and @xmath22 can be obtained by examining the image itself , but more accurate numbers can be calculated by assuming radial symmetry and minimizing the imaginary flux components of the deprojected visibilities ( a face - on radially symmetric disk at the center of the field should have zero imaginary flux components at all spatial frequencies ) . using this latter approach , we obtained positional offsets of @xmath23 and @xmath24 in @xmath21 and @xmath22 , respectively . we estimate the pa to be 100@xmath255 deg west of north from the orientation of the continuum image and the inclination to be 50@xmath255 deg from the aspect ratio of simulated images ( see section [ empty ] ) . the deprojected visibility profile , shown in figure 2 , has a null characteristic of disks with sharp inner holes ( i.e. , a large drop in the surface density over a small radial distance ) . the location of this null is set by the spatial frequency associated with the inner hole and directly constrains its radius . we binned the data using 40 k@xmath26 bins . this bin size is somewhat arbitrary , but it results in 13 visibility values , a number similar to that of the sed points . we have verified that the shape of the visibility profile and , in particular , the location of the null are robust to the choice of bin size . in [ modeling ] , we model the deprojected visibility profile , together with the sed , to place constraints on several disk structure parameters . we constructed the optical to millimeter wavelength sed for rxj1633.9 - 2442 from the following sources . the r - band flux comes from the usno - b1 catalog ( monet et al . 2003 ) , and the ground based near - ir fluxes are from the _ 2mass _ survey ( skrutskie et al . the 12 and 22 @xmath0 m fluxes come from the wide - field infrared survey explorer ( wise ) . the _ spitzer _ and 1.3 mm fluxes are from paper i , while the 850 @xmath0 m flux is from nutter et al . unfortunately , no _ spitzer _ spectrum is available for this source . our target is embedded in the ophiuchus molecular cloud , and thus is strongly affected by extinction . taking advantage of the lack of near - ir excess , we estimate an extinction a@xmath27 = 5.6 mag from the j - k@xmath28 color excess , adopting a@xmath29 = 5.88 @xmath8 ( ( j - k@xmath28)-(j - k@xmath28)@xmath30 ) , where ( j - k@xmath30 ) is the expected color of a dwarf main sequence star ( kenyon @xmath31 hartmann , 1995 ) of the same spectral type as rxj1633.9 - 2442 , a k5 star ( see [ magellan ] ) . the extinction at other wavelengths were estimated from the extinction relations listed in cieza et al . extinction becomes negligible at 24 @xmath0 m and beyond , so the long wavelength fluxes have not been corrected for it . the photometry data from the literature and the adopted uncertainties , are listed in table 1 . the resulting sed is shown in figure 3 and is characterized by the complete lack of detectable excess emission at _ 2mass _ ( 1.2 to 2.2 @xmath0 m ) and _ spitzer_-irac ( 3.6 - 8.0 @xmath0 m ) wavelengths . since a very small amount of dust ( @xmath32 m@xmath33 ) is needed to produce detectable near - ir excess , the sed alone indicates extreme levels of dust depletion in the inner disk ( r @xmath3 1 au ) of rxj1633.9 - 2442 . the 22/24 @xmath0 m fluxes are somewhat below the lower quartile of the ctts population ( furlan et al . 2006 ) , but the 70 @xmath0 m and ( sub)millimeter fluxes are typical of a massive primordial disk . non - redundant aperture masking ( nrm ) has been well - established as a means of achieving the full diffraction limit of a single telescope ( tuthill et al . 2000 , 2006 ; kraus et al . nrm uses a pupil - plane mask to block most of the light from a target , resampling the primary mirror into set of smaller subapertures that form a sparse interferometric array . nrm allows for superior calibration of the stellar primary s point spread function and elimination of speckle noise by the application of interferometric analysis techniques , specifically the measurement of closure phases . we observed rx j1633.9 - 2442 on april 22 - 23 , 2011 , using the keck - ii 10-m telescope with laser - guide star adaptive optics . all observations were conducted with the facility ao imager , nirc2 , which has aperture masks installed in the cold filter wheel near the pupil stop . we used a 9-hole aperture mask , which yields 28 independent baseline triangles about which closure phases are measured . all nrm observations operate in a subarray mode of the narrow camera , which has a pixel scale of 9.963 mas / pix , and we conducted our observations using the broadband @xmath34 filter . each observing sequence consisted of multiple `` visits '' of rx j1633.9 - 2442 , alternating with observations of independent calibrator stars . many of these calibrators were other transitional disk hosts in ophiuchus listed in paper i. each visit consisted of a sequence of 12 exposures that were each 20s , and there were 14 and 17 visits per night . the data analysis was identical to that used in previous papers ( e.g. , ireland et al . 2008 ; kraus et al . 2008 , 2011 ) , combined with the new calibration technique described in kraus & ireland ( 2012 ) . to briefly summarize , the images were flat - fielded and bad pixels were removed by interpolating between neighboring pixels . the image was then multiplied by a super - gaussian window function of the form @xmath35 , with @xmath36 the radius in pixels from the center of the interferogram . a two - dimensional fourier transform was then made of each exposure in a visit , and this fourier transform was point - sampled at the positions corresponding to the baseline vectors in the aperture mask . for each visit we then computed the vector of mean uncalibrated closure - phases and the standard error of the mean . finally , we calibrated the closure - phases for each visit using an optimal linear combination of the calibrators observed in the same sequence of visits . our analysis found no statistically significant signal in the calibrated closure phases for rx j1633.9 - 2442 , and hence that it is single to within the detection limits of the observations . using the same procedures as in our previous nrm work mentioned above ( i.e. , a monte - carlo method that simulates random closure phase datasets of a point source with closure - phase errors and covariances that match those of the real data ) , we found contrast limits ( @xmath37 ) of 5.9 mag at 20 - 40 mas , 6.9 mag at 40 - 80 mas , and 6.8 mag at @xmath3880 mas . the corresponding mass detection limits , based on the 1 myr dusty models of chabrier et al . ( 2000 ) and the assumed distance of 120 pc , are 6 @xmath39 at 2.4 - 4.8 au , and 3.5 @xmath39 at @xmath40 4.8 au . if any planetary companions are brightened by significant accretion luminosity , as seems likely ( see 4.4 ) , then the mass detection limits could be even lower . as part of our recent survey of ophiuchus transition disks ( paper i ) , we obtained high resolution spectra of rx j1633.9 - 2442 using the 2.5-m du pont telescope in las campanas observatory . from these data , we derived a k7 spectral type and an accretion rate of @xmath210@xmath41(m@xmath42yr@xmath43 ) . we reobserved rx j1633.9 - 2442 with the magellan inamori kyocera echelle ( mike ) spectrograph on the 6.5-m clay telescope , also at las campanas observatory , on june 25@xmath6 , 2011 in order to obtain a second - epoch accretion rate . we used the red arm of the spectrograph and a 1@xmath1 slit to obtain the complete optical spectrum between 4900 and 9500 at a resolution of 22,000 . this resolution corresponds to @xmath20.3 at the location of the h@xmath44 line and to a velocity dispersion of @xmath214 km / s . we obtained a set of 3 spectra , with a exposure time of 3 minutes each . the data were reduced using the standard iraf packages imred : cddred and echelle : doecslit . since the final clay spectrum has a better signal to noise ratio than our previous du pont observations , we have revisited the spectral type classification of rx j1633.9 - 2442 by comparing the new data against the elodie high - resolution spectral library ( prugniel & soubiran , 2001 ) . figure 4 ( * left panel * ) shows the rx j1633.9 - 2442 spectra in the narrow 6197 to 6203 region , containing v i and fe i lines , whose shapes , depths , and relative strengths are highly sensitive to effective temperature ( padgett et al . we find that the k5 template is a much better match to the rx j1633.9 - 2442 spectrum than the k7 template is and hence adopt this new and slightly revised spectral type for the modeling work . figure 4 ( * right panel * ) shows the continuum subtracted velocity profile of the h@xmath44 line for our du pont and clay observations . the vertical lines mark a velocity width ( @xmath45 , measured at 10@xmath12 of the peak value ) of 200 km / s , the boundary between accreting and non - accreting objects suggested by jayawardana et al . the h@xmath44 line is broader than 200 km / s , asymmetric and variable in both shape and intensity as expected from magnetospheric accretion . we find @xmath45 @xmath2300 km / s from our du pont observations and @xmath2230 km / s in our new clay data . for accreting objects , @xmath45 correlates with accretion rates derived from models of the magnetospheric accretion process . the relation given by natta et al . ( 2004 ) , and adopted in paper i , @xmath46 , translate the @xmath45 values into accretion rates of 10@xmath41 and 10@xmath47 m@xmath42yr@xmath43 , for the du pont and clay data , respectively . in order to constrain the structure of the rx j1633.9 - 2442 disk based on the observed sma visibilities and sed , we use the mcfost radiative transfer code ( pinte et al . mcfost adopts a monte carlo approach to follow `` photon packets '' propagated through the disk ( i.e. , a parametrized dust density structure ) . mcfost outputs synthetic seds and monochromatic raytraced images . the raytraced images can be used to simulate synthetic sma visibilities with the same _ u - v _ sampling as the actual observations . the deprojected synthetic visibilities can then be directly compared against the real data , using the same radial binning , as described in the following section . we follow the model fitting procedure described by mathews et al . ( 2012 ) , which uses the levenberg - marquardt @xmath48 minimization algorithm to perform an efficient exploration of the parameter space . we start by using mcfost to create a small grid of models and the corresponding seds and 880 @xmath0 m raytraced images . the raytraced images are input to the ft ( fourier transform ) task in the common astronomy software applications ( casa ) package , which outputs visibility data sets with the same _ u - v _ sampling as the real sma observations . then , for each model , the @xmath49 of the model is calculated as the sum of the @xmath49 of the sed and the @xmath49 of the visibility profile . we use the idl routine mpfit ( markwardt , 2009 ) to implement the levenberg - marquardt @xmath48 minimization algorithm to calculate the numerical gradients of the @xmath49 function and determine the next point in the parameter space to be sampled until the algorithm converges to a @xmath49 minimum . to better sample the parameter space and in order to avoid local minima , we carried out the search algorithm several times using different starting values . each of the runs provide a set of best - fit parameters . the distribution of the best - fit values for each parameter can be used to calculate the mean and an associated uncertainty ( see 4.2 in mathews et al . 2012 for a discussion on estimating the uncertainties ) . transition disks are complex systems that have been modeled with a wide range of structures . some objects such as coku tau/4 and dm tau have inner holes that seem to be completely depleted of ir emitting grains ( dalessio et al . 2005 ; calvet et al . 2005 ) , while others such as gm aur have cavities filled with optically thin dust ( calvet et al . 2005 ) . yet other systems , like t cha and lkca 15 are best described as having optically thin gaps separating optically thick inner and outer disk components ( olofsson et al . 2011 ; espaillat et al . 2010 ) . in what follows , we explore all 3 possibilities for the structure of the rx j1633.9 - 2442 disk . we begin our modeling by adopting the simplest possible structure for a transition disk : a disk with an empty cavity . motivated by physical models of viscous accretion disks ( e.g. , hartmann et al . 1998 , lynden - bell & pringle 1974 ) and the discrepancy in disk sizes obtained from continuum and co line images ( hughes et al . 2008 ; isella et al . 2007 ; petu , guilloteau & dutrey , 2005 ) , we follow mathews et al . ( 2012 ) and adopt the following description for the surface density profile and the size of the disk : @xmath50 $ ] where @xmath51 is the surface density at a characteristic radius , @xmath52 , and @xmath53 is the radial dependence of the disk viscosity , @xmath54 . in this prescription , the surface density is proportional to r@xmath55 in the inner disk , but it quickly becomes dominated by the exponential taper at large radii . @xmath56 = 0 within the cavity of radius @xmath57 . we set the outer radius of the disk , @xmath58 , to 200 au . however , we note that our model is not very sensitive to the exact value of @xmath58 because the exponential taper of the outer disk implies very low surface densities ( below the sensitivity of our sma observations ) for radii @xmath59 @xmath60 . the vertical distribution of the dust is given by a gaussian with a scale height @xmath61 , where @xmath62 is the scale height at @xmath63 , and @xmath64 is the power law describing the flaring of the disk . the dust content is described by a differential power - law for the grain size distribution ( d@xmath65(@xmath66 ) @xmath67 @xmath68d@xmath66 ) , between 0.005 @xmath0 m and 3900 @xmath0 m . the size of the longest wavelength in the sed , which approximates the maximum grain sizes the data are sensitive to ( draine 2006 ) . ] we adopt the porous grains from mathis @xmath31 whiffen ( 1989 ) for the grain composition . the stellar parameter are those of the best matched photospheric model ( kurucz 1979 , 1993 ) for a k5 star ( t@xmath69=4350 k , log g = 4.0 ) at the relatively well established distance of 120 pc to the ophiuchus molecular cloud ( loinard et al . we initially created seds and visibility profiles assuming an inclination of 60 deg for the disk , but quickly found that adopting an inclination of 50 deg resulted in synthetic images with aspect ratios that are closer to the observed image and hence use this latter value for all models . table 2 lists all the parameters that are fixed in our model . the 6 free parameters for the empty cavity " model are listed in table 3 . m@xmath70 is not a free parameter . it is obtained by integrating the surface density profile over radius and assuming a gas to dust mass ratio of 100 . for this parameterization , we ran the search algorithm 5 times . the visibility profile and the sed of the overall best - fit model from all 5 runs are indicated in figures 2 and 3 , respectively . the probability weighted means and uncertainties ( calculated as in mathews et al . 2012 ) for each parameter are listed in table 3 in the empty cavity " column . while the model reproduces the visibility profile very well , the observed 22 and 24 @xmath0 m fluxes are factors of @xmath22 - 3 ( i.e. , @xmath210 to 20@xmath71 ) higher then predicted by the model . since the 22 and 24 @xmath0 m measurements are independent ( the former is from wise and the latter is from _ spitzer _ ) , photometric problems can be ruled out . the empty cavity " model does not reproduce the significant 12 @xmath0 m excess either . the large discrepancy between the observations and the best - fit model in the mid - ir can be understood considering that the search algorithm was most likely driven by the @xmath49 of the visibility profile , which is very sensitive to the value of r@xmath72 . in other words , the visibility profile very strongly constrains the size of the inner cavity to be 22.7@xmath251.6 au , but such a large cavity is incompatible with the observed 12 and 22/24 @xmath0 m excesses . we find that the sed and sma data can _ not _ be reconciled even adopting a hotter k1-type central star ( which is clearly ruled out by the optical spectrum , see figure 4 ) . it is thus unavoidable to conclude that the @xmath223 au cavity imaged at submillimeter wavelengths is not completely depleted of mid - ir emitting grains . to try to reproduce the observed mid - ir excesses , we partially fill the cavity by incorporating 2 additional free parameters that result in a 2-component disk : @xmath73 and @xmath74 . following andrews et al . ( 2011a ) and mathews et al . ( 2012 ) , the surface density profile of the disk is modified such that @xmath75 = @xmath73 @xmath56 , for @xmath74 @xmath76 r @xmath76 @xmath57 . that is , @xmath56 no longer drops to zero at @xmath57 , but is sharply reduced to a lower value between @xmath57 and @xmath74 . this sharp reduction in the surface density profile at @xmath57 is meant to reproduce the inner hole seen at submillimeter wavelengths . @xmath56 remains zero for r @xmath76 r@xmath77 . we ran the search algorithm 5 times for this new parameterization . the visibility profile and the sed of the overall best - fit 2-component model are indicated in figures 2 and 3 , respectively . the weighted means and uncertainties for each parameter are listed in table 3 in the 2-component disk " column . the inner component of this model is characterized by a surface density reduction of @xmath2100 with respect to the outer disk and an inner radius of @xmath27 au . as shown by figures 2 and 3 , the 2-component parametrization allow us to simultaneously obtain satisfactory fits for both the visibility profile and the sed . however , as the properties of the inner disk component are effectively controlled by only 3 sed points ( i.e , the 8.0 , 12 , and 22/24 @xmath0 m fluxes ) , the solution is unlikely to be unique . we next explore an alternative geometry for the inner disk , a narrow ring . as an alternative of the 2-component model , we modify the empty cavity model with two additional parameters , @xmath78 and @xmath79 , describing the location and width of a ring within the cavity . the description of the outer disk ( i.e. , beyond r@xmath72 ) remains unchanged . with this parametrization , we also ran the search algorithm 5 times . the results are shown in figures 2 , 3 and listed in table 3 , together with the other 2 earlier parameterizations . we find that a narrow ring at @xmath210 au fits the visibility profile and the sed almost as well as the 2-component model does ( see also @xmath49 of the visibilities and the sed in table 3 ) . the small width of the ring ( 2 au ) can be understood from the fact that the ring is optically thick and its mid - ir emission is dominated by the inner rim facing the star . [ mod - results ] table 3 shows that the outer disk parameters for all three model structures agree remarkably well . all the values are well within 1-@xmath80 or 2-@xmath80 , which give us confidence in both our modeling results and uncertainty estimates . also , because the sma data is only sensitive to the properties of the outer disk , the visibility profiles of the three models match the observations comparably well ( see figure 2 and @xmath49 of the visibilities in table 3 ) . we find that the disk is relatively massive ( @xmath215 m@xmath5 ) and rather flat . the scale height at r@xmath81 is given by @xmath82 , corresponding to @xmath3 0.8 au at a radius of @xmath240 au for all 3 models . this flat geometry is in agreement with the result by orellana et al . ( 2012 ) , who found a scale height of 2 au at 100 au for rx j1633.9 - 2442 , and is most likely due to significant grain growth and dust settling . our models , driven by the ( sub)millimeter colors , do favor a grain size distribution extending beyond the millimeter size scale . as for the the inner disk , its structure is not well constrained by current data . the narrow optically thick ring and the more extended optically thin region are only a subset of possible solutions , and the radial symmetry assumed is not necessarily correct ( see [ dynamical ] ) . however , we can say that the mid - ir emission originates beyond @xmath25 au , from a dust component that is distinct from the outer disk imaged at submillimeter wavelengths . similarly , the extreme depletion of dust grains within a few au of the star is well established by the lack of detectable excess at irac wavelengths ( 3.6 to 8.0 @xmath0 m ) . as mentioned in the introduction , orellana et al . ( 2012 ) successfully modeled the sed of rx j1633.9 - 2442 with a 7.9 au inner hole , which was virtually empty . in other words , the larger submillimeter cavity imaged by the sma is _ not _ detectable from the sed alone . the surface density profile of the 3 models we considered are shown in figure [ sigma ] . by simultaneously modeling the visibility and sed data , we are able to demonstrate that the circumstellar environment of rx j1633.9 - 2442 presents _ at least _ three distinct radial regions . the innermost region ( r @xmath3 5 au ) is depleted of small grains . the middle region contains _ some _ mid - ir emitting grains , with an unknown configuration . the outermost region of the disk starts at @xmath225 au , with a sudden increase in the surface density . this complex structure is certainly intriguing , but is not unique to rx j1633.9 - 2442 . similar structures have already been proposed to reconcile the submillimeter images and seds of several other transition disks , including dm tau , rx j1615.3 - 3255 ( andrews et al . 2011a ) , and rx j1604.3 - 2130 ( mathews et al . 2012 ) . in the next section , we discuss the physical processes that could potentially explain our modeling results and the overall properties of rx j1633.9 - 2442 . most of our conclusions are also applicable to the 3 other similar objects listed above . as discussed in [ intro ] , multiple mechanisms have been proposed to explain the inner opacity holes of transition disks , including photoevaporation , grain growth , and dynamical clearing . in what follows , we consider how the predictions from models of each process compare to the properties of rx j1633.9 - 2442 . toward the end of the section , we assess the likelihood of a connection between transition disks and each of the two leading theories of giant planet formation , core accretion and gravitational instability . photoevaporation by the central star is currently believed to play an important role on the dissipation of circumstellar disks . photoevaporation can be driven by energetic photons in the fuv ( 6ev @xmath83 13.6ev ) , euv ( 13.6ev @xmath83 0.1kev ) and x - ray ( @xmath84kev ) energy range . photons in each energy domain operate in different ways . euv photons can not penetrate far into the disk and drive relatively weak photovaporation winds ( 10@xmath41 m@xmath42/yr ) . thus , euv photoevaporation only becomes important once most of the disk mass has been depleted and the accretion rate drops below @xmath210@xmath41 m@xmath42/yr ( alexander et al . 2006a,2006b ) , at which point the outer disk is no longer able to resupply the inner disk with material and the inner disk drains on a viscous timescale ( @xmath85yr ) . once the inner disk is drained , a hole is formed , the disk edge is directly exposed to the euv radiation , and the disk rapidly photoevaporates from the inside out . the very large disk mass of rx j1633.9 - 2442 and the presence of accretion are inconsistent with euv - induced photoevaporation being the formation mechanism for its inner hole . more recent studies have incorporated x - ray ( owen et al . 2011 ; 2012 ) and/or fuv irradiation ( gorti , dullemond & hollenbach 2009 ; gorti & hollenbach 2009 ) into photoevaporation models . according to these models , x - ray and fuv photons can penetrate deeper into the disk , and drive much higher photoevaporation rates ( @xmath210@xmath86 m@xmath42/yr ) . as a result , the hole is expected to form earlier in the evolution of the disk , while the disk is still relatively massive . since the surface density of the inner disk at the time it starts to drain is high , x - ray / fuv photoevaporation models can in principle explain the presence of massive disks with inner holes and moderate accretion ( i.e. , they would represent the inner disk draining stage ) . however , these models also predict that accretion onto the star should quickly drop as the size of the inner cavity grows . in the context of these models , whether the inner hole of a transition disk can potentially be explained by x - ray / fuv photoevaporation depends on the size of the inner hole , the observed accretion rate , the stellar mass , and the x - ray luminosity . in particular , the region in the hole size versus accretion rate plane that is consistent with x - ray / fuv photoevaporation is a very strong function of stellar mass , m@xmath87 , ( see fig . 17 in owen et al . 2012 ) because : 1 ) the location at which the x - ray heated gas becomes unbound and opens a gap in the disk is proportional to m@xmath87 and 2 ) the x - ray luminosity is also a strong function of stellar mass ( l@xmath88 @xmath67 m@xmath89 ; preibisch et al . we thus estimate the stellar mass of rx j1633.9 - 2442 by comparing its temperature and luminosity to the predictions of theoretical evolutionary tracks . a t@xmath69 of 4350 k is directly derived from the k5 spectral type . we estimate a luminosity of 0.70 l@xmath42 applying a bolometric correction ( from hartigan et al . 1994 ) to the extinction corrected j - band magnitude and adopting a distance of 120 pc . according to the models by d@xmath90antona @xmath31 mazzitelli ( 1998 ) and siess et al . ( 2000 ) the temperature and luminosity of rx j1633.9 - 2442 correspond to those of a 2 myr old 0.7 m@xmath42 star and a 6 myr 1.0 m@xmath42 star , respectively . the large discrepancy in stellar age highlights the uncertainty of these models . however , with that caveat , rx j1633.9 - 2442 can be considered to be a @xmath3 1 m@xmath42 star , for which disk holes larger than 20 au around accreting objects can not be explained by photoevaporation . also , photoeveporation can not easily account for the 3-region structure we found in 3 . we thus conclude that the inner hole of rx j1633.9 - 2442 is unlikely to be due to any kind of photoevaporation process . dust opacity , @xmath91 ( @xmath92/g ) , is a very strong function of particle size . as soon as primordial sub - micron dust grains grow into larger bodies ( r @xmath59 @xmath26 ) , most of the solid mass never interacts with the radiation , and @xmath91 plunges . observational support for grain growth in disks is robust and comes from at least two independent lines of evidence : the shapes of the silicate features around 10 and 20 @xmath0 m ( kessler - silacci ; 2006 , olofsson et al . 2010 ) and the spectral slopes of disks at ( sub)millimeter wavelengths ( andrews & williams , 2005 , 2007 ; wilner et al . 2005 ; ricci et al . grain growth has been proposed as one of the possible explanations for the opacity holes of transition disks because it might be a strong function of radius ( it is expected to be more efficient in the inner regions where the surface density is higher and the dynamical timescales are shorter ) . idealized dust coagulation models , ignoring fragmentation and radial drift , do in fact predict extremely efficient grain growth in the inner disk and can produce seds similar to those of rx j1633.9 - 2442 ( dullemond & dominik , 2005 ) . however , dust fragmentation and radial drift result in the efficient replenishment of micron size grains ( brauer , dullemond & henning 2008 ; birnstiel , ormel & dullemond 2011 ) and a smooth and _ shallow _ dependence of @xmath91 on disk radius . in contrast , both the sed and sma visibilities of rx j1633.9 - 2442 are consistent with a _ steep _ discontinuity in the optical depth ( i.e. , @xmath93 ) at r@xmath72 and r@xmath77 ( or r@xmath94 ) . furthermore , if the optical depth discontinuities are mainly due to a reduction in @xmath91 instead of @xmath95 , this would favor the onset of the magneto - rotational instability ( chiang & murray - clay , 2007 ) and would exacerbate accretion . the very low accretion rate of the rx j1633.9 - 2442 disk ( @xmath3 10@xmath41 m@xmath42yr@xmath43 ; see [ magellan ] ) contradicts this scenario . we thus consider grain growth to be a very unlikely explanation for the inner opacity reductions . unlike grain growth , the dynamical interaction of a ( sub)stellar companion embedded within the disk can produce a sharp inner hole ( artymowicz & lubow ( 1994 ) . ireland @xmath31 kraus ( 2008 ) showed that the famous transition disk coku tau 4 is in fact a near - equal mass binary system , which naturally explains the hole that had been inferred from its sed . this immediately raised the question of whether most transition disks were close binaries . however , it is now clear that most sharp holes are _ not _ due to binarity . the transition objects dm tau , gm aur , lkca 15 , ux tau , and ry tau have all been observed with the keck interferometer ( pott et al . 2010 ) . for these objects , stellar companions with flux ratios @xmath76 20 can be ruled out down to sub - au separations . our keck aperture masking observations discussed in [ masking ] rule out the presence of even a brown dwarf companion down to a projected separation of 2.4 au ( corresponding to a maximum physical separation of 3.8 au ) . therefore , a stellar ( or brown dwarf ) companion can not explain neither the @xmath225 au hole seen in our sma image nor the inner most cavity indicated by the sed . instead , the 3-region structure of the rx j1633.9 - 2442 disk discussed in [ mod - results ] suggests the presence of _ at least _ two low - mass objects dynamically sculpting the disk : an object at @xmath96 au from the star that creates the discontinuity in the surface density seen in the submillimeter image and another object at @xmath3 7 au that clears up the inner disk of material . we thus argue that the dynamical interaction of multiple planets embedded within the disk is the most likely explanation for the overall properties of rx j1633.9 - 2442 . the need for multiple planets to explain the properties of rx j1633.9 - 2442 is very strongly supported by recent hydrodynamical simulations of giant planets embedded in primordial disks by dodson - robinson & salyk ( 2011 ) and zhu et al . ( 2011 ) showing that multiple planets are in fact _ required _ to produce inner holes and gaps wide enough to have a noticeable effect in the emerging sed . both studies find that a single giant planet can not explain wide optically thin gaps and holes . multiple forming planets also help explaining the low accretion rates _ onto the star _ ( for a given disk mass ) of many transition objects ( najita et al . 2007 ; espaillat et al . 2012 ) as each planet accretes a significant fraction of the material being transported across the disk . _ kepler _ observations have demonstrated that systems with multiple planets with small semi - major axes are common ( lissauer et al . the lack of near - ir excess ( @xmath97 8.0 @xmath0 m ) in the sed of rx j1633.9 - 2442 combined with variable and detectable levels of accretion ( @xmath210@xmath4110@xmath98 m@xmath42yr@xmath43 ) could be an indication of a densely packed planetary system resulting in multiple optically thick tidal streams that transport a significant amount of material onto the star but cover a small area of the inner disk . we ran some tests and found that _ axisymmetrically distributed _ circumstellar material would produce a detectable near - ir excess unless the surface density of the inner disk ( r @xmath3 5 au ) is reduced by a factor of 10@xmath99 with respect to that of a typical " ctts disk . in the absence of planets , the accretion rate onto the star should be proportional to the surface density of the inner disk . therefore , if the surface density is reduced by a 10@xmath99 factor from typical levels , one would expect an accretion rate of 10@xmath10010@xmath101 m@xmath42yr@xmath43 for rx j1633.9 - 2442 . such low rates are undetectable and 2 orders of magnitude lower than the observed value . nevertheless , the lack of near - ir excess and the accretion rate could be reconciled if the inner disk contains optically thick tidal streams with a geometric filling factor of a few percent , which is in agreement with the results of hydrodynamic simulations ( dodson - robinson & salyk , 2011 ) . since these hydrodynamic simulations predict very complex disk structures , the 2-component " and ring within cavity " models presented herein are by necessity a crude oversimplification that reflects the lack of resolved data at the appropriate resolution ( i.e. , at a few au scale ) . the notion that the properties of some transition disks are signposts of ongoing planet formation ( e.g. , najita et al . 2007 ; paper i ; dodson - robinson & salyk 2011 ; espaillat et al . 2012 ) has gained credence from the recently identified companions to t cha and lkca 15 . using the aperture masking technique on the very large telescope , hulamo et al . ( 2011 ) detected a faint object within the inner cavity of the t cha disk . the object is located at 62 mas ( @xmath27 au ) from the primary and has a luminosity ratio of 5.1 mag in the l@xmath90-band ( 3.8 @xmath0 m ) . based on the upper limits from similar k@xmath102-band ( 2.2 @xmath0 m ) observations , the authors derived a k@xmath103-l@xmath90 color @xmath38 1.25 for the companion , suggesting the object must be surrounded by dust . also using the aperture masking technique , but in the keck telescope , kraus & ireland ( 2012 ) , identified a similar object inside the inner hole of lkca 15 . in this case , the companion has been detected in multiple epochs and at multiple wavelengths . the object seems to be a point source at 2.1 @xmath0 m ( 6.8 mag fainter than the primary ) , but is extended at 3.7 @xmath0 m . since the inclination of the lkca 15 disk is known from resolved submillimeter observations ( andrews et al . 2011b ) , a deprojected separation of @xmath215 au from the primary can be derived for the companion , assuming it is coplanar with the disk . kraus & ireland interpreted their observations as a young planet surrounded by warm dust . rx j1633.9 - 2442 shares an intriguing property with t cha and lkca 15 : a very low accretion rate for a given disk mass . lkca 15 has a disk mass of @xmath255 m@xmath5 ( andrews et al . 2011b ) and an accretion rate of @xmath210@xmath104 m@xmath42/yr ( hartmann et al . 1998 ) . similarly , t cha has a disk mass of @xmath217 m@xmath5 ( olofsson et al . 2011 ) and it seems to be accreting only very weakly and sporadically onto the star ( alcala et al . 1993 ; schisano et al . 2009 ) . in the absence of a planet , the mass accretion rate onto the star should be roughly proportional to the mass of the disk ( najita et al . 2007 ) ; however , a planet massive enough to open a gap in the disk is expected to divert most of the material accreting from the outer disk onto itself . as a result , in the presence of a jupiter mass planet , the accretion _ onto the star _ is reduced by a factor of @xmath210 with respect to the mass transport across the outer disk ( lubow @xmath31 d@xmath90angelo , 2006 ) . in [ modeling ] we derived a disk mass of @xmath215 m@xmath5 for rx j1633.9 - 2442 . despite this large disk mass , we estimate an accretion rate of @xmath3 10@xmath105 m@xmath42yr@xmath43 based on the velocity dispersion of its h@xmath106 line ( see [ magellan ] ) . for comparison , the median disk mass and accretion rate for classical t tauri stars in ophiuchus are 5 m@xmath5 ( andrews @xmath31 williams , 2005 ) and 10@xmath107 m@xmath42yr@xmath43 ( natta et al . 2006 ) , respectively . as discussed in the previous section , the unusually low accretion rate onto the star for the given disk mass observed in rx j1633.9 - 2442 is consistent with the presence of actively accreting objects embedded within the disk . moderate to low accretion rates seem to be a general feature of transition disks . while the distributions of mass accretion rates of transition and non - transition disks are very wide and clearly overlap , the accretion rates of disks with evidence for holes and gaps tend to be a factor of 5 lower than those of full `` disks '' ( espaillat et al . all things considered , the properties of rx j1633.9 - 2442 , t cha , and lkca 15 are best explained by the runaway gas accretion phase in models of giant planet formation through core accretion ( lissauer @xmath31 stevenson , 2007 ) . according to the core accretion model , gas giant planets form by first accreting a solid core , which later attracts a massive gaseous envelope . initially the gas accretion rate onto the core is much lower than the accretion rate of solids . as the mass of the core increases , the rate of gas accretion accelerates . once the mass of gas in the core matches the mass of solids , runaway gas accretion occurs and is sustained for as long as there is material in the gas feeding zone of the planet ( i.e. , until the forming planet clears a gap in the disk ) . due to accretion shock luminosity , the envelope accretion phase corresponds to the highest luminosity state a giant planet will ever have ( marley et al . this luminosity spike could be in fact what makes the detection of forming planets even possible with current instrumentation . the duration of the runaway gas accretion phase is estimated to be of the order of 10@xmath108 years for a 1 m@xmath5 planet and somewhat longer for a larger planet ( marley et al . this timescale is also in agreement with the incidence of transition disks sharing the properties of rx j1633.9 - 2442 ( combining a sharp inner hole , a large disk mass , and a low accretion rate ) . we note that , while the occurrence rate of objects that can be broadly defined as transition disks is of the order of 20@xmath12 , they are _ not _ all consistent with giant planet formation . accreting objects with steeply rising mid - ir seds , such as rx j1633.9 - 2442 , t cha , and lkca 15 are an order of magnitude less common ( cieza et al . 2010 , 2012 ; romero et al . 2012 ) . if our interpretation is correct , massive accreting transition disks with rising mi - ir seds around single stars are _ by far _ the best places for direct imaging searches of forming planets since they are not only the sites of ongoing planet formation , but also the places where forming planets should be the brightest ! as shown by hulamo et al . ( 2011 ) and kraus @xmath31 ireland ( 2012 ) , non - redundant aperture masking is the most promising technique for such searches as it delivers the highest contrast ratio at the diffraction limit of the telescope . in addition to core accretion , gravitational instability ( gi ) has also been proposed as a formation mechanism for giant planets ( boss 1997 ; durisen et al . nevertheless , gi seems to be less relevant to the transition disks discussed herein for several reasons . first , gi planets are expected to form at evolutionary stages much earlier ( age @xmath76 1 myr ) than those of transition disks , when the disk is still extremely massive ( m@xmath70/m@xmath109 @xmath15 0.1 ) and deeply embedded within an extended envelope , while the age distribution of transition disks likely to harbor forming planets favors a @xmath15 2 - 3 myr formation timescale ( cieza et al . second , gi should operate mostly at large radii where the cooling times are shorter than the local orbital periods , a condition needed for fragmentation ( gammie 2001 , rafikov 2007 ) . gi is believed to be much less effective at disk radii @xmath3 40 au ( boley @xmath31 durinsen , 2008 ) or even @xmath3 100 au ( boley 2009 ) , which makes gi less consistent with the inner hole sizes of most transition disks . finally , since the timescale for the formation of planets through gi is @xmath210@xmath110 yr , it is statistically unlikely that the formation event itself would be observed in a nearby molecular cloud with an age of few million years and hundreds , not thousands , of young stellar objects . since gi models do not run long enough to predict the long term evolution of the disk after the formation of the planet , it could be argued that a gi planet could remain embedded in a massive disk for a relative long period of time after it has formed . however , as discussed above , the high disk masses and low accretion rates of rx j1633.9 - 2442 , tcha and lkca 15 , as well as the properties of the tcha , and lkca 15 companions , suggest that the putative planets are accreting most of their mass at the current epoch ( i.e. , we are watching them form ! ) . even though some planets might form through gi , the properties and incidence of the transition objects mentioned here are in much better agreement with planet formation through core accretion . we present continuum high resolution sma observations of the transition disk rxj1633.9 - 2442 and simultaneously model the entire optical to millimeter wavelength sed and sma visibilities in order to constrain the structure of its disk . the submillimeter image reveals that the disk is highly inclined ( @xmath11150 deg ) and has an inner cavity @xmath225 au in radius . this cavity is not empty as some warm dust is needed to explain the excess emission observed at 12 , 22 , and 24 @xmath0 m . the mid - ir excess can be reproduced with either a narrow , optically thick ring at @xmath210 au or an optically thin region extending from @xmath27 to 25 au . the lack of near - ir excess emission indicates that the inner disk ( r @xmath3 5 au ) is mostly depleted of ir - emitting grains . since rx j1633.9 - 2442 is a single star , the properties of the disk ( a complex structure , a relative massive outer disk , and weak accretion ) favors dynamical clearing by multiple planets as the hole formation mechanism . this conclusion can be extended to objects with similar properties and disk structures , such as the three transition objects mentioned in section [ mod - results ] : dm tau , rx j1615.3 - 3255 , and rx j1604.3 - 2130 . the filamentary structures predicted by hydrodynamical models of multiple planets embedded within a disk can reconcile the accretion rate and sed of rx j1633.9 - 2442 as they can transport significant amounts of material to the inner disk without overproducing the observed ir excess . the properties and occurrence rate of objects such as rx j1633.9 - 2442 , t cha , and lkca 15 ( and those of the companions recently identified to these latter objects ) are in good agreement with the runaway gas accretion phase of the core accretion model , when giant planets gain their envelopes and suddenly become massive enough to dynamically clear a gap in the disk . if the inner holes of rx j1633.9 - 2442 , dm tau , rx j1615.3 - 3255 , rx j1604.3 - 2130 , t cha , and lkca 15 are in fact due to ongoing giant planet formation through core accretion , these types of systems would represent ideal laboratories to study this complex process in detail and place much needed observational constraints . for instance , the location of the lkca 15 companion and the age of the system would already imply that core accretion can actually form giant planets at @xmath215 au within @xmath23 myr , which is a difficult challenge for current models ( dodson - robinson @xmath31 bodenheimer , 2010 ) . similarly , the sizes of the inner holes in the disks of rx j1633.9 - 2442 , rx j1615.3 - 3255 , and rx j1604.3 - 2130 also suggest the presence of young giant planets at @xmath15 20 au orbital separations . in the near future , the unprecedented sensitivity and resolution of the atacama large millimeter / submillimeter array ( alma ) will revolutionize the fields of disk evolution and planet formation . most studies of nearby circumstellar disks will soon be based on high resolution images of both thermal emission and molecular gas tracers rather than on sed observations . alma will provide new insights on the structure of disks and their dynamics and teach us about turbulence , grain growth and dust settling , and the evolution of the dust to gas mass ratio , the understanding of all of which are key to planet formation theory . detailed alma studies of disks hosting forming planets is the most direct and promising approach to learn about the planet formation process and the conditions in which planets form . support for this work was provided by nasa through the _ sagan _ fellowship program under an award from the california institute of technology . g.s.m . and acknowledge nasa / jpl and nsf for funding support through grants rsa-1369686 and ast08 - 08144 respectively . a.l.k . was supported by nasa through the _ hubble _ fellowship program . m.r.s and f.d.m acknowledge support from millenium science initiative , chilean ministry of economy , nucleus p10 - 022-f . crrlrr 0.65 & 3.53e+00 & 14.85 & 30@xmath12 & usno - b1 & 1 + 1.2 & 1.04e+02 & 10.46 & 15@xmath12 & 2mass & 2 + 1.6 & 1.85e+02 & 9.36 & 15@xmath12 & 2mass & 2 + 2.2 & 2.02e+02 & 8.80 & 15@xmath12 & 2mass & 2 + 3.6 & 9.67e+01 & 8.66 & 10@xmath12 & _ spitzer _ & 3 + 4.5 & 7.10e+01 & 8.51 & 10@xmath12 & _ spitzer _ & 3 + 5.8 & 5.12e+01 & 8.37 & 10@xmath12 & _ spitzer _ & 3 + 8.0 & 3.28e+01 & 8.23 & 10@xmath12 & _ spitzer _ & 3 + 12 & 2.41e+01 & 7.80 & 10@xmath12 & wise & 4 + 22 & 2.54e+02 & 3.79 & 10@xmath12 & wise & 4 + 24 & 2.28e+02 & 3.74 & 10@xmath12 & _ spitzer _ & 3 + 70 & 7.13e+02 & & 15@xmath12 & _ spitzer _ & 3 + 850 & 2.10e+02 & & 15@xmath12 & jcmt & 5 + 1300 & 8.18e+01 & & 15@xmath12 & sma & 3 + lr + stellar t@xmath69 [ k ] & 4350 + log g & 4 + distance [ pc ] & 120 + inclination [ deg ] & 50 + grain size distribution slope , p & @xmath103.5 + a@xmath112 [ @xmath0 m ] & 0.005 + a@xmath113 [ @xmath0 m ] & 3900 + [ fix_table ] @xmath60 [ au ] & 39.7 @xmath25 5.6 & 38.2 @xmath25 5.3 & 42.7 @xmath25 3.7 + @xmath57 & 22.7 @xmath25 1.6 & 25.6 @xmath25 1.4 & 27.3 @xmath25 2.1 + log(@xmath114)[g/@xmath92 ] & @xmath100.23 @xmath116 & @xmath100.31 @xmath25 0.31 & @xmath100.39 @xmath25 0.24 + @xmath53 & -0.49 @xmath25 0.24 & @xmath100.30 @xmath25 0.41 & @xmath100.81@xmath25 0.16 + @xmath117 & 0.014 @xmath25 0.003 & 0.018 @xmath25 0.002 & 0.019 @xmath25 0.002 + @xmath64 & 0.37 @xmath25 0.04 & 0.30 @xmath25 0.09 & 0.22 @xmath25 0.14 + @xmath74 [ au ] & & 6.9 @xmath25 1.6 & + log(@xmath73 ) & & @xmath102.0 @xmath25 0.4 & + @xmath78 [ au ] & & & 10.0 @xmath25 2.3 + @xmath79 [ au ] & & & 2.2 @xmath25 0.15 + @xmath118m@xmath5]@xmath120 & 19 & 13 & 14 + @xmath49 visibility ( range)@xmath121 & 1438 & 1227 & 2650 + @xmath49 sed ( range ) & 89109 & 2433 & 2733 + @xmath49 visibility ( adopted)@xmath122 & 36 & 21 & 28 + @xmath49 sed ( adopted ) & 103 & 24 & 31 + @xmath49 total ( adopted ) & 139 & 45 & 59 +
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we present continuum high resolution submillimeter array ( sma ) observations of the transition disk object rxj1633.9 - 2442 , which is located in the ophiuchus molecular cloud and has recently been identified as a likely site of ongoing giant planet formation .
the observations were taken at 340 ghz ( 880 @xmath0 m ) with the sma in its most extended configuration , resulting in an angular resolution of 0.3@xmath1 ( 35 au at the distance of the target ) .
we find that the disk is highly inclined ( _ i _
@xmath2 50 deg ) and has an inner cavity @xmath225 au in radius , which is clearly resolved by our observations .
we simultaneously model the entire optical to millimeter wavelength spectral energy distribution ( sed ) and sma visibilities of rxj1633.9 - 2442 in order to constrain the structure of its disk .
we find that an empty cavity @xmath225 au in radius is inconsistent with the excess emission observed at 12 , 22 , and 24 @xmath0 m . instead
, the mid - ir excess can be modeled by either a narrow , optically thick ring at @xmath210 au or an optically thin region extending from @xmath27 au to @xmath225 au .
the inner disk ( r @xmath3 5 au ) is mostly depleted of small dust grains as attested by the lack of detectable near - ir excess .
we also present deep keck aperture masking observations in the near - ir , which rule out the presence of a companion up to 500 times fainter than the primary star ( in @xmath4-band ) for projected separations in the 5 - 20 au range .
we argue that the complex structure of the rxj1633.9 - 2442 disk is best explained by _ multiple _
planets embedded within the disk .
we also suggest that the properties and incidence of objects such as rxj1633.9 - 2442 , t cha , and lkca 15 ( and those of the companions recently identified to these two latter objects ) are most consistent with the runaway gas accretion phase of the core accretion model , when giant planets gain their envelopes and suddenly become massive enough to open wide gaps in the disk .
| 17,164 | 574 |
in this section the definitions of the basic concepts and the notation to be used throughout this contribution shall be presented . differentiable manifolds are denoted by italic capital letters @xmath0 and , to our purposes , all such manifolds will be connected causally orientable lorentzian manifolds of dimension @xmath1 . the signature convention is set to @xmath2 . @xmath3 and @xmath4 will stand respectively for the tangent and cotangent spaces at @xmath5 , and @xmath6 ( resp . @xmath7 ) is the tangent bundle ( cotangent bundle ) of @xmath8 . similarly the bundle of @xmath9-contravariant and @xmath10-covariant tensors of @xmath11 is denoted @xmath12 . if @xmath13 is a diffeomorphism between @xmath11 and @xmath14 , the push - forward and pull - back are written as @xmath15 and @xmath16 respectively . the hyperbolic structure of the lorentzian scalar product naturally splits the elements of @xmath3 into timelike , spacelike , and null , and as usual we use the term _ causal _ for the vectors ( or vector fields ) which are non - spacelike . to fix the notation we introduce the sets : @xmath17 the simplest example ( leaving aside $ \r^+$ ) of causal tensors are the causal 1-forms ( $ \equiv \dp_{1}(v)$ ) \cite{s - e}\footnote{see also bergqvist 's and senovilla 's contributions to this volume . } , while a general characterization of $ \dp^{+}_r\equiv \dp^{+}_r(v)$ is the following ( see \cite{sup } for a proof)$^1 $ : \begin{prop } $ { \bf t}\in \dp^{+}_r$ if and only if the components $ t_{i_1\dots i_r}$ of $ { \bf t}$ in all orthonormal bases fulfill $ t_{0\dots 0}\geq|t_{i_1\dots i_r}|$ , $ \forall i_1\dots i_r$ , where the $ 0$-index refers to the temporal component . \label{orthonormal } \end{prop } % \p see \cite{sup}.\n we are now ready to present our main concept , which tries to capture the notion of some kind of relation between the causal structure of $ v$ and $ w$. \begin{defi } let $ \f : v\rightarrow w$ be a global diffeomorphism between two lorentzian manifolds . we shall say that $ w$ is properly causally related with $ v$ by $ \f$ , denoted $ v\prec_{\f}w$ , if for every $ \x\in\z^{+}(v)$ we have that $ \f^{'}\x$ belongs to $ \z^{+}(w)$. $ w$ is said to be properly causally related with $ v$ , denoted simply as $ v\prec w$ , if $ \exists \f$ such that $ v\prec_{\f}w$. \label{prec } \end{defi } { \bf remarks } \begin{enumerate } \item this definition can also be given for any set $ \zeta\subseteq v$ by demanding that $ ( \f^{'}\x)_{\f(x)}\in \z^{+}(\f(x))\,\,\,\ , \forall \x\in\z^{+}(x)$ , $ \forall x\in\zeta$. \item two diffeomorphic lorentzian manifolds may fail to be properly causally related as we shall show later with explicit examples . \end{enumerate } \begin{defi } two lorentzian manifolds $ v$ and $ w$ are called causally isomorphic if $ v\prec w$ and $ w\prec v$. this shall be written as $ v\sim w$. \label{equiv } \end{defi } we claim that if $ v\sim w$ then their causal structure are somehow the same . let $ \g$ and $ \tilde{\g}$ be the lorentzian metrics of $ v$ and $ w$ respectively . by using \be \tilde{\g}(\f^{'}\x,\f^{'}\y)=\f^{*}\tilde{\g}(\x,\y ) , \label{pull - back } \ee we immediately realize that $ v\prec_{\f } w$ implies that $ \f^{*}\tilde{\g}\in\dp^{+}_2(v)$. conversely , if $ \f^{*}\t\in\dp^{+}_2(v)$ then for every $ \x\in\z^{+}(v)$ we have that $ ( \f^{*}\t)(\x,\x)=\t(\f^{'}\x,\f^{'}\x)\geq 0 $ and hence $ \f^{'}\x\in\z(w)$. however , it can happen that $ \z^{+}(v)$ is actually mapped to $ \z^-(w)$ , and $ \z^{-}(v)$ to $ \z^+(w)$. this only means that $ w$ { \em with the time - reversed orientation } is properly causally related with $ v$. keeping this in mind , the assertion $ \f^{*}\t\in\dp^{+}_2(v)$ will be henceforth taken as equivalent to $ v\prec_{\f}w$. \section{mathematical properties } let us present some mathematical properties of proper causal relations . \begin{prop } if $ v\prec_{\f}w$ then : \begin{enumerate } \item $ \x\in\z^{+}(v)$ is timelike $ \longrightarrow$ $ \f^{'}\x\in \z^{+}(w)$ is timelike . \item $ \x\in\z^{+}(v)$ and $ \f^{'}\x\in \z^{+}(w)$ is null $ \longrightarrow$ $ \x$ is null . \end{enumerate } \label{caus } \end{prop } \p for the first implication , if $ \x\in\z^{+}(v)$ is timelike we have , according to equation ( \ref{pull - back } ) , that $ \f^{*}\tilde{\g}(\x,\x)=\t(\f^{'}\x,\f^{'}\x)$ which must be a strictly positive quantity as $ \f^{*}\tilde{\g}\in\dp^{+}_2(w)$ \cite{s - e}. for the second implication , equation ( \ref{pull - back } ) implies $ 0=\f^{*}\t(\x,\x)$ which is only possible if $ \x$ is null since $ \f^{*}\t\in\dp^{+}_2(v)$ and $ \x\in\z^{+}(v)$ ( see again \cite{s - e}).\n \begin{prop } $ v\prec_{\f}w \hspace{2 mm } \longleftrightarrow \hspace{2 mm } \f^{'}\x\in\z^{+}(w)$ for all null $ \x\in\z^{+}(v)$. \label{null } \end{prop } \p for the non - trivial implication , making again use of ( \ref{pull - back } ) we can write : \ [ \f^{'}\x\in\z^{+}(w)\ \forall\x\ \mbox{null in}\ \z^{+}(v)\leftrightarrow\f^{*}\tilde{\g}(\x,\y)\geq 0\ \ \forall\ \x,\y\ \mbox{null in}\ \z^{+}(v ) \ ] which happens if and only if $ \f^{*}\tilde{\g}$ is in $ \dp^{+}_2(v)$ ( see \cite{s - e } property 2.4).\n \begin{prop}[transitivity of the proper causal relation]\hspace{0.1 cm } \\ if $ v\prec_{\f } w$ and $ w\prec_{\psi } u$ then $ v\prec_{\psi\circ\f}u$ \label{order } \end{prop } \p consider any $ \x\in \z^{+}(v)$. since $ v\prec_{\f } w$ , $ \f^{'}\x\in \z^{+}(w)$ and since $ w\prec_{\psi } u$ we get $ \psi^{'}[\f^{'}\x]\in\z^{+}(u)$ so that $ ( \psi\circ\f)^{'}\x\in\z^{+}(u)$ from what we conclude that $ v\prec_{\psi\circ\f } u$.\n therefore , we see that the relation $ \prec$ is a preorder . notice that if $ v\sim w$ ( that is $ v\prec w$ and $ w\prec v$ ) this does not imply that $ v = w$. nevertheless , one can always define a partial order for the corresponding classes of equivalence . next , we identify the part of the boundary of the null cone which is preserved under a proper causal relation . a lemma is needed first . recall that $ \x$ is called an `` eigenvector '' of a 2-covariant tensor $ { \bf t}$ if $ { \bf t}(\cdot , \x ) = \lambda \g ( \cdot , \x ) $ and $ \lambda$ is then the corresponding eigenvalue . \begin{lem } if $ { \bf t}\in \dp^{+}_2(x)$ and $ \x\in\z^{+}(x)$ then $ { \bf t}(\x,\x)=0\ \longleftrightarrow\x$ is a null eigenvector of $ { \bf t}$. \label{null - eigen } \end{lem } \p let $ \x\in\z^{+}(x)$ and assume $ 0={\bf t}(\x,\x)=t_{ab}x^{a}x^{b}$. then since $ t_{ab}x^{b}\in\dp^{+}_1(x)$ \cite{s - e } we can conclude that $ x_a$ and $ t_{ab}x^{b}$ must be proportional which results in $ x^{a}$ being a null eigenvector of $ t_{ab}$. the converse is straightforward.\n \begin{prop } assume that $ v\prec_{\f } w$ and $ \x\in\z^{+}(x),\ x\in v$. then $ \f^{'}\x$ is null at $ \f(x)\in w$ if and only if $ \x$ is a null eigenvector of $ \f^{*}\tilde{\g}(x)$. \label{cone } \end{prop } \p let $ \x$ be in $ \z^{+}(x)$ and suppose $ \f^{'}\x$ is null at $ \f(x)$. then , according to proposition \ref{caus } , $ \x$ is also null at $ x$. on the other hand we have $ 0=\tilde{\g}(\f^{'}\x,\f^{'}\x)=\f^{*}\tilde{\g}(\x,\x)$ and since $ \f^{*}\t|_{x}\in\dp^{+}_2(x)$ , lemma \ref{null - eigen } implies that $ \x$ is a null eigenvector of $ \f^{*}\t$ at $ x$.\n the vectors which remain null under the causal relation $ \f$ are called its { \em canonical null directions}. on the other hand , the null eigenvectors of $ { \bf t}\in\dp^{+}_2 $ can be used to classify this tensor , as proved in \cite{s - e}. as a result we have \begin{prop } if the relation $ v\prec_{\f}w$ has n linearly independent canonical null directions then $ \f^{*}\t=\lambda\g$. \label{conf } \end{prop } \p if there exist $ n$ independent canonical null directions , then $ \f^{*}\t$ has $ n$ independent null eigenvectors which is only possible if $ \f^{*}\t$ is proportional to the metric tensor $ \g$ ( \cite{s - e , sup}.)\n proposition \ref{conf } has an interesting application in the following theorem \begin{theo } suppose that $ v\prec_{\f}w$ and $ w\prec_{\f^{-1}}v$. then $ \f^{*}\t=\lambda\g$ and $ ( \f^{-1})^{*}\g=\fr{1}{(\f^{-1})^{*}\lambda}\t$ for some positive function $ \lambda$ defined on $ v$. \label{inv } \end{theo } \p under these hypotheses , using proposition \ref{caus } , we get the following intermediate results \bea \f^{'}\x\in \z^{+}(w)\ \mbox{null and $ \x\in\z^{+}(v)$}\longrightarrow\x\ \mbox{is null,}\nonumber\\ ( \f^{-1})^{'}\y\in\z^{+}(v)\ \mbox{null and $ \y\in\z^{+}(w)$}\longrightarrow\y\ \mbox{is null.}\nonumber \eea now , let $ \x\in \z^{+}(v)$ be null and consider the unique $ \y\in t(v)$ such that $ \x=(\f^{-1})^{'}\y$. then $ \y=\f^{'}\x$ and $ \y\in\z^{+}(w)$ as $ \f$ sets a proper causal relation and $ \x$ is in $ \z^{+}(v)$. hence , according to the second result above $ \y$ must be null and we conclude that every null $ \x\in \z^{+}(v)$ is push - forwarded to a null vector of $ \z^{+}(w)$ which in turn implies that $ \f^{*}\t=\lambda\g$. in a similar fashion , we can prove that $ ( \f^{-1})^{*}\g=\mu\t$ and hence $ ( \f^{-1})^{*}\lambda = 1/\mu$.\n \begin{coro } $ v\prec_{\f}w$ and $ w\prec_{\f^{-1}}v \,\ , \longleftrightarrow \,\ , \f$ is a conformal relation . \end{coro } \section{applications to causality theory } in this section we will perform a detailed study of how two lorentzian ma-\\ nifolds $ v$ and $ w$ such that $ v\prec_{\f}w$ share common causal features . to begin with , we must recall the basic sets used in causality theory , namely $ i^{\pm}(p)$ and $ j^{\pm}(p)$ for any point $ p\in v$ ( these definitions can also be given for sets ) . one has $ q\in j^+(p)$ ( respectively $ q\in i^+(p)$ ) if there exists a continuous future directed causal ( resp.\ timelike ) curve joining $ p$ and $ q$. recall also the cauchy developments $ d^{\pm}(\zeta)$ for any set $ \zeta\subseteq v$ \cite{ff , w , cond}. another important concept is that of future set : $ \a\subset v$ is said to be a future set if $ i^{+}(\a)\subseteq \a$. for example $ i^{+}(\zeta)$ is a future set for any $ \zeta$. all these concepts are standard in causality theory and are defined in many references , see for instance \cite{ff , w , cond}. \begin{prop } if $ v\prec_{\f } w$ then , for every set $ \zeta\subseteq v$ , we have $ \f(i^{\pm}(\zeta))\subseteq i^{\pm}(\f(\zeta))$ and $ \f(j^{\pm}(\zeta))\subseteq j^{\pm}(\f(\zeta))$. \label{set } \end{prop } \p it is enough to prove it for a single point $ p\in v$ and then getting the result for every $ \zeta$ by considering it as the union of its points . for the first relation , let $ y$ be in $ \f(i^{+}(p))$ arbitrary and take $ x\in i^{+}(p)$ such that $ \f(x)=y$. choose a future - directed timelike curve $ \g$ joining $ p$ and $ x$. from proposition \ref{caus } , $ \f(\g)$ is then a future - directed timelike curve joining $ \f(p)$ and $ y$ , so that $ y\in i^{+}(\f(p))$. the second assertion is proven in a similar way using again proposition \ref{caus}. the proof for the past sets is analogous.\n the converse of this proposition does not hold in general unless we impose some causality conditions on the spacetime . \begin{defi } a lorentzian manifold $ v$ is said to be distinguishing if for every neighbourhood $ u_p$ of $ p\in v$ there exist another neighbourhood $ b_p\subset u_p$ containing $ p$ which intersects every causal curve meeting $ p$ in a connected set . \label{distinguishing } \end{defi } we need some concepts of standard causality theory . for any $ p\in v$ one can introduce normal coordinates in a neighbourhood $ { \cal n}_p$ of $ p$ ( see , e.g. \cite{cond } ) . then the exponential map provides a diffeomorphism $ \exp:{\cal o}\subset t_p(v)\rightarrow { \cal n}_p$ where $ { \cal o}$ is an open neighbourhood of $ \vec{0}\in t_p(v)$. the interior of the future ( past ) light cone of $ p$ is defined by $ c^{\pm}_p=\exp(\mbox{int}(\z^{\pm}(p))\cap { \cal o})$ , and obviously $ c^{\pm}_p\subseteq i^{\pm}(p)$ \cite{cond}. other important issue deals with the chronology relation $ < \!\!<$ between two points . we have $ p<\!\!<q$ if there exist a future timelike curve joining $ p$ and $ q$. see \cite{kronheimer } for an axiomatic study of this relation . \begin{prop } let $ \g$ be a piecewise continuous curve of a distinguishing lorentzian manifold $ ( v,\g)$. then , $ \g$ is total with respect to $ < \!\!<$ if and only if $ \g$ is timelike . \end{prop } \p clearly if $ \g$ is timelike then $ \g$ must be a total set for the relation $ < \!\!<$ ( this is true for every spacetime ) . for the converse consider a curve $ \g$ which is total with respect to $ < \!\!<$ and let $ q\in\g$ be an arbitrary point of the curve . if we take a normal neighbourhood of $ q$ , $ { \cal n}_q$ then we can find a neighbourhood $ u_q$ of $ q$ which is intersected in a connected set by every causal curve meeting $ q$. now , if we pick up a point $ z\in\g\cap u_q$ we have that either $ q<\!\!<z$ or $ z<\!\!<q$. assuming the former we deduce that there exists a timelike curve $ \tilde{\g}$ joining $ q$ and $ z$ which implies that $ \tilde{\g}\cap u_q$ is a connected set . this property together with the distinguishability of $ v$ implies that $ \tilde{\g}$ must be a subset of $ u_q$ and hence $ \tilde{\g}\subset { \cal n}_q$ from what we conclude that $ \tilde{\g}\subset c_p$ ( \cite{cond } ) and hence $ z\in c_p$ $ \forall z\in\g\cap u_q$ which is only possible if $ \g\cap u_q$ is timelike . by covering $ \g$ with sets of the form $ \g\cap u_q$ , $ q\in \g$ we arrive at the desired result.\n \begin{prop } let $ \f : v\rightarrow w$ be a diffeomorphism with the property $ \f(i^{+}(p))\subseteq i^{+}(\f(p))\ \forall p\in v$. then if $ w$ is distinguishing , $ \f$ is a proper causal relation . a similar result holds replacing $ i^{+}$ by $ i^{-}$. \label{chronological } \end{prop } \p from the statement of this proposition is clear that $ \forall$ $ p , q$ of $ v$ such that $ p<\!\!<q$ then $ \f(p)<\!\!<\f(q)$. therefore every timelike curve $ \g$ of $ v$ is mapped onto a continuous curve in $ w$ total with respect to $ < \!\!<$ and hence timelike due to the distiguishability of $ w$. furthermore if the curve $ \g$ is future directed then $ \f(\g)$ must be also future directed since $ < \!\!<$ is preserved which is only possible if every timelike future - pointing vector is mapped onto a future - pointing timelike vector . as a consequence , if $ \k$ is a null vector , $ \f^{'}\k$ must be a causal vector ( to see it just construct a sequence of timelike future directed vectors converging to $ \k$ ) which proves that $ \f$ is a proper causal relation.\n the results for the cauchy developments are the following : \begin{prop } if $ v\prec_{\f } w$ then $ d^{\pm}(\f(\zeta))\subseteq \f(d^{\pm}(\zeta ) ) , \,\ , \forall \zeta\subseteq v$. \label{cauch } \end{prop } \p it is enough to prove the future case . let $ y\in d^{+}(\f(\zeta))$ arbitrary and consider any causal past directed curve $ \g^{-}_{\f^{-1}(y)}\subset v$ containing $ \f^{-1}(y)$. since the image curve by $ \f$ of $ \g^{-}_{\f^{-1}(y)}$ is a causal curve passing through $ y$ , ergo meeting $ \f(\zeta)$ , we have that $ \g^{-}_{\f^{-1}(y)}$ must meet $ \zeta$ from what we conclude that $ y\in\f(d^{+}(\zeta))$ due to the arbitrariness of $ \g^{-}_{\f^{-1}(y)}$.\n \begin{coro } if $ \s\subset w$ is a cauchy hypersurface then $ \f^{-1}(\s)$ is also a cauchy hypersurface of $ v$. \label{hyp } \end{coro } \p if $ \s$ is a cauchy hypersurface then $ d(\s)=w$ , and from proposition \ref{cauch } $ d(\s)\subseteq\f(d(\f^{-1}(\s)))$. since $ \f$ is a diffeomorphism the result follows.\n one can prove the impossibility of the existence of proper causal relations sometimes . for instance , from the previous corollary we deduce that $ v\prec w$ is impossible if $ w$ is globally hyperbolic but $ v$ is not . other impossibilities arise as follows . let us recall that , for any inextendible causal curve $ \g$ , the boundaries $ \partial i^{\pm}(\g)$ of its chronological future and past are usually called its future and past event horizons , sometimes also called particle horizons \cite{ff , w , cond}. of course these sets can be empty ( then one says that $ \g$ has no horizon ) . \begin{prop } suppose that every inextendible causal future directed curve in $ w$ has a non - empty $ \partial i^{-}(\g)$ ( $ \partial i^{+}(\g)$ ) . then any v such that $ v\prec w$ can not have inextendible causal curves without past ( future ) event horizons . \label{hor } \end{prop } \p if there were a future - directed curve $ \g$ in $ v$ with $ \partial i^{-}(\g)=\emptyset$ , $ i^{-}(\g)$ would be the whole of $ v$. but according to proposition \ref{set } $ \f(i^{-}(\g))\subseteq i^{-}(\f(\g))$ from what we would conclude that $ i^{-}(\f(\g))=w$ against the assumption . \n the class of future ( or past ) sets characterize the proper causal relations for distinguishing spacetimes as it is going to be shown next ( every statement for future objects has a counterpart for the past ) . \begin{lem } if $ \a$ is a future set then $ p\in\overline{\a}\longleftrightarrow i^{+}(p)\subseteq\a$. \label{closure } \end{lem } \p suppose $ i^{+}(p)\subseteq a$. then since $ c_p^{+}\subseteq i^{+}(p)$ and $ p\in \overline{c_p^{+}}$ we have that $ u_p\cap c_p^{+}\neq\emptyset$ for every neighbourhood $ u_p$ of $ p$ which in turn implies that $ u_p\cap\a\neq\emptyset$ and hence $ p\in\overline{\a}$. conversely , let $ p$ be any point of $ \overline{\a}$ then $ i^+(p)\subseteq i^+(\overline{a})=i^+(a)\subseteq a$.\n \begin{theo } suppose that $ ( w,\t)$ is a distinguishing spacetime . then a diffeomorphism $ \f:(v,\g)\rightarrow ( w,\t)$ is a proper causal relation if and only if $ \f^{-1}(\a)$ is a future set for every future set . $ \a\subseteq w$. \label{key } \end{theo } \p suppose $ \a\subseteq w$ is a future set , $ v\prec_{\f } w$ and take $ \f^{-1}(\a)\subseteq v$. proposition \ref{set } implies $ \f(i^{+}(\f^{-1}(\a)))\subseteq i^{+}(\f(\f^{-1}(\a)))=i^{+}(\a)\subseteq\a$ which shows that $ i^{+}(\f^{-1}(\a))\subseteq \f^{-1}(\a)$. conversely , for any $ p\in v$ take the future set $ i^{+}(\f(p))$ and consider the future set $ \f^{-1}(i^{+}(\f(p)))$. as $ \f(p)\in\overline{i^{+}(\f(p))}$ then $ p\in\overline{\f^{-1}(i^{+}(\f(p)))}$ and according to lemma \ref{closure } $ i^{+}(p)\subseteq\f^{-1}(i^{+}(\f(p)))$ so that $ \f(i^{+}(p))\subseteq i^{+}(\f(p))$. since this holds for every $ p\in v$ and $ w$ is distinguishing , proposition \ref{chronological } ensures that $ \f$ is a proper causal relation.\n this theorem has important consequences . \begin{prop } if $ v\sim w$ and both manifolds are distinguishing , then there is a one - to - one correspondence between the future ( and past ) sets of $ v$ and $ w$. \label{conserv } \end{prop } \vspace{-0.5 cm } \p if $ v\sim w$ then $ v\prec_{\f}w$ and $ w_{\prec\psi}v$ for some diffeomorphisms $ \f$ and $ \psi$. by denoting with $ { \cal f}_v$ and $ { \cal f}_w$ the set of future sets of $ v$ and $ w$ respectively , we have that $ \f^{-1}({\cal f}_w)\subseteq{\cal f}_v$ and $ \psi^{-1}({\cal f}_v)\subseteq{\cal f}_w$ , due to theorem \ref{key}. since both $ \f$ and $ \psi$ are bijective maps we conclude that $ { \cal f}_v$ is in one - to - one correspondence with a subset of $ { \cal f}_w$ and vice versa which , according to the equivalence theorem of bernstein \cite{bern } , implies that $ { \cal f}_v$ is in one - to - one correspondence with $ { \cal f}_w$.\n \section{causal transformations } in this section we will see how the concepts above generalize , in a natural way , the group of conformal transformations in a lorentzian manifold $ v$. \begin{defi } a transformation $ \f : v\longrightarrow v$ is called { \em causal } if $ v\prec_{\f}v$. \label{gg } \end{defi } the set of causal transformations of $ v$ will be denoted by $ \c(v)$. this is a subset of the group of transformations of $ v$ which is closed under the composition of diffeomorphisms , due to proposition \ref{order } , and contains the identity map . this algebraic structure is well - known , see e.g. \cite{semigroup } , and called subsemigroup with identity or submonoid . thus , $ \c(v)$ is a { \em submonoid } of the group of diffeomorphisms of $ v$. nonetheless , $ \c(v)$ usually fails to be a group . in fact we have , \begin{prop } every subgroup of causal transformations is a group of conformal transformations . \label{group } \end{prop } \p let $ g\subseteq \c(v)$ be a subgroup of causal transformations and consider any $ \f\in g$ , so that both $ \f$ and $ \f^{-1}$ are causal transformations . then $ \f$ is necessarily a conformal transformation as follows from theorem \ref{inv}.\n from standard results , see \cite{semigroup } , we know that $ \c(v)\cap\c(v)^{-1}$ is just the group of conformal transformations of $ v$ and there is no other subgroup of $ \c(v)$ which contains $ \c(v)\cap\c(v)^{-1}$. the causal transformations which are not conformal transformations are called proper causal transformations . it is now a natural question whether one can define infinitesimal generators of one - parameter families of causal transformations which generalize the `` conformal killing vectors '' , and in which sense . notice , however , that if $ \{\f_{s}\}_{s\in \r}$ is a one - parameter group of causal transformations , from the previous results the only possibility is that $ \{\f_{s}\}$ be in fact a group of conformal motions . on the other hand , things are more subtle if there are no conformal transformations in the family $ \{\f_s\}$ other than the identity , in which case it is easy to see that the ` best ' one can accomplish is that either $ g^{+}\equiv \{\f_{s}\}_{s\in \r^{+}}$ or $ g^{-}\equiv \{\f_{s}\}_{s\in \r^{-}}$ is in $ \c(v)$. if this happens one talks about maximal one - parameter submonoids of proper causal transformations . of course , it is also possible to define local one - parameter submonoids of causal transformations $ \{\f_s\}_{s\in i}$ for some interval $ i=(-\epsilon,\epsilon)$ of the real line assuming that $ \{\f_s\}_{s\in ( 0,\epsilon)}$ consists of proper causal transformations . in any of these cases , we can define the infinitesimal generator of $ \{\f_s\}$ as the vector field $ \xiv = d\f_{s}/ds|_{s=0}$. given that $ \f_{s}^{*}\g\in\dp_{2}$ for all $ s\geq 0 $ ( or for all non - positive $ s$ ) , one can somehow control the lie derivative of $ \g$ with respect to $ \xiv$. for instance , it is easy to prove that $ \lie\g ( \k,\k ) \geq 0 $ ( or $ \leq 0 $ ) for all null $ \k$ , clearly generalizing the case of conformal killing vectors . an explicit example of this will be shown in the next section . \section{examples } { \bf example 1 einstein static universe and de sitter spacetime . } let us take $ v$ as the einstein static universe \cite{ff } and $ w=\ss$ as de sitter spacetime . in both cases the manifold is $ \r\times s^{3}$ and hence they are diffeomorphic . by proposition \ref{hor } we know that $ v\not\prec w$ because every causal curve in de sitter spacetime possesses event horizons . however , the proper causal relation in the opposite way does hold as can be shown by constructing it explicitly . the line element of each spacetime is ( with the notation $ d\o^2=d\z^2+\sin^2\z d\phi^2 $ ) : \bea v : & & ds^{2}=dt^{2}-a^2(d\x^{2}+\sin^{2}\x d\o^{2})\ \nonumber\\ w : & & d\tilde{s}^{2}=d\bar{t}^{2}-\alpha^{2}\cosh^{2}(\bar{t}/\alpha ) ( d\bar{\x}^{2}+\sin^{2}\bar{\x } d\bar{\o}^{2})\ \nonumber \eea where $ \x,\z,\phi$ ( and their barred versions ) are standard coordinates in $ s^{3}$ and $ a,\alpha$ are constants . the diffeomorphism $ \p : w\rightarrow v$ is chosen as $ \{t = b\bar{t } , \x={\bar\x } , \z={\bar\z } , \phi={\bar\phi}\}$ for a constant $ b$. one can readily get $ \p^{*}\g$ \begin{eqnarray * } ( \p^{*}\g)_{ab}dx^{a}dx^{b}= b^{2}d\bar{t}^2-a^2(d\bar{\x}^{2}+\sin^{2}\bar{\x } d\bar{\o}^{2})\end{aligned}\ ] ] which on using proposition [ orthonormal ] shows that @xmath18 if @xmath19 and therefore @xmath20 is proper causal relation for those @xmath21 . + consider the following spacetimes : @xmath22 is the region of lorentz - minkowski spacetime with @xmath23 in spherical coordinates @xmath24 ; @xmath25 is the outer region of schwarzschild spacetime with @xmath26 in schwarzschild coordinates @xmath27 . define the diffeomorphism @xmath28 given by @xmath29 for an appropriate positive constant @xmath21 , so that we have @xmath30 by choosing @xmath21 and @xmath31 one can achieve @xmath32 _ whenever _ @xmath33 , while for @xmath34 @xmath13 fails to be a proper causal relation . actually @xmath35 due to corollary [ hyp ] as @xmath36 is globally hyperbolic but @xmath22 is not . take now the diffeomorphism @xmath37 defined by @xmath38 , so that @xmath39 reads @xmath40 from where we immediately deduce that @xmath41 for every @xmath42 as long as @xmath43 . we have thus proved that @xmath44 if @xmath33 , but not for @xmath34 . this is quite interesting and clearly related to the null character of the event horizon @xmath45 in schwarzschild s spacetime . + let us take as @xmath46 the flat friedman - robertson - walker ( frw ) spacetimes in standard frw coordinates @xmath47 with line element given by @xmath48 and assume that the source of einstein s equations is a perfect fluid with equation of state given by @xmath49 ( @xmath50 pressure , @xmath51 density , @xmath52 constant ) . then the scale factor is @xmath53 with constant @xmath54 . by straightforward calculations , it can be proven the following causal equivalences : w~_0 + w~v these causal equivalences are rather intuitive if we have a look at the penrose diagram of each spacetime ( figure [ steady ] ) . , b ) @xmath55 and c ) @xmath56 . notice the similar shape of diagram c ) with that of @xmath57 , and of the steady state part of @xmath58 with a ) and b ) @xcite . ] * example 4 ( vaidya s spacetime . ) * let us show finally an example of a submonoid of causal transformations . consider the vaidya spacetime whose line element is @xcite @xmath59 where @xmath60 is a null coordinate ( that is , @xmath61 is a null 1-form ) , and @xmath62 is a non - increasing function of @xmath60 interpreted as the mass . take the diffeomorphisms @xmath63 . then @xmath64 can be cast in the form @xmath65 hence , @xmath66 iff @xmath67 , which implies that @xmath68 are causal transformations , so that @xmath69 is a maximal submonoid of causal transformations . the differential equation for the infinitesimal generator @xmath70 of this submonoid is easily calculated and reads @xmath71 this is a particular case of a proper kerr - schild vector field , recently studied in @xcite . notice that schwarzschild spacetime is included for the case @xmath72const . , in which case @xmath73 is a killing vector . this may lead to a natural generalization of symmetries . in this work a new relation between lorentzian manifolds which keeps the causal character of causal vectors has been put forward . with the aid of this relation , we have introduced the concepts of causal relation and causal isomorphism of lorentzian manifolds which allow us to establish rigorously when two given lorentzian manifolds are causally indistinguishable regardless their metric properties . this tools could be also useful in order to find out the global causal structure of a given spacetime by just putting it in causal equivalence with another known spacetime . finally a new transformation for lorentzian manifolds , called causal transformation has been defined . these transformations are a natural generalization of the group of conformal transformations and their actual relevance is one of our main lines of future research . this research has been carried out under the research project upv 172.310-g02/99 of the university of the basque country . 99 g. bergqvist and j. m. m. senovilla _ null cone preserving maps , causal tensors and algebraic rainich theory . quantum grav.*18 * , 5299 - 5326 , ( 2001 ) j. m. m. senovilla _ super - energy tensors . _ quantum grav . * 17 * , 2799 - 2842 , ( 2000 ) . s. w. hawking and g. f. r. ellis _ the large scale structure of spacetime . _ cambridge university press , cambridge , ( 1973 ) . r. m. wald _ general relativity . _ the university of chicago press , ( 1984 ) . j. m. m. senovilla _ singularity theorems and their consequences . _ . grav . * 30 * , 701 - 848 , ( 1998 ) . e. h. kronheimer and r. penrose proc . . soc . * 63 * 481 - 501 ( 1967 ) see e. g. f. hausdorff _ set theory _ chelsea n.y . j. hilgert , k. h. hofmann and j. d. lawson _ lie groups , convex cones and semigroups . _ oxford sciencie publications , ( 1989 ) . vaidya _ proc . indian acad . _ * a*33 , 264 ( 1951 ) . b. coll , s. r. hildebrandt and j. m. m. senovilla . _ kerr schild symmetries . * 33 * , 649 - 670 , ( 2001 ) .
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in this work we define and study the relations between lorentzian manifolds given by the diffeomorphisms which map causal future directed vectors onto causal future directed vectors . this class of diffeomorphisms , called _ proper causal relations _ , contains as a subset the well - known group of conformal relations and are deeply linked to the so - called causal tensors of ref.@xcite .
if two given lorentzian manifolds are in _ mutual _ proper causal relation then they are said to be causally isomorphic : they are indistinguishable from the causal point of view .
finally , the concept of causal transformation for lorentzian manifolds is introduced and its main mathematical properties briefly investigated .
alfonso garca - parrado and jos m. m. senovilla 0.5 cm _ departamento de fsica terica .
universidad del pas vasco .
+ apartado 644 , 48080 bilbao , spain _
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a crucial ingredient in quantum information processing based on solid state systems is the transfer of quantum information . assuming that there are quantum registers for computing and storing information , the ability to transfer this information reliably and efficiently from one register to another is vital for the construction of larger , distributed and networked systems . a solution to this challenge has been proposed through the use of spin chains @xcite . the mathematical framework underpinning spin chains can be applied to various physical devices ; these could be made of any components whose states can be mapped onto spin @xmath0 particles interacting with their neighbors . electrons or excitons trapped in nanostructures form explicit examples @xcite , as do nanometer scale magnetic particles @xcite or a string of fullerenes @xcite . another representation is the encoding into a soliton - like packet of excitations @xcite . within spin chains , a single - site excitation is defined as an up " spin in a system that is otherwise prepared to have all spins down " . a discussion about unmodulated spin chains has been given in @xcite whereas in @xcite the couplings were chosen to be unequal . there has also been research on wire - like chains with controlled coupling strength at either end @xcite and transfer through parallel spin chains @xcite , to name but a few closely related areas . here we only consider linear spin chains whose coupling strength @xmath1 between two neighboring sites @xmath2 and @xmath3 has been pre - engineered to ensure perfect state transfer ( pst ) along the chain @xcite . for a chain of length @xmath4 with characteristic coupling constant @xmath5 , the pst coupling strength sequence is defined as @xcite @xmath6 for devices based on excitons in self - assembled quantum dots , @xmath5 is mainly governed by frster coupling @xcite , which in turn depends on the distance between the dots as well as the overlap between the electron and hole wavefunctions in each dot . in gate - defined quantum dots , however , @xmath5 will depend on tunnelling and thus on parameters such as the width and height of the barriers which separate the different dots , as well as on the overlap of electronic wavefunctions centered in different dots . for chains of fullerenes or actual atoms @xmath5 will represent some hopping " parameter describing the propensity of the excitation to transfer from one site to the other . the natural dynamics of a spin chain can then be described by a time independent hamiltonian as follows @xmath7.\end{aligned}\ ] ] in a perfect system ( to which perturbations will then be applied ) we will assume the single excitation energies @xmath8 to be independent of the site @xmath2 , and therefore only concentrate on the second term of eq . ( [ hami ] ) . in some physical systems such as quantum dot strings , @xmath8 could naturally differ according to position , but may be tuned to be the same at all sites via application of local fields @xcite . the fidelity @xmath9 , corresponding to mapping the initial state @xmath10 over a time @xmath11 into the desired state @xmath12 by means of the chain natural dynamics , is given by @xmath13 and pst is realized when the evolution is arranged to achieve @xmath14 . we use the fidelity of state vectors to determine the transfer quality of information for unentangled states , as detailed for example in @xcite . for entangled states , we measure instead the entanglement of formation ( eof ) as defined in ref . + the time evolution of a system is dependent on its characteristic coupling constant @xmath5 . in particular , the time scale for pst from one end of a chain to the other , also known as _ mirroring time _ , is @xmath15 so that the periodicity of the system evolution is given by @xmath16 . as the hamiltonian ( [ hami ] ) preserves the excitation number , the evolution of the initial state will remain within the original excitation subspace . we will now consider the influence of general fabrication defects on linear spin chains with multiple excitations . + * ( a ) random noise * + we model the effect of fabrication errors ( random , but fixed in time ) for the energies and couplings in the system by adding to all non - zero entries in the hamiltonian matrix a random energy @xmath17 for @xmath18,@xmath19 . the scale is fixed by @xmath20 which we set to 0.1 and for each @xmath21 the different random number @xmath22 is generated with a flat distribution between zero and unity . for the other side of the diagonal with @xmath23 , @xmath24 , preserving the hermiticity of the hamiltonian . this method of including fabrication defects means that we could observe effects of a reasonable magnitude although clearly other distributions could also be modeled ; for specific tests , the weight of the noise would have to be determined according to the individual experiment being simulated . + * ( b ) site - dependent `` single - particle '' energies * + as a further possible fabrication defect , we consider the effect of the first term of ( [ hami ] ) that we previously dismissed under ideal conditions @xmath25 @xmath26 may represent external perturbations , such as local magnetic fields , or additional single site fabrication imperfections . we thus assume here that @xmath8 is not independent of the site _ i _ any more . + * ( c ) excitation - excitation interactions * + in spin chains with multiple excitations , we also consider a perturbation term @xmath27 which represents the interaction between excitations in nearby sites . for example , this may correspond to a biexcitonic interaction in quantum dot - based chains @xcite . + * ( d ) next - nearest neighbor interactions * + finally , we also investigate the effect of unwanted longer range interactions , which could be an issue when considering pseudospins based on charge degrees of freedom . for this we add to ( [ hami ] ) the perturbative term @xmath28.\end{aligned}\ ] ] the expression for @xmath29 will depend on the type of interaction between spin chain sites . here we explicitly consider three cases . the first and more general approximates the next - nearest neighbor interaction as proportional to the average of the related interactions between nearest neighbor sites , @xmath30 with the parameter @xmath31 defining the strength of the interaction . this expression simulates the original coupling modulation of the chain . secondly we explicitly consider dipole - dipole interactions , which are relevant , e.g. to chains of quantum dots with exciton qubits and frster coupling @xcite . in this case the coupling between sites scales as @xmath32 , with @xmath33 the distance between the two sites considered . for roughly equidistant sites , we then expect the next - nearest neighbor couplings to be about a tenth of the nearest neighbor couplings . by using this and eq . ( [ pst ] ) we obtain @xmath34^{-\frac{1}{6}}+[(i+1)(n - i-1)]^{-\frac{1}{6}}\right\}^{-3}.\ ] ] finally we consider the case of coupling due to tunnelling , relevant e.g. to graphene quantum dots with spin qubits @xcite . here the coupling scales as @xmath35 , with @xmath36 the on - site coulomb energy and @xmath37 the tunnelling ( hopping ) parameter , with @xmath38 the forbidden momentum in the barrier @xcite . in this case there is no explicit expression for the next - nearest neighbor couplings in terms of @xmath39 , but @xmath40 can be determined numerically by using the expression for @xmath11 and eq . ( [ pst ] ) . we note that we expect @xmath40 to be very small , as the interaction decays exponentially with the distance . if we consider chains of quantum dots with exciton qubits , we can assume @xmath41 , where @xmath42 , and @xmath43 . results in fig . [ fig : nnnicomp ] then show that this is in very good agreement with considering eq . ( [ ji+2 ] ) with @xmath44 , a value that one would expect from dipole - dipole interaction and basically equidistant sites ( see discussion above ) . in this case the effect of next - nearest neighbor interaction is extremely detrimental to the system as even the fidelity of the first state transfer at @xmath45 is reduced by almost 50% . on the example of an 8-spin chain with initial input state @xmath46 , fidelity vs. rescaled time @xmath47 , according to both eq . ( [ ji+2 ] ) and eq . ( [ ji+2dip - dip ] ) ( dipole - dipole coupling ) . the peak at @xmath48 is approximately 0.52 . as the model of eq . ( [ ji+2 ] ) matches the data derived from eq . ( [ ji+2dip - dip ] ) extremely well , the lines of the respective plots are nearly indistinguishable . + inset : as for main panel but for graphene quantum dots and spin qubits , with tunnelling coupling ( thin lines ) and coupling according to eq . ( [ ji+2 ] ) , @xmath49 ( bold lines ) . again results from the two models for the coupling constants are almost indistinguishable from each other.,scaledwidth=45.0% ] by contrast , the tunnelling mechanism case leads to very different results . using the parameters in @xcite for graphene dots and spin qubits , we obtain the results shown in the inset of fig . [ fig : nnnicomp ] : after @xmath50 the system has lost less than 2% of its fidelity . these results are in very good agreement with eq . ( [ ji+2 ] ) with @xmath51 as can be seen in the inset of fig.1 , where the plots are virtually indistinguishable . it is very encouraging that in this case realistic parameters point to such small values of @xmath31 and thus generate very high fidelity transfer . in the following we will use the expression in eq . ( [ ji+2 ] ) to further discuss the effects of @xmath52 . as values of @xmath31 smaller than 0.01 have a minor detrimental effect on the system , we will from here onwards focus on the range @xmath53 , where the upper bound may be of interest to some experimental implementations , e.g. as discussed in the dipole - dipole interaction case . we will now consider the effect of the fabrication defects ( a ) to ( d ) first on the transfer of factorisable states , i.e. unentangled chains , and then on entanglement creation and entanglement transfer along a spin chain . for unentangled states , we will consider a 6-spin chain for all investigations of fabrication defects while in the case of entanglement , we also consider an 8-spin chain . as we will explicitly see later for @xmath26 , the influence of fabrication defects does depend on the chain length , but not on the parity of the chain . the device we consider in this paper is a linear spin chain with couplings fixed such that the conditions for pst are satisfied . one of the properties of these devices , which we make heavy use of , is the mirroring rule @xcite . the mirroring rule is such that any state injected into a linear spin chain , subject to the constraints of eq . ( [ pst ] ) and site - independent @xmath8 , evolves into its `` mirror state '' , where the symmetry center of a chain is the middle point in chains with @xmath4 even , or middle site @xmath54 in chains with @xmath4 odd . this mirroring property is independent of the number of excitations a state comprises and also of the length of the chain . furthermore the mirroring rule holds for states spread across the whole chain : all excited sites are mirrored across to their `` twin '' with respect to the chain center of symmetry . we call this mirrored state the `` twin state '' . when investigating the quality of a device with fabrication defects of any sort , we are primarily interested in the effect on the `` twin state '' at @xmath55 but it is also important to consider the effects on the next few periods as some quantum information protocols may require periodicity of their systems @xcite . as can be seen in fig . [ fig:6spinnoise ] , the effect of random noise leads to a continuous and definite decline in the transfer fidelity of both the input state @xmath56 and its twin state @xmath57 at the mirroring times . naturally , this means an increased probability of the occurrence of other possible states ( not shown on the graph ) but this happens on a relatively unpredictable basis , with no one state ever becoming and remaining particularly prominent . on a 6-spin chain with two excitations , fidelity vs. rescaled time @xmath47 . states other than the input state and its twin state are not shown . the first peak at @xmath58 is 0.9975.,scaledwidth=45.0% ] ( eq . ( [ interc ] ) ) on a 6-spin chain , fidelity vs. rescaled time @xmath47 . + ( a ) @xmath59 : the first peak at @xmath58 is 0.9988 . + ( b ) @xmath60 : the first peak at @xmath58 is 0.9954.,scaledwidth=45.0% ] in comparison , when looking at the evolution over a few periods , even a relatively large value of @xmath59 in eq . ( [ interc ] ) has less effect on the 6-spin chain , as shown in fig . [ fig : interc ] ( a ) . similarly , the on - site energies represented by eq . ( [ enerc ] ) also lead to an unrecoverable decay in the state transmission fidelity , with the excitations being ultimately entirely spread out along the chain ( not shown ) . a combination of any of these three perturbation factors simply accelerates the decay trend of the desired states . + however , if we simply analyze the first revival peak at @xmath61 , we see that for varying @xmath62 and @xmath63 , even when perturbing the system by as much as 20% of @xmath5 , the system suffers less than a 10% loss in fidelity ( fig . [ fig:3d ] ) . here , to represent the fact that @xmath8 is site - dependent , we used a value @xmath64 for all sites _ i _ , with @xmath65 a random number generated from a flat distribution for each @xmath2 . an average over 200 realizations for every value of @xmath63 was then taken . in a linear 6-spin chain , measured on the first revival peak at time @xmath61 , vs @xmath62 and @xmath63 .,scaledwidth=45.0% ] finally , we also note that next - nearest neighbor interactions perturb the system similarly to noise . this can be seen in fig . [ fig:110000nnni ] where even for a reasonable value of @xmath66 , the transfer peaks of both the initial state of a 6-spin chain and its mirror twin quickly decay . in order to achieve the same fidelity loss at @xmath67 as for @xmath66 ( fig . [ fig:110000nnni ] ) , we have to consider a value of @xmath62 _ one order of magnitude bigger _ , as is shown in fig . [ fig : interc ] ( b ) . ( eq.([hami2 ] ) ) with @xmath66 : fidelity of the state @xmath68 in a 6-spin chain vs. rescaled time @xmath47 . the first peak at @xmath58 is 0.9966.,scaledwidth=45.0% ] in our simulations we have kept the value of @xmath69 constant as @xmath4 is varied : this models the physical constraint that in any realistic system the coupling strength is capped by a maximum characteristic value . as a result @xmath70 for even ( odd ) chains . to avoid this implicit dependence on @xmath4 , we have here set @xmath71 ( where @xmath72 is a random number from a uniform distribution within 0 and 1 ) and averaged every point on the graph from 200 random realizations . as we see in part ( a ) of fig . [ fig:110000_nnnienerc ] , the effect of @xmath52 becomes very detrimental to the system even for the relatively small value of @xmath73 , although the loss of fidelity for @xmath66 is very small even for long chains . on the other hand , the effect of on - site energies may be tolerable for values of @xmath63 up to @xmath74 , where long chains of @xmath75 suffer less than a 20% loss in fidelity . a simple estimate of the effect of errors in the energy level spectrum @xcite suggests an overall error , or loss in fidelity for pst that scales as an exponential decay in @xmath4 with gaussian dependence on the characteristic noise parameters . we compare this with numerical results in fig . [ fig:110000_nnnienerc ] , where the loss of fidelity due to @xmath26 scales as @xmath76 and the loss due to @xmath52 scales as @xmath77 with increasing chain length @xmath4 , where @xmath78 and @xmath79 so that @xmath80 and @xmath81 characterize the impact of the noise . the comparison in fig . [ fig:110000_nnnienerc ] shows that this simple analytical form can reproduce the numerical results to a high degree of accuracy : the loss of fidelity scales indeed as an exponential decay with gaussian damping in the noise parameters . at @xmath61 vs. chain length @xmath4 for three values of @xmath31 , as labelled . fits are according to @xmath82 with @xmath83 . + ( b ) average fidelity of initial vector @xmath84 at @xmath61 vs. chain length @xmath4 for three values of @xmath63 , as labelled . fits are according to @xmath85 with @xmath86.,scaledwidth=48.0% ] we conclude from these results that the transfer of unentangled states across the device is very robust against the perturbations @xmath26 and @xmath87 , and less so for @xmath52 . the effect of noise in the system , as we have implemented it , is slightly more noticeable at the transfer time @xmath88 but still allows for excellent transfer at a loss rate of just over 10% over the course of 4 periods . with regards to longer term periodicity , it is the next - nearest neighbor interaction term @xmath52 which perturbs the system most for the values of @xmath8 , @xmath62 and @xmath31 shown . one of the most outstanding properties of spin chains is their ability to not just transfer reliably factorisable states , but also to transfer information encoded as entangled states . this is again based on the mirroring rule @xcite and was first mentioned and discussed under various aspects in refs . entanglement is one of the key resources in quantum computing , and is crucial to some quantum cryptography protocols , and to quantum teleportation . being able to reliably transfer entanglement from one place to another is therefore a core interest that the device we are analyzing should be able to respond to . this property follows from the fact that the set of states @xmath89 , @xmath90 to which the mirroring rule applies , is a basis set so that any state , and in particular entangled states , can be written as the superposition @xmath91 . here @xmath92 is the total number of excitations in the state , and @xmath93 the ensemble of indices @xmath94 different from zero . the above relation implies that after @xmath88 has passed the following twin state is reached : @xmath95 , so that in particular any entangled state is transferred into its mirror entangled state . when considering spin chains with ( i ) an entangled initial state or ( ii ) an initial state that leads to entanglement , it is more useful to observe the evolution of the eof in the system . we use chains with an initial bell state on spins 1 and 2 to represent scenario ( i ) , as for example @xmath96 . accordingly , we monitor the eof in the reduced density operator of spins @xmath97 , tracing out the rest of the chain . to monitor the twin state " entanglement we calculate the eof for the reduced density operator of spins @xmath98 . scenario ( ii ) is different in that the initial state of the chain is not entangled , but will lead to entanglement through natural dynamics . as an example of this , we use a linear chain with input @xmath99 on both spins 1 and @xmath4 . this is equivalent to an initial state @xmath100 , where the subscripts designate the spin site . a system set up in this way will then show a maximally entangled state in spins 1 and @xmath4 at time @xmath55 @xcite . the effect of noise on entangled chains shows a similar trend to that of unentangled chains . however , we notice that the loss of eof in fig . [ fig : gatenoise ] which represents case ( ii ) is nearly 10% bigger than the loss of eof in fig . [ fig:0110noise ] ( case ( i ) ) over the course of the shown 7 periods . as a comparison , the unentangled state of fig . [ fig:6spinnoise ] loses a similar amount of fidelity over the same amount of time as the chain in fig . [ fig : gatenoise ] . on the entanglement of formation between the end spins of a linear 8-spin chain with input state @xmath101 vs. rescaled time @xmath47 . the first peak at @xmath58 is 0.9983.,scaledwidth=45.0% ] on the entanglement of formation between the end spins of an 8-spin chain with input state @xmath102 vs. rescaled time @xmath47 . the first peak at @xmath58 is 0.9993.,scaledwidth=45.0% ] there is no effect from perturbation @xmath87 , ( eq . ( [ interc ] ) ) , in case ( i ) , as there is only one excitation in the system in both amplitudes . obviously this would not be so for a bell state with a doubly excited amplitude . similarly , the effect of @xmath87 in case ( ii ) is restricted to the two excitation subspace populated by the evolution of the @xmath103 component of @xmath10 only and is thus not very prominent , as is shown in fig . [ fig : gateinterc ] . ( eq.([interc ] ) ) with @xmath59 on the entanglement of formation between the end spins of a linear 8 spin chain with input state @xmath101 vs. rescaled time @xmath47 . the first peak at @xmath58 is 0.9996.,scaledwidth=45.0% ] the effect of on - site energies on the other hand remains , regardless of the system . we demonstrate this in part ( b ) of figs . [ fig:10 + 01_nnnienerc ] and [ fig : gate_nnnienerc ] , which show the detrimental effect of @xmath26 on entangled systems ( types ( i ) and ( ii ) respectively ) . similar to fig . [ fig:110000_nnnienerc ] , the loss in eof scales as an exponential in @xmath4 with gaussian damping in the noise parameters . again , the influence of @xmath26 has been averaged over 200 random realization using random numbers @xmath8 from a flat distribution , such that @xmath104 and @xmath65 . ): + ( a ) eof of qubits 1 and 2 at @xmath61 vs. chain length @xmath4 for three values of @xmath31 , as labelled . fits are according to @xmath82 with @xmath105 . + ( b ) average eof of qubits 1 and 2 at @xmath61 vs. chain length @xmath4 for three values of @xmath63 , as labelled . fits are according to @xmath85 with @xmath106.,scaledwidth=48.0% ] ): + ( a ) eof of qubits 1 and @xmath4 at @xmath55 vs. chain length @xmath4 for three values of @xmath31 , as labelled . fits are according to @xmath82 with @xmath107 . + ( b ) average eof of qubits 1 and @xmath4 at @xmath55 vs. chain length @xmath4 for three values of @xmath63 , as labelled . fits are according to @xmath85 with @xmath108.,scaledwidth=48.0% ] figs . [ fig:10 + 01_nnnienerc ] ( b ) and [ fig : gate_nnnienerc ] ( b ) also show that for @xmath109 , eof close to unity can still be achieved for all chain lengths considered . however , we note that for larger values of @xmath110 , chains with type ( ii ) entanglement suffer significantly more entanglement loss than those of type ( i ) which are already entangled at @xmath111 . as with unentangled states , next - nearest neighbor interaction is a relevant issue for entangled states . in fig . [ fig:110000nnni ] , we showed that a 6-spin chain suffered serious fidelity loss ( @xmath112 ) for relatively small @xmath113 after about 4 periods , but still reached a fidelity of 0.9966 at @xmath88 ; similarly we see in fig . [ fig : gate_nnnienerc ] that for the same value of @xmath31 , a 6-spin chain with case ( ii ) entanglement would reach eof of 0.99 at @xmath55 and performs thus equally well . eof of almost unity persists for all @xmath4 considered and small @xmath31 . more than 90% of the eof is maintained even for @xmath31 as large as 0.05 for long chains with type ( ii ) entanglement while chains with type ( i ) suffer significantly more and long chains lose over 30% of their eof for the same value of @xmath31 . we note that for values of @xmath114 , in chains with type ( ii ) entanglement , whilst eof is fairly well maintained at @xmath115 , _ its subsequent periodicity is completely lost for any chain length _ [ fig : nnnicon ] , for a demonstration of the same effect in an unentangled chain see fig . [ fig : nnnicomp ] ) . : + ( a ) eof of @xmath10 vs rescaled time @xmath47 for @xmath73 : the periodicity of the entanglement of formation is completely lost after the second peak at @xmath116 . + ( b ) eof of @xmath10 vs rescaled time @xmath47 for @xmath117 : the periodicity of the entanglement of formation is completely lost after the first peak at @xmath45.,scaledwidth=45.0% ] as noted above the effect of @xmath52 on entangled states is different from case ( i ) to case ( ii ) . in fig . [ fig:0110nnni ] we see that for @xmath66 a state that is initially already entangled does not suffer very much and retains an eof of over 90% at @xmath118 , after 4 periods , whereas a chain where the entanglement is created through the system dynamics suffers a loss in eof of about 15% after 4 periods , at @xmath119 , as shown in fig . [ fig : gatennni ] . ( [ hami2 ] ) ) with @xmath66 on the entanglement of formation between the end spins of an 8 spin chain with input state @xmath120 vs. rescaled time @xmath47 . the first peak at @xmath58 is 0.9986.,scaledwidth=45.0% ] ( eq . ( [ hami2 ] ) ) with @xmath66 on the entanglement of formation between the end spins of a linear 8 spin chain with input state @xmath101 vs. rescaled time @xmath47 . the first peak at @xmath58 is 0.9976.,scaledwidth=45.0% ] overall , we observe that the perturbative influence of next - nearest neighbor interaction ( eq . ( [ hami2 ] ) ) is the main cause for loss in fidelity in both unentangled states as well as states whose entanglement results only from their dynamics and which are initially unentangled , although unentangled chains suffer slightly more . entangled chains with an initially entangled state on the other hand have been shown to be more robust against this type of defect . despite these variations , our study clearly demonstrates that next - nearest neighbor interactions are the most damaging form of perturbation overall , as seen in state transfer fidelity or eof . the reason for this is that the next - nearest neighbor interaction is the only fabrication defect or limitation in the set ( a)-(d ) that effectively opens up new ` channels ' for the system dynamics . the hamiltonian @xmath52 of eq . ( [ hami2 ] ) connects chain sites which would otherwise be disconnected ( at the same order in perturbation ) . it therefore facilitates a more efficient ( in a detrimental sense ) ` spread ' of excitations . the general consequence of this is more damage to transfer fidelity or eof , when compared to defects ( a)-(c ) with the same level of noise . numerical simulations ( not shown ) support this explanation , as addition of new perturbative ` channels ' for the dynamics ` by hand ' ( as opposed to via @xmath52 ) into the full hamiltonian can lead to similar results to those in , for example , fig . [ fig : nnnicon ] . it has been shown in ref . @xcite that opening new channels even beyond next - nearest neighbor interaction can be compensated for if local control within the chain is possible by adjusting the nearest neighbor coupling . this degree of local control may not always be available and/or might not be desirable , so in our work we consider the longer range interactions as a potential perturbation on nearest neighbor systems designed to produce pst . in this section we will discuss the effect of imperfect excitation injection into the device when multiple excitation states are considered . a possible device configuration for input / output of multiple excitations is sketched in fig . [ diag ] , where each site in the chain is associated with a register which can act as input / output device . we assume that there exists a clock to which the machinery at both ends of the chain have reference . ( without such a clock even simple pst could not operate , as extraction has to be timed with respect to injection . ) the timing errors we consider are with respect to this reference clock . and @xmath121 registers and the information is transferred to the @xmath122 and @xmath123 via the mirroring rule.,scaledwidth=44.5% ] we will first consider the case of unentangled input states @xmath124 and then analyze the effects on entangled input states . let us consider the injection and transfer of a state @xmath124 . an important question to ask is how and to what extent a time delay in input operations would alter the transmission fidelity . this question is particularly important for multiple excitations : for a single excitation a delayed injection would not alter the overall state evolution , but there would be merely an overall time shift . for multiple excitations however , it may occur that during the preparation of a state @xmath124 , with @xmath125 , not all input sites are accessed at exactly the same time . as a consequence a system which is supposed to be prepared in a two - excitation state may start evolving as a one excitation system if the two required excitations are not injected in synchrony . + let us focus on the latter case . when considering such a delayed input there is a finite probability that the second spin @xmath38 we want to inject an excitation into is already occupied . the result of this scenario is dependent on the injection mechanism , so we will consider the two main possibilities : ( i ) spin excitation via a rabi - flopping control pulse on spin @xmath38 applicable for example to systems in which excitations correspond to ground state excitons confined within a quantum dot chain or to flipping the spin of an electron already confined within the chain and ( ii ) injection via swap operation or injection of an additional particle in the chain . the latter may correspond e.g. to the scenario in which the state in the qubit of register @xmath38 closest to the spin chain ( see fig . [ diag ] ) is swapped with the state in the chain site @xmath38 e.g. via a train of laser pulses in the case of exciton qubits ( see @xcite ) or to the scenario in which the main computation occurs via coherent electron transport in quantum wires ( such as in @xcite ) , each connected to a spin chain site . in case ( i ) injecting an excitation corresponds to applying a @xmath126-pulse using the hamiltonian @xmath127 with @xmath128 the rabi frequency ( we assume that qubits can be manipulated on an individual basis ) . accordingly , at the delayed injection of the second excitation , the system state will evolve as @xmath129 with @xmath130 for @xmath131 and @xmath132 . here we have explicitly displayed the set of indices @xmath93 . the last term in eq . ( [ evol_1 ] ) corresponds to the error induced by having a non - zero probability @xmath133 of an excitation present in spin @xmath38 , and translates , after injection , in the probability of having _ no - excitations _ at all present in the chain ( see fig . [ fig : delay_6_rs](a ) ) . if the desired dynamics is such that a certain site @xmath121 has unit probability to contain an excitation at a later time @xmath134 though , the system can be refocused by measuring site @xmath121 at @xmath134 : a result of excitation present " would collapse the system state into two - excitation dynamics , which is the closer to the desired dynamics the smaller the delay ; a result of no - excitation present " would imply that the chain contains indeed no excitation at all . we underline that the latter result could be used as a protocol to reinitialize the chain itself . in case ( ii ) the scenario is very different . should the first excitation already have a non - zero probability of occupying site @xmath38 , the second excitation will have a related probability of remaining in the register . the register would then become entangled with the spin chain . the injection process can in fact be described as follows ( where we assume that the presence of the excitation in site @xmath38 is the only cause of failed injection ) @xmath135 with @xmath130 for @xmath131 . the simplest way to destroy this unwanted entanglement is to measure , after injection , whether the second excitation is still present in the register or wire . by this measure , we remove the entanglement between register and device , but we also get to know exactly what state the device itself is in : if the measure outcome is no excitation in the register " the injection has been successful and the spin chain now follows a two excitation evolution which is the closer to the desired dynamics the smaller the injection delay has been . this is described in fig . [ fig : delay_6_rs](b ) . if the outcome of the measurement is excitation in the register " the chain will continue to evolve as a one excitation system . however in the latter case we have also re - set the chain ready for trying again the injection of the second excitation at the earliest convenient time . delay vs. rescaled time @xmath47 : + ( a ) shows injection by rabi - flopping , where the resulting error remains in the system as the zero vector @xmath136 . at @xmath61 the state of the first spin is measured and an excitation is found . this allows for refocusing the system in the two excitation subspace only . the peak of @xmath56 at multiples of @xmath137 plus delay is 0.7807 . if at @xmath61 the second spin is measured instead , the peak fidelity becomes 0.7203 . + ( b ) shows injection by swap operation where the error remains in the system in the one excitation subspace . at @xmath61 the state of the register is measured and no excitation is found . this projects the system in the two excitation subspace only and disentangles the chain from the register . the peak of @xmath56 at multiples of @xmath137 plus delay is 0.9870.,scaledwidth=45.0% ] in fig . [ fig:6spin2exdelays ] , we give an overview of various possible delay scenarios , assuming input by swap operation with error correction by measurement of the injecting register or wire performed immediately after the attempted injection of the second excitation . in panel ( a ) , the delay of the second excitation is equal to @xmath138 . even with the error correction measurement we see an impact on the system as the delay between the two injections puts a cap on the fidelity of the revivals of the desired two - excitation input state . the larger the delay ( up to the mirror time @xmath88 ) , the more serious the impact . frame ( b ) demonstrates that even if timely simultaneous injection were not possible , the ability to inject the second excitation an integer number of periods @xmath137 after the first one would allow for perfect revivals . in contrast , the third frame ( c ) shows that a second injection exactly @xmath88 away from the ideal case would result in the total decay of the desired state . however , we have to keep in mind that at @xmath55 where @xmath139 has the lowest fidelity , the corresponding twin state will be achieving perfect fidelity and that therefore a new state will emerge , we illustrate this in fig . [ fig : newborn ] : at a time equal to an odd multiple of @xmath88 , the initial input state @xmath140 has zero fidelity , while its twin state @xmath141 has unit fidelity . injection of the second excitation into the second site ( which would have resulted in the ideal state @xmath56 if there had been no delay ) results in @xmath142 . this state then continues to evolve with its twin state @xmath143 , both of them alternately reaching perfect fidelity . the additional features on the plots of vectors with two excitations , as well as the narrowing of the main peak are due to excitation of the particular superpositions of energy eigenstates in the two - excitation subspace that correspond to our initial state . this shows how we can use delayed input to transfer states which are different from the `` twin '' sites of the input sites while still assuring pst . this may be useful if e.g. not all sites are accessible for input operations . . + ( a ) input delayed by @xmath138 . + ( b ) input delayed by an integer number of @xmath137 ( here : @xmath144 ) . + ( c ) input delayed by an odd multiple of @xmath88 ( here : @xmath145 ) . + for ( a ) the maximum recurring fidelity of @xmath57 is 0.9313 , at odd integer multiples of @xmath146 plus the delay . for ( b ) , due to the complete period delay , @xmath57 emerges with unit fidelity , whereas for ( c ) the occurrence is negligible.,scaledwidth=45.0% ] ( here : @xmath147 ) vs. rescaled time @xmath47 . the peak of @xmath148 at @xmath149 reaches unity.,scaledwidth=45.0% ] imperfect injection can occur even without delays between different excitations injections due to other device imperfections . a typical example could be the possibility that in the swap injection scenario one of the two excitations is partially reflected into the wire . this may occur in a device in which the spin chain is formed by gate - defined quantum dots where electrons are injected via ( computational ) wires , with the reflection being caused by imperfect lowering of a potential barrier between the wire and the selected spin chain site @xmath38 . in this case the injection would be described by @xmath150 where the first spin has been perfectly injected at site @xmath121 , while there is a reflection probability @xmath151 associated with spin @xmath38 . in this scenario measuring the absence ( presence ) of excitation in the register after injection would ensure that the chain is undergoing exactly the desired dynamics ( or that the chain is ready for re - injecting the second excitation at the closest suitable time ) . the above scenario can be straightforwardly extended to the case in which a finite probability of reflection is associated with both injection sites . + while in this paper we are only discussing delays at the input stage , similar problems may arise of course at read - out , should the extraction of a state covering multiple sites not be as timely as we would hope it to be . it is also worth noting that any peak following delayed input is shifted forward in time linearly with increasing delay . when considering imperfect or non - synchronous input and entanglement transfer , we see a similar effect to that on unentangled states . again , as the overall desired system state is more complicated , we monitor the evolution of the amount of entanglement in the relevant sub - system , instead of individual vectors . in fig . [ fig : gatedelay ] we show the evolution of the first eof peak after the delayed input has taken place , for case ( ii ) where entanglement is created from an initial product state . in this setting it is not possible to implement the refocusing protocol for the rabi flopping , but we are assuming that the confirmed injection protocol for the swap operation is implemented instantaneously , just after the injection attempt . due to the fact that the two injection sites in the state described in this figure are at opposite chain ends , there is virtually no difference between injection by rabi flopping and injection by swap operation . as long as the second injection is only delayed by a small fraction of @xmath88 , the first excitation will not have spread out far enough yet to affect the second injection site . we see that even for delays as large as @xmath152 , the system retains over 90% of its possible entanglement . vs. input delay for an 8-spin chain with initial input state @xmath101 for both rabi flopping and swap operation type of injections.,scaledwidth=45.0% ] when we consider two adjacent injection sites , there is instead a clear discrepancy between the decay in entanglement of formation in a chain with input via rabi flopping or swap operation ( fig [ fig:0110delay ] ) . again , we assume that there is an instantaneous swap correction protocol applied , while refocussing after the rabi - flopping is not possible . as in the case of non - synchronous input in unentangled states , a delay in a swap type injection perturbs the system far less than using rabi flopping , and shows virtually no loss in the amount of entanglement in the system , even for delays as large as @xmath153 . if instead rabi flopping is used , the decay in eof is similar to that of fig . [ fig : gatedelay ] . for both swap operation and rabi flopping type injections.,scaledwidth=45.0% ] in this paper we have considered a variety of physically relevant perturbative factors in spin chains . we have investigated in detail their effects on information transfer and entanglement generation and transfer . for different forms of perturbation , the results of our extensive numerical studies on the quality of transfer can be captured with a straightforward analytic expression that demonstrates exponential damping with the number of spins on the chain and gaussian dependence on the relevant perturbation parameter . this expression provides a simple tool for estimating the efficiency of a chain under the action of perturbations . we have also related our dimensionless perturbation parameter scales to actual parameters for candidate spin chain realizations such as quantum dots with exciton qubits and graphene dots with spin qubits , providing a calibration of our estimator tool for these experimental systems . we have considered the transport of both unentangled states as well as entangled states via spin chains , subject to various forms of perturbation , and a general conclusion is that next - nearest neighbor interactions are the most damaging to transfer fidelity or eof . the reason for this can be traced back to the fact that the next - nearest neighbor interaction is the only perturbation that effectively opens up new ` channels ' for the system dynamics . by connecting sites which would otherwise be disconnected ( at the same order in perturbation ) , it allows a more efficient ( in a detrimental sense ) ` spread ' of excitations , and consequently drastically diminishes the occurrence of quantum coherent effects such as revivals and pst . in particular , we have seen that the introduction of next - nearest neighbor coupling may lead to non - negligible quality loss after only a few periods . systems with simpler input states are generally slightly less affected than those of a slightly more complex nature but as different perturbation factors affect different excitation subspaces , there is no clear advantage of one particular type of state in terms of robustness . however , for all fabrication defects considered we have found that the transport during the first period remains of high quality for perturbation amplitudes of the order of few percent , while the periodicity itself may be destroyed for next - nearest neighbor couplings above about @xmath154 . this is of particular concern for schemes based on dipole - dipole interactions , where , for roughly equidistant sites , the next - nearest neighbor interactions are of the order of @xmath155 . nevertheless , due to the gaussian dependence of transport efficiency on perturbation amplitudes , provided that the perturbations are kept below these few percent thresholds , spin chains are demonstrated to be very good candidates for the implementation of solid state quantum information processing devices , robust against various forms of perturbation . additionally , we have seen that the effect of imperfect input or injection operations leads to a permanent ( but constant ) loss in information transport quality and entanglement generation . for input delayed by a time approaching the mirroring time @xmath88 the transport of the intended state is replaced by a new input state , which is then subject to the system dynamics as per usual . furthermore , there is a fundamental difference between the two input methods we have considered . if rabi flopping is used and the injection of the second excitation is delayed , the error induced by the imperfect input remains in the system , except if a site is expected to be in state @xmath156 at a known time , when measurement of this site can recover dynamics close to the ideal case . in the case of input by swap operation , the error can be dramatically reduced by subsequent measuring of the environment . when considering swap type injection , unentangled states and type ( i ) entanglement states are remarkably robust against non - synchronous injection , with hardly any loss in fidelity even for large delays . type ( ii ) dynamically - generated entangled states are instead more affected due to their more complex set - up . even so the loss in eof remains less than 10% for delay values up to 10% of the mirroring time @xmath88 . our studies demonstrate quantitatively the criteria that need to be met , in terms of perturbation scales and injection errors , for imperfect spin chains to work efficiently as quantum information transfer and entanglement transfer / generation devices . with modest errors at or below the few percent level , spin chains prove to be good and robust devices , which is very encouraging for future experiments on these systems . + rr was supported by epsrc - gb and hp . ida acknowledges partial support by hp . ida and rr acknowledge the kind hospitality of the hp research labs bristol .
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spin chains have been proposed as a reliable and convenient way of transferring information and entanglement in a quantum computational context .
nonetheless , it has to be expected that any physical implementation of these systems will be subject to several perturbative factors which could potentially diminish the transfer quality . in this paper , we investigate a number of possible fabrication defects in the spin chains themselves as well as the effect of non - synchronous or imperfect input operations , with a focus on the case of multiple excitation / qubit transfer .
we consider both entangled and unentangled states , and in particular the transfer of an entangled pair of adjacent spins at one end of a chain under the mirroring rule and also the creation of entanglement resulting from injection at both end spins .
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it is now established that the collision of heavy - ions at the relativistic heavy - ion collider have led to the formation of an entirely new form of matter @xcite . while the underlying degrees of freedom prevalent in the hot plasma are , as yet , unknown @xcite , various constraints may be imposed through a study of its partonic substructure . the foremost tool in this study is the modification of the hard jets , usually referred to as jet quenching @xcite . the number of hadrons with transverse momentum @xmath0 gev ( which , necessarily originate in the fragmentation of hard jets ) is reduced by almost a factor of 5 in central @xmath1-@xmath1 collisions , compared to that expected from elementary nucleon nucleon encounters enhanced by the number of expected binary collisions @xcite . jet modification is a probe with a wide range of complexity in terms of experimental observables . by now , measurements on single inclusive observables have been extended to very high @xmath2 ( @xmath3 gev ) . there also exist , a large number of multi - particle jet - like correlation observables , photon - jet , jet - medium and heavy flavor observables @xcite . in these proceedings , we attempt a very brief review of the underlying theory and some of the new jet correlation observables which may be used to understand the underlying space - time and momentum space structure of the produced matter . most current calculations of the in - medium modification of light partons may be divided into four major schemes , often referred to by the names of the original authors . all schemes utilize a factorized approach , where the final cross section to produce a hadron @xmath4 with high transverse momentum @xmath2 and a pseudo - rapidity between @xmath5 and @xmath6 may be expressed as an integral over the product of the nuclear structure functions [ @xmath7 , to produce partons with momentum fractions @xmath8 , a hard partonic cross section to produce a hard parton with a transverse momentum @xmath9 and a medium modified fragmentation function for the final hadron [ @xmath10 , the modification of the partonic jet is encoded in the calculation of the medium modified fragmentation function . the four schemes of energy loss are in principle a different set of approximation schemes to estimate this quantity from perturbative qcd calculations . the reaction operator approach in opacity , often referred to as the gyulassy - levai - vitev ( glv ) scheme @xcite , assumes the medium to be composed of heavy , almost static , colour scattering centers ( with debye screened yukawa potentials ) which are well separated in the sense that the mean free path of a jet @xmath11 , the colour screening length of the medium . the opacity of the medium @xmath12 quantifies the number of scattering centers seen by a jet as it passes through the medium , _ i.e. _ , @xmath13 , where @xmath14 is the thickness of the medium . at leading order in opacity , a hard jet , produced locally in such a plasma with a large forward energy @xmath15 , scatters off one such potential and in the process radiates a soft gluon . multiple such interactions in a poisson approximation are considered to calculate the probability for the jet to lose a finite amount of its energy . the path integral in opacity approach , referred to as the armesto - salgado - wiedemann ( asw ) approach @xcite , also assumes a model for the medium as an assembly of debye screened heavy scattering centers . a hard , almost on shell , parton traversing such a medium will engender multiple transverse scatterings of order @xmath16 . it will in the process split into an outgoing parton and a radiated gluon which will also scatter multiply in the medium . the propagation of the incoming ( outgoing ) partons as well as that of the radiated gluon in this background colour field may be expressed in terms of effective green s functions , which are obtained in terms of path integrals over the field . also similar to the glv approach , a poisson approximation is then used to obtain multiple emissions and a finite energy loss . in the finite temperature field theory scheme referred to as the arnold - moore - yaffe ( amy ) approach @xcite , the energy loss of hard jets is considered in an extended medium in equilibrium at asymptotically high temperature @xmath17 ( and as a result @xmath18 ) . in this limit , one uses the effective theory of hard - thermal - loops ( htl ) to describe the collective properties of the medium . a hard on - shell parton undergoes soft scatterings with momentum transfers @xmath19 off other hard partons in the medium . such soft scatterings induce collinear radiation from the parton , with a transverse momentum of the order of @xmath20 . multiple scatterings of the incoming ( outgoing ) parton and the radiated gluon need to be considered to get the leading order gluon radiation rate . this is obtained from the imaginary parts of infinite order ladder diagrams . these rates are then used to evolve an initial distribution of hard partons through the medium in terms of a fokker - plank equation . in the higher - twist scheme @xcite , one directly computes the modification to the fragmentation functions due to multiple scattering in the medium by identifying and re - summing a class of higher twist contributions which are enhanced by the length of the medium . the initial hard jet is assumed to be considerably virtual , with @xmath21 . the propagation and collinear gluon emissions from such a parton are influenced by the multiple scattering in the medium . one assumes that , on exiting the medium , the hard parton has a small , yet perturbative scale @xmath22 . one evolves this scale back up to the hard scale of the original produced parton , @xmath23 , by including the effect of multiple emissions in the medium . the multiple scatterings introduce a destructive interference for radiation at very forward angles and as such modify the evolution of the fragmentation functions in the medium . in any scheme , the magnitude of the modification is controlled by a single space - time dependent parameter which may be related to the well known transport coefficient @xmath24 . this is defined as the mean transverse momentum squared per unit length , transferred by the medium to the hard jet . in actual computations , a model of the space - time dependence is invoked , and a maximum for @xmath24 is set to best fit with experimental results . values of the maximum of @xmath24 , in the vicinity of the center of a central collision , at a time of @xmath25fm / c , range from 1 gev@xmath26/fm up to 20 gev@xmath26/fm @xcite , depending on the scheme , as well as , the model of the medium used . while the suppression of single inclusive hadrons may be used to determine the maximum value of @xmath24 , corrrelations between the leading hadron and the medium may be used to test the space - time profile which is used as the ansatz @xcite . once considers the nuclear modification factor @xmath27 as a function of the angle with the reaction plane . the results of such an analysis , from the higher twist approach , for a medium profile taken from a full @xmath28 hydrodynamics simulation at @xmath29 % centrality are presented in the left panel of fig . the plots clearly demonstrate that the modification is maximal when the jet propagates through the thickest part of the medium in a direction perpendicular to the reaction plane . the spread between the lines is directly dependent on the particular space - time ansatz chosen for the evolution of the produced matter . to determine the momentum structure of the plasma , one generalizes @xmath24 from a scalar to a tensor @xcite , where the scalar @xmath30 . an example where such a situation may arise is in the presence of large turbulent colour fields , which may be generated in the early plasma due to anisotropic parton distributions @xcite . these large fields , transverse to the beam , tend to deflect radiated gluons from a transversely traveling jet , preferentially , in the longitudinal directions . while such effects influence the solid angle distributions of the radiated gluons around the originating parton , they do not have a considerable effect on the total energy lost . such phenomena may yield an explanation for the ridge like structure seen in the near side correlations @xcite . in the right panel of fig . [ fig1 ] , results from a quantitative estimate are presented where the trigger quark ( with @xmath31 gev ) radiates a 4 gev gluon , which is then subjected to such transverse fields ( over a distance of 3 fm ) . using a time dependent @xmath24 , with an initial value consistent with fits to experiment , one obtains a noticeable ridge like structure , _ i.e. _ , a considerable broadening in @xmath5 but not in @xmath32 . 99 k. adcox _ et al . _ , nucl . phys . a * 757 * , 184 ( 2005 ) ; j. adams _ et al . _ , nucl . phys . a * 757 * , 102 ( 2005 ) . e. v. shuryak and i. zahed , phys . c * 70 * , 021901 ( 2004 ) ; phys . d * 70 * , 054507 ( 2004 ) . v. koch _ et . * 95 * , 182301 ( 2005 ) ; a. majumder and b. muller , phys . c * 74 * , 054901 ( 2006 ) . m. gyulassy and m. plumer , phys . b * 243 * , 432 ( 1990 ) ; x. n. wang and m. gyulassy , phys . lett . * 68 * , 1480 ( 1992 ) ; r. baier _ et . b * 483 * , 291 ( 1997 ) ; nucl . b * 484 * , 265 ( 1997 ) . k. adcox _ et al . _ , phys . rev . lett . * 88 * , 022301 ( 2002 ) ; c. adler _ et al . _ , phys . lett . * 89 * 202301 ( 2002 ) . x. f. guo and x. n. wang , phys . lett . * 85 * , 3591 ( 2000 ) ; nucl . a * 696 * , 788 ( 2001 ) ; b. w. zhang and x .- wang , nucl . a * 720 * , 429 ( 2003 ) ; a. majumder _ et . lett . * 99 * , 152301 ( 2007 ) . a. majumder , j. phys . g * 34 * , s377 ( 2007 ) . a. majumder , phys . c * 75 * , 021901 ( 2007 ) ; a. majumder _ et . c * 76 * , 041902 ( 2007 ) ; t. renk _ et . al . _ , phys . c * 75 * , 031902 ( 2007 ) ; g. y. qin _ et . c * 76 * , 064907 ( 2007 ) . a. majumder _ et . al . _ , phys . lett . * 99 * , 042301 ( 2007 ) . m. asakawa _ et . lett . * 96 * 252301(2006 ) . p. arnold _ et . _ , phys . d * 72 * , 054003 ( 2005 ) ; p. romatschke and m. strickland , phys . d * 68 * , 036004 ( 2003 ) ; s. mrowczynski , phys . b * 314 * ( 1993 ) 118 .
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the use of jet modification to study the properties of dense matter is reviewed .
different sets of jet correlations measurements which may be used to obtain both the space - time and momentum space structure of the produced matter are outlined .
| 3,253 | 49 |
chemical elements heavier than lithium are synthesized in stars . such `` metals '' are observed at times when the universe was only @xmath6% of its current age in the inter galactic medium ( igm ) as absorption lines in quasar spectra ( see ellison et al . 2000 , and references therein ) . hence , these heavy elements not only had to be synthesized but also released and distributed in the igm within the first billion years . only supernovae of sufficiently short lived massive stars are known to provide such an enrichment mechanism . this leads to the prediction that _ the first generation of cosmic structures formed massive stars ( although not necessarily only massive stars ) . _ in the past 30 years it has been argued that the first cosmological objects form globular clusters ( ) , super massive black holes ( ) , or even low mass stars ( ) . this disagreement of theoretical studies might at first seem surprising . however , the first objects form via the gravitational collapse of a thermally unstable reactive medium , inhibiting conclusive analytical calculations . the problem is particularly acute because the evolution of all other cosmological objects ( and in particular the larger galaxies that follow ) will depend on the evolution of the first stars . nevertheless , in comparison to present day star formation , the physics of the formation of the first star in the universe is rather simple . in particular : * the chemical and radiative of processes in the primordial gas are readily understood . * strong magnetic fields are not expected to exist at early times . * by definition no other stars exist to influence the environment through radiation , winds , supernovae , etc . * the emerging standard model for structure formation provides appropriate initial conditions . in previous work we have presented three dimensional cosmological simulations of the formation of the first objects in the universe ( , ) including first applications of adaptive mesh refinement ( amr ) cosmological hydrodynamical simulations to first structure formation ( , , abn hereafter ) . in these studies we achieved a dynamic range of up to @xmath7 and could follow in detail the formation of the first dense cooling region far within a pre galactic object that formed self consistently from linear density fluctuation in a cold dark matter cosmology . here we report results from simulations that extend our previous work by another 5 orders of magnitude in dynamic range . for the first time it is possible to bridge the wide range between cosmological and stellar scale . we employ an eulerian structured adaptive mesh refinement cosmological hydrodynamical code developed by bryan and norman ( , ) . the hydrodynamical equations are solved with the second order accurate piecewise parabolic method (; ) where a riemann solver ensures accurate shock capturing with a minimum of numerical viscosity . we use initial conditions appropriate for a spatially flat cold dark matter cosmology with 6% of the matter density contributed by baryons , zero cosmological constant , and a hubble constant of 50 km / s / mpc ( ) . the power spectrum of initial density fluctuations in the dark matter and the gas are taken from the computation by the publicly available boltzmann code cmbfast ( ) at redshift 100 ( assuming an harrison zeldovich scale invariant initial spectrum ) . we set up a three dimensional volume with 128 comoving kpc on a side and solve the cosmological hydrodynamics equations assuming periodic boundary conditions . this small volume is adequate for our purpose , because we are interested in the evolution of the first pre galactic object within which a star may be formed by a redshift of @xmath8 . first we identify the lagrangian volume of the first proto galactic halo with a mass of @xmath9 in a low resolution pure n body simulation . then we generate new initial conditions with four initial static grids that cover this langrangian region with progressively finer resolution . with a @xmath10 top grid and a refinement factor of 2 this specifies the initial conditions in the region of interest equivalent to a @xmath11 uni grid calculation . for the adopted cosmology this gives a mass resolution of @xmath12 for the dark matter ( dm , hereafter ) and @xmath13 for the gas . the small dm masses ensure that the cosmological jeans mass is resolved by at least ten thousand particles at all times . smaller scale structures in the dark matter will not be able to influence the baryons because of their shallow potential wells . the theoretical expectation holds , because the simulations of abn which had 8 times poorer dm resolution led to identical results on large scales as the simulation presented here . during the evolution , refined grids are introduced with twice the spatial resolution of the parent ( coarser ) grid . these child ( finer ) meshes are added whenever one of three refinement criteria are met . two langrangian criteria ensure that the grid is refined whenever the gas ( dm ) density exceeds 4.6 ( 9.2 ) its initial density . additionally , the local jeans length is always covered by at least 64 grid cells cells . ] ( 4 cells per jeans length would be sufficient , ) . we have also carried out the simulations with identical initial conditions but varying the refinement criteria . in one series of runs we varied the number of mesh points per jeans length . runs with 4 , 16 , and 64 zones per jeans length are indistinguishable in all mass weighted radial profiles of physical quantities . no change in the angular momentum profiles could be found , suggesting negligible numerical viscosity effects on angular momentum transport . a further refinement criterion that ensured the local cooling time scale to be longer than the local courant time also gave identical results . this latter test checked that any thermally unstable region was identified . the simulation follows the non equilibrium chemistry of the dominant nine species species ( h , h@xmath14 , h@xmath15 , e@xmath15 , he , he@xmath14 , he@xmath16 , h@xmath17 , and h@xmath18 ) in primordial gas . furthermore , the radiative losses from atomic and molecular line cooling , compton cooling and heating of free electrons by the cosmic background radiation are appropriately treated in the optically thin limit ( , ) . to extend our previous the studies to higher densities three essential modifications to the code were made . first we implemented the three body molecular hydrogen formation process in the chemical rate equations . for temperatures below 300 k we fit to the data of orel ( ) to get @xmath19 . above 300 k we then match it continuously to a powerlaw ( ) @xmath20 . secondly , we introduce a variable adiabatic index for the gas ( ) . the dissipative component ( baryons ) may collapse to much higher densities than the collisionless component ( dm ) . the discrete sampling of the dm potential by particles can then become inadequate and result in artificial heating of the baryons ( cooling for the dm ) once the gas density becomes much larger than the local dm density . to avoid this , we smooth the dm particles with a gaussian of width 0.05 for grids with cells smaller than this length . at this scale , the enclosed gas mass substantially exceeds the enclosed dm mass . the standard message passing library ( mpi ) was used to implement domain decomposition on the individual levels of the grid hierarchy as a parallelization strategy . the code was run in parallel on 16 processors of the sgi origin2000 supercomputer at the national center for supercomputing applications at the university of illinois at urbana champaign . we stop the simulation at a time when the molecular cooling lines reach an optical depth of ten at line center because our numerical method can not treat the difficult problem of time dependent radiative line transfer in multi dimensions . at this time the code utilizes above 5500 grids on 27 refinement levels with @xmath21 computational grid cells . an average grid therefore contains @xmath22 cells . [ colorplate ] our simulations ( fig . [ colorplate ] , fig . [ 5panel ] ) , identify at least four characterisic mass scales . from the outside going in , one observes infall and accretion onto the pre galactic halo with a total mass of @xmath23 , consistent with previous studies ( , , , abn , and for discussion and references ) . at a mass scale of about 4000 solar mass ( @xmath24 ) rapid cooling and infall is observed . this is accompanyed by the first of three valleys in the radial velocity distribution ( fig . [ 5panel]e ) . the temperature drops and the molecular hydrogen fraction increases . it is here , at number densities of @xmath25 , that the high redshift analog of a molecular cloud is formed . although the molecular mass fraction is not even 0.1% it is sufficient to cool the gas rapidly down to @xmath26 . the gas can not cool below this temperature because of the sharp decrease in the cooling rate below @xmath27 . at redshift 19 ( fig . [ 5panel ] ) , there are only two mass scales ; however , as time passes the central density grows and eventually passes @xmath28 , at which point the ro - vibrational levels of are populated at their equilibrium values and the cooling time becomes independent of density ( instead of inversely proportional to it ) . this reduced cooling efficiency leads to an increase in the temperature ( fig . [ 5panel]d ) . as the temperature rises , the cooling rate again increases ( it is 1000 times higher at 800 k than at 200 k ) , and the inflow velocities slowly climb . in order to better understand what happens next , we examine the stability of an isothermal gas sphere . the critical mass for gravitational collapse given an external pressure @xmath29 ( be mass hereafter ) is given by ebert ( ) and bonnor ( ) as : @xmath30 here @xmath29 is the external pressure and @xmath31 , @xmath32 , and @xmath33 the gravitational constant , the boltzmann constant and the sound speed , respectively . we can estimate this critical mass locally if we set the external pressure to be the local pressure to find @xmath34 where @xmath35 is the mean mass per particle in units of the proton mass . using an adiabatic index @xmath36 , we plot the ratio of the enclosed gas mass to this modified be mass in figure [ bemass ] . our modeling shows ( fig . [ bemass ] ) , that by the fourth considered output time , the central 100 exceeds the be mass at that radius , indicating unstable collapse . this is the third mass scale and corresponds to the second local minimum in the radial velocity curves ( fig . [ 5panel]e ) . the inflow velocity is @xmath37 is still subsonic . although this mass scale is unstable , it does not represent the smallest scale of collapse in our simulation . this is due to the increasing molecular hydrogen fraction . when the gas density becomes sufficiently large ( @xmath38 ) , three - body molecular hydrogen formation becomes important . this rapidly increases the molecular fraction ( fig . [ 5panel]c ) and hence the cooling rate . the increased cooling leads to lower temperatures and even stronger inflow and . at a mass scale of @xmath39 , not only is the gas nearly completely molecular , but the radial inflow has become supersonic ( fig . [ 5panel]e ) . when the mass fraction approaches unity , the increase in the cooling rate saturates , and the gas goes through a radiative shock . this marks the first appearance of the proto stellar accretion shock at a radius of about 20 astronomical units from its center . when the cooling time becomes independent of density the classical criterion for fragmentation @xmath40 ( ) can not be satisfied at high densities . however , in principal the medium may still be subject to thermal instability . the instability criterion is @xmath41 where @xmath42 denotes the cooling losses per second of a fluid parcel and @xmath43 and @xmath44 are the gas temperature and mass density , respectively . at densities above the critical densities of molecular hydrogen the cooling time is independent of density , i.e. @xmath45 where @xmath46 is the high density cooling function ( e.g. ) . fitting the cooling function with a power - law locally around a temperature @xmath47 so that @xmath48 one finds @xmath49 . hence , under these circumstances the medium is thermally stable if @xmath50 . because , @xmath51 for the densities and temperatures of interest , we conclude that the medium is thermally stable . the above analysis neglects the heating from contraction , but this only strengthens the conclusion . if heating balances cooling one can neglect the @xmath52 term in equation ( [ eq : ic ] ) and find the medium to be thermally stable for @xmath53 . [ bemass ] however , here we neglected the chemical processes . the detailed analysis for the case when chemical processes occur on the collapse time scale is well known ( ) . this can be applied to primordial star formation ( ) including the three body formation of molecular hydrogen ( ) which drives a chemo thermal instability . evaluating all the terms in this modified instability criterion ( , equation 36 ) one finds the simple result that for molecular mass fractions @xmath54 the medium is expected to be chemo thermally unstable . these large molecular fractions illustrate that the strong density dependence of the three body formation dominates the instability . examining the three dimensional temperature and density field we clearly see this chemo thermal instability at work . cooler regions have larger fractions . however , no corresponding large density inhomogeneities are found and fragmentation does not occur . this happens because of the short sound crossing times in the collapsing core . when the formation time scale becomes shorter than the cooling time the instability originates . however , as long as the sound crossing time is much shorter than the chemical and cooling time scales the cooler parts are efficiently mixed with the warmer material . this holds in our simulation until the final output where for the first time the formation time scale becomes shorter than the sound crossing time . however , at this point the proto stellar core is fully molecular and stable against the chemo thermal instability . consequently no large density contrasts are formed . because at these high densities the optical depth of the cooling radiation becomes larger than unity the instability will be suppressed even further . interestingly , rotational support does not halt the collapse . this is for two reaons . the first is shown in panel a of fig . [ angtrans ] , which plots the specific angular momentum against enclosed mass for the same seven output times discussed earlier . concentrating on the first output ( fig . [ angtrans ] ) , we see that the central gas begins the collapse with a specific angular momentum only @xmath55% as large as the mean value . this type of angular momentum profile is typical of halos produced by gravitational collapse ( e.g. ) , and means that the protostellar gas starts out without much angular momentum to lose . as a graphic example of this , consider the central one solar mass of the collapsing region . it has only an order of magnitude less angular momentum at densities @xmath56 than it had at @xmath57 although it collapsed by over a factor 100 in radius . the remaining output times ( fig . [ angtrans ] ) indicate that there is some angular momentum transport within the central @xmath58 ( since l plotted as a function of enclosed mass should stay constant as long as there is no shell crossing ) . in panel c , we divide @xmath42 by @xmath59 to get a typical rotational velocity and in panels b and d compare this velocity to the keplerian rotational velocity and the local sound speed , respectively . we find that the typical rotational speed is a factor two to three below that required for rotational support . furthermore , we see that this azimuthal speed never significantly exceeds the sound speed , although for most the mass below @xmath58 it is comparable in value . we interpret this as evidence that it is shock waves during the turbulent collapse that are responsible for much of the transported angular momentum . a collapsing turbulent medium is different from a disk in keplarian rotation . at any radius there will be both low and high angular momentum material , and pressure forces or shock waves can redistribute the angular momentum between fluid elements . lower angular momentum material will selectively sink inwards , displacing higher angular momentum gas . this hydrodynamic transport of angular momentum will be suppressed in situation where the collapse proceeds on the dynamical time rather on the longer cooling time as in the presented case . this difference in cooling time and the widely different initial conditions may explain why this mechanism has not been observed in simulations of present day star formation ( e.g. , and references therein ) . however , such situations may also arise in the late stages of the formation of present day stars and in scenarios for the formation of super massive black holes . to ensure that the angular momentum transport is not due to numerical shear viscosity ( ) we have carried out the resolution study discussed above . we have varied the effective spatial resolution by a factor 16 and found identical results . furthermore , we have run the adaptive mesh refinement code with two different implementations of the hydrodynamics solver . the resolution study and the results presented here were carried out with a direct piecewise parabolic method adopted for cosmology (; ) . we ran another simulation with the lower order zeus hydrodynamics ( ) and still found no relevant differences . these tests are not strict proof that the encountered angular momentum transport is not caused by numerical effects ; however , they are reassuring . the strength of magnetic fields generated around the epoch of recombination is minute . in contrast , phase transitions at the qantum chromo dynamic ( qcd ) and electro weak scales may form even dynamically important fields . while there is a plethora of such scenarios for primordial magnetic field generation in the early universe they are not considered to be an integral part of our standard picture of structure formation . this is because not even the order of these phase transitions is known ( ) , and references therein ) . unfortunately , strong primordial small - scale ( @xmath60 comoving ) magnetic fields are poorly constrained observationally ( ) . the critical magnetic field for support of a cloud ( ) allows a rough estimate up to which primordial magnetic field strengths we may expect our simulation results to hold . for this we also assume a flux frozen flow with no additional amplification of the magnetic field other than the contraction ( @xmath61 ) . for a comoving b field of @xmath62 on scales @xmath63 the critical field needed for support may be reached during the collapse possibly modifying the mass scales found in our purely hydrodynamic simulations . however , the ionized fraction drops rapidly during the collapse because of the absence of cosmic rays ionizations . consequently ambipolar diffusion should be much more effective in the formation of the first stars even if such strong primordial magnetic fields were present . previously we discussed the formation of the pre galactic object and the primordial `` molecular cloud '' that hosts the formation of the first star in the simulated patch of the universe ( ) . these simulations had a dynamic range of @xmath64 and identified a @xmath0 core within the primordial `` molecular cloud '' undergoing renewed gravitational collapse . the fate of this core was unclear because there was the potential caveat that three body formation could have caused fragmentation . indeed this further fragmentation had been suggest by analytic work ( ) and single zone models ( ) . the three dimensional simulations described here were designed to be able to test whether the three body process will lead to a break up of the core . _ no fragmentation due to three body formation is found . _ this is to a large part because of the slow quasi hydrostatic contraction found in abn which allows sub sonic damping of density perturbations and yields a smooth distribution at the time when three body formation becomes important . instead of fragmentation a single fully molecular proto star of @xmath65 is formed at the center of the @xmath66 core . however , even with extraordinary resolution , the _ final _ mass of the first star remains unclear . whether all the available cooled material of the surroundings will accrete onto the proto star or feedback from the forming star will limit the further accretion and hence its own growth is difficult to compute in detail . within @xmath67 about @xmath68 may be accreted assuming that angular momentum will not slow the collapse ( fig . [ acrete ] ) . the maximum of the accretion time of @xmath69 is at @xmath70 . however , stars larger than @xmath71 will explode within @xmath72 . therefore , it seems unlikely ( even in the absence of angular momentum ) that there would be sufficient time to accrete such large masses . a one solar mass proto star will evolve too slowly to halt substantial accretion . from the accretion time profile ( fig . [ acrete ] ) one may argue that a more realistic minimum mass limit of the first star should be @xmath73 because this amount would be accreted within a few thousand years . this is a very short time compared to expected proto stellar evolution times . however , some properties of the primordial gas may make it easier to halt the accretion . one possibility is the destruction of the cooling agent , molecular hydrogen , without which the acreting material may reach hydrostatic equilibrium . this may or may not be sufficient to halt the accretion . one may also imagine that the central material heats up to @xmath74 k , allowing lyman-@xmath75 cooling from neutral hydrogen . that cooling region may expand rapidly as the internal pressure drops because of infall , possibly allowing the envelope to accrete even without molecular hydrogen as cooling agent . additionally , radiation pressure from ionizing photons as well as atomic hydrogen lyman series photons may become significant and eventually reverse the flow . the mechanisms discussed by haehnelt ( 1993 ) on galactic scales will play an important role for the continued accretion onto the proto star . this is an interplay of many complex physical processes because one has a hot ionized strmgren sphere through which cool and dense material is trying to accrete . in such a situation one expects a raleigh taylor type instability that is modified via the geometry of the radiation field . at the final output time presented here there are @xmath76 hydrogen molecules in the entire protogalaxy . also the formation time scale is long because there are no dust grains and the free electrons ( needed as a catalyst ) have almost fully recombined . hence , as soon as the the first uv photons of lyman werner band frequencies are produced there will be a rapidly expanding photo dissociating region ( pdr ) inhibiting further cooling within it . this photo - dissociation will prevent further fragmentation at the molecular cloud scale . i.e. no other star can be formed within the same halo before the first star dies in a supernova . the latter , however , may have sufficient energy to unbind the entire gas content of the small pre galactic object it formed in ( ) . this may have interesting feedback consequences for the dispersal of metals , entropy and magnetic field into the intergalactic medium ( , ) . smoothed particle hydrodynamics ( sph , e.g. ) , used extensively in cosmological hydrodynamics , has been employed ( ) to follow the collapse of solid body rotating uniform spheres . the assumption of coherent rotation causes these clouds to collapse into a disk which developes filamentary structures which eventually fragment to form dense clumps of masses between @xmath77 and @xmath78 solar masses . it has been argued that these clumps will continue to accrete and merge and eventually form very massive stars . these sph simulation have unrealistic initial conditions and much less resolution then our calculations . however , they also show that many details of the collapse forming a primordial star are determined by the properties of the hydrogen molecule . we have also simulated different initial density fields for a lambda cdm cosmology . there we have focused on halos with different clustering environments . although we have not followed the collapse in these halos to proto - stellar densities , we have found no qualitative differences in the `` primordial molecular cloud '' formation process as discussed in abn . also other amr simulations ( ) give consistent results on scales larger than @xmath79 . in all cases a cooling flow forms the primordial molecular cloud at the center of the dark matter halo . we conclude that the molecular cloud formation process seems to be independent of the halo clustering properties and the adopted cdm type cosmology . also the mass scales for the core and the proto star are determined by the local bonnor ebert mass . consequently , we expect the key results discussed here to be insensitive to variations in cosmology or halo clustering . the picture arising from these numerical simulations has some very interesting implications . it is possible that all metal free stars are massive and form in isolation . their supernovae may provide the metals seen in even the lowest column density quasar absorption lines ( , and references therein ) . massive primordial stars offer a natural explanation for the absence of purely metal free low mass stars in the milky way . the consequences for the formation of galaxies may be even more profound in that the supernovae provide metals , entropy , and magnetic fields and may even alter the initial power spectrum of density fluctuations of the baryons . interestingly , it has been recently argued , from abundance patterns , that in low metallicity galactic halo stars seem to have been enriched by only one population of massive stars ( ) . these results , if confirmed , would represent strong support for the picture arising from our ab initio simulations of first structure formation . to end on a speculative note there is suggestive evidence that links gamma ray bursts to sites of massive star formation ( e.g. ) . it would be very fortunate if a significant fraction of the massive stars naturally formed in the simulations would cause gamma ray bursts ( e.g. ) . such high redshift bursts would open a remarkably bright window for the study of the otherwise dark ( faint ) ages . 1 . [ pd68 ] peebles , p. j. e. & dicke , r. h. 1968 , apj * 154 * , 891 2 . [ h69 ] hirasawa , t. 1969 , prog . . phys . * 42 * , 523 3 . [ pss83 ] palla , f. , salpeter , e.e . , stahler , s.w . 1983 , apj * 271 * , 632 4 . [ a95 ] abel , t. 1995 , thesis , university of regensburg . [ aanz ] abel , t. , anninos , p. , norman , m.l . , zhang , y. 1998 , apj * 508 * , 518 . [ abn99 ] abel , t. , bryan , g. l. , & norman , m. l. 1999 , in `` evolution of large scale structure : from recombination to garching '' , eds . banday , t. , sheth , r. k. and costa , l. n. 7 . [ abn00 ] abn : abel , t. , bryan , g.l . , norman , m.l . 2000 , apj * 540 * , 39 8 . [ cosmo ] friedmann models with a cosmological constant which currently seem to fit various observational test better differ from the standard cdm model considered here only slightly at the high redshifts modeled . [ bn97 ] bryan , g.l . , norman , m.l . 1997 , in _ computational astrophysics _ , eds . clarke and m. fall , asp conference # 123 10 . [ bn99 ] bryan , g.l . , norman , m.l . 1999 , in _ workshop on structured adaptive mesh refinement grid methods _ , i m a volumes in mathematics no . 117 , ed . n. chrisochoides , p. 165 [ wc84 ] woodward , p. r. , & colella , p. 1984 , j. comput 54 * , 115 12 . [ betal95 ] bryan , g.l . , cen , r. , norman , m.l . , ostriker , j.p . & stone , j.m . 1994 , apj * 428 * , 405 13 . [ sz96 ] seljak , u. , & zaldarriaga , m. 1996 , apj * 469 * , 437 14 . [ tkmh97 ] truelove , j. k. , klein , r. i. , mckee , c. f. , holliman , j. h. , howell , l. h. & greenough , j. a. 1997 , apjl * 489 * , l179 15 . [ aazn97 ] abel , t. , anninos , p. , zhang , y. , norman , m.l . 1997 , newa * 2 * , 181 16 . [ azan97 ] anninos , p. , zhang , y. , abel , t. , and norman , m.l . 1997 , newa * 2 * , 209 . [ orel ] orel , a.e . 1987 , j.chem.phys . * 87 * , 314 18 . [ on98 ] omukai , k. & nishi , r. 1998 , apj * 508 * , 141 19 . [ tegmark97 ] tegmark , m. , silk , j. , rees , m.j . , blanchard , a. , abel , t. , palla , f. 1997 , apj * 474 * , 1 . [ bcl99 ] bromm , v. , coppi , p. s. , & larson , r. b. 1999 , apjl * 527 * , l5 21 . [ ebert ] ebert , r. 1955 , zs . [ bonnor ] bonnor , w. b. 1956 , mnras * 116 * , 351 23 . [ field65 ] field , g. b. 1965 , apj * 142 * , 531 24 . [ gp98 ] galli , d. & palla , f. 1998 , a&a * 335 * , 403 25 . [ sy77 ] sabano , y. & yoshi , y. 1977 , pasj * 29 * , 207 26 . [ silk83 ] silk , j. 1983 , mnras * 205 * , 705 27 . [ qz88 ] quinn , p.j . & zurek , w.h . 1988 , apj * 331 * , 1 28 . [ bb93 ] burkert , a. & bodenheimer , p. 1993 , mnras * 264 * , 798 29 . [ nwb80 ] norman , m.l . , wilson , j.r . & barton , r.t . 1980 , apj * 239 * , 968 30 . [ sn92 ] stone , j.m . & norman , m.l . 1992 , apjs * 80 * , 791 31 . [ soj97 ] sigl , g. , olinto , a. v. & jedamzik , k. 1997 , phys.rev.d * 55 * , 4582 32 . [ bfs97 ] barrow , j. d. , ferreira , p. g. and silk , j. 1997 , physical review letters * 78 * , 3610 33 . [ ms76 ] mouschovias , t. ch . , spitzer , l. 1976 , apj * 210 * , 326 34 . [ haehnelt ] haehnelt , m.g . 1995 , mnras * 273 * , 249 35 . [ mf99 ] mac low , m. & ferrara , a. 1999 , apj * 513 * , 142 36 . [ f98 ] ferrara , a. 1998 , apjl * 499 * , l17 37 . [ cb00 ] cen , r. & bryan , g.l . 2000 , apjl * 546 * , l81 38 . [ m92 ] monaghan , j. j. 1992 , araa * 30 * , 543 39 . [ mba01 ] machacek , m.e . , bryan , g.l . , & abel , t 2001 , apj * 548 * , 509 40 . [ essp00 ] ellison , s. l. , songaila , a. , schaye , j. & pettini , m. 2000 , , 1175 41 . [ wq00 ] wasserburg , g. j. & qian , y. 2000 , apjl * 529 * , l21 42 . [ r99 ] reichart , d. e. 1999 , apjl * 521 * , l111 43 . [ cl00 ] ciardi , b. & loeb , a. 2000 , apj * 540 * , 687 44 . t.a . happily acknowledges stimulating and insightful discussions with martin rees and richard larson . glb was supported through hubble fellowship grant hf-0110401 - 98a from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . under nasa contract nas5 - 26555 .
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we describe results from a fully self consistent three dimensional hydrodynamical simulation of the formation of one of the first stars in the universe .
dark matter dominated pre - galactic objects form because of gravitational instability from small initidal density perturbations . as they assemble via hierarchical merging , primordial gas cools through ro - vibrational lines of hydrogen molecules and sinks to the center of the dark matter potential well .
the high redshift analog of a molecular cloud is formed .
when the dense , central parts of the cold gas cloud become self - gravitating , a dense core of @xmath0 undergoes rapid contraction . at densities
@xmath1 a @xmath2 proto - stellar core becomes fully molecular due to three body formation .
contrary to analytical expectations this process does not lead to renewed fragmentation and only one star is formed .
the calculation is stopped when optical depth effects become important , leaving the final mass of the fully formed star somewhat uncertain . at this stage
the protostar is acreting material very rapidly ( @xmath3 ) .
radiative feedback from the star will not only halt its growth but also inhibit the formation of other stars in the same pre galactic object ( at least until the first star ends its life , presumably as a supernova ) .
we conclude that at most one massive ( @xmath4 ) metal free star forms per pre galactic halo ,
consistent with recent abundance measurements of metal poor galactic halo stars .
# 1_[#1 ] _
# 110^#1 # 1n _ # 1 # 1k_#1 # 1@xmath5 # 1_#1 _ # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
| 8,953 | 420 |
the early calculations by svane and gunnarson showed that when self - interaction corrections were included in the local density approximation , density functional theory ( dft ) predicted a first order phase transition located near the wigner - seitz radius @xmath1 where @xmath2 , @xmath3 is the density , and @xmath4 is the bohr radius @xcite . on the contrary , dft calculations using either the generalized gradient approximation ( gga ) or local spin density approximation ( lsda ) without the self - interaction correction have predicted a second - order phase transition at @xmath5 and @xmath6 and an itenerant anti - ferromagnetic phase up to @xmath7 and @xmath8 respectively @xcite . g@xmath9w@xmath9 , using the lda or gga orbitals to compute the initial green s function , finds the same transition order as their underlying dft functionals , though the phase transition density is shifted upwards to @xmath10 @xcite . the most recent set of g@xmath9w@xmath9 calculations begin with lda+u and gga+u single particle orbitals for the initial green s function @xcite . the `` + u '' methods include an on - site repulsion for the two different spin densities to penalize double occupancy and pushes the system towards an anti - ferromagnetic state . using g@xmath9w@xmath9 on top of these methods , researchers find a continuous metal to insulator phase transition and locate it close to @xmath11 . this phase transition has also been investigated using dynamical mean field theory ( dmft ) by approximating the coulomb interaction as a strictly short ranged on - site interaction between two electrons on the same hydrogen ion @xcite . using this method it was found to be a first - order phase transition at @xmath12 . this transition location is an extrapolation from their finite temperature data to the ground state @xcite . a highly accurate benchmark is required to disambiguate these results . previous efforts to produce such a benchmark have been performed using variational quantum monte carlo@xcite . this calculation was consistent with either a very weak first order or a second order transition at @xmath13 . the error estimates in these measurements are sufficiently large to include a number of the previous results . our goal in this work is to provide a benchmark with improved accuracy . in this section we will discuss the method we use , the hamiltonian for the system , and some computational aspects particular to our calculation . in this work we use dmc to generate all of our results . this method has been used to produce benchmark results for light elements such as hydrogen and the electron gas and has been increasingly used for solid state systems @xcite . this variational stochastic projector method filters out the ground state component of a trial wave function to sample the ground state probability distribution @xcite . by using a trial wave function we are able to avoid the notorious `` sign problem '' which plagues exact monte carlo calculations of fermions but introduce error which raises the energy . the nodes or phase of the trial wave function serves as a boundary condition on the the random walk . the error introduced by this approximation is referred to as the `` fixed - node error '' @xcite . in rydberg units , the hamiltonian for hydrogen is , @xmath14 where capital letters , @xmath15 , correspond to ion coordinates and lower case letter , @xmath16 , correspond to electronic coordinates . this is a zero temperature calculation and does not include the kinetic energy of the protons ; they are clamped to the bcc lattice . in this work we will refer to the two atoms in the bcc unit cell as the a and b simple cubic sublattices . our trial wave function is a single slater jastrow wave function , @xmath17 where @xmath18 where @xmath19 and similarly for the down spin electrons , @xmath20 . for the ground state it is always the case that @xmath21 . for the quasiparticle calculation they differ by 1 . the jastrow consists of two terms : a one - body term , @xmath22 , and a two - body term , @xmath23@xcite and are of the form , @xmath24 where @xmath25 refer to ionic coordinates , @xmath16 refer to electron coordinates , @xmath26 and @xmath27 are the electron spins , and @xmath28 and @xmath29 are bspline@xcite functions whose parameters are variational degrees of freedom . both the one body and two body terms include a cusp condition which , in conjunction with the determinant , exactly cancels the divergent coulomb potential energy contribution when an ion and electron or two electrons coincide@xcite . we optimize the parameters in the trial wave function using a variant of the linear method of umrigar and coworkers@xcite . instead of rescaling the eigenvalues found during the generalized eigenvalue problem , we perform a line minimization on them using a @xmath30-point fit to a quadratic function . we find that this can increase the rate of convergence to the optimal set of variational parameters@xcite . we parameterize the two - body jastrow function so that it is symmetric under exchange of up and down electron labels . this requires the same parameterization for @xmath23 between up - up and down - down pairs , @xmath31 , but allows for a separate set of parameters for up - down @xmath23 terms,@xmath32 . the one - body jastrow is parameterized differently in the paramagnetic and anti - ferromagnetic phases . in the paramagnetic phase we use a one body jastrow which is not a function of electron spin or ion sublattice . in the anti - ferromagnetic phase we use a jastrow that is the same for up - a / down - b , @xmath33 , and for up - b / down - a , @xmath34 , electron spin - ion sublattice pairs . this ensures that the wave function is unchanged if up and down electron labels are swapped at the same time as the a and b sublattice labels are . for a slater - jastrow wave function , the magnitude of the fixed node error is affected by the quality of the single particle orbitals used in the slater determinant . there are many different orbitals that can be used in the determinant : analytic forms such as plane waves or gaussians , orbitals from density function theory , or orbitals derived from another quantum chemistry method . it has been found that , for solids , the best trial wave functions are often obtained using orbitals from dft . however , different functionals produce different orbitals , so a careful choice of functional is necessary @xcite . in this work we consider single particle orbital sets generated using three different techniques all of which are based on dft . in the paramagnetic phase we use orbitals generated using the local density approximation ( lda)@xcite . in the anti - ferromagnetic phase , as we will describe below , we use two different sets of orbitals , one generated using a lda+u functional , and the other generated by performing two lda calculations , each of which includes only the ions of one sublattice . we generate these orbitals using quantum espresso @xcite and a lda pseudopotential generated using opium @xcite . ) . the horizontal line is the energy obtained using the split sublattice orbitals , and dashed line is a single standard deviation in energy . the minimum energy orbitals are generated using u@xmath35 . the line passing through the lda+u energies is a quadratic fit for all u@xmath36.[fig : pol_u ] ] . the minimum energy lda+u orbitals ( circles ) are slightly less polarized than the split sublattice ones ( diamond ) . the line is a guide to the eye . [ fig : pol_e ] ] for the anti - ferromagnetic phase , we obtain the split sublattice orbitals by performing a dft calculation on the simple cubic lattice and translating the orbitals according to the bcc basis vector , @xmath37 where @xmath38 is the lattice constant . we choose to use this basis because , for the ground state , it explicitly breaks spin symmetry and creates a set of anti - ferromagnetic orbitals centered on each sublattice and has no adjustable parameters . we also use lda+u orbitals to be able to tune the degree of spin polarization in the system . the lda+u functional can interpolate between the paramagnetic and anti - ferromagnetic phases by tuning u. for small values of u the spin polarization is small and for large u it is complete . though this is a simple single parameter single particle orbital optimization technique , it can be very expensive to converge . the sensitivity of the orbitals to the choice of u is illustrated in figure [ fig : pol_u ] . the resulting energy vs. polarization curve is presented in figure [ fig : pol_e ] . to find the optimal value of u we must perform qmc calculations on large systems , 432 electrons , for several values of u and orbital occupations . the orbitals may be occupied according to the lda+u energy prediction , or by filling the lowest band regardless of lda+u occupation . these partially polarized phases exhibit large finite size effects , and require large simulation cells to accurately determine the optimal u , its polarization and energy . near the phase transition the lda+u basis provides a better description of the anti - ferromagnetic state as evidenced by lower energies . because the dmc uses the full coulomb potential for the electron - ion potential , the only effect the choice of pseudopotential has on our dmc results is through the quality of the single particle orbitals . to check the effect of the pseudopotential , we compared the dmc energy of the single particle orbitals generated using our lda pseudopotential with those generated using the trail - needs pseudopotential in the paramagnetic phase at @xmath39 @xcite . for the @xmath40 hydrogen supercell we found no statistical difference in their energies . as described above , we use trial wave functions for the anti - ferromagnetic phase that explicitly break spin symmetry . we compute the polarization of the phase using the sublattice spin density , @xmath41 where @xmath42 is the spin density for the up or down electrons , @xmath43 , on the a sublattice whose protons are located at @xmath44 , and the step function is zero outside a sphere of radius @xmath45 . @xmath46 is defined identically but with a and b indices exchange . we are free to adjust @xmath45 between zero and @xmath47 , the wigner - seitz radius . as @xmath48 the statistics of the observable improve due to increased sampling volume . as @xmath49 the estimator projects out just the spin at the sublattice proton positions . we find that @xmath50 is a reasonable compromise and use it for all densities . using these spin densities we define the a sublattice polarization , @xmath51 , as , @xmath52 in this work we only investigate phases where the polarization is symmetric , @xmath53 . in the paramagnetic phase @xmath54 and in the anti - ferromagnetic phase @xmath55 . this estimator works in trial wave function based qmc when the up / down symmetry of the state has explicitly been broken . if the trial wave function is a linear combination of up - a / down - b and up - b / down - a spin electrons assigned to sublattices , then the previous estimator yields zero even for anti - ferromagnetic phases . in that case it is necessary to compute the correlation function using the polarization operator , @xmath56 we converge the dft calculations using a @xmath57 k - point grid for the metallic phases and @xmath58 grid for the insulating or partially polarized phases . we find that using an energy cutoff of @xmath59 rydberg is sufficient to converge the dft energy as well as the orbitals used in the quantum monte carlo . in the paramagnetic phase , we perform dmc for a @xmath60 supercell of @xmath61 twists and @xmath62 total ions , and a @xmath63 supercell of @xmath64 twists and @xmath65 total ions using both the ewald and modified periodic coulomb ( mpc ) potential @xcite for the electrons @xcite . these runs were performed with more than two thousand configurations and extrapolated to zero time step using a three point extrapolation . the smallest time steps had accept ratios greater than @xmath66 . the energy is then averaged over twist points and the two supercells are extrapolated linearly as volume@xmath67 to infinite volume . we find that both the ewald and mpc extrapolate to similar values , and forgo larger simulation cells in the extrapolation . we put more effort into computing the properties of the anti - ferromagnetic phase because it is relatively more complex than the paramagnetic phase . because the anti - ferromagnetic phase is insulating , we use a smaller number of twists . the finite size and brillouin zone sampling are done using a @xmath60 supercell of @xmath63 twists ( @xmath62 protons ) , a @xmath63 supercell of @xmath60 twists ( @xmath65 protons ) , and a @xmath64 supercell of @xmath40 twists ( @xmath68 protons ) . we locate the transition density between the paramagnetic and anti - ferromagnetic phases by computing the energy of each phase as a function of wigner - seitz radius and finding the crossing point . by carefully extrapolating finite size effects , sampling the fermi surface using twist averaging , and controlling for time step error , we determine the transition density within a small statistical error @xcite . we control for systematic error , due to the fixed node approximation , by comparing the energies for several different sets of single particle orbitals in the anti - ferromagnetic phase and by using a slater - jastrow - backflow wave function in the paramagnetic phase . .fixed - phase dmc energy per hydrogen atom in rydberg for bcc hydrogen in the anti - ferromagnetic ( afm ) , and paramagnetic ( pm ) phases.[tbl : h_eos ] [ cols="^,^,^",options="header " , ] + the gaps we compute and reference values from previous studies are presented in table [ table : gaps ] . the excitonic gap extrapolates to zero gap while the quasiparticle gap projects to a small but finite gap , @xmath69 . both of these gaps are consistent with a phase transition location closer to @xmath39 than previous g@xmath9w@xmath9 studies . we note that the quasiparticle gap suffers from much larger finite size effects than the excitonic gap which results in a more difficult extrapolation and larger error bar estimate . at this density , an estimate for the excitonic binding energy , the difference between the excitonic and quasiparticle gap extrapolated to the thermodynamic limit , is @xmath70 . we have presented benchmark calculations on the phase transition density and order for bcc hydrogen . as shown in table [ tbl : heos_prev ] , the phase transition is likely to be continuous and is located at @xmath71 . the most recent g@xmath9w@xmath9 calculations@xcite show the same transition order and agree very well with our transition location . if these calculations were to choose a slightly different value of u , they may be brought into perfect agreement . previous publications@xcite using g@xmath9w@xmath9 resulted in slightly less accurate transition densities , likely due to a worse dft starting point . our results provide confidence in the g@xmath9w@xmath9 method when starting from a high quality dft calculation . previous dft results identify the same phase transition order , but greatly overestimate the transition location@xcite . previous qmc results@xcite were performed on small systems , of @xmath62 ions , which is under converged with respect to system size . they are also performed using vmc which is much more sensitive to the form and optimization of the trial wave function . however , these results predict a similar transition density as the current ones . the remainder of the methods , dmft @xcite and sic - dft @xcite get the phase transition order and location incorrect . jm acknowledges useful conversations with sarang gopalakrishnan , norm tubman , and lucas wagner . this work was performed in part under the auspices of the u.s . department of energy ( doe ) by llnl under contract de- ac52 - 07na27344 . jm and dmc were supported by nsf oci-0904572 . jm , dmc , mm , and jk were supported by the network for ab initio many - body methods supported by predictive theory and modeling program of basic energy sciences , department of energy ( doe ) . this work used the extreme science and engineering discovery environment ( xsede ) , which is supported by national science foundation grant number oci-1053575 , and resources provided by the innovative and novel computational impact on theory and experiment ( incite ) at the oak ridge leadership computing facility at the oak ridge national laboratory , which is supported by the office of science of the u.s . department of energy under contract no . de - ac05 - 00or22725 . llnl release number llnl - jrnl-638420 . 47ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) link:\doibase 10.1016/0009 - 2614(84)85513-x [ * * , ( ) ] link:\doibase 10.1021/ct301044e [ * * , ( ) ] link:\doibase " 10.1016/0038 - 1098(90)90641-n [ * * , ( ) ] link:\doibase 10.1103/physrevlett.65.2414 [ * * , ( ) ] link:\doibase 10.1103/physrevb.58.12680 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.195123 [ * * , ( ) ] link:\doibase 10.1103/physrevb.77.155114 [ * * , ( ) ] link:\doibase 10.1103/physrevb.66.035102 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.73.33 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.086403 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.70.1952 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1073/pnas.1007309107 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.68.13 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.45.566 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.84.1607 [ * * , ( ) ] link:\doibase 10.1063/1.443766 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.71.2777 [ * * , ( ) ] link:\doibase 10.1103/physrev.98.1479 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1002/cpa.3160100201 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.110201 [ * * , ( ) ] link:\doibase 10.1063/1.2437215 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1063/1.3665391 [ * * , ( ) ] link:\doibase 10.1021/ct3003404 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.185502 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.73.33 [ * * , ( ) ] link:\doibase 10.1103/physrevb.45.13244 [ * * , ( ) ] http://www.quantum-espresso.org [ * * , ( ) ] @noop `` '' link:\doibase 10.1063/1.1888569 [ * * , ( ) ] link:\doibase 10.1063/1.1829049 [ * * , ( ) ] link:\doibase 10.1103/physrevb.53.1814 [ * * , ( ) ] link:\doibase 10.1103/physreve.64.016702 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.125106 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.72.2438 [ * * , ( ) ] @noop * * , ( ) _ _ , @noop ph.d . thesis , ( ) link:\doibase 10.1103/physrevb.51.4014 [ * * , ( ) ]
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solid atomic hydrogen is one of the simplest systems to undergo a metal - insulator transition . near the transition
, the electronic degrees of freedom become strongly correlated and their description provides a difficult challenge for theoretical methods . as a result ,
the order and density of the phase transition are still subject to debate . in this work
we use diffusion quantum monte carlo to benchmark the transition between paramagnetic and anti - ferromagnetic body centered cubic atomic hydrogen in its ground state .
we locate the density of the transition by computing the equation of state for these two phases and identify the phase transition order by computing the band gap near the phase transition .
these benchmark results show that the phase transition is continuous and occurs at a wigner - seitz radius of @xmath0 .
we compare our results to previously reported density functional theory , hedin s gw approximation , and dynamical mean field theory results . the metal - insulator transition in body centered cubic ( bcc ) hydrogen
was first considered by mott @xcite . at high density , the system is metallic : the electrons are in the paramagnetic , conducting phase . as the lattice spacing
is increased the electrons localize on the hydrogen ions forming an anti - ferromagnetic , insulating phase .
predicting the order of the phase transition allows us to identify the qualitative nature of the transition : whether the transition is due to strong particle correlations , a mott insulator transition , or due to long range coulomb interactions , the band - insulator transition @xcite .
locating the precise transition density is a difficult test for an electronic structure method . since the original study ,
several closely related systems , finite lattices or 1d chains of hydrogen atoms , have become a standard test used to benchmark quantum chemistry methods @xcite .
unfortunately , current electronic structure methods such as density functional theory , dynamical mean field theory , variational quantum monte carlo , and hedin s gw disagree over both the phase transition order and density @xcite . in this work
, we determine the phase transition density and order using diffusion quantum monte carlo ( dmc ) .
dmc is one of the most accurate methods for computing electronic structure properties of atomic , molecular , and condensed phases @xcite .
it has been used extensively to characterize hydrogen and the alkali metals at ambient and at high pressure @xcite .
we use dmc to compute the ground state equation of state in both the paramagnetic and anti - ferromagnetic phase as well as the quasiparticle and excitonic band gaps near the transition .
we begin by reviewing previous calculations and the details of the current calculation .
next we discuss our results for the density of the phase transition and the magnitude of the band gap .
finally , we conclude with a comparison of results obtained with other methods and how they compare to those obtained using dmc .
| 5,905 | 706 |
hera is an @xmath0 collider which has especially high sensitivity to new particles coupling to lepton - quark pairs . in 1994 - 97 hera collided 27.5 gev positrons on 820 gev protons . in 1998 the proton energy was raised to 920 gev increasing the center - of - mass energy @xmath1 from 300 gev to 318 gev . in 1998 and in the first months of 1999 , hera ran with electrons . in may 1999 hera switched back to @xmath2 collisions . the three main colliding periods as well as the corresponding luminosities for each experiment are summarized in table [ tab : lumit ] . * * . _ _ luminosities collected by h1 and zeus for each colliding period . _ _ [ cols="^,^,^,^,^",options="header " , ] hera resulting limits on @xmath3 coupling are the following : * h1 : @xmath4 * zeus : @xmath5 figure [ fig : final ] summarizes the limits on the anomalous coulings @xmath6 ( @xmath7 vectorial coupling ) and @xmath8 ( @xmath3 magnetic coupling ) obtained at hera , lep and tevatron . hera sensitivity to @xmath8 is competitive with other colliders . hera is the unique collider to test direct @xmath10 interactions . about @xmath11 of @xmath2 data and @xmath12 of @xmath13 data have been collected per experiment at hera-1 . no evidence of new physics has been observed in various models in inclusive analyses ( contact interactions , extra - dimensions , leptoquarks ) and exclusive analyses ( lepton - flavour violation , @xmath14-violating susy , excited fermions ) , therefore new constraints have been set . hera limits are seen to be competitive with and complementary to the lep and tevatron searches . the status of isolated lepton events with missing @xmath15 is still intriguing and will become clearer with the new hera-2 data . hera has been shutdown since fall 2000 for a general upgrade : new focussing magnets have been installed in order to increase the luminosity and many improvements have been performed in the detectors in order to increase their sensitivity . moreover , the lepton beam will be longitudinally polarised in the h1 and zeus interaction regions . the first luminosity runs are predicted for beginning 2002 and hera-2 is expected to accumulate 1 @xmath16 in the next 5 years . the anticipated factor of ten increase in the integrated luminosity will give an outstanding discovery potential for hera . i which to thank my h1 and zeus colleagues who contributed to the results presented here as well as people who helped me in preparing this talk .
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recent results on searches for physics beyond the standard model obtained by the h1 and zeus experiments are reported here .
after a brief introduction to the hera collider , indirect searches for contact interactions and extra - dimensions are presented as well as direct searches for new physics including leptoquarks , lepton - flavour violation , squarks produced by r - parity violation and excited fermions .
new results from isolated lepton events and single top searches are also presented .
finally the future prospects of hera-2 are shown .
* search for new particles at hera * + mireille schneider + cppm , 163 , avenue de luminy , case 907 , + 13288 marseille cedex 9 , france
| 709 | 180 |
the in - spiral and coalescence of binary neutron star systems is a topic of increasingly intensive research in observational and theoretical astrophysics . it is anticipated that the first direct detections of gravitational wave ( gw ) will be from compact binary mergers . binary neutron star ( bns ) mergers are also thought to produce short - hard gamma - ray bursts ( sgrb s ) @xcite . simultaneous detections of a prompt gravitational wave signal with a spatially coincident electromagnetic ( em ) counterpart dramatically increases the potential science return of the discovery . for this reason , there has been considerable interest as to which , if any , detectable em signature may result from the merger @xcite . other than sgrbs and their afterglows , including those viewed off - axis @xcite , suggestions include optical afterglows associated with the radio - active decay of tidally expelled r - process material@xcite ( though detailed calculations indicate they are faint @xcite ) , radio afterglows following the interaction of a mildly relativistic shell with the interstellar medium @xcite , and high - energy pre - merger emission from resistive magnetosphere interactions @xcite . merging neutron stars possess abundant orbital kinetic energy ( @xmath4ergs ) . a fraction of this energy is certain to be channelled through a turbulent cascade triggered by hydrodynamical instabilities during merger . turbulence is known to amplify magnetic fields by stretching and folding embedded field lines in a process known as the small - scale turbulent dynamo @xcite . amplification stops when the magnetic energy grows to equipartition with the energy containing turbulent eddies @xcite . an order of magnitude estimate of the magnetic energy available at saturation of the dynamo can be informed by global merger simulations . these studies indicate the presence of turbulence following the nonlinear saturation of the kelvin - helmholtz ( kh ) instability activated by shearing at the ns surface layers @xcite . the largest eddies produced are on the @xmath5 km scale and rotate at @xmath6 , setting the cascade time @xmath7 and kinetic energy injection rate @xmath8 at @xmath9ms and @xmath10 respectively . when kinetic equipartition is reached , each turbulent eddy contains @xmath11 of magnetic energy , and a mean magnetic field strength @xmath12 whether such conditions are realized in merging neutron star systems depends upon the dynamo saturation time @xmath13 and equipartition level @xmath14 . in particular , if @xmath15 then turbulent volumes of neutron star material will contain magnetar - level fields throughout the early merger phase . once saturation is reached , a substantial fraction of the injected kinetic energy , @xmath16 , is resistively dissipated @xcite at small scales . magnetic energy dissipated by reconnection in optically thin surface layers will accelerate relativistic electrons @xcite , potentially yielding an observable electromagnetic counterpart , independently of whether the merger eventually forms a relativistic outflow capable of powering a short gamma - ray burst . in this letter we demonstrate that the small - scale turbulent dynamo saturates quickly , on a time @xmath15 , and that @xmath17 g magnetic fields are present throughout the early merger phase . this implies that the magnetic energy budget of merging binary neutron stars is controlled by the rate with which hydrodynamical instabilities randomize the orbital kinetic energy . our results are derived from simulations of the small scale turbulent dynamo operating in the high - density , trans - relativistic , and highly conductive material present in merging neutron stars . we have carefully examined the approach to numerical convergence and report grid resolution criteria sufficient to resolve aspects of the small - scale dynamo . our letter is organized as follows . the numerical setup is briefly described in section 2 . section 3 reports the resolution criterion for numerical convergence of the dynamo completion time and the saturated field strength . in section 4 we asses the possibility that magnetic reconnection events may convert a sufficiently large fraction of the magnetic energy into high energy photons to yield a prompt electromagnetic counterpart detectable by high energy observatories including _ swift _ and _ fermi_. and @xmath18 . lower resolutions are shown in red and graduate to black with higher resolution . _ top _ : the root mean square magnetic field strength in units of @xmath19 . when a turbulent volume is resolved by @xmath20 zones , the small - scale dynamo proceeds so slowly that almost no amplification is observed in the first 1ms . _ middle _ : the magnetic energy in units of the rest mass @xmath21 shown on logarithmic axes . it is clear that the linear growth rate increases at each resolution . _ bottom _ : the kinetic energy ( upper curves ) shown again the magnetic energy ( lower curves ) again in units of @xmath21 . for all resolutions , the kinetic energy saturates in less than 1 @xmath7.,width=326 ] the equations of ideal relativistic magnetohydrodynamics ( rmhd ) have been solved on the periodic unit cube with resolutions between @xmath20 and @xmath18 . [ eqn : rmhd - system ] @xmath22 here , @xmath23 is the magnetic field four - vector , and @xmath24 is the total specific enthalpy , where @xmath25 is the total pressure , @xmath26 is the gas pressure and @xmath27 is the specific internal energy . the source term @xmath28 includes injection of energy and momentum at the large scales and the subtraction of internal energy ( with parameter @xmath29mev ) to permit stationary evolution . vortical modes at @xmath30 are forced by the four - acceleration field @xmath31 which smoothly decorrelates over a large - eddy turnover time , as described in @xcite . we have employed a realistic micro - physical equation of state ( eos ) appropriate for the conditions of merging neutron stars . it includes contributions from high - density nucleons according a relativistic mean field model @xcite , a relativistic and degenerate electron - positron component , neutrino and anti - neutrino pairs in beta equilibrium with the nucleons , and radiation pressure . for our conditions , all the components make comparable contributions to the pressure . we have also employed a simpler gamma - law eos and found close agreement for the conditions explored in this paper , indicating that the mhd effects are insensitive the eos for trans - sonic conditions . the models presented in our resolution study use the far less expensive gamma - law equation of state . all of the simulations presented in this study use the hlld approximate riemann solver @xcite , which has been demonstrated as crucial in providing the correct spatial distribution of magnetic energy in mhd turbulence @xcite . the solution is advanced with an unsplit , second - order muscl - hancock scheme . spatial reconstruction is accomplished with the piecewise linear method configured to yield the smallest possible degree of numerical dissipation . the divergence constraint on the magnetic field is maintained to machine precision at cell corners using the finite volume ct method of @xcite . full details of the numerical scheme may be found in @xcite . magnetic fields are amplified in our simulations by the small - scale turbulent dynamo . turbulent fluid motions stretch and fold the magnetic field lines , causing exponential growth of magnetic energy ( e.g. * ? ? ? this growth is attributed to the advection and diffusion of @xmath32 through the mhd induction equation ( eq . [ eqn : rmhd - system]c ) . when the magnetic field is weak ( @xmath33 ) @xmath32 evolves passively , and the turbulence is hydrodynamical . this limit is referred to as small - scale kinematic turbulent dynamo , and is well described by kazentzev s model @xcite . this model predicts that the power spectrum of magnetic energy peaks at the resistive scale @xmath34 and obeys a power law @xmath35 at longer wavelengths . the kinematic phase ends when the magnetic field acquires sufficient tension to modify the hydrodynamic motions , after which time a dynamical balance between kinetic and magnetic energy is established . numerical simulations of mhd turbulence are typically limited to magnetic prandtl numbers @xmath36 . however , neutron star material is characterized by @xmath37 , with the viscous cutoff due to neutrino diffusion occurring at around 10 cm , while the resistive scale is significantly smaller @xcite . however , the disparity between true and simulated magnetic prandlt number does not influence our conclusions . this is because dynamos are generically easier to establish in the high pm regime than the small @xcite . , together with the empirical model ( equation [ eqn : magnetic - fit ] ) with best - fit parameters . the horizontal dashed line indicates the magnetic energy , @xmath38 at the dynamo completion . from left to right , the vertical dashed lines mark the end of the startup , kinematic , and saturation phases.,width=326 ] we use an initially uniform , pulsar - level ( @xmath39 g ) seed magnetic field . this field is sub - dominant to the kinetic energy by 10 orders of magnitude , so that the initial field amplification is expected to be well - described by kazentsev s model . indeed , we find that during this phase the power spectrum of magnetic energy follows @xmath35 ( fig . [ fig : pspec - time - devel - res ] ) , peaking at around 10 grid zones , which we identify as the effective scale of resistivity . the saturation process begins at ever - earlier times with increasing numerical resolution . this reflects the fact that during the kinematic phase , magnetic energy exponentiates on a time scale controlled by shearing at the smallest scales . in numerically converged runs , full saturation occurs with @xmath40 and is characterized by scale - by - scale super - equipartition , with @xmath41 at all but the largest scale . the same driven turbulence model was run through magnetic saturation at resolutions @xmath20 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , and @xmath46 . another model at @xmath18 was run through the end of the kinematic phase , but further evolution was computationally prohibitive . [ fig : energy - growth ] shows the development of @xmath47 , @xmath48 , and @xmath49 as a function of time at each resolution . we find that sufficiently resolved runs ( @xmath50 ) attain _ mean _ magnetic field strengths of @xmath51 g within two large eddy rotations . all models with resolutions @xmath52 eventually attain mean fields of @xmath53 g . the saturated field strength increases until resolution @xmath45 . we find that the kinematic growth rate is higher at each higher resolution , while the time - scale for the non - linear saturation converges at @xmath45 to roughly five large - eddy rotation times , or about 0.5ms for the physical parameters of binary neutron star mergers . lcc @xmath54 & startup time - scale & none , artifact @xmath55 & hydrodynamic cascade fully developed & @xmath7 @xmath56 & kinematic growth time - scale & none , @xmath57 @xmath58 & end of kinematic phase & @xmath59 @xmath60 & non - linear saturation time - scale & @xmath61 , @xmath45 @xmath38 & saturated magnetic energy & @xmath62 , @xmath46 [ tab : magnetic - fit ] in order to quantitatively describe the time development of magnetic energy @xmath63 , we describe it with an empirical model , @xmath64 where the 6 parameters ( summarized in table [ tab : magnetic - fit ] ) are obtained by a least - squares optimization . [ fig : magnetic - fit ] shows the empirical model given in equation [ eqn : magnetic - fit ] applied to a representative run at @xmath44 . the first phase , @xmath65 is a startup transient , and lasts until the hydrodynamic cascade is fully developed at @xmath66 . the kinematic dynamo phase lasts between @xmath55 and @xmath58 , during which the magnetic energy exponentiates on the time scale @xmath56 . at @xmath58 , the smallest scales reach kinetic equipartition and the growth rate slows . in the final stage , @xmath48 asymptotically approaches @xmath38 on the time - scale @xmath60 . we define the dynamo completion time @xmath13 as @xmath67 . [ fig : convergence ] shows the best - fit @xmath38 , @xmath13 , and @xmath56 as a function of the resolution . the magnetic energy @xmath38 at dynamo completion shows signs of converging to a value of @xmath62 by resolution @xmath46 . the time scale @xmath60 on which the magnetic energy asymptotically approaches @xmath38 is consistently @xmath68 at different resolutions . the dynamo completion time @xmath13 is numerically converged at @xmath69 by @xmath45 . the best - fit kinematic growth time follows a power law in the resolution , @xmath70 . this is consistent with the value of @xmath71 expected if the dynamo time is controlled by shearing at the smallest scale , the cascade is kolmogorov ( i.e. @xmath72 ) , and the viscous cutoff @xmath73 occurs at a fixed number of grid zones . in that case , @xmath74 . ( _ blue _ ) and the dynamo completion time @xmath13(_green _ ) defined as @xmath75 . _ bottom _ : convergence study of the best - fit model parameter @xmath38 expressed as the ratio of magnetic to kinetic energy @xmath76 . the converged value of the volume - averaged @xmath77 . nevertheless , at intermediate wavelengths @xmath78 . as shown in figure [ fig : pspec - time - devel - res ] , the largest scales remain kinetically dominated . this indicates the suppression of coherent magnetic structure formation near the integral scale of turbulence.,width=326 ] the time development of kinetic and magnetic energy power spectra has been studied for a single run with resolution @xmath46 . we present three - dimensional , spherically integrated power spectra with the dimensions of @xmath79 , defined as @xmath80\right|^2}\\ p_m(k_i ) & = \frac{1}{\delta k_i}\sum _ { { \mathbf k } \in \delta k_i } { \left|\mathcal{f } _ { { \mathbf k } } \left [ { \mathbf b } /\sqrt{8\pi}\right]\right|^2 } \end{aligned}\ ] ] [ eqn : pspec ] where the newtonian versions of kinetic and magnetic energy are appropriate since the conditions are only mildly relativistic . the definitions in equations [ eqn : pspec ] satisfy @xmath81 for @xmath82 and @xmath83 . figure [ fig : pspec - time - devel - res ] shows the power spectrum of kinetic and magnetic energy at various times throughout the growth and saturation of magnetic field . during the kinematic phase , the kinetic energy has a power spectrum @xmath84 consistent with the kolmogorov theory for incompressible hydrodynamical turbulence , while @xmath85 consistent with kazenstev s model . @xmath86 maintains the same shape , but exponentiates in amplitude at the time scale @xmath56 which is controlled by shearing at the resistive scale . according to kazentsev s model , @xmath86 should peak at the resistive scale . this is consistent with the observed peak in the magnetic energy at roughly 10 grid zones , the same scale at which we observe the viscous cutoff . this is also consistent with @xmath87 expected from the numerical scheme employed . when the magnetic energy at the resistive scale surpasses the level of the kinetic energy at that scale , @xmath86 changes shape . the equipartition scale @xmath88 moves into the inertial range , and migrates to larger scale until full saturation occurs with @xmath89 . the movement of @xmath88 to larger scale is associated with the formation of coherent and dynamically substantial magnetic structures of increasing size . the time - dependence has been suggested to be @xmath90 @xcite . in the fully saturated state , the magnetic field is in scale - by - scale super - kinetic equipartition throughout the inertial range , with @xmath91 . the largest scales remain kinetically dominated so that the numerically converged saturation level is @xmath92 . in this letter we have determined the time scale and saturation level of the small - scale turbulent dynamo operating in the conditions of binary neutron star mergers . we have presented numerically converged simulations showing that magnetic fields are amplified to the @xmath93 g level within a small fraction of the merger dynamical time ( @xmath15 ) , independently of the seed field strength . if hydrodynamical instabilities create fluctuating velocities on the order of @xmath94 as indicated by global simulations , then each @xmath95 turbulent volume dissipates @xmath96 erg of magnetic energy per 0.1 ms . if @xmath97 represents the fraction of the merger remnant that contains such turbulence , the magnetic energy dissipated during the merger is at least @xmath98 a fraction of that dissipation will occur through magnetic reconnection in optically thin surface layers , supplying relativistic electrons which synchrotron radiate in the merger remnant magnetosphere . if 5% of that magnetic energy dissipation creates radiation in the 15 - 150 kev band , then the fluence at 200 mpc would be @xmath3 , potentially rendering most merging neutron stars in the advanced ligo and virgo detection volumes detectable by _ swift _ bat . if so , then merging neutron stars are accompanied by a prompt electromagnetic counterpart , independently of whether a later merger phase yields a collimated outflow capable of powering a short gamma - ray burst . we suggest that merger flares may be present in the current sample of short grbs and may be roughly isotropic on the sky since they are seen to distances where the cosmological matter distribution becomes homogeneous . searches for merger flares should seek to identify short flares , not unlike soft - gamma repeaters , among the short burst population . if mergers also produce short grbs short - hard grbs , then merger flares may constitute a precursor component of the emission . the presence of strong magnetic fields may also aid in the ejection of neutron - rich material from surface layers of the merger remnant , possibly enhancing the enrichment of inter - stellar medium by r - process nuclei @xcite . enhanced production of r - process nuclei also increases the likelihood of em detection by radio - active decay powered afterglows , or `` kilonovae '' @xcite . finally , it has been shown that magnetic fields will significantly influence the gravitational wave signature and remnant disk mass , if they exist at the @xmath99 g level @xcite . such strong fields are unlikely in older neutron star binaries , but our results suggest they may be revived , albeit at small scales , during the merger . to have significant influence , those fields would have to fill a considerable fraction of the merger volume . as we have shown in this letter , the overall magnetic energy budget is controlled by the prevalence ( @xmath97 ) and vigor ( @xmath100 ) of the turbulent volumes . this fact motivates the use of higher resolution global simulations aimed at measuring @xmath97 and @xmath100 . this research was supported in part by the nsf through grant ast-1009863 and by nasa through grant nnx10af62 g issued through the astrophysics theory program . resources supporting this work were provided by the nasa high - end computing ( hec ) program through the nasa advanced supercomputing ( nas ) division at ames research center .
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the simultaneous detection of electromagnetic and gravitational wave emission from merging neutron star binaries would aid greatly in their discovery and interpretation . by studying turbulent amplification of magnetic fields in local high - resolution simulations of neutron star merger conditions ,
we demonstrate that magnetar - level ( @xmath0 g ) fields are present throughout the merger duration .
we find that the small - scale turbulent dynamo converts 60% of the randomized kinetic energy into magnetic fields on a merger time scale .
since turbulent magnetic energy dissipates through reconnection events which accelerate relativistic electrons , turbulence may facilitate the conversion of orbital kinetic energy into radiation .
if @xmath1 of the @xmath2 erg of orbital kinetic available gets processed through reconnection , and creates radiation in the 15 - 150 kev band , then the fluence at 200 mpc would be @xmath3 , potentially rendering most merging neutron stars in the advanced ligo and virgo detection volumes detectable by _ swift _ bat .
| 5,406 | 252 |
oscillatory integrals play an important role in the theory of pseudodifferential operators . they are also a useful tool in mathematical physics , in particular in quantum field theory , where they are used to give meaning to formal fourier integrals in the sense of distributions . for phase functions which are homogeneous of order one , this also leads to a characterization of the wave front set of the resulting distribution , as it is known to be contained in the manifold of stationary phase of the phase function . in these applications , the restriction to phase functions that are homogeneous of order one is often obstructive . in many cases , this restriction can be overcome by shifting a part of the would - be phase function to the symbol , cf . example [ ex : delta+ ] below . however , such a shift is not always possible , for instance if the would - be phase function contains terms of order greater than one . such phase functions are present in the twisted convolutions that occur in quantum field theory on moyal space , cf . examples [ ex : ncqft ] and [ ex : ncqftb ] below . up to now , a rigorous definition of these twisted convolution integrals could be given only in special cases and in such a way that the information on the wave front set is lost . thus , it is highly desirable to generalize the notion of oscillatory integrals to encompass also phase functions that are not homogeneous of order one . such generalizations were proposed by several authors . however , to the best of our knowledge , the wave front sets of the resulting distributions were not considered , except for one very special case . we comment on these settings below , cf . remark [ rem : asadafujiwara ] . it is shown here that the restriction to phase functions that are homogeneous of order one can indeed be weakened , without losing information on the wave front set . the generalization introduced here not only allows for inhomogeneous phase functions , but also for phase functions that are symbols of any positive order . however , one has to impose a condition that generalizes the usual nondegeneracy requirement . it is also shown that the wave front sets of the distributions thus obtained are contained in a set that generalizes the notion of the manifold of stationary phase . we conclude with a discussion of some applications . throughout , we use the following notation : for an open set @xmath0 , @xmath1 means that @xmath2 is a compact subset of @xmath0 . @xmath3 stands for @xmath4 . for a subset @xmath5 , @xmath6 stands for the projection on the first component @xmath7 denotes the @xmath8 times continuously differentiable functions supported on @xmath0 and @xmath9 the set of elements of @xmath7 with compact support in @xmath0 . the dual space of @xmath10 is denoted by @xmath11 and @xmath12 . the pairing of a distribution @xmath13 and a test function @xmath14 is denoted by @xmath15 . the dot @xmath16 stands for the scalar product on @xmath17 . @xmath18 denotes the angle between two vectors @xmath19 . as usual , cf . @xcite , we define a symbol as follows : let @xmath20 be an open set . a function @xmath21 is called a _ symbol of order @xmath22 _ if for each @xmath1 and multiindices @xmath23 , we have @xmath24 the set of all such functions , equipped with these seminorms will be denoted by @xmath25 . furthermore , we denote @xmath26 and @xmath27 . for simplicity , we restrict ourselves to these symbols . the generalization to the symbols @xmath28 is straightforward . one then has to restrict to @xmath29 , @xmath30 , where @xmath31 is the order of the phase function introduced below . also the generalization to asymptotic symbols as discussed in @xcite is straightforward . the following proposition is a straightforward consequence of the definition of @xmath25 : [ prop : cont ] the maps @xmath32 , @xmath33 and the multiplication @xmath34 are continuous . the following proposition is proven in ( * ? ? ? 1.7 ) : [ prop : dense ] if @xmath35 , then @xmath36 is dense in @xmath25 for the topology of @xmath37 . now we introduce our new definition of a phase function . [ def : phase ] a _ phase function _ of order @xmath31 on @xmath38 is a function @xmath39 such that 1 . @xmath40 is a symbol of order @xmath41 . [ enum : phase ] for each @xmath1 there are positive @xmath42 such that @xmath43 [ rem : phase ] condition [ enum : phase ] generalizes the usual nondegeneracy requirement and ensures that @xmath40 oscillates rapidly enough for large @xmath44 . in particular it means that @xmath40 is not a symbol of order less than @xmath31 . it also means that one can choose @xmath45 such that @xmath46 is well - defined and a symbol of order @xmath47 . here @xmath48 can be chosen such that @xmath49 is compact for each @xmath50 . [ rem : asadafujiwara ] our definition of a phase function is a generalization of a definition introduced by hrmander ( * ? ? ? 2.3 ) in the context of pseudodifferential operators . he considered phase functions of order 1 ( in the nomenclature introduced above ) and characterized the singular support of the resulting distribution , but not its wave front set . our characterization of the singular support ( cf . corollary [ cor : m ] ) coincides with the one given by hrmander ( * ? ? ? inhomogeneous phase functions were also considered by asada and fujiwara @xcite in the context of pseudodifferential operators on @xmath51 . in their setting , @xmath52 , @xmath53 and there must be a positive constant @xmath54 such that @xmath55 furthermore , all the entries of this matrix ( and their derivatives ) are required to be bounded . thus , the phase function is asymptotically at least of order 1 and at most of order 2 . the admissible amplitudes are at most of order 0 . the wave front set of such operators on @xmath51 is not considered by asada and fujiwara . the same applies to the works of boulkhemair @xcite and ruzhansky and sugimoto @xcite , who work in a similar context . coriasco @xcite considered a special case of hrmander s framework , where again @xmath52 , @xmath53 and @xmath56 with @xmath57 , a subset of the symbols of order 1 . furthermore , he imposed growth conditions on @xmath58 that are more restrictive than condition [ enum : phase ] . the resulting operators on @xmath59 can then be extended to operators on @xmath60 . if a further condition analogous to is imposed , then also the wave front set , which is there defined via @xmath61-microregularity , can be characterized ( at least implicitly , by the change of the wave front set under the action of the operator ) . [ prop : diff ] if @xmath40 is a phase function of order @xmath31 and @xmath62 and there is a @xmath63 such that @xmath64 , then @xmath65 and the map @xmath66 is continuous . for @xmath67 we have @xmath68 so that @xmath69 is continuous . differentiation gives @xmath70 the expression in curly brackets is a symbol of order @xmath71 . with the same argument as before one can thus differentiate @xmath72 times . we formulate the main theorem of this section analogously to ( * ? ? ? * thm . the proof is a straightforward generalization of the proof given there . [ thm : osc ] let @xmath73 be a phase function of order @xmath31 on @xmath38 . then there is a unique way of defining @xmath74 for @xmath75 such that @xmath76 coincides with when @xmath62 for some @xmath67 and such that , for all @xmath22 , the map @xmath77 is continuous . moreover , if @xmath63 and @xmath78 , then the map @xmath79 is continuous . to prove this , we need the following lemma : [ lemma : v ] let @xmath40 be a phase function of order @xmath31 on @xmath38 . then there exist @xmath80 , @xmath81 and @xmath82 such that for the differential operator @xmath83 with adjoint @xmath84 we have @xmath85 furthermore , @xmath86 is a continuous map from @xmath25 to @xmath87 . we choose @xmath88 as in definition [ def : phase ] and @xmath48 as in remark [ rem : phase ] . we set @xmath89 then we have @xmath90 it is easy to see that @xmath91 , @xmath92 and @xmath93 are symbols in the required way . the last statement follows from proposition [ prop : cont ] . the uniqueness is a consequence of proposition [ prop : dense ] . for @xmath94 and @xmath95 , we have , with @xmath86 as in lemma [ lemma : v ] , @xmath96 { \mathrm{d}}^nx { \mathrm{d}}^s\theta,\end{aligned}\ ] ] for any @xmath63 and thus @xmath97 \rvert } { \mathrm{d}}^nx { \mathrm{d}}^s\theta.\ ] ] now the multiplication @xmath98 is continuous . thus , if @xmath62 , then @xmath99 $ ] is a symbol of order @xmath100 and in particular we have , for each @xmath1 , @xmath101 \rvert } ( 1+{\lvert \theta \rvert})^{p\mu - m } \leq f_{p , k}(a ) \sum_{{\lvert \alpha \rvert } \leq p } \sup_{x \in k } { \lvert d^\alpha f \rvert},\ ] ] where @xmath102 is a seminorm on @xmath25 . for @xmath62 , we may thus choose @xmath72 such that @xmath103 and define @xmath104 { \mathrm{d}}^nx { \mathrm{d}}^s\theta.\ ] ] as the sum on the r.h.s . of is a seminorm on @xmath105 , and due to the continuity properties discussed above , the map @xmath106 is continuous . for @xmath107 , we have @xmath108 by . thus , we can unambiguously define @xmath109 . we may now further characterize the distributions that result from a generalized oscillatory integral . [ def : sp ] let @xmath40 be a phase function of order @xmath31 on @xmath38 . we define @xmath110 we call @xmath111 the _ asymptotic manifold of stationary phase_. by definition , it is conic . [ lemma : sp ] @xmath112 is a closed conic subset of @xmath113 . @xmath111 is a closed subset of @xmath114 . from the definition of @xmath112 it follows that if @xmath115 , then @xmath116 for all @xmath117 , so @xmath112 is conic . we now show that @xmath118 is open in @xmath113 . let @xmath119 be such that there are positive @xmath42 such that @xmath120 we set @xmath121 . by taylor s theorem we have @xmath122 where @xmath123 and @xmath124 fulfill the bounds @xmath125 here @xmath126 are chosen such that @xmath127 , @xmath128 . furthermore , we restrict to @xmath129 . then we may use that @xmath13 is a symbol of order @xmath130 to conclude that there are positive constants @xmath131 , which are bounded for @xmath132 , for which @xmath133 holds . as the zeroth order term in the taylor expansion grows faster than @xmath134 for a fixed positive constant @xmath54 , we can make @xmath135 so small that @xmath136 grows faster than @xmath137 for some positive @xmath138 . thus , @xmath112 is closed in @xmath113 . in order to prove the closedness of @xmath111 , we first note that if @xmath139 , then by the above there is a neighborhood of @xmath140 that does not intersect @xmath141 . thus , it suffices to show that for @xmath142 , @xmath143 there is a neighborhood that does not intersect @xmath111 . by condition [ enum : phase ] of definition [ def : phase ] , there must be positive constants @xmath42 such that @xmath144 by the same argument as above , such a bound holds true in a conic neighborhood @xmath145 of @xmath146 . by the definition of @xmath111 , there are posititve @xmath147 such that @xmath148 we now want to show that one can choose a conic neighborhood @xmath149 of @xmath119 , contained in @xmath145 , such that an analogous bound holds , i.e. , there are positive @xmath147 so that @xmath150 by the above construction , @xmath151 grows as @xmath152 in @xmath145 . the deviations that occur by varying @xmath50 and @xmath44 also scale as @xmath152 , as @xmath40 is a symbol of order @xmath31 . recalling @xmath153 and again using taylor s theorem , one shows that by making @xmath149 small enough , one still retains an inequality of the form . by suitably restricting @xmath149 in @xmath0 , we can ensure that for @xmath154 we have @xmath155 whenever @xmath115 . then no new direction @xmath44 for which we would have to check the bound can appear while varying @xmath50 in @xmath156 . given , it is clear that we can also take a conic neighborhood @xmath157 of @xmath158 , by tilting it by angles less than @xmath159 . choosing @xmath160 gives a neighborhood of @xmath161 that does not intersect @xmath111 . [ prop : m ] if the support of the symbol @xmath162 does not intersect @xmath112 , then @xmath76 is smooth . we choose a neighborhood @xmath163 of @xmath164 whose closure does not intersect @xmath112 . we choose a smooth function @xmath48 that is equal to one in a neighborhood of @xmath112 and vanishes on @xmath165 . we choose another smooth function @xmath166 on @xmath38 with support in @xmath163 which is identical to one whenever @xmath167 for @xmath168 . by definition of @xmath112 , the set @xmath169 is bounded , so we can choose @xmath166 such that @xmath170 is compact for each @xmath50 . then we define @xmath171 by the definition of @xmath112 and @xmath48 , we have @xmath80 and @xmath82 . with these definitions , we have @xmath172 here we used that @xmath48 and @xmath166 have nonoverlapping supports . as @xmath173 and @xmath86 differentiates only w.r.t . @xmath44 , we thus have @xmath174 for arbitrary integer @xmath72 . as @xmath86 maps symbols of order @xmath22 to symbols of order @xmath175 , @xmath76 is smooth by proposition [ prop : diff ] . [ cor : m ] the singular support of @xmath76 is contained in @xmath141 . [ thm : sp ] the wave front set of @xmath76 is contained in @xmath111 . [ lemma : sp1 ] let @xmath176 , @xmath143 . then there is a conic neighborhood @xmath145 of @xmath119 , a conic neighborhood @xmath86 of @xmath158 and positive constants @xmath177 such that @xmath178 furthermore , there is a positive @xmath138 such that @xmath179 condition is fulfilled for @xmath119 by condition [ enum : phase ] of definition [ def : phase ] . that it is also fulfilled in a neighborhood of @xmath119 can be shown analogously to the proof of the closedness of @xmath112 in lemma [ lemma : sp ] . condition is fulfilled for @xmath180 . that it is also fulfilled in a neighborhood of @xmath180 can again be shown as in lemma [ lemma : sp ] . in order to prove the last statement , we note that by we have @xmath181 where @xmath182 has length @xmath183 and lies on the cone with angle @xmath159 around @xmath8 ( see the figure , where @xmath184 is denoted by @xmath72 ) . for the distance of @xmath8 and @xmath185 we have the bound ( see the dashed lines in the figure ) @xmath186 using , we then obtain the above statement . by corollary [ cor : m ] , it suffices to consider points @xmath187 . let @xmath143 . due to proposition [ prop : m ] , we may assume that @xmath162 is supported in an arbitrarily small closed conic neighborhood @xmath145 of @xmath112 . we thus need to show that there is a @xmath188 , identically one near @xmath140 and a conic neighborhood @xmath86 of @xmath158 such that for each @xmath189 there is a seminorm @xmath190 on @xmath25 such that @xmath191 for all @xmath162 supported in @xmath145 . as in the proof of theorem [ thm : osc ] , it suffices to construct such a bound for @xmath94 and then make use of proposition [ prop : dense ] . let @xmath192 be as in lemma [ lemma : sp1 ] . choose @xmath193 that is identically one near @xmath140 and whose support is contained in @xmath194 . choose a @xmath195 that is identical to one on @xmath196 . now we set @xmath197 then for @xmath198 we have @xmath199 now we choose @xmath200 , identical to one near @xmath140 and with support in @xmath201 . we also choose @xmath202 which is identically one on @xmath203 . we consider @xmath204 . then by proposition [ prop : diff ] , the second term yields a smooth function , so that the above bound is fulfilled . it remains to consider @xmath205 { \mathrm{d}}^n x { \mathrm{d}}^s \theta \rvert } \\ & \leq \int { \lvert v_k^p[\psi(x ) ( 1-\xi(\theta ) ) a(x , \theta ) ] \rvert } { \mathrm{d}}^n x { \mathrm{d}}^s \theta.\end{aligned}\ ] ] here we used that @xmath206 is identically one on the support of @xmath207 . by lemma [ lemma : sp1 ] , one now has @xmath208 \rvert } \leq c_{m , p}(a ) ( 1+{\lvert \theta \rvert})^m ( { \lvert \theta \rvert}^\mu + { \lvert k \rvert})^{-p},\ ] ] where @xmath209 is a seminorm on @xmath25 . by using @xmath210 one can make the integral convergent and assure by choosing @xmath72 large enough . [ ex : delta+ ] we consider the two - point function @xmath211 of a free massive scalar relativistic field , where one has @xmath212 and @xmath213 with @xmath214 here , we use the notation @xmath215 , with @xmath216 . note that @xmath40 is not homogeneous . in @xcite this problem is circumvented by using @xmath217 as phase function and multiplying the symbol with the function @xmath218 . it is then no longer a symbol ( as it is not smooth in @xmath219 ) , so one has to allow for so - called asymptotic symbols . furthermore , one has to show that the multiplication with such a term does not spoil the fall - off properties , in particular that differentiation w.r.t . @xmath44 lowers the order . in the present approach , this is not necessary . @xmath40 is a phase function in the sense of definition [ def : phase ] and therefore , by theorem [ thm : osc ] , it defines an oscillatory integral for every symbol @xmath162 . in order to find the wave front set , we compute @xmath220 it is easy to see that its modulus is bounded from below by a positive constant unless @xmath221 or @xmath222 and @xmath223 . thus , we have @xmath224 furthermore , @xmath225 for large @xmath44 , this behaves as @xmath226 where the remainder term @xmath227 scales as @xmath228 . thus , only in the directions @xmath229 the bound on the angle of @xmath8 and @xmath230 can not be fulfilled . hence , we obtain the well - known result . in that convention , the sign of the zeroth component in the cotangent bundle has to be reversed . ] @xmath231 a variant of this example is obtained by considering phase functions of the form @xmath232 with @xmath14 a positive function that is a symbol of order @xmath233 . such expressions occur for example in quantum field theory on the moyal plane with hyperbolic signature and signify a distortion of the dispersion relations , cf . the above trick to put @xmath234 into the symbol still works , but then the symbol will be of type @xmath235 , where @xmath236 is original type of the symbol . it is straightforward to check that still defines a phase function of order 1 in the sense defined here , and that its stationary phase is as above . if the function @xmath14 in is a symbol of order @xmath237 , of corresponds to solutions of the hyperbolic wave equation @xmath238 . the modification suggested here means that the underlying pde is no longer hyperbolic . ] then the shift of @xmath234 to the symbol is not possible , as this would no longer give a symbol of type @xmath239 . thus , a treatment of in the context of phase functions that are homogeneous of order 1 is not possible . however , one can still interpret as a phase function of order @xmath240 and easily computes @xmath241 [ ex : ncqft ] in quantum field theory on the moyal plane of even dimension @xmath242 with euclidean signature , one frequently finds phase funtions of the form @xmath243 here @xmath244 is some real antisymmetric @xmath245 matrix of maximal rank @xmath242 . the above is clearly a symbol of order 2 , and we have @xmath246 as @xmath244 has rank @xmath242 , condition [ enum : phase ] of definition [ def : phase ] is fulfilled . from the above it follows that @xmath247 and thus also @xmath248 , so that the resulting distributions are smooth . we note that up to now such integrals could only be treated in the so - called adiabatic limit @xcite . but then one loses the information about the singular behaviour in position space , contrary to the present case , where the wave front set is known completely . [ ex : ncqftb ] in quantum field theory on the moyal plane with hyperbolic signature , one frequently finds phase functions of the form @xmath249 where @xmath250 with @xmath251 as in and @xmath244 as in example [ ex : ncqft ] . the above is a symbol of order 2 , but it is not a phase function as defined here , as can most easily be seen in the case @xmath252 . then with @xmath253 one obtains @xmath254 if the signs of @xmath255 and @xmath256 coincide , then the above derivatives tend to a constant as a function of @xmath44 , so that condition [ enum : phase ] of definition [ def : phase ] is not fulfilled . the rigourous treatment of such integrals is an open problem , which we plan to address in future work . it is a pleasure to thank dorothea bahns for helpful discussions and her detailed comments on the manuscript . i would also like to thank ingo witt for valuable comments . this work was supported by the german research foundation ( deutsche forschungsgemeinschaft ( dfg ) ) through the institutional strategy of the university of gttingen . d. bahns , s. doplicher , k. fredenhagen and g. piacitelli , _ field theory on noncommutative spacetimes : quasiplanar wick products _ , phys . d * 71 * , 025022 ( 2005 ) . c. dscher and j. zahn , _ dispersion relations in the noncommutative @xmath258 and wess - zumino model in the yang - feldman formalism _ , annales henri poincare * 10 * , 35 ( 2009 ) . r. wulkenhaar , _ field theories on deformed spaces _ , j. geom . phys . * 56 * , 108 ( 2006 ) . c. dscher , _ yang - feldman formalism on noncommutative minkowski space _ , ph.d . thesis , hamburg ( 2006 ) .
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a generalized notion of oscillatory integrals that allows for inhomogeneous phase functions of arbitrary positive order is introduced . the wave front set of the resulting distributions
is characterized in a way that generalizes the well - known result for phase functions that are homogeneous of order one .
| 7,254 | 72 |
a significant theory regarding inequalities and exponential convexity for real valued functions , has been developed @xcite . the intention to generalize such concepts for the @xmath0-semigroup of operators , is motivated from @xcite . + in the present article , we shall derive a jessen s type inequality and the corresponding adjoint - inequality , for some @xmath0-semigroup and the adjoint - semigroup , respectively . + + the notion of banach lattice was introduced to get a common abstract setting , within which one could talk about the ordering of elements . therefore , the phenomena related to positivity can be generalized . it had mostly been studied in various types to spaces of real - valued functions , e.g. the space @xmath1 of continuous functions over a compact topological space @xmath2 , the lebesque space @xmath3 or even more generally the space @xmath4 constructed over measure space @xmath5 for @xmath6 . we shall use without further explanation the terms , order relation ( ordering ) , ordered set , supremum , infimum . + firstly , we shall go through the definition of vector lattice . any ( real ) vector space @xmath7 with an ordering satisfying ; @xmath8 : : : @xmath9 implies @xmath10 for all @xmath11 @xmath12 : : : @xmath13 implies @xmath14 for al @xmath15 and @xmath16 is called an _ ordered vector space_. the axiom @xmath8 , expresses the translation invariance and therefore implies that the ordering of an ordered vector space @xmath7 is completely determined by the positive part @xmath17 of @xmath7 . in other words , @xmath9 if and only if @xmath18 . moreover , the other property @xmath12 , reveals that the positive part of v is a convex set and a cone with vertex @xmath19 ( mostly called the _ positive cone _ of v ) . * an ordered vector space @xmath7 is called a _ vector lattice _ , if any two elements @xmath20 have a supremum , which is denoted by @xmath21 and an infimum denoted by @xmath22 . + it is trivially understood that the existence of supremum of any two elements in an ordered vector space implies the existence of supremum of finite number of elements in @xmath7 . furthermore , @xmath23 implies @xmath24 , so the existence of finite infima therefore implied . * few important quantities are defined as follows @xmath25 * some compatibility axiom is required , between norm and order . this is given in the following short way : @xmath26 the norm defined on a vector lattice is called a lattice norm . now , we are in a position , to define a banach lattice in a formal way . a _ banach lattice _ is a banach space @xmath7 endowed with an ordering @xmath27 , such that @xmath28 is a vector lattice with a lattice norm defined on it . + a banach lattice transforms to _ banach lattice algebra _ , provided @xmath29 implies @xmath30 . @xmath31 + a linear mapping @xmath32 from an ordered banach space @xmath7 into itself is _ positive _ ( denoted by : @xmath33 ) if @xmath34 , for all @xmath35 . the set of all positive linear mappings forms a convex cone in the space @xmath36 of all linear mappings from @xmath7 into itself , defining the natural ordering of @xmath36 . the absolute value of @xmath32 , if it exists , is given by @xmath37 thus @xmath38 is positive if and only if @xmath39 holds for any @xmath15 . * [ @xcite , p.249 ] * a bounded linear operator @xmath32 on a banach lattice v is a positive contraction if and only if @xmath40 for all @xmath15 . @xmath31 + an operator @xmath41 on @xmath7 satisfies the positive minimum principle if for all @xmath42 , @xmath43 @xmath44 a ( one parameter ) @xmath0-semigroup ( or strongly continuous semigroup ) of operators on a banach space @xmath7 is a family @xmath45 such that ( i ) : : @xmath46 for all @xmath47 . ( ii ) : : z(0)=i , the identity operator on v. ( iii ) : : for each fixed @xmath15 , @xmath48(with respect to the norm on v ) as @xmath49 . where @xmath50 denotes the space of all bounded linear operators defined on a banach space v. the ( infinitesimal ) generator of @xmath51 is the densely defined closed linear operator @xmath52 such that @xmath53 @xmath54 where , for @xmath55 , @xmath56f}{t}\,\,\,\,(f\in v).\ ] ] @xmath31 + let @xmath51 be the strongly continuous positive semigroup , defined on a banach lattice v. the positivity of the semigroup is equivalent to @xmath57 where for positive contraction semigroups @xmath51 , defined on a banach lattice v we have ; @xmath58 the literature presented in @xcite , guarantees the existence of the strongly continuous positive semigroups and positive contraction semigroups on banach lattice v , with some conditions imposed on the generator . the very important amongst them is , that it must always satisfy ( [ pmp ] ) . + a banach algebra @xmath59 , with the multiplicative identity element @xmath60 , is called the _ unital banach algebra_. we shall call the strongly continuous semigroup @xmath51 defined on @xmath59 , a _ normalized semigroup _ , whenever it satisfies @xmath61 the notion of normalized semigroup is inspired from normalized functionals @xcite . the theory presented in next section , is defined on such semigroups of positive linear operators defined on a banach lattice @xmath7 . in 1931 , jessen @xcite gave the generalization of the jensen s inequality for a convex function and positive linear functionals . see ( @xcite , pp-47 ) . we shall prove this inequality for a normalized positive @xmath0-semigroup and convex operator , defined on a banach lattice . + throughout the present section , @xmath7 will always denote a unital banach lattice algebra , endowed with an ordering @xmath27 . let @xmath62 be a nonempty open convex subset of @xmath7 . an operator @xmath63 is convex if it satisfies @xmath64 whenever @xmath65 and @xmath66 . @xmath31 + let @xmath67 denotes the set of all differentiable convex functions @xmath68 . * ( jessen s type inequality ) * let @xmath51 be the positive @xmath0-semigroup on @xmath7 such that it satisfies ( [ nsg ] ) . for an operator @xmath69 and @xmath70 ; @xmath71 * proof : * since @xmath68 is convex and differentiable , by considering an operator - analogue of [ theorem a , pp-98,@xcite ] , we have for any @xmath72 , there is a fixed vector @xmath73 such that @xmath74 using the property ( [ nsg ] ) along with the linearity and positivity of operators in a semigroup , we obtain @xmath75 in this inequality , set @xmath76 and the assertion ( [ jti ] ) follows . @xmath31 + the existence of an identity element and condition ( [ nsg ] ) , imposed in hypothesis of the above theorem is necessary . we shall elaborate the said , by following examples . let @xmath77 , @xmath51 be the left shift semigroup defined on x and @xmath78 taking the mirroring along @xmath79-axis . the identity function does not contain a compact support and therefore is not in @xmath59 . if we now take a bell - shaped curve like @xmath80 , @xmath81 . then f is positive , @xmath82 , and @xmath83 has maximum at @xmath84 , and it is between 0 and 1 elsewhere . on the other hand , @xmath85 has a maximum at @xmath86 and it is immediate that we can not compare the two functions in the usual ordering . see figure 1(a ) . let @xmath87 , and @xmath88 . the rotation semigroup @xmath51 is defined as , @xmath89 , @xmath90 . the identity element @xmath91 , s.t . for all @xmath92 , @xmath93 . then @xmath94 . or we can say that any complex number @xmath95 is mapped to @xmath96 . @xmath97 satisfies ( [ nsg ] ) , only when @xmath98 is a multiple of @xmath99 . let @xmath100 , then @xmath101 , hence @xmath102 . on the other hand , @xmath103 , and @xmath104 . hence , the equality holds in ( [ jti ] ) when @xmath98 is a multiple of @xmath99 , but the two sides are not comparable in general . it can easily be verified that @xmath105 is a subgroup of @xmath106 , as @xmath107 . therefore @xmath108 is a normalized semigroup . see figure 1(b ) . + @xmath31 + in previous section , a jessen s type inequality has been derived , for a normalized positive @xmath0-semigroup @xmath51 . this gives us the motivation towards knowing the behaviour of its corresponding adjoint semigroup @xmath109 on @xmath110 . as the theory for dual spaces gets more complicated , we do not expect to have the analogous results . it may ask for a detail introduction towards a part of the dual space @xmath110 , for which an adjoint of jessen s type inequality makes sense . given two banach spaces @xmath59 and @xmath111 and a bounded linear operator @xmath112 , recall that the adjoint @xmath113 is defined by @xmath114 for a strongly continuous positive semigroup @xmath51 on a banach space @xmath59 , by defining @xmath115 for every @xmath98 , we get a corresponding adjoint semigroup @xmath109 on the dual space @xmath116 . in @xcite , it is obtained that , the adjoint semigroup @xmath109 fails in general to be strongly continuous . the investigation @xcite , shows that @xmath109 acts in a strongly continuous way on ; @xmath117 this is the maximal such subspace on @xmath116 . the space @xmath118 was introduced by philips in 1955 , and latter has been studied extensively by various authors . at the present moment , we do not necessarily require the strong continuity of the adjoint semigroup @xmath109 on @xmath116 . + if @xmath59 is an ordered vector space , we say that a functional @xmath119 on @xmath59 is positive if @xmath120 , for each @xmath121 . by the linearity of @xmath119 , this is equivalent to @xmath119 being order preserving . i.e. @xmath122 implies @xmath123 . the set @xmath124 of all positive linear functionals on @xmath59 , is a cone in @xmath116 . + we are mainly interested in the study of the space @xmath110 , where in our case @xmath7 is a banach lattice algebra . let us consider the regular ordering among the elements of @xmath110 . i.e. @xmath125 , whenever @xmath126 , for each @xmath127 . + consider the convex operator ( [ co ] ) . in case of the equality , @xmath128 is simply a linear operator and the adjoint @xmath128 can be defined as above . but how can it be defined in other case ? this question has already been answered . + in @xcite , some kind of adjoint has been associated to a nonlinear operator @xmath128 . in fact , this is possible for lipschitz continuous operators only . consider the banach space @xmath129 of all lipschitz continuous operators @xmath130 satisfying @xmath131 , equipped with the norm @xmath132_{lip}= sup_{x_1\neq x_2}\frac { \|f(x_1)-f(x_2)\|}{\|x_1-x_2\|},\quad x_1,x_2\in x.\ ] ] where @xmath133 is the identity . it is easy to see that the space @xmath134 of all bounded linear operators from x to y is a closed subspace of @xmath129 . in particular , we set @xmath135 and call @xmath136 the pseudo - dual space of @xmath59 ; this space contains the usual dual space @xmath116 as closed subspace . + for @xmath137 , the pseudo - adjoint @xmath138 of @xmath128 is defined by ; @xmath139 this is of course a straightforward generalization of ( [ co ] ) ; in fact , for linear operators @xmath140 we have @xmath141 . i.e. the restriction of the pseudo - adjoint to the dual space is the classical adjoint . + for the sake of convenience , we shall denote the adjoint of the operator @xmath128 by @xmath142 , throughout the present section . either it s a classical adjoint or the pseudo - adjoint ( depending upon the operator @xmath128 ) . + similarly , the considered dual space of the vector lattice algebra @xmath7 will be denoted by @xmath110 , which can be the intersection of the pseudo - dual and classical dual spaces in case of a nonlinear convex operator . let @xmath128 be the convex operator on a banach space @xmath59 , then the adjoint operator @xmath142 on the dual space @xmath116 is also convex . * * proof:**for @xmath121 and @xmath143@xmath144 where @xmath145 . by putting @xmath146 , for @xmath147 and using the convexity of the operator @xmath128 we finally get @xmath148 hence , @xmath142 is convex on @xmath116 . * ( adjoint - jessen s inequality ) * let @xmath109 be the adjoint semigroup on @xmath110 such that the original semigroup @xmath51 , the operator @xmath78 and the space @xmath7 are same as in theorem ( 1 ) . for a convex operator @xmath149 and @xmath150 @xmath151 * proof : * for @xmath15 and @xmath150 , consider @xmath152,f ) & = & ( z^\ast(t)f^\ast,\phi(f ) ) \\ & = & ( f^\ast , z(t)(\phi f ) ) \\ & \geq & ( f^\ast,\phi(z(t)f ) ) \\ & = & ( \phi^\ast ( f^\ast),z(t)f ) \\ & = & ( z^\ast(t)[\phi^\ast f^\ast],f ) \end{aligned}\ ] ] therefore , the assertion ( [ ajti ] ) is satisfied . in this section we shall define the exponential convexity of an operator . moreover , few complex structures , involving the operators from a semigroup , will be proved to be exponentially convex . let @xmath7 be a banach lattice endowed with ordering @xmath27 . an operator @xmath153 is exponentially convex if it is continuous and for all @xmath154 @xmath155 where @xmath156 such that @xmath157 , @xmath158 . * * proof:**@xmath163 + take any @xmath156 and @xmath164 , @xmath162 . since the interval @xmath165 is convex , the midpoints , @xmath166 . now set @xmath167 , for @xmath162 . then we have , @xmath168 , for all @xmath158 . therefore , for all @xmath154 , we can apply @xmath169 to get , @xmath170 @xmath171 + let @xmath172 , such that @xmath173 , for @xmath158 . define @xmath174 , so that @xmath175 . therefore , for all @xmath154 , we can apply @xmath176 to get , @xmath177 @xmath31 + let @xmath159 be an exponentially convex operator . writing down the fact for @xmath178 , in ( 10 ) , we get that @xmath179 , for @xmath180 and @xmath15 . for @xmath181 , we have @xmath182 hence , for @xmath183 and @xmath184 , we have @xmath185 i.e. @xmath153 , does indeed satisfy the condition of convexity . @xmath31 + for @xmath186 , let us assume that @xmath63 is continuously differentiable on @xmath62 . i.e. the mapping @xmath187 , is continuous . moreover @xmath188 , will be a continuous linear transformation from @xmath7 to @xmath189 . a bilinear transformation @xmath190 defined on @xmath191 is symmetric if @xmath192 for all @xmath20 . such a transformation is * positive definite [ nonnegative definite ] * , if for every nonzero @xmath15 , @xmath193 [ @xmath194 . then , @xmath188 is symmetric wherever it exists . see [ @xcite , pp-69 ] . [ condif ] let @xmath128 be continuously differentiable and suppose that second derivative exists throughout an open convex set @xmath186 . then @xmath128 is convex on @xmath62 if and only if @xmath188 is nonnegative definite for each @xmath195 . and if @xmath188 is positive definite on @xmath62 , then @xmath128 is strictly convex . let @xmath7 be a unital banach algebra . for @xmath15 , a family of operators @xmath199 is defined as @xmath200 then @xmath201 . whenever , @xmath35 , @xmath202 , therefore by theorem 3 , the mapping @xmath203 is convex . [ t4 ] let @xmath51 be the positive @xmath0-semigroup , defined on a unital banach lattice algebra @xmath7 , such that it satisfies ( [ nsg ] ) . let @xmath15 , such that @xmath204 , for @xmath205 , @xmath206 , if @xmath207 and @xmath208 , if @xmath209 . let us define @xmath210 then ( i ) : : for every @xmath154 and for every @xmath211 , @xmath212 , @xmath213_{i , j=1}^n\geq 0.\ ] ] ( ii ) : : if the mapping @xmath214 is continuous on @xmath215 , then it is exponentially convex on @xmath215 . * * proof:**consider the operator @xmath216 for @xmath217 , @xmath218 and @xmath219 where @xmath220 . then @xmath221 so , @xmath222 is a convex operator . therefore by applying ( [ jti ] ) we get @xmath223 and the assertion ( [ a ] ) follows . assuming the continuity and using the * proposition 1 * we have also exponential convexity of the operator @xmath214 . @xmath31 + * proof : * similar to the proof of theorem ( [ t4 ] ) . @xmath31 + + * competing interests * + the authors declare that they have no competing interests . + * author s contribution * + all authors contributed equally and significantly in writing this paper . all authors read and approved the final manuscript . + * acknowledgement * + authors of this paper are grateful to prof . andrs btkai for his generous help in construction of examples . 20 , _ convexity , subadditivity and generalized jensen s inequality _ , vol 4 , no . 2 , 183194 . 2013 . , _ exponential convexity , positive semi - definite matrices and fundamental inequalities _ , j. math . , vol 4 , no 2 , 171 - 189 . 2010 . , _ nonlinear spectral theory _ , walter de gruyter . new york , 2004 . , _ inequalities _ , springer - verlag , berlin , 1961 . , _ interpolation and approximation _ , dover , new york , 1975 . , _ ber die entwicklung realen funktionen in reihen mittelst der methode der kleinsten quadrate _ , j. reine angewendte math . , no 94 , 4173 . 1883 . , _ cauchy type means on one - parameter @xmath0-group of operators _ , j. math . vol 9 , no . 2 , 631639 . 2015 . , _ bemaerkinger om konvekse funktioner og uligheder imellem middelvaerdier i _ , muf . tidsskrifr , 17 - 28 . 1931 . , _ konano dimenzionnalni vektorski prostori i primjene _ , tehnike knjige , zagreb , 1990 . , _ classical and new inequalities in analysis _ , kluwer academic publishers , the netherlands , 1993 . , _ one - parameter semigroups of positive operators _ , lect . notes in math . 1184 , springer - verlag , 1986 . , _ convex functions , partial orderings , and statistical applications _ , academic press , inc . new york , 1992 . , _ the adjoint semi - group _ , pac . j. math . no 5 , 269 - 283 , 1955 . , _ convex functions _ , academic press , new york and london , 1973 . , _ the exponential and logarithmic functions on commutative banach algebras _ , int . journal of math . analysis , vol . 42 , 2079 - 2088 . 2010 . , _ the adjoint of a positive semigroup _ , compositio mathematica , no 90 , 99 - 118 , 1994
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in this paper the jessen s type inequality for normalized positive @xmath0-semigroups is obtained .
an adjoint of jessen s type inequality has also been derived for the corresponding adjoint - semigroup , which does not give the analogous results but the behavior is still interesting .
moreover , it is followed by some results regarding positive definiteness and exponential convexity of complex structures involving operators from a semigroup .
example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
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the old - new interest in the problem of soliton - soliton intertaction and soliton molecules has been increasingly accumulating particularly over the past few years . this is mainly motivated by the application of optical solitons as data carriers in optical fibers @xcite and the realization of matter - wave solitons in bose - einstein condensates @xcite . one major problem limiting the high - bit rate data transfer in optical fibers is the soliton - soliton interaction . on the one hand , soliton - soliton interaction is considered as a problem since it may destroy information coded by solitons sequences . on the other hand , it is part of the problem s solution , since the interaction between solitons leads to the formation of stable soliton molecules which can be used as data carriers with larger `` alphabet '' @xcite . the interaction force between solitons was first studied by karpman and solovev using perturbation analysis @xcite , gordon who used the exact two solitons solution @xcite , and anderson and lisak who employed a variational approach @xcite . it was shown that the force of interaction decays exponentially with the separation between the solitons and depends on the phase difference between them such that in - phase solitons attract and out - of - phase solitons repel . this feature was demonstrated experimentally in matter - wave solitons of attractive bose - einstein condensates @xcite where a variational approach accounted for this repulsion and showed that , in spite of the attractive interatomic interaction , the phase difference between neighboring solitons indeed causes their repulsion @xcite . for shorter separations between the solitons , malomed @xcite used a perturbation approach to show that stationary solutions in the form of bound states of two solitons are possible . however , detailed numerical analysis showed that such bound states are unstable @xcite . stable bound states were then discovered by akhmediev @xcite and a mechanism of creating robust three - dimensional soliton molecules was suggested by crasovan _ recently , soliton molecules were realized experimentally by stratmann _ _ in dispersion - managed optical fibers @xcite and their phase structure was also measured @xcite . perurbative analysis was used to account theoretically for the binding mechanism and the molecule s main features @xcite . quantization of the binding energy was also predicted numerically by komarov _ _ @xcite . in refs.@xcite , a hamiltonian is constructed to describe the interaction dynamics of solitons . the mechanism by which the relative phase between the solitons leads to their force of interaction , and hence the binding mechanism , is understood only qualitatively as follows . for in - phase ( out - of - phase ) solitons , constructive ( destructive ) interference takes place in the overlap region resulting in enhancement ( reduction ) in the intensity . as a result , the attractive intensity - dependent nonlinear interaction causes the solitons to attract ( repel ) @xcite . a more quantitative description is given in refs . @xcite . in view of its above - mentioned importance from the applications and fundamental physics point of views , we address here the problems of soliton - soliton interaction and soliton molecule formation using the exact two solitons solution . this approach has been long pioneered by gordon @xcite where he used the exact two solitons solution of the homogeneous nonlinear schr@xmath0dinger equation to derive a formula for the force of interaction between two solitons , namely @xmath1 where @xmath2 is the solitons separation and @xmath3 is their phase difference . this formula was derived in the limit of large solitons separation and for small difference in the center - of - mass speeds and intensities , which limits its validity to slow collisions . with appropriately constructed hamiltonian , wu _ et al . _ have derived , essentially , a similar formula that gives the force between two identical solitons and reliefs the condition on slow collisions @xcite . here , we present a more comprehensive treatment where we derive the force between two solitons for arbitrary solitons intensities , center - of - mass speeds , and separation . we also generalize gordon s formula to inhomogeneous cases corresponding to matter - wave bright solitons in attractive bose - einstein condensates with time - dependent parabolic potentials @xcite and to optical solitons in graded - index waveguide amplifiers @xcite . many interesting situations can thus be investigated . this includes the various soliton - soliton collision regimes with arbitrary relative speeds , intensities , and phases . most importantly , soliton - soliton interaction at short solitons separations will now be accounted for more quantitatively than before . specifically , soliton molecule formation is clearly shown to arise from the time - dependence of the relative phase which plays the role of the restoring force . in this case , the force between the two solitons is shown to be composed of a part oscillating between attractive and repulsive , which arises from the relative phase , and an attractive part that arises from the nonlinear interaction . the time - dependence of the relative phase results in a natural oscillation of the molecule s bond length around an equilibrium value . the various features of the soliton molecule , including its equilibrium _ bond length _ , _ spring constant _ , _ frequency _ and _ amplitude _ of oscillation , and _ effective mass _ , will be derived in terms of the fundamental parameters of the solitons , namely their intensities and the nonlinear interaction strength . the two solitons solution is derived here using the inverse scattering method @xcite . although the two solitons solution of the homogeneous nonlinear schr@xmath0dinger equation is readily known @xcite , here we not only generalize this solution to inhomogeneous cases , but also present it in a new form that facilitates its analysis . the solution will be given in terms of the four fundamental parameters of each soliton , namely the initial amplitude , center - of - mass position and speed , and phase . the main features of the solution will be shown clearly such as the contribution of the nonlinear interaction to the actual separation and phase difference between solitons where it turns that the separation between the two solitons increases with logarithm of the difference between the amplitudes of the two solitons . furthermore , the general statement that _ a state of two equal solitons with zero relative speed and finite separation does not exist as a stationary state for the homogeneous nonlinear schr@xmath0dinger equation _ , will be transparently and rigorously proved . stability of soliton molecules is an important issue since , in real systems , perturbations caused by various sources such as losses , raman scattering , higher order dispersion , and scattering from local impurities , tend to destroy the molecules . to investigate the stability of the soliton molecules described by our formalism , we have considered three situations . first , we studied the reflection of the molecule from a hard wall and a softer one . while for the hard wall the molecule preserves its molecular structure after reflection , it generally breaks up for the softer ones due to energy losses at the interface . secondly , the scattering of the molecule by a potential barrier was also investigated . we show that the molecular structure is maintained only for some specific heights of the barrier . this suggests a quantization in the binding energy as predicted by komarov _ the oscillation period of the reflected molecule is noticed to be smaller than for the incident one . in addition , the outcome of scattering depends on the phase of the molecule s oscillation at the interface of the barrier . for instance , a dramatic change in the scattering outcome takes place if the _ coalescence _ point of the molecule lies exactly at the interface . in such a case , the otherwise totally transmitting molecule will now split into reflecting and tunneling solitons . thirdly , we have considered the collision between a single soliton with a stationary soliton molecule . the effects of different initial speeds , amplitudes , and phases of the scatterer soliton were studied . it turns out that for slower collisions , it is easier for the scatterer soliton to break up the soliton molecule , while for fast collisions the scatterer soliton expels and then replaces one of the solitons in the molecule . the phase of the scatterer soliton plays also a crucial rule in preserving or breaking the bond of the molecule , which can be used as _ key _ tool to code or uncode data in the molecule . the rest of the paper is organized as follows . in section [ exact_sec ] , we use the inverse - scattering method to derive the two solitons solution of the inhomogeneous nonlinear schr@xmath0dinger equation and present the solution in the above - mentioned appealing form . the main features of the solution will be discussed in subsection [ mainsec ] . the center - of - mass positions and relative phases will be derived in subsections [ com_sec ] and [ phases_sec ] , respectively . the force between solitons will be derived in section [ force_sec ] where gordon s formula will be extracted as a special case in subsection [ gordon_sec ] and our more general formula will be derived in subsection [ our_sec ] . in subsection [ num_sec ] , we compare our formula with the numerical calculation . in section [ molsec ] , we show the possibility of forming soliton molecules , derive their main features in subsection [ formation_sec ] , and investigate their stability in subsection [ stability_sec ] . we end in section [ conc_sec ] with a summary of results and conclusions . the details of the derivation of the two solitons solution and the center - of - mass positions are relegated to appendices a and b , respectively . matter - wave solitons of trapped bose - einstein condensates and optical solitons in optical fibers can be both described by the dimensionless gross - pitaevskii equation @xmath4 , time to @xmath5 , and the wavefunction @xmath6 to @xmath7 , where @xmath8 and @xmath9 are the characteristic frequencies of the quasi one - dimensional ( @xmath10 ) trapping potential in the axial and radial directions , respectively . in these units , the strength of the interatomic interaction will be given by the ratio @xmath11 , where @xmath12 is the @xmath13-wave scattering length . for the case of optical solitons , the function @xmath6 represents the beam envelope , @xmath14 is the propagation distance , @xmath15 is the radial direction , and the intensity - dependent term represents the kerr nonlinearity . in this case , scaling is in terms of the characteristic parameters of the fiber , as for instance in ref . the specific form of the prefactors of the inhomogeneous and nonlinear terms guarantees the integrability of this equation @xcite . for the special case of @xmath16 , the homogeneous case is retrieved . other interesting special cases have also been considered @xcite . as outlined in appendix [ appa ] , we use the darboux transformation method to derive the two solitons solution of this gross - pitaevskii equation , which can be put in the form @xmath17\nonumber\\ & + & \sqrt{\frac{n_2 \alpha_{22}(t)}{2}}e^{i ( \phi_{01}+\phi_{02}+\phi_2(x , t)+ \tan^{-1}({\alpha_2\over\alpha_1 } ) + \tan^{-1}({\alpha_4\over\alpha_3}))}\nonumber\\ & \times&{\rm{sech}}\left[\alpha_{22}(t)(x - x_{\rm cm2}(t ) ) + \frac{1}{2}\log\left(\frac{\alpha_1 ^ 2+\alpha_2 ^ 2}{\alpha_3 ^ 2 + \alpha_4 ^ 2}\right)\right]\label{exactsol},\end{aligned}\ ] ] where + + @xmath18 , @xmath19 , + + @xmath20 , + + @xmath21 , + + @xmath22 , + + @xmath23 , + + @xmath24 , + + @xmath25 , + + @xmath26 , + + @xmath27 , + + @xmath28 , + + @xmath29 , + + @xmath30 , + + @xmath31 , + + @xmath32 , + + @xmath33 , + + @xmath34 , + + @xmath35 , + + @xmath36 . + + the solution is put in this suggestive form to facilitate its analysis . the first sech part corresponds to the exact single soliton solution with center - of - mass position @xmath37 , width @xmath38 , phase @xmath39 , and normalization @xmath40 . hence , @xmath41 and @xmath42 correspond to the initial center - of - mass position and speed , respectively . the second sech term contains the same features in addition to a shift in both the center - of - mass position and phase . it should be noted , however , that @xmath43 , @xmath44 , @xmath45 , and @xmath46 correspond to the center - of - mass position and speed , normalization , and phase of the single noninteracting solitons . due to the interaction between solitons , these four characteristic quantities may not correspond exactly to the values of the same physical quantities as they did for the single soliton solution . for instance , @xmath37 will not correspond to the center - of - mass of one of the solitons . instead , the soliton may be shifted from that position due to the interaction with the other soliton . in the following , we present a detailed analysis of the locations and phases of the two solitons . inspection shows that there are two main regimes for the two solitons solution , namely the regime of resolved solitons and the regime of overlapping solitons . in the former case , the center - of - mass concept is well defined and analysis of the relative dynamics becomes feasible . the solitons are considered resolved as long as the two main peaks are not overlapping , which means that partial overlap may occur in this regime . the analysis in this section assumes the resolved solitons regime . in the resolved solitons regime , the argument of the second sech term of eq . ( [ exactsol ] ) , namely @xmath47 $ ] , simplifies to a function with three roots . the fact that the sech function is peaked at the roots of its argument , leads to that the second sech term corresponds to three `` solitons '' . this is shown in fig [ fig1 ] . we denote these solitons as the `` left '' , `` central '' , and `` right '' solitons with peak locations at @xmath48 , @xmath49 , and @xmath50 , respectively . we notice that the central soliton is located at the position of the soliton of the first sech term , namely near @xmath37 . further inspection shows that the two solitons at this location are out of phase and interfere destructively such that they do not appear in the total profile . therefore , the two solitons that our solution of eq . ( [ exactsol ] ) describes are in fact the left and right solitons arising from the second sech term . this is different from what one aught to conclude from the form of the exact solution , namely that the first sech term corresponds to one soliton and the second sech term corresponds to the other soliton . in general , the center - of - mass locations of the left and right solitons , @xmath48 and @xmath50 , do not match @xmath37 and @xmath51 . typically , @xmath48 will be shifted to the left of @xmath37 , @xmath50 will be shifted to the right of @xmath51 , while @xmath52 remains near @xmath37 . the amount of shift depends mainly on the normalization difference @xmath53 and the relative speed @xmath54 , as will be shown in the next subsection . an interesting general and exact result is that , for the homogeneous case @xmath16 , a state of two equal solitons , @xmath55 , with zero relative speed , @xmath56 , and finite separation , does not exist as an exact solution of the gross - pitaevskii equation . this can be proven by substituting @xmath55 and @xmath56 in @xmath57 to find that the right and left solitons migrate to @xmath58 and @xmath59 , respectively , while the center - of - mass of the central soliton matches exactly @xmath37 . furthermore , we show in section [ phases_sec ] that the phase difference between this central soliton and that of the first sech term equals , in this case , @xmath60 ; guaranteeing their destructive interference . thus , the three solitons disappear in such a special case and @xmath6 becomes the trivial solution . in view of the above , a need arises to derive formulae for the center - of - mass positions , @xmath48 and @xmath50 in terms of the solitons parameters , which will then be used to derive the force between the two solitons . in this section , we derive formulae for the three roots of @xmath57 which correspond to the locations of the left , central , and right solitons , @xmath48 , @xmath52 , and @xmath50 , respectively . to facilitate the derivation , we define @xmath61}$ ] , @xmath62}$ ] , @xmath63 , @xmath64 , @xmath65 , and @xmath66 . the equation @xmath67 is a third - order polynomial in both @xmath68 and @xmath69 . in principle , this equation can be solved algebraically for @xmath68 or @xmath69 . however , extracting @xmath15 from the resulting three roots will not be possible analytically for general @xmath70 . alternatively , and as can be seen from fig . [ fig1 ] , we can exploit the simple linear behavior of @xmath57 near its roots . assuming , without loss of generality , @xmath71 , and noting that in the resolved solitons regime the solitons separation @xmath72 is large , we argue in appendix [ appb1 ] that @xmath73 for all @xmath15 , @xmath74 for @xmath75 , @xmath76 for @xmath77 , and @xmath78 for @xmath79 . based on this , the center - of - mass position @xmath50 can be derived from a taylor expansion of @xmath57 in powers of large @xmath68 and @xmath69 . expanding @xmath57 in powers of large @xmath68 only and leaving @xmath69 arbitrary , accounts for @xmath52 and @xmath48 simultaneously . keeping terms up to first order of @xmath80 , we show in appendix [ appb1 ] that the position of the right soliton is given by @xmath81 where @xmath82 and the position of the the left soliton is given by @xmath83 where @xmath84 , @xmath85 , and @xmath86 and @xmath87 are given in appendix [ appb1 ] . the second and third terms on the right hand side of eqs . ( [ xreq2 ] ) and ( [ xleq2 ] ) account for the shift in the center - of - mass position with respect to the single soliton ones . the third terms are much smaller than the second ones since they decay exponentially with the solitons distance @xmath88 . in the limit @xmath89 and @xmath90 , both @xmath91 and @xmath92 take the form @xmath93}$ ] . thus , it is obvious that for @xmath94 , @xmath95 and @xmath96 . this agrees with our earlier result that two equal solitons with zero relative speed and finite separation , do not exist as an exact solution . in this section , we calculate the phases of the left , central , and right solitons in reference to the phase of the soliton of the first sech part in the two solitons solution . for simplicity , the special case of @xmath97 and @xmath98 will be assumed . the phase difference between the left , central , and right solitons on one hand and the soliton of the first sech term on the other hand is generally given by @xmath99 which by observing that for all @xmath15 @xmath100 reduces to @xmath101 this expression gives the phases of the right soliton @xmath102 , the central soliton @xmath103 , and the left soliton @xmath104 , for @xmath75 , @xmath52 , and @xmath48 , respectively . to calculate these phases we express the parameters @xmath105 in terms of @xmath68 and @xmath69 , as follows @xmath106 @xmath107 @xmath108 @xmath109 in general , @xmath73 for all @xmath15 , but @xmath74 only for @xmath110 , and @xmath111 for @xmath112 , as shown in appendix [ appb1 ] . + + * i. phase of the right soliton @xmath102 : * + + in this case @xmath113 which leads to @xmath114 and thus @xmath115 , @xmath116 , @xmath117 , @xmath118 . therefore + + + @xmath119 + + + and @xmath120 , which gives @xmath121 where @xmath122 is a function that depends on the ratio @xmath123 for nonzero @xmath124 and @xmath125 . + + * ii . phase of the central soliton @xmath103 : * + + in this case @xmath126 which gives @xmath127 , and hence @xmath128 , and @xmath129 . finally , we get @xmath130 the last result shows that , for @xmath131 , i.e. , two equal solitons with zero relative speed , the central soliton and the soliton of the first sech term of the two solitons solution are out of phase and therefore interfere destructively . + + + * iii . phase of the left soliton @xmath104 : * + + in this case @xmath132 which results in @xmath78 and @xmath133 , @xmath134 , @xmath135 , @xmath136 . therefore , @xmath128 and @xmath137 , which gives @xmath138 the phase difference between the left and right solitons is thus given by @xmath139 noting that @xmath140 for @xmath141 and small but finite @xmath125 , and @xmath142 for @xmath143 and small but finite @xmath124 , we finally conclude that the phase difference between the two solitons is given by @xmath144 the above results are verified in fig . [ fig2 ] , where we plot @xmath145 , @xmath102 , @xmath104 , and @xmath103 versus @xmath15 . the agreement between our estimated values and the exact curve is evident . in this section , we use the results of the previous two subsections to derive the force between the left and right solitons . the force is proportional to the acceleration of the solitons separation @xmath146\,e^{-\frac{1}{4 } g_0 { n_s } { x_d}(t ) e^{\gamma ( t)}}\label{deltaeq},\end{aligned}\ ] ] where @xmath147 and the coefficients @xmath148 are given in appendix [ appb2 ] . the first two terms on the right hand side of eq . ( [ deltaeq ] ) are the dominant ones since they correspond to the noninteracting solitons separation , @xmath149 , and their logarithmic shifts @xmath150 arising from the interaction between solitons . the time - dependence of @xmath151 originates from @xmath152 , @xmath153 , and @xmath149 . the acceleration can thus be derived @xmath154 \ , e^{-\frac{1}{4 } g_0 n_s { \delta}(t ) e^{\gamma ( t)}+3 \gamma ( t)}\label{force2},\ ] ] where the coefficients @xmath155 are given in appendix [ appb2 ] . in the last equation , we have used eq . ( [ deltaeq ] ) with only the first two terms of its right hand side to substitute for @xmath149 in terms of @xmath2 in the exponential factor . the first term on the right hand side of eq . ( [ force2 ] ) corresponds to the force due to the external potential which vanishes for the homogeneous case . the rest of the terms correspond to the force of interaction between the solitons . the interaction force depends , as expected , on the phase difference of the two solitons and decays exponentially with their separation . it should be noted that this equation is a generalization of gordon s formula @xcite in two aspects . first , it is derived for a time - dependent inhomogeneous medium . secondly we have , essentially , no restriction on the difference between the two solitons amplitudes and speeds ; apart from some extreme cases which were mentioned in appendix [ appb1 ] and will be discussed further below . for the homogeneous case , @xmath16 , and in the limits @xmath89 and @xmath90 , the acceleration formula , eq . ( [ force2 ] ) , simplifies considerably . an apparent inconsistency occurs when switching the order of these two limits , namely @xmath156 while @xmath157 which differ by an overall minus sign . the conflict is resolved by invoking eq . ( [ phasediff ] ) where it is shown that in the first case : @xmath158 , while in the second case : @xmath159 . therefore , the two approaches agree on the following result @xmath160 this is essentially gordon s result , eq . ( [ gordoneq ] ) , since in his derivation gordon took @xmath161 and soliton amplitude @xmath162 , @xmath163 , as can be seen in eqs.(1 ) and ( 6 ) of ref . @xcite . substituting @xmath161 and @xmath164 in the last equation , it becomes identical to eq . ( [ gordoneq ] ) . it should be mentioned here that eq . ( [ gordoneq ] ) was also derived in ref . @xcite using a perturbation analysis based on the inverse scattering method , and in ref . @xcite using a variational calculation . for nonzero @xmath152 and in the limits @xmath89 and @xmath90 , the acceleration formula takes the form @xmath165 this is a generalization to gordon s formula for the inhomogeneous case as modeled by eq . ( [ gp ] ) . depending on the specific form of @xmath152 , the two force terms , namely the external ( first term ) and the interaction ( second term ) , can be repulsive , attractive , or oscillatory . in addition , the phase difference @xmath3 also depends on @xmath152 . it is established in the homogeneous case , as will also be shown in section [ molsec ] , that the time - dependence of the phase difference is responsible for binding the two solitons in the soliton molecule . here , in addition to the possibility of forming soliton molecules , the dependence of @xmath3 on @xmath152 allows for controlling the parameters of the molecule such as its equilibrium bond length , period , and spring constant . this and possibly other interesting phenomena will be left for future investigation . to obtain an estimate of the accuracy of the general acceleration formula , eq . ( [ force2 ] ) , we calculate numerically the acceleration from the exact two solitons solution , eq . ( [ exactsol ] ) , and compare the two results . the distance between the two solitons of the function @xmath166 is determined using a numerical algorithm that employs our formulae for @xmath48 and @xmath50 given by eqs . ( [ xreq2 ] ) and ( [ xleq2 ] ) to calculate seed values . the distance is then differentiated numerically twice at @xmath167 . in fig . [ fig3 ] , we compare the two results . good agreement is obtained for @xmath168 . the analytical solution diverges at @xmath169 . the value at which divergence takes place is set by the specific choice of parameters in fig . as pointed out in section [ com_sec ] , the divergence occurs due to the merging of the central and left solitons . this artifact divergency can be remedied by associating the location of the local maximum of @xmath57 to @xmath48 once this maximum has reached the @xmath15-axis from above . restricting our study to the region where agreement is obtained , we interestingly notice that the acceleration is oscillating between positive and negative values . this means that the force between the solitons is oscillating between attractive and repulsive . the possibility of having attractive forces for finite @xmath124 is particularly interesting ; for two solitons with nonzero relative positive speed , i.e. the solitons are initially diverging from each other , the force between them is attractive . this suggests that , if the force remains attractive for sufficient time , the two solitons will slow down and eventually converge at some point . if true , this should occur at small distances since the force decays exponentially with distance and when the two solitons are allowed to diverge even for a short while , the force might be weakened such that the two solitons can not return back . to be able to judge on such a possibility , we need to know what happens to the acceleration at later times . to that end , we calculate numerically the acceleration in terms of @xmath124 and @xmath14 . the result is plotted in fig . [ fig4 ] , where it is clear that the acceleration indeed decays with time for all @xmath124 . this leads to that any nontrivial effect of the oscillating force is most likely to take place at short solitons separations . this is what we find in the next section where the possibility of forming stable soliton molecules is pointed out . we have shown in the previous section that , as a result of the solitons time - dependent relative phase , the force of interaction between solitons is oscillating between repulsive and attractive . since the force decays exponentially with the solitons separations , this oscillation will have a tangible effect only when the two solitons are close to each other . in this section , we investigate the force of interaction between solitons for short solitons separation . in such a special case , eq . ( [ deltaeq ] ) takes a simple form that accounts for the solitons separation in terms of their relative phase . using this formula , we show that the solitons will be bound to oscillate around some equilibrium distance where the phase plays the role of the restoring force . comparison with exact numerical calculations shows that this formula is accurate for almost the full range of the solitons separation , except at the coalescence point ( if any ) . in subsection [ formation_sec ] , we discuss the main features of the resulting soliton molecules , and in subsection [ stability_sec ] , we investigate numerically their stability in different scattering regimes . to focus on the role of relative phase , we simplify the analysis by restricting our treatment to the homogeneous case , @xmath16 , and zero relative speed , @xmath170 . we also set @xmath171 so that any separation between the solitons to be as small as possible which in this case arises only from the logarithmic shifts ( @xmath172 in eq . ( [ deltaeq ] ) ) . in this case , the solitons separation @xmath151 , given by eq . ( [ deltaeq ] ) , simplifies in the limit @xmath173 to @xmath174-\frac{4}{g_0 n_s}\log \left(\frac{g_0 n_d^2}{n_s}\right)\label{delmol},\ ] ] and the acceleration is given by @xmath175 it is noted that for @xmath176 or @xmath177 , gordon s formula is retrieved , but here with an explicit time - dependence of the phase , @xmath178 . this acceleration formula deviates considerably from gordon s formula for @xmath179 . specifically , for @xmath180 , @xmath151 diverges to @xmath59 at @xmath181 , which indicates that the two solitons coalesce . this is confirmed below by examining the exact solution at this condition . we note here that an approximate expression for the solitons separation was also derived in refs . in addition , our predicted molecule s oscillation frequency ( see eq . ( [ freqeq ] ) below ) agrees with these references . to verify this feature , we calculate numerically the distance between the two solitons directly from the exact solution , eq . ( [ exactsol ] ) , for different values of @xmath182 . for @xmath183 , the density plot in fig . [ fig5]a , shows a soliton molecule of two clearly resolved solitons with a separation oscillating around some nonzero equilibrium distance . approaching the solitons coalescence point with @xmath184 , the density plot in fig . [ fig6]a , shows the two solitons approaching each other more than the previous case . furthermore , this figure shows a slight bounce back by one of the solitons in the region of collision . approaching further the coalescence condition with @xmath185 , we indeed observe in fig . [ fig7]a that the two solitons merge almost completely . for a more quantitative comparison , we calculate numerically the center - of - mass trajectories of the two solitons . we show the trajectory curves in the density plots of figs . [ fig5]a-[fig7]a . in figs . [ fig5]b-[fig7]b , we plot the solitons separation obtained from formula ( [ delmol ] ) and the numerical trajectories obtained from the exact solution . it is clear from these figures that this formula agrees well with the exact soliton separation except near the collision region . in fig . [ fig5]b , the two solitons remain away from each other during the collision , and therefore good agreement is obtained with the exact result even in the collision region . in fig . [ fig6]b the two solitons approach each other further such that formula ( [ delmol ] ) does not account for the above - mentioned slight bounce of one of the solitons . in fig . [ fig7]b , agreement with the exact solution in the collision region is qualitative . we found that at the condition @xmath180 and for @xmath173 , the analytic curve overlaps with that of the exact solution ; apart from the horizontal segments where formula ( [ delmol ] ) diverges to @xmath59 . further insight is obtained by plotting the density profile of the soliton molecule at some specific times , as shown in figs . [ fig5]c-[fig7]c . in fig . [ fig5]c , we observe that the initial amplitude imbalance is never removed during the dynamics . instead , it becomes maximum when the two solitons are closest to each other . in addition , we notice that the oscillation amplitude of the larger soliton around its equilibrium position is larger . figure [ fig6]c shows clearly the soliton bounce which takes place in the time interval @xmath186 to @xmath187 . in these subfigures we plot two vertical dashed lines that indicate the position of the solitons at the closest approach . it is clear that after first closest approach at @xmath186 , the right soliton bounces back with a maximum displacement at @xmath188 . in fig . [ fig7]c , it is shown that , although the two solitons coalesce , two small symmetric @xmath189 appear . a detailed examination of these wings shows that they are the remnants of the two solitons after they coalesce and they both bounce back in the collision region similar to the case of fig . [ fig6 ] . it is also instructive to show the dynamics of the phase profile during the molecule s oscillation . this is shown with the contour plots of fig . [ fig8 ] which correspond to the molecules of figs . [ fig5]-[fig7 ] . in fig . [ fig8]a , which corresponds to fig . [ fig5 ] , the two solitons start initially in phase . by time the phase of the right soliton , which is the one with higher intensity amplitude and larger oscillation displacement , starts to exceed that of the left soliton . at the point of closest approach , the phase difference is exactly @xmath60 . after that point , the two solitons diverge again , the phase difference starts to decrease , and the cycle is repeated . similar behavior is seen in fig . however , in fig . [ fig8]c , where the two solitons coalesce for a considerable amount of time , the phase difference during the coalescence time is zero . it is thus not completely understood why , in this case , the two solitons still repel each other and eventually split . having established the existence of the soliton molecule from the exact two solitons solution and derived a formula that describes its bond length , here we use this formula to examine more closely the propertie of the soliton molecule and its mechanism of binding . it is clear from eq . ( [ delppmol ] ) that the sinusoidal time - dependence of the solitons relative phase leads to a force of interaction that oscillates between attractive and repulsive and hence allowing for soliton molecule formation . further details of the mechanism of binding will be uncovered by expressing the acceleration , @xmath190 in terms of @xmath2 by substituting for @xmath191 from eq . ( [ delmol ] ) into eq . ( [ delppmol ] ) , to get @xmath192 this shows that the interaction force between the two solitons is the resultant of an attractive part and a repulsive part . the equilibrium bond length , defined by @xmath193 , is given by @xmath194 in consistence with our previous result , the equilibrium bond length diverges as @xmath195 . solving the last equation for @xmath196 and then substituting in eq . ( [ delpp2mol ] ) , @xmath197 simplifies to @xmath198.\ ] ] for small amplitude oscillations , @xmath199 , the last equation gives @xmath200 the restoring force ( @xmath201 ) originates from the phase - dependent terms , @xmath202 . this appealing form of the force of interaction shows that the force between the solitons is of hooke s law type with a spring constant @xmath203 where @xmath204 is the _ bare mass _ of the molecule . expressed in terms of @xmath205 and @xmath125 , the spring constant takes the form @xmath206 which shows that @xmath207 for @xmath143 ; corresponding to a soliton molecule of infinite bond length . furthermore , @xmath208 diverges for @xmath180 , which signifies soliton coalescence , as we have pointed out in the previous subsection . since the frequency of the soliton molecule is given by @xmath209 and the spring constant is given in eq . ( [ keq ] ) , the effective mass @xmath210 will be given by @xmath211 which again diverges at the soliton coalescence condition , @xmath180 . having determined the main properties of the soliton molecule , we can now return back to eq . ( [ delmol ] ) to express @xmath151 as @xmath212 where @xmath213 is the initial value of @xmath151 , which is given by the solitons parameters through @xmath214 the amplitude of the oscillation @xmath215 is thus given by @xmath216 which gives an elastic potential energy @xmath217 it should be noted here that this is equal to the mechanical energy since the initial speed vanishes , @xmath218 . the fact that the potential energy diverges at the coalescence condition @xmath180 is a gain an artifact of the calculation , but it at least indicates that the bond is tighter than cases where @xmath176 or @xmath177 . here , we investigate the stability of the soliton molecule against break up in the following three collision regimes : @xmath219 reflection by a hard wall , @xmath220 crossing a finite potential barrier , and @xmath221 collision with a single soliton . to that end we solve the gross - pitaevskii equation , eq . ( [ gp ] ) , numerically . as an initial state , we use , for cases @xmath219 and @xmath220 the two solitons solution , eq . ( [ exactsol ] ) , which represents the soliton molecule . for case @xmath221 , we use the superposition of the exact single soliton , eq . ( [ psi1_exact ] ) , with the two solitons solution . before starting the discussion of results , we point out that in figs . [ fig9]-[fig14 ] , we present the results of this section using spacio - temporal density plots . since the solitons are too thin compared to the spacial range that we consider , a density plot with full spacial and time ranges will not show a clear solitons peak density or center - of - mass path , as fig . [ fig9]c shows . to solve this problem , we restricted the density plotting to a finite range of @xmath166 , namely between 0.025 and 0.15 corresponding to the upper part of the solitons peaks . this results in an easier tracking of both the solitons peak density and center - of - mass path , as shown in fig . [ fig9]a , b , d and the rest of subsequent figures . for reflection from a hard wall we solve eq . ( [ gp ] ) with a potential step of the form @xmath222 where @xmath223 and @xmath224 are the hight and location of the potential wall , respectively . the result of reflection from this hard wall , with @xmath225 , is shown in fig . the soliton molecule preserves its molecular structure but with different characteristics . the solitons in the reflected molecule do not coalesce as in the incident molecule . in other words , the equilibrium bond length becomes larger . the density plot shows that initially the two solitons are of comparable intensities . after reflection , the brighter color of the left soliton and darker color of the right soliton indicate that the left soliton acquires higher intensity on the expense of the right soliton . we also notice that the left soliton performs two reflections from the potential interface . after the first reflection , it collides with the right soliton and then collides with the potential interface for the second time . the picture becomes different when the hight of the wall is reduced to @xmath226 , as shown in fig . the soliton molecule breaks up after reflection . this is due to loss of energy at the interface of the potential . part of the soliton molecule transmits as a nonsolitonic pulse that broadens and decays in intensity by time . by plotting @xmath166 in fig . [ fig9]c with its full range , we can see the nonsolitonic part as the left- and right - going two red ejections corresponding to the transmitted and reflected nonsolitonic pulses , respectively . in fig . [ fig9]d , we combine fig . [ fig9]b and fig . [ fig9]c to show the locations of the nonsolitonic ejections with respect to the solitons centers . in the case of reflection from a hard wall , the nonsolitonic ejections are essentially not present which results in the stability of the molecular structure . for reflection from a potential barrier we solve eq . ( [ gp ] ) with the potential @xmath227 where @xmath223 , @xmath228 , and @xmath224 , are the height , width , and location of the right side of the barrier , respectively . in fig . [ fig10 ] , we show the many different possibilities that result when the height of the barrier is changed . the free evolution case with @xmath229 is shown as a reference plot . the full reflection case is shown for @xmath225 , which is similar to the previous case of reflection from a hard wall . reducing the height of the barrier to @xmath230 , we notice that the soliton molecule breaks up after reflection . as pointed above , this is due to the nonsolitonic ejections taking place at the interfaces of the potential . reducing the height of the barrier to @xmath231 , a sign of solitons recombining appears in the form of a soliton molecule of a short lifetime . at @xmath232 a stable molecule is remarkably formed with a considerably shorter period than for the incident molecule . we have confirmed numerically that this molecule remains stable for much longer time provided that the soliton molecule remains sufficiently far from the boundaries of the spacial grid . this unique structure remains for some small domain around @xmath232 , but is lost for @xmath233 , where the molecule breaks for long evolution times . decreasing the height of the barrier to @xmath234 , the molecule breaks at the interface and splits into a reflected and transmitted solitons . for @xmath226 , the molecule breaks at the interface , but both solitons transmit through the barrier . for @xmath235 , the transmitted solitons show a sign of recombining again but with shorter period than for the free evolution case and larger than in the case of @xmath232 . motivated by the fact that at the coalescence point the intensity of the molecule is considerably higher than at other instants , we expect to find different scattering dynamics when the soliton molecule meets the interface of the potential at different phases of its periodic oscillation . in fig . [ fig11 ] , this is investigated by fixing the height of the potential barrier at @xmath232 and changing the initial launching position of the molecule . starting at @xmath236 , the molecule breaks after reflection . a shortly - lived molecule is obtained at @xmath237 , and a stable molecule is found for @xmath238 , which corresponds to the @xmath232 case of fig . [ fig10 ] . at @xmath239 , the soliton splits at the interface into transmitted and reflected solitons . the coalescence point is , in this case , located at the interface . transmission takes place due to the high intensity of the soliton at the coalescence point . at @xmath240 , the two solitons still split as in the previous subfigure but with a weak transmitted soliton intensity less than 0.025 and hence will not be shown in our density plots which are restricted to the intensities between 0.025 and 0.15 , as pointed out previously . for @xmath241 , the coalescence point takes place before the molecule reaches the interface and both solitons reflect but the molecule breaks up . for @xmath242 the two reflected solitons start to recombine forming a stable molecule at @xmath243 . at @xmath244 the reflected molecule starts to break up since the second coalescence point becomes close to the interface . thus , the conclusion from this figure is that the soliton molecule is more vulnerable to break up when it meets the interface at the coalescence point . equivalently , soliton molecules with larger equilibrium bond length , such that coalescence does not occur , will be more stable against breakup post reflection from barriers . finally , we present in figs . [ fig12]-[fig14 ] the results of scattering of a soliton molecule by a single soliton described by eq . ( [ newsol1 ] ) with normalization @xmath245 , center - of - mass position and speed @xmath246 and @xmath247 , respectively . the effects of the phase , speed , and amplitude of the injected soliton are investigated separately . in fig . [ fig12]a , a soliton initially at @xmath248 is launched towards a stationary soliton molecule near @xmath249 . at the impact , the molecule brakes up , its right soliton is ejected in the direction of the positive @xmath15-axis , and the left soliton combines with the scatterer soliton to form a new stationary molecule shifted by about a bond length to the left . we point out here that for such an outcome to occur , it is essential that the amplitude of the scatterer soliton is nearly equal to that of the right soliton of the molecule . otherwise , a different outcome , as that of fig . [ fig14 ] , will be obtained . in fig . [ fig12]b , the same numerical experiment is repeated but with adding a @xmath60 to the phase of the scatterer soliton . clearly , this phase addition prevents the formation of a new molecule resulting in three solitons diverging from each other . from applications point of view , the phase of @xmath60 could be used as a `` key '' to `` unlock '' the molecule for the purpose of extracting stored data . in fig . [ fig13 ] , we show the effect of the initial speed of the injected soliton . in contrary to one s first judgment , the molecule preserves its structure for fast collisions , as in fig . [ fig13]a , and breaks up for slower collisions , as in fig . [ fig13]b . in fig . [ fig14 ] , the injected soliton has an amplitude that is approximately two times larger than any of the two solitons of the molecule . the injected soliton penetrates the molecule leaving it almost unchanged apart from a center - of - mass shift to the left . we have used the inverse scattering method to derive the two solitons solution of a nonlinear schr@xmath0diner equation with a parabolic potential and cubic nonlinearity with time - dependent coefficients , as given by eq . ( [ gp ] ) . the solution was then simplified and put in a suggestive form in terms of the fundamental parameters of the two solitons , namely their amplitudes , center - of - mass positions and speeds , and their phases . in this form , two different regimes of the solution , namely the resolved solitons and overlapping solitons , were distinguished and the main features such as the solitons separation and relative phase were extracted . from the expression for the solitons separation we find that for the homogeneous case and zero solitons relative speed , the solitons separation diverges logarithmically with the solitons amplitude difference such that , for equal solitons , the trivial solution is obtained . the force of interaction was then derived , essentially , for arbitrary solitons parameters . this resulted in generalizing gordon s formula @xcite to _ i _ ) the generalized inhomogeneous case considered here , _ ii _ ) arbitrary solitons relative speed and amplitudes , and _ iii _ ) short solitons separations ( compared to their width ) . with this formula , the possibility of forming soliton molecules emerged naturally , where the force at short distance was shown to be composed of an attractive part , resulting from the nonlinearity , and another part that oscillates between repulsive and attractive resulting from the time - dependent relative phase . the main features of the soliton molecule , including its equilibrium bond length and bond spring constant , frequency and amplitude of oscillation , effective mass , and its elastic potential energy , where then calculated in terms of the solitons parameters . it turns out that the amplitudes difference @xmath63 plays an important role in determining these quantities . furthermore , we show that at the condition @xmath250 , the two solitons coalesce while away from this condition , the solitons approach each other but remain resolved . at this condition , the molecule s effective mass and spring constant have maximum values . in our expressions eqs . ( [ keq ] ) and ( [ masseq ] ) diverge because these formulae were derived assuming the solitons remain resolved . to have a sense of its stability we investigated numerically _ i _ ) the collision of the soliton molecule with a hard wall and softer one , _ ii _ ) scattering by a potential barrier , and _ iii _ ) collision with a single molecule . the first case showed that while the molecular structure is preserved after reflection from a hard wall , it breaks when reflecting from a softer one . reflection from a finite barrier showed that the molecular structure is preserved only for specific heights of the barrier . for an incident molecule with the coalescence condition satisfied , the molecule will be most vulnerable to break up when the coalescence point takes place at the interface of the barrier . this is simply understood by the fact that the intensity of the soliton molecule is maximum when the two solitons coalesce such that tunneling becomes possible . stability of the molecule was also investigated by scattering the molecule with a single soliton . it turned that slower collisions tend to break up the molecule more easily than faster ones . in addition , the outcome of the collision depends on the phase of the incoming soliton such that a scatterer soliton which is in - phase with the molecule will typically preserve its molecular structure , but for an out - of - phase soliton , the molecule breaks up . the two solitons solution presented here and the analysis that shows how to extract the solitons separation and relative phase may constitute the basis for a more accurate and detailed investigation of the origin of the soliton - soliton force , especially for short separations and at coalescence . the results of this paper will be hopefully of relevance to possible future applications of soliton molecules as data carriers or memories . the author acknowledges helpful discussions with vladimir n. serkin . the lax pair associated with the gross - pitaevskii equation , eq . ( [ gp ] ) , is obtained using our lax pair search method @xcite and reads @xmath251 @xmath252 where + + @xmath253 , @xmath254 , @xmath255 , @xmath256 , + + + @xmath257 , + + + @xmath258 , @xmath259 , and @xmath260 and @xmath261 are arbitrary constants . here , @xmath6 is the solution of eq . ( [ gp ] ) and @xmath262 is the _ auxiliary _ field . the _ compatibility condition _ @xmath263 of the linear system , eqs . ( [ phix ] ) and ( [ phit ] ) , is equivalent to the gross - pitaevskii equation , eq . ( [ gp ] ) , and its complex conjugate . for a known _ seed _ solution , @xmath264 , of eq . ( [ gp ] ) the linear system will have the solution @xmath265 . the darboux transformation is defined as @xmath266={\bf\phi}\cdot{\lambda}-{\bf\sigma}\,{\bf\phi}$ ] , where @xmath266 $ ] is the transformed field and @xmath267 . requiring the linear system to be _ covariant _ under the darboux transformation imposes the transformation @xmath268={\bf p}+{\bf j}\cdot{\bf\sigma}-{\bf\sigma}\cdot{\bf j}$ ] , where @xmath268 $ ] is the transformed @xmath269 . this gives the new solution in terms of the seed solution as @xmath270 using the trivial solution @xmath271 as a seed , the darboux transformation generates the well - known sech - shaped single soliton solution @xcite @xmath272 where @xmath273 @xmath274 @xmath275 @xmath276 , @xmath277 , @xmath278 , and the constant @xmath279 corresponds to an arbitrary overall phase . this solution corresponds to a soliton density profile that is localized at @xmath37 , moving with center - of - mass speed @xmath280 , and normalized to @xmath40 . using this single soliton solution as a seed , the darboux transformation generates a two solitons solution . the solution of the linear system , eqs . ( [ phix ] ) and ( [ phit ] ) , can in this case be derived and simplified to the following form @xmath281 @xmath282 @xmath283 @xmath284 @xmath285 where , @xmath286 @xmath287 @xmath288 @xmath289 finally , the two solitons solution is obtained by substituting for @xmath290 and @xmath291 in eq . ( [ newsol1 ] ) , which upon substituting @xmath292 and further simplification can then be put in the form of eq . ( [ exactsol ] ) . since @xmath68 and @xmath69 are functions of @xmath15 , we start by examining their values near the roots of @xmath57 . this task can be simplified by rewriting @xmath68 as @xmath293 . for @xmath294 , we get @xmath295 , and for @xmath296 , we have @xmath297 . for @xmath113 the first term in the exponent of @xmath68 is larger than the second one , provided that @xmath40 is not much larger than @xmath298 , as remarked at the end of this section , hence @xmath73 . for @xmath132 the magnitude of the first term in the exponent of @xmath68 is smaller than the magnitude of the second one , which again leads to @xmath73 . for the region @xmath299 , the condition @xmath73 is always satisfied . in conclusion , @xmath73 for all @xmath15 , apart from situations with extreme values of @xmath300 . the situation is simpler for @xmath69 : @xmath74 for @xmath113 , @xmath301 for @xmath302 , and @xmath303 for @xmath132 . thus , to find @xmath50 , we expand @xmath57 in powers of large @xmath68 and @xmath69 and to find @xmath48 and @xmath52 , we expand @xmath57 in powers of large @xmath68 . expanding @xmath57 in powers of @xmath68 and @xmath69 around @xmath58 up to first order in @xmath80 and @xmath304 , we find @xmath305 where @xmath306 the root of this equation gives the center - of - mass position of the right soliton . the first two log terms equal @xmath307 . the third log term is constant and diverges for @xmath94 . the last term is small since it is proportional to @xmath80 , but it is needed because it contains the phase - dependent contributions . due to the combination of @xmath308 and @xmath68 , finding an algebraic expression for the root of this equation will not be possible . instead , we ignore at first the @xmath80 term to obtain the dominant contribution which will then be used to find the @xmath80 contribution . this gives @xmath309 substituting back in eq . ( [ q1 ] ) and solving for @xmath15 , we finally get @xmath310 where @xmath311 . for the left and central solitons , we expand @xmath57 in powers of @xmath68 ; leaving @xmath69 arbitrary . in this manner , we account for the two roots , @xmath48 and @xmath52 , simultaneously . expanding @xmath57 in powers of large @xmath68 , we get @xmath312 where @xmath313\nonumber\\&/&{\left(g_0 ^ 2 e^{2 \gamma_0 } ( { n_d}+{n_s } { y } ) ^2 + 4 ( { y } + 1)^2 { \eta_d}^2\right)}.\end{aligned}\ ] ] similar to the above procedure for @xmath50 , we solve first eq . ( [ q2 ] ) without the @xmath80 term , which gives @xmath314 - 8 { \eta_d}^2}{2 g_0 ^ 2 { n_s}^2 e^{2 \gamma_0}+8 { \eta_d}^2},\ ] ] where @xmath315 in the limit @xmath89 and @xmath90 , the solution @xmath87 approaches 0 , which corresponds to @xmath316 . the solution @xmath317 approaches 1 , which corresponds to @xmath318 . thus , the solutions @xmath87 and @xmath317 correspond the left and central solitons , respectively . since we will be interested only in the left and right solitons , we take for the rest of this section the @xmath87 . to find the contribution of the @xmath80 term , we substitute for @xmath87 in the @xmath80 term of eq . ( [ q2 ] ) , and then solve for @xmath69 to get a corrected expression for @xmath87 @xmath319 -8 { \eta_d}^2}{2 g_0 ^ 2 { n_s}^2 e^{2 \gamma_0}+8 { \eta_d}^2},\ ] ] where @xmath320 , @xmath321 , @xmath322 , @xmath323 , and @xmath324 . expanding for large @xmath325 and then solving for @xmath15 , we finally get @xmath326 final remarks about the validity of the above derivation are in order . the condition @xmath73 will be met in the region @xmath113 only for @xmath327 . note that the numerator of the right hand side of this inequality is less than the denominator by at least @xmath328 . for @xmath132 , the condition @xmath73 will be met only for @xmath329 . here , the numerator of the right hand side of this inequality is larger than the denominator by at least @xmath328 . a more quantitative estimate for the ratio @xmath300 can be obtained using the above results for @xmath330 and @xmath331 . furthermore , the above - derived formula for @xmath48 is limited to values of the parameters for which the quantities @xmath332 and @xmath333 are positive . at @xmath334 , the two roots @xmath48 and @xmath52 coincide . in fig . [ fig1 ] , this corresponds to the maximum of the @xmath57-curve lying on the @xmath15-axis . for @xmath335 , the maximum of the curve is below the @xmath15-axis and the two roots become nonreal . noting that : + @xmath341 + @xmath342 + @xmath343 + @xmath344 + @xmath345 + @xmath346 , + + the acceleration , @xmath190 , can be calculated to take the form of eq . ( [ force2 ] ) with coefficients @xmath347 @xmath348 @xmath349 @xmath350 a. hasegawa and y. kodama , _ solitons in optical communications _ ( oxford university press , new york , 1995 ) ; l.f . mollenauer and j.p . gordon , _ solitons in optical fibers _ ( acadamic press , boston , 2006 ) ; g.p . agrawal , _ nonlinear fiber optics _ ( academic press , san diego , 2001 ) , 3rd ed . ; n.n . akhmediev and a. ankiexicz , _ solitons : nonlinear pulses and beams _ ( chapman and hall , london , 1997 ) . l. khaykovich , f. schreck , g. ferrari , t. bourde , j. cubizolles , l.d . carr , y. castin , c. salomon , science * 296 * , 1290 ( 2002 ) . m. stratmann , t. pagel , and f. mitschke , phys . * 95 * , 143902 ( 2005 ) . ) , @xmath47 $ ] . thick ( green ) curve : the soliton intensity @xmath351 . light ( red ) curve : the soliton intensity with only the second sech term of eq . ( [ exactsol ] ) . the parameters used are : @xmath16 , @xmath352 , @xmath353 , @xmath354 , @xmath355 , @xmath356 , @xmath357 , @xmath358 , and @xmath167.,width=566 ] ) . dotted ( blue ) curve corresponds to the phase of the right soliton , @xmath102 , calculated from eq . ( [ phaser ] ) . thick dashed curve corresponds to the phase of the central soliton , @xmath103 , calculated from eq . ( [ phasec ] ) . dashed - dotted curve corresponds to the phase of the left soliton , @xmath104 , calculated from eq . ( [ phasel ] ) . the solid curves correspond to the density profiles as in fig . the parameters used are : @xmath16 , @xmath352 , @xmath353 , @xmath354 , @xmath355 , @xmath359 , @xmath360 , @xmath357 , @xmath358 , and @xmath167.,width=566 ] , versus @xmath124 . thick ( blue ) curve is calculated from eq . ( [ force2 ] ) . light ( red ) curve is calculated numerically from the exact two solitons solution , eq . ( [ exactsol ] ) . the parameters used are : @xmath16 , @xmath352 , @xmath353 , @xmath354 , @xmath355 , @xmath359 , @xmath357 , @xmath358 , and @xmath167.,width=566 ] ) . * ( b ) * thick ( green ) curve is solitons separation calculated from the exact solution ( [ exactsol ] ) . light ( red ) curve is the solitons separation calculated from formula ( [ delmol ] ) . * ( c ) * density profile at some specific times . the parameters used are : @xmath16 , @xmath361 , @xmath362 , @xmath363 , @xmath356 , @xmath364 , @xmath358.,width=566 ] -[fig7 ] . subfigures * ( a ) * , * ( b ) * , and * ( c ) * , correspond to fig . [ fig5 ] , fig . [ fig6 ] , and fig . [ fig7 ] , respectively . the blue ( lower ) and green ( upper ) curves correspond to the center - of - mass trajectories of the left and right solitons , respectively.,width=566 ] ) . the dashed vertical line represents the interface of the potential step . the parameters used are : @xmath364 , @xmath366 , @xmath367 , @xmath368 , @xmath369 , @xmath370 , @xmath371 . * ( a ) * @xmath225 . * ( b),(c),(d ) * @xmath226 . ,
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the problem of soliton - soliton force is revisited . from the exact two solitons solution of a nonautonomous gross - pitaevskii equation
, we derive a generalized formula for the mutual force between two solitons .
the force is given for arbitrary solitons amplitude difference , relative speed , phase , and separation .
the latter allows for the investigation of soliton molecules formation , dynamics , and stability .
we reveal the role of the time - dependent relative phase between the solitons in binding them in a soliton molecule .
we derive its equilibrium bond length , spring constant , frequency , effective mass , and binding energy of the molecule .
we investigate the molecule s stability against perturbations such as reflection from surfaces , scattering by barriers , and collisions with other solitons .
| 18,432 | 211 |
since its discovery by kageyama _ et al_.@xcite , the spin dimer compound srcu@xmath4(bo@xmath5)@xmath4 has attracted much attention as a suitable material for frustrated spin systems in low dimension . srcu@xmath4(bo@xmath5)@xmath4 exhibits various interesting phenomena , such as a quantum disordered ground state @xcite and a complex shape of magnetization curve@xcite , because of its unique crystal structure . in consideration of the structure , miyahara and ueda suggested that the magnetic properties of the spin dimer compound srcu@xmath4(bo@xmath5)@xmath4 can be described as a spin-@xmath6 two - dimensional ( 2d ) orthogonal - dimer model @xcite , equivalent to the shastry - sutherland model on square lattice with some diagonal bonds @xcite . the ground state of the shastry - sutherland model in dimer phase is exactly represented by a direct product of singlets . the low - energy dispersions possess six - fold degeneracy and are almost flat reflecting that the triplet tends to localize on vertical or horizontal bonds . recent experiments by esr @xcite and neutron inelastic scattering ( nis ) have observed splitting of degenerate dispersions of srcu@xmath4(bo@xmath5)@xmath4 , which can not be explained by the _ isotropic _ shastry - sutherland model . hence c ' epas _ et al . _ pointed out that the dzyaloshinski - moriya ( dm ) interaction @xcite must be added between vertical and horizontal dimers in the isotropic shastry - sutherland model in order to explain the splitting . @xcite in this paper , as a simple model to clarify effects of the dm interaction to low - energy excitations in orthogonal - dimer systems , one - dimensional ( 1d ) orthogonal - dimer model with the dm interaction is studied by using the perturbation theory and the numerical exact - diagonalization method . in the absence of the dm interactions , properties of ground state , low - energy excitations , and magnetization processes of the 1d orthogonal dimer model has been studied by several authors . the hamiltonian of the 1d orthogonal - dimer model with the dm interaction is given by @xmath7 where @xmath8 here @xmath9 is the number of unit cells in the system , as shown by a broken rectangle in fig . the unit cell includes two dimers along vertical and horizontal direction , which are designated by the index , @xmath10 and @xmath11 , respectively . @xmath12 ( @xmath13 and @xmath14 ) denotes a spin-@xmath6 operator on @xmath15-spin in @xmath10-th dimer . @xmath16 and @xmath17 severally indicate the exchange coupling in intra - dimer and in inter - dimer . due to the structure of system , the dm exchange interaction , @xmath18 , exists only on inter - dimer bonds and has only a component perpendicular to two kinds of dimer in the unit cell . the periodic boundary condition is imposed to the system . , that is @xmath19 . the unit cell includes a vertical and horizontal dimer . the former dimers are at @xmath10-site and the latter at @xmath20-site.,width=283 ] in this section , let us discuss the ground state and low - energy excitations of the 1d orthogonal dimer model with the dm interaction . we can expect that the ground state is in the dimer phase in the limit of strong intra - dimer coupling ( @xmath21 ) , even when the dm interaction is switched on the isotropic system . therefore , it is reasonable to treat the intra - dimer hamiltonian ( [ eq : intra ] ) as an unperturbated one and the others as perturbation . the inter - dimer interaction @xmath17 creates two adjacent triplets from a pair of a singlet and triplet and vice versa , and besides shows scatterings between two triplets . the dm interaction not only causes the former process but also creates or annihilates two adjacent singlets . therefore the dm interaction can play a crucial role in the ground state and the low - energy excitations in the dimer phase . first , we discuss the ground - state energy of hamiltonian ( [ eq : hamiltonian ] ) . in the absence of the dm interaction , the ground state is exactly represented by a direct product of singlets and its energy is given as @xmath22 . on the other hands , the ground - state energy of total hamiltonian ( [ eq : hamiltonian ] ) is estimated as @xmath23 from the perturbation expansion up to the third order in @xmath24 and @xmath25 . the result means that the ground state can not be exactly described by the direct product of singlets owing to the dm interaction . next , we argue the low - energy excitations in the system . since the ground state belongs to the dimer phase in the region of strong-@xmath16 , the lowest excited states will be well described by @xmath26 here , @xmath27 and @xmath28 are the total magnetization and the wave number , respectivery . @xmath29 and @xmath30 in ket severally denote a singlet and a triplet with @xmath31 at @xmath10-site and , @xmath32 ( @xmath33 ) is defined as an operator to create a triplet propagating on vertical ( horizontal ) dimers . by using two states of eqs . ( [ eq : vfourier ] ) and ( [ eq : pfourier ] ) , the hamiltonian ( 1 ) is projected on following ( @xmath34)-matrix : @xmath35 where @xmath36,~ { \mbox{\boldmath $ v$}}_m(k)\equiv \left [ \begin{array}{c } t_{m , k}^{\rm ver } \\ t_{m , k}^{\rm hor } \\ \end{array } \right].\end{aligned}\ ] ] the eq . ( [ eq : hm ] ) for @xmath1 has no off - diagonal elements within perturbation up to the third order . therefore the excitation energies for @xmath1 are given by @xmath37 in contrast to the 2d orthogonal dimer model , two excitation energies , @xmath38 and @xmath39 , split in the case of 1d system . it is also interesting to note that the curvature of @xmath39 appears in the third ordered correction in eq . ( [ eq : excitede1 ] ) . on the other hand , the projected hamiltonian with @xmath40 has an off - diagonal element . the perturbation calculation up to the third order leads to the matrix : @xmath41 , \label{eq : apm1}\end{aligned}\ ] ] where @xmath42 by diagonalizing eq . ( [ eq : apm1 ] ) , the excitation energies with @xmath40 are obtained as @xmath43 the curvature of @xmath44 is dominant by the first ordered correction with regard to @xmath25 in the off - diagonal element @xmath45 . the correction derives from the scattering between a singlet and a triplet with @xmath2 due to the dm interaction . subtracting the ground - state energy of eq . ( [ eq : grounde ] ) from excited - state energies of eq . ( [ eq : excitede0 ] ) , ( [ eq : excitede1 ] ) , and ( [ eq : excitede3 ] ) , the low - energy dispersions , @xmath46 , are estimated as @xmath47 figure 2 shows the low - energy dispersions for @xmath48 and @xmath49 . the low energy spectra @xmath50 and @xmath51 are severally represented by the lower and upper solid lines , and then the upper and lower dotted lines denote @xmath52 and @xmath53 in eqs . ( [ eq : omega2 ] ) . the full and open circles represent the low - energy spectra with @xmath1 and @xmath2 . the perturbation theory is in agreement with the numerical diagonalization , as to low - energy excitations . the dispersions with same @xmath31 are not degenerate , which happens even if the dm interaction is not taken account of . therefore the inter - dime coupling @xmath17 is also important for splitting of the low - energy dispersions with same @xmath31 . this is because the parity on the vertical dimer is conserved in the one - dimension system without the dm interaction . the dm interaction not only splits into branches with @xmath1 and @xmath2 , but also makes triplets move more easily . and @xmath49 . the solid and dotted curves are of @xmath1 and @xmath40 , respectively . the full ( open ) circles indicates the excitation energies for @xmath1 ( @xmath2 ) calculated by numerical diagonalizations.,width=264 ] we investigated the low - energy excitations in the 1d orthogonal - dimer model with the dm interaction using the perturbation theory and numerical exact - diagonalization method . the dm interaction allows a triplet to propagate in singlet sea as seen in the 2d system @xcite , while the triplet is localized on vertical or horizontal dimer in the absence of the dm interaction . this curvature effect happens in a two - dimensional orthogonal - dimer model , but the splitting of spectra reflects a stringent constraint for the motion of a triplet due to one dimensionality as well as the dm interaction . we gratefully acknowledge helpful discussions and comments with tetsuro nikuni , masaaki nakamura , and takahiro yamamoto on several points in this work . we would also like to thank k@xmath54ichir@xmath54 ide for valuable advice on numerical techniques . 9 h. kageyama , et al . * 82 * ( 1999 ) 3168 . k. onizuka , et al . : j. phys . * 69 * ( 2000 ) 1016 . s. miyahara and k. ueda : phys . 87 * ( 2001 ) 3701 . b. s. shastry and b. sutherland : physica b * 108 * ( 1981 ) 1089 . h. nojiri , et al . : j. phys . * 68 * ( 1999 ) 2906 . i. dzyaloshinski : j. phys . solids * 4 * ( 1958 ) 241 ; + t. moriya : phys . rev . * 120 * ( 1960 ) 91 . o. cpas , et al . : phys . rev . lett . * 87 * ( 2001 ) 167205 . a. koga , k. okunishi and n. kawakami : phys . b. * 62 * ( 2000 ) 5558 . n. b. ivanov and j. richter : phys . a * 232 * ( 1997 ) 308 . j. richter , n. b. ivanov and j. shulenburg : phys . * 88 * ( 2002 ) 201601 . s. miyahara and k. ueda : j. phys . ( _ suppl . _ ) b * 70 * ( 2001 ) 180 .
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effects of the dzyaloshinski - moriya ( dm ) interaction on low - energy excitations in a one - dimensional orthogonal - dimer model are studied by using the perturbation expansions and the numerical diagonalization method . in the absence of the dm interaction ,
the triplet excitations show two flat spectra with three - fold degeneracy , which are labeled by magnetization @xmath0 . these spectra split into two branches with @xmath1 and with @xmath2 by switching - on of the dm interaction and besides the curvature appears in the triplet excitations with @xmath3 more strongly than those of @xmath1 . ,
, quantum spin ; low - energy excitation ; dzyaloshinski - moriya interaction ;
| 3,201 | 207 |
the parton model , according to which hadrons consist of quarks antiquarks and gluons ( partons ) , bound together in different ways , has been very successful in reproducing experiment . this provides a relatively explicit and transparent technique for the description of high energy particle interactions . the distributions of partons inside hadrons are characterized by the structure functions satisfying the dokshitzer gribov lipatov altarelli parisi ( dglap ) equations @xcite or ones that are basically similar . such a function is the probability density of finding a parton in a hadron carrying a fraction of the total hadron s momentum . numerical solutions of the equations are in a remarkable agreement with experimental measurements , especially for the nucleon @xcite . photons being involved in high energy interactions are also able to manifest hadronic structure . one can intuitively comprehend this since the photon directly couples to quarks and therefore may split into quark antiquark pairs . the parton contributions in two - photon processes and some crucial peculiarities of the kinematic behavior of the photon structure function have been described by walsh and zerwas @xcite . the first work in studying quantum - chromodynamics corrections to the naive pointlike structure of the photon belongs to witten @xcite . this problem was also studied in refs . @xcite . introducing the evolution equations , similar to the dglap ones , for photons as well as the properties of the corresponding solutions were under scrutiny , for instance , in a series of papers by glk , reya , grassie and vogt @xcite . a formulation of high energy @xmath2 interactions taking into account the hadronic properties of the photon was proposed in ref . @xcite . today , we possess convincing experimental evidence for the fact that the real photon has non - trivial parton structure @xcite . on the other hand , the cosmic microwave background ( cmb ) photons may play an important role in the formation of cosmic rays ( cr ) . one of the brightest representatives is the greisen zatsepin kuzmin ( gzk ) limit on the energy of cr @xcite . for example , protons of energies of over about @xmath3 ev would be decelerated by interaction with the cmb photons , mostly due to resonant pion production , @xmath4 . other interesting processes , the @xmath5 reactions and their possible astrophysical implications , were extensively discussed in the literature ( see , _ e.g. _ , refs . @xcite and the references cited therein ) . in this context , to invoke the parton content of the real photon could be convenient for calculations of the probabilities of such processes . here , we attempt to show the example of @xmath0 boson production in the @xmath5 scattering which may have important consequences for astrophysics @xcite . studying this reaction could also provide a test of the universality of the parton distribution functions of the photon . let us first consider inclusive on - shell @xmath0 boson production in the reaction @xmath6 at the resonance region using the parton language . we will view it from the center - of - mass ( cms ) frame of the @xmath7 system . here , for example , a substantial fraction of the cmb photons will be of energies of about @xmath8 , where @xmath9 is the neutrino energy in the laboratory frame defined as the frame in which the cmb is isotropic , @xmath10 is the cmb photon energy ( typical value @xmath11 ev @xcite ) . this reaction is standardly factorized into two subprocesses : the emission of a positron by the photon and annihilation of the neutrino with the positron into @xmath0 ( see fig . then the corresponding cross section may be written as here @xmath13 is the total cms energy squared ( @xmath14 ) , @xmath15 is the probability density function to find the positron in the photon carrying the fraction @xmath16 of the total photon s momentum , and @xmath17 is the cross section of the annihilation subprocess . note that we explicitly write the @xmath13 dependence of the function instead of the more traditional @xmath18 one ( 4-momentum transfer squared ) since we deal with an @xmath13-channel subprocess . where @xmath20 is the mass of the @xmath0 boson , @xmath21 is the partial width of the initial channel ( the partial width for the decay @xmath22 ) , and @xmath23 is the total decay width of @xmath0 . in the leading order one can find that @xcite to determine the function @xmath15 , we adopt the formalism given in ref . it is fair to expect @xmath15 to satisfy , up to factors associated with the quark colors and fractional electric charges , the same evolution equation as the quark distributions in the photon do , provided the gluons are excluded and one takes into account only the electromagnetic interaction . then , in the leading order we write the following equation for the positron distribution @xcite : the dependence of the cross section on @xmath13 is displayed in fig . 2a in comparison with calculations of the closely related process @xmath32 carried out by seckel @xcite . here @xmath33 gev , @xmath34 gev@xmath35 , @xmath36 @xcite . one can see that the values given by eq . ( [ eq:6 ] ) are about two times higher than those of ref . @xcite . let us turn now to the charged current interaction of the neutrino with the quark content of the photon ( see fig . the corresponding cross section can be obtained in the same way as it is done for neutrino proton scattering @xcite : where @xmath39 is the probability density to find a quark @xmath40 ( antiquark @xmath41 ) in the photon carrying the fraction @xmath16 of the total photon s momentum . taking into account only the densities of the lightest quarks @xmath42 and @xmath43 from ref . @xcite , we found that here @xmath45 and @xmath46 are the electric charge and mass of the quark @xmath40 respectively ( for antiquarks the equation is analogous ) . note that eq . ( [ eq:9 ] ) is valid in the limit @xmath47 . we set @xmath48 gev and @xmath49 . the dependence of the cross section thus determined on @xmath13 in the range 20 gev@xmath50 @xmath51 1000 gev@xmath50 at some values of @xmath18 is shown in fig . this reaction may have interesting astrophysical implications because the struck quark may fragment into hadrons . the latter can be highly boosted and on decaying ( if unstable ) may produce particles with energies exceeding their gzk limit . if it occurs in the vicinity of the earth the decay products may reach us without significant energy loss , provided the incident quark momentum pointed in the direction of the earth . a similar idea has been proposed , for example , in ref . @xcite , when photons would appear beyond the gzk limit from decays of highly boosted @xmath52 , which , in turn , were the decay products of real @xmath53 bosons excited in @xmath54 annihilation ( the so - called `` @xmath55-burst '' mechanism ) . but there are problems here , mainly associated with the origin of such high energy neutrinos , @xmath56 ( see , _ e.g. _ , ref . @xcite ) . in our case , the minimal neutrino energy required to produce hadrons is smaller than the latter one by about a factor of 400 , and the corresponding cross section is also suppressed by a factor @xmath57 . anyway , one may expect that these processes were important for high energy neutrino absorption in the early universe . throughout this paper we implicitly used the assumption that the parton distributions are process independent , which has been experimentally justified for the nucleon . for example , the functions phenomenologically derived from electron nucleon and neutrino nucleon deep inelastic scattering data are close to each other . using them one can correctly predict the probabilities of inclusive production of @xmath58 pairs in @xmath59 collisions ( drell yan process ) @xcite . we have discussed only the @xmath7 interactions . meanwhile , all the things we said above may be straightforwardly applied to the reactions involving the antineutrino . likewise , heavier charged leptons can be considered . one may also include neutral current interactions in the neutrino quark scattering . other processes involving the cmb photons can be treated in similar way . 99 v. n. gribov , l. n. lipatov , sov . j. nucl . phys . * 15 * , 438 ( 1972 ) . g. altarelli , g. parisi , nucl . b * 126 * , 298 ( 1977 ) . y. l. dokshitzer , sov . jetp * 46 * , 641 ( 1977 ) . zeus collaboration , z. phys . c * 72 * , 399 ( 1996 ) . t. f. walsh and p. zerwas , phys . lett . b * 44 * , 195 ( 1973 ) . e. witten , nucl . b * 120 * , 189 ( 1977 ) . r. j. dewitt _ et al . _ , phys . d * 19 * , 2046 ( 1979 ) . w. a. bardeen , a. j. buras , phys . d * 20 * , 166 ( 1979 ) . d. w. duke and j. f. owens , phys . d * 22 * , 2280 ( 1980 ) . m. glk and e. reya , phys . d * 28 * , 2749 ( 1983 ) . m. glk , k. grassie , and e. reya , phys . d * 30 * , 1447 ( 1984 ) . m. glck , e. reya , and a. vogt , phys . d * 45 * , ( 1992 ) 3986 . g. a. schuler and t. sjstrand , phys . b * 300 * , 169 ( 1993 ) . j. cvach , nucl . b ( proc . suppl . ) * 79 * , 501 ( 1999 ) . k. greisen , phys . * 16 * , 748 ( 1966 ) . g. t. zatsepin and v. a. kuzmin , sov . jetp lett . * 4 * , 78 ( 1966 ) . d. seckel , phys . lett . * 80 * , 900 ( 1998 ) . a. abbasabadi _ et al . _ , d * 59 * , 013012 ( 1999 ) . a. abada , j. matias , r. pittau , nucl . phys . b * 543 * , 255 ( 1999 ) . e. masso , f. rota , phys . b * 488 * , 326 ( 2000 ) . e. v. bugaev , int . j. mod . a * 20 * , 6909 ( 2005 ) . m. haghighat , m. m. ettefaghi , m. zeinali , phys . d * 73 * , 013007 ( 2006 ) . t. k. gaisser , f. halzen , t. stanev , phys . * 258 * , 173 ( 1995 ) . p. bhattacharjee , g. sigl , phys . rept . * 327 * , 109 ( 2000 ) . _ , _ particles and nuclei . an introduction to the physical concepts _ , 5th edn . ( springer , berlin , heidelberg 2006 ) . l. n. lipatov , _ standard model of elementary particle interactions _ ( lectures given at saint petersburg state university , 2006 ) . w. furmanski and r. petronzio , phys . b * 97 * , 437 ( 1980 ) . e. g. floratos , c. kounnas , and r. lacaze , nucl . b * 192 * , 417 ( 1981 ) . chen and p. zerwas , phys . d * 12 * , 187 ( 1975 ) . ( particle data group ) , j. phys . g * 33 * , 1 ( 2006 ) . t. j. weiler , astropart . phys . * 11 * , 303 ( 1999 ) . f. e. close , _ an introduction to quarks and partons _ ( academic press , london 1979 ) .
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we possess convincing experimental evidence for the fact that the real photon has non - trivial parton structure .
on the other hand , interactions of the cosmic microwave background photons with high energy particles propagating through the universe play an important role in astrophysics . in this context , to invoke the parton content could be convenient for calculations of the probabilities of different processes involving these photons . as an example , the cross section of inclusive resonant @xmath0 boson production in the reaction @xmath1 is calculated by using the parton language .
neutrino
photon deep inelastic scattering is considered .
| 3,543 | 147 |
let @xmath5 be the quantum enveloping algebra associated to an affine algebra @xmath6 without derivation . let @xmath7 be finite - dimensional @xmath5-modules . suppose @xmath8 is irreducible and @xmath7 have crystal bases @xmath9 . then it is known @xcite that there exists a unique map @xmath0 from @xmath10 to @xmath11 commuting with any crystal operators @xmath12 and @xmath13 . there also exists an integer - valued function @xmath14 on @xmath10 , called energy function , satisfying a certain recurrence relation under the action of @xmath12 ( see ) . combinatorial @xmath0-matrices or energy functions play an important role in the affine crystal theory . in the kyoto path model @xcite , that realizes the affine highest weight crystal in terms of a semi - infinite tensor product of perfect crystals , the energy function is an essential ingredient for the computation of the affine weight . in the box - ball system @xcite or its generalizations @xcite in the formulation of crystal bases , the time evolution of the system is defined by using the combinatorial @xmath0-matrix . energy functions are also crucial in the calculation of conserved quantities . in @xcite a new connection was revealed between the energy function and the kkr or kss bijection @xcite that gives a one - to - one correspondence between highest weight paths and rigged configurations . recently , for all nonexceptional affine types , all kr crystals , crystal bases of kirillov reshetikhin ( kr ) modules ( if they exist ) , were shown to exist and their combinatorial structures were clarified @xcite . hence , it is natural to consider the problem of obtaining a rule to calculate the combinatorial @xmath0-matrix and energy function . in this paper , for type @xmath2 we calculate the image of the combinatorial @xmath0-matrix for any classical highest weight element in the tensor product of kr crystals @xmath1 ( theorem [ th : main ] ) . ( note that the first upper index of the second component is 1 . ) we also obtain the value of the energy function for such elements . although we get the rule only for highest weight elements , there is an advantage from the computational point of view , since it is always easy to calculate the action of crystal operators @xmath15 for @xmath16 not only by hand but also by computer . to identify highest weight elements in the image @xmath4 the notion of @xmath3-diagrams , introduced in @xcite , is used effectively . the paper is organized as follows . in section 2 we briefly review crystals and @xmath3-diagrams . in section 3 we recall the kr crystal @xmath17 for type @xmath18 and @xmath19 , and the notion of combinatorial @xmath0-matrix and energy function . the condition for an element of @xmath1 or @xmath4 to be classically highest is also presented . the main theorem is given in section 4 . in section 5 we prove a special case of the theorem , and reduction to this case is discussed in section 6 according to whether @xmath20 is odd or even . mo was supported by grant jsps 20540016 . the work of rs is supported by the core research for evolutional science and technology of japan science and technology agency . let @xmath6 stand for a simple lie algebra or affine kac moody lie algebra with index set @xmath21 and @xmath22 the corresponding quantized enveloping algebra . axiomatically , a @xmath6-crystal is a nonempty set @xmath23 together with maps @xmath24 where @xmath25 is the weight lattice associated to @xmath6 . the maps @xmath12 and @xmath13 are called kashiwara operators and @xmath26 is the weight function . to each crystal one can associate a crystal graph with vertices in @xmath23 and an arrow colored @xmath27 from @xmath28 to @xmath29 if @xmath30 or equivalently @xmath31 . for @xmath32 and @xmath27 , let @xmath33 in this paper we only consider crystal bases coming from @xmath22-modules . for a complete definition of crystal bases see for example @xcite . let @xmath34 be crystals . then @xmath35 can be endowed with the structure of crystal . the actions of kashiwara operators and the value of the weight function are given by @xmath36 the multiple tensor product is defined inductively . in order to compute the action of @xmath15 on multiple tensor products , it is convenient to use the rule called signature rule " @xcite . let @xmath37 be an element of the tensor product of crystals @xmath38 . one wishes to find the indices @xmath39 such that @xmath40 to do it , we introduce ( @xmath41-)signature by @xmath42 we then reduce the signature by deleting the adjacent @xmath43 pair successively . eventually we obtain a reduced signature of the following form . @xmath44 then the action of @xmath12 ( resp . @xmath13 ) corresponds to changing the rightmost @xmath45 to @xmath46 ( resp . leftmost @xmath46 to @xmath45 ) . if there is no @xmath45 ( resp . @xmath46 ) in the signature , then the action of @xmath12 ( resp . @xmath13 ) should be set to @xmath47 . the value of @xmath48 ( resp . @xmath49 ) is given by the number of @xmath45 ( resp . @xmath46 ) in the reduced signature . consider , for instance , an element @xmath50 of the 3 fold tensor product @xmath51 . suppose @xmath52 . then the signature and reduced one read @xmath53 thus we have @xmath54 we denote by @xmath55 the highest weight crystal of highest weight @xmath56 , where @xmath56 is a dominant integral weight . let @xmath57 with @xmath27 be the fundamental weights associated to a simple lie algebra . in this paper , we consider the types of @xmath58 and @xmath59 . as usual , a dominant integral weight @xmath60 is identified with a partition or young diagram with columns of height @xmath61 for @xmath62 , except when @xmath63 is a spin weight , namely , @xmath64 for type @xmath65 and @xmath66 and @xmath64 for type @xmath59 . to represent elements of @xmath55 we use kashiwara nakashima ( kn ) tableaux , a generalization of semistandard young tableaux for type @xmath67 . for kn tableaux refer to @xcite . ( see also @xcite for a summary . ) contrary to the original one , we use the french notation where parts are drawn in increasing order from top to bottom . to calculate the actions of @xmath15 on a kn tableau it is convenient to use so - called the japanese reading word of a tableau . for a kn tableau @xmath68 move from right to left and on each column move from bottom to top . during this process we read letters , thereby obtaining a word @xmath69 . a letter can be identified with an element of @xmath70 , crystal of the vector representation . hence @xmath69 can be viewed as an element of @xmath71 with @xmath72 being the number of nodes in @xmath68 or length of @xmath69 . then the action of @xmath12 or @xmath13 is calculated by using the signature rule . we still need to remember the crystal graph of @xmath70 for type @xmath73 , but it is easy as described in @xcite . let @xmath74 be @xmath58 or @xmath59 . for a subset @xmath75 , we say that @xmath32 is @xmath76-highest if @xmath77 for all @xmath78 . we set @xmath79 . we describe @xmath76-highest elements in terms of a notion of @xmath3-diagram @xcite . a @xmath3-diagram @xmath25 of shape @xmath80 is a sequence of partitions @xmath81 such that @xmath82 and @xmath83 are horizontal strips . we depict this @xmath3-diagram by the skew tableau of shape @xmath80 in which the cells of @xmath83 are filled with the symbol @xmath46 and those of @xmath82 are filled with the symbol @xmath45 . write @xmath84 and @xmath85 for the outer and inner shapes of the @xmath3-diagram @xmath25 . for type @xmath86 we have a further requirement : the outer shape @xmath56 contains columns of height at most @xmath87 , but the inner shape @xmath88 is not allowed to be of height @xmath87 ( hence there are no empty columns of height @xmath87 ) . as we have discussed we identify a young diagram with a weight . @xcite [ p : branch ] let @xmath56 be an @xmath74 weight that does not contain spin weights . then there is an isomorphism of @xmath89-crystals @xmath90 that is , the multiplicity of @xmath91 in @xmath92 , is the number of @xmath3-diagrams of shape @xmath80 . there is a bijection @xmath93 from @xmath3-diagrams @xmath25 of shape @xmath80 to the set of @xmath76-highest elements @xmath28 of @xmath89-weight @xmath94 . for any columns of height @xmath87 containing @xmath46 , place a column @xmath95 otherwise , place @xmath96 in all positions in @xmath25 that contain a @xmath45 , and fill the remainder of all columns by strings of the form @xmath97 . we move through the columns of @xmath28 from top to bottom , left to right . each @xmath46 in @xmath25 ( starting with the leftmost moving to the right ignoring @xmath46 at height @xmath87 ) will alter @xmath28 as we move through the columns . suppose the @xmath46 is at height @xmath98 in @xmath25 . if one encounters a @xmath96 , replace @xmath96 by @xmath99 . if one encounters a @xmath100 , replace the string @xmath97 by @xmath101 . [ ex : pm - diag ] let us consider the following @xmath3-diagram . @xmath102 to obtain @xmath103 we first draw the tableau @xmath104 reading from left there are @xmath46 s at height 4,3,2,2,1 . each @xmath46 alter the above tableau as follows . the 1st @xmath46 changes the first column as @xmath105 ( reading from bottom ) , the 2nd and 3rd change the second column as @xmath106 , the 4th changes the third column as @xmath107 and the 5th changes the fourth column as @xmath108 . therefore , @xmath103 is given by @xmath109 for a word @xmath110 let @xmath111 . we use this convention also for @xmath112 . note that the order in @xmath113 is reversed from @xmath114 . next proposition shows how we get to the highest element from a @xmath3-diagram by applying @xmath12 s . [ prop : to highest ] let @xmath25 be a @xmath3-diagram whose outer shape has depth @xmath20 . suppose @xmath115 for @xmath65 , @xmath116 for @xmath86 , @xmath117 for @xmath59 . let @xmath118 be the number of columns of the outer shape with height @xmath41 . let @xmath119 ( resp . @xmath120 ) be the number of @xmath45 ( resp . @xmath46 ) at height @xmath41 . define a word @xmath114 by @xmath121 where @xmath122 where @xmath123 for @xmath65 , @xmath124 for the other cases , and @xmath125 for @xmath59 , @xmath126 for the other cases . then @xmath127 is the hightest weight element with highest weight given by its outer shape . moreover , at each step when we apply @xmath128 or @xmath129 , including @xmath130 , the action is maximal , namely , if we apply @xmath131 or @xmath132 , the outcome turns out @xmath133 . suppose @xmath134 for type @xmath86 . we first prove the claim when there is no @xmath46 in @xmath25 . set @xmath135 . then the japanese reading word of the tableau corresponding to @xmath25 is given by @xmath136 the 1-signature is just given by @xmath137 , where @xmath138 , and there is no need to reduce . hence one can apply @xmath139 . calculating similarly for @xmath140 one always has a simple @xmath41-signature of the form @xmath141 , and we arrive at the highest weight element as desired . next we consider the general case . we prove by induction on @xmath72 , the number of @xmath46 . if @xmath142 , the claim is proven . suppose @xmath143 and let @xmath98 be the height of the lowest @xmath46 in @xmath25 . let @xmath144 be the same @xmath3-diagram as @xmath25 except that there are one less @xmath46 s at height @xmath98 . compare the japanese reading word of the corresponding tableaux of @xmath25 and @xmath144 . the difference is : @xmath145 apart from this difference in two words , there are subwords of the form @xmath146 or letters @xmath96 on the left and subwords of the form @xmath147 or @xmath148 for some @xmath149 on the right . let us calculate the 1-signatures of both words . they are @xmath150 for @xmath25 and @xmath151 for @xmath144 . ( there are no @xmath43 pairs . ) after applying @xmath152 on both @xmath25 and @xmath144 , the 2-signatures also turn out of the form @xmath153 for @xmath25 and @xmath154 for @xmath144 . the difference is that there is @xmath155 or @xmath99 in @xmath25 but @xmath156 or @xmath157 in @xmath144 . similar situations continue until we apply @xmath158 , and after applying @xmath159 , the two results coincide . hence we should have the desired result . the proof in the case of @xmath160 for type @xmath86 is almost the same . the only difference is that we first treat the case when there is no @xmath46 in @xmath25 except at height @xmath87 , since there is no empty column of height @xmath87 . hence we omit the proof . for a @xmath3-diagram given in example [ ex : pm - diag ] set @xmath161 . then , according to the previous proposition @xmath127 is a highest weight element . later in this paper we will need to apply @xmath162 to a @xmath3-diagram @xmath25 . since @xmath163 is no longer @xmath76-highest , we have to use a pair of @xmath3-diagrams @xmath164 to consider @xmath165-highest elements . namely , @xmath25 represents a @xmath76-highest element and @xmath166 represents a @xmath165-highest element in the @xmath89-component whose highest weight vector correponds to @xmath25 . under this bijection we identify a @xmath165-highest element @xmath28 with a pair of @xmath3-diagram @xmath164 . to describe the action of @xmath162 on @xmath164 perform the following algorithm : 1 . successively run through all @xmath46 in @xmath166 from left to right and , if possible , pair it with the leftmost yet unpaired @xmath46 in @xmath25 weakly to the left of it . 2 . successively run through all @xmath45 in @xmath166 from left to right and , if possible , pair it with the rightmost yet unpaired @xmath45 in @xmath25 weakly to the left . 3 . successively run through all yet unpaired @xmath46 in @xmath166 from left to right and , if possible , pair it with the leftmost yet unpaired @xmath45 in @xmath166 . * lemma 5.1 ) [ prop : e1 action ] if there is an unpaired @xmath46 in @xmath166 , @xmath162 moves the rightmost unpaired @xmath46 in @xmath166 to @xmath25 . else , if there is an unpaired @xmath45 in @xmath25 , @xmath162 moves the leftmost unpaired @xmath45 in @xmath25 to @xmath166 . else @xmath162 annihilates @xmath164 . let @xmath6 be an affine lie algebra of type @xmath167 , @xmath168 , or @xmath169 with the underlying finite - dimensional simple lie algebra @xmath170 of type @xmath171 , or @xmath86 , respectively . we label the vertices of the corresponding dynkin diagram according to @xcite , so the index set of @xmath6 ( resp . @xmath170 ) is @xmath172 ( resp . @xmath173 ) . in this section we review kr crystals @xmath17 of type @xmath6 given in @xcite for @xmath174 and @xmath175 for @xmath167 , @xmath176 for @xmath168 and @xmath177 for @xmath169 . as an @xmath74-crystal , @xmath17 is given by @xmath178 here @xmath55 is the @xmath74-crystal of highest weight @xmath56 and the sum runs over all dominant weights @xmath56 that can be obtained from @xmath179 by the removal of vertical dominoes , where @xmath57 are the @xmath41-th fundamental weights of @xmath74 . in order to define the actions of @xmath180 and @xmath181 we first consider an automorphism @xmath182 on the kr crystal @xmath17 . the dynkin diagrams of type @xmath183 , and @xmath169 all have an automorphism interchanging nodes @xmath133 and @xmath184 . @xmath182 corresponds to this dynkin diagram automorphism . by construction @xmath182 commutes with @xmath185 and @xmath186 for @xmath187 . hence it suffices to define @xmath182 on @xmath76-highest elements . because of the bijection @xmath188 from @xmath3-diagrams to @xmath76-highest elements as described in section [ subsec : pm diag ] , it suffices to define the corresponding map @xmath189 on @xmath3-diagrams . let @xmath25 be a @xmath3-diagram of shape @xmath80 . let @xmath190 be the number of columns of height @xmath41 in @xmath94 for all @xmath191 with @xmath192 . if @xmath193 , then in @xmath25 , above each column of @xmath94 of height @xmath41 , there must be a @xmath46 or a @xmath45 . interchange the number of such @xmath46 and @xmath45 symbols . if @xmath194 , then in @xmath25 , above each column of @xmath94 of height @xmath41 , either there is no sign or a @xmath195 pair . suppose there are @xmath196 @xmath195 pairs above the columns of height @xmath41 . change this to @xmath197 @xmath195 pairs . the result is @xmath198 , which has the same inner shape @xmath94 as @xmath25 but a possibly different outer shape . let @xmath199 and @xmath200 be such that @xmath201 is a @xmath76-highest element . then , @xmath202 is given by @xmath203 where @xmath204 . the affine crystal operators @xmath205 and @xmath206 are then defined as @xmath207 if @xmath208 , the structure of the kr crystal turns out simple . an crystal element of @xmath209 can be identified with one - row kn tableau of length @xmath210 with letters from @xmath211 ( @xmath212 ) ( and @xmath133 for @xmath168 ) . denoting the number of letters @xmath211 or @xmath133 by @xmath213 or @xmath214 , we have the so - called coordinate representation of @xmath209 @xcite . @xmath215 the action of @xmath15 for @xmath216 can be calculated as we explained in the last paragraph of section [ subsec : crystals ] . the action of @xmath217 is given by @xmath218 we list the values of @xmath219 below . @xmath220 where @xmath221 . let us now consider a tensor product of kr crystals @xmath222 . it is known @xcite that there exists a unique bijection @xmath0 , called combinatorial @xmath0-matrix , commuting with kashiwara operators @xmath15 for any @xmath223 . since @xmath0 preserves the weight , @xmath224 should be sent to @xmath225 by @xmath0 , where @xmath226 ( resp . @xmath227 ) is the @xmath228-highest elements of @xmath229 ( resp . @xmath230 ) in @xmath17 ( resp . @xmath231 ) . for the other elements the image is uniquely determined , since @xmath222 is known to be connected @xcite . next we explain the energy function @xmath14 . let @xmath232 correspond to @xmath233 by @xmath0 . suppose @xmath234 . applying @xmath12 on both sides of @xmath235 , we are led to consider the following four cases : @xmath236 then the function @xmath14 is uniquely determined , up to adding a constant , by @xmath237 although it is not obvious that such a function exists , it is shown to exist @xcite . next we investigate conditions for an element of @xmath1 or @xmath4 to be @xmath228-highest . recall the following fundamental fact : @xmath238 in particular , if @xmath239 is @xmath228-highest , then @xmath28 has to be @xmath228-highest . [ prop : ht cond 1 ] let @xmath240 be a dominant integral weight that appears in as highest weight . by abuse of notation let @xmath241 also stand for the highest kn tableau of weight @xmath241 . let @xmath242 be an element of @xmath209 represented by coordinates . then , an element @xmath243 of @xmath1 is @xmath228-highest , if and only if @xmath244 for some @xmath241 as above and the following conditions for @xmath242 are satisfied . * @xmath245 if @xmath246 , * @xmath247 if @xmath246 or @xmath248 , * @xmath249 if @xmath250 and @xmath251 , * @xmath252 if @xmath253 and @xmath251 . in the case of @xmath160 where @xmath254 , @xmath255 appearing in ( iii ) should be understood as @xmath133 . apply for @xmath216 and use the formula for @xmath256 in . in what follows , for a @xmath3-diagram @xmath25 we use the following notation . let @xmath257 be one of @xmath258 ( @xmath259 stands for emptiness ) . we denote by @xmath260 the number of columns of the outer shape of @xmath25 of height @xmath41 that contain @xmath257 . ( 30,10 ) ( 0,0)(1,0)30 ( 5,6)(1,0)20 ( 0,0)(5,0)4 ( 5,6)(5.3,6.5)(6.7,6.8 ) ( 8.3,6.8)(9.7,6.5)(10,6 ) ( 5,6)(0,1)2 ( 25,6)(0,-1)2 ( 25.3,3.9)(0.3,-0.1)16 ( 4.7,8.1)(-0.3,0.1)16 ( 7.1,6.8)@xmath261 ( 12.1,6.8)@xmath262 ( 17.1,6.8)@xmath263 ( 22.1,6.8)@xmath264 ( 5,3.7)(0,1)2.3 ( 5,2.3)(0,-1)2.3 ( 4.8,2.7)@xmath41 ( 10,5.2)@xmath46 ( 14.2,5.2)@xmath46 ( 15,5.2)@xmath45 ( 19.2,5.2)@xmath45 ( 20,5.2)@xmath45 ( 24.2,5.2)@xmath45 ( 20,4.3)@xmath46 ( 24.2,4.3)@xmath46 ( 0,0)(5,0)3 ( 10.95,5.42)(0.3,0)11 ( 20.95,4.52)(0.3,0)11 [ prop : ht cond 2 ] an element @xmath239 of @xmath4 is @xmath228-highest , if and only if @xmath265 and @xmath29 is a @xmath76-highest element whose corresponding @xmath3-diagram @xmath25 satisfies @xmath266 apply for @xmath216 and use proposition [ prop : e1 action ] to calculate @xmath267 of the @xmath3-diagram @xmath25 . we consider the combinatorial @xmath0-matrix @xmath268 let @xmath269 be @xmath228-highest and @xmath270 . then , @xmath271 is also @xmath228-highest , and from propositions [ prop : ht cond 1 ] and [ prop : ht cond 2 ] @xmath244 for some dominant integral weight @xmath241 , @xmath272 and there exists a @xmath3-diagram @xmath25 such that @xmath273 . thus we have @xmath274 for an element of @xmath209 we use both the coordinate representation and the japanese reading word of the corresponding one - row tableau . let @xmath260 ( @xmath275 ) be data corresponding to @xmath25 as in the previous section . then our main result is : [ th : main ] with the notations above we have the following formulas . @xmath276 here @xmath277 , and @xmath278 . we also note @xmath279 ( mod 2 ) . moreover , the value of the energy function is given by @xmath280 if we normalize @xmath14 in such a way as @xmath281 . here @xmath282 is the first part of the partition corresponding to the weight of @xmath283 . solving the formulas for @xmath260 with respect to @xmath241 and @xmath242 , we obtain the coordinates of the image @xmath283 of the inverse of @xmath0 for an element @xmath284 of @xmath4 are given by @xmath285 here we should understand @xmath286 . in what follows in this section we prove theorem [ th : main ] by assuming technical propositions in later sections . we give a proof only for type @xmath169 , since the difference from the other cases is very small as we have seen in proposition [ prop : to highest ] . suppose we need to apply @xmath113 with such a word @xmath114 as @xmath287 for type @xmath169 ( see e.g. ) . then for type @xmath168 we replace it with @xmath288 and for type @xmath167 @xmath289 consider first the case when @xmath20 is odd . suppose @xmath290 and @xmath25 are related as in the statement of the theorem . we are to show @xmath291 by proposition [ reduction_odd3 ] showing is reduced to the case where @xmath242 is of the form @xmath292 . applying this proposition again to this case , it is then reduced to the case where @xmath293 , since there is no @xmath100 or @xmath96 in @xmath242 of the previous case . hence proposition [ prop_special ] completes the proof of . ( notice that when @xmath294 for any odd @xmath41 and the other @xmath260 are all zero . ) using these propositions we can calculate @xmath14 as @xmath295 since @xmath296 . the case when @xmath20 is even can be proven similarly by using propositions [ reduction_even3 ] and [ prop_special ] . let @xmath297 be a dominant integral weight whose corresponding young diagram is depicted as follows . ( 25,11.5)(-1,0 ) ( -2.3,5)@xmath298 ( 0,0)(1,0)25 ( 0,0)(0,1)10 ( 0,10)(1,0)5 ( 5,10)(0,-1)2 ( 5,8)(1,0)5 ( 10,8)(0,-1)1 ( 10.3,7)(0.5,-0.1)20 ( 20,4)(0,1)1 ( 25,4)(-1,0)5 ( 25,0)(0,1)4 ( 2.3,4.8)@xmath299 ( 2.5,6)(0,1)4 ( 2.5,4)(0,-1)4 ( 7.3,3.8)@xmath300 ( 7.5,5)(0,1)3 ( 7.5,3)(0,-1)3 ( 22.3,1.8)@xmath301 ( 22.5,3)(0,1)1 ( 22.5,1)(0,-1)1 ( 0,10)(1,10.6)(1.8,10.7 ) ( 5,10)(4,10.6)(3.2,10.7 ) ( 2.1,10.6)@xmath302 ( 5,8)(6,8.6)(6.8,8.7 ) ( 10,8)(9,8.6)(8.2,8.7 ) ( 7.1,8.6)@xmath303 ( 20,4)(21,4.6)(21.8,4.7 ) ( 25,4)(24,4.6)(23.2,4.7 ) ( 22.0,4.6)@xmath304 here one can assume @xmath305 and @xmath306 . we also assume that @xmath307 then the claim of this section is the following special version of the main theorem : [ prop_special ] let @xmath241 be as above . then we have @xmath308 the condition ( [ s : eq : sumc_i = k ] ) for @xmath309 is necessary . for example , in type @xmath310 , the image of the combinatorial @xmath0-matrix and the value of the energy function for @xmath311 are @xmath312 and @xmath313 . we divide the proof of this proposition into three parts . let us define two words that will be used in the proof . @xmath314 where underbraces are introduced to show a unit of repetitions , and @xmath315 are defined as follows . set @xmath316 , then @xmath317 where @xmath318 and for @xmath319 , @xmath320 @xmath321 is defined by @xmath322 and @xmath323 where for @xmath324 , @xmath325 and @xmath326 is @xmath327 finally , @xmath328 in the process of proof , we use a dominant integral weight @xmath329 given by @xmath330 . we assume that @xmath304 is even . the proof for odd @xmath304 is similar . during the proof , we often identify a kn tableau with its japanese reading word . the goal of this subsection is the following lemma : [ lem_special1 ] @xmath331 this is a direct consequence of the following three sublemmas . @xmath332 [ lem : special1 ] @xmath333 to begin with we apply @xmath180 on @xmath334 for maximal times . define a word @xmath335 by @xmath336 then the @xmath76-highest element of @xmath334 is @xmath337 by the map @xmath338 we get the corresponding @xmath3-diagram and @xmath189 acts on it as follows : ( 31.2,8 ) ( 0,0)(18,0)2 ( 0,0)(1,0)13.2 ( 0,0)(0,1)8 ( 0,8)(1,0)3.3 ( 3.3,8)(0,-1)2 ( 3.3,6)(1,0)3.3 ( 6.6,6)(0,-1)2 ( 6.6,4)(1,0)3.3 ( 13.2,2)(0,-1)2 ( 9.9,4)(0,-1)2 ( 9.9,2)(1,0)3.3 ( 27.9,4)(1,0)3.3 ( 31.2,4)(0,-1)2 ( 0.15,7.2)@xmath339 ( 3.45,5.2)@xmath339 ( 6.75,3.2)@xmath339 ( 14,3)@xmath340 ( 18.15,7.2)@xmath341 ( 21.35,5.2)@xmath341 ( 24.65,3.2)@xmath341 ( 28.1,3.2)@xmath341 ( 28.1,2.2)@xmath339 note that there are @xmath304 @xmath46 s at height @xmath342 of the right @xmath3-diagram . assume that @xmath304 satisfies @xmath343 . then @xmath344 is @xmath345 and @xmath346 is @xmath347 from this expression , we get @xmath348 . applying @xmath349 we get @xmath350 to convert the action of @xmath351 into that of @xmath352 , we need to define the words @xmath353 as follows . @xmath354 where the subwords @xmath355 are @xmath356 define @xmath357 and @xmath358 where the subwords @xmath359 are @xmath360 then , starting from @xmath361 , one calculates @xmath362 here @xmath363 means @xmath364 . with @xmath338 , this corresponds to the following @xmath3-diagram and @xmath189 acts on it as follows : ( 25,8 ) ( 0,0)(15,0)2 ( 0,0)(1,0)9.9 ( 0,0)(0,1)7 ( 0,7)(1,0)3.3 ( 3.3,7)(0,-1)2 ( 3.3,5)(1,0)3.3 ( 6.6,5)(0,-1)2 ( 6.6,3)(1,0)3.3 ( 9.9,3)(0,-1)3 ( 0.1,6.3)@xmath339 ( 3.4,4.3)@xmath339 ( 6.7,2.3)@xmath339 ( 10.9,3)@xmath340 ( 15.1,6.3)@xmath341 ( 18.4,4.3)@xmath341 ( 21.7,2.3)@xmath341 thus , starting from @xmath365 , we calculate @xmath366 where the final formula gives @xmath367 . from @xmath368 and @xmath369 , we see that the 0-signature of @xmath370 is @xmath371 . therefore we get @xmath372 which gives the desired expression . [ lem : special2 ] starting from @xmath373 , we have @xmath374 where the final expression is equal to @xmath375 . since @xmath376 , we have finished the proof of lemma [ lem_special1 ] . the goal of this subsection is the following lemma : [ lem_special2 ] @xmath377 to begin with , we have @xmath378 here , we need to divide the calculation into two cases . + _ case 1 : _ if @xmath379 , we have @xmath380 _ case 2 : _ if @xmath381 , we have @xmath382 to begin with we remark that in this case we have @xmath383 assume that @xmath210 satisfies @xmath384 . then @xmath385 define a word @xmath386 by @xmath387 . then we have @xmath388 by the map @xmath338 , @xmath389 corresponds to the following @xmath3-diagram , @xmath189 acts on it as follows : ( 35,10 ) ( 0,1)(20,0)2 ( 0,0)(1,0)15 ( 0,0)(0,1)8 ( 0,8)(1,0)3.3 ( 3.3,8)(0,-1)2 ( 3.3,6)(1,0)3.3 ( 11.6,4)(1,0)3.4 ( 15,4)(0,-1)4 ( 7.7,-1)@xmath390 ( 13.2,-1)@xmath210 ( 6.6,-1)(0,0.4)18(0,1)0.2 ( 11.6,-1)(0,0.4)13(0,1)0.2 ( 15,-1)(0,0.4)5(0,1)0.2 ( 6.6,7)(0,-1)2 ( 6.6,5)(1,0)5 ( 26.6,7)(1,0)5 ( 31.6,7)(0,-1)2 ( 0.15,8.3)@xmath339 ( 3.45,6.3)@xmath339 ( 11.85,4.3)@xmath339 ( 15.8,4)@xmath391 ( 20.1,8.3)@xmath341 ( 23.4,6.3)@xmath341 ( 26.95,5.3)@xmath392 ( 26.95,6.3)@xmath393 ( 31.8,4.3)@xmath341 there are @xmath394 @xmath46 s at height @xmath342 in the right @xmath3-diagram . assume that @xmath394 satisfies @xmath395 . we also assume that this @xmath41 satisfies @xmath396 for the sake of simplicity . then @xmath397 is @xmath398 and @xmath399 is @xmath400 from this expression , we have @xmath401 . applying @xmath402 we get @xmath403 note that the length of the string @xmath404 is @xmath301 whereas that of the string @xmath405 is @xmath406 . in order to convert the action of @xmath351 into that of @xmath352 , we define the word @xmath407 as follows . @xmath408 , @xmath409 where subwords are defined by @xmath410 @xmath411 , @xmath412 where subwords are defined by @xmath413 and @xmath414 . computation of @xmath415 proceeds as follows : @xmath416 by @xmath338 , this corresponds to the following @xmath3-diagram , and @xmath189 acts on it as follows : ( 35,11 ) ( 0,1)(20,0)2 ( 0,0)(1,0)15 ( 0,0)(0,1)9 ( 0,9)(1,0)3.3 ( 3.3,9)(0,-1)2 ( 3.3,7)(1,0)3.3 ( 6.6,7)(0,-1)2 ( 6.6,5)(1,0)5 ( 15,3)(0,-1)3 ( 7.7,-1)@xmath390 ( 13.1,-1)@xmath210 ( 6.6,-1)(0,0.4)15(0,1)0.2 ( 11.6,-1)(0,0.4)15(0,1)0.2 ( 15,-1)(0,0.4)5(0,1)0.2 ( 11.6,6)(0,-1)2 ( 11.6,4)(1,0)3.4 ( 35,6)(0,-1)2 ( 31.6,6)(1,0)3.4 ( 0.2,9.3)@xmath339 ( 3.5,7.3)@xmath339 ( 7.0,5.3)@xmath392 ( 15.8,4)@xmath391 ( 20.1,9.3)@xmath341 ( 23.5,7.3)@xmath341 ( 27.0,5.3)@xmath393 ( 31.9,5.3)@xmath341 ( 31.9,4.3)@xmath339 let us assume that @xmath417 . then the right @xmath3-diagram corresponds to the expression ( [ eq : w_3 ] ) with @xmath418 and @xmath419 in ( [ eq : w_3 ] ) being replaced with @xmath420 and @xmath421 . application of @xmath422 is similar to that of @xmath423 on ( [ eq : w_3 ] ) and we obtain @xmath424 as @xmath425 the remaining computation of @xmath426 is almost the same as the computation of @xmath427 given in the final part of the proof of lemma [ lem : special1 ] . the only difference in @xmath428 is caused by the fact that letters @xmath184 and @xmath429 appear @xmath210 times in @xmath424 . as for @xmath430 , the beginning two steps @xmath431 gives @xmath432 by comparing this with @xmath433 , we see that the rest of the computation of @xmath430 is almost the same as that given in lemma [ lem : special2 ] . this completes the proof for case 1 . note that in this case we have @xmath434 . action of @xmath352 is obtained by formally setting @xmath435 in case 1 . therefore we have @xmath436 where @xmath437 is given in ( [ eq : special3 ] ) . when we further apply @xmath430 on this formula , we realize that there are extra exponents originating from @xmath438 in the first tensor component of the right hand side . these extra contributions coincide with the exponents @xmath439 in @xmath440 . we have completed the proof of lemma [ lem_special2 ] . now we can prove proposition [ prop_special ] . we prove the first relation by descending induction on @xmath301 . if @xmath441 ( this is the maximal possible value ) , we have @xmath442 . in this case we see @xmath443 by weight consideration . the induction proceeds by using lemmas [ lem_special1 ] and [ lem_special2 ] . as for the energy function , we have to look carefully the action of @xmath180 in lemmas [ lem_special1 ] and [ lem_special2 ] . if @xmath180 acts on the second component of the tensor product , we write @xmath0 , and @xmath444 on the first component . we summarize actions of @xmath180 to get @xmath375 and @xmath445 in two lemmas as follows ( proceeds from left to right ) : @xmath446 the diagram is drawn in the case of @xmath447 . including the other inequality case , we see that we have exactly the same number of @xmath448 and @xmath449 cases ( see ) . therefore we have @xmath450 . using the same induction as above we obtain @xmath451 . this completes the proof of proposition [ prop_special ] . let @xmath453 be @xmath228-highest . recall that we defined @xmath454 by @xmath240 . ( readers are warned that it is not the multiplicity of @xmath41 in the corresponding partition @xmath241 but its conjugate @xmath455 . ) note that @xmath456 unless @xmath457 and @xmath41 is odd . we also know that the coordinates other than @xmath458 are all @xmath133 by proposition [ prop : ht cond 1 ] . let us define a word @xmath459 by @xmath460 where the exponents are defined as follows . for @xmath461 , @xmath462 define @xmath463 . then @xmath464 . for @xmath465 , @xmath466 set @xmath467 and define other @xmath468 by @xmath469 for @xmath470 . note that @xmath471 . set @xmath472 , @xmath473 and define other @xmath474 by @xmath475 for @xmath476 . since @xmath482 is the @xmath3-diagram of outer shape @xmath241 such that all the columns have @xmath46 as symbol , we see @xmath483 . thus one has @xmath484 . we have @xmath485 . to convert the result into that for @xmath486 we define a word @xmath487 as follows : @xmath488 where @xmath489 , @xmath490 , @xmath491 . then we have @xmath492 applying @xmath493 further , we obtain @xmath494 finally , applying @xmath495 we obtain the desired relation . let us consider the operation of @xmath502 in @xmath503 . since @xmath504 , @xmath505 acts on the second component at most @xmath506 times and the rest goes to the first . the 2-signature of the second component of ( [ tochu6_2 ] ) is @xmath507 . from the highest condition for @xmath508 we have @xmath509 , thus @xmath505 acts on @xmath510 only . we can continue similarly and obtain the desired result . finally , we consider the action of @xmath511 . the 1-signature of eq.@xmath501 is @xmath512 . by applying @xmath513 , we get @xmath514 the 2-signature of the above element is @xmath515 . from the highest condition for @xmath508 we have @xmath516 , thus @xmath505 does not act on @xmath517 . therefore @xmath518 acts on the first component and obtain @xmath519 we can continue the computation and arrive at proposition [ reduction_odd ] . in this subsection , let @xmath25 and @xmath144 be the @xmath3-diagrams . as before , corresponding to @xmath25 and @xmath144 , we use the parametrization @xmath260 and @xmath521 ( @xmath522 ) respectively . note that by definition @xmath523 . define a word @xmath524 by @xmath525 here the exponents for @xmath526 are @xmath527 the exponents for @xmath528 are @xmath529 we use proposition [ prop : e1 action ] . schematically , the pair of @xmath3-diagrams corresponding to @xmath198 looks as follows : @xmath541{0,0,0,0.2 } \put(2,9){\rule{20pt}{20pt } } \put(6,8){\rule{20pt}{20pt } } \put(13,5){\rule{20pt}{20pt } } \put(17,4){\rule{20pt}{20pt } } \put(21,3){\rule{20pt}{20pt } } \put(28,0){\rule{20pt}{20pt } } \color{black } \thicklines \put(0,0){\line(1,0){36 } } \put(0,0){\line(0,1){11 } } \put(0,11){\line(1,0){8 } } \put(8,11){\line(0,-1){2 } } \put(8,9){\line(1,0){2 } } \multiput(10,9)(0.3,-0.1){10}{\circle*{0.1 } } \put(13,8){\line(1,0){2 } } \put(15,8){\line(0,-1){2 } } \put(15,6){\line(1,0){8 } } \put(23,6){\line(0,-1){2 } } \put(23,4){\line(1,0){2 } } \multiput(25,4)(0.3,-0.1){10}{\circle*{0.1 } } \put(28,3){\line(1,0){2 } } \put(30,3){\line(0,-1){2 } } \put(30,1){\line(1,0){6 } } \put(36,1){\line(0,-1){1 } } \thinlines \put(2,11){\line(0,-1){1 } } \put(2,10){\line(1,0){4 } } \put(6,10){\line(0,-1){1 } } \put(6,9){\line(1,0){2 } } \put(13,6){\line(1,0){2 } } \put(17,6){\line(0,-1){1 } } \put(17,5){\line(1,0){4 } } \put(21,5){\line(0,-1){1 } } \put(21,4){\line(1,0){2 } } \put(28,1){\line(1,0){2 } } \put(32,1){\line(0,-1){1 } } \put(0.2,10.3){$+$ } \put(1.2,10.3){$+$ } \put(2.2,10.3){$+$ } \put(3.2,10.3){$+$ } \put(4.2,10.3){$-$ } \put(5.2,10.3){$-$ } \put(6.2,10.3){$-$ } \put(7.1,10.3){$-$ } \put(2.2,9.3){$+$ } \put(3.2,9.3){$+$ } \put(4.2,9.3){$+$ } \put(5.2,9.3){$+$ } \put(6.2,9.3){$+$ } \put(7.1,9.3){$+$ } \put(6.2,8.3){$+$ } \put(7.1,8.3){$+$ } \put(8.2,8.3){$+$ } \put(9.2,8.3){$+$ } \put(13.2,7.3){$-$ } \put(14.1,7.3){$-$ } \put(13.2,6.3){$+$ } \put(14.1,6.3){$+$ } \put(13.2,5.3){$+$ } \put(14.1,5.3){$+$ } \put(15.2,5.3){$+$ } \put(16.2,5.3){$+$ } \put(17.2,5.3){$+$ } \put(18.2,5.3){$+$ } \put(19.2,5.3){$-$ } \put(20.2,5.3){$-$ } \put(21.2,5.3){$-$ } \put(22.2,5.3){$-$ } \put(17.2,4.3){$+$ } \put(18.2,4.3){$+$ } \put(19.2,4.3){$+$ } \put(20.2,4.3){$+$ } \put(21.2,4.3){$+$ } \put(22.2,4.3){$+$ } \put(21.2,3.3){$+$ } \put(22.2,3.3){$+$ } \put(23.2,3.3){$+$ } \put(24.2,3.3){$+$ } \put(28.2,2.3){$-$ } \put(29.2,2.3){$-$ } \put(28.2,1.3){$+$ } \put(29.2,1.3){$+$ } \put(28.2,0.3){$+$ } \put(29.2,0.3){$+$ } \put(30.2,0.3){$+$ } \put(31.2,0.3){$+$ } \put(32.2,0.3){$+$ } \put(33.2,0.3){$+$ } \put(34.2,0.3){$-$ } \put(35.2,0.3){$-$ } \put(0.5,11.6){$p_r^\cdot$ } \put(2.5,11.6){$p_r^-$ } \put(4.5,11.6){$p_r^+$ } \put(6.0,11.6){$p_{r-2}^\cdot$ } \put(8.5,9.6){$p_{r}^\mp$ } \put(13.5,8.6){$p_i^\cdot$ } \put(15.2,6.6){$p_{i+2}^\mp$ } \put(17.5,6.6){$p_i^-$ } \put(19.5,6.6){$p_i^+$ } \put(21.1,6.6){$p_{i-2}^\cdot$ } \put(23.5,4.6){$p_i^\mp$ } \put(28.6,3.6){$p_1^\cdot$ } \put(30.5,1.6){$p_3^\mp$ } \put(32.6,1.6){$p_1 ^ -$ } \put(34.6,1.6){$p_1^+$ } \end{picture } \nonumber\end{aligned}\ ] ] here the thick lines represent outer shape of @xmath198 and the thin lines represent the inner @xmath3-diagram . ( since we are to consider the @xmath162 action , we need such a pair of @xmath3-diagrams . ) the numbers @xmath260 represent the numbers of columns which have the same pattern of @xmath46 and @xmath45 indicated below @xmath260 . according to proposition [ prop : e1 action ] , we make pairs of two @xmath46 symbols which we indicate by gray squares in the diagram . then we see that we can apply @xmath162 up to @xmath542 times , which gives the value for @xmath543 . the pair of @xmath3-diagrams corresponding to @xmath544 looks as follows : @xmath545 note that the numbers of columns of height 1 have changed from @xmath546 , @xmath547 , @xmath548 to @xmath548 , @xmath546 , @xmath547 . in order to compute @xmath549 , we usually make @xmath544 into @xmath550-highest by applying suitable @xmath113 , apply @xmath189 and then apply @xmath551 ( see ) . however , since @xmath189 commutes with the action of @xmath12 @xmath552 , we can apply @xmath189 on the pair of @xmath3-diagrams directly . namely , @xmath189 changes the outer @xmath3-diagram only . the pair of @xmath3-diagrams corresponding to @xmath553 looks as follows : @xmath554 note that the outer shape has also been changed at @xmath548 . we use proposition [ prop : to highest ] . the quantities @xmath118 , @xmath119 and @xmath120 there should be used for the corresponding numbers of the inner @xmath3-diagram of ( [ pair+-diagram ] ) . since we are considering the inner @xmath3-diagram , we have to understand the word @xmath114 there as follows : @xmath556 and the formula for @xmath557 and @xmath558 are the same in terms of @xmath118 , @xmath119 and @xmath120 . then , @xmath559 and differences @xmath560 are @xmath561 and differences @xmath562 are @xmath563 we see that the word @xmath114 computed here coincides with @xmath526 . again , we use proposition [ prop : to highest ] . in this case , the quantities @xmath118 , @xmath119 and @xmath120 there mean those for the outer @xmath3-diagram of ( [ pair+-diagram ] ) . let us compute the word @xmath565 there in the case of our ( [ pair+-diagram ] ) . to begin with , @xmath566 is @xmath567 and differences @xmath560 are @xmath568 and differences @xmath562 are @xmath569 we see that the word @xmath114 computed here coincides with @xmath528 except for @xmath570 which does not appear in @xmath528 . let @xmath571 be the @xmath228-highest weight element whose outer shape coincides with ( [ pair+-diagram ] ) . then the above lemma shows that @xmath572 . since there are exactly @xmath573 columns of height 1 in @xmath571 , we see that the content of columns of height 1 in the tableau @xmath574 are all 2 and that the other columns are the same as @xmath571 . from the shape of ( [ pair+-diagram ] ) we see that @xmath574 coincides with @xmath144 given in proposition [ reduction_odd2 ] . to summarize , we have @xmath575 , hence we complete the proof of proposition [ reduction_odd2 ] . \(i ) we have @xmath586 where we have used @xmath587 in the final line . thus @xmath588 . + ( ii ) to begin with let us show @xmath589 . we compute @xmath590 thus @xmath591 , which shows the coincidence of the first letters of @xmath592 and @xmath593 . as for the other @xmath594 and @xmath595 , note that @xmath596 when @xmath41 is odd , we see @xmath597 . when @xmath41 is even , we have @xmath598 and thus we have @xmath597 , i.e. , @xmath599 for all @xmath41 . we have @xmath600 , i.e. , @xmath601 . similarly , we have @xmath602 . as for other @xmath603 and @xmath604 , we have @xmath605 thus we have @xmath606 for all @xmath41 and obtain @xmath589 . similarly we can show @xmath607 . we compute @xmath608 thus @xmath609 , i.e. , the coincidence of the first letters of @xmath592 and @xmath593 . next , we have @xmath610 . on the other hand , we have @xmath467 and @xmath611 , thus @xmath612 , i.e. , @xmath613 . similarly , we can recursively show @xmath614 for all @xmath41 , @xmath615 , @xmath616 for all @xmath41 . so we have @xmath607 , and therefore we get the final result @xmath617 . + ( iii ) the 0-signature of @xmath508 is @xmath618 for some @xmath619 and that of @xmath579 is @xmath620 for some @xmath621 . here we divide into two cases . let us first assume @xmath622 . then the actions of @xmath180 on two tensor products look as follows ( proceed from left to right ) : @xmath623 thus we have @xmath624 ( rr ) pairs and @xmath210 ( ll ) pairs . therefore we have @xmath625 which gives the desired relation . next assume @xmath626 . then we have @xmath627 ( rr ) pairs and @xmath628 ( ll ) pairs and again we obtain @xmath625 . let @xmath453 be @xmath228-highest and @xmath240 . @xmath456 unless @xmath457 and @xmath41 is even . we also know that the coordinates other than @xmath629 are @xmath133 by proposition [ prop : ht cond 1 ] . let us set @xmath630 throughout this subsection . let us define a word @xmath631 by @xmath632 where the exponents are defined as follows . for @xmath633 , @xmath634 define @xmath635 . then @xmath464 . for @xmath636 , @xmath637 set @xmath638 and define other @xmath468 by @xmath639 for @xmath640 . note that @xmath471 . set @xmath472 and define other @xmath474 by @xmath641 for @xmath642 . in this subsection , let @xmath25 and @xmath144 be the @xmath3-diagrams . as before , corresponding to @xmath25 and @xmath144 , we use the parametrization @xmath260 and @xmath521 ( @xmath522 ) respectively . define a word @xmath649 by @xmath650 here the exponents for @xmath651 are @xmath652 the exponents for @xmath653 are @xmath654 s. v. kerov , a. n. kirillov , n. yu . reshetikhin , _ combinatorics , the bethe ansatz and representations of the symmetric group _ , zap.nauchn . ( lomi ) * 155 * ( 1986 ) 5064 ( english translation : j. sov . math . * 41 * ( 1988 ) 916924 ) .
|
we calculate the image of the combinatorial @xmath0-matrix for any classical highest weight element in the tensor product of kirillov reshetikhin crystals @xmath1 of type @xmath2 .
the notion of @xmath3-diagrams is effectively used for the identification of classical highest weight elements in @xmath4 .
| 16,726 | 89 |
the recognised convention for representing the overall galaxy population is the luminosity distribution or space density of galaxies ( see review by binggeli , sandage & tammann 1988 and more recently @xcite ) . this distribution is typically derived from a magnitude limited redshift survey which is corrected for malmquist bias ( e.g. , efstathiou , ellis & peterson 1988 ) and fitted with the three parameter schechter function @xcite . generally , the schechter function is found to be a formally good representation , although some surveys ( e.g. , @xcite ; @xcite ) have hinted at an upturn at the faintest limits , where the statistics become poor ( typically @xmath25 mag , see also @xcite ) . the process of fitting a luminosity function ( lf ) reduces the galaxy population to three crucial numbers : the characteristic luminosity , @xmath10 ( or @xmath26 ) , the normalisation , @xmath27 , and the faint - end slope , @xmath28 . since the pioneering work of @xcite numerous measurements of these three crucial parameters have been made ( see @xcite ) . among the most recent are those derived from the two - degree field galaxy redshift survey ( 2dfgrs ; @xcite ) and the sloan digital sky survey ( sdss1&2 ; @xcite ; 2003a ) . however , although individual surveys yield schechter function parameters to high accuracy , a comparison between independently published values shows excessively large variations , indicative of strong systematic errors ( see @xcite ; @xcite ; @xcite ) . @xcite discuss the impact of surface brightness selection effects on measurements of the lf ( see also @xcite ; @xcite ) and find that such effects _ may _ provide an explanation for the variation seen . in @xcite we compared a number of recent @xmath24-band schechter function values and used the precision number counts of the millennium galaxy catalogue with a uniform isophotal detection limit of @xmath29 mag arcsec@xmath1 in the @xmath24-band to revise the normalisation parameters . this resulted in a reasonably consistent global schechter function with : @xmath30 mag , @xmath31 mpc@xmath7 and @xmath32 ( taken from @xcite ) . despite the apparent convergence brought about by @xcite there are two strong reasons to now go beyond the monovariate schechter function : \(1 ) numerical and semi - analytic simulations are sufficiently developed that it is now possible to predict not just the luminosity distribution but also the size distribution ( see e.g. @xcite ; buchalter , jimenez & kamionkowski 2001 ; @xcite ) and hence the _ bivariate _ distribution of luminosity and size ( or luminosity and surface brightness , see e.g. @xcite ; @xcite ; @xcite ; @xcite ) . this bivariate brightness distribution ( bbd ) can be fitted by a 6-parameter chooniewski function @xcite essentially a schechter function with a luminosity dependent gaussian distribution in surface brightness ( see e.g. @xcite ; @xcite ) or compared directly without confinement by any functional form . either way , a multivariate distribution provides a more stringent constraint than the monovariate schechter function ( see also @xcite ) ; \(2 ) @xcite essentially circumvent the surface brightness issues raised by @xcite by allowing the normalisation of the lfs to accommodate the missing flux from low surface brightness galaxies . this approximation succeeds because the galaxies that dominate the bright galaxy number counts are all around the @xmath26 value , where the surface brightness distribution appears well defined ( see @xcite , @xcite ) . we also know from studies of giant ellipticals ( @xcite ; @xcite ) , disk galaxies ( @xcite ) , dwarf galaxies in the local group ( @xcite ) , rich clusters ( @xcite ; @xcite ) and from early work on the global field bbd ( @xcite ; @xcite ; @xcite ; @xcite ) that there exists a luminosity surface brightness relation ( see also @xcite for the near - ir relation and a discussion of its variation with wavelength ) . together these imply that the impact of surface brightness selection effects will be a function of luminosity . therefore recovering the full luminosity distribution can only be done by simultaneously considering luminosity _ and _ surface brightness a conclusion reached many times over the past half - century ( @xcite ; @xcite ; @xcite ) . an additional motivation is that given the size of contemporary datasets , it becomes logical to start to explore multivariate distributions to quantify trends such as the luminosity - surface brightness relation @xcite or the colour - luminosity relation ( e.g. , @xcite ; see also @xcite ) . here we reconstruct the bivariate luminosity - surface brightness distribution with careful attention to the impact of surface brightness and size selection effects . in section [ data ] we describe the imaging and redshift data . in section [ cosmoke ] we describe the conversion of our catalogue from apparent to absolute properties , the various selection boundaries and our methodology for deriving the joint luminosity - surface brightness distribution . we apply it to our data in section [ mgcbbd ] and compare our results to previous estimates of the bbd ( @xcite ; @xcite ) , the lf ( @xcite ; @xcite ; @xcite ; @xcite ) and the surface brightness or size distribution ( @xcite ; @xcite ) . we summarise our key results in section [ conclusions ] . throughout we assume @xmath33 as defined in section [ cosmology ] . the millennium galaxy catalogue ( mgc ) is a deep ( @xmath34 mag arcsec@xmath1 ) , wide - field ( @xmath35 deg@xmath36 ) @xmath24-band imaging survey obtained with the wide field camera on the 2.5-m isaac newton telescope . the survey region is a long , 35-arcmin wide strip along the equator , covering from @xmath37 to @xmath38 ( j2000 ) , and is fully contained within the regions of both the 2dfgrs and sdss data release 1 ( dr1 ; @xcite ) . in @xcite we gave a detailed description of the observations , reduction , object detection , galactic extinction correction and classification ( using sextractor ; @xcite ) and presented precision galaxy number counts over the range @xmath39 mag . all non - stellar sources to @xmath40 mag were visually inspected and if an object was found to be incorrectly deblended or if the object s parameters were obviously wrong , the object was re - extracted by manually changing the sextractor extraction parameters until a satisfactory result was achieved . in addition , all low - quality regions in the survey ( e.g. near ccd defects ) were carefully masked out ( see @xcite and @xcite ) . we demonstrated that the internal photometric accuracy of the mgc is @xmath41 mag and that the astrometric accuracy is @xmath42 arcsec . in @xcite we then compared the photometric accuracy , completeness and contamination of the mgc to the 2dfgrs and sdss - edr and dr1 datasets . being deeper and of higher resolution , we found that the mgc is both more accurate and more complete than either of these surveys . the mgc is currently the highest quality and most complete representation of the nearby galaxy population . [ mgcexamples ] shows images of three galaxies from the digital sky survey ( left ) , the sdss ( centre ) and the mgc ( right ) , illustrating the superior resolution and greater depth of the mgc data . the mgc - bright catalogue of @xcite was bounded at the bright end by @xmath43 mag , where stars begin to flood in our best seeing fields . we now extend this catalogue brightwards to @xmath44 mag by carefully examining all non - stellar objects in the range @xmath45 mag , repairing objects that had been over - deblended by sextractor and checking completeness with ned at @xmath46 mag , where sextractor optimised for faint galaxy detection and photometry may occasionally miss very large objects . a second change to mgc - bright with respect to @xcite is the deletion of some exclusion regions . liske et al . had placed exclusion regions around very bright objects ( @xmath47 mag ) to avoid spurious halo detections and problems with background estimation in their vicinity . @xmath48 such regions had been erroneously placed around galaxies fainter than @xmath49 mag and these have been deleted , resulting in an increase of the good - quality survey area from @xmath50 to @xmath35 deg@xmath36 . further minor changes were introduced to mgc - bright , each affecting @xmath51tens of objects : different deblending decisions ; spectroscopic identification of galaxies previously misclassified as stars ( compact emission line galaxies ) and vice versa ( stars which appear extended because of almost perfect alignment with small , faint background galaxies ) ; and identification of asteroids previously misclassified as galaxies ( discovered by comparison with sdss - dr1 imaging data ) . the final mgc - bright sample now comprises @xmath2 galaxies . by design the mgc survey region is fully contained within the 2dfgrs and sdss survey regions . hence we first turned to these surveys in order to obtain redshifts for mgc - bright . in addition we obtained redshifts from the 2qz @xcite , ned ( excluding 2dfgrs and sdss objects ) and the smaller surveys of francis , nelson & cutri ( 2004 ) ( pf ) and @xcite ( lsbg ) . in turn each survey , @xmath52 , was matched against mgc galaxies and stars with @xmath53 mag . this magnitude limit and the following matching procedure were chosen in order to be able to account for every @xmath52 object and hence to verify the completeness of the mgc . objects with multiple redshift measurements . the distribution has an rms dispersion of @xmath54 km s@xmath55 . the solid line is a gaussian with the same median and rms as the data . ] first we found all pairs of @xmath52 and mgc objects where each is identified as the other s nearest neighbour in that catalogue . a pair with separation @xmath56 smaller than some @xmath57 is classified as a match unless one or more other mgc objects have also identified the @xmath52 object as their nearest neighbour in @xmath52 with a separation @xmath58 ( where @xmath56 refers to the separation of the closest pair ) . these @xmath52 objects as well as objects with @xmath59 and those which have not been identified as a nearest neighbour by any mgc objects are classified as initially unmatched . the value of @xmath60 is chosen as the separation where the invariably occurring tail in the distribution of separations begins to dominate . the above definition of initially unmatched objects is designed to conservatively flag all possible mismatches . almost all mismatches fall into one of three categories , which are a priori difficult to distinguish from one another but for which one needs to proceed differently : ( i ) unusually large positional disagreement ; ( ii ) two ( or more ) objects in one catalogue where the other catalogue only has one object , because of different deblending choices or different resolution ; ( iii ) one object in @xmath52 but none in mgc because of inaccurate photometry in either @xmath52 or mgc , or because the @xmath52 object is an asteroid or a spurious noise detection , or because the object is genuinely missing from the mgc . all initially unmatched objects are visually inspected in order to reliably identify those objects that were in fact correctly matched . the few cases where the mismatch was due to poor object extraction in the mgc were fixed . each redshift of a matched object is assigned a redshift quality , @xmath61 , which has the same meaning as in the 2dfgrs @xcite : @xmath62 means no redshift could be measured ; @xmath63 means possible but doubtful redshift ; @xmath64 means probable redshift ; and @xmath65 or @xmath66 means good or very good redshift . @xmath62 and @xmath67 are considered failures and we will use only redshifts with @xmath68 in this paper . in addition to the 2dfgrs this system is also already used by pf . we translate the various quality parameters of the other surveys to our system in the following manner : sdss - dr1 redshifts are assigned @xmath65 unless the sdss zstatus or zwarning flags indicate a serious problem ; for the 2qz we use @xmath69 ; and ned and lsbg redshifts are assigned @xmath64 . in total the public data provided good quality redshifts for @xmath70 mgc galaxies . in order to complete the redshifts for the full mgc - bright sample we conducted our own redshift survey ( which we label mgcz ) , targeting those galaxies which did not already have publicly available redshifts . these observation were mainly carried out with the two degree field ( 2df ) facility on the anglo australian telescope . the data were collected , reduced and redshifted in an identical manner to that of the 2dfgrs ( see @xcite for full details ) . in particular we again used the same redshift quality parameter , @xmath61 , as the 2dfgrs ( see above ) . the redshift incompleteness of the 2df data showed a clear bias against low - surface brightness galaxies . hence we undertook additional single - object long - slit observations , targeting low - surface brightness galaxies ( @xmath71 mag arcsec@xmath1 , where @xmath72 is the apparent effective surface brightness within the half - light radius , see section [ mgcsb ] ) as well as gaps in our 2df coverage , with the double beam spectrograph ( dbs ) on the australian national university s @xmath73-m , the low resolution spectrograph ( lrs ) on the @xmath74-m telescopio nazionale galileo on la palma , the eso multi - mode instrument ( emmi ) on the new technology telescope in la silla and the gemini multiple object spectrograph ( gmos , long - slit nod & shuffle mode ) on gemini north . in all cases we observed between @xmath75 and @xmath76 s through a @xmath77 to @xmath67-arcsec wide slit , aligning the slit with the major axis of the object to collect as much flux as possible . all the data were reduced in a similar manner using standard iraf procedures . the resulting spectra are of similar resolution as our 2df data , but of higher quality : they increased our redshift completeness at @xmath78 mag and @xmath71 mag arcsec@xmath1 from @xmath79 to @xmath80 per cent . using all available data and mgcz data qthe overall redshift completeness of mgc - bright is @xmath81 per cent . in fig . [ completeness ] ( third row from top ) we show the final incompleteness as a function of apparent effective surface brightness and apparent magnitude , ( @xmath82 ) and ( @xmath83 ) model colours . in table [ alldata ] we list the numbers of redshifts contributed by each of the surveys . for @xmath84 galaxies we have more than one @xmath85 redshift measurement , totaling @xmath86 redshifts . these duplications are mainly due to overlaps between the 2dfgrs and sdss - dr1 , but @xmath87 of these galaxies have at least one mgcz redshift . for these objects we pick a ` best ' redshift by sorting first by @xmath61 , then by signal - to - noise ratio ( where available ) and then by the order of the surveys in table [ alldata ] . fig . [ multiplezs ] shows the distribution of differences between the multiple redshifts which has an rms of @xmath54 km s@xmath55 . to calculate the mgc bbd we first need to adopt a cosmological framework , appropriate @xmath5-corrections [ @xmath88 and an evolutionary model [ @xmath89 . it is of course possible to determine a lf while simultaneously solving for one or more of these quantities see e.g. @xcite , who derive a lf while simultaneously solving for both the number - density evolution and the luminosity evolution . however , our concern with this approach is that it can lead to erroneous results _ if _ the incompleteness is significant and biased ( as indicated in @xcite and fig . [ completeness ] ) . in this case the evolutionary functions would be fitting not only evolution but also the uncorrected selection biases . instead , our philosophy is to adopt reasonable assumptions for the evolution and concentrate on a strategy to account for selection bias . -corrections as defined in section [ kcorrections ] . the solid curves show the bright and faint magnitude limits assuming the mean @xmath5-correction and a universal evolution of @xmath90 . the dashed lines again show the magnitude limits but now assuming the bluest and reddest @xmath5-corrections used in the mgc . the upper and lower redshift limits are marked as vertical lines and the @xmath91 value from norberg et al . ( 2002 ) is shown as a horizontal dotted line . note that the lower redshift limit defines the minimum absolute magnitude of @xmath92 mag . the locations of known abell clusters are indicated and clearly show up as redshift over - densities . ] we use @xmath93 and @xmath94 broadly confirmed by the wmap mission combined with the 2dfgrs power spectrum but adopt @xmath95 km s@xmath55 mpc@xmath96 for ease of comparison with previous results . we also define maximum and minimum redshift limits . the maximum limit is chosen as the point at which red @xmath26-galaxies fall below our imposed @xmath40 mag limit ( this occurs at @xmath97 , see fig . [ mz ] , excluding @xmath98 galaxies ) . the lower limit is determined by the point at which the redshifts are significantly influenced by the local velocity field . from @xcite we find that the pairwise velocity dispersion , as measured from the 2dfgrs , is @xmath99 km s@xmath55 . hence to ensure minimal impact from peculiar velocities ( i.e. , @xmath100 ) we must set @xmath101 ( excluding 44 galaxies ) . fig . [ mz ] shows the absolute magnitude versus redshift distribution for our data along with the upper and lower redshift limits ( vertical lines ) and the typical value of @xmath91 ( @xcite ; dotted line ) . note that the lower @xmath102 limit , combined with the faint apparent magnitude limit ( @xmath40 mag ) ultimately defines how far down the luminosity distribution one can probe ( @xmath103 mag ) . one can only _ reliably _ push fainter by extending the faint apparent magnitude limit of the survey and not by covering a wider area ( unless direct distances are measured or the detailed local peculiar velocity field is modeled ) . in total there are @xmath104 galaxies within these redshift limits . -band @xmath5-correction . ] @xmath5-corrections , or bandpass corrections , are galaxy specific ultimately galaxy component specific and given the known variation in galaxy types , a mean @xmath5-correction ( as adopted in @xcite and in most previous studies ) may be overly simplistic ( see also @xcite who reach a similar conclusion ) . here we identify a best fit synthetic spectrum from the combined broad - band mgc and sdss - dr1 colours for each individual galaxy and then directly measure the @xmath5-correction from this optimal template . we use the extinction - corrected mgc kron ( @xmath105 kron radii ) and extinction - corrected sdss - dr1 petrosian magnitudes for this purpose . the petrosian magnitudes are corrected to the ab system using @xmath106 ; @xmath107 and @xmath108 as recommended on the sdss - dr2 website . the @xmath109 magnitudes are adjusted for the known zeropoint offset of @xmath110 mag between the mgc and sdss - dr1 @xcite and converted from vega to ab . this conversion was found to be @xmath111 for the mgc system . -probability distribution from our @xmath112 fitting algorithm ( histogram ) as compared to the expected distribution ( for five degrees of freedom , solid line ) . note that because we re - analyse galaxies with probabilities less than @xmath66 per cent no values with @xmath113 occur . ] we now wish to derive the optimal spectral template match to these six filters in the ab system . we elect to use the spectrum library of @xcite which includes ellipticals and early and late - type spirals , with ages ranging from @xmath114 gyr to @xmath115 gyr ( @xmath116 synthetic templates in all ) . to determine the best template we calculate the ab magnitudes as if observed through the @xmath109 and sdss @xmath117 filters for a single template and iteratively rescale the amplitude of the spectral template to minimise @xmath118 : @xmath119 ^ 2 $ ] where the errors are given by : @xmath120 and @xmath121 mag . these error values were derived as follows . firstly , using just the sdss filters , we determined the @xmath122-distribution for the optimal spectral fits based on the errors listed on the sdss website ( @xmath123 [ red leak ] , @xmath124 ) . we then scaled the @xmath125 errors until the recovered @xmath126-distribution matched the expected distribution for four degrees of freedom . we then repeated this process now including the @xmath109 filter ( see fig . [ prob ] ) . these final errors are in good agreement with the study of @xcite . the @xmath118-minimisation was repeated for each of the @xmath116 spectral templates and the smallest of the @xmath116 @xmath126 values used to identify the most appropriate template . finally if the @xmath126-probability for the optimal template was less than @xmath66 per cent we repeated the process , first rejecting the @xmath109 filter and then the @xmath127 and @xmath102 filters in turn ( hence the truncation at @xmath128 on fig . [ prob ] ) . magnitudes fainter than the quoted @xmath129 per cent completeness limit were not used ( @xmath130 , @xmath131 , @xmath132 , @xmath133 , @xmath134 mag , see @xcite ) . [ kcorrexamples ] shows two example galaxies with the best - fit spectral templates overlaid on the measured magnitudes . [ prob ] shows the resulting @xmath122 distribution ( histogram ) and the expected @xmath118-distribution ( for five degrees of freedom , solid line ) . the mgc @xmath5-corrections trace out the 27 tracks of the synthetic spectra of poggianti ( 1997 ) . the filled circles indicate the mean of our data and the vertical dashed lines show the adopted redshift limits used in the bbd analysis . ] shown in fig . [ kcorr ] are the resulting @xmath5-corrections for the mgc , which trace out the @xmath116 different spectral templates , as well as the mean @xmath5-correction in @xmath135 redshift intervals ( large dots ) . finally , in the few cases ( @xmath136 galaxies in total ) where a galaxy did not have an sdss - dr1 match we assigned the mean @xmath5-correction value at the galaxy s redshift . we adopt a global evolution of the form @xmath137 with an initial value of @xmath138 ( i.e. , @xmath139 , see @xcite ) . given the redshift limits this implies an evolutionary correction in the range @xmath140 mag . this is a passive luminosity evolution model with no merging over this redshift range ( @xmath141 ) . in section [ evolmstar ] we solve for @xmath142 and find @xmath143 ( see fig . [ beta ] ) . more complex evolutionary scenarioes will be considered in future papers where we sub - divide the galaxy population by type and component . however we do note that recent studies ( e.g. , @xcite ) including our own ( @xcite ) indicate a very low low - z merger rate based on counts of dynamically close pairs . it is also worth noting that the step - wise maximum likelihood method used assumes no number density evolution . we adopt the effective surface brightness , i.e. the average surface brightness within the half - light radius , as our structural measurement . the effective surface brightness is chosen as it makes no assumption of the radial flux profile and can be measured empirically from the data for any galaxy shape . hence it is a direct measure reflecting the global compactness regardless of whether the galaxy is smooth or lumpy . the half - light radius is the semi - major axis of the ellipse containing half the flux of sextractor s best magnitude . the ellipse s centre , ellipticity and position angle are taken from sextractor . for compact objects the half - light radius is affected by the seeing . through simulations of both exponential and de vaucouleurs profile galaxies using iraf artdata ( see appendix [ appsim ] for details ) we find that we can correct the observed half - light radius for this blurring and recover its true value , @xmath144 , to an accuracy of @xmath145 per cent ( see fig . [ selection ] ) by : @xmath146 where @xmath147 is the observed half - light radius and @xmath148 is the fwhm of the seeing profile , all in arcsec . work is currently underway to determine 2d bulge - disk decompositions from the mgc data which will be presented in a future paper . however at this stage it is unclear whether the effective radius or independent bulge and disk parameters are the more fundamental . here , we define our absolute effective surface brightness , @xmath149 , as : @xmath150 - 10 \log ( 1+z ) - k(z ) - e(z),\ ] ] we note that @xmath144 is defined along the semi - major axis and hence the above prescription assumes galaxies are optically thin , i.e. , derived parameters are independent of inclination . the role of dust will be explored in a future paper . like all surveys our catalogue will suffer from some level of incompleteness in both the imaging and spectroscopic surveys . as our data are the deepest available over this region of sky we can not empirically quantify our imaging incompleteness ( see @xcite for an assessment of the incompleteness of the supercosmos sky survey , 2dfgrs and sdss - edr / dr1 relative to the mgc ) . we can , however , define the selection limits at any redshift , following @xcite , and use this information later ( section [ swml ] ) to quantify the selection boundary in our final space density distribution . and the solid line shows the fit . the low surface brightness and maximum size limits are defined as limits of reliable photometry by requiring errors in the photometry of less than @xmath151 mag and @xmath151 mag arcsec@xmath1 . the dashed line is the fundamental detection limit . ] we begin by noting the mgc s apparent survey limits : a bright and faint apparent magnitude limit of @xmath152 mag and @xmath153 mag , a limiting apparent effective surface brightness of @xmath154 mag arcsec@xmath1 , a minimum size of @xmath155 arcsec ( where @xmath148 is the seeing fwhm ) and a maximum size of @xmath156 arcsec . the last three limits refer to the seeing - corrected quantities ( see section [ mgcsb ] ) and were derived from simulations of @xmath157-inclined optically thin exponential disks using iraf s artdata package ( see appendix [ appsim ] for details ) . it is important to note that the surface brightness and maximum size limits do _ not _ define the _ absolute detection _ limits but the boundary of _ reliable photometry _ ( where ` reliable ' is arbitrarily defined as @xmath151 mag and @xmath151 mag arcsec@xmath1 accuracy , see fig . [ selection ] ) . [ maxhlr ] shows these limits in the apparent magnitude - apparent effective surface brightness plane , along with the galaxy distribution with ( dots ) and without ( triangles ) redshifts . -corrections ) and the full ( @xmath141 ) mgc distribution ( dots ) . the figure illustrates that the selection boundaries vary dramatically with redshift ( bottom to top ) such that each redshift slice , within the survey limitations , only samples a restricted region of the luminosity - surface brightness plane . note that some points may lie outside the box if their @xmath5-corrections are significantly different from the galaxy for which the selection lines have been drawn . figure degraded see http://www.eso.org/@xmath51jliske/mgc/ for full pdf copy of this paper . ] for each galaxy @xmath158 with known redshift , @xmath159 , we can now determine the parameter space in absolute magnitude , @xmath160 , and absolute effective surface brightness , @xmath149 , over which this galaxy could have been observed , given the survey limits . the appropriate limits are ( following @xcite ) : @xmath161 note that @xmath162 is defined by both the minimum effective surface brightness for reliable detection , @xmath163 , and the maximum reliable size limit , @xmath164 . the maximum reliable size limit is imposed by the smoothing box size when calculating the background sky levels . these limits are introduced in more detail in @xcite . one can also envisage a sixth limit ( also @xmath165 ) due to the dynamic range of the ccd detector . in our case none of the galaxies have their central regions flooded and so this limit can be ignored . [ limfig ] shows the selection limits for three galaxies at redshifts @xmath166 ( lower panel ) , @xmath167 ( middle ) and @xmath168 ( upper ) along with the location of the galaxy ( large solid circle ) , galaxies with similar redshifts ( open squares ) and the full ( @xmath169 ) mgc distribution ( dots ) . the selection boundary consistently forms a five - sided figure which glides through the full distribution as the redshift increases . because of the individual @xmath5-corrections each galaxy has its own unique selection window . it is worth noting that large luminous galaxies are undetectable , or at least photometrically unreliable , at very low redshift . hence the methods of @xcite which use @xmath170-style corrections , _ may _ underestimate this population because they implicitly assume selection boundaries only cut into the distribution at some upper redshift limit . @xcite accounted for this effect in their analysis . for each galaxy with known redshift we now define an observable window function , @xmath171 , as : @xmath172 the functions @xmath173 will be used in section [ swml ] in the construction of the bbd . we now construct the mgc bbd using a bivariate brightness step - wise maximum likelihood ( swml ) estimator similar to that described by @xcite . this is an extension of the standard swml method defined by @xcite in the sense that the sample is now divided into bins of both absolute magnitude _ and _ absolute surface brightness as opposed to just absolute magnitude alone . note that @xcite formulated their method in terms of absolute magnitude and a physical diameter . however , the mgc is primarily magnitude and surface brightness limited with a fixed isophotal detection limit of @xmath29 mag arcsec@xmath1 . for this reason we elect to work in @xmath160-@xmath149 space rather than @xmath160-@xmath174 space . essentially , for @xmath175 objects the volume - corrected relative bbd in @xmath176 absolute magnitude bins and @xmath177 surface brightness bins with widths @xmath178 and @xmath179 can be evaluated by ( cf . @xcite ) : @xmath180},\ ] ] where the relative space density of galaxies , @xmath181 , is given in terms of the space density of the previous iteration , @xmath182 , the weighting function , @xmath183 , and the visibility function , @xmath184 . the weighting function , @xmath183 , corrects for the incompleteness in our redshift survey . this incompleteness is demonstrably not random ( see fig . [ completeness ] ) and we are preferentially missing faint dim galaxies with extreme colours . when constructing lfs this bias is often ignored ( e.g. , @xcite ) or considered a function of magnitude only ( e.g. , @xcite ) . here , we account for luminosity and surface brightness dependent incompleteness and define @xmath183 as : @xmath185 where @xmath186 is the total number of galaxies lying in the same apparent magnitude - apparent surface brightness bin as galaxy @xmath158 and @xmath187 is the number of galaxies with known redshifts ( i.e. , those with @xmath85 , see section [ data ] ) in the same bin . hence each galaxy is weighted by the inverse of the redshift completeness in the galaxy s apparent magnitude - surface brightness bin . the visibility function @xmath184 is given by : @xmath188 where @xmath189 is defined in section [ sellim ] . hence @xmath184 is the fraction of the @xmath190-@xmath191 bin which lies inside the observable window of galaxy @xmath158 . [ matrices ] shows example @xmath183 and @xmath184 matrices for @xmath192 mgc16504 ( upper and lower left respectively ) as well as the summed @xmath193 and @xmath194 matrices ( upper right and lower right respectively ) . @xmath195 is essentially the observed galaxy distribution corrected for incompleteness and @xmath196 is analogous to the visibility surface of visibility theory ( phillipps , davies & disney 1990 ) , although not precisely as it does not account for the variation of physical cross - section with redshift or for isophotal corrections . the steep sides of the @xmath196 surface ( fig . [ matrices ] ) highlights the narrow range over which surface brightness selection effects go from negligible to extreme and hence the importance of understanding a survey s selection limits . we now apply the above methodology to our mgc sample . to recap , we extract galaxies in the range @xmath197 mag , determine individual @xmath5-corrections , determine seeing - corrected surface brightness estimates , attach an incompleteness weight to each galaxy , reject galaxies which lie outside of their designated reliably observable regions ( @xmath198 galaxies too large , @xmath199 too compact and @xmath200 too dim ) , generate weighting and visibility matrices for each galaxy and then iteratively apply equation ( [ phijk ] ) until a stable set of @xmath201-values is found . fig . [ bbd ] shows the result of this procedure after renormalisation ( described below ) . it is worth noting that the majority of the rejected galaxies are extremely compact and have entered our sample only through spectroscopy of apparently stellar objects ( 2qz , sdss qso survey , ned and mgcz ) . mpc@xmath7 mag@xmath55 ( mag arcsec@xmath1@xmath202.,width=377,height=302 ] the swml method , by its construction , loses all information regarding the absolute normalisation ( cf . @xcite ) . here the normalisation issues are compounded by the consideration of selection bias . we overcome this by calculating the number of galaxies in the absolute magnitude and absolute surface brightness interval @xmath203 mag and @xmath204 mag arcsec@xmath1 . this parameter space is observable for any @xmath112 over the redshift range , @xmath205 ( equivalent to a co - moving volume of @xmath206 mpc@xmath207 for the @xmath208 deg@xmath36 field - of - view ) . within these redshift , absolute magnitude and surface brightness ranges there are @xmath209 galaxies with a summed weight , due to incompleteness , of @xmath210 . the @xmath211 values are now rescaled to reproduce the correct space density for these galaxies ( i.e. , @xmath212 mpc@xmath7 ) . the error includes both possion and large scale structure considerations and was derived from mock 2df ngp catalogues ( @xcite ; http://star-www.dur.ac.uk/@xmath51cole/mocks/main.html ) extracted from the virgo consortium hubble volume . volumes equivalent in shape to the mgc normalisation volume were extracted and the standard deviation of @xmath213 mag galaxies determined . the final error , inclusive of large scale structure , is roughly twice that of the poisson error alone . [ bbdcont ] shows the final normalised bbd on a logarithmic scale . the thick dotted line shows the region within which the statistical errors are @xmath136 per cent or less . the thick solid line shows the effective detection limit defined as the bbd region sampled by at least @xmath214 galaxies ( equivalent to a volume limit of @xmath215 mpc@xmath216 ) . the detection limit bounds the bbd at the low surface brightness and faint limits and clearly starts to impinge on the distribution from around @xmath217 mag ( coincidentally the usually adopted boundary between the dwarf and giant galaxy populations ) . this confirms that even the mgc , the deepest local imaging survey to date , remains incomplete for both high ( @xmath218 mag ) and low surface brightness galaxies . to sample these regions requires both deeper ( @xmath219 mag arcsec@xmath1 ) _ and _ higher resolution imaging data ( @xmath220 arcsec ) over a comparable or larger area ( @xmath221 deg@xmath36 ) with high spectroscopic completeness . following @xcite and @xcite we attempted to fit the mgc bbd with a chooniewski function @xcite , given by : @xmath222 ^ 2\right\rbrace\end{aligned}\ ] ] the parameters @xmath27 , @xmath10 and @xmath28 are the conventional schechter function parameters @xcite . the three additional parameters , @xmath223 , @xmath224 and @xmath142 are the characteristic surface brightness , the gaussian width of the surface brightness distribution and the luminosity - surface brightness relation respectively . this function has been derived analytically by @xcite , @xcite and @xcite . note that the additional normalisation factor , @xmath225 , ensures that @xmath226 and @xmath28 are directly equivalent to the more familiar schechter function parameters . the fit is achieved using the downward simplex method @xcite and the 1-@xmath227 errors are derived from a one - dimensional @xmath118-minimisation across each parameter ( @xmath228 ) . for the @xmath229 degrees of freedom ( @xmath230 data values minus @xmath231 model parameters ) we find an unreduced @xmath126 of @xmath232 . this suggests that the chooniewksi function is a poor fit to the data ( see fig . [ bbdfit ] ) . the main reason appears to be the intrinsic assumption of a constant @xmath224 , whereas the distribution shown in fig . [ bbd ] clearly broadens , a feature also observed by @xcite in their sdss data . we discuss this further in section [ sbd ] . chooniewski parameters and associated errors are tabulated in the upper half of table [ lfdata ] along with other published data including our previous measurement based on the 2dfgrs @xcite . mpc@xmath7 mag@xmath55 ( mag arcsec@xmath1@xmath202 . the greyscale image and the thick solid line are the same as those in fig . [ bbdcont].,width=377,height=302 ] of particular interest over the past few decades has been the galaxy luminosity distribution . this can be derived from the bbd by integrating over the surface brightness axis , ( the associated errors are the root sum squares of the individual errors for each bin ) . [ lf ] ( left panel ) shows the result : @xmath233 mpc@xmath7 , @xmath234 mag and @xmath9 . note that the error in the normalisation parameter , @xmath27 , is the combination ( in quadrature ) of the error from the fitting algorithm combined with the error from the number of galaxies within the normalisation volume . for completeness we also show the implied lf from our chooniewski function fit ( dotted line ) . however recall the poor @xmath126 value from this fit which highlighted the fit s inability to model the dwarf population correctly ( mainly due to the broadening of the surface brightness distribution ) . the right panel of fig.[lf ] shows the formal 1,2,3-@xmath227 error contours for the schechter function fit to the collapsed mgc bbd luminosity distribution . contours of the schechter function fit . ] fig . [ lfcomp ] compares our lf based on the bbd analysis of the deeper , higher quality , higher completeness mgc data to a number of other recent lf estimates ( tabulated in the lower half of table [ lfdata ] ) . note that these schechter functions are shown with their original @xmath27-values as opposed to the @xcite corrected values . we find our lf recovers a comparable @xmath10 value to most previous surveys , but a slightly higher normalisation ( as one might expect from a bbd - style analysis which recovers more light from the low surface brightness population than a non - bbd style analysis ) , and an @xmath28 value consistent with the median of these other surveys . in comparison to the esp lf ( @xcite ) we find consistent schechter function parameters within the quoted errors of the two surveys . in comparison to the 2dfgrs lf ( @xcite ) we find consistent @xmath10 and @xmath27 values but a flatter ( move positive ) @xmath28 . this discrepancy in @xmath28 is formally significant at the @xmath235-level , however we do note that the 2dfgrs has undergone a major overhaul of its photometry since this result ( see discussion in @xcite ) . the final 2dfgrs post - recalibration lf has yet to be determined . our result are consistent with the @xmath10 value only for sdss1 and inconsistent with all three schechter function values for sdss2 . a detailed discussion of this discrepancy follows in the next section . and @xmath28 together with the mgc @xmath236 , @xmath67 and @xmath237-@xmath227 contours ( right panel ) . ] -@xmath227 error ellipses for the schechter function fits along with the data points from the right panel of fig . [ lfcomp ] . ] the most noticeable outliers in fig . [ lfcomp ] are the two sdss results , sdss1 @xcite and sdss2 @xcite . the discrepancy in the normalisation of sdss1 has been comprehensively discussed by both @xcite and @xcite and ascribed to the lack of an evolutionary model in the sdss1 analysis , coupled with their method of normalisation . an over - density in the sdss commissioning data region also contributes . the sdss1 normalisation has been revised by @xcite : @xmath238 ) . once corrected , the sdss1 lf agrees well with the mgc ( 2@xmath227-discrepancy in @xmath28 ) , the 2dfgrs and the eso slice project ( esp ) estimates . the more recent sdss2 result has a significantly fainter @xmath10 to @xmath109 is given by @xcite as @xmath239 . ] and a much flatter faint - end slope than any of the other surveys . @xcite argued that the difference in the @xmath240-band lfs of sdss1 and sdss2 was simply due to the inclusion of luminosity evolution in the latter and that the luminosity densities measured by the 2dfgrs and sdss2 agreed to within the errors once the differing amounts of evolution had been taken into account . however , in the @xmath241-band the discrepancy in @xmath28 between sdss1 and sdss2 is more than twice that in the @xmath240-band . furthermore , even though the luminosity evolution derived in the sdss2 fitting process is very strong ( @xmath242 mag per unit redshift , where @xmath243 ) it can not explain away the difference between the mgc and sdss2 values . our evolutionary correction given in section [ evolution ] corresponds to @xmath244 . if we replace this with the sdss2 evolution , @xmath245 , and re - derive our collapsed mgc lf we find @xmath246 mag and @xmath247 ( or @xmath248 mag and @xmath249 if we restrict the lf fitting to the equivalent sdss2 limit of @xmath250 mag ) . hence some tension remains between the sdss2 and mgc lfs irrespective of evolution . the same argument holds when comparing sdss2 to the 2dfgrs since the latter adopted an evolution equivalent to @xmath251 . to test whether the surface brightness selection limits might be responsible , fig . [ lfcomp2 ] shows a re - analysis of the mgc data , following the same procedures as before , but now imposing progressively shallower surface brightness limits ( as indicated ) . the left panel again shows the lfs and the right panel the @xmath236-@xmath227 error ellipses of the schechter function fits . it is quite apparent that shallower surveys progressively miss a greater fraction of the dwarf population(s ) and ultimately giants recovering both a fainter @xmath10 and a more positive @xmath28 . in fact , fig.[lfcomp2 ] based on mgc data mirrors the expectation from the simulations of @xcite remarkably well ( see their fig . 5 ) . on fig . [ lfcomp2 ] the lf with an imposed selection limit of @xmath252 mag arcsec@xmath1 appears to match the sdss2 lf closely and initially suggests that surface brightness selection might explain the sdss2 result . ( note that sdss1 implemented a bbd - style analysis ) . however fig . [ completeness ] ( left panels ) do not show a significant difference in the low surface brightness spectroscopic completeness between the 2dfgrs and sdss surveys suggesting limits closer to @xmath253 mag arcsec@xmath1 . we note that the sdss2 surface brightness @xmath254 per cent completeness limit ( including consideration of deblending and flux loss issues , etc . ) is quoted as @xmath255 \approx 23.4 $ ] mag arcsec@xmath1 ( @xcite , their fig . 3 ) , where @xmath256 $ ] indicates that the sdss surface brightness measures are derived from ` circular ' half - light radii . conversely the mgc ( and most contemporary datasets ) use major axis half - light radii . these are derived by growing the optimal ellipse until it encloses @xmath254 per cent of the flux . the use of a circular aperture will always result in a lower half - light radius than an elliptical aperture ( except for purely face - on systems ) , and hence in a higher effective surface brightness . the two effective surface brightnesses are related by : @xmath257 = \mu^e[{\rm circular}]-2.5\log \left(\frac{b}{a } \right),\ ] ] where @xmath258 is the axis ratio . for a random distribution of infinitely thin disks the expected median axis ratio @xmath259 is @xmath260 . in reality the galaxy distribution is more diverse consisting of spheroids , bulges and thick disk components / systems . circular half - light radii measurements are also somewhat resolution dependent , introducing both seeing and redshift dependencies into the conversion . empirically we find for our mgc data : @xmath261 , implying a correction of @xmath262 mag arcsec@xmath1 . although we can not directly compare our mgc elliptical half - light radii to the circular s ' ersic half - light radii used by sdss2 , we can compare them to the standard sdss - dr1 circular petrosian half - light radii . we find @xmath263 , independently confirming the correction of @xmath262 mag arcsec@xmath1 . hence the quoted sdss2 @xmath254 per cent completeness limit of @xmath255=23.4 $ ] mag arcsec@xmath1 equates to @xmath264=23.77 $ ] mag arcsec@xmath1 . the colour correction is more complex as it depends strongly on galaxy type and luminosity . the low luminosity population is the bluest ( @xmath265 mag ) and hence surface brightness incompleteness will preferentially affect the faint end of the lf . evidence for such a colour completeness bias can be seen in the observed sdss incompleteness within the mgc region . this is shown as a function of @xmath266 or @xmath267 versus @xmath72 in the middle row ( centre and right panels ) of fig . [ completeness ] . as the sdss spectroscopic survey is @xmath240-band selected it should not be surprising that the @xmath109 completeness is a more complex surface in magnitude , surface brightness and colour . in the worst case scenario we find a 50 per cent completeness limit for the bluest systems of @xmath268 mag arcsec@xmath1 which is consistent with fig . [ completeness ] ( middle left ) , but more importantly significantly fainter than the surface brightness limit implied by fig . [ lfcomp2 ] . that a colour selection is obvious in both the @xmath266 and @xmath267 sdss completeness maps of fig . [ completeness ] argues that the colour selection bias is not a feature of the @xmath127 band data but more a general trend across all filters . adopting an appropriate @xmath269 selection criterion is clearly non - trivial , however a simple way forward is to construct a lf from the available sdss2 redshifts within the mgc region ( see table . 1 ) . this will have the sdss selection function inherently built in . we adopt @xmath270 mag , @xmath271 mag arcsec@xmath1 , the sdss2 @xmath241-band evolutionary correction and apply a conventional monovariate step - wise maximum likelihood @xcite . we now recover @xmath272 mag and @xmath273 ( or @xmath274 mag and @xmath275 for @xmath276 mag ) and consistent at the @xmath277-@xmath227 level with sdss2 . we therefore conclude that the discrepancy between the mgc and sdss2 lfs is complex , but predominantly due to the colour bias within the sdss ( cf . [ completeness ] middle right ) . this highlights the additional complexity in measuring lfs for filters other than the spectroscopic selection filter . finally we note the recent preprint by @xcite in which an attempt is made to sample the very low luminosity regime from their new york university value added catalogue @xcite . this incorporates a strategy for correcting the surface brightness incompleteness within the sdss2 dataset resulting in a significantly steeper faint - end slope ( @xmath278 to @xmath279 in @xmath240 ) . the premise is that the surface brightness distribution of the dwarf population can be extrapolated from that of the giant population . this results in correction factors for the space density of dwarf systems of up to @xmath280 ( see @xcite , their fig . 6 ) . in a similar spirit of producing a ` best guess ' lf at low luminosities fig.[guesstimate ] shows the mgc lf with the selection boundaries and lower redshift bounds removed . this undoubtedly represents an underestimate of the true space density at low luminosities ( @xmath281 mag ) , but in so far as a comparison can be made it is consistent with the sdss result @xcite , in the sense that the sdss data lie above the mgc lower limits . our lf is also consistent with a compilation of @xmath282 galaxies within the local sphere ( @xmath283 mpc ) taken from @xcite . apart from indicating the amount of work that is yet to be done , fig.[guesstimate ] does suggest that the low redshift global luminosity function must steepen beyond @xmath284 at some intermediate magnitude ( @xmath285 mag ) . however it would seem unlikely that it could steepen enough for the dwarf systems to make a significant contribution to the total local luminosity density ( as indicated by the dashed line ) . mpc ) data taken from the catalogue of @xcite . the schechter function from fig . [ lfcomp ] is shown as the solid line and its extrapolation as the dotted line . the dashed line shows the contour of equal contribution to the overall luminosity density . ] as we found earlier the chooniewski function adopted by previous studies ( @xcite ; @xcite ) provides a poor fit to the joint luminosity - surface brightness distribution . this is mainly due to the apparent broadening of the surface brightness distribution towards fainter luminosities . [ sbcomp ] highlights this by showing the surface brightness distribution at progressively fainter absolute magnitude intervals along with the selection limits taken from fig . [ bbdcont ] . we attempt to fit each of these distributions ( fig . [ sbcomp ] , solid lines ) with a gaussian function defined as : @xmath286,\ ] ] where , @xmath223 is the characteristic ( peak ) surface brightness and @xmath224 is the dispersion ( analogous to those values in the chooniewski function ) . the errors reflect @xmath287-statistics from the original observed number of galaxies contributing to each histogram element . for elements with zero values ( i.e. , no galaxies detected ) we adopt the error appropriate for a detection of one galaxy so that the significance of these ` null ' detections can be used to help constrain the fits . the number of degrees of freedom ( listed in the final column of table [ gfdata ] ) is the number of data values plus any zero values either side of the distribution within the selection boundaries minus the number of fitting parameters ( three ) . [ sbgauss](a ) shows the @xmath223 ( dark grey line ) and 1-@xmath224 values ( shaded region ) as a function of absolute magnitude . these results are also tabulated in table [ gfdata ] . [ sbgauss](a ) shows a trend of an initially invariant luminosity - surface brightness relation until @xmath288 mag , and then a steady decline to the limits of detection at @xmath289 mag . the dispersion of the surface brightness distributions grows steadily broader as the luminosity decreases . the changes in both the luminosity - surface brightness relation and its dispersion explain the poor fit of the chooniewski function which can not accommodate such features . at this stage it is also worth noting that the gaussian fits return poor @xmath126 values at intermediate luminosities with even a hint of bimodality ( at @xmath290 mag ) . this may imply that the low luminosity distribution consists of two overlapping populations : rotationally supported disks ( lower surface brightnesses ) and pressure supported spheroids ( higher surface brightnesses ) . if borne out by more detailed structural studies then one might expect the rotationally supported disk systems to exhibit both a steeper @xmath160-@xmath291 relation and a narrower surface brightness distribution than the overall population . three luminosity - surface brightness trends ( @xmath292-@xmath293 ) are well known ( see table [ lfdata ] ) : the kormendy relation for spheroids ( @xcite ) ; freeman s law for disks @xcite ; and the virgo / local group dwarf relation ( @xcite ; @xcite ) . more recently @xcite identified a relation for late - type disks similar to that seen for dwarf systems in virgo and the local group . the general trends for these established relations are overlain on fig . [ sbgauss](a ) . one can speculate that the combination of these trends might well lead to the overall shape of the global @xmath292-@xmath293-distribution shown on fig . [ sbgauss](a ) , as the spheroid - to - disk ratio typically declines with decreasing luminosity . hence the global trend initially follows a kormendy relation , switching to a more constant freeman law and finally into a declining dwarf relation . in comparison to the @xcite sample for late - type disks we find a shallower luminosity - surface brightness relation and a broader surface brightness distribution this may have two reasons : firstly , the djl sample is @xmath294-band selected and our use of a global filter conversion may introduce this error ; secondly , the djl sample is far more local and both inclination and dust - corrected . an obvious extension to the mgc analysis would be to also implement these corrections . obviously , detailed structural analysis is required before a meaningful comparison to these ad hoc local studies can be made . however , we can highlight two key questions : ( i ) are there two distinct disk populations , i.e. a giant disk population adhering to freeman s law and a dwarf - disk population which does not , or just a single dwarf - disk relation ? ( ii ) is the dwarf surface brightness distribution bimodal reflecting the two dominant classes of dwarf galaxies ( dwarf ellipticals and dwarf irregulars ) ? there are three significant studies of the global @xmath292-@xmath293 distribution ( see table [ lfdata ] ) : a 2dfgrs study by @xcite , an sdss study by @xcite and a local galaxy sdss study by @xcite . in comparison to @xcite we find that the mgc data follow a similar @xmath292-@xmath293 relation but show significantly broader surface brightness distributions . [ sbgauss](b ) shows the comparison where we have taken the data from table c2 of @xcite and fitted gaussian distributions in a similar manner as for the mgc data . the obvious discrepancy is in the @xmath224 values and this reflects a number of significant improvements . firstly , the deeper mgc isophote coupled with the improved redshift completeness of mgcz ( see fig . [ completeness ] ) enables the inclusion of lower surface brightness systems . as these preferentially occur at lower luminosity the effect acts to broaden the surface brightness distribution at fainter luminosities . secondly , the 2dfgrs sample was not seeing - corrected and hence the sizes of the high surface brightness galaxies were overestimated . thirdly , the cross et al.analysis derived the surface brightness measurements under the assumption that all galaxies were perfect exponential disks . these factors conspire to push the 2dfgrs bbd into a narrower distribution . a fourth factor , hard to qualitatively assess , is the major revisions to the 2dfgrs photometry which have taken place since @xcite ( see @xcite for further details on the 2dfgrs recalibration history ) . in fig . [ sbgauss](c ) we overlay the recent sdss results by @xcite . the shen et al . study is @xmath295-band selected with an imposed surface brightness cut at @xmath296=23 $ ] mag arcsec@xmath1 . shen et al . divide their sample into a number of sub - groupings and we elect to show the data from their fig . 13 which separates galaxies according to concentration index , @xmath297 ( see @xcite for details ) . this divides the population into galaxies earlier or later then s0/a ( with an @xmath298 per cent success rate ) . to convert the shen et al . @xmath240-band results to @xmath109 we derive the early and late - type @xmath299 colour in each of the @xmath240-band absolute magnitude bins specified by shen et al . to do this we assume our eyeball e / s0 type corresponds to ` early ' and our eyeball sabcd / irr to ` late ' ( a detailed account of the eyeball morphologies and morphological luminosity functions is presented in driver et al . , in preparation ) . hence each data point is adjusted independently . in addition we must also correct the sdss effective surface brightness measurements from circular to major axis measurements as discussed in section [ sdssdiff ] , i.e. @xmath300 = \mu^e[{\rm circular}]+0.37 $ ] . [ sbgauss](c ) shows the comparison between the shen et al . data and the mgc and we see a uniform offset of @xmath301 mag arcsec@xmath1 . despite extensive experimentation with various data subsets and selection boundaries we have been unable to reproduce this result from the mgc data . [ sbgauss](d ) shows a comparison between the recent very low redshift sdss @xmath240 results @xcite and the mgc . we implement the same colour and surface brightness corrections as for the shen et al . data . the ridge line of the very local sdss data shows good agreement with the mgc data , albeit with a narrower distribution ( i.e. , lower @xmath227 values ) . however this appears to be a further manifestation of the sdss use of circular apertures . for example , if we derive the surface brightness distribution around the @xmath10-point for the mgc data using circular and major axis derived surface brightness measures we find a peak offset of @xmath302 ( slightly larger than our estimated median offset of @xmath262 ) with @xmath303 and @xmath304 . this latter value is now consistent with the recent sdss estimate ( @xcite ) . note that the dashed lines on fig . [ sbgauss ] indicated the sdss extrapolated data used to correct the faint - end of the luminosity function . to the mgc limit these extrapolated distributions appear to match our data well providing some vindication for the extrapolation process ( see @xcite , section [ llfe ] and fig . [ guesstimate ] ) . the above two comparison between the mgc and the @xcite and @xcite studies must imply an inconsistency between the two sdss results . this is perhaps more obvious if one bypasses the mgc data and directly compare fig . 14 of shen et al with fig . 5 of blanton et al . the peaks of the late - type / low sersic index sb distributions ( both measured in @xmath240 ) are clearly distinct . given the consistency with the 2dfgrs and the traditional structural studies discussed earlier we conclude that the problem most likely lies with the shen et al . analysis . if we disregard the shen et al study , we can conclude that the ridge line of the effective surface brightness distribution is well constrained and consistent between the mgc , 2dfgrs and the most recent sdss results . however the mgc distribution is significantly broader than either the 2dfgrs or sdss results which is due to improvements in the analysis ( in comparison to the 2dfgrs ) and the use of elliptical as opposed to circular aperture photometry ( in comparison to the sdss ) . in section [ evolution ] we described our adopted evolutionary parameter , @xmath142 , where the luminosity evolution follows the form : @xmath305 $ ] . in section [ sdssdiff ] we saw that this evolution differs from that adopted by the 2dfgrs and sdss2 and discussed the possible impact . [ beta ] ( upper panels ) highlights this dependency by showing the derived values for @xmath10 and @xmath28 for various values of @xmath142 . taking this one step further we can attempt to constrain @xmath142 directly by adopting the additional constraint that the recovered @xmath10 parameter should be invariant to redshift . we divide our sample into two redshift intervals : @xmath306 and @xmath307 ( with median redshifts of @xmath308 and @xmath309 respectively ) and redetermine the collapsed lf distributions using our bbd analysis for regular intervals of @xmath142 . we hold @xmath28 fixed to the value derived from the full redshift range for each @xmath142 ( i.e. , the value indicated in the middle panel of fig . [ beta ] ) and solve for @xmath10 and @xmath27 . it is necessary to hold @xmath28 fixed because the high-@xmath102 sample has too few low - luminosity galaxies to adequately constrain it . [ beta ] ( main panel ) shows the resulting @xmath310 versus @xmath142 and we find the weak constraint : @xmath311 . while this is not particularly stringent , it is consistent with our adopted value of @xmath312 . this constraint on beta is marginally inconsistent ( @xmath313 ) with the evolution found by @xcite for the @xmath241-band ( @xmath314 ) , but consistent with the evolution adopted by the 2dfgrs ( @xmath315 ) . perhaps more importantly this range of @xmath142 implies a potential systematic uncertainty on the quoted @xmath10 ( @xmath28 ) values of @xmath316 ( @xmath317 ) . as noted at the start of section [ cosmoke ] , it is also equally plausible that this error does not actually reflect evolution per se but could instead be interpreted as a redshift dependent photometric error or similar . from the variation of @xmath10 between low and high redshift samples . upper panels : the @xmath10 and @xmath28 values for the entire dataset for different beta values . ] in this paper we have attempted to recover the space density and surface brightness distributions of galaxies , paying careful attention to surface brightness selection biases . we have used the millennium galaxy catalogue which , although smaller in size , has significantly higher resolution , greater depth and higher redshift completeness than either the 2dfgrs or sdss datasets , hence probing to fainter intrinsic luminosities . in comparison to most earlier studies this work includes the following enhancements . \1 . imaging data which probes @xmath277@xmath67 mag arcsec@xmath1 deeper than the 2dfgrs and sdss surveys . \2 . exceptionally high spectroscopic completeness ( @xmath4 per cent ) . individual @xmath5-corrections derived from @xmath318 fits to spectral templates . seeing - corrected half - light radius measurements without assumption of profile shape . modeling of the mgc surface brightness detection and reliability limits which includes the effects of the analysis software . \6 . a weighting system which accounts for apparent magnitude and apparent surface brightness dependent redshift incompleteness . \7 . a joint luminosity - surface brightness step - wise maximum likelihood method which incorporates the selection boundaries . having incorporated these additional improvements to the classical measurement of galaxy luminosity functions , we generally find close agreement with previous results . in particular , our collapsed luminosity distribution is marginally higher in normalisation and marginally flatter than the 2dfgrs @xcite , esp @xcite and sdss1 ( @xcite ; after renormalisation by @xcite ) results . it is significantly brighter and steeper than the more recent sdss2 luminosity function @xcite . we infer that this discrepancy arises because of a colour - selection bias in the sdss . the mgc survey extends to lower luminosity than these previous surveys , providing more leverage and hence more reliability at the faint end . our final schechter function parameters are : @xmath319 mpc@xmath7 , @xmath320 mag and @xmath9 with a @xmath321 of @xmath322 indicating a respectable fit . overall we find that surface brightness selection effects do not play a significant role at bright luminosities . however , the combination of the luminosity - surface brightness relation and the broadening of the surface brightness distribution is likely to lead to the loss of both low surface brightness and compact dwarf systems in contemporary surveys hence the variation in @xmath28 ( cf . [ lfcomp ] & [ lfcomp2 ] and discussion in section [ spacedensity ] ) . at the very faint end we provide lower limits which suggest that the luminosity function must turn up at some point faintwards of @xmath323 mag . the final mgc schechter function parameters , which include compensation for the effects of selection bias , imply a luminosity density of @xmath325 mpc@xmath7 ( adopting @xmath326 mag and @xmath327 , cf . this is fully consistent with the mgc revised values for the 2dfgrs , esp and sdss1 surveys ( see table 3 of @xcite ) . it is also worth noting that this value is also consistent with the original 2dfgrs ( @xcite ) and esp ( @xcite ) values . the more recent sdss2 result ( @xcite ) finds @xmath328 mpc@xmath7 and @xmath329 mpc@xmath7 . hence while the @xmath12 value is consistent the @xmath13 value is not ( @xmath330 ) . the sdss @xmath13 value is derived by adopting the evolutionary parameters derived by @xcite . if we use our evolution instead ( @xmath138 , see section [ evolution ] ) the sdss @xmath13 luminosity density becomes @xmath331 mpc@xmath7 which although closer is also inconsistent with our result . from section [ sdssdiff ] we concluded that a colour bias may have resulted in an underestimate of the sdss @xmath241-band lf and hence to an underestimate of the sdss luminosity densities . based on the consistency between the mgc , 2dfgrs and esp results we can also find that while the more sophisticated bbd analysis does recover some additional flux from the low surface brightness population , it does not constitute a significant change ( consistent with the findings of @xcite ) . our recovered bivariate brightness distribution exhibits a clear ` ridge ' of data well separated from the selection boundaries down to @xmath332 mag at which point both very compact and highly extended dwarf systems will be missed . this indicates that the recovered schechter function is generally robust to surface brightness selection effects for luminous galaxies but that the dwarf population(s ) remain elusive . we attempted to fit a chooniewski function @xcite , essentially a schechter function with a gaussian distribution in surface brightness , to the overall distribution . it introduces three additional parameters : a characteristic surface brightness ; a gaussian dispersion ; and a luminosity surface brightness relation . we find that this function fails to provide a satisfactory fit due to the broadening of the surface brightness distribution with decreasing luminosity and a distinct change in slope of the luminosity - surface brightness relation at @xmath333 mag . we find that the surface brightness distribution of galaxies exhibits interesting behaviour , with a well bounded gaussian - like distribution at @xmath26 , qualitatively consistent with the combination of the kormendy relation for spheroids and a @xmath334 broader freeman s law for bright disks . at @xmath26 we find that the surface brightness distribution is well described by a gaussian with parameters : @xmath335 mpc@xmath7 , @xmath336 mag arcsec@xmath1 , @xmath337 mag arcsec@xmath1 with a @xmath321 of @xmath338 . at fainter luminosities the normalisation increases , the characteristic surface brightness becomes fainter and the dispersion broadens until the distribution appears almost flat ( or possibly bimodal ) . the schechter function fit to the gaussian normalisations ( column 4 of table 3 ) provides a basic completeness correction and yields consistent schechter function parameters to those quoted in section [ lfresult ] , i.e. @xmath339 mpc@xmath7 , @xmath340 mag and @xmath341 with a @xmath321 of @xmath342 . in comparison to the 2dfgrs study of @xcite we find a consistent luminosity - surface brightness relation but with a significantly broader dispersion in surface brightness . this is predominantly due to the implementation of seeing corrected size , direct ( i.e. , profile - independent ) half - light radius measurements and improved completeness for extended systems . a comparison to the sdss results of @xcite and @xcite are complex because of the sdss standard of circular aperture photometry . however we find that our results can be reconciled with those of @xcite but not with those of @xcite . we also note that the sdss use of circular apertures also leads to significantly narrower surface brightness distributions . various studies ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) relate the gaussian dispersion of the surface brightness distribution , or equivalently of the @xmath174 is the half - light radius measured in kpc . it is related to surface brightness by @xmath343 and hence @xmath344 . ] distribution , to the dimensionless spin parameter of the dark matter halo , @xmath345 @xcite . the basic concept is that the baryons in the disk reflect the angular momentum state ( rotational or dispersive ) of the dark matter halo . while the luminosity relates to the mass , the disk scale - length relates to the specific angular momentum via the spin parameter . numerical simulations generally predict values for @xmath21 ( @xcite ; see also @xcite ) . this is most consistent with our observations at fainter luminosities . however , at brighter luminosities we find a significantly narrower dispersion @xmath346 . this may imply that brighter , more mature galaxies have evolved via mechanisms currently beyond the standard numerical simulations ( i.e. , the baryons have decoupled from the dark matter halo and evolved away from the halo properties ) . alternatively it may argue for significant non - directional baryon infall onto the halo , muddying any coupling between the primordial baryons and the halo @xcite . further possibilities might also include the presence of bulges in high luminosity systems whose properties depend more critically on a central supermassive black hole @xcite . one intriguing result of the simulations is the lack of variation of the surface brightness or size distribution with mass ( @xcite . however , the more detailed studies which actually follow individual halos _ do _ imply a greater variation in @xmath347 for low - mass systems . for example , @xcite trace the history of three halos through the merging process . they clearly show that @xmath347 initially varies significantly as the halos grow but that it becomes more resilient to mergers once the halos reach sufficient mass . ultimately all three halos converge towards a common value at high mass . this seems plausible as one would expect smaller systems to be more affected by mergers than more massive systems . the surface brightness distribution of galaxies has been hotly debated for over thirty years since the advent of freeman s law and the conjecture that it is wholly a selection bias @xcite . while in many minds the debate is resolved through deep hi studies ( e.g.@xcite ) and ly-@xmath28 absorption line studies ( @xcite ; @xcite ) , others remain loyal to disney s conjecture ( e.g. oneil , andreon & culliandre 2003 ) and still argue for a flat surface brightness distribution at all luminosities ( @xcite ; @xcite ; @xcite ) implying a significant amount of hidden mass . at least in oneil et al.s case this stance derives from neglecting the clear luminosity - surface brightness relation seen in their data . the global surface brightness distribution may indeed be flat but the low surface brightness systems are predominantly dwarfs , providing little enhancement to the global luminosity , baryon or mass densities . we have attempted to address this debate through careful consideration of _ most _ selection issues . we probe to depths where the distribution is clearly bounded and where we robustly demonstrate we could reliably recover giant low surface brightness galaxies were they to exist in significant numbers . they do not . while we address the selection bias in the redshift survey as a function of apparent magnitude and apparent surface brightness other potential selection effects remain unexplored . for example , the redshift completeness may well be a function of spectral type , in which case the assumption that all galaxies of similar apparent magnitude and surface brightness are drawn from the same redshift distribution will not be correct . in particular , inert low surface brightness galaxies which may have insufficient features to readily yield a redshift and may be missed entirely . we also do not at this stage decompose galaxies into bulge and disk components . this is clearly required by both the observations ( which show two , or possibly three , distinct luminosity - surface brightness relations for spheroid and disk components ) as well as by theory and simulations which argue for distinct formation mechanisms for these components ( merger and accretion respectively ; e.g. pierani , mohayaee & de freitas pacheco 2004 ) . we hence intend to extend our analysis in three ways : @xmath214 per cent spectroscopic completeness ; bulge - disk decompositions ; and extension to longer wavelengths and in particular the near - ir which promises smoother , dust - free profiles . we thank shiyin shen , eric bell and michael blanton for useful discussions and exchange of data , steve phillipps , alister graham and stefan andreon for comments on the early drafts and the referee for a number of detailed scientific and stylistic improvements . the millennium galaxy catalogue consists of imaging data from the isaac newton telescope and spectroscopic data from the anglo australian telescope , the anu 2.3 m , the eso new technology telescope , the telescopio nazionale galileo , and the gemini telescope . the survey has been supported through grants from the particle physics and astronomy research council ( uk ) and the australian research council ( aus ) . the data and data products are publicly available from http://www.eso.org/@xmath51jliske/mgc/ or on request from j. liske or s.p . driver . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . funding for the sloan digital sky survey ( sdss ) has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , university of pittsburgh , princeton university , the united states naval observatory , and the university of washington .
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we recover the joint and individual space density and surface brightness distribution(s ) of galaxies from the millennium galaxy catalogue .
the mgc is a local survey spanning 30.9 deg@xmath0 and probing approximately one two mag arcsec@xmath1 deeper than either the two - degree field galaxy redshift survey ( 2dfgrs ) or the sloan digital sky survey ( sdss ) .
the mgc contains @xmath2 galaxies to @xmath3 mag with @xmath4 per cent spectroscopic completeness . for each galaxy
we derive individual @xmath5-corrections and seeing - corrected sizes .
we implement a joint luminosity - surface brightness step - wise maximum likelihood method to recover the bivariate brightness distribution ( bbd ) inclusive of most selection effects . integrating the bbd over surface brightness
we recover the following schechter function parameters : @xmath6 mpc@xmath7 , @xmath8 mag and @xmath9 .
compared to the 2dfgrs @xcite we find a consistent @xmath10 value but a slightly flatter faint - end slope and a higher normalisation , resulting in a final luminosity density @xmath11 mpc@xmath7 marginally higher than , but consistent with , the earlier 2dfgrs ( @xcite ) , esp ( @xcite ) , and sdss @xmath12 @xcite results .
the mgc is inconsistent with the sdss @xmath13 result ( @xmath14 ) if one adopts the derived sdss evolution ( @xcite ) .
the mgc surface brightness distribution is a well bounded gaussian at the @xmath10 point with @xmath15 mpc@xmath7 , @xmath16 mag arcsec@xmath1 and @xmath17 .
the characteristic surface brightness for luminous systems is invariant to @xmath18 mag faintwards of which it moves to lower surface brightness .
the surface brightness distribution also broadens ( @xmath19 ) towards lower luminosities .
the luminosity dependence of @xmath20 provides a new constraint for both the theoretical development ( dalcanton , spergel & summers 1997 ; mo , mao & white 1998 ) and numerical simulations ( e.g. , @xcite ) which typically predict a mass - independent @xmath21 ( see @xcite and @xcite ) . higher resolution ( fwhm @xmath22 arcsec ) and
deeper ( @xmath23 mag arcsec@xmath1 in the @xmath24-band ) observations of the local universe are now essential to probe to lower luminosity and lower surface brightness levels .
[ firstpage ] galaxies : general galaxies : fundamental parameters galaxies : luminosity function , mass function galaxies : formation galaxies : evolution astronomical data bases : catalogues
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the current picture of galaxies is that they are composed of baryons ( stars , gas , dust ) and non baryonic dark matter . while observationally the distribution of the baryons can be studied , it is difficult to probe how the dark matter is distributed compared to the baryons . traditionally , the study of galaxy dynamics has been the strongest proof of the existence of dark matter in galaxies ( e.g. , @xcite ; @xcite ; @xcite ) . but given the inherent degeneracies in the inversion of dynamical data to obtain density profiles , it is hard to measure how the dark matter is distributed . another approach is to use gravitational lens systems . for strongly lensed quasars ( qsos ) , the geometry and photometric properties of the lens system depend on the projected mass inside the lensed qso images and therefore can be used to constrain the mass distribution . for the particular case of spiral galaxies , generally composed of a bulge , a disk and a dark matter halo , the rotational velocity curves can not disentangle the relative contributions of the different mass components in the inner ( luminous ) parts of the galaxy . on the other hand , gravitational lenses offer the possibility of doing so , especially in the case of edge - on spirals . there are only five spiral gravitational lenses known so far ( b0218 + 357 : @xcite ; pks 1830 - 211 : @xcite ; q2237 + 0305 : @xcite ; b1600 + 434 : @xcite ; pmn j2004 - 1349 : @xcite ) . only one of them is seen edge - on , b1600 + 434 . however , none of them is ideal for studying the relative mass contribution of the different components . here , we report the discovery of an edge - on spiral lens galaxy as part of the cyder survey , which may be the best spiral lens system to separate the relative mass contributions of its constituents . the caln - yale deep extragalactic research ( cyder ) survey ( @xcite ; @xcite , hereafter t05 ) is an optical and near - infrared imaging and spectroscopic program carried out in archived , moderately deep _ chandra _ fields . cxocy j220132.8 - 320144 ( hereafter cy 2201 - 3201 ) is one of the faint x - ray sources detected by cyder in its d1 field . optical follow - up of this field seemed to suggest that cy 2201 - 3201 was an edge - on spiral with a very bright nucleus . however , optical spectroscopy and imaging of this source in good seeing conditions revealed its true nature . cy 2201 - 3201 is a lensing system , showing two images of a distant @xmath1 quasar being lensed by an edge - on spiral galaxy at @xmath0 . the paper is structured as follows . in 2 we present in chronological order the observational data gathered to characterize this system . in 3 we analyze the observational constraints obtained , and discuss them in 4 . finally we draw our conclusions in 5 . throughout , we assume @xmath4 , @xmath5 and @xmath6 km s@xmath3 mpc@xmath3 . cy 2201 - 32 was observed by the @xmath7 x - ray observatory with the acis - i instrument for 50.16 ks in 2000 july as part of the observation of the hcg90 field ( pi bothum ) . we retrieved this image from the archive and analyzed it using standard techniques with the ciao package ( @xcite , t05 ) . cy 2201 - 32 was detected in the soft band ( 0.5 - 2.0 kev ) with 11.6 @xmath8 3.9 counts and undetected in the hard band ( 2.0 - 8.0 kev ) . we computed an x - ray flux of @xmath9 erg @xmath10 s@xmath3 and x - ray luminosity of @xmath11 erg s@xmath3 in the soft band ( t05 ) . as part of the cyder survey we imaged the cyder d1 field ( t05 ) with the cerro tololo inter - american observatory ( ctio ) 4 m telescope using the mosaic ii camera in 2001 august . we took images for a total of 6600 and 1800s in the @xmath12 and @xmath13 bands , reaching limiting magnitudes of @xmath14 and @xmath15 and effective seeings of 1.0 and 0.9 , respectively . cy 2201 - 3201 appears in these ctio images as an edge - on spiral with a very bright nucleus / bulge with a total measured magnitude of @xmath16 and @xmath17 . on 2003 october 31 , we took spectra of the x - ray optical counterparts in the cyder d1 field with the ut4 vlt fors2 instrument . we used the 300v grism which gives a resolution of @xmath18 ( 10.5 ) for our 1 slits . cy2201 - 3201 was observed in one of our masks . given the multi - object purpose of the masks , all slits were oriented north - south . we took five exposures of 1800 s in seeing condition of 0.50-0.75 and bright / grey sky conditions . we reduced the spectra using standard iraf routines . surprisingly , for this source we found two spectra of a qso at a redshift of z=3.90 in our best seeing spectra ( see fig . [ fig1 ] ) . the two qso spectra were blended in our worse seeing exposures . our 30 s mask acquisition image taken with a seeing of 0.45 confirmed the existence of two point sources close to the center of the edge - on spiral galaxy , and therefore confirmed the lensing nature of the system . we observed the cy 2201 - 3201 system with the magellan clay telescope using the magic instrument on 2004 september 8 . magic has a pixel scale of 0.0691 pixel@xmath3 . we took a series of 300 s exposures in three filters ( sdss @xmath19 , @xmath20 and @xmath21 ) for a total of 1800 , 2400 , and 1800 s in the @xmath19 , @xmath20 and @xmath21 filters respectively . we reduced the images using standard procedures in iraf . the resulting combined images have effective seeings of 0.68 , 0.65 and 0.62 in @xmath19 , @xmath20 , and @xmath21 , respectively . figure [ fig2 ] shows the combined @xmath21-band image . the system configuration is composed of an edge - on spiral with two images of the qso at each side of the disk . this is one of the expected configurations produced by an edge - on spiral galaxy lens @xcite , except that it is missing a third image that should be located close to the disk . to obtain tight constraints in the lens modeling it is important to know as accurately as possible the position and the fluxes of the observed components . in our images the seeing is only slightly smaller than the qso images separation . therefore the overlap between the qso images and the galaxy light distribution makes it hard to separate their relative fluxes and determine their positions . we have thus used a monte carlo markov chain method ( mcmc ; @xcite ; @xcite , @xcite ) to try to determine the positions and fluxes ( and their errors ) of the three components as accurately as possible . we model the system with three components , two point sources ( qso_a and qso_b ) and a spiral galaxy . each point source is modeled with three parameters : the @xmath22 and @xmath23 of its center and its flux . the spiral galaxy is modeled as an elliptical exponential with six parameters : @xmath22 and @xmath23 of the galaxy center , flux , axis ratio , position angle of the major axis and effective radius . our model thus has 12 parameters . in order to obtain the best values we generate images with varying values of these 12 parameters and convolve them with the point spread function as given by bright nearby stars . we then compare the resulting model image with the observed one and compute a @xmath24 goodness of fit . we run a mcmc to obtain the best values for 11 of the 12 parameters . we fix one parameter , the position angle of the major axis , because the large elongation of the galaxy allows a robust direct determination . table [ tab1 ] presents the parameters used to constrain the lens model which result from a combination of the parameters found in the three images ( @xmath19 , @xmath20 , and @xmath21 ) . the errors on the fluxes have been artificially increased ( see below ) . lccc[b ] parameter & qso a & qso b & galaxy + @xmath25 ( arcsec ) & @xmath26 & @xmath27 & @xmath28 + @xmath29 ( arcsec ) & @xmath26 & @xmath30 & @xmath31 + relative flux & @xmath32 & @xmath33 & ... + axis ratio ( b / a ) & ... & ... & @xmath34 + position angle ( degrees ) & ... & ... & @xmath35 + @xmath36_e ( arcsec ) & ... & ... & @xmath37 + @xmath38 ( km s@xmath3 ) & ... & ... & @xmath39 + table [ tab2 ] gives the measured magnitudes of the two qso images and the lens galaxy . on 2004 september 7 , 8 and 9 , we took spectra of the lens galaxy using the magellan clay boller & chivens spectrograph ( b&c ) . we use a 1 wide slit aligned along the major axis of the lens galaxy . we use two grisms : 600 and 1200 lines mm@xmath3 , giving dispersions of 1.6 and 0.8 pixel@xmath3 , respectively . we observed the lens for 30 minutes in 0.8 conditions and 3 hr in 1.2 conditions with the lower resolution setting and for 3.5 hrs in 0.7 conditions with the higher resolution set - up . unfortunately , the slit position moved with respect to the object during the observations and the effective on - source exposure is shorter . we reduced the images using standard procedures in iraf . figure [ fig3 ] shows the two dimensional low - resolution spectrum at the wavelength of the observed [ oii ] doublet . we measure the rotational velocity of the disk using the [ oii ] doublet . we manage to detect signal out to a radius of 3.8 equivalent to 2.7 the effective radius where the rotational velocity appears to have already flattened . in fact for an exponential mass distribution the maximum of the rotational velocity is at approximately twice the effective radius @xcite . in order to obtain a good estimate of the rotational velocity we use an mcmc . we model the two - dimensional spectrum assuming an exponential distribution for the galaxy light and an arc - tangent function for the rotational velocity curve . we generate two - dimensional spectra with the expression @xmath40 where @xmath41 is the normalization , @xmath42 and @xmath43 are the pixel coordinates of the center of the galaxy in the two dimensional spectrum , @xmath44 is the scale radius of the exponential profile , @xmath45 is the `` turnover '' rotational velocity radius and @xmath46 is the asymptotic rotational velocity . we then convolve the model image with a gaussian of the same width of the seeing in the spatial direction and with a gaussian of the same width of the spectral resolution in the spectral direction . we generate four model images : one for the low - resolution data taken on september 7 , one for the low resolution data taken on september 9 , one for the high resolution data taken on september 7 and one for the high resolution data taken on september 8 . we compare the model images to the observed ones and compute a global @xmath24 of goodness of fit . we run an mcmc with two free parameters , @xmath41 and @xmath46 , to obtain the asymptotic rotational velocity that best fit the observations . we fix the rest of the parameters ( @xmath42 , @xmath43 @xmath44 and @xmath45 ) to the values directly measured from the two dimensional spectra ( @xmath42 and @xmath43 ) or the magic images ( @xmath44 ) . we fix @xmath45 to the same value in pixels of @xmath44 . we obtain a best value for the asymptotic rotational velocity of @xmath47 km s@xmath3 . with the arc - tangent rotational velocity model used , we compute a rotational velocity at 2.7 times the scale radius of @xmath48 km s@xmath3 . most of the signal in the fit comes from data interior to this radius and therefore we will use the value at this radius to compare to the lens models that best fit the qso image positions and fluxes . lccc filter & qso a & qso b & galaxy + @xmath19 & 25.06@xmath49 & 25.22@xmath50 & 22.11@xmath51 + @xmath20 & 23.09@xmath52 & 23.20@xmath53 & 21.06@xmath51 + @xmath21xs & 22.62@xmath49 & 22.74@xmath51 & 20.61@xmath54 + @xmath55 & 18.75@xmath56 & 18.96@xmath57 & 17.18@xmath58 + in addition we measure the galaxy and qso redshifts to be @xmath59 and @xmath60 , respectively . we observed cy2201 - 3201 with the eso new technology telescope ( ntt ) on 2004 october 30 in service mode . we took an exposure of 4185 s in the @xmath55 band . the reduced combined image provided by eso had an effective seeing of 0.75 . we calibrated it with standard stars . we use the same mcmc method used for the magellan images to calculate the positions and fluxes of the three modeled components in this @xmath55 image . the calculated positions are much more uncertain than the ones calculated with the magellan images . the fluxes nevertheless help us constrain the spectral energy distributions of the qso and lens galaxy ( see table [ tab2 ] ) . we model the cy 2201 - 3201 system using the positions and fluxes of the two qso images together with the parameters measured for the lens galaxy . we use the gravlens softwareckeeton / gravlens/ ] @xcite developed by c. keeton . the models used are described in @xcite . table [ tab1 ] summarizes the observational constraints used for the modeling . note that we have artificially increased the error in the measured fluxes to allow for micro - lensing and/or qso variability effects . we would like to test the hypothesis that mass is distributed similarly to light , and therefore start modeling the system with one ellipsoidal mass component forced to have the same ellipticity as the one observed in the light . we then add a spherical component meant to represent the galaxy halo , which we then allow to be slightly elongated . we also add external shear to see the role it plays in the overall modeling . in this section we discuss only the constraints enforced by the observed qso positions and fluxes . we ignore the constraints imposed by the third and sometimes fourth images predicted but not seen and the predicted circular velocity of the model . we discuss those constraints in the next section . for simplicity , we start with a singular isothermal ellipsoid ( sie ) mass model which gives a flat rotation curve . we follow @xcite and use his `` alpha '' model . the projected mass surface density is given by : @xmath61 , where @xmath62^{1/2}$ ] . @xmath63 is related to the axis ratio @xmath64 by : @xmath65 . we fix the position and ellipticity of the galaxy and only allow the normalization to vary to get the best fit to the observed qso positions and fluxes . figure [ fig4 ] shows the basic configuration of the best fit . three images of the qso are produced at one side of the rotation axis of the galaxy . this is a typical spiral galaxy lens configuration @xcite , with the two brighter images at each side of the disk and the fainter image closer to the disk . figure [ fig4 ] shows that the position of the qso in the source plane ( denoted by the plus sign , @xmath66 ) is very close and outside of the radial caustic ( @xmath67 ) . if the qso were closer to the galaxy center and inside this caustic then five images would be produced ( with the central one highly demagnified ) . next , we try an elliptical navarro - frenk - white ( nfw ) model @xcite . the spherical mass density in the nfw model is given by @xmath68 , where @xmath69 . the projected surface mass density is @xmath70 , where @xmath71 is given by equation ( 48 ) in @xcite . see also @xcite ; @xcite ; @xcite ; @xcite ; @xcite . keeton s gravlens code defines an elliptical model from this spherical one replacing the polar radius @xmath20 in the projected surface mass density expression by the ellipse coordinate @xmath72^{1/2}$ ] . this model produces the same `` disk '' configuration as the previous one . we fit the qso positions and fluxes allowing only the normalization and characteristic radius of the model ( @xmath73 ) to vary . we find that a large range of values produce acceptable fits . the values of the characteristic radius and normalization that produce acceptable fits are degenerate . the trends in this degeneracy are that , when all other parameters are fixed , the larger the characteristic radius , the lower the normalization and the smaller the radial caustic . that is , the central concentration of the model dictates the size of the radial caustic . figure [ fig4 ] shows a typical configuration of this model . finally , we model cy2201 - 3201 with an exponential mass component @xcite . the projected surface mass density of this model is given by @xmath74 where @xmath75 is the same as above , @xmath64 is the axis ratio and @xmath76 is the scale radius of the exponential distribution . we fix the scale radius , the position angle and the axis ratio to the values obtained with the light distribution ( table [ tab1 ] ) and solve for the normalization that best reproduces the position and fluxes . the model shows the disk configuration as before and reproduces the position and fluxes of the two qso images fairly well ( see fig . [ fig4 ] ) . given that the three models produce acceptable fits , one would like to find another constraint to differentiate between them . comparing the three models , we find that the position and relative magnification of the third image depend on the concentration of the mass models . for the most concentrated model ( sie ) , the position of the third image is further away from the galaxy center and the relative magnification compared to the brightest qso image is the lowest ( 30% ) . for the least concentrated model ( exponential ) , the position is closer to the galaxy center and the relative magnification is the largest ( 59% ) . we start with a simple two component model : one sie , as in the previous section , to which we add a single isothermal sphere ( sis ) . the sis model uses the alpha model of @xcite ( the same as the sie , see above ) in which @xmath77 , the axis ratio @xmath78 , and @xmath79^{1/2}$ ] . we vary the normalizations of the sis and the sie models , fixing all other parameters , to find the best fitting model for the observed data . a wide range of normalizations in the models produce acceptable fits to the position and fluxes . all acceptable fits produce five images of the qso ( the central one strongly de - magnified ) . only models in which the rotational velocity of the sis is approximately 50% larger than the rotational velocity of the sie do not give acceptable fits . figure [ fig5 ] shows an example of an acceptable fit in which both components have similar rotational velocities . best fits are produced for models with larger sie contribution . in general when the contribution from the spherical component ( sis ) gets larger in the fits , then the following trends are observed : the area enclosed by the radial caustic increases and the area enclosed by the tangential caustic decreases ; the magnification of the qso images gets larger and the relative magnification of the fourth brightest image compared to the brightest gets larger . on the other hand , the relative magnification of the third image compared to the brightest is almost independent ( staying around 30% ) of the ratio of the sie and sis contributions . the fourth and fifth images disappear at almost the point where the sis contribution is negligible and the model is dominated by the sie component . we next model the system with an expectedly more realistic mass distribution with two components : an exponential distribution ( as in 3.1 ) to account for the disk and a spherical nfw ( same as above but with the axis ratio @xmath78 ) representing the halo . we search for models that best fit the data , varying the two normalizations and the characteristic scale @xmath73 of the nfw model , and fixing all other parameters . again , acceptable fits to the positions and fluxes are produced for a wide range of model parameters . the only models that are rejected are those in which the spherical nfw component dominates . this occurs when the rotational velocity of the nfw is equal to or 25% larger ( depending on the concentration parameter of the nfw ) than the rotational velocity of the exponential component at 2.7 disk scale radii . these rejected models produce five images , with the fifth central image strongly demagnified . the only exception is when the combination of the two mass components is such that the qso needs to be close to the caustic to fit the observed position and fluxes . in this case the fifth image moves away from the center towards the fourth image in the image plane . the trends with respect to the relative strengths of the exponential and nfw components are the following : the larger the nfw spherical component , the worse the fit to the positions and the larger the magnification of the qso images . the relative magnification of the third brightest image compared to the brightest image goes from @xmath8050% when the nfw dominates to @xmath8059% when the exponential dominates . if we model the halos with some ellipticity , the results present general trends that are between the previous one mass component and two mass components cases discussed . the most significant difference in the case of an exponential plus elliptical nfw model is the magnification of the resulting images . the total magnification of the qso images is in general lower and the relative magnification of the third image compared to the brightest becomes larger than in the previous cases . in the previous section we restricted ourselves to modeling the system taking into account the positions and fluxes of the two observed qso images as constraints . we have also measured the rotational velocity of the lens galaxy and can use it as an additional constraint . for each model we compute the rotational velocity predicted at 2.7 @xmath44 , where @xmath44 is the scale radius of the exponential distribution of the galaxy light . this is the value of the radius for which we can measure the rotational velocity with our spectroscopic data ( see 2.5 ) . all rotational velocity comparisons are performed at this radius , neglecting the rotational velocity dependence with radius . for the one mass component models , the best fitting sie model to the qso image positions and fluxes predicts a lower rotational velocity than observed , but consistent within the 1 @xmath81 error . the exponential and elliptical nfw models predict larger rotational velocities than observed , which are consistent only at the 2 @xmath81 level . if we include the rotational velocity in the overall minimization of the two component models , we find that for the sie+sis model the best fit occurs when the rotational velocities of the sis is approximately 20% lower than the rotational velocity of the sie . unacceptable models happen when the rotational velocity of the sis is @xmath80 5% larger than the rotational velocity of the sie . for the exponential+nfw case , we do not find acceptable models that satisfy the three constraints : positions and fluxes of the qso images and rotational velocity of the galaxy . the general trend is that the larger the relative contribution of the spherical component ( halo ) , the lower the rotational velocity predicted to fit the positions and fluxes . cy 2201 - 3201 is an edge - on galaxy lens . such systems offer the possibility of decomposing the relative mass contributions of the disk , bulge and halo . in our optical images cy 2201 - 3201 appears as a bulgeless edge - on spiral galaxy , producing two images of a background @xmath1 qso . we have therefore modeled the system with one ( disk ) and two ( disk+halo ) mass components , neglecting the bulge . all viable models explored produce either three or four visible images . however , our optical images only show two images of the qso ( fig [ fig2 ] ) . we observe the two images that are farther away from the disk but miss the third ( and in some configurations the fourth ) image located very close to the disk . we have investigated whether dust extinction could be responsible for this missing image . the colors of the qso images a and b ( table [ tab2 ] ) are in fact redder than expected for a typical qso at that redshift . although low in signal - to - noise ratioand covering a small wavelength range , the qso spectrum does not reveal indication of strong intrinsic absorption in the qso itself . one is then led to conclude that the redder colors are due to dust extinction between the qso and the observer . the most probable source of attenuation comes from the disk itself . we have computed the amount of dust extinction at the galaxy redshift ( @xmath82 ) necessary to explain the observed colors assuming a mean qso spectrum at this redshift , no intrinsic absorption at the qso itself and the value of the local galactic extinction @xmath83 @xcite . we find that the qso image a requires an extinction by the galaxy disk of @xmath84 and the image b an extinction of @xmath85 . images a and b are seen at a projected distance of 0.35 and 0.47 ( 1.7 and 2.2 @xmath86 kpc ) of the disk plane respectively . the inferred values of the extinction by the disk are then likely taking into account thee projected distances to the disk . our @xmath21 band image is the one with the highest signal - to - noise detection of the qso images . we have inserted a third image of the qso in our model images at the position given by our lens models and verify through our mcmc method the maximum flux it could have without being significantly detected . if the ratio of expected flux to maximum observed flux are completely due to absorption by dust at the lens galaxy , then there are at least 3 mag of extinction more at the position of the third image than at the brightest . taking into account the possible extinction at the brightest image , the third image would suffer @xmath87 of extinction going through the lens galaxy . in our models of the lens , we have first explored the assumption that the mass distribution has the same ellipticity as the light . we have tried three one - mass component models fixing the galaxy center , axis ratio and position angle to those observed in the optical . we have fit an sie , an exponential and an nfw model . all three models reproduce the two qso images and predict a third image that is unobserved . the predicted rotational velocities are consistent with the observed value at the 1 @xmath81 level for the sie model and at the 2 @xmath81 level for the exponential and elliptical nfw models . we have also tried other methods to estimate the galaxy mass . we have fit the observed galaxy photometry with the pegase synthesis evolutionary models @xcite . we obtain an absolute magnitude of @xmath88 , adding an evolutionary correction . ] which is a factor 3 fainter than l@xmath89 @xcite . the stellar mass - to - light ratio(@xmath90/@xmath91 ) of the best fitting model is @xmath92 , so its stellar mass would be @xmath93 @xmath94 @xmath95 . the expected rotational velocity for such a mass at 2.7 times the scale radius is @xmath96 km s@xmath3 ( depending on the mass model ) . the stellar mass by itself is thus insufficient to produce the observed ( or predicted ) rotational velocity . if we assume the local tully - fisher relation @xcite neglecting evolution and the measured absolute magnitude , we obtain a value for the expected rotational velocity of @xmath97 km s@xmath3 , which is consistent with the measured rotational velocity . we have also studied more realistic models . the galaxy does not appear to have a bulge in the optical images , and therefore we have modelled the system with two mass components : one for the disk and one for the halo . we have tried an sie+sis model and an exponential+nfw model . these models reproduce the positions and fluxes of the two observed qso images . however , the sie+sis model predicts four visible images and the best fitting exponential+nfw predicts three visible images . none of these predicted additional images are seen in our images ( see above ) . we measure a rotational velocity of @xmath2 km s@xmath3 at 2.7 disk scale radius for the lens galaxy . the sie+sis model predicts values of the rotational velocity consistent with this value for certain combinations of the relative contributions of the sie and sis components ( see 3.3 ) . however , the exponential+nfw model predicts higher values of the rotational velocity if we fit the observed position and fluxes of the qso images . our magellan spectroscopic data were obtained on three different nights with two different settings . as explained in 2.5 , we have four sets of same night / grating data . the individual fits to the rotational velocity at 2.7@xmath44 for each set of data are @xmath98 , @xmath99,@xmath100 , and @xmath101 ( low - resolution september 7 , low - resolution september 9 , high - resolution september 7 and high - resolution september 8 , respectively ) . the relative dispersion of these values may hint at a possible underestimation of the errors . it this were the case , the range of allowed @xmath46 values would be larger and the exponential+nfw model prediction would still be viable . we have also explored other effects that can affect our modeling . our system is likely to be influenced by some external shear which will contribute to the image separation but not to the rotational velocity of the lens galaxy . in fact , cy2201 - 3201 lies 7 away from the hickson compact group hcg 90 . we have computed what would be the external shear produced by the group assuming it is modeled with a sie with the same velocity dispersion as measured from the galaxy members . hcg 90 is relatively small and the external shear induced in the cy 2201 - 3201 system is negligible for our purposes . apart from hcg 90 , cy 2201 - 3201 appears to be isolated and not in any group , cluster , or obvious large - scale structure . in fact , as part of the cyder survey we have obtained spectra of several sources in the field and , with the limited spectroscopic data we have , not found any sign of a massive structure . another possible flaw in our modeling could be that the galaxy center is miscalculated . if the galaxy center is much closer to the qso images than the position we have measured then most of the discussed configurations would no longer apply and the system would display other image configurations , which can in fact place tighter constraints on the relative contribution of the halo and disk to the total mass budget in the central regions of the lens galaxy . however , the seeing and pixel size of our images and the consistency of the galaxy center in our different filter images make us believe that the true center if different from the one measured should not be very far off . we have presented the discovery and subsequent follow up observations of the cy2201 - 3201 system composed of an edge - on spiral at @xmath82 splitting a background @xmath102 qso into two observed images each at opposite sides of the disk . we have modeled the system with one ( disk ) and two ( disk+halo ) mass components . the most likely configuration is the `` disk''configuration with three images of the qso one at each side of the disk and the fainter one approximately behind the disk . there are also possible configurations that produce four or even five ( this one unlikely ) observable images . however we only observe two images . we have discussed the possibility that the third ( and fourth , if existent ) image is extincted by the disk . we estimate that an @xmath87 at th epredicted position of the third image at the galaxy lens redshift is required to be consistent with our observations . we have measured the rotational velocity of the lens galaxy to be @xmath103 km s@xmath3 at a radius 2.7 times the scale radius of the galaxy exponential light distribution . if we use an sie+sis model to fit the qso image positions and fluxes and this value of the rotational velocity , we find that the contribution by the sie ( disk ) to the @xmath46 at this radius is required to be the same or larger than the contribution of the sis ( halo ) . if we use an exponential+nfw model then we are unable to reproduce this value of the rotational velocity if we fit the positions and fluxes of the qso images . we have speculated whether we have underestimated the error in our determination of the @xmath46 which would make the exponential+nfw model viable . cy 2201 - 3201 is the best lensing spiral galaxy known to date that can be used to disentangle the contributions of its different mass components . unfortunately our current follow up observations are not constraining enough to elucidate between different possible models . more accurate image source positions , the discovery or not of the predicted third image and the measurement of its properties and a precise rotational velocity would make this system fulfill its potential . cy 2201 - 3201 has been awarded _ hubble space telescope _ ( hst ) acs time in cycle 13 . we have also been awarded more spectroscopic time at magellan . we expect that the higher quality images and extra spectra will help us improve our modeling and place strong constraints on the relative contribution of the disk and halo mass components . we thank c. keeton for making public his gravlens code and replying to our questions . we thank paul schechter for his encouragement and helpful discussions . we thank all observatory staff for their help during observations . we thank the anonymous referee for his / her comments that have helped us improve the paper . f.j.c . acknowledges support from the spanish ministerio de educacin y ciencia ( mec ) , project aya2005 - 09413-c02 - 01 with ec - feder funding and from the research project 2005sgr00728 from the generalitat de catalunya . j.m . gratefully acknowledges support from the chilean centro de astrofsica fondap 15010003 . f.j.c . and j.m . acknowledge support from a `` convenio bilateral csic - universidad de chile '' . e.g. is supported by the national science foundation under grant ast 02 - 01667 .
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we present the cxocy j220132.8 - 320144 system , which is composed of an edge - on spiral galaxy at @xmath0 lensing a @xmath1 background quasar .
two images of the quasar are seen .
the geometry of the system is favorable to separate the relative mass contribution of the disk and halo in the inner parts of the galaxy .
we model the system with one elliptical mass component with the same ellipticity as the light distribution and manage to reproduce the quasar image positions and fluxes .
we also model the system with two mass components , disk and halo .
again , we manage to reproduce the quasar image positions and fluxes .
however , all models predict at least a third visible image close to the disk that is not seen in our images . we speculate that this is most likely due to extinction by the disk .
we also measure the rotational velocity of the galaxy at 2.7 disk scale radius to be @xmath2 km s@xmath3 from the [ oii ] emission lines . when adding the rotational velocity constraint to the models , we find that the contribution to the rotational velocity of the disk is likely to be equal to or larger than the contribution of the halo at this radius .
the detection of the third image and a more accurate measurement of the rotational velocity would help to set tighter constraints on the mass distribution of this edge - on spiral galaxy .
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the aim of this work is to present more evidence that excitation spectra of complex open - shell atoms , and probably any other atom at sufficient excitation energies , display clear quantum chaotic features . this phenomenon is caused by strong mixing of many - electron excited states by the residual two - body coulomb interaction . it manifests in particular in a gaussian statistics of the @xmath0 amplitudes for these states . since the time of bohr s hydrogen atom theory atoms were considered as perfectly regular dynamical systems . as the classical theory of chaos evolved it became apparent that highly excited atomic states in the rydberg range could become chaotic if an external field is applied @xcite , as long as the underlying classical motion is chaotic . on the other hand , it was also due to bohr that the notion of compound nuclei was introduced in physics . the behaviour of these highly excited nuclear states is essentially quantum - mechanical . nevertheless , they display a number of chaotic properties . for example , the statistics of their energy spectra show certain universal features , and transition amplitudes involving compound states obey gaussian statistics @xcite . to describe these properties it was suggested by wigner that the hamiltonian of a compound nucleus could be modeled by a random matrix , and different characteristics found by averaging over ensembles of such matrices ( see reviews @xcite ) . the first insight into quantum chaotic properties of complex atoms was given by rosenzweig and porter @xcite who analyzed experimental spectra of some neutral atoms and showed that in heavy open - shell atoms the spectral statistics are similar to those of compound nuclei . that analysis was later extended and refined in @xcite . of course , the study of eigenvalues provides valuable information about the system . on the other hand , the spectral statistics observed in heavy open - shell atoms are similar to those of the hydrogen atom in a strong magnetic field @xcite , or even a particle in a 2-dimensional classically ergodic billiard @xcite . however , the eigenstates of these quantum systems must be completely different , and it is clear that the eigenvalue statistics can not really tell us much about the origin of chaotic behaviour , or indeed the structure of the chaotic eigenstates . the first inquiry into the possibility of chaos in the eigenstates of complex atoms was done by b. chirikov @xcite . he studied configuration compositions of eigenstates of the ce atom using data from the tables @xcite , and came to the conclusion that the ` eigenfunctions are random superpositions of some few basic states ' . inspired by that work we conducted an extensive numerical study of the spectra and eigenstates of complex open - shell atoms , using the rare - earth atom of ce as an example @xcite . this allowed us to investigate many - body quantum chaos in a real system . we showed that atomic excited states are in fact similar to nuclear compound states and developed a statistical approach for analyzing their properties . unlike eigenvalues , the eigenfunctions are not observable directly . to probe the structure of the chaotic eigenstates one can look at the transition probabilities or matrix elements of some external perturbation coupling them to each other , or to regular , simple eigenstates ( like the ground state ) . the matrix elements involving chaotic eigenstates must have gaussian statistics . we showed that its main characteristics the mean squared value of the matrix element between the chaotic multiparticle states ( compound states ) , can be calculated in terms of statistical parameters of the eigenstates and single - particle amplitudes and occupation numbers of the orbitals present in the compound states @xcite . in this work we have chosen the quantity most easily accessible experimentally the @xmath0 amplitudes . it also gives us an opportunity to look for experimental signatures of chaos in the ce atom using the work by bisson _ et al _ @xcite , where over 200 line strengths were measured for transitions between a large number of levels within 3.5 ev of the ground state . it should be mentioned that there are many other possible atomic systems to search for quantum chaos , e.g. , in doubly excited states and inner - shell excitation spectra of alkaline - earth atoms @xcite , or even multiply excited states of light atoms @xcite . let us now recall briefly what chaotic many - electron atomic eigenstates are . suppose one uses a basis of some single - electron orbitals ( e.g. , the hartree - fock ones ) to construct many - electron basis states @xmath3 . the states @xmath4 can be taken as single - determinant states corresponding to certain configurations of a few valence electrons , or constructed from them through some coupling scheme to be of definite total angular momentum @xmath5 . the true atomic eigenstates @xmath6 and eigenvalues @xmath7 are obtained by diagonalizing the hamiltonian matrix @xmath8 . the coefficients @xmath9 describe mixing of the basis states by the residual coulomb interaction . in the multi - electron excitation range the number of basis states @xmath3 formed by distributing several electrons among a few open orbitals is large . many of these states are nearly degenerate and the mean spacing between the basis state energies @xmath10 is likely to be smaller than the typical value of the off - diagonal matrix element @xmath11 . in this situation the basis states are strongly mixed together @xcite . apart from a few lowest levels , each of the eigenstates is a superposition of a large number of basis states . of course , by a simple perturbation theory argument , the mixing must be weak for distant basis states ( large @xmath12 ) . the strong mixing takes place within a certain energy range @xmath13 , where @xmath14 is the mean level spacing , @xmath15 , and @xmath16 is called the _ spreading width _ , since it characterizes the spread of the eigenstates to which a given basis state contributes noticeably . one can estimate the number of _ principal components _ , i.e. those that contribute significantly to a given eigenstate ( [ a ] ) , as @xmath17 . the coefficients @xmath9 corresponding to the principal components have typical values @xmath18 . their statistics is close to that of independent random variables , and tends towards gaussian when the mixing is strong . in this case even the single - electron orbital occupancies are far from integer and only the total angular momentum , parity and the energy itself remain good quantum numbers @xcite . thus , we can talk about _ quantum chaos _ in the system . this situation is similar to that in compound nuclei and the corresponding chaotic eigenstates can be called atomic compound states . the model configuration interaction calculations performed for ce produced a value of @xmath19 ev , and demonstrated the existence of a dense spectrum of chaotic compound excited states with @xmath20 ( @xmath21 ev ) just few ev from the ground state @xcite . consider two chaotic many - body states ( compound states , for short ) that are superpositions of large numbers of basis states , @xmath22 and @xmath23 . if the expansion coefficients @xmath9 are random , the matrix element of some operator @xmath24 @xmath25 is a sum of a large number of almost uncorrelated random items @xcite . therefore , one should expect that such matrix elements display gaussian statistics with zero mean . hence , the probability distribution of the matrix elements between compound states can be characterized by their mean squared value alone . if @xmath24 is a single - particle operator , e.g. , the electric dipole moment @xmath26 ( @xmath27 and @xmath28 are single - particle states ) , it is convenient to express its matrix elements in terms of the matrix elements of the density matrix operator @xmath29 , @xmath30 where @xmath31 . in @xcite and @xcite a statistical approach to calculation of mean squared matrix elements between compound states has been developed . it is first based on the assumption that contributions from different single - particle transitions @xmath32 in the matrix element ( [ mel ] ) are uncorrelated . the mean squared value is then given by @xmath33 where averaging is done over a number of compound states around @xmath34 and/or @xmath35 . the mean squared value of the density matrix operator @xmath36 is expressed in terms of the parameters of the compound states 1 and 2 ( i.e. their energies and spreading widths ) , and the average occupation numbers of the single - particle states @xmath27 and @xmath28 . in a spherically symmetric system where the states 1 and 2 are characterized by their total angular momenta @xmath37 and projections @xmath38 , the wigner - eckhart theorem applies , and it is convenient to deal with the reduced matrix elements @xmath39 independent of the projections @xmath38 . for example , the mean squared value of the zero - rank reduced density matrix operator ( @xmath40 then ) is obtained in the following two forms @xcite : @xmath41 where @xmath42 are the mean level spacings near the states 1 and 2 , @xmath43 and @xmath44 are the orbital occupation numbers , and @xmath45 is a `` finite - width @xmath46 function '' . it depends on the spreading widths @xmath47 of the compound states and on the energy difference @xmath48 between the transition frequency for the compound many - electron states @xmath49 and the frequency @xmath50 of the single - particle transition between the orbitals @xmath51 and @xmath52 . the function @xmath45 has a maximum at @xmath53 and describes the energy conservation for the compound states . its width is determined by the spreading widths @xmath54 . note that @xmath55 in eq . ( [ answ0 ] ) denote averaging of the occupation - number factors over the compound states 1 or 2 . note also that the exact form of the function @xmath56 depends of the spreading of the compound states over the basis components , i.e. on the `` shapes '' of the eigenstates . in the simplest approximation this spreading is described by the breit - wigner formula ( see numerical studies in @xcite ) and @xmath45 is also a breit - wigner profile @xmath57 to calculate the mean squared value of the @xmath0 amplitude we now need a formula for the reduced density matrix operator of the first rank . starting from the definition @xcite @xmath58 for @xmath59 ( linear polarization along the quantization axis ) and assuming that transitions between different magnetic sublevels @xmath60 are uncorrelated we can derive a formula for the mean square of ( [ rank1 ] ) , and then use it to obtain the mean - squared @xmath0 amplitude , @xmath61 analogous to the lower formula in eq . ( [ answ0 ] ) , or an alternative form with @xmath62 and @xmath63 on the right - hand side . the factor @xmath64 on the right hand side of eq . ( [ answe1 ] ) is due to the fact that there are three final - state momenta @xmath65 accessible from a given @xmath66 by means of a dipole transition . in deriving this expression an additional assumption has been made that the occupancies of the @xmath67 and @xmath68 states are statistically independent , and the states with different @xmath60 within the same @xmath51 shell are equally populated . this supposition influences only the `` emptiness '' factors @xmath69 , which are close to unity anyway when the number of single - electron states available is much greater than the number of active electrons . the square of the reduced dipole matrix element @xmath70 is called the strength of the line @xmath71 , so eq . ( [ answe1 ] ) allows one to estimate _ mean line strengths _ for transitions involving compound states . it is interesting to note that the statistical theory expression ( [ answe1 ] ) satisfies the dipole sum rule @xcite ( in atomic units ) , @xmath72 is the number of active valence electrons included in the configuration space of the problem . to obtain this result one should replace summation over the final states @xmath73 with integration over @xmath74 , take into account that @xmath75 [ see eq . ( [ bw ] ) ] , neglect the `` emptiness '' factor @xmath76 and use @xmath77 , and rely on the single - particle sum rules for the orbitals @xmath78 occupied in the initial state @xmath34 , @xmath79 numerical results for the ce atom --------------------------------- cerium , @xmath80 , is the second of the lanthanide atoms . its electronic structure consists of the xe - like @xmath81 core and four valence electrons . the atomic ground state is described by the @xmath82 configuration with @xmath83 @xcite . the origin of the extremely complex and dense excitation spectra of the rare - earth atoms is the existence of several open orbitals near the ground state , namely @xmath84 , @xmath85 , @xmath86 , and @xmath87 , or , in relativistic notation , @xmath88 , @xmath89 , @xmath90 , @xmath91 , @xmath92 , @xmath93 , and @xmath94 . these make a total of @xmath95 single - electron states . for ce with @xmath96 valence electrons there are about @xmath97 possible many - electron states constructed of them . if we allow for the two possible parities , about ten possible total angular momenta @xmath5 , and @xmath98 different projections ( another factor of ten ) , there will be still hundreds of energy levels within a given @xmath99 manifold . in the present work we perform relativistic configuration interaction calculations in the hartree - fock - dirac basis analogous to those in @xcite . in that work we limited ourselves to just 7 nonrelativistic configurations constructed of the @xmath84 , @xmath85 , @xmath86 , and @xmath87 orbitals , for both odd and even states , which produced 260 and 276 states with @xmath83 and @xmath100 , respectively . to make the results more realistic we have extended the configuration basis set by 9 odd and 23 even nonrelativistic configurations . basically , the additional configurations were obtained by exciting one of the four electrons of an `` old '' configuration into the next orbital , e.g. , the lowest even @xmath101 configuration would produce @xmath102 , @xmath103 , @xmath104 , and @xmath105 configurations . to keep the size of the configuration space reasonable we included only the configurations with mean energies within about 10 ev from the ce ground state . this increased the total number of @xmath106 and @xmath100 states to 862 and 1433 , respectively . note that @xmath2 states have been chosen because these manifolds are among the most abundant . as a result , the level density @xmath107 has increased greatly above 4 ev and become closer to that observed experimentally . of course , to be meaningful the level density must be averaged over some small energy interval to obtain a smooth function rather than a set of spikes . an alternative procedure is to look at the cumulative number of levels @xmath108 which we present in fig . [ fdens ] for @xmath109 states . each @xmath110 plot is a staircase of steps of the unit height occurring at successive excited state energies . the level density can be easily estimated from the slope of the @xmath110 plot . the experimental data for the 132 even levels with @xmath2 known from @xcite is shown by the solid - line staircase , and the energies are given with respect to either experimental , or calculated ground state energy . they can be compared with the dashed line that shows @xmath110 for our earlier small - basis calculation @xcite ( 276 states ) , and the dotted line for the present calculation ( 1433 states ) . the improvement is obvious , however , the agreement is not perfect . we believe that the remaining disagreement is not due to some missing configurations in the ci calculation , but rather due to an overall `` softening '' of the spectra due to screening of the coulomb repulsion between the valence electrons by the electrons of the core @xcite . in the ci language this effect is produced by the high - energy excitations of the valence electrons into the continuum together with the electron excitations from the core . two typical features can be observed in the spectra of complex atoms @xcite . the first clearly seen in fig . [ fdens ] , is the rapid increase of the level density @xmath111 with energy @xcite . its origin is purely combinatorial the larger the excitation energy , the greater the number of ways it can be distributed among a few single - particle excitations . in the independent - particle model this dependence is described by the following exponent @xcite : @xmath112 where @xmath113 and @xmath114 are some constants , and @xmath115 is the ground state energy of the system . this dependence also follows from the thermodynamic definition of the temperature , @xmath116\ } /de$ ] , combined with the estimates of the average number of excited fermi particles , @xmath117 , and that of the excitation energy , @xmath118 . the experimental spectra of rare - earth atoms and their ions examined in @xcite are in agreement with eq . ( [ rho ] ) . figure [ fdens ] shows that the calculated cumulative level number plot is fitted well by @xmath119 with @xmath120 ev@xmath121 , @xmath122 ev@xmath123 , and the `` ground state '' energy of the @xmath100 sequence @xmath115 shifted by 0.25 ev up from the true @xmath1 ground state of ce . thus , eq . ( [ rho ] ) gives a good overall fit of the calculated level density below 6 ev . the second feature typical for the spectra of complex many - body systems is level repulsion . it is a basic quantum mechanics fact that two levels with identical quantum numbers can not be degenerate if they are coupled by a non - zero matrix elements they `` repel '' each other . in quantum chaotic systems this repulsion is characterized by the wigner level spacing distribution @xmath124 where @xmath125 is the nearest - neighbour level spacing normalized so that @xmath126 . equation ( [ wigner ] ) shows that the probability of finding small level spacings is indeed vanishingly small . as we pointed out in the introduction , spectral statistics do not tell much about the eigenstates of the system . however , eq . ( [ wigner ] ) is still a good test for some possible hidden quantum numbers , e.g. , the total spin or orbital momentum , which might characterize atomic eigenstates besides @xmath127 . if these do exist , small level spacings ( `` degeneracies '' ) will be more abundant than predicted by eq . ( [ wigner ] ) . these statistics were checked for many experimental @xcite and calculated @xcite complex atomic spectra , as well as for molecular vibronic spectra @xcite . as seen from fig . [ fdens ] the level density changes significantly for the first 500 levels of the calculated spectrum . to analyze the distribution of the corresponding level spacings we use the analytical density fit @xmath128 to normalize the spacings : @xmath129 their distribution shown on the inset in fig . [ fdens ] is in reasonable agreement with the wigner formula . the deviations are probably due to the long - range fluctuations of the level density , not accounted for by the simple exponential ( [ rho ] ) . in the previous calculation @xcite , where only the lowest orbitals of each symmetry were included , we also observed the wigner distribution . when orbitals with higher principal quantum numbers become involved ( as seen from fig . [ fdens ] above 3.5 ev ) the spatial extent of the eigenstates increases . this should cause a decrease of the residual coulomb interaction between the electrons . on the other hand , the level spacings also become smaller . as a result , the state mixing at these excitation energies remains strong , which is confirmed by the agreement with the wigner distribution , and the eigenstates are chaotic . our estimate of the number of principal components @xmath130 shows that it becomes even greater as the energy increases , in accord with the estimate @xmath131 ( @xmath132 ev , and the mean level spacing @xmath133 ev at @xmath134 ev ) . in sec . [ mael ] we explained that matrix elements involving chaotic compound states should have gaussian statistics , and the mean squared value of the matrix elements could be estimated in terms of some average characteristics of the compound states . in this section we concentrate on the dipole matrix elements ( @xmath0 amplitudes ) @xmath135 between the 14 lowest states with @xmath136 and 80 consecutive @xmath100 states obtained numerically in our ci calculations of ce . we have chosen this energy region to cover the range explored in the experiment @xcite , where absolute values were derived for 228 of the most intense lines of neutral ce between 10706 and 22184 cm@xmath121 . of course , low - lying atomic states , e.g. , the ground state , have well - defined configuration composition and are not chaotic , hence , the @xmath0 amplitudes between them should not be distributed in any particular statistical way . however , the matrix elements ( [ m21 ] ) will become random ( and close to gaussian ) as soon as at least one of the states involved , the initial or the final , moves into the compound - state energy range and becomes a superposition of many random components . besides that , the mean squared value of the matrix element is expected to show some smooth secular variation with the energy of the states involved . for these reasons we skip the first 20 states with @xmath138 and analyze the statistics of the @xmath139 @xmath0 amplitudes for the following 80 even states by grouping them in bunches of twenty : 2140 , 4160 , 6180 , and 81100 , which correspond to the mean excitation energies of 2.49 , 2.95 , 3.40 , and 3.70 ev above the atomic ground state ( the mean energy of the lowest 14 odd states is 0.68 ev ) . thus , each plate in fig . [ fcomp ] shows the distribution of the 280 reduced dipole matrix elements together with their root - mean - square ( r.m.s . ) value . also shown in fig . [ fcomp ] are the gaussian distributions @xmath140 , where the root - mean - square parameter @xmath141 has been adjusted to minimize @xmath142 around the center of the histogram . the values of @xmath141 and @xmath142 are given in table [ comp ] . two effects can be seen in fig . [ fcomp ] . first , the distributions of the matrix elements are indeed close to gaussian . second , the width of the distributions ( the mean squared value of the matrix elements ) varies with the energy of the even states . it is mostly this effect that is responsible for the visible discrepancies between the histograms and the gaussian fits . to eliminate it we can use a running average procedure to normalize the amplitudes : @xmath143 where @xmath144 is the r.m.s . value over the 14 odd states , calculated for every even state @xmath145 . figure [ fnorm ] confirms that the 1120 normalized @xmath0 amplitudes for the 21100 even states are distributed according to the normal law . the inset shows the dependence of the r.m.s . @xmath0 amplitude @xmath144 on the energy of the even state @xmath7 . fluctuations aside , it is in agreement with the r.m.s . values calculated from the statistical theory , eq . ( [ answe1 ] ) , at the energies of the 30th , 50th , 70th and 90th even states . the numerical values of the r.m.s . @xmath0 amplitudes are listed in table [ comp ] . note that we have chosen eq . ( [ answe1 ] ) with 1 standing for the odd states and 2 for the even ones . in our numerical example we consider the dependence of the r.m.s . @xmath0 amplitude on the energy of the even states , and keep the odd states the same . therefore , as in eq . ( [ answe1 ] ) , we only need to know the average occupation numbers for the lowest 14 odd states , and the result depends on the final even state via its energy @xmath146 , mean level spacing @xmath147 and spreading width @xmath148 . as we saw in our previous calculations @xcite , the even states of ce with @xmath2 become very much chaotic at excitation energies of just 2 ev , i.e. from the 20th level up . also , as earlier in @xcite , we use average configuration energies rather than single - particle hartree - fock energies to determine the transition frequencies @xmath149 needed for calculation of @xmath150 in eq . ( [ answe1 ] ) . the ground state of ce is described as @xmath82 , however , the dominant configuration among the 14 lowest odd states is @xmath151 , and we used it to calculate the transition energies . for example , the energy of the @xmath152 transition @xmath153 was determined as the difference between the average energies of the @xmath154 and @xmath151 configurations . physically , this corresponds to choosing a particular mean field close to that of the low - lying odd states of ce for calculation of the transition energies . it should be mentioned though that the results obtained with the hartree - fock frequencies @xmath155 were not too different . gaussian statistics of the dipole matrix elements result in the porter - thomas ( pt ) distribution of the line strengths @xmath156 @xmath157 where @xmath158 is the mean line strength . divergence of this function at small @xmath159 means that if the @xmath0 amplitudes are gaussian , there should be many weak lines in the spectrum . earlier evidence of the pt statistics of line strengths can be found in calculations of dipole excitations in complex atoms @xcite , and transitions between the vibronic levels in molecules measured in @xcite . in@xcite absolute values of @xmath160 were obtained for 228 of the most intense observed lines between 10706 and 22184 cm@xmath121 in ce . it is interesting to analyze these data to see whether they support our theoretical and numerical considerations . the values of @xmath160 listed in @xcite are defined as @xmath161 , where @xmath162 is the @xmath0 transition rate from the upper level @xmath163 into the lower level @xmath145 @xcite . we use the experimental values of @xmath160 , @xmath164 and transition frequencies @xmath165 to extract values of the line strengths @xmath166 in fig . [ port ] the probability distribution of the 228 experimental line strengths is shown . compared to the expected pt formula ( [ pt ] ) , there is a clear lack of small line strengths . nevertheless , the decreasing part of the histogram can be fitted well by a pt distribution with an additional normalization factor @xmath167 , @xmath168 shown in fig . [ port ] by a solid line for @xmath169 and @xmath170 a.u . that minimize @xmath142 for the 22 bins with @xmath171 a.u . it would be tempting to say that the excellent agreement between the pt curve and the histogram is a confirmation of the gaussian statistics of the @xmath0 amplitudes in ce . the value of @xmath167 would then indicate that about one half of all lines are missing in the experimental data . however , the value of @xmath170 a.u . corresponds to the r.m.s . @xmath0 amplitude of 1.8 , which is more than 2 times greater than our numerical results in fig . [ fcomp ] and fig . [ fnorm ] ( inset ) and in table [ comp ] . on the other hand , the experimentally observed 228 lines include transitions between levels with various total angular momenta between @xmath172 and 8 ( @xmath173 , of course ) whereas we have about 500 hundred lines with just @xmath174 in our calculation in the analogous energy range . this means that in @xcite only the strongest 10% or less of all lines have in fact been measured . the very suggestive agreement with the pt distribution in fig . [ port ] should then be considered as merely fortuitous . it is worth noting that in experiment the lines are selected by their intensities proportional to @xmath160 , rather then strengths . hence , even lines with large strengths can be omitted if their frequencies are small . let us look at the simplest model of this effect and see how it influences the observed strengths distribution . assume that transitions in a certain frequency range @xmath175 are studied , and different values within this interval are equally probable . the observed intensities of the lines are proportional to @xmath176 . if we assume that there is a minimal threshold intensity that can be registered , the original pt distribution of strengths would be modified as follows : @xmath177 ~,\quad s > s_0\cr } \end{aligned}\ ] ] where @xmath178 is the minimal strength that can be observed at @xmath179 , and @xmath167 is the normalization factor . as seen from fig . [ port ] , eq . ( [ mpt ] ) also gives a very good fit of the experimental data with @xmath180 , @xmath181 and @xmath182 @xcite . note , however , that the new value of the mean line strength is 1.5 times smaller than the one we had from the pure pt fit . therefore , the assumptions used in our processing of the experimental data affect the estimates of the experimental r.m.s . @xmath0 amplitudes , and we should not be too concerned about the apparent disagreement with our numerical calculations . besides that , extraction of absolute line strengths from the experimental data is not free from uncertainties estimated in @xcite at 1020% . for 30 transitions in ce the @xmath160 values were obtained more accurately from branching ratios and delayed photoionization measurements of lifetimes ( @xcite , table 2 ) . when we look at the statistics of the corresponding line strengths ( fig . [ port ] , inset ) and compare it with the pt distribution ( [ pt ] ) , a value of @xmath183 is obtained , much smaller than the estimates of @xmath158 from the statistics of the 228 lines . thus , it appears that to make firm conclusions about gaussian statistics of the @xmath0 amplitudes a much more thorough experimental survey is needed . on the other hand , even relative measurements of a large number of line strengths could be very valuable for examining these statistics @xcite . in this work we have extended the configuration interaction approach of @xcite to calculate large numbers of eigenstates in ce . in agreement with our earlier studies the energy level statistics indicate that the simple configurational basis states are strongly mixed together by the residual electron interaction , and the only good quantum numbers in the spectrum are parity and the total angular momentum . the total orbital momentum @xmath184 and spin @xmath159 are not conserved due to the spin - orbit interaction , whose effect is dynamically enhanced , just as that of any other perturbation in a chaotic many - body system @xcite . the strong configuration mixing makes multielectron atomic eigenstates chaotic . this in turn results in a gaussian statistics of the matrix elements for chaotic atomic eigenstates ( compound states ) . this understanding is fully confirmed by our numerical calculations of the 1120 @xmath0 amplitudes between the 14 lowest @xmath83 states and 80 @xmath138 states above 2 ev . it is important that the parameter of the gaussian , the r.m.s . @xmath0 amplitude , varies slowly with the excitation energy . this effect should be taken into account when analyzing the statistics of the matrix elements . we also show that a statistical theory can be used to estimate mean squared matrix elements involving compound states . it enables one to express the answer in terms of the single - particle matrix elements and occupation numbers , and parameters of the compound states , namely the number of principal components and the spreading width . this approach has already been applied to calculation of matrix elements between compound states in nuclei @xcite . it could be useful in various other many - body systems , e.g. , atomic clusters or quantum dots , where direct diagonalization of the hamiltonian matrix is not feasible because of a huge size of the hilbert space of the problem . an attempt has been made to analyze existing experimental data for the line strengths in ce @xcite . it appears that the statistics of the measured line strengths is compatible with the porter - thomas distribution , with allowance for the missing weak lines . however , the discrepancy between the calculated r.m.s . @xmath0 amplitudes and those inferred from the experimental data does not allow us to say that the existence of quantum chaos in the ce eigenstates has been confirmed experimentally . to make this statement one would have to do a much more complete survey and statistical analysis of the line strengths in the ce spectrum . on the other hand , this means that a comparison between the experimental and theoretical line strengths in ce is not yet possible , even at the level of their mean values . theoretically , to calculate precisely the dipole matrix elements between particular levels in the compound - state energy range of complex atoms like ce looks a prohibitively difficult problem . experimentally , identification of specific lines in enormously complicated spectra is also a very difficult task . however , we would like to suggest that extraction of mean characteristics from the experiment and comparison with the corresponding theoretical estimates is a meaningful way of exploring such complex systems . as a result , one might hope to get a deeper insight into the existence of quantum chaos in many - body systems on the whole , and in complex open - shell atoms , in particular . 10 gribakin@newt.phys.unsw.edu.au , a selection of papers compiled and introduced by g. casati and b. v. chirikov ( university press , cambridge , 1995 ) , part 2 ; see also m. courtney and d. kleppner , phys . rev . a * 53 * , 178 ( 1996 ) and references therein . a. bohr , and b. mottelson , _ nuclear structure _ ( benjamin , new york , 1969 ) , vol . 1 . t. a. brody , j. flores , j. b. french , p. a. mello , a. pandey and s. s. m. wong , rev . mod 53 * , 385 ( 1981 ) . t. guhr , a. mller - groeling , and h. a. weidenmller , phys . ( to be published ) , preprint cond - mat/9707301 . n. rosenzweig and c. e. porter , phys . rev . * 120 * 1698 ( 1960 ) . s. camarda and p. d. georgopulos , phys . lett . * 50 * , 492 ( 1983 ) . d. delande and j. c. gay , phys . lett . * 57 * , 2006 ( 1986 ) . m. v. berry , ann . phys . * 131 * 163 ( 1981 ) ; o. bohigas , m. j. giannoni , and c. schmit , phys . * 52 * , 1 ( 1983 ) . b. v. chirikov , phys . lett . * 108a * , 68 ( 1985 ) . w. c. martin , r. zalubas and l. hagan , _ atomic energy levels - the rare - earth elements _ , natl . bur . stand . ref . ( u.s . ) , nbs-60 , ( u.s . , gpo , washington , dc , 1978 ) . v. v. flambaum , a. a. gribakina , g. f. gribakin , and m. g. kozlov , phys . a * 50 * , 267 ( 1994 ) . a. a. gribakina , v. v. flambaum and g. f. gribakin , phys . e , * 52 * , 5667 ( 1995 ) . v. v. flambaum , a. a. gribakina and g. f. gribakin , phys . a , * 54 * ( 1996 ) . v. v. flambaum and o. k. vorov , phys . * 70 * , 4051 ( 1993 ) ; v. v. flambaum , in _ time reversal and parity violation in neutron reactions _ , edited by c. r. gould , j. d. bowman and yu . p. popov ( world scientific , singapore , 1994 ) , p. 39 . s. e. bisson , e. f. worden , j. g. conway , b. comaskey , j. a. d. stockdale , and f. nehring , j. opt . 8 * , 1545 ( 1991 ) . j. p. connerade , m. a. baig , and m. sweeney , j. phys . b * 23 * , 713 ( 1990 ) . j. p. connerade , i. p. grant , p. marketos , and j. oberdisse , j. phys . b * 28 * , 2539 ( 1995 ) . j. p. connerade , j. phys . b * 30 * , l31 ( 1997 ) . n. vaeck and n. j. kylstra , in xx icpeac , abstracts of contributed papers , edited by f. aumayr , g. betz , and h. p. winter ( unpublished ) , vol . 1 , th128 . if the number of active electrons in the system is more than two , the hamiltonian matrix @xmath11 appears to be sparse , i.e. , there is a certain fraction of zero off - diagonal matrix elements in it ( only the basis states that differ by the positions of no more than two electrons can be coupled by the two - body coulomb interaction ) . in this case one should compare the magnitude of @xmath11 with the mean `` distance '' @xmath185 between the basis states directly coupled by non - zero @xmath11 . see , e.g. p. jacquod and d. l. shepelyansky , phys . lett . * 79 * , 1837 ( 1997 ) ; b. georgeot and d. l. shepelyansky , _ ibid _ * 79 * , 4365 ( 1997 ) and references therein . there are some specific correlations caused by the two - body character of the electron interaction , but they do not change the statistics of the matrix elements . however , they can influence the estimate of the mean square value of the matrix element , see v. v. flambaum , g. f. gribakin , and f. m. izrailev , phys . e * 53 * , 5729 ( 1996 ) . see , e.g. v. a. dzuba , v. v. flambaum , and m. g. kozlov , phys . a , * 54 * , 3948 ( 1996 ) , and references therein . note that there are no rydberg series in our ci calculation for ce , and only the @xmath125 , @xmath186 , @xmath187 , and @xmath188 orbitals with the two lowest principal quantum numbers are considered . of course , the inclusion of single - particle rydberg excitation series would give infinite density peaks at the positive ion thresholds . however , the extended large - radius rydberg states are physically different from the compact compound states , and they decouple from the compound state spectra at high @xmath189 , a. a. gribakina and g. f. gribakin , j. phys . b , * 29 * , l809 ( 1996 ) . th . zimmermann , h. @xmath190 , and l. s. cederbaum , phys . * 61 * , 3 ( 1988 ) . r. karazija , _ sums of atomic quantities and mean characteristics of spectra _ ( mokslas , vilnius , 1991 ) . i. s. sobelman , _ atomic spectra and radiative transitions _ ( springer - verlag , berlin , 1992 ) . the normalization integral @xmath191 for this set of values is equal to 0.902 , as this fit , just as the pure pt fit , evidently fails to describe the large number of line strengths greater than 15 a.u . observed in @xcite .
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using a configuration interaction approach we study statistics of the dipole matrix elements ( @xmath0 amplitudes ) between the 14 lower states with @xmath1 and 21st to 100th even states with @xmath2 in the ce atom ( 1120 lines ) .
we show that the distribution of the matrix elements is close to gaussian , although the width of the gaussian distribution , i.e. the root - mean - square matrix element , changes with the excitation energy .
the corresponding line strengths are distributed according to the porter - thomas law which describes statistics of transition strengths between chaotic states in compound nuclei .
we also show how to use a statistical theory to calculate mean squared values of the matrix elements or transition amplitudes between chaotic many - body states .
we draw some support for our conclusions from the analysis of the 228 experimental line strengths in ce [ j. opt .
. am . *
8 * , 1545 ( 1991 ) ] , although direct comparison with the calculations is impeded by incompleteness of the experimental data .
nevertheless , the statistics observed evidence that highly excited many - electron states in atoms are indeed chaotic .
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if the topology of space - time is @xmath5 then the only known exact solution representing a black object is the uniform black string with horizon topology @xmath6 @xcite . though this solution exists for all values of the mass it is unstable below a critical value , @xmath7 , as shown by gregory and laflamme @xcite . horowitz and maeda @xcite argued that a uniform black string can not change its topology into a black hole in finite affine time , making the possibility of such a transition questionable . they suggested the possibility of a transition to a nonuniform black string . gubser @xcite showed the existence of nonuniform black string solutions . the non - uniform black string solution in 6 dimensions was investigated numerically @xcite for a range of the mass values above @xmath7 . nonuniform black string configurations do not exist for masses below the gregory - laflamme point , in the region where the uniform black string solution is unstable @xcite . a natural candidate for a black object in this mass range is a black hole . unfortunately , no exact black hole solutions are known in a 5 ( or more ) dimensional space with a compactified dimension . still , on the basis of physical intuition , a black hole solution should exist for very small values of the mass . when the radius of the horizon is much smaller than the size of the compactified dimension , i.e. when @xmath8 , the black hole should be unaware of the compactification . then a myers - perry @xcite solution should become asymtotically exact . indeed , a numerical solution , extending the myers - perry solution to larger values of the mass was found by harmark and obers @xcite and kudoh and wiseman @xcite . using general arguments kol @xcite suggested that the black hole branch and the nonuniform black string branch meet at a point when the black hole fills the compact dimension . further arguments for such a transition were presented in @xcite . recently , we have studied a related problem , namely the existence of black holes in randall - sundrum @xcite theories . we used @xcite an approximation scheme based on the expansion of solutions in the ratio of the radius of the horizon of the black hole to the ads curvature . in this paper we employ a similar strategy by expanding the metric and other relevant quantities in the ratio of the two natural lengths associated with a black hole configuration , ( i ) the five dimensional schwarzschild radius , @xmath1 , associated with the mass @xmath9 and ( ii ) the compactification length , @xmath2 , defined as the proper circumference of the compact dimension in the region far away from the mass . the dimensionless ratio of these two quantities serves as an excellent expansion parameter . as the solution is an even function of @xmath1 and @xmath2 we use @xmath10 as our expansion parameter . such an expansion for add black holes has recently been proposed and harmark @xcite and gorbonos and kol @xcite evaluated the leading term of the expansion . to find an unique solution we must fix boundary conditions , both at infinity and at the horizon of the black hole . we find solutions in two different regions . in the asymptotic region , @xmath11 an ` asymptotic solution ' , and in the near horizon region , @xmath12 a ` near solution ' is found . using the asymptotic solution we satisfy asymptotic boundary conditions but the boundary conditions at the horizon can not be satisfied . the near solution suffers from the opposite problem , as it can not be used to satisfy asymptotic boundary conditions . to solve those problems we combine the two solutions . we match the asymptotic solution with the near solution in the region @xmath13 where both are valid . this method was used in @xcite to find small black holes in randall - sundrum scenario , and in @xcite to find small black holes on cylinders . the work in this paper is parallel to and agrees with previous results @xcite , but here we calculate the metric up to fourth order in @xmath14 ( second order in @xmath4 ) . the organization of this paper is as follows : in section [ sec : method ] we describe the general method of our calculations , and list the parameters that will be calculated from the metric . in sections [ sec : zeroth]-[sec : second ] we present the detailed calculations up to second order . in section [ sec : summary ] we summarize our results for the metric , up to second order , and discuss the possible scenarios for black holes of increasing mass . a reader not interested in technical details should read the next section , ( [ sec : method ] ) and then proceed directly to the summary and discussions ( [ sec : summary ] ) . as we indicated in the introduction we will investigate black hole solutions of the einstein equation in 5 dimensional space when one of the dimensions is compactified . recently , this problem has attracted the attention of several groups . harmark and obers@xcite introduced the relative tension of black holes as an order parameter , wrote down a generalized smarr formula @xcite , investigated the phase diagram for black objects , and studied the solutions numerically . gorbonos and kol @xcite proposed an analytic approximation scheme based on the expansion in the ratio of the radius of the horizon and the compactification length . this method is similar to the expansion method we used @xcite to investigate black holes in the randall - sundrum @xcite scenario . we will follow a similar path here and calculate further terms of the expansion . by necessity , the perturbation method is applied in two overlapping regions . in the asymptotic region , @xmath11 , @xmath1 is the smallest scale . therefore , an expansion in @xmath14 is actually an expansion in @xmath1 and in every order the metric is a general function of the dimensionless coordinates @xmath15 , @xmath16 we refer to this solution as the ` asymptotic solution . ' in the near horizon region , @xmath12 , @xmath17 is the smallest scale . therefore , an expansion in @xmath14 is actually an expansion in @xmath17 . in every order , the metric is a general function of the dimensionless coordinates @xmath18 , @xmath19 we refer to this solution as the ` near solution . ' in each region , when we calculate the @xmath20-th order terms , einstein s equations is solved in terms of two functions : a gauge function and a wave function , which satisfies a linear , ( inhomogeneous ) , master equation . the differential operator for the master equation in the asymptotic region is @xmath21 where @xmath22 is a coordinate in the compact dimension and @xmath23 is a radial coordinate in 3-dimensional space . in the near region the operator is @xmath24 where @xmath25 is a radial coordinate and @xmath26 is an angular coordinate in 4-dimensional space . the driving term in the master equation for the @xmath20th order contribution depends on the solution in lower orders . it determines an inhomogeneous solution in the wave function . the additional homogeneous solution introduces new parameters that should be fixed by boundary conditions . in empty space , or for the uniform black string configuration , the cylinder is invariant under translations in the compact direction . when a point mass is introduced translation invariance is broken , but a @xmath27 reflection symmetry remains around the location of the mass in the compact direction . so , if we endow the compact direction with the coordinate @xmath28 $ ] and put the mass at @xmath29 then the solution is symmetric about @xmath30 and periodic in @xmath22 with period @xmath2 . asymptotically ( far away from the mass ) , we assume that the metric is minkowski . the leading order correction to the minkowski metric is of order @xmath31 . such a term provides , among others , the four - dimensional newtonian potential . we must impose constraints on the near solution , as well . the horizon of a small black hole has @xmath32 topology . we require that there is no black string attached to the horizon . consequently , the kretchmann scalar must be regular on the @xmath22 axis everywhere , when @xmath33 . furthermore , the surface gravity must be constant , otherwise the horizon is not regular @xcite . to match the asymptotic and the near solutions we must work in the intermediate region @xmath34 , where @xmath35 is a four - space - dimensional radial coordinate . in this region the functions @xmath36 can be expanded in the coordinates . owing to the @xmath37 symmetry the components depend only on @xmath35 and @xmath26 , the angle between the compact direction and the four - dimensional sub - manifold . then one can use a double expansion in @xmath38 and @xmath39 to write the metric as @xmath40 the expansion includes only even powers due to the @xmath27-symmetry as will be explained later . globally , our expansion parameter is @xmath41 . in the two regions @xmath4 has a different meaning . the asymptotic solution is constructed as an expansion in @xmath1 , such that @xmath42 , so one keeps @xmath20 constant and solves the equations for all values of @xmath43 . for example , minkowski metric corresponds to the zeroth order ( @xmath44 , @xmath45 ) approximation to our problem , with appropriate coefficients , @xmath46 . the first order , @xmath47 , @xmath45 , provides linearized gravity ( newtonian potential ) , etc . the near solution is constructed as an expansion in @xmath17 , such that @xmath48 , so at fixed @xmath43 we solve for all values of @xmath20 . here , @xmath49 , with appropriately chosen coefficients , @xmath46 , corresponds to the myers and perry five dimensional black hole ( mps ) @xcite . the matching is done in the intersecting points denoted by circles , as can be seen in fig.([gridfig ] ) . for example , first order of the asymptotic solution intersects zeroth order of the near solution at @xmath50 . we use this information to fix parameters in the asymptotic solution . terms as points in the @xmath51 grid . the horizontal ( vertical ) lines describe the asymptotic ( near ) solution in increasing order of @xmath4 . the matching is done at the intersecting points denoted by circles . ] in the intermediate region the asymptotic solution is expanded in @xmath39 , while the near solution is expanded in @xmath38 . the table shows the sequence of calculating terms in the asymptotic and near solutions . terms connected by an arrow are identified by the double expansion . a similar table has been presented in @xcite ] the calculation is carried out order by order . both solutions ( asymptotic and near ) must be known up to order @xmath52 before calculating order @xmath53 . within order @xmath53 , one should calculate the asymptotic solution first and then the near solution . the sequence of calculation of terms in the two solutions is depicted in fig.([tablefig ] ) . two parameters can be measured in the asymptotic region , the adm mass and the relative tension around the compact dimension . @xcite expanding the asymptotic metric around the minkowski metric we write @xmath54 , then the adm mass , @xmath55 , and the tension , @xmath56 , are defined as @xmath57dw~ , \label{defadmmass}\\ b&=&\int d^{4}x t^{ww}=-\frac{1}{4g_{5}}\lim_{r\rightarrow\infty}r^{2 } \int_{-l/2}^{l/2}\left[\frac{2}{r}h_{rr}+h_{tt , r}\right]dw~. \label{deftension}\end{aligned}\ ] ] it is useful to define the relative tension , @xmath58 . in @xcite the pair @xmath59 is used to describe a phase diagram of all possible black objects . the near solution provides the thermodynamic parameters of the horizon , i.e. entropy and temperature . the entropy is defined as the three dimensional area of the horizon divided by @xmath60 . the temperature is defined through the zeroth law of black holes thermodynamics as the surface gravity divided by @xmath61 . a smarr formula @xcite relates the entropy , temperature , mass , and relative tension as @xcite @xmath62 in the asymptotic region zeroth order in @xmath4 means that we neglect the mass . therefore , the configuration is asymptotically flat ( since the mass is confined to a small region ) . so , the asymptotic form of the metric is minkowski on a cylinder @xmath63 where @xmath22 is the compact coordinate such that @xmath28 $ ] . in the near region zeroth order in @xmath4 means that @xmath64 . the mass point is in five dimensional asymptotic flat infinite manifold . therefore , the near solution is given by the myers - perry @xcite solution ( mps ) , @xmath65 where @xmath1 is the five dimensional schwarzschild radius . in order to match the metrics ( [ a0 ] , [ n0 ] ) one needs to specify a coordinate transformation between @xmath66 and @xmath67 . we choose to work with a specific coordinate transformation throughout the calculation of all orders . ( one can , however , choose to redefine the transformation in each order ) . the coordinates @xmath68 , @xmath69 , and @xmath70 are shared by the coordinate systems . furthermore , we want to preserve the periodic structure in @xmath22 so we choose the transformation [ transformation ] @xmath71~ , \label{z}\\ r&=&\rho\sin\psi~,\label{r}\\ \rho&=&\sqrt{r^{2}+v^{2}}=\sqrt{r^{2}+\frac{l^{2}}{\pi^{2}}\sin^{2}\frac{\pi w}{l } } ~,\label{rho}\\ \psi&=&\arctan\frac{r}{v}=\arctan\frac{\pi r}{l\sin(\pi w / l)}~,\label{psi}\end{aligned}\ ] ] one can verify that the metrics ( [ a0],[n0 ] ) agree up to order @xmath72 . we show now that the expansion contains only terms with integer powers of @xmath73 . it is sufficient to establish that the solution must be an even function of @xmath1 and of @xmath2 ( at fixed @xmath74 ) . the solution is an even function of @xmath1 , because the driving term , the zeroth order near metric ( [ n0 ] ) depends only on @xmath75 . the zeroth order near metric , ( [ n0 ] ) , combined with ( [ transformation ] ) is also an even function of @xmath2 . the conditions imposed on the metric also force it to be even in @xmath2 . the mass is located at @xmath76 , ( @xmath77 ) . the configuration of a mass point on a cylinder has a @xmath27 symmetry about @xmath29 , ( @xmath78 ) , requiring that the terms of the metric depend on @xmath22 and @xmath2 through functions of the form @xmath79 . in the asymptotic region , @xmath11 , we expand the metric as @xmath80 where @xmath81 is the flat metric ( [ a0 ] ) . the general form of the solution for @xmath82 appears in many places in the literature @xcite . we re - derive it here using the matching method . we expand the einstein equation to first order in @xmath4 and get a set of homogeneous linear equations for @xmath82 . using the coordinate transformation @xmath83 we choose the gauge @xmath84 . next we solve @xmath85 for @xmath86 , and @xmath87 for @xmath88 , where @xmath89 is the einstein tensor . the solution is given by [ h1 ] @xmath90 where the gauge function satisfies the symmetry properties @xmath91 and @xmath92 is a constant . the function @xmath93 satisfies the equations @xmath94 the solution to eq.([htteq ] ) can be written as a fourier series @xmath95 the constants @xmath96 are determined by matching ( [ httmodes ] ) to the zeroth order near solution ( [ n0 ] ) . then it follows that up to leading order in @xmath17 the two solutions must be identical , @xmath97 . the constants @xmath96 are given by the fourier expansion of @xmath98 . [ fourier1 ] @xmath99 using ( [ fourier1 ] ) we are able to sum eq.([httmodes ] ) as @xmath100 for @xmath101 the metric is given in terms of the ` zero mode , ' independent of @xmath22 , and can be summarized as [ h1asymp ] @xmath102~,\\ g_{rw}&=&\epsilon w_{,r}(r , w)~.\end{aligned}\ ] ] the constant @xmath92 , which appears in the metric ( [ h1asymp ] ) is related to the tension in the compact dimension @xmath103dw = -\frac{\alpha\epsilon l}{2g_{5}}~. \label{tension}\ ] ] the tension @xmath56 should be zero because we deal here with linearized gravity , with no interaction between the mass and its periodic images . consequently , we must set @xmath104 . dimensional analysis also requires @xmath104 . first , we can deduce from eqs.([h1 ] ) that @xmath105 . the asymptotic solution is an expansion in the mass , so @xmath92 must be independent of the mass and proportional to @xmath2 , the only quantity of the correct dimension at our disposal . as we have discussed earlier , function ( [ htt1final ] ) is even in @xmath2 . if @xmath106 then the functions @xmath86 and @xmath88 , in ( [ h1 ] ) , would have a mixed symmetry under @xmath107 . note that this symmetry argument can and will be used in higher ( @xmath20th ) orders of the expansion to eliminate terms of the form @xmath108 from @xmath109 and @xmath110 . next , we turn to evaluate the conserved adm mass using eqs.([h1asymp ] ) @xmath111dw = \frac{3\pi\mu^{2}}{8g_{5}}~. \label{admmass}\ ] ] the effective four - dimensional newton constant is defined by @xmath112 . using eqs.([h1asymp ] , [ admmass ] ) we find that @xmath113 . we should emphasize that the physically measurable quantities , the relative tension , the mass , and newton s constant , are independent of the gauge function @xmath114 . as we mentioned earlier , in higher orders of the expansion the relative tension in the compact dimension is not zero . however , we require that the adm mass is completely determined by the first order , such that higher orders do not change eq.([admmass ] ) . this way , we make sure that the expansion in @xmath4 is also an expansion in @xmath55 and that thermodynamic properties of the black hole are well defined in each order . we will return to this issue in section [ sec : second ] when we discuss the second order contributions . the gauge function @xmath114 , which appears in ( [ h1 ] ) , can be partially determined by matching to the zeroth order of the near solution . however , we prefer to determine the function completely by matching to the first order of the near solution , as well . in the near region it is convenient to use the coordinates @xmath67 just like in eq.([n0 ] ) . we use the following ansatz for the metric : @xmath115 the functions @xmath116 are expanded to first order in @xmath4 as [ n1 ] @xmath117 the first step is to use a gauge transformation of the form @xmath118 , @xmath119 , which leaves the last term of ( [ spherical ] ) unchanged to leading order , to choose a gauge where @xmath120 this gauge choice simplifies the equations for the rest of the functions to be determined . next , one can solve the equation @xmath121 for @xmath122 . the rest of the equations can be solved in terms of a single function , @xmath123 , obeying a second order partial differential equation . the solution is [ n1solution ] @xmath124\nonumber\\ & & -\frac{2\rho^{2}h_{1,\rho}(\rho,\psi)}{\sin^{2}\psi\,\cos\psi } + \frac{\rho^{3}}{\rho^{2}-\mu^{2}}f_{1,\psi}(\rho,\psi ) -\rho^{2}\tan\psi f_{1,\rho}(\rho,\psi ) ~. \label{vn1}\end{aligned}\ ] ] the wave function @xmath125 satisfies the differential equation @xmath126 the solution of eq.([h1equation ] ) can be written as @xmath127h_{\nu}(\psi)~ , \label{h1nu}\\ h_{\nu}(\psi)&=&\frac{4}{\sqrt{2\pi}}\sin^{3/2}2\psi\left [ \pi\cos\alpha_{\nu}p^{3/2}_{\nu}(\cos2\psi ) -2\sin\alpha_{\nu}q^{3/2}_{\nu}(\cos2\psi)\right]\nonumber\\ & & = ( 2\nu-1)\cos[(2\nu+3)\psi-\alpha_{\nu } ] -(2\nu+3)\cos[(2\nu-1)\psi-\alpha_{\nu}]~ , \label{hpsi}\end{aligned}\ ] ] where @xmath128 and @xmath129 are associated legendre functions of the first and second kind . the functions @xmath130 and @xmath131 will be fixed below using the constraints on function @xmath132 . these constraints come from symmetries , from regularity requirements , and from boundary conditions . @xmath27 symmetry about the direction @xmath78 ( @xmath29 ) implies the conditions [ z2brane ] @xmath133 it can be verified that condition ( [ h1,psi ] ) is automatically satisfied by ( [ h1nu ] ) . for a black hole configuration , unlike for a black string configuration , the mass is localized at the origin . then the components of the metric , ( [ n1 ] ) , are finite at @xmath134 and arbitrary @xmath35 . this will only be true if @xmath135 it can be verified that ( [ nostring ] ) implies @xmath136 . then ( [ hpsi ] ) simplifies to @xmath137 where we have omit a possible overall sign . as we mentioned earlier , the expansion in @xmath41 and the fact that the zeroth order of the near solution ( [ n0 ] ) depends only on @xmath75 imply that the metric is even in @xmath1 . this means that the function @xmath132 should be even in @xmath138 . for large @xmath25 the legendre functions in ( [ h1nu ] ) behave as @xmath139 where @xmath140 are analytic functions . since we require that @xmath132 is even in @xmath25 , @xmath141 should be integer . the integral in ( [ h1nu ] ) is reduced to a sum over integer values of @xmath141 . @xmath142\sin^{3}\psi + 2r\sqrt{r^{2}-1}\sum_{n=1}^{\infty } \left[a_{n } p^{1}_{n}(2r^{2}-1)+b_{n } q^{1}_{n}(2r^{2}-1)\right]\nonumber\\ & & \hspace{2in}\times\left[(2n-1)\sin(2n+3)\psi -(2n+3)\sin(2n-1)\psi\right]~. \label{h1sum}\end{aligned}\ ] ] the case @xmath44 requires special attention , since the @xmath143 . the metric ( [ spherical ] ) is static . therefore , the surface @xmath144 is a killing horizon . the normal vector @xmath145 should be null on the horizon . this implies that on the horizon @xmath146 , thus the horizon is located at constant @xmath147 . using metric ( [ n1 ] ) we expand @xmath148 in @xmath4 as @xmath149 . the conditions @xmath150 and @xmath151 restrict the gauge function @xmath152 @xmath153 the surface gravity for metric ( [ spherical ] ) is defined as @xmath154 the surface gravity should be constant on the horizon , otherwise the horizon is singular @xcite . when we expand the metric in @xmath4 , the surface gravity must be constant in every order of the expansion . we use metric ( [ n1 ] ) and conditions ( [ f1horizon ] ) to evaluate the surface gravity in order @xmath4 @xmath155 the requirement of a constant surface gravity constrains the function @xmath132 . at @xmath156 the legendre functions in eq.([h1sum ] ) take the values @xmath157 , @xmath158 . we use the representation ( [ h1sum ] ) to evaluate the surface gravity ( [ kappah1 ] ) @xmath159 the set @xmath160 is complete on the interval @xmath161 $ ] , therefore , the solution to eq.([chi1 ] ) is @xmath162 in other words , the sum in eq.([h1sum ] ) contains legendre functions of the first kind only . the functions @xmath163 are polynomials of order @xmath164 in @xmath165 . the remaining free parameters of the near solution , @xmath166 , must be determined from matching the two gauge functions , @xmath114 in the asymptotic solution ( [ a1 ] ) and @xmath167 in the near solution ( [ n1 ] ) . we use ( [ transformation ] ) to transform the asymptotic solution to @xmath168 coordinates and then we expand it in @xmath17 . the gauge function @xmath114 is also expanded as @xmath169 we keep the explicit factor @xmath170 in ( [ wn ] ) to insure that @xmath171 . we start with zeroth order in @xmath17 . we transform the asymptotic metric using ( [ transformation ] ) and expand it to zeroth order in @xmath17 . [ asymprhopsi0 ] @xmath172~ , \label{vasymp0}\end{aligned}\ ] ] where the superscript @xmath173 stands for asymptotic and the functions are defined in ansatz ( [ spherical ] ) . matching ( [ asymprhopsi0 ] ) to zeroth order of the near solution ( [ n0 ] ) determines the function @xmath174 @xmath175 the asymptotic solution is [ asymprhopsi1 ] @xmath177~ , \label{aasymp1}\\ u^{a}&=&1+\frac{\pi^{2}}{l^{2}}\left[\rho^{2}\cos^{2}\psi\sin^{2}\psi -\frac{\mu^{2}\sin\psi\cos^{2}\psi(5\sin3\psi-37\sin\psi)}{12 } -2\mu^{2}\frac{\sin\psi}{\rho } w_{2,\psi}(\rho,\psi)\right]~ , \label{uasymp1}\\ v^{a}&= & -\frac{\pi^{2}}{l^{2}}\left[\rho^{3}\sin\psi\cos^{3}\psi + \mu^{2}\rho\sin\psi\cos^{3}\psi(\sin^{2}\psi+2 ) + \mu^{2}\rho\sin\psi w_{2,\rho}(\rho,\psi ) -\mu^{2}\cos\psi w_{2,\psi}(\rho,\psi)\right ] ~. \label{vasymp1}\end{aligned}\ ] ] the near solution ( [ n1 ] ) should be expanded to second order in @xmath1 , which appears in @xmath41 and in the rescaled coordinate @xmath138 . the large @xmath25 behavior of the terms of @xmath132 , ( [ h1sum ] ) , is @xmath178 these expressions contribute by terms of order @xmath179 in metric ( [ spherical ] ) . we deduce that @xmath180 terms contribute by negative orders in @xmath1 and must be eliminated . therefore we impose @xmath181 if @xmath180 . for similar reasons , gauge function @xmath152 should also be a polynomial of order @xmath182 in @xmath35 . thus , functions @xmath132 and @xmath152 are @xmath183\sin^{3}\psi~,\label{h10}\\ f_{1}&=&(\mu^{-2}\rho^{2}-1)f_{2}(\psi)-\zeta_{1}~,\label{f1asymp}\end{aligned}\ ] ] where we have already imposed @xmath184 , ( [ f1horizon ] ) . the exact form of the periodic function @xmath185 will be fixed below . then the near metric , expanded to second order in @xmath1 , is [ nearmu2 ] @xmath186~ , \label{anear2}\\ u^{n}&=&1-\frac{2\tan^{2}\psi}{l^{2}}\left[\rho^{2}[2a_{0}+f_{2}(\psi ) + \cot\psi f_{2}'(\psi)]-\mu^{2}[4a_{0}-6b_{0}+f_{2}(\psi)+\zeta_{2 } + \cot\psi f_{2}'(\psi)]\right]~ , \label{unear2}\\ v^{n}&= & \frac{\rho^{3}\tan\psi}{l^{2 } } \left[-4a_{0}-2f_{2}(\psi)+\cot\psi f_{2}'(\psi)\right ] ~. \label{vnear2}\end{aligned}\ ] ] comparing ( [ asymprhopsi1 ] ) and ( [ nearmu2 ] ) we find that @xmath187 ~.\label{w2}\end{aligned}\ ] ] at this point the near solution still contains two free parameters , @xmath188 and @xmath189 . the @xmath27-symmetry condition , ( [ f1brane ] ) , imposes one constraint on these parameters @xmath190 a constraint , determining @xmath189 , is derived from the first law of black hole thermodynamics . it will be discussed in the next section . the zeroth law of black hole thermodynamics states that the temperature of a black hole is @xmath191 where @xmath192 is the surface gravity , which is constant on the horizon . to calculate the temperature for the near solution we use eqs.([kappa ] , [ kappah1 ] , [ bn ] ) to find that @xmath193 the first law of black hole thermodynamics is @xmath194 , where the entropy is proportional to the area of the horizon @xmath195 we calculate the entropy for the near solution and find that @xmath196 according to the first law the temperature can be found as @xmath197 . the mass appears in the entropy only through @xmath9 and @xmath41 , so if we combine the first law and the zeroth law we get @xmath198=2\pi\mu[1 + 5\zeta_{1}\epsilon]~. \label{t2}\ ] ] this fixes the last parameter . if we combine eqs.([b0],[t2 ] ) we find that @xmath199 to summarize , the near metric , to first order in @xmath4 , is [ near1final ] @xmath200~ , \label{b1final}\\ a^{n}&=&1+\frac{\pi^{2}\epsilon}{3}\left[(3\frac{\rho^{2}}{\mu^{2}}-1)\cos^{4}\psi-1 \right]~ , \label{a1final}\\ u^{n}&=&1+\pi^{2}\epsilon(\frac{\rho^{2}}{\mu^{2}}-1)\sin^{2}\psi\cos^{2}\psi~ , \label{u1final}\\ v^{n}&= & -\pi^{2}\epsilon\frac{\rho^{3}}{\mu^{2}}\sin\psi\cos^{3}\psi ~. \label{v1final}\end{aligned}\ ] ] the location of the horizon is at @xmath201 . the entropy and the temperature are @xmath202 these expressions are in agreement with previous results @xcite . the only freedom left in the first order metric is the @xmath203 terms of the gauge function in the asymptotic solution ( [ wn ] ) . the @xmath44 and @xmath204 terms are fixed in the region @xmath205 by our matching procedure , as given in eqs.([w0],[w2 ] ) . in addition , in the asymptotic region , @xmath101 , the metric should be minkowski , therefore , we require that @xmath206 . a form of the gauge function which is consistent with these conditions appears in appendix [ app : asymp2 ] . prior to completing calculations in second order we describe our general procedure for calculating higher order contributions . first we consider the asymptotic solution , which is fully determined ( up to gauge freedom ) by the lower order contributions . in the asymptotic region we expand the metric in @xmath4 as @xmath207 assume that we know the solution up to order @xmath208 and we intend to obtain the @xmath20th order solution . einstein s equation is linear in the @xmath209 but due to lower order contributions it is inhomogeneous . the solution is similar to ( [ h1 ] ) but , in addition to the solution of the homogeneous equation includes extra terms corresponding to a particular solution ( the particular solution is denoted by @xmath210 ) , as follows [ hn ] @xmath211 where @xmath92 is a constant , and @xmath212 is a gauge function which is periodic and antisymmetric in @xmath22 . the function @xmath213 is also periodic and symmetric in @xmath22 . it satisfies the equation @xmath214 where @xmath215 is the deriving term , which depends on lower orders , @xmath216 , @xmath217 . the homogeneous solution to eq.([httneq ] ) can be written as a fourier series . consequently , we have @xmath218 the particular solution @xmath210 is completely determined by the lower orders of the asymptotic solution ( without any use of the near solution ) . the free parameters are @xmath92 and the set @xmath219 . these can be determined by matching to the lower orders of the near solution as follows . take @xmath220 from the near solution up to order @xmath52 and expand it in @xmath1 . take the term of order @xmath221 and find its fourier series just like in eq.([fourier1 ] ) . compare that fourier series with the forier series of @xmath213 to lowest order in @xmath17 , and determine the constants @xmath219 . just like in first order , we can show that @xmath222 . solution ( [ hn ] ) should be even in @xmath2 . the functions @xmath223 and @xmath213 are even functions of @xmath2 since these functions are determined by lower orders but for dimensional reasons @xmath224 is odd . therefore , @xmath225 must vanish . next , we have to make sure that the definition of the adm mass does not change . so , we require that @xmath226dw=0~. \label{fixm}\ ] ] in general , the tension in the compact dimension , ( [ tension ] ) , does not vanish . as a result , the effective four - dimensional newton s constant , @xmath227 , acquires a correction of order @xmath53 . this means that @xmath227 depends on the mass and the _ equivalence principle is violated . _ at this point the asymptotic solution is determined up to order @xmath53 , except for the gauge function @xmath212 , which is ( partially ) determined by matching to the near solution to orders up to @xmath53 . near the horizon it is convenient to use the metric in form ( [ spherical ] ) and expand it in @xmath4 as follows [ emetric ] @xmath228 we need the near solution up to order @xmath208 and the asymptotic solution up to order @xmath20 to calculate the @xmath20th order near solution . again , the @xmath20th order einstein s equation is a linear inhomogeneous equation in @xmath229 . the inhomogeneity depends on @xmath230 . we solve the equations , in a way , similar to that for the first order correction . the first step is to apply a gauge transformation of the form @xmath231 , @xmath232 to arrive at a gauge , in which @xmath233 next , one can solve the inhomogeneous equation @xmath121 for @xmath234 . the rest of the equations can be solved in terms of a single function , @xmath235 , obeying a second order partial differential equation , which is similar to ( [ h1equation ] ) , @xmath236 where @xmath237 depends on the lower order corrections . the solution of eq.([hnequation ] ) can be found by the method of separation of variables , just like it has been done when we have calculated the first order correction ( [ h1nu ] ) . the eigenfunctions of eq.([hnequation ] ) are the functions @xmath238 , which appear in eq.([hpsi ] ) . the driving term , @xmath237 , can also be expanded in the set @xmath238 . in addition , the boundary conditions at @xmath78 and at @xmath134 in @xmath20th order are the same as in first order , eqs.([h1,psi ] , [ nostring ] ) . and just like in first order the function @xmath239 should be even in @xmath35 . so , the general solution of eq.([hnequation ] ) is @xmath240\sin^{3}\psi\nonumber\\ & & + \sum_{n=1}^{\infty } \left[2r\sqrt{r^{2}-1}\left(a_{n } p^{1}_{n}(2r^{2}-1)+b_{n } q^{1}_{n}(2r^{2}-1)\right)+p_{n}(r)\right]\nonumber\\ & & \hspace{2in}\times\left[(2n-1)\sin(2n+3)\psi -(2n+3)\sin(2n-1)\psi\right]~ , \label{hnsum}\end{aligned}\ ] ] where @xmath241 is a particular solution of the inhomogeneous equation ( [ hnequation ] ) . the particular solutions are completely determined by the lower orders of the near solution ( without using the asymptotic solution ) . the homogeneous part should be completely fixed by matching to the asymptotic solution . the matching is done as follows : take the asymptotic solution up to order @xmath53 and transform it to the @xmath168 coordinates , using ( [ transformation ] ) . expand the solution in @xmath17 , take the term of order @xmath242 and compare it to the near solution of order @xmath53 to fix the parameters @xmath243 , and the gauge functions @xmath244 and @xmath212 . at this point the function @xmath212 is determined only up to order @xmath242 and one should fix it completely before proceeding to the next order . in the next section we apply this method to calculate the second order contributions . up to first order , the asymptotic solution is given by equations ( [ a0 ] ) , ( [ a1 ] ) , ( [ h1 ] ) , and ( [ htt1final ] ) . the second order solution is described in the appendix [ app : asymp2 ] . it contains an inhomogeneous part , @xmath245 , which is completely determined by the first order solution , and a homogeneous part , which includes the wave function @xmath246 , and the gauge function @xmath247 [ h2 ] @xmath248 the wave function @xmath249 satisfies the homogeneous part of eq.([httneq ] ) , and can be written in the form of eq.([httnmodes ] ) @xmath250 to fix the constants , @xmath219 , we follow the prescription that appears after eq.([httnmodes ] ) . we take @xmath220 from the first order near solution , ( [ b1final ] ) , and expand it in @xmath1 , up to order @xmath251 , to get @xmath252 + \mu^{4}\frac{\pi^{2}(4\cos^{4}\psi-3)}{12l^{2}\rho^{2}}~. \label{gttn1}\ ] ] then we take @xmath220 from the second order asymptotic solution and transform it to the @xmath168 coordinates , using ( [ transformation ] ) . after expanding it in @xmath17 , up to order @xmath253 , we obtain @xmath254 + \mu^{4}\left[\frac{2\cos2\psi+1}{4\rho^{4 } } -\frac{\pi^{2}(3\cos6\psi+124\cos^{4}\psi+21\cos2\psi+4 ) } { 96l^{2}\rho^{2}}\right]+\epsilon^{2}h_{tt}^{(h)}~. \label{gtta2}\ ] ] we find @xmath249 from @xmath255 and calculate its fourier coefficients @xmath256\nonumber\\ & & = \frac{\pi^{3}l\sinh\frac{2\pi r}{l}\left(-\cosh\frac{4\pi r}{l } + 16\cosh\frac{2\pi r}{l}\cos\frac{2\pi w}{l}+5\cos\frac{4\pi w}{l}-20\right ) } { 24r\left(\cosh\frac{2\pi r}{l}-\cos\frac{2\pi w}{l}\right)^{3}}~. \label{htth}\end{aligned}\ ] ] the asymptotic ( @xmath101 ) form of the metric is [ h2asymp ] @xmath257 the contribution to the adm mass vanishes indeed , @xmath258dw = \frac{\pi^{3}l^{2}}{4}-2r^{2}\left.w_{,r}^{(2)}\right|_{w =- l/2}^{w = l/2}=0 ~ , \label{deltam2}\ ] ] where the last equality follows from boundary condition ( [ w2l2 ] ) . the tension ( [ tension ] ) is now @xmath259dw = -m\frac{\pi^{2}\epsilon}{6}~. \label{tension2}\ ] ] following section [ subsec : nearn ] , we find that the second order solution is [ emetric2 ] @xmath260~ , \label{b2}\\ a_{2}&=&\frac{4h_{2,\psi}(\rho,\psi)}{\sin^{2}\psi\,\cos\psi } + 2f_{2}(\rho,\psi)+2\rho f_{2,\rho}(\rho,\psi)-\frac{\pi^{4}\cos^{2}\psi}{864\mu^{4 } } \left[\mu^{2}(105\rho^{2}+11\mu^{2})\cos6\psi\right.\nonumber\\ & & \left.+(-64\rho^{4}+138\mu^{2}\rho^{2}+58\mu^{4})\cos4\psi + ( 1472\rho^{4}+1383\mu^{2}\rho^{2}+237\mu^{4})\cos2\psi -768\rho^{4}+774\mu^{2}\rho^{2}-2\mu^{4}\right]~ , \label{a2}\\ u_{2}&=&-2 \frac{6\rho h_{2}(\rho,\psi ) -(2\rho^{2}-\mu^{2})h_{2,\rho}(\rho,\psi)}{\rho\sin\psi\,\cos^{2}\psi } -2\tan^{2}\psi f_{2}(\rho,\psi)-2\tan\psi f_{2,\psi}(\rho,\psi)~ , \label{u2}\\ v_{2}&= & \frac{\rho(2\rho^{2}-\mu^{2})}{(\rho^{2}-\mu^{2})\sin^{2}\psi\,\cos\psi } \left[3h_{2}(\rho\psi)-\tan\psi h_{2,\psi}(\rho\psi)\right ] -\frac{2\rho^{2}h_{2,\rho}(\rho\psi)}{\sin^{2}\psi\,\cos\psi}\nonumber\\ & & + \frac{\rho^{3}}{\rho^{2}-\mu^{2}}f_{2,\psi}(\rho,\psi ) -\rho^{2}\tan\psi f_{2,\rho}(\rho,\psi ) + \frac{\pi^{4}\rho(2\rho^{2}-\mu^{2})\sin2\psi}{864\mu^{4}(\rho^{2}-\mu^{2})}\left [ \mu^{2}(15\rho^{2}+\mu^{2})\cos6\psi\right.\nonumber\\ & & \left . + ( -8\rho^{4}+29\mu^{2}\rho^{2}+4\mu^{4})\cos4\psi + ( 184\rho^{4}+185\mu^{2}\rho^{2}+11\mu^{4})\cos2\psi -96\rho^{4}+99\mu^{2}\rho^{2}-16\mu^{4}\right]~. \label{v2}\end{aligned}\ ] ] the function @xmath261 satisfies the differential equation ( [ hnequation ] ) with @xmath262 where the functions @xmath263 are given in eq.([hpsisin ] ) . the solution of eq.([hnequation ] ) with inhomogeneity ( [ s2 ] ) can be written in form eq.([hnsum ] ) with @xmath264 . for each of the non vanishing @xmath265 one can add combinations of the homogeneous solution which includes the legendre functions as appear in eq.([hnsum ] ) . these should be chosen such that @xmath265 is regular at @xmath266 , and is a polynomial in @xmath35 of the lowest order possible ( in principle , the homogeneous solution is of order @xmath267 ) . in fact , the particular solution for eq.([hnequation ] ) with inhomogeneity ( [ s2 ] ) includes polynomials of order @xmath268 in @xmath35 , only , [ h2p ] @xmath269 in addition , to avoid negative powers of @xmath1 , the homogeneous part of @xmath261 and the gauge function @xmath270 should be polynomials of order @xmath268 in @xmath271 . @xmath272\sin^{3}\psi + 8a_{1}r^{2}(r^{2}-1)(1+\cos^{2}\psi)\sin^{3}\psi~,\label{h2}\\ f_{2}(\rho,\psi)&\equiv & ( r^{2}-1)^{2}f_{4}(\psi ) + ( r^{2}-1)f_{2}(\psi)+f_{0}(\psi)~.\label{f2}\end{aligned}\ ] ] we now turn to imposing boundary conditions . the horizon is located at a constant @xmath147 . we set @xmath273 and solve @xmath150 for @xmath274 . we find @xmath275~.\label{f0}\ ] ] the surface gravity is constant ( due to the fact that the legendre functions of the second kind are not included in ( [ h2 ] ) ) , @xmath276~.\label{kappa2}\ ] ] we match the near solution , ( [ emetric2 ] ) , to the asymptotic solution , ( [ asymp2 ] ) , expanded to order @xmath277 . comparing the expressions for @xmath220 , we find that @xmath278~,\label{2f4}\\ f_{2}(\psi)&=&\frac{\pi^{4}}{17280}\left[15\cos8\psi+188\cos6\psi + 1248\cos4\psi+3156\cos2\psi+2993\right]~.\label{2f2}\end{aligned}\ ] ] matching the other components , as well , we find the constants @xmath279 the gauge function , @xmath280 , is @xmath281~. \label{w2lexpand}\end{aligned}\ ] ] the metric ( [ emetric2 ] ) should be symmetric about the plane @xmath78 . as a result we find that @xmath282 we calculate the entropy of the black hole @xmath283~. \label{entropy2}\ ] ] we compare the zeroth law @xmath284 and the first law @xmath197 to find that @xmath285 to summarize , the location of the horizon , the entropy , and the temperature are @xmath286~. \label{rhoh2}\\ s&=&\frac{\pi^{2}\mu^{3}}{2g_{5}}\left[1+\frac{\pi^{2}\epsilon}{8 } + \frac{\pi^{4}\epsilon^{2}}{384}\right]~. \label{s2final}\\ t&=&\frac{1}{2\pi\mu}\left[1-\frac{5\pi^{2}\epsilon}{24 } + \frac{43\pi^{4}\epsilon^{2}}{1152}\right]~. \label{t2final}\end{aligned}\ ] ] the near metric is given by [ near4final ] @xmath287 + \frac{\pi^{4}\epsilon^{2}}{5760\mu^{2}}\left[1570\mu^{2}+17\rho^{2 } -4(14\rho^{2}-395\mu^{2})\cos2\psi\right.\right.\nonumber\\ & & \left.\left.+(52\rho^{2}+580\mu^{2})\cos4\psi -4(2\rho^{2}-25\mu^{2})\cos6\psi -(\rho^{2}-2\mu^{2})\cos8\psi\right]\frac{}{}\right)~ , \label{b4final}\\ a^{n}&=&1+\frac{\pi^{2}\epsilon}{3\mu^{2}}\left[(3\rho^{2}-\mu^{2})\cos^{4}\psi-1 \right]+\frac{\pi^{4}\epsilon^{2}}{5760\mu^{4}}\left[1800\rho^{4}+1324\mu^{2}\rho^{2 } -307\mu^{4 } + 4(675\rho^{4}+422\mu^{2}\rho^{2}-325\mu^{4})\cos2\psi\right.\nonumber\\ & & \left . + 8(135\rho^{4}+88\mu^{2}\rho^{2}-55\mu^{4})\cos4\psi + 4(45\rho^{4}+26\mu^{2}\rho^{2}-15\mu^{4})\cos6\psi + 5\mu^{2}(4\rho^{2}-\mu^{2})\cos8\psi\right]~ , \label{a4final}\\ u^{n}&=&1+\frac{\pi^{2}\epsilon}{\mu^{2}}(\rho^{2}-\mu^{2})\sin^{2}\psi\cos^{2}\psi -\frac{\pi^{4}\epsilon^{2}\sin^{2}\psi}{1440\mu^{4 } } \left[-540\rho^{4}-970\mu^{2}\rho^{2}+385\mu^{4}\right.\nonumber\\ & & \left . -5(144\rho^{4}+229\mu^{2}\rho^{2}-385\mu^{4})\cos2\psi -5(\rho^{2}-\mu^{2})(36\rho^{2}+82\mu^{2})\cos4\psi -55\mu^{2}(\rho^{2}-\mu^{2})\cos6\psi\right]~ , \label{u4final}\\ v^{n}&= & -\frac{\pi^{2}\epsilon\rho^{3}\sin\psi\cos^{3}\psi}{\mu^{2 } } -\frac{\pi^{4}\epsilon^{2}\rho^{3}\sin2\psi}{2880\mu^{4 } } \left[990\mu^{2}+540\rho^{2 } + ( 720\rho^{2}+1207\mu^{2})\cos2\psi\right.\nonumber\\&+&\left.2(90\rho^{2}+89\mu^{2})\cos4\psi + 25\mu^{2}\cos6\psi\right ] ~. \label{v4final}\end{aligned}\ ] ] in this paper we have calculated the metric of a small black hole in a five dimensional cylinder . the metric is found perturbatively , with an expansion parameter @xmath288 . @xmath55 is the physical adm mass and @xmath2 is the circumference of the compact dimension , both measured at infinity . we calculate the metric , up to second order in @xmath4 ( fourth order in the ratio @xmath14 ) . the asymptotic solution is given in eqs.([a1],[h1],[htt1final ] ) and in the appendix [ app : asymp2 ] . the near solution is provided in eqs . ( [ spherical],[near4final ] ) . we obtained the following expressions for the entropy and the temperature of the black hole @xmath289~ , \label{sfinal}\\ t&=&\frac{1}{2\pi\mu}\left[1-\frac{5\pi^{2}\epsilon}{24 } + \frac{43\pi^{4}\epsilon^{2}}{1152}\right]~. \label{tfinal}\end{aligned}\ ] ] these , and other results , presented below , were previously obtained in first order of @xmath4 by gorbonos and kol @xcite and by harmark @xcite . using the smarr formula we find the relative binding energy @xmath290 we have chosen the coordinate system such that the horizon is independent of the angle , @xmath26 , between the three dimensional space and the @xmath22 axis . as the scale of the five dimensional radial coordinate , @xmath35 , is fixed by the term of the metric , @xmath291 , the radius of the horizon is uniquely determined . the horizon is located at @xmath147 as in eq.([spherical ] ) . we find that @xmath292~. \label{rhohfinal}\ ] ] the question of transition from black hole to black string , as a function of the mass parameter , has been discussed at length in the literature @xcite , @xcite . one scenario proposed was the transition at the intersection of the black hole and nonuniform black string lines in the @xmath293 ( mass - relative tension ) phase diagram . we use the second order results for the small black hole to extrapolate to such a transition . we study three aspects of the transition ; the mass - relative tension phase diagram , comparison of entropies , and the change of topology ( using two different approaches ) . a uniform black string is described by the metric @xmath294 the relative tension is constant , @xmath295 , and the entropy is @xmath296 the entropy and relative tension of the non - uniform black string were have not been found numerically yet.@xcite . such a configuration exists for masses larger than the gregory - laflamme mass , which is , in terms of our variables , @xmath297 . in fig.([phasediagram ] ) we draw the @xmath298 phase diagram for the uniform black string , the non - uniform black string ( just the leading order term near the gl transition ) , and the black hole branches . the black hole turns into a nonuniform black string beyond the transition point , @xmath299 . we did not include recent numerical results @xcite , which , due to numerical difficulties , have large errors @xcite in fig.([entropies ] ) we draw the entropies of the black objects in units of @xmath300 . comparing ( [ sfinal ] ) and ( [ bsentropy ] ) we find that the enropies of the uniform black string and the black hole are equal for @xmath301 . below that value the black hole has higher entropy , and above that value the black string has higher entropy . this simple check suggests that the black hole will be unstable for @xmath302 . note again that our second order expansion in @xmath4 shows good convergence . using the first order expansion only the intersection of entropies would occur at @xmath303 . a transition between a black hole and a black string requires a topology change of the horizon . this change is likely to happen when the horizon of the black hole fills the compact dimension and touch itself . at this point a small pinch of the horizon will turn it into a non - uniform black string . in the notation we use , eq.([transformation ] ) , this will happen when @xmath304 the first order formula would give @xmath305 . again , we see a fairly good convergence , making our result reasonably reliable . we will see a further indication for this point in another , independent , calculation of the critical value of @xmath4 . the apparent agreement of @xmath306 with @xmath307 ( equality of the entropies of the uniform black string and the black hole ) is somewhat problematic . we see no mathematical reason for such a agreement , though it would be interesting if these points also coincided . note that for geometric reasons the black hole branch should cease to exist above the critical value of @xmath4 so the stable state of the system is the uniform black string , which has a higher entropy than the nonuniform black string . in terms of the coordinates we chose the horizon is located at constant @xmath35 . however , this does not mean it is spherical . compactification breaks the @xmath308 symmetry . the horizon can be an oblate ( elongated perpendicular to the compact direction ) or a prolate ( elongated parallel to the compact dimension ) ellipsoid . there is no generally accepted method for distinguishing between the oblate and the prolate configurations . some authors @xcite define the eccentricity of the black hole as @xmath309 where @xmath310 is the maximal area of a cross section of the horizon parallel to the compact dimension , and @xmath311 is the maximal area of a cross section of the horizon perpendicular to the compact dimension . @xmath312 a prolate ( oblate ) horizon has positive ( negative ) eccentricity . we use the near solution ( [ emetric2 ] ) to find that @xmath313 . we see that the horizon is prolate . another possible measure of the eccentricity is given by the intrinsic ricci scalar of the horizon , which is calculated in terms of the three metric @xmath314 ~.\label{horizonmetric}\ ] ] using the near solution , the intrinsic curvature of ( [ horizonmetric ] ) is @xmath315~.\label{intrinsicr}\ ] ] it is maximal at @xmath134 , which indicates that the horizon is indeed prolate . one might speculate that a prolate horizon will tend to grow in the periodic direction and turn into a nonuniform black string . however , the circumference of the compact direction grows as well @xcite and the transition into a black string is not obvious . in @xcite it is suggested to measure the inter polar distance , which is the proper distance between the poles of the prolate horizon along the compact dimension . in principle , to calculate this distance we need to break the integral defining this distance into two parts . in the first part we should use the near solution and in the other we should use the asymptotic solution . in the near horizon region one should integrate along @xmath134 from @xmath147 to some arbitrary point @xmath316 . in the asymptotic region we should integrate along @xmath317 from @xmath318 to @xmath319 @xmath320 in @xcite it was found that the zeroth order approximation is @xmath321 , which means that the circumference grows enough to make room for the black hole . if the mass ( @xmath4 ) value , at which the black hole fills the compact dimension , is small enough then it is possible that the integral over the near solution is sufficient in ( [ lpoles ] ) , or , in other words , we can choose @xmath322 . then we obtain @xmath323 first of all , we see that the distance between the poles of the horizon is a decreasing function of @xmath4 . if we solve the equation @xmath324 we obtain @xmath325 , which is in a rough agreement with values we obtained for the critical value , @xmath326 , above . taking the first order term only we would obtain @xmath327 . clearly , the convergence of @xmath328 is not as good as those of @xmath326 and @xmath306 . one reason for the poorer convergence is that for very small @xmath4 ( [ lpoles2 ] ) must fail , as a significant contribution coming from the asymptotic solution was omitted . it should work , however in the region where @xmath329 . fortunately , in this region the @xmath4-expansion is still fairly reliable and the rough agreement of @xmath330 is encouraging . this work is supported in part by the u.s . department of energy grant no . de - fg02 - 84er40153 . we thank richard gass and cenalo vaz for fruitful discussions . we also thank barak kol , hideaki kudoh , evgeny sorkin , and toby wiseman for their comments , which helped us improving this work . the second order asymptotic solution is given by eqs.([h2 ] ) [ h2anh ] @xmath332-\frac{r}{8}\left[h\,h_{,r } + 4w\,h_{,rw}\right]+(w_{,r})^{2}-r\,k_{,w } ~,\\ \bar{h}_{ww}^{(2)}&=&\frac{r^{2}}{16}\left[(h_{,r})^{2 } + ( h_{,w})^{2}\right]-\frac{r}{2}\left[2w_{,w}h_{,r } + w\,h_{,rw}\right]+\frac{3}{16}h^{2}+(w_{,w})^{2}-r\,k_{,w } ~,\\ \bar{h}_{rw}^{(2)}&=&k-\frac{r}{16}\left[h\,h_{,w}+8w_{,r}h_{,r}\right ] + w_{,w}w_{,r}~,\end{aligned}\ ] ] where @xmath333 , @xmath334 , and @xmath335 , @xmath336~ , \label{w1final}\\ k&=&\frac{\pi l^{3}}{16r^{3}}\coth\frac{2\pi r}{l } \left[1-\frac{2\pi r}{l}\coth\frac{2\pi r}{l}\right ] \arctan\left(\tan\frac{\pi w}{l}\coth\frac{\pi r}{l}\right ) + \frac{\pi l^{3}\sin\frac{2\pi w}{l } \left[1-\frac{2\pi r}{l}\coth\frac{2\pi r}{l}\right ] } { 32r^{3}\left[\cosh\frac{2\pi r}{l}-\cos\frac{2\pi w}{l}\right]}~.\end{aligned}\ ] ] the function @xmath337 was chosen such that @xmath338 . it satisfies eqs . ( [ w0],[w2 ] ) , and it has a structure similar to @xmath339 . @xmath340 is an odd function of @xmath22 , but @xmath341 . however , @xmath27-symmetry implies that @xmath342 . this means that the gauge function , @xmath280 , has nontrivial boundary conditions at @xmath343 @xmath344~. \label{w2l2}\ ] ] to match the asymptotic and the near solution we have to transform the asymptotic solution to the @xmath168 coordinates , using ( [ transformation ] ) , and expand to order @xmath277 . we also expand the gauge function @xmath280 as @xmath345~. \label{w2near}\ ] ] the asymptotic metric in the @xmath168 coordinates , expanded to order @xmath277 is [ asymp2 ] @xmath346~ , \label{uasymp2}\\ v^{a}&= & -\frac{\pi^{2}}{l^{2}}\left[\rho^{3}\sin\psi\cos^{3}\psi + \mu^{2}\rho\sin\psi\cos^{3}\psi(\sin^{2}\psi+2 ) + \mu^{2}\rho\sin\psi w_{2,\rho}(\rho,\psi ) -\mu^{2}\cos\psi w_{2,\psi}(\rho,\psi)\right ] ~. \label{vasymp2}\end{aligned}\ ] ] t. harmark and n.a . obers , jhep 0205,032 ( 2002 ) , e - print archive : hep - th/0204047 ; t. harmark , phys.rev.d69:104015,2004 e - print archive : hep - th/0310259 t. harmark and n.a . obers , e - print archive : hep - th/0309230 h. kudoh and t. wiseman , prog.theor.phys.111:475-507,2004 e - print archive : hep - th/0310104 ; e - print archive : hep - th/0409111 ; b. kol , e - print archive : hep - th/0206220 ; b. kol and t. wiseman , e - print archive : hep - th/02304070 ; e. sorkin , b. kol , t. piran _ phys.rev _ .d 69:064032 ( 2004 ) e - print archive : hep - th/0310096 ; d. karasik , c. sahabandu , p. suranyi , and l.c.r . wijewardhana , _ phys . rev . _ d70:064007 ( 2004 ) , e - print archive : gr - qc/0404015 t. harmark , phys.rev.d69:104015,2004 e - print archive : hep - th/0310259 ; d. gorbonos and b. kol , jhep 0406:053,200 , e - print archive : hep - th/0406002 l. smarr , j.w . york _ phys . _ d17 , 2529 ( 1978 ) i. rcz and r. m. wald , _ class . _ * 9*:2643 - 2656 ( 1992 ) .
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we study black hole solutions in @xmath0 space , using an expansion to fourth order in the ratio of the radius of the horizon , @xmath1 , and the circumference of the compact dimension , @xmath2 .
a study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at @xmath3 . at the same value of @xmath4 the entropies of the uniform black string and of the black hole are approximately equal
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the physics of quantum dots continuously attracted a lot of experimental and theoretical interest . @xcite within the assumption that the thouless energy ( @xmath0 ) is much larger than mean single - particle level spacing ( @xmath1 ) , @xmath2 , an effective zero - dimensional hamiltonian has been derived . @xcite in this so - called universal hamiltonian the electron - electron interaction that involves a set of matrix elements in the single - particle basis is reduced to just three parameters : the charging energy ( @xmath3 ) , the ferromagnetic exchange ( @xmath4 ) and the interaction in the cooper channel . the single particle energies are random quantities with wigner - dyson statistics . thus the universal hamiltonian provides a convenient framework for the theoretical description of quantum dots . the charging energy ( typically @xmath5 ) restricts the probability of real electron tunneling through a quantum dot at low temperatures @xmath6 . @xcite this phenomenon of the coulomb blockade leads to suppression of the tunneling density of states in quantum dots at low temperatures @xcite . it was also understood that a small enough exchange interaction @xmath7 is important for a quantitative description of the experiments on low temperature ( @xmath8 ) transport through quantum dots fabricated in a two - dimensional electron gas . @xcite for a quantum dot of size @xmath9 ( @xmath10 stands for the fermi wave length ) the exchange interaction can be estimated by bulk value of the fermi - liquid interaction parameter ( @xmath11 ) : @xmath12 . as it is well - known , strong enough exchange interaction in bulk materials leads to a stoner instability at @xmath13 and a corresponding quantum phase transition between a paramagnet and a ferromagnet . in quantum dots it is possible to realize an interesting situation in which the ground state has a finite total spin . @xcite in the case of the equidistant single - particle spectrum it occurs for @xmath14 . as @xmath15 increases towards @xmath1 , the total spin in the ground state increases and at @xmath16 all electrons in a quantum dot become spin polarized . this phenomenon of mesoscopic stoner instability is specific to finite size systems and disappears in the thermodynamic limit @xmath17 . due to the entanglement of the charge and spin degrees of freedom in the universal hamiltonian , the mesoscopic stoner instability affects the electron transport through a quantum dot . for example , it leads to an additional nonmonotonicity of the energy dependence of the tunneling density of states @xcite and to the enhancement of the shot noise . @xcite the cooper channel interaction in the description within the universal hamiltonian framework is responsible for superconducting correlations in quantum dots . @xcite we shall assume throughout the paper that the cooper channel interaction is repulsive and , therefore , omit it . @xcite we also neglect corrections to the universal hamiltonian due to the fluctuations in the matrix elements of the electron - electron interaction . @xcite they are small in the regime @xmath18 but lead to interesting physics beyond the universal hamiltonian . @xcite in the presence of a spin - orbit coupling the description of a quantum dot in the framework of the universal hamiltonian breaks down . even for a weak spin - orbit coupling ( large spin - orbit length , @xmath19 ) fluctuations of the matrix elements of the electron - electron interaction can not be neglected in spite of the condition @xmath18 . @xcite for a quantum dot in a two - dimensional electron gas the orbital degrees of freedom are coupled to in - plane components of the spin . then in the regime @xmath20 the low energy description is again possible in terms of the universal hamiltonian but with the ising exchange interaction ( @xmath21 ) . @xcite in this case mesoscopic stoner instability is absent for the equidistant single - particle spectrum . @xcite as a consequence , the tunneling density of states is almost independent of @xmath22 while the longitudinal spin susceptibility @xmath23 is independent of @xmath24 as in a clean fermi liquid . @xcite the experiments on tunneling spectra in nanometer - scale ferromagnetic nanoparticles revealed the presence of an exchange interaction with significant anisotropy . @xcite the simplest model which allows to explain the main features of experimentally measured excitation spectra of ferromagnetic nanoparticles resembles the universal hamiltonian with uniaxial anisotropy in exchange interaction . @xcite such modification of exchange interaction can arise due to shape , surface , or bulk magnetocrystalline anisotropy . in addition , in the presence of spin - orbit scattering the anisotropic part of the exchange interaction can experience large mesoscopic fluctuations . @xcite the alternative reason for appearance of anisotropy in the exchange interaction in quantum dots is the presence of ferromagnetic leads . @xcite the universal hamiltonian with an anisotropic exchange interaction ( albeit it is not microscopically justified ) is interesting on its own as the simplest model interpolating between the cases of the heisenberg and ising exchange interactions . since in the latter case there is no mesoscopic stoner instability for the equidistant single - particle spectrum , it is interesting to understand how it disappears as the exchange develops anisotropy . does the spin of the ground state vanish continuously or discontinuously as the anisotropy increases ? for the ising exchange interaction transverse dynamical spin susceptibility @xmath25 is nontrivial . its imaginary part is odd in frequency with maxima and minima at @xmath26 , respectively . @xcite in the case of the heisenberg exchange @xmath27 reduces to a delta - function . but how does this reduction occur with decrease in anisotropy ? in low dimensions @xmath28 interaction and disorder can induce a transition between paramagnetic and ferromagnetic phases at a finite temperature @xmath24 . @xcite in @xmath29 the stoner instability can be promoted by disorder and occurs at smaller values of exchange interaction @xcite . in the universal hamiltonian the disorder remains in randomness of the single - particle levels . as it is known , @xcite level fluctuations affect the temperature dependence of the average static spin susceptibility @xmath23 in the case of the heisenberg exchange . in the case of the ising exchange the role of disorder is even more dramatic . for the equidistant single - particle levels @xmath23 is temperature independent . due to level fluctuations the average spin susceptibility acquires a curie type @xmath24-dependence dominating at low enough @xmath24 and for @xmath30 . @xcite in this regime of strong ( with respect to the small distance @xmath31 to the average position of the stoner instability at @xmath32 ) level fluctuations a quantum dot is in the paramagnetic phase on average but it can be fully spin - polarized for a particular realization of the single - particle levels . these fully spin - polarized realizations should affect the tails of the distribution functions for @xmath23 and dynamical transverse spin susceptibility @xmath25 , but how exactly ? can it be possible that at zero temperature the level fluctuations shift the position of the stoner instability from its average position , @xmath32 , and lead to the existence of a finite temperature transition between the paramagnetic and the ferromagnetic phases in quantum dots ? of course , the very same questions can be asked for the case of the heisenberg exchange . in this paper we address these questions within the universal hamiltonian framework extended to the case of exchange interaction with uniaxial anisotropy . we compute the temperature and magnetic field dependence of the static longitudinal spin susceptibility @xmath23 for equidistant single - particle spectrum . except the case of the ising exchange it always has a non - zero temperature - dependent contribution of curie type ( @xmath33 ) or of @xmath34 type . this indicates that destruction of the mesoscopic stoner instability by uniaxial anisotropy is not abrupt . for equidistant single - particle levels we also compute the transverse spin susceptibility . it always has a maximum and a minimum whose positions tend to zero frequency with decrease of anisotropy . we show that at low temperatures and for @xmath30 the statistical properties of the longitudinal spin susceptibility ( both for the ising and heisenberg exchanges ) are determined by the statistics of the extrema of a certain gaussian process with a drift . this random process resembles locally a fractional brownian motion with the hurst exponent @xmath35 where @xmath36 . we recall that the fractional brownian motion with the hurst exponent @xmath37 is the gaussian process @xmath38 with zero mean @xmath39 and the two - point correlation function @xmath40 ^ 2 } = |t - t^\prime|^{2h}$ ] . we rigorously prove that in the case of ising ( heisenberg ) exchange all moments of static longitudinal spin susceptibility @xmath23 are finite for @xmath41 ( @xmath42 ) . for the ising exchange we argue also that all moments of dynamic transverse spin susceptibility @xmath25 do not diverge for @xmath41 . we estimate the tail of the complementary cumulative distribution function for @xmath23 for both ising and heisenberg exchange interactions . we demonstrate that the average static longitudinal spin susceptibility @xmath23 has nonmonotonous dependence on magnetic field in the case of ising exchange . our results mean that the level fluctuations do not shift the stoner instability from its average position and do not induce a finite temperature transition between the paramagnetic and the ferromagnetic phases . the outline of the paper is as follows . in sec . [ sec : formalism ] we introduce the model hamiltonian , derive exact analytical expressions for the corresponding grand canonical partition function and longitudinal static spin susceptibility . in sec . [ sec : eqspectrum ] we analyze the temperature and magnetic field dependence of longitudinal static spin susceptibility in the case of equidistant single - particle spectrum and anisotropic exchange interaction . in sec . [ sec : lf : ic ] we present a detailed analysis of the effect of level fluctuations on the longitudinal static spin susceptibility for the cases of ising and heisenberg exchange interactions . in sec . [ sec : trss ] we compute the transverse dynamical spin susceptibility in the case of equidistant single - particle spectrum and anisotropic exchange interaction and analyze the effect of level fluctuations in the case of ising exchange interaction . we conclude the paper with summary of the main results and discussion of how our predictions can be experimentally verified ( sec . [ sec : dc ] ) . some of the results were published in a brief form in ref . we consider the following hamiltonian with direct coulomb and anisotropic exchange interactions : @xmath43 the noninteracting hamiltonian , @xmath44 is given as usual in terms of the single - particle creation ( @xmath45 ) and annihilation ( @xmath46 ) operators . it involves the spin - dependent ( @xmath47 ) single - particle energy levels @xmath48 . in what follows , we assume that they depend on applied magnetic field @xmath49 via the zeeman splitting , @xmath50 . here @xmath51 and @xmath52 stand for the land g - factor and the bohr magneton , respectively . the charging interaction part of the hamiltonian , @xmath53 describes the direct coulomb interaction in a quantum dot in the zero - dimensional approximation , @xmath54 . here @xmath55 denotes the particle number operator , and @xmath56 is the background charge . the term @xmath57 represents the anisotropic exchange interaction within the qd . the total spin operator @xmath58 is defined in terms of the standard pauli matrices @xmath59 . in the case of isotropic heisenberg exchange , @xmath60 , the hamiltonian reduces to the universal hamiltonian which describes a quantum dot in the limit @xmath2 . @xcite in this case the single - particle levels @xmath61 are random . their statistics ( in the absence of magnetic field , @xmath62 ) is described by the orthogonal wigner - dyson ensemble . the hamiltonian with the ising exchange , @xmath63 , and @xmath62 can be used for description of lateral quantum dots with spin - orbit coupling . @xcite in this case the statistics of @xmath61 is described by the unitary wigner - dyson ensemble . the grand canonical partition function for the hamiltonian is defined as @xmath64 ( @xmath65 denotes the chemical potential ) . it can be found by using the following trick . let us separate @xmath66 into the heisenberg and ising parts : @xmath67 then the time evolution operator in the imaginary time can be rewritten as @xmath68 where @xmath69 . the exponent in the second line of eq . indicates that the grand canonical partition function for the hamiltonian can be found in two steps . at first , one can use well - known results for the partition function for the case of isotropic exchange and effective magnetic field @xmath70 . @xcite secondly , one needs to integrate over the effective magnetic field @xmath71 with the kernel given in the first line of eq . . thus we obtain the following exact result for the grand canonical partition function of hamiltonian : @xmath72 here @xmath73 . the integers @xmath74 and @xmath75 represent the number of spin - up and spin - down electrons , respectively . the total number of electrons is @xmath76 , and @xmath77 . we note that for a configuration with given @xmath74 and @xmath75 electrons the total spin equals @xmath78 . the integers @xmath79 denote @xmath80 projection of the total spin @xmath81 . the factors @xmath82 and @xmath83 are canonical partition functions for @xmath74 and @xmath75 noninteracting spinless electrons , respectively . the canonical partition function takes into account the contributions from the single - particle energies and is given by darwin - fowler integral : @xmath84 for the heisenberg exchange interaction , @xmath60 our result coincides with the result known in the literature . @xcite in the case of purely ising exchange interaction , @xmath63 , our result agrees with the result obtained in ref . [ ] . we note that the result can be also derived directly from hamiltonian with the help of wei - norman - kolokolov transformation ( see appendix [ wnk - der ] ) . in order to analyze the exact result for the grand canonical partition function , it will be convenient to use the following completely equivalent integral representation : @xmath85 the grand canonical partition function for non - interacting spinless electrons is defined in a standard way @xmath86 the variables @xmath87 and @xmath88 have the meaning of the zero - frequency matsubara components of an electric potential and a magnetic field which can be used to decouple the direct coulomb @xcite and exchange interaction @xcite terms , respectively . the general expressions and for the grand partition function @xmath89 allow us to extract the results for the longitudinal spin susceptibility : @xmath90 it is worthwhile to mention that in zero magnetic field one can use the equivalent formula @xmath91 to simplify calculations . as it is well - known,@xcite at @xmath92 ( the regime we are interested in ) we can perform integration over @xmath87 in eq . in the saddle - point approximation . then the grand canonical partition function is factorized into two multipliers : @xmath93 where @xmath94 describes the effect of charging energy . here @xmath95 is the solution of the saddle - point equation @xmath96 and @xmath97 stands for the thermodynamic density of states at the fermi level . we note that in the regime @xmath6 ( which we are interested in ) one can approximate @xmath95 by @xmath98 . the term @xmath99 } \notag \\ & \hspace{1cm}\times \frac{\sinh ( h ) \sinh \bigl ( ( b+\eta \mathcal{b})h / j_\perp\bigr ) } { \sinh \bigl ( \beta(b+\eta \mathcal{b})/2\bigr ) } e^{-\frac{\beta\mathcal{b}^2}{4 |j_z - j_\perp| } ] } \notag \\ & \hspace{1 cm } \times \prod_{\sigma}e^{\beta \omega_0(\tilde\mu)-\beta \omega_0(\tilde\mu+h \sigma/\beta)}\label{zs1}\end{aligned}\ ] ] describes the contribution due to exchange interaction . the function @xmath100 = \int\limits_{-\infty}^\infty de\ , \nu_0(e)\ , \notag \\ \times \ln \left [ 1+\frac{\sinh^2(h/2)}{\cosh^2(e/2t)}\right ] \end{gathered}\ ] ] that appears in eq . depends on a particular realization of the single - particle spectrum via the single - particle density of states @xmath101 . provided @xmath102 , we can write @xmath103 = \frac{h^2}{\beta \delta}-v(h ) , \label{zzeq2}\ ] ] where @xmath104 .\label{vh_def}\ ] ] here @xmath105 stands for the deviation of the single - particle density of states @xmath106 from its average ( over realizations of the single - particle spectrum ) value : @xmath107 . the charging energy contribution @xmath108 is independent of the magnetic field and therefore does not affect the spin susceptibility . we note that the normalization is such that @xmath109 for @xmath110 . in what follows we will discuss @xmath111 only . we start our analysis from the case of the equidistant single - particle spectrum , i.e. we completely neglect the effect of level fluctuations ( we set @xmath112 in eq . to zero ) . we discuss the role of level fluctuations in sec . [ sec : lf : ic ] . using the integral representation and we can perform integration over @xmath88 and find @xmath114 here @xmath115 is the energy scale specific for the anisotropic problem that interpolates between @xmath116 ( for @xmath117 ) and @xmath118 ( for @xmath63 ) . the function @xmath119 is defined as follows @xmath120 using eq . , the zero field longitudinal spin susceptibility can be written as @xmath121 at high temperatures @xmath122 , the result for the zero - field longitudinal static spin susceptibility can be simplified ( cf . . then we obtain @xmath123 away from the isotropic case ( @xmath117 ) a set of temperature intervals with different temperature behavior of the longitudinal spin susceptibility exists . below we use the asymptotic result from appendix [ app : f1:zzf1 ] . at temperatures @xmath124 , we find @xmath125 for the temperature range @xmath126 , we obtain @xmath127 if the temperature is within the interval @xmath128 , the zero field longitudinal static spin susceptibility becomes @xmath129 finally , for the lowest temperature range @xmath130 we find ( cf . eq . ) @xmath131 we mention that @xmath23 consists of two contributions ( see eqs . - and ): the one which resembles the fermi - liquid result for spin susceptibility , @xmath132 , and the other which is of curie type , @xmath133 . such behavior is illustrated in fig . [ fig : plot_chi_zz ] where the dependence of longitudinal spin susceptibility on temperature and @xmath22 at a fixed ratio @xmath134 is shown . we emphasize that longitudinal spin susceptibility diverges at @xmath32 regardless of the value of @xmath135 . to understand the origin of such interesting behavior of the zero field longitudinal spin susceptibility it is useful to rewrite eq . in terms a series form again : @xmath136 here we used the following result @xmath137 which is valid provided the following conditions hold : @xmath138 and @xmath139 ( see appendix [ app : znzn ] ) . $ ] to the fermi - liquid - like result on dimensionless magnetic field and inverse temperature , @xmath140 and @xmath141 . we chose @xmath142 and @xmath143.,width=264 ] in the case of large temperatures @xmath144 our results - imply the fermi - liquid behavior of @xmath145 . in this temperature range all terms except the first one with @xmath146 in the sum over @xmath147 in eq . cancel each other . then we find @xmath148 and , consequently , @xmath149 $ ] . this result implies that the average value of @xmath150 is of the order of @xmath151 \gg 1 $ ] regardless of @xmath135 . at the same time the average value of the squared total spin @xmath152 is of the order of @xmath151 + 1/[\beta(\delta - j_\perp)]$ ] . therefore , at @xmath153 the total spin strongly fluctuates in all three directions so that @xmath154 whereas for @xmath155 the total spin fluctuates along the @xmath80 axis only so that @xmath156 . we mention the unusual ( inverse square - root ) temperature dependence of the longitudinal spin susceptibility in eq . . however , the result is valid in a temperature range that exists only if @xmath157 . then the restrictions for the temperature become @xmath158 . therefore we can use the arguments from the previous paragraph . in order to explain the @xmath159 dependence of @xmath23 , one needs to perform the perturbation expansion in @xmath160 for eq . . at low temperatures @xmath161 our results - and imply a curie type longitudinal spin susceptibility . in this case the second term in brackets in the right hand side of eq . can be neglected . the sum over @xmath147 can be estimated by the integral which is dominated by @xmath162 . then we find @xmath163 this estimate yields the typical value of @xmath164 $ ] and , thus , the curie type behavior of the longitudinal spin susceptibility : @xmath165 ^ 2 $ ] . therefore , at relatively low temperatures @xmath166 the configuration with a non - zero total spin @xmath167 $ ] gives the main contribution to the thermodynamic quantities . } { d ( \delta - j_z)}$ ] on @xmath168 and @xmath169 for @xmath170 . in the left upper region the curie - like behavior dominates the fermi - liquid - like result . in the right lower region the curie - like correction to the fermi - liquid - like result , @xmath171\propto \frac{1}{(\delta - j_z)^2}$ ] is small . red region corresponds to the intermediate regime in which there is a correction @xmath172\propto \frac{1}{t^{1/2}(\delta - j_z)^{3/2}}$ ] to the fermi - liquid result due to transverse degrees of freedom.,width=264 ] for small magnetic fields @xmath173 the longitudinal spin susceptibility @xmath174 can be well approximated by the zero field result . for larger magnetic fields @xmath175 there are two regions of temperature with different behavior . in the range of temperatures @xmath176 the longitudinal spin susceptibility becomes linear in temperature ( cf . ): @xmath177 at higher temperatures @xmath178 the temperature dependence of the longitudinal spin susceptibility saturates : @xmath179 in the limit of large magnetic fields the ground state energy for the configuration with the total spin projection @xmath180 is equal to @xmath181 . thus the projection of the total spin in the ground state is @xmath182 $ ] . it allows us to estimate the longitudinal spin susceptibility as @xmath183 $ ] in agreement with eq . . using the integral representation and , we integrate over @xmath88 and obtain @xmath185 here @xmath186 stands for the jacobi theta function . since @xmath187 , the jacobi theta function @xmath188 becomes equal to unity . then for @xmath189 we find @xmath190 where @xmath191 at temperatures @xmath192 , with the help of eq . we obtain that the longitudinal spin susceptibility is given by eq . . in the temperature range @xmath193 , the behavior of @xmath23 is described by eq . . in the case of an easy plane anisotropy the interplay between fermi - liquid and curie - like temperature dependencies of the longitudinal spin susceptibility can be explained in exactly the same way as it was done for the case of an easy axis anisotropy . the longitudinal static spin susceptibility is almost insensitive to the presence of a small magnetic field @xmath194 . in the opposite case @xmath195 , one can neglect @xmath196 in the @xmath197 s arguments in eq . . then at @xmath195 we find @xmath198 the result implies that for magnetic fields in the range @xmath199 the longitudinal spin susceptibility is described by eq . whereas for @xmath200 , @xmath23 is given by eq . . as it was explained above , the hamiltonian describes a quantum dot in the zero - dimensional limit for the ising and heisenberg exchange interactions only . therefore , it is reasonable to study the effect of level fluctuations on the results obtained above for @xmath63 and @xmath60 . we start with the case of ising exchange . to simplify the general result in the case of the ising exchange , it is convenient to make a change of variable @xmath201 , to take the limit @xmath202 , and then to integrate over @xmath71 . thus we find @xmath203 where @xmath204 the information on fluctuations of single - particle levels is encoded in the even random function @xmath112 via the density of states ( see eq . ) . we remind that the single - particle density of states @xmath205(e ) has non - gaussian statistics . @xcite however , for @xmath206 the function @xmath112 is a gaussian random variable with zero mean value . @xcite the two - point correlation function of @xmath207 can be written as follows ( see appendix [ app : lh ] ) : @xmath208 . \label{corrvv}\end{gathered}\ ] ] here @xmath209 is the euler digamma function and @xmath210 is the euler mascheroni constant . in the case of the ising exchange , the parameter @xmath211 in eq . is equal to @xmath212 since the energy levels @xmath61 in the hamiltonian are described by the unitary wigner - dyson ensemble ( class a ) . @xcite the asymptotics of @xmath213 are as follows : @xcite @xmath214 according to eq . , the average longitudinal spin susceptibility @xmath215 is determined by the quantity @xmath216 . although @xmath112 is a gaussian random variable , exact evaluation of @xmath216 for arbitrary values of @xmath217 and @xmath218 is a complicated problem . we start from the perturbation theory in the correlation function @xmath219 . expanding expression for @xmath220 to the second order in @xmath207 and performing the averaging of @xmath221 with the help of eq . , we find @xmath222 . \label{averlnxi1}\end{aligned}\ ] ] there exist four regions of different behavior of @xmath216 . they are shown in fig . [ figureregions ] . it is convenient to introduce the renormalized exchange @xmath223 . in the region i , @xmath224 , the arguments of @xmath220 satisfy the condition @xmath225 . the latter allows one to use the asymptotics of @xmath213 for @xmath226 ( see eq . ) . then we find @xmath227 .\label{eq : xi : regi}\ ] ] hence we obtain the following result for the average longitudinal spin susceptibility at temperatures @xmath228 : @xmath229 .\quad \label{eq : chizz : regi}\end{aligned}\ ] ] in the region i the corrections to the longitudinal spin susceptibility are always small and , therefore , the perturbation theory is well justified . we present a more transparent way for derivation of eq . . at first , one can substitute @xmath230 for @xmath231 in the expression ( with @xmath63 ) for the equidistant spectrum . secondly , we expand @xmath23 to the second order in the deviation @xmath232 . finally , one can perform averaging with the help of the relation @xcite @xmath233 and obtain the result ( with @xmath189 ) . in the region ii , @xmath234 , one can perform an expansion in @xmath235 in the right hand side of eq . since the condition @xmath236 holds . however , the argument of @xmath237 is typically large and we need to use its asymptotics for @xmath238 ( see eq . ) . then we obtain @xmath239 therefore , the average longitudinal spin susceptibility in the region ii ( @xmath240 is as follows : @xmath241 at zero magnetic field we check that the contribution of the second order in @xmath237 to @xmath242 is of order of @xmath243 ( see appendix [ app:2dorder ] ) . therefore the perturbation theory in the two - point correlation function of @xmath207 is justified for @xmath244 only . in this regime the variance of @xmath23 is small @xmath245/\overline{\chi}_{zz}^2 \sim \bar{j_z}/(\pi^2 \bm{\beta } t ) \ll 1 $ ] ( see appendix [ app:2dorder ] ) . therefore , at @xmath244 one can expect the normal distribution of @xmath23 . finally , in the region iii , @xmath246 , the typical value of @xmath247 contributing to the integral in the right hand side of eq . can be not only of the order unity but also of the order of @xmath248 . in the latter case , since @xmath249 one needs to use the asymptotics of @xmath213 for @xmath238 ( see eq . ) . then we find @xmath250 we thus obtain the average longitudinal spin susceptibility in the region iii ( @xmath251 ) : @xmath252 for magnetic fields @xmath253 the effect of level fluctuations is suppressed and the perturbation theory is justified . at @xmath254 the result agrees with the result whereas at @xmath255 the corrections due to level fluctuations in and become of the same order . due to fluctuations for the case of ising exchange in the plane of dimensionless magnetic field and inverse temperature , @xmath140 and @xmath256 . note that in our analysis we assume @xmath92.,width=226 ] results and imply non - monotonous behavior of the average longitudinal spin susceptibility with magnetic field @xmath257 in the temperature range @xmath258 ( see fig . [ fig : chi_zz_on_b ] ) . the susceptibility @xmath259 as a function of @xmath257 has a minimum at @xmath260 . in the region of strong fluctuations @xmath261 we expect similar behavior of the average longitudinal spin susceptibility . although the result is derived for @xmath92 , for @xmath262 it can be obtained from the following zero - temperature arguments . the difference in the ground state energies for the state with projections @xmath263 and @xmath180 of the total spin can be estimated as @xmath264 here @xmath265 is the fluctuation of the energy window in which there are @xmath266 levels on average . it can be expressed as @xmath267 where @xmath268 is the fluctuation of the number of single - particle levels in the strip with @xmath266 levels in average . from the random matrix theory it is well known that @xcite @xmath269 . \label{eq : crit2}\ ] ] comparing the energies of the ground states with total spin projections @xmath263 and @xmath180 , we find from eq . that @xmath270 . \label{eq : crits3}\ ] ] hence the average longitudinal spin susceptibility can be estimated as @xmath271 , \ ] ] where @xmath272 . using eq . , we reproduce the result . $ ] due to fluctuations on @xmath140 ( see eqs . and ) . the temperature @xmath273$].,width=226 ] the average longitudinal spin susceptibility is mostly affected by the level fluctuations in the region ii ( @xmath274 ) . the perturbative result loses its validity at @xmath275 . such a regime is realized in the close vicinity of the stoner instability @xmath276 . in this case of strong fluctuations it is useful to know the distribution function of @xmath23 rather than the average value . in the range of temperatures @xmath277 , the integral in the right hand side of eq . is dominated by large values of @xmath278 . then , using the asymptotic expression , one can check that for @xmath279 the two - point correlation function is homogeneous of degree two : @xcite @xmath280 with the help of eq . , at zero magnetic field @xmath189 and for @xmath281 , eqs . and can be simplified to @xmath282 we remind that the normalization is such that @xmath283 at @xmath284 . according to eq . for @xmath285 , the grand canonical partition function increases as @xmath22 increases . hence it follows that @xmath286 . according to eq . , the statistics of the zero field longitudinal spin susceptibility is determined by the single parameter @xmath287^{1/2}$ ] . the gaussian random process @xmath288 has zero mean and is even in @xmath88 , @xmath289 . its two - point correlation function reads @xmath290 hence we find that @xmath291 ^ 2 } = - 2 u^2 \ln |u| + o(u^2)=o(u^{2h } ) \label{eq : local}\ ] ] for any @xmath292 . thus the trajectories of @xmath288 are continuous and its increments are strongly positively correlated ( see fig . [ fig : process ] ) . in fact the process @xmath288 is in many aspects close to the ballistic one @xmath293 with @xmath294 being a gaussian random variable ( recall that @xmath295 is the unique process with @xmath296 ) . the process @xmath288 has arisen before in a seemingly unrelated context . @xcite ; dashed lines @xmath297 are guides for the eye.,width=226 ] we are interested in the complementary cumulative distribution function @xmath298 , i.e. the probability that @xmath299 exceeds @xmath300 : @xmath301 . it has the following properties : @xmath302 , @xmath303 and @xmath298 is monotonously decreasing as @xmath300 increases . the average moments of @xmath299 can be conveniently written as @xmath304^k}= k \int_{0}^{\infty}dw w^{k-1 } \mathcal{p}(w)$ ] . although a closed analytical expression for the complementary cumulative distribution function is not known , we bound @xmath298 from above to prove that all moments of @xmath299 ( and consequently all moments of @xmath23 ) are finite for @xmath41 . at first , we split the gaussian weight @xmath305 in the integral in the right hand side of eq . and obtain ( @xmath306 is an arbitrary splitting parameter ) @xmath307 the inequality allows us to reduce the problem of finding an upper bound for @xmath298 to the statistics of the maxima of the gaussian process @xmath308 which locally resembles a fractional brownian motion with a drift . indeed , from eq . we find @xmath309 to give an upper bound for the probability @xmath310 we employ the slepian s inequality . @xcite let us consider an auxiliary gaussian process @xmath311 where @xmath312 is the standard brownian motion ( @xmath313 ; the hurst exponent @xmath314 ) . for any interval @xmath315 the sample paths @xmath316 and @xmath317 are bounded . the following relations hold : @xmath318 ^ 2 } \geqslant \overline{[y_\gamma(h_1)-y_\gamma(h_2)]^2 } .\end{gathered}\ ] ] the first two equalities are trivially satisfied while the last inequality follows from an easily verifiable inequality @xmath319 ^ 2}\leqslant 8 r \ln 2 $ ] for @xmath320 . then the processes @xmath321 and @xmath322 satisfy the slepian s inequality : @xmath323 for all real @xmath324 . using a well - known result for the brownian motion with a linear drift ( see e.g. , ref . [ ] ) @xmath325 we find the following upper bound for the complementary cumulative distribution function : @xmath326 \right\ } . \label{eq : tail}\ ] ] from eq . it follows that for @xmath327 all moments of @xmath299 ( and hence all moments of @xmath23 ) are finite for @xmath41 . therefore even in the presence of the strong level fluctuations the stoner instability occurs at @xmath32 only . for @xmath41 and for temperatures @xmath92 the quantum dot is in the paramagnetic state . for @xmath328 the saddle - point approximation in eq . becomes exact and the statistics of @xmath299 reduces to the statistics of maxima of the process @xmath329 . as it can be seen from rescaling of @xmath88 , the probability that the maximum of @xmath330 exceeds @xmath324 equals the probability that the maximum of @xmath331 defined on @xmath332 exceeds @xmath333 . from the results of hsler and piterbarg@xcite it follows that the large-@xmath324 tail of @xmath334 is determined by a small vicinity of the point @xmath335 where the variance of @xmath336 attains its maximum @xmath337 . furthermore , should we have a finite limit @xmath338 ^ 2 } } { k^2(s - t ) } > 0 \label{eq : kdef}\ ] ] for some function @xmath339 regularly varying at @xmath116 with index @xmath340 , the precise asymptotics would read @xmath341 , \quad w / z^2 \gg 1 . \label{eq : probtail}\end{gathered}\ ] ] here @xmath342 stands for the functional inverse of @xmath339 . in our case . translates into @xmath343 which is regularly varying with index @xmath344 [ recall that a function @xmath345 is regular varying at @xmath346 with index @xmath347 if @xmath348 for any @xmath349 . the result of ref . [ ] is therefore not directly applicable , but we believe this to be a technicality . in analogy with a similar situation for fractional brownian motion , we expect the asymptotics to hold with only the @xmath300-independent factor @xmath350 modified . note that the exponential part can be tracked to be the tail of a normal distribution with variance @xmath337 taken at @xmath351 , and that it had been correctly reproduced by our initial estimate . therefore we find with logarithmic accuracy that the tail of the complementary cumulative distribution function is given by ( @xmath352 \delta j_z/[t ( \delta - j_z)]$ ] ) @xmath353 this result is valid in the temperature range @xmath327 and is consistent with the upper bound . to illustrate the result we approximate the gaussian process @xmath288 by a degenerate one @xmath354 , where @xmath294 is the gaussian random variable with zero mean @xmath355 and variance @xmath356 . substituting the process @xmath357 for @xmath288 into the right hand side of eq . , we estimate the partition function as @xmath358 $ ] . the large values of @xmath111 correspond to large negative values of @xmath294 such that @xmath359 . therefore , the tail of distribution of @xmath299 is simple exponential . hence we find that for @xmath328 the tail of the complementary cumulative distribution function @xmath298 is given by eq . without the logarithm in the pre - exponent . as shown in fig . [ fig : ccdf ] the overall behavior of @xmath298 for @xmath328 is well enough approximated by the complementary cumulative distribution function for the degenerate process @xmath357 . also we mention that the behavior of @xmath298 for @xmath328 is very different from its behavior at @xmath360 . for the later , @xmath298 is given by the complementary cumulative distribution function of the normal distribution ( see fig . [ fig : ccdf ] ) . equation implies that the average moments of @xmath299 scale as @xmath361 for @xmath328 . hence for @xmath261 the @xmath362-th moment of the spin susceptibility is given by @xmath363^k , \quad k=1 , 2 , \dots . \label{eq : chi : kth}\ ] ] the result can be obtained from the saddle - point analysis of the integral in the right hand side of eq . , i.e. , in essence , by larkin - imry - ma type arguments . @xcite the scaling of the average spin susceptibility ( eq . with @xmath364 ) was proposed in ref . [ ] using arguments of larkin - imry - ma type . on @xmath365 at @xmath366 computed numerically for @xmath367 ( @xmath368 ) ( upper solid curve ) and @xmath369 ( @xmath370 ) ( lower solid curve ) . the black dotted curve is the complementary cumulative distribution function for the normal distribution with mean and variance as one can find from the lowest order perturbation theory in @xmath207 for @xmath366 and @xmath367 ( cf . eqs . and ) . the red dashed curve is the complementary cumulative distribution function of the degenerate process @xmath357 for @xmath366 and @xmath369 . inset : comparison of the tail of @xmath298 computed numerically for @xmath369 ( @xmath370 ) and asymptotic result .,width=226 ] for the case of the isotropic exchange , @xmath371 , the integration over @xmath71 in eq . becomes trivial . then for @xmath92 we obtain @xcite @xmath372 where @xmath373 since in the absence of magnetic field @xmath89 grows with increase of @xmath15 ( see eq . ) , one can check that for the heisenberg exchange @xmath286 . the detailed results of the perturbative expansion in @xmath207 for the longitudinal spin susceptibility can be found in ref . [ ] . similarly to the case of the ising exchange , the effect of fluctuations is important at @xmath189 and @xmath374 $ ] . in this range of parameters the typical value of @xmath278 in the integral in the right hand side of eq . is large , @xmath375 where @xmath376 . then for @xmath189 eq can be rewritten as @xmath377 where @xmath378^{1/2}$ ] . here @xmath379 which corresponds to the orthogonal wigner - dyson ensemble . the complementary cumulative distribution function @xmath380 can be estimated in a similar way as in the previous section . writing @xmath381 \notag \\ \times \max\limits_{h\geqslant 0 } \bigl \{e^{-h^2-(z/\sqrt{\gamma})v(h)}\bigr \ } , \label{eq : split : iso}\end{gathered}\ ] ] with arbitrary splitting parameter @xmath382 ( @xmath306 ) , we obtain the following upper bound : @xmath383 \biggr \ } , \label{eq : tail : iso}\end{aligned}\ ] ] where @xmath384\}$ ] . this upper bound implies that all moments of @xmath299 ( and of @xmath23 ) are finite for @xmath42 . at @xmath385 the integral in the right hand side of eq . can be evaluated in the saddle - point approximation , reducing the statistics of @xmath299 to the statistics of maxima of the process @xmath386 . then using as in the previous section the results of hsler and piterbarg@xcite , we find the tail of the complementary cumulative distribution function at @xmath387 $ ] is given by @xmath388 ( see eq . ) . we note that for this tail the drift term @xmath389 in the process @xmath330 is not important . the typical value of @xmath88 contributing to the integral in the right hand side of eq . is @xmath390 . then for @xmath328 we find , with logarithmic accuracy , @xmath391 . hence for @xmath392 the average @xmath362-th moment of the longitudinal spin susceptibility can be estimated as @xmath393 where @xmath394 $ ] is the spin susceptibility in the absence of level fluctuations . we note that for @xmath392 the scaling of the average longitudinal spin susceptibility similar to eq . with @xmath364 was derived in ref . the transverse spin susceptibility is defined as follows ( see , e.g. , ref . [ ] ) @xmath395e^{-\beta h}\bigr ) , \label{eq : chi_perp_def}\ ] ] where @xmath396 . since , in contrast with @xmath397 , the operators @xmath398 , @xmath399 of the total spin do not commute with the hamiltonian @xmath37 ( for @xmath400 ) , the transverse spin susceptibility can acquire non - trivial frequency dependence . in order to find the dynamic transverse spin susceptibility we use the heisenberg equations of motion for the spin operators : @xmath401 $ ] . since the operator @xmath180 commutes with the hamiltonian , it has no dynamics , @xmath402 . for the other components of the total spin we find @xmath403 using expressions , we integrate over time in eq . and obtain the following operator expression for the transverse spin susceptibility : @xmath404-\hat{s}_z\bigr ) e^{-\beta h } } { \omega+b+(j_\perp - j_z)(2\hat{s}_z+\sigma)+i0^+ } . \label{eq : chi_perp_2}\end{aligned}\ ] ] since operators @xmath397 and @xmath405 commute with @xmath37 , one easily evaluates the trace in eq . with the help of eq . . thus we derive the exact result for the dynamic transverse spin susceptibility : @xmath406-l}{\omega+b+(j_\perp - j_z)(2l+\sigma)+i0^+ } . \label{eqzb2}\end{aligned}\ ] ] in what follows we will be interested in the imaginary part of @xmath25 . the real part can be restored from the kramers - kronig relations . using eq . , the imaginary part of the dynamic transverse spin susceptibility can be written as @xmath407 here we introduce the fourier transform of the partition function @xmath408 in the complex magnetic field @xmath409 : @xmath410 as it follows from eq . , the imaginary part of the transverse spin susceptibility obeys the sum rule @xmath411 where the magnetization @xmath412 . since at @xmath189 the function @xmath413 is even , the imaginary part of the zero - field transverse spin susceptibility is odd in frequency : @xmath414 , so the sum rule is trivially satisfied . we mention that in the case of an isotropic exchange , @xmath117 , eq . becomes trivial , @xmath415 . in this case the behavior of the transverse spin susceptibility is fully determined by the behavior of the magnetization @xmath416 . therefore , in what follows we shall not discuss the transverse spin susceptibility for the isotropic exchange . at first we consider the case of an equidistant single - particle spectrum and , therefore , neglect effects related to the level fluctuations . as it was discussed in sec . [ sec : longsusc ] , for @xmath138 the partition function can be factorized in accordance with eq . . since the factor @xmath108 does not depend on the magnetic field , it does not influence the results for @xmath25 and we omit it below in this section . it implies that @xmath111 , @xmath417 , and @xmath418 should be substituted for @xmath89 , @xmath413 , and @xmath408 in eqs . - , respectively . using eq . for the equidistant single - particle spectrum we can rewrite @xmath417 in the following way : @xmath419 next , performing integration over @xmath420 , we obtain the following result : @xmath421 . \label{im_general}\end{aligned}\ ] ] in the case @xmath422 , applying the euler - maclaurin formula to estimate the sums over @xmath423 , we find @xmath424 in the opposite case @xmath425 , the term with @xmath426 in the right hand side of eq . provides the main contribution . then we obtain @xmath427 we note that for @xmath63 , both expressions and coincide and are valid , in fact , for arbitrary @xmath428 . according to eq . , @xmath429 is represented as the sum of delta - peaks . since their positions are independent of the realization of single - particle levels , the delta - peaks survive averaging of @xmath429 over level fluctuations . therefore , in order to discuss the frequency dependence of the transverse spin susceptibility in a form of a smooth curve , we assume some natural broadening @xmath430 for these delta - peaks . then the sum over @xmath428 in eq . can be replaced by an integral and we obtain @xmath431 where @xmath432 $ ] . in the limit of large frequencies or large magnetic fields , @xmath433 , the imaginary part of the transverse spin susceptibility is exponentially small : @xmath434 . \label{eq : imagchi : tail}\end{gathered}\ ] ] in the absence of magnetic field , @xmath189 , @xmath435 is an odd function of the frequency @xmath436 . for @xmath437 the imaginary part of the transverse spin susceptibility has linear behavior : @xmath438\end{aligned}\ ] ] where @xmath439 of the traverse spin susceptibility on @xmath134 for @xmath440.,width=226 ] the slope of @xmath429 at @xmath441 has different behaviors for @xmath442 and for @xmath443 . in the interval @xmath444 the slope grows monotonously with the increase of @xmath135 and diverges at @xmath60 . in the range @xmath445 the slope has a minimum ( see fig . [ fig : imchislope ] ) . the imaginary part of the zero field transverse spin susceptibility has two extrema ( a minimum at a negative frequency and a maximum at a positive frequency ) . in the case @xmath446 and @xmath447 the positions of the extrema can be estimated as @xmath448 . \label{eq : ext : gen}\end{gathered}\ ] ] the behavior of @xmath25 as a function of frequency is shown in fig . [ fig : dssplots ] . in the presence of a magnetic field @xmath429 is shifted along the frequency axis and becomes asymmetric ( see fig . [ fig : dssplots ] ) . on @xmath436 for @xmath449 and several values of @xmath135 : @xmath450 ( red solid line ) , @xmath451 ( blue dashed line ) and @xmath63(green dotted line ) . the curves shrink to @xmath441 as one moves closer to the isotropic case.,width=226 ] it is worthwhile to discuss the case of the ising exchange ( @xmath63 ) in more detail . in the regime of small frequencies and magnetic fields , @xmath452 , the imaginary part of the dynamic spin susceptibility reads @xmath453 ^ 2}{4j_z^2(\delta - j_z)}\right \ } . \label{eqnising2s}\end{aligned}\ ] ] although @xmath27 is asymmetric in the presence of magnetic field , it still vanishes at zero frequency , @xmath454 . in the opposite limit @xmath455 , from eq . we find @xmath456 ^ 2}{4j_z^2(\delta - j_z)}\right \ } . \label{eqnising2l}\end{aligned}\ ] ] in the case @xmath189 , our results and coincide with the small and large frequency asymptotics of the result obtained in ref . the presence of a magnetic field leads to a shift of the extrema of the imaginary part of the dynamic spin susceptibility according to @xmath457 the above results for the dynamic spin susceptibility have been obtained without taking the level fluctuations into account . below we consider how the level fluctuations affect the dynamic spin susceptibility in the case of the ising exchange . as we shall demonstrate , the effect of level fluctuations on @xmath27 is small in most cases . since the effect of level fluctuations is suppressed by the magnetic field , below we consider only the case @xmath189 . we start from a generalization of eq . to an arbitrary spectrum ( see appendix [ app : znzn ] ) : @xmath458 with the help of eqs . , , and we rewrite the imaginary part of the dynamic spin susceptibility as follows : @xmath459 \biggl |_{n=(\sigma j_z-\omega)/(2j_z ) } .\end{gathered}\ ] ] here the random function @xmath460 is equal to unity in the absence of level fluctuations ( for @xmath461 ) . expanding the right hand side of eq . to the second order in @xmath207 we find @xmath462 in the high temperature regime , @xmath463 , and for @xmath464 , all three integrals in the right hand side of eq . are of the same order . using the asymptotic expression for the function @xmath213 at @xmath465 , we obtain the following result for the imaginary part of the average dynamic spin susceptibility at low frequencies @xmath466 and high temperatures @xmath463 : @xmath467 . \label{eq : ising : fluc1}\end{aligned}\ ] ] here @xmath468 is given by eq . with @xmath189 . we mention that eq . can be obtained from eq . if one substitutes @xmath230 for @xmath231 and performs averaging with the help of eq . . in the regime of low frequencies and high temperatures the effect of level fluctuations is small . in the case of high frequencies and high temperatures , @xmath469 , the first and second lines in the right hand side of eq . provide the main contribution . then with the help of the asymptotic expression for @xmath213 at @xmath470 we find that for @xmath471 the imaginary part of the average dynamic spin susceptibility can be written as @xmath472 here @xmath468 is given by eq . with @xmath189 . we note that the result is valid provided @xmath473 ^ 2\ll \pi^2\bm{\beta}$ ] so that the perturbation theory in @xmath207 is justified . we emphasize that although the result eq . is valid at high temperatures @xmath474 , it can not be obtained from eq . by a substitution of @xmath230 for @xmath231 and averaging with the help of eq . . in the case of low temperatures @xmath475 , the @xmath423-independent contributions in the right hand side of eq . vanish in the leading order . using the asymptotic result for @xmath213 at @xmath470 ( see eq . ) we obtain @xmath476 hence we find the following result for the imaginary part of the average dynamical spin susceptibility at low frequencies , @xmath477 : @xmath478 here @xmath468 is given by eq . with @xmath189 . in the temperature range @xmath477 the effect of level fluctuations is suppressed by an additional small factor @xmath479 . thus we expect that the perturbation theory is valid even for @xmath480 $ ] . in the high frequency regime , @xmath481 , and at low temperatures @xmath475 we obtain from eq . the following result for the average dynamical spin susceptibility : @xmath482 here @xmath468 is given by eq . for @xmath189 . the perturbation theory is justified for @xmath483 ^ 2 , \delta j_z/[t(\delta - j_z ) ] \right \ } \ll \pi^2\bm{\beta}$ ] . we remind that the maximum of @xmath468 is close to the frequency @xmath484 . then , as it follows from eq . , the fluctuations yield an enhancement of the maximal value of the average dynamical spin susceptibility of the relative order @xmath485 $ ] . due to fluctuations there is a small shift of the maximum towards zero frequency , @xmath486 . since @xmath487 , we can bound the function @xmath488 from above as @xmath489 therefore @xmath488 remains finite for @xmath41 . thus , in spite of the level fluctuations , the stoner instability in @xmath27 emerges only at @xmath32 . according eq . , averaging over level fluctuations keeps @xmath27 finite . however , the form of the curve can be changed drastically in the regime of strong fluctuations . to estimate @xmath27 at @xmath490 $ ] we substitute the degenerate process @xmath357 for @xmath112 into eq . . then a straightforward calculation yields @xmath491\ ] ] for @xmath492 . we recall that @xmath493 $ ] . this result implies that @xmath494 has a minimum and a maximum at frequencies @xmath495 due to strong fluctuations of the single - particle levels the frequency of the extremum shifts towards higher frequencies ( in comparison with the corresponding result without fluctuations ) and becomes temperature independent . the fluctuations do not affect considerably the values of @xmath494 at the extrema . therefore the slope at @xmath441 becomes smaller , @xmath496 . in this paper we have addressed the spin fluctuations and dynamics in quantum dots and ferromagnetic nanoparticles . within the framework of the model hamiltonian which is an extension of the universal hamiltonian to the case of uniaxial anisotropic exchange interaction , we have derived exact analytic expressions for the static longitudinal and dynamic transverse spin susceptibilities for arbitrary single - particle spectrum . for the equidistant single - particle levels we analyzed the temperature and magnetic field dependence of @xmath23 . for @xmath497 the zero - field longitudinal spin susceptibility has temperature dependence of type @xmath33 ( curie - like ) or @xmath34 . this indicates that the destruction of the mesoscopic stoner instability by uniaxial anisotropy is not abrupt . the magnetic field suppresses the temperature dependence of @xmath23 making spins aligned along the field . for the case of the ising exchange interaction we study the effect of single - particle level fluctuations on @xmath23 in detail the temperature dependence of @xmath23 appears only due to level fluctuations . we showed that at low temperatures and for @xmath30 ( where fluctuations are strong ) the statistical properties of the longitudinal spin susceptibility are determined by the statistics of the extrema of a gaussian process with a drift . this random process resembles locally a fractional brownian motion . we rigorously prove that in this regime of strong fluctuations all moments of zero - field static longitudinal spin susceptibility @xmath23 are finite for @xmath41 and temperatures @xmath498 . this means that the stoner instability is not shifted by the level fluctuations away from its average position at @xmath499 . also , our results imply that randomness in the single - particle levels does not lead to a transition at finite @xmath500 between a paramagnetic and a ferromagnetic phase . we expect that these conclusions hold also for temperatures @xmath501 . however , we can not argue it within our approach ; a separate ( perhaps numerical ) analysis is needed . we found that the magnetic field suppresses the effect of level fluctuations on the average longitudinal spin susceptibility . interestingly , the dependence of @xmath215 on @xmath257 is non - monotonous with a minimum . we extended the analysis of the effect of strong level fluctuations to the case of heisenberg exchange . we demonstrated that in this case the very same conclusions as for the ising exchange hold . for equidistant single - particles levels we computed the temperature and magnetic field dependence of the imaginary part of the transverse spin susceptibility @xmath27 . we found that it always has a maximum and a minimum whose positions tend to zero frequency with the decrease of anisotropy . the height of the maximum and the depth of the minimum increase with the decrease of anisotropy . for the ising exchange we took into account the effect of single - particle level fluctuations on @xmath27 . we argued that all moments of the dynamic transverse spin susceptibility @xmath25 do not diverge for @xmath41 . we found that at @xmath502 the positions of the extrema of @xmath503 have a @xmath504-type dependence at high temperatures and become independent of @xmath24 at low temperatures ( in the regime of strong level fluctuations ) . interestingly , the level fluctuations do not change the minimal and maximal values of @xmath503 significantly . our results , in principle , can be checked in quantum dots and nanoparticles made of materials close to the stoner instability , such as co impurities in a pd or pt host , fe or mn dissolved in various transition - metal alloys , ni impurities in a pd host , and co in fe grains , as well as nearly ferromagnetic rare - earth materials . @xcite however , to test our most interesting results on spin susceptibility in the regime of strong level fluctuations one needs to explore the regime @xmath505 . the closest material to the stoner instability we are aware of , yfe@xmath506zn@xmath507 , @xcite has the exchange interaction @xmath508 which is near the border of the regime with strong level fluctuations at low temperatures . we acknowledge useful discussions with y. fyodorov , y. gefen , a. ioselevich , a. shnirman and m. skvortsov . the research was funded in part by rfbr grant no . 14 - 02 - 00333 , the council for grant of the president of russian federation ( grant no . mk-4337.2013.2 ) , dynasty foundation , ras programs `` quantum mesoscopics and disordered systems '' , `` quantum physics of condensed matter '' and `` fundamentals of nanotechnology and nanomaterials '' , and by russian ministry of education and science . in this appendix we present a derivation of the partition function for the hamiltonian . for simplicity , we consider the case of zero magnetic field . we use the notation of ref . [ ] . we start from the hamiltonian @xmath510 . then the corresponding partition function can be written as @xmath511 , where @xmath66 is given by eq . and @xmath512 denotes the averaging over all many - particle states with the weight @xmath513 . to get rid of terms of the fourth order in electron operators in the exponent we apply the hubbard - stratonovich transformation , @xmath514 } = \lim_{n\rightarrow \infty}\int \bigl[\prod_{n=1}^{n}d\bm\theta_n\bigr ] \prod_\alpha \mathcal{t}e^{it{\bm\theta_n \bm s_\alpha}/n}\notag \\ \times \exp \left [ -\frac{i \delta}{4}\sum\limits_{n=1}^{n}\left ( \frac{\theta^2_{x , n}+\theta^2_{y , n}}{j_\perp}+\frac{\theta^2_{z , n}}{j_z}\right ) \right ] , \label{evol}\end{gathered}\ ] ] where @xmath515 . here and further we omit the normalization factors . we restore the correct normalization factor ( depending on @xmath516 , and @xmath22 ) in the final result . to calculate the time - ordered exponent ( @xmath315 ) of non - commuting operators it is useful to apply the wei - norman - kolokolov transformation @xcite allowing us to rewrite @xmath315-exponent as a product of usual exponents : @xmath517 where @xmath518 and @xmath519 . we employ the initial condition @xmath520 . the new variables @xmath521 and @xmath522 are related to the variables @xmath523 as follows : @xmath524 the vector variables @xmath525 are real but the transformation assumes that the contour of integration in eq . has been rotated . in order to preserve the number of variables we impose the following constraints on the new variables : @xmath526 and @xmath527 . we mention that the transformation assumes such a discretization of time that the quantity @xmath528 corresponds to @xmath529 in the continuous limit . in general , there are a lot of discrete representations of @xmath529 , e.g. of the form @xmath530 with @xmath531 . however , the choice of the symmetric one ( with @xmath532 ) is optimal since it allows us to work within the first order in @xmath533 in eq . . we note that the jacobian of the transformation is equal to @xmath534 . having in mind the further usage of the results , we rewrite @xmath535 as the product @xmath536 with @xmath537 . now , rewriting two exponents in terms of two sets of new variables , we obtain @xmath538 ^ 2}\biggr \ } \prod\limits_\alpha \operatorname{tr}\prod_{p=\pm } \left [ e^{-ipt_p \epsilon_\alpha n_\alpha } e^{p s^{-p}_\alpha \kappa_{p , n_p}^p } e^{i s_\alpha^z \delta \sum_{n=1}^{n_p } \rho_{p , n } } e^{i s_\alpha^p \delta \sum_{n=1}^{n_p } \kappa_{p , n}^{-p } \exp(- i p \delta \sum_{j=1}^n \rho_{p , j } ) } \right ] , \label{eq2}\end{aligned}\ ] ] where @xmath539 and @xmath540 . let us introduce a set of auxiliary variables @xmath541 to get rid of terms of the fourth order in @xmath542 s : @xmath543 ^ 2}=\int d\eta_{p , n_p } e^ { \frac{ip \delta\varkappa}{4 j_\perp } \eta_{p , n_p}^2 } \notag \\ \times e^ { -\frac{ip \delta\varkappa}{2 j_\perp } \eta_{p , n_p}\kappa_{p , n_p}^{-p}(\kappa_{p , n_p}^p+\kappa_{p , n_p-1}^p ) ] } .\end{gathered}\ ] ] to proceed with the evaluation of @xmath544 we need to calculate the following integrals over @xmath545 s : @xmath546 following ref . [ ] , we introduce the new variables @xmath547 where @xmath548 such choice of @xmath549 and @xmath550 allows us to get rid of terms of the third order ( second order in @xmath551 s and first order in @xmath552 ) in eq . within the first order in @xmath533 . the last term in the right hand side of the second equation in determines the jacobian @xmath553 of the transformation , @xmath554 $ ] . we emphasize that it can be missed in the continuous representation . evaluating the single - particle trace @xmath555 in the expression explicitly , one can obtain ( the limit @xmath556 is assumed ) @xmath557 } e^{-\frac{i p \delta}{4j_z } [ \rho_{p , n_p}^2+\frac{\varkappa}{1-\varkappa } \eta_{p , n_p}^2 ] } e^{-\frac{1}{j_\perp}\chi_{p , n_p}^{-p}(\chi_{p , n_p}^p-\chi_{p , n_p-1}^p ) } \biggr \ } \notag \\ \times \prod\limits_\alpha \biggl \{1+e^{-2i\epsilon_\alpha(t_+-t_- ) } + 2e^{-i\epsilon_\alpha(t_+-t_-)}\cos\left ( \frac{\delta}{2}\sum_{p=\pm}\sum_{n_p=1}^{n_p}\rho_{p , n_p}\right ) + \prod_{p=\pm } e^{-ip \epsilon_\alpha t_p } \exp\left ( \frac{i p \delta}{2 } \sum_{n_p=1}^{n_p}\rho_{p , n_p}\right ) \notag \\ \times \biggl ( p \chi_{p , n_p}^p e^{- i p \delta \varkappa \sum_{n_p=1}^{n_p } ( \rho_{p , n_p}-\eta_{p , n_p } ) } + i \delta \sum_{n_{-p}=1}^{n_{-p } } \chi_{-p , n_{-p}}^p e^{- i p \delta \varkappa \sum_{n=1}^{n_{-p } } ( \rho_{-p , n}-\eta_{-p , n } ) } e^{i p \delta \sum_{n=1}^{n_{-p } } \rho_{-p , n } } \biggr ) \biggr \ } .\end{gathered}\ ] ] now the integration over variables @xmath558 can be performed ( see details in appendix b of ref . [ ] ) . then we find @xmath559 } e^{-\frac{i p \delta}{4j_z } [ \rho_{p , n_p}^2+\frac{\varkappa}{1-\varkappa } \eta_{p , n_p}^2 ] } \biggr \ } \prod_\alpha\left ( \oint\limits_{|z_\alpha|=1}\frac{i dz_\alpha}{2\pi z_\alpha^2}\right ) e^{-w } \notag \\ \times \exp \biggl ( - 2 v \cos\left [ \frac{\delta}{2}\sum_{p=\pm } \sum_{n_p=1}^{n_p } \rho_{p , n_p}\right ] \biggr ) \int\limits_0^\infty dy \ , e^{-y } \exp \biggl \ { -i j_\perp v y \left ( \prod \limits_{p=\pm } e^{i \frac{p \delta}{2 } \sum_{n_p=1}^{n_p } \rho_{p , n_p}}\right ) \notag \\ \times \left ( \sum_{p=\pm } p\ , e^{- i p \delta \varkappa \sum_{n_p=1}^{n_p } ( \rho_{p , n_p}-\eta_{p , n_p } ) } \delta \sum_{n_p=1}^{n_p } e^{- i p \delta \sum_{n=1}^{n_p } [ ( 1-\varkappa)\rho_{p , n}+\varkappa\eta_{p , n } ] } \right ) \biggr \ } . \label{eq : zj1}\end{gathered}\ ] ] here we introduce the following notation @xmath560 let us introduce new variables to make the expression more standard : @xmath561 + \xi_{p}(0 ) . \ ] ] here we switch to continuous representation . we obtain @xmath562 \notag \\ \times e^{\frac{1}{2}[\xi_p(t_p)-\xi_p(0 ) ] } e^{-\frac{i p}{4j_z } \int_0^{t_p } dt [ \frac{(ip \dot{\xi}_p+\varkappa \eta_p)^2}{(1-\varkappa)^2 } + \frac{\varkappa \eta_p^2}{1-\varkappa } ] } \biggr \ } \notag \\ \times e^{- 2 v \cosh [ \frac{1}{2(1-\varkappa)}\sum_{p=\pm}p ( \xi_{p}(t_p)-\xi_p(0 ) - i p \varkappa\int_0^{t_p } dt^\prime \eta_p(t^\prime ) ) ] } \notag \\ \times \exp \biggl \ { -i j_\perp v y \left ( \prod \limits_{p=\pm } e^{\frac{1}{2(1-\varkappa ) } [ \xi_p(t_p)-\xi_p(0)-ip\varkappa \int_0^{t_p } dt \eta_p(t ) ] } \right ) \notag \\ \times \left ( \sum_{p=\pm } p\ , e^{\frac{1}{1-\varkappa } [ \xi_p(0)-\varkappa \xi_p(t_p)+ip \varkappa \int_0^{t_p } dt \eta_p(t ) ] } \int\limits_0^{t_p}\!\ ! dt \ , e^{- \xi_p(t ) } \right ) \biggr \ } . \label{eq : zj2}\end{gathered}\ ] ] there is some freedom in choosing the initial conditions for field variables @xmath563 . it is convenient to choose them such that the following relations hold : @xmath564 = 0 . \label{eq : bc}\end{gathered}\ ] ] then eq . can be rewritten as @xmath565 \notag \\ \times \int\limits_{-\infty}^\infty dx\ , e^{\frac{1}{2}[\xi_p(t_p)-\xi_p(0 ) ] } e^{-\frac{i p}{4j_z } \int_0^{t_p } dt [ \frac{(ip \dot{\xi}_p+\varkappa \eta_p)^2}{(1-\varkappa)^2 } + \frac{\varkappa \eta_p^2}{1-\varkappa } ] } \notag \\ \times e^{\frac{i x p}{1-\varkappa } [ \xi_p(0)-\varkappa \xi_p(t_p ) + i p \varkappa \int_0^{t_p}dt \eta_p(t ) ] } \biggr \ } e^{- 2 v \cosh\frac{\xi_{+}(t_+)-\xi_{-}(t_-)}{2 } } \notag \\ \times e^{-\frac{i j_\perp}{4 } \sum_{p=\pm } p \int_0^{t_p } dt\ , e^{- \xi_p(t ) } } \delta \left ( \sum_{p=\pm } \xi_p(t_p ) + 2 \ln ( 4v y)\right ) . \label{eq : zj3}\end{gathered}\ ] ] integrating over the variables @xmath566 we find @xmath567 \,e^{i p \int_0^{t_p } dt \mathcal{l}_p } e^{-(1 - 2i p x ) \xi_p(0)/2 } \biggr \ } \notag \\ \times \int\limits_0^\infty \frac{dy}{4 y v } \ , e^{-y - w } \ , \exp \left ( - 2 v \cosh\bigl [ \frac{1}{2}\sum_{p=\pm}p \xi_{p}(t_p ) \bigr ] \right ) \notag \\ \times \delta \left ( \sum_{p=\pm } \xi_p(t_p ) + 2 \ln ( 4v y)\right ) . \label{eq : zj4}\end{gathered}\ ] ] the functional integral is of feynman - kac type with the lagrangian @xmath568 then the calculation of the partition function can be reduced to an evaluation of two matrix elements : @xmath569 } \notag \\ \times \langle \xi_+ | e^{-i \mathcal{h}_j t_+ } | \xi_+^\prime \rangle \langle \xi_-^\prime | e^{i \mathcal{h}_j t_- } | \xi_- \rangle . \label{eq : zj5}\end{gathered}\ ] ] here the one - dimensional quantum mechanical hamiltonian @xmath570 its eigenfunctions are given by the modified bessel functions @xmath571 where @xmath572 is a real number : @xmath573 the eigenvalues of @xmath574 are equal to @xmath575 : @xmath576 . after integration over @xmath218 we obtain @xmath577 here we use the following result ( see formula 6.794.11 on p. 794 of ref . [ ] ) @xmath578 integration over @xmath579 can be now easily performed , and we obtain @xmath580 using the identity ( see formula 6.521.3 on p. 658 of ref . [ ] ) @xmath581 we can perform the integration over @xmath582 . with the help of the integral representation of the modified bessel function @xmath583 we integrate over @xmath572 . finally , integration over @xmath217 yields @xmath584 here we restored the correct numerical factor using the normalization condition @xmath585 at @xmath586 . in order to derive the partition function for the hamiltonian from eq . one needs to make the substitution @xmath587 and to integrate over the variable @xmath87 : @xmath588 then we obtain eq . with @xmath189 . it is easy to obtain the partition function with non - zero magnetic field . the field shifts the @xmath80 projection of the total spin in the evolution operator : @xmath589 . this shift affects only the boundary conditions on @xmath294 in . at @xmath591 , the value of the integral in eq . is determined by the region @xmath592 . thus one can expand @xmath197 in the denominator into a series in @xmath593 and obtain @xmath594 here we performed the expansion to the second order in @xmath595 , having the further calculation for the spin susceptibility in mind . at @xmath596 , the argument of the @xmath197 in the denominator is large and one can make the following replacement : @xmath597 . then we find @xmath598 . \label{eq : app : f1:zzf1r2}\ ] ] the same simplification for @xmath599 can be used in the limit @xmath600 and @xmath601 . then we obtain @xmath602 . \label{eq : app : f1:zzf1r3}\ ] ] at @xmath600 and @xmath603 the relevant region of integration in eq . is determined by the denominator . thus one can omit @xmath604 , expand @xmath605 in the numerator and find @xmath606 the only relevant case for our values of @xmath217 and @xmath218 in eq . is the case with @xmath607 . in this regime the denominator can be substituted by @xmath608 and the region of integration can be extended to infinity . then the following result can be obtained from by replacement @xmath609 : @xmath610 in this appendix we present brief arguments why eqs . and are correct . since eq . can be obtained from eq . , we consider only the latter . we start from the following exact expression @xmath611 where @xmath612 . as usual , at @xmath138 the integral over @xmath613 can be performed in the saddle - point approximation . this yields @xmath614 where @xmath615 the function @xmath616 can be rewritten as @xmath617 now we bound @xmath618 from above . using that random function @xmath619 depends , in fact , on @xmath620 ( cf . , we obtain the following set of inequalities : @xmath621 hence we demonstrate that @xmath622 is independent of @xmath423 . therefore we can write @xmath616 at @xmath603 as follows : @xmath623 in this appendix we present a brief derivation of eq . . the correlation function of the single - particle density of states is given by @xcite @xmath625 . \label{app_dnu1}\ ] ] here the function @xmath626 depends on the statistics of the ensemble of single - particle energies . using eq . , the identity @xmath627 and the definition of @xmath112 we obtain @xmath628 , \label{eqc_app}\end{aligned}\ ] ] where @xmath629 . \label{eq : app : g}\ ] ] the function @xmath630 has the following fourier transform with respect to variable @xmath631 : @xmath632 since the function @xmath630 is even in @xmath631 , the function @xmath633 is even in @xmath196 . the function under the integral in the r.h.s . of eq . has poles at @xmath634 where @xmath428 and @xmath423 are integers . computation of the residues yields @xmath635 substitution into eq . leads to @xmath636 .\end{aligned}\ ] ] at @xmath637 the function @xmath626 has the following asymptotic behavior : @xmath638 recall that @xmath379 for the orthogonal wigner - dyson ensemble , @xmath212 for the unitary wigner - dyson ensemble and @xmath639 for the symplectic wigner - dyson ensemble . then at @xmath206 we find @xmath640[1-\cos(h_2 t)]}{t \sinh^2(\pi t ) } \notag \\ & = \sum_{\sigma=\pm } l(h_1+\sigma h_2)- 2 l(h_1 ) - 2l(h_2 ) , \end{aligned}\ ] ] where @xmath641 is even in @xmath88 . next , for @xmath642 @xmath643 e^{-2\pi n t } \notag \\ & = \frac{2 h}{\pi^2\bm{\beta } } \left [ \operatorname{re}\psi\left ( 1+\frac{i h}{2\pi}\right ) - \psi(1)\right ] .\end{aligned}\ ] ] this is the eq . of the paper . using the well - known asymptotic expressions for the euler digamma function @xmath644 at small and large values of its argument one arrives at eq . . in this appendix we present the derivation of the perturbative results and for @xmath189 . in addition , we compute the next order in @xmath237 for the correction to @xmath215 . the contribution of the second order in @xmath207 is given by @xmath647 and @xmath648 . we find @xmath649 . \label{eq : f2_11}\ ] ] here we remind @xmath650 . it is instructive to compare the second order contribution with the second order contribution to the variance of @xmath299 : @xmath651 . \label{eq : var}\end{aligned}\ ] ] in the regime @xmath652 the arguments of @xmath237 in the right hand side of eqs . and are small . using the asymptotic expression for @xmath213 at @xmath465 , we obtain @xmath653 the result for @xmath654 is translated into eq . of the paper . from eq . we find that @xmath655 at low temperatures @xmath656 the asymptotic expression of @xmath213 for @xmath470 must be used in eq . . we find @xmath657 from eq . it follows that @xmath658 in view of the result we can expect that @xmath299 has a normal distribution with mean @xmath659/2 $ ] and variance @xmath648 in the regime @xmath660 . for @xmath661 and @xmath662 the complementary cumulative distribution function for the normal distribution and the complementary cumulative distribution function obtained numerically for the process @xmath112 are compared in fig . [ fig : ccdf ] . we note that for @xmath661 and @xmath663 numerical integration of eqs . and yields @xmath664 and @xmath665 . these values are still different from the asymptotic estimates . in the regime @xmath652 the fourth order contributions are proportional to @xmath666 and therefore negligible . for low temperatures @xmath656 the contributions of the fourth order in @xmath207 are listed below : @xmath667 ^ 2 = -36 \ln^2 2 \ , z^4 , \ ] ] @xmath668 ^ 2 \notag \\ & + 2 \int_{-\infty}^\infty \frac{dh_1d h_2}{\pi } \ , e^{-h_1 ^ 2-h^2_2 } \bigl [ \overline{v(h_1\sqrt{y})v(h_2\sqrt{y})}\bigr ] ^2 \notag \\ & = \left ( 4 \ln^2 2 + 8 b_{2,2 } \right ) \ , z^4 , \\ b_{2,2 } & = \frac{1}{2 } \int_0^{2\pi } \frac{d\phi}{2\pi } \bigl ( \overline{v(\cos\phi)v(\sin\phi)}\bigr ) ^2 \approx 0.35 , \end{aligned}\ ] ] @xmath669 @xmath670 \notag \\ = & - \left ( 2\ln^2 2 + 2 b_{2,1,1 } \right ) \ , z^4 , \\ b_{2,1,1 } = & \frac{15}{4 } \int_0^{2\pi}\frac{d\phi}{4\pi}\int_0^\pi d\theta \ , \sin^3\theta \,\ , \overline{v(\cos\phi)v(\sin\phi)}\notag \\ & \times \overline{v(\cos\theta)v(\sin\theta\cos\phi)}\approx 0.79 , \end{aligned}\ ] ] @xmath671 ^ 2 = 3 \ln^2 2 \ , z^4 .\ ] ] here we recall that @xmath672 . summing up , for @xmath656 we obtain @xmath673 where @xmath674 using eq . and the definition of the spin susceptibility we obtain for @xmath189 @xmath675 .\ ] ] j. a. folk , c. m. marcus , r. berkovits , i. l. kurland , i. l. aleiner , b. l. altshuler , phys . t*90 * , 26 ( 2001 ) ; g. usaj , h.u . baranger , phys . b * 67 * , 121308 ( 2003 ) ; y. alhassid , t. rupp , phys . rev . lett . * 91 * , 056801 ( 2003 ) ; y. alhassid , t. rupp , a. kaminski , l. i. glazman , phys . b * 69 * , 115331 ( 2004 ) . m. schechter , phys . b * 70 * , 024521 ( 2004 ) ; zu - jian ying , m. cuoco , c. noce , huan - qiang zhou , phys . rev . b * 74 * , 012503 ( 2006 ) ; zu - jian ying , m. cuoco , c. noce , huan - qiang zhou , phys . rev . b * 74 * , 214506 ( 2006 ) ; s. schmidt , y. alhassid , k. van houcke , europhys . lett . * 80 * , 47004 ( 2007 ) ; s. schmidt , y. alhassid , phys . * 101 * , 207003 ( 2008 ) ; k. van houcke , y. alhassid , s. schmidt , s. m. a. rombouts , arxiv:1011.5421 ; y. alhassid , k. n. nesterov , s. schmidt , phys . t * 151 * , 014047 ( 2012 ) ; k. n. nesterov , y. alhassid , phys . rev . b * 87 * , 014515 ( 2013 ) . s. guron , m.m . deshmukh , e.b . myers , and d.c . ralph , phys . lett . * 83 * , 4148 ( 1999 ) ; m.m . deshmukh , s. kleff , s. guron , e. bonet , a.n . pasupathy , j. von delft , and d.c . ralph , phys . * 87 * , 226801 ( 2001 ) . canali and a.h . macdonald , phys . 85 * , 5623 ( 2000 ) ; s. kleff , j. von delft , m.m . deshmukh , and d.c . ralph , phys . b * 64 * , 220401 ( 2001 ) ; s. kleff and j. von delft , phys . rev . b * 65 * , 214421 ( 2002 ) . l.d . graham and d. s. schreiber , j. appl . phys . * 39 * , 963 ( 1968 ) ; l. shen , d. s. schreiber , and a. j.arko , phys . rev . * 179 * , 512 ( 1969 ) ; p. gambardella , s. rusponi , m. veronese , s. s. dhesi , c. grazioli , a. dallmeyer , i. cabria , r. zeller , p. h. dederichs , k. kern , c. carbone , h. brune , science * 300 * , 1130 ( 2003 ) ; a.m.clogston , b. t. matthias , m. peter , h. j. williams , e. corenzwit , and r. c. sherwood , phys . rev . * 125 * , 541 ( 1962 ) ; d. shaltiel , j. h.wrenick , h. j.williams , and m. peter , phys . rev . * 135 * , a1346 , ( 1964 ) ; j. w. loram and k. a. mirza , j. phys . phys . * 15 * , 2213 ( 1985 ) ; j. w. loram , k. a. mirza , and z. chen , j. phys . * 16 * , 233 ( 1986 ) ; g. mpourmpakis , g.e . froudakis , a.n . andriotis , m. menon , phys . b * 72 * , 104417 ( 2005 ) . i. v. kolokolov , phys . lett . * a * 114 , 99 ( 1986 ) ; ann . 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we explore the static longitudinal and dynamic transverse spin susceptibilities in quantum dots and nanoparticles within the framework of the hamiltonian that extends the universal hamiltonian to the case of uniaxial anisotropic exchange . for the limiting cases of ising and heisenberg exchange interactions
we ascertain how fluctuations of single - particle levels affect the stoner instability in quantum dots .
we reduce the problem to the statistics of extrema of a certain gaussian process .
we prove that , in spite possible strong randomness of the single - particle levels , the spin susceptibility and all its moments diverge simultaneously at the point which is determined by the standard criterion of the stoner instability involving the mean level spacing only .
= 2000
| 26,907 | 175 |
there has been much recent interest in the properties of active galactic nuclei ( agn ) and the interaction between these objects and their host galaxies . observationally , strong evidence supporting such interaction exists . up to 70% of cd galaxies contain radio sources at their centres @xcite . these radio sources drive shocks into the intra - cluster medium ( icm ) , heating and transporting outward the cluster gas . the spatial coincidence of radio emission with cavities in the hot x - ray emitting gas @xcite agrees well with analytical and numerical models @xcite , and is required to explain the lack of catastrophic cooling expected in the absence of a feedback mechanism in dense cluster cores @xcite . another piece of evidence comes from the field of galaxy formation . the growth of structure through the gravitational instability provides a simple and accurate description of the build - up of dark matter scaffolding through cosmic time , however agn feedback is required to prevent runaway cooling from forming too many massive galaxies @xcite . the tight correlation between galaxy and black hole masses @xcite provides further evidence for the tight coupling between agn and galaxy evolution . broadly speaking , two types of agn are often discussed . powerful outbursts observable at both optical and radio wavelengths are believed to be triggered by gas - rich mergers , facilitating a sudden influx of large quantities of fuel to the agn . these are often observed in the local volume as seyfert agn , and as quasars at higher redshifts . they are almost invariably accompanied by strong radio emission . by contrast , the vast majority of radio - loud agn at @xmath1 do not have a corresponding seyfert phase . the number counts of these objects are consistent with low - luminosity radio agn activity being triggered by the cooling out of hot gas from the galactic halo @xcite . in a seminal work , fanaroff & riley showed that the luminosity of a radio agn is intrinsically tied to its morphology . the most powerful radio sources are invariably edge - brightened ( fanaroff & riley referred to these as type ii objects , or fr - iis ) , propagating bow shocks through the surrounding gas as they expand . the less powerful fr - i sources , on the other hand , have radio emission that is core - dominated . these objects dominate radio source counts in the local volume , and are typically interpreted as agn that start out as fr - iis but are not powerful enough to traverse the dense galaxy or cluster core without being disrupted by rayleigh - taylor and kelvin - helmholtz instabilities . agn feedback influences galaxy formation and evolution by affecting the gas available for star formation . recently , kaviraj et al . showed that agn feedback is required to explain the rapid transition of early - type galaxies from the blue cloud to the red sequence . there are a number ways in which agn feedback can affect the gas . kinetic feedback , typically associated with powerful radio sources , can shock heat and uplift the gas to large radii . the agn radiation field can also heat the gas and drive outflows . finally , turbulence induced by disruption of the jet - inflated radio cocoons , a process mostly associated with fr - i sources , can either trigger star formation ( by aiding gravitational collapse ; e.g. silk & nusser 2010 ) or suppress it ( by heating the gas further ) . the relative importance of these processes is unclear . in particular , it is not clear how important feedback from the powerful but rare fr - iis is compared with the less powerful but omnipresent fr - is . we aim to address this issue in the present paper . the paper is structured as follows . in section [ sec : sample ] we describe our sample . results are presented in section [ sec : results ] , and discussed in section [ sec : discussion ] . we conclude in section [ sec : summary ] . in this section we describe the construction of a sample of galaxies around radio sources . the radio source list was compiled from a combination of catalogues . the largest radio agn in terms of angular size were selected from the 3crr sample of laing et al . , and complemented by additional data from machalski et al . . smaller radio sources were identified from two large - area 1.4 ghz surveys , faint images of the radio sky at twenty centimetres ( first ; becker et al . 1995 ) and the nrao vla sky survey ( nvss ; condon et al . 1998 ) . these are complementary in the sense that first has higher sensitivity , but nvss is better at picking up extended objects . shabala et al . identified optical sloan digital sky survey ( sdss ; strauss et al . 2002 ) counterparts for first / nvss radio sources to @xmath2 . we limit the present work to @xmath0 , which provides a good balance between statistically meaningful number counts of galaxies and relatively low contamination in radio / optical positional matching due to line - of - sight effects ( which increases with redshift ) . for @xmath3 , we selected radio sources from the config sample @xcite . this approach allows construction of a sample containing both the more powerful , edge - brigtened fr - ii sources , as well as the less powerful edge - darkened fr - is . details are given in tables [ tab : fr2s ] and [ tab : fr1s ] . we note that while the sample is not complete , we expect it to be representative as it spans a wide range of radio source sizes , luminosities , morphologies and ages . lllllll iau & other & redshift & frequency & las & axial & @xmath4 + name & name & & ( ghz ) & ( arcsec ) & ratio & ( myr ) + + iau & other & redshift & frequency & las & axial & @xmath4 + name & name & & ( ghz ) & ( arcsec ) & ratio & ( myr ) + 0010 - 1108 & & 0.077 & 1.4 & 200 & @xmath5 & + 0057 - 0052 & & 0.044 & 1.4 & 17 & @xmath6 & + 0131 + 0033 & & 0.079 & 1.4 & 46 & @xmath7 & + 0739 + 3947 & & 0.098 & 1.4 & 33 & @xmath5 & + 0745 + 3357 & & 0.063 & 1.4 & 14 & @xmath8 & + 0756 + 3703 & & 0.077 & 1.4 & 48 & @xmath9 & + 0758 + 3747 & & 0.041 & 1.4 & 43 & @xmath10 & + 0805 + 2409 & 3c192 & 0.060 & 1.41 & 201 & @xmath11 & + 0819 + 5232 & & 0.189 & 1.4 & 49 & @xmath12 & + 0821 + 4702 & & 0.130 & 1.4 & 38 & @xmath13 & + 0902 + 5203 & & 0.099 & 1.4 & 28 & @xmath14 & + 0911 + 3724 & & 0.105 & 1.4 & 60 & @xmath15 & @xmath16 + 0921 + 4538 & 3c219 & 0.174 & 1.52 & 190 & @xmath17 & @xmath18 + 0930 + 0348 & & 0.089 & 1.4 & 33 & @xmath19 & + 0939 + 3553 & 3c223 & 0.138 & 1.50 & 306 & @xmath20 & @xmath21 + 0941 + 3944 & & 0.107 & 1.4 & 190 & @xmath22 & + 0947 + 0725 & & 0.087 & 1.4 & 310 & @xmath23 & + 0949 - 0050 & & 0.081 & 1.4 & 25 & @xmath24 & + 0955 + 0135 & & 0.099 & 1.4 & 44 & @xmath25 & + 1001 + 2847 & 3c234 & 0.185 & 1.41 & 113 & @xmath26 & @xmath27 + 1006 + 3454 & & 0.099 & 1.4 & 1896 & @xmath28 & + 1007 + 0030 & & 0.095 & 1.4 & 17 & @xmath29 & + 1016 + 6014 & & 0.031 & 1.4 & 100 & @xmath30 & + 1016 + 4046 & & 0.128 & 1.4 & 139 & @xmath31 & + 1020 + 4832 & & 0.053 & 1.4 & 675 & @xmath32 & + 1032 + 5644 & & 0.045 & 1.4 & 389 & @xmath33 & + 1036 + 0006 & & 0.097 & 1.4 & 50 & @xmath34 & + 1039 + 0510 & & 0.068 & 1.4 & 24 & @xmath35 & + 1059 + 0517 & & 0.035 & 1.4 & 239 & @xmath36 & + 1111 + 4050 & & 0.074 & 1.4 & 322 & @xmath37 & + 1116 + 2915 & & 0.049 & 1.4 & 130 & @xmath38 & + 1137 + 6120 & & 0.111 & 1.4 & 230 & @xmath39 & + 1144 + 3710 & & 0.115 & 1.4 & 75 & @xmath40 & + 1153 + 0329 & & 0.079 & 1.4 & 13 & @xmath41 & + 1155 + 5454 & & 0.050 & 1.4 & 216 & @xmath42 & + 1214 + 0528 & & 0.078 & 1.4 & 25 & @xmath43 & + 1217 + 0336 & & 0.077 & 1.4 & 72 & @xmath44 & + 1217 + 0339 & & 0.078 & 1.4 & 24 & @xmath45 & + 1218 + 5026 & & 0.200 & 1.4 & 210 & @xmath46 & @xmath47 + 1252 + 0315 & & 0.099 & 1.4 & 24 & @xmath48 & + 1254 + 5305 & & 0.054 & 1.4 & 33 & @xmath49 & + 1254 + 2737 & & 0.086 & 1.4 & 69 & @xmath50 & + 1302 + 6229 & & 0.076 & 1.4 & 18 & @xmath51 & + 1321 + 4235 & 3c285 & 0.079 & 1.65 & 184 & @xmath52 & @xmath53 + 1328 - 0307 & & 0.085 & 1.4 & 289 & @xmath54 & + 1330 - 0206 & & 0.087 & 1.4 & 20 & @xmath55 & + 1331 - 0252 & & 0.087 & 1.4 & 13 & @xmath56 & + 1350 + 2816 & & 0.072 & 1.4 & 60 & @xmath57 & @xmath58 + 1352 + 3126 & 3c293 & 0.045 & 1.52 & 256 & @xmath59 & @xmath60 + 1354 + 0528 & & 0.077 & 1.4 & 14 & @xmath61 & + 1400 + 3021 & & 0.206 & 1.4 & 120 & @xmath62 & @xmath63 + 1443 + 5201 & 3c303 & 0.141 & 1.45 & 47 & @xmath64 & + 1454 + 1620 & & 0.046 & 1.4 & 660 & @xmath65 & + 1504 + 2600 & 3c310 & 0.054 & 1.45 & 305 & @xmath66 & @xmath67 + 1508 + 5415 & & 0.096 & 1.4 & 15 & @xmath68 & + 1516 + 0015 & & 0.052 & 1.4 & 328 & @xmath69 & + 1524 + 5428 & 3c319 & 0.192 & 1.65 & 109 & @xmath70 & @xmath71 + 1531 + 2404 & 3c321 & 0.096 & 1.51 & 307 & @xmath72 & + 1552 + 2005 & 3c326 & 0.089 & 1.40 & 1206 & @xmath73 & + 1557 + 5440 & & 0.047 & 1.4 & 200 & @xmath74 & + 1602 + 0158 & & 0.104 & 1.4 & 606 & @xmath13 & + 1617 + 3222 & 3c332 & 0.152 & 1.4 & 800 & @xmath75 & @xmath76 + 1700 + 3008 & & 0.035 & 1.4 & 135 & @xmath15 & @xmath77 + 1713 + 6402 & & 0.079 & 1.4 & 13 & @xmath78 & + 1728 + 3146 & 3c357 & 0.166 & 1.4 & 60 & @xmath79 & @xmath80 + 1729 + 5415 & & 0.084 & 1.4 & 28 & @xmath81 & + 2153 - 0711 & & 0.059 & 1.4 & 183 & @xmath82 & + 2157 - 0750 & & 0.062 & 1.4 & 28 & @xmath83 & + 2214 + 1350 & 3c442a & 0.026 & 1.38 & 597 & @xmath84 & + 2231 - 0824 & & 0.083 & 1.4 & 111 & @xmath56 & + 2315 - 0026 & & 0.091 & 1.4 & 25 & @xmath85 & + 2333 - 0027 & & 0.059 & 1.4 & 81 & @xmath86 & + llllll iau & other & redshift & frequency & las & @xmath4 + name & name & & ( ghz ) & ( arcsec ) & ( myr ) + + iau & other & redshift & frequency & las & @xmath4 + name & name & & ( ghz ) & ( arcsec ) & ( myr ) + 0318 + 4151 & ngc 1265 & 0.025 & 1.38 & @xmath87 & + 0319 + 4130 & 3c84 & 0.017 & 1.38 & @xmath88 & + 0847 + 5352 & & 0.045 & 1.4 & @xmath89 & + 0901 + 2901 & & 0.194 & 1.4 & @xmath90 & + 1002 + 1951 & & 0.168 & 1.4 & @xmath91 & + 1031 + 5225 & & 0.166 & 1.4 & @xmath92 & + 1140 + 1203 & & 0.081 & 1.4 & @xmath93 & + 1145 + 1936 & 3c264 & 0.022 & 1.40 & @xmath94 & @xmath95 + 1225 + 1253 & 3c272.1 & 0.005 & 1.41 & @xmath96 & + 1229 + 1144 & a1552 & 0.086 & 1.49 & @xmath97 & + 1230 + 1223 & 3c274 & 0.005 & 1.46 & @xmath98 & + 1236 + 1632 & & 0.068 & 1.4 & @xmath99 & + 1316 + 0702 & & 0.051 & 1.4 & @xmath100 & + 1342 + 0504 & & 0.136 & 1.4 & @xmath92 & + 1416 + 1048 & 3c296 & 0.025 & 1.45 & @xmath101 & + 1449 + 6316 & 3c305 & 0.042 & 1.42 & @xmath102 & + 1504 + 2835 & & 0.043 & 1.4 & @xmath103 & + 1513 + 2607 & 3c315 & 0.108 & 1.42 & @xmath104 & @xmath105 + 1516 + 0701 & & 0.034 & 1.4 & @xmath92 & + 1617 + 3500 & ngc 6109 & 0.030 & 0.609 & @xmath106 & + 1628 + 3933 & 3c338 & 0.030 & 4.89 & @xmath107 & @xmath71 + galaxies located near radio sources were selected from sdss . in this paper we are interested in the effects of radio sources on photometric properties of galaxies , and a photometric ( rather than spectroscopic ) sample was therefore employed . while this greatly increases sample statistics , care must be taken in assessing the accuracy of photometric redshift estimates . in figure [ fig : photoz ] we plot spectroscopic and photometric redshifts for a subsample of 161 galaxies , derived from the sdss pipeline . the photometric redshifts are estimated using template fitting . these spectral energy distribution templates are obtained using galaxies with spectroscopic identifications , and extended with spectral synthesis models @xcite . photometric redshifts are in good agreement with spectroscopic values for @xmath108 , with most values agreeing within the uncertainties of photometric measurements . this agreement disappears at low redshifts . in our sample , 12 of 21 fr - i radio sources have @xmath109 , compared to only 11 of 72 fr - iis . this is a known issue with the sdss pipeline @xcite , and is related to the inherent difficulty in getting spectroscopic redshifts to be more accurate than @xmath110 . it is therefore important to consider whether any systematic effects are introduced into the analysis by adopting photometric redshifts . specifically , there are three crucial considerations : completeness of the sample , contamination , and selection biases . , but diverge at lower redshift.,scaledwidth=45.0% ] _ completeness _ an estimate of how many bona fide group members are not picked up by our photometric identification procedure comes from comparing galaxy counts obtained using spectroscopic and photometric redshifts . we selected all spectroscopically identified galaxies projected to be in or near the 26 largest radio sources in our sample . the true number of galaxies associated with the group was estimated by requiring that the spectroscopic redshifts of group members were not offset from the redshift of the radio source host by more than @xmath111 . this @xmath112 is the optimal tolerance value : relaxing the redshift restriction ( by increasing @xmath112 ) in an individual group does not change the number counts appreciably ; however , decreasing @xmath112 gives an appreciable decrease in the number counts , indicating that some group members are being excluded . the number of photometric matches was then obtained by requiring that the spectroscopic redshift of each galaxy lay within the photometric @xmath113 uncertainties . figure [ fig : spectrophotoratio ] shows the redshift dependence of the photometric - to - spectroscopic matching rates . at @xmath114 the completeness is close to unity . at @xmath115 the completeness can be as low as 30 percent . . the photometric samples can be significantly incomplete ( as low at 30% ) at lower redshift . only groups with more than one spectroscopic identification are shown.,scaledwidth=45.0% ] _ contamination _ incorrect redshifts can also result in inclusion of unrelated galaxies seen in projection . the largest radio source in our sample , 3c236 , is 32 arcmin in size . at a redshift of 0.1 , and with a redshift uncertainty of 0.02 , this corresponds to a co - moving volume of 960 mpc@xmath116 . for comparison , the local number density of groups has been estimated by yang et al . to be around @xmath117 mpc@xmath118 . a similar number is obtained from simulations for the number density of local dark matter haloes with masses in excess of @xmath119 @xmath120 . most of these groups contain only a single member , with only 3 percent harbouring four or more galaxies @xcite , and thus the expected contamination is of order one to a few galaxies . since most groups subtend a much smaller solid angle on the sky than 3c236 , contamination is not expected to be a major issue in the present work . _ selection effects _ it is important to examine whether adopting photometric redshifts preferentially selects against galaxies with certain properties . galaxy properties are often a strong function of mass , and galaxy colours are of particular interest in the present work . we therefore investigated whether the offset between photometric and spectroscopic redshift values correlates with either colour or @xmath121-band magnitude ( a proxy for stellar mass ; e.g. shabala et al . we found no such correlations , suggesting that despite the lack of completeness at low redshift , photometric redshift identifications are suitable for our analysis . it is further worth noting that because in our analysis below we compare similar galaxies inside and outside the radio contours , any major redshift - dependent systematic effects are unlikely . in the following we therefore include all galaxies for which the redshift of the target radio source lies within the uncertainty of the galaxy photometric redshift . sdss photometry was obtained for objects within a radius equal to 1.5 times the maximum angular extent of the radio source ( column 5 in tables [ tab : fr2s ] and [ tab : fr1s ] ) from the centre of the radio source . galaxies with photometric redshifts matching the spectroscopic redshift of the radio source host ( i.e. with @xmath122 ) were retained for analysis . radio maps were used to detemine whether individual galaxies are ( in projection ) located inside or outside the radio contours . where available , high resolution radio source maps at 1.4 - 1.6 ghz ( leahy , bridle & strom 2000 ) were used . first / nvss maps at 1.4 ghz were adopted for the rest of the sources , with the exception of fr - i sources 3c338 ( 4.89 ghz map used ) and ngc6109 ( 608 mhz ) . for these two objects , the lower resolution first maps at 1.4 ghz show similar source size and structure to the above maps , and the higher resolution versions were therefore used . examples are shown in figure [ fig : contoursexample ] . _ fr - ii sources _ galaxies lying within the radio source contours are subject to projection effects due to the uncertainty in radio source orientation . in other words , these galaxies could lie either immediately behind or in front of the radio emission , but appear to be coincident with this emission . this problem can be mitigated for the fr - ii ( classical double ) radio sources . these objects typically have cylindrical morphologies , at least in regions away from the hotspots . the aspect ratio @xmath123 between semi - major and semi - minor axes is given for each source in column 5 of table [ tab : fr2s ] . for @xmath124 being the maximum linear extent of the fr - ii source , galaxies lying within @xmath125 of the radio cource centre will be immune from such a projection effect , since they fall within the radio cocoon regardless of the projection angle . projection effects due to uncertainty in galaxy position in redshift space are much harder to quantify . however , as most radio agn reside in cluster centres and are quite extended in the direction perpendicular to the jet ( i.e. @xmath126 is comparable with the cluster core radius ) , we expect the majority of the galaxies that appear to lie within @xmath126 of the cluster centre to do so . motivated by these considerations , we separated galaxies associated with fr - ii radio sources into three categories . galaxies lying outside the radio emission ( in projection ) were allocated to the `` outside '' group . galaxies projected to lie within the radio source contours but outside @xmath126 of the cluster centre were placed in the `` mixed '' group . some of these would presumably be physically coincident with the radio emission , while others wo nt . finally , galaxies located within @xmath126 of the cluster centre were placed in the `` inside '' group . these correspond to objects that are very likely to have been overrun by the radio source- only galaxies located well away from the cluster core but seen close to the line of sight to the core can contaminate this sample . _ fr - i sources _ it is much more difficult to perform a similar analysis for the edge - darkened fr - i sources . these primarily consist of two types of objects which are often referred to in the literature as twin - jet radio agn and relaxed doubles . the former initially appear as two jets emanating from the central engine . the jets are eventually disrupted ( usually at a well - defined flare point ) , and at larger radii the radio emission turns into a plume due to interaction with the intra - cluster medium . 3c296 is a classical example of this type of source . we include narrow and wide - angled tail galaxies ( e.g. 3c264 ) in this subclass of fr - is , since these are most likely twin - jet radio sources that have been swept up by interaction with the icm through which they are travelling . relaxed doubles are sources in which radio emission declines gradually with radius , in a quasi - radially symmetric fashion . these objects do not exhibit any compact structure , with 3c84 being a famous example . we identify @xmath126 with the flare point for twin - jet sources , and with half the maximal radial extent ( i.e. @xmath127 ) for relaxed doubles . the galaxies associated with fr - i radio sources are then split into `` outside '' , `` inside '' and `` mixed '' regions in exactly the same way as those near fr - iis . as we show below , fr - is do not appear to exhibit any difference in properties between the three classes , suggesting our classification is adequate . the major purpose of this work is to study the effects of radio sources on photometric properties of galaxies , and in particular galaxy colours . we therefore stacked all galaxies into the three groups outlined above . around fr - ii radio sources this yielded 58 galaxies in the `` inside '' category , 6678 in the `` outside '' group , and 318 in the `` mixed '' . for fr - is , 27 galaxies were found in the `` inside '' group , 2149 in the `` outside '' , and 36 in `` mixed '' . galaxy properties such as colour are a strong function of mass , with massive galaxies preferentially being redder . we therefore restricted our analysis to objects within a well - defined mass range @xmath128 @xmath120 . stellar mass was approximated via the @xmath121-band absolute magnitude , assuming that the solar bolometric correction is applicable ( see shabala et al . 2008 for details ) . in figure [ fig : coloursall ] we plot the cumulative distribution of @xmath129 @xmath130-corrected colours for the three groups , stacking galaxies around all radio sources in our sample . there is a clear offset between the `` inside '' and `` outside '' groups , with the `` inside '' galaxies exhibiting redder colours . colours , stacked for all radio sources . the three regions are defined in the text . galaxies within the path of the radio source show a clear offset to redder colours.,scaledwidth=45.0% ] splitting up the sample into fr - i and fr - ii radio sources ( figure [ fig : coloursfrtypes ] ) , it can be seen that this colour difference is driven exclusively by the fr - ii population . in other words , fr - iis appear to affect their environment , while fr - is do not . these conclusions are confirmed by komogorov - smirnov ( ks ) tests , with the observed fr - ii offset being statistically significant at the 2.5% level , and no statistically significant differences found for fr - is . care must be taken in interpreting these results . all radio sources in our sample are located at the centres of groups or clusters . mass segregation means that galaxies near the cluster centres ( and therefore more likely to have been classified in the `` inside '' category ) will be more massive , and therefore redder . figure [ fig : massdist ] shows that this is indeed the case for both the fr - i and fr - ii samples . we correct for this effect in two ways . firstly , our analysis is restricted to galaxies with @xmath131 @xmath120 . in other words , the most massive , red galaxies that would drive redder colours for the `` inside '' category are excluded . furthermore , we break up the @xmath128 @xmath120 mass range into bins and weigh the colours by the ratio of number counts between the category of interest ( `` inside '' or `` mixed '' ) and the control sample ( `` outside '' ) . to put it another way , we mimic the mass profile of the `` outside '' category . figure [ fig : coloursfrtypescorr ] , confirmed by ks tests , shows that our findings are robust to these corrections . the results of section [ sec : results ] appear to suggest that fr - ii and fr - i radio sources interact with their environments in entirely different ways . a large body of observational evidence suggests that fr - ii sources grow by driving large - scale bow - shocks ( e.g. fabian et al . 2003 , croston et al . 2009 ) , sweeping up the ambient gas as they proceed . shock heating and gas uplifting away from the centre of the potential well are seen in analytical models and numerical simulations @xcite , and are in fact necessary to solve the cooling flow problem @xcite and reconcile observations of the local stellar mass function with galaxy formation models @xcite . while the impact of powerful ( i.e. fr - ii ) radio source feedback on cluster gas is well understood , it is not immediately obvious that such feedback can affect individual galaxies in a similar way . a propagating bow shock will compress and shock heat both the atomic and molecular gas reservoirs that it overruns . if the local gas density is above some critical value , the gas clumps will become radiative ( e.g. sutherland & bicknell 2007 ) and the shock can in fact _ enhance _ star formation , rather than suppress it @xcite . it is a different story if one is concerned with diffuse hi gas reservoir , however , which is more susceptible to both shock heating ( due to longer cooling times ) and uplifting . fr - i radio sources start their lives as fr - iis , but have their jets disrupted by interaction with the dense interstellar / intergalactic medium . as a result , the bow shocks typically do not propagate far outside the host galaxy , and it is difficult for fr - is to affect their environment other than by turbulent mixing of the radio plasma with the cluster gas . the findings of the previous section strongly suggest that it is the propagation of the bow shock that is responsible for the redder colours of galaxies . the exact feedback mechanism ( i.e. heating or gas expulsion ) is not important to the present discussion . either way , fr - iis provide a way of suddenly depleting a reservoir of gas previously available for star formation . we set out to test whether gas heating / removal associated with an fr - ii radio source can explain the observed distribution of galaxy colours . the first step is to construct a star formation history for a galaxy unaffected by agn feedback . our modelling follows the prescriptions of kaviraj et al . , which have proved very successful at describing the colour evolution of the early - type galaxy population . the star formation histories of galaxies are represented by an instantaneous burst of star formation at high redshift , followed by more gradual recent star formation . the old burst , in which most of the stars are formed , is characterized by its age @xmath132 , typically @xmath133 gyrs ( e.g. kaviraj 2009 ) . recent star formation is parametrized by the age of the burst @xmath134 , the timescale associated with star formation @xmath135 and the initial gas fraction @xmath136 available for recent star formation at time @xmath134 . the star formation rate is given by the schmidt - kennicutt law , @xmath137 with star formation efficiency @xmath138 and dynamical timescale @xmath139 gyrs . apart from being physically sensible , kaviraj et al . find that these values reproduce the observed number counts of blue cloud and red sequence galaxies . we note that , being an empirical relation , the schmidt - kennicutt law implicitly includes the effects of supernova feedback on star formation . equating the star formation rate with the rate of change of gas mass yields @xmath140 where @xmath136 is the initial gas fraction in the galaxy at time @xmath134 . stellar population synthesis models can then be used to infer the colour evolution of the galaxy . we adopt the bruzual & charlot models with a salpeter initial mass function ( imf ) . galaxy colours are computed from luminosities , adding up contributions from the old underlying stellar population and the more recently formed stars . figure [ fig : examplecolourtracks ] shows example tracks for @xmath129 colour evolution with @xmath141 gyrs , @xmath142 gyrs , @xmath143 and metallicities of @xmath144 and @xmath145 . colour evolution for a star - forming galaxy with an underlying 9 gyr old population , and recent star formation described by the schmidt - kennicutt law ( equation [ eqn : sk_sfr ] ) with @xmath146 gyrs and @xmath143 . solid lines are for metallicity of 0.2 z@xmath147 , and dashed for 1 z@xmath147.,scaledwidth=45.0% ] comparison with figure [ fig : coloursfrtypes ] shows that the model and observations span a similar range in colour . in reality , the parameters @xmath132 , @xmath134 , @xmath148 and @xmath149 will differ from galaxy to galaxy . we follow the study of local lirgs by kaviraj in adopting distributions for @xmath132 and @xmath134 . the probability distribution in the old burst timescale @xmath132 spans the range @xmath150 gyrs , rising linearly from these values towards a peak at @xmath151 gyrs . the timescale @xmath134 associated with the onset of recent star formation is modelled with a flat prior to an age of @xmath152 gyrs . while it is true that these distributions found for lirgs may not apply to our more general galaxy population , we believe that we can be justified in adopting these . the @xmath129 colours are primarily driven by the @xmath153-band luminosity evolution of the young stellar population , and are therefore not very sensitive to the old burst age . as figure [ fig : examplecolourtracks ] shows , the colour evolution is relatively slow for @xmath154 gyr , and we therefore do not expect the exact form of the prior on @xmath134 to affect our results either . importantly , while we adopt a log - normal form for the distribution of the initial gas graction @xmath136 , the normalization is left as a free parameter used to match the observed colours in unaffected ( i.e. `` outside '' ) galaxies in our sample . model colour distributions without agn feedback . thick black solid line corresponds to observations of galaxies lying outside the fr - ii radio sources . thick red solid line shows observations of galaxies that have been overrun by fr - ii radio sources . dashed and dotted lines denote various models , with mean initial gas fraction fixed at @xmath155 but varying scatter about this value and metallicity . while a good fit can be obtained for the bulk of the `` outside '' galaxy population , no reasonable gas fraction distribution can reproduce the colours of galaxies affected by fr - ii feedback.,scaledwidth=45.0% ] figure [ fig : outsidefit ] shows the best - fit models to @xmath129 colours for the `` outside '' galaxies near fr - ii radio agn . a range of metallicities is considered . reasonable gas fractions of @xmath156 and @xmath157 reproduce the bulk of the observations . these values are consistent with the observed gas fraction @xmath158 in local early - type galaxies @xcite . the very blue and very red galaxies are not very well reproduced by the models , most likely due to a number of contributing factors . it is possible that we have not encompassed the full range of metallicities or old starburst ages . as figure [ fig : outsidefit ] shows , up to a 0.5 dex offset in @xmath129 colour can arise due to a change in metallicity . alternatively , tidal stripping effects not considered here may be important : accreting galaxies would exhibit enhanced star formation , while the stripped companions would appear redder . an intriguing possibility for the origin of the reddest galaxies , discussed in section [ sec : feedbackmodes ] , is that these carry an imprint of previous agn outbursts . we note that , overall , the models fit the bulk of the sample well . modifying the scatter changes the overall slope as well as shape in the middle of the distribution , but does not systematically shift all galaxies to either redder or bluer colours . the redder colours of `` inside '' galaxies in our sample can only be achieved with very low gas fractions , a requirement that is both at odds with the observed gas fraction values in the local volume @xcite and the gas fractions required to explain the colours of unaffected galaxies ( lying in the `` outside '' group ) in the same clusters . in our simple model we assume that the effect of a poweful agn ( such as an fr - ii ) is to starve the galaxy of gas previously available for star formation . the galaxy star formation history is therefore identical to that described above , except for a sudden truncation a time @xmath159 ago , where @xmath159 is the time since an agn - driven shock has overrun the galaxy . since typical lobe expansion speeds for classical double radio sources are of the order of a few percent of the speed of light @xcite , and most galaxies that are definitely affected by the agn lie relatively close to cluster centre by construction ( see section [ sec : separatesamples ] ) , @xmath159 approximately corresponds to the current age of the radio source . but with agn feedback . colours of galaxies overrun by expanding fr - ii shocks ( red line ) are well explained by the model.,scaledwidth=45.0% ] in figure [ fig : insidefit ] we show the impact of including this mode of feedback in our best - fit models . models with agn ages of @xmath160 myrs do a good job of reproducing the data . these timescales are consistent with a median fr - ii age of 44 myrs in our sample , and provide strong evidence that it is the depletion of the gas reservoir by an expanding fr - ii radio source that drives the colour evolution of affected galaxies . in preceding sections we have shown that powerful fr - ii radio sources can ( and do ) affect the evolution of galaxies outside their hosts . it is important to understand how long - lasting such effects are , and whether they are important to the interplay between gas heating and cooling . to address this question , we need to know how often powerful radio sources are triggered . shabala et al . studied a complete radio / optical sample of local galaxies , and concluded that around 1 percent of the most massive ( @xmath161 @xmath120 ) galaxies host agn with 1.4 ghz luminosity in excess of @xmath162 @xmath163 , which approximately corresponds to the separation luminosity between fr - is and fr - iis @xcite . given maximum fr - ii lifetimes of @xmath160 myrs , this suggests that a typical massive galaxy will launch between one and a few fr - ii outbursts in a hubble time , with the typical time between outbursts of order a few gyrs . the passing bow shock will both sweep up and heat the gas . the time delay before this gas is once again available for star formation is therefore equal to the sum of the cooling time @xmath164 and the timescale associated with reincorporation of this gas in the galaxy @xmath165 . the cooling time is a function of gas temperature and density , @xmath166 myrs . the reincorporation timescale is typically longer than this , being comparable to the dynamical timescale for the dark matter halo hosting the cluster , @xmath167 gyrs at the present epoch . these numbers suggest that fr - ii outbursts in clusters can indeed prevent galaxies from forming stars despite the relatively infrequent , short - term nature of these outbursts . the timescales of star formation suppression can also be examined observationally . for this purpose we employed a sample of 16 clusters with no detectable radio emission at 1.4 ghz , as described by dunn & fabian and dunn et al . . we split these clusters into regions in an attempt to mimic the process for fr - ii host clusters . this is done by weighing the @xmath168 and @xmath124 values ( see section [ sec : sample ] ) for each cluster hosting an fr - ii by the number of galaxies contained within that radius , yielding mass - weighted averages for these quantities . colours for clusters with no detected 1.4 ghz radio agn or bubbles . the three regions are chosen to match the fr - ii sample , and mass corrections made . there is still a clear offset to redder colours for innermost galaxies.,scaledwidth=45.0% ] figure [ fig : coloursquiescent ] shows the resultant colour distribution for the three regions . there is a clear difference between galaxies located close to cluster centre and those on the periphery , even after the usual mass corrections are performed . it is interesting to compare the radio source and quiescent samples . in figure [ fig : coloursquiescentvsfrii ] we plot colour distributions for the same regions in fr - i , fr - ii and quiescent clusters , adjusting the mass distributions in clusters with radio sources to mimic those for the appropriate quiescent samples . galaxies in clusters which contain fr - i radio sources are consistently bluer than those in the quiescent cluster sample . by contrast , fr - ii galaxies are marginally redder that the quiescent sample ( and both much redder than the fr - i sample ) in the `` inside '' region ; bluer in the `` mixed '' region ; and bluer still in the `` outside '' region , where the fr - i and fr - ii distributions are statistically indistinguishable . at face value , these findings are somewhat surprising . if galaxies in the quiescent cluster sample are unaffected by agn feedback , we would expect these to have bluer colours that the fr - ii sample at all radii , and perhaps even bluer colours than the fr - i galaxies if these suffer any ( perhaps low - level ) feedback . the results make perfect sense , however , if quiescent clusters represent a population which has undergone past agn outbursts and has not yet regained all of the ejected / heated gas . in this picture , an fr - ii agn is triggered , heating and ejecting gas from galaxies within the cluster . this gas cools and returns on timescale @xmath169 . since fr - ii lifetimes are at least an order of magnitude less than the gas return timescale ( section [ sec : recurrence ] ) , it is possible that the agn activity terminates long before the cool gas is returned to the cluster . the cluster can then appear quiescent , but still be very much affected by the recent agn outburst , consistent with the idea that radio sources can affect their host cluster on timescales much longer than the duration of the active phase ( e.g. fabian et al . 2003 , shabala & alexander 2009a ) . alternatively , the observable agn signature could disappear while still undergoing feedback . by contrast , clusters hosting radio sources must have had sufficient fuel to feed the central agn for the past @xmath159 . since @xmath170 , the radio source triggering- and , by extension , the return of the cool gas- must have happened recently . this explains the bluer colours of galaxies in clusters which host a radio source in the `` outside '' and `` mixed '' regions . the `` inside '' region of fr - ii lobes is by definition affected by feedback ( since it is the volume which has definitely been overrun by the radio lobes ) , and is therefore devoid of gas . it is likely that these galaxies simply have not had the time to form new stars before the gas was blasted out again by the latest agn outburst , explaining why their colours are similar to those for the quiescent sample . the marginally redder colours of galaxies in fr - ii clusters are likely due to a slightly longer time since the last burst of star formation than in the quiescent sample . the quiescent cluster sample was selected on the basis of a lack of 1.4 ghz radio emission . it is not immediately clear what physical condition this corresponds to . radio sources undergo significant dynamical evolution , increasing in size and typically decreasing in luminosity after a few myrs @xcite . radio telescopes are surface brightness limited , a quantity that depends on both source size and luminosity , @xmath171 where @xmath172 is the beam fwhm . at some point in its evolution a radio source can become dim and diffuse enough to no longer be detected in surveys . this can happen both as the agn jet causes expansion of the radio cocoon , and after the jet terminates . many clusters host radio bubbles ( e.g. fabian et al . 2003 , forman et al . these are remnants of past agn activity , formed as the radio lobes are pinched by kelvin - helmholtz and rayleigh - taylor instabilties . the radio plasma in the bubbles is underdense , and these therefore rise buoyantly through the cluster at a speed comparable with the sound speed , sweeping up cluster gas as they do so @xcite . if undisturbed , these bubbles can take a few gyrs to reach cluster periphery . it is therefore important to ask is over what fraction of their lifetime the active radio agn and bubbles are detectable in radio emission . the electron population emitting at 1.4 ghz is subject to energy losses via synchrotron emission , adiabatic expansion , and inverse - compton upscattering of cmb photons . radio bubbles are typically @xmath173 myrs old , a time when inverse - compton losses begin to dominate the radio luminosity . kaiser & best give @xmath174 in this regime , where the density profile of the icm goes as @xmath175 . the largest cocoon / bubble that can be detected in nvss ( @xmath176 arcsec , @xmath177 mjy ) is then @xmath178 . using the luminosity - size relation , @xmath179^{6/(10+\beta ) } \label{eqn : rbubble}\ ] ] where @xmath180 and @xmath181 are the radius and luminosity at reference time @xmath182 . source age is related to size via an appropriate dynamical model ; this age can be written as @xmath183 . in the case of an active radio source , kaiser & alexander give @xmath184 , while for an inactive bubble @xmath185 @xcite . this yields the maximum observable lifetime of @xmath186^{\frac{6}{x(10+\beta ) } } \label{eqn : tmax}\ ] ] shabala et al . derive ages and jet power for a complete sample of local radio galaxies . here , we identify the largest , oldest sources with the jet termination phase , and therefore set @xmath187 myrs , @xmath188 kpc , @xmath189 @xmath163 . this yields @xmath190 for the active , and @xmath191 for the bubble phases . in other words , the radio source would become undetectable after a few hundred myrs . this is significantly shorter than the gas return timescale , and therefore it is entirely possible that the `` quiescent '' clusters in our sample contain radio cocoons and/or bubbles undetectable in current large - scale surveys , explaining the red colours of their galaxies . the bubbles can be disrupted by a number of instabilities . however , even if they are not long - lived enough to fade and avoid detection , the long gas return timescales imply that galaxies will appear red in such clusters for an appreciable fraction of the hubble time . in considering fr - ii and fr - i radio sources separately , we have shown that two distinct modes of agn feedback exist . powerful events , which we identify with the fr - ii phase , are relatively rare , but can affect their environment on cosmological timescales through gas heating and expulsion . on the other hand , the less poweful fr - i events are much more frequent ( by as much as two orders of magnitude ; shabala et al . 2008 ) , but can not affect the evolution of galaxies outside their own host . this is consistent with a picture in which fr - i sources start out as fr - iis and are simply disrupted within the dense galaxy core . agn counts in the local volume are dominated by these low - luminosity events , and are consistent with the agn being triggered by cooling of gas out of the hot halo @xcite . recently shabala & alexander have shown that this low - level mode of feedback is sufficient to stop star formation in massive galaxies at late times and match the local stellar mass function . observations of the agn fraction are insensitive to the duration of the active and quiescent phases of radio activity , instead only providing information on the ratio @xmath192 . as discussed above , both these timescales will be significantly longer for fr - ii events than for fr - is . galaxy colours , on the other hand , are sensitive to the _ absolute _ timescales , rather than just this ratio . it appears likely that the few very red galaxies in figure [ fig : insidefit ] that can not be explained by our simple model may reside in clusters which have hosted more than one powerful fr - ii outburst . in this scenario , if re - triggering of the agn in the fr - ii host is facilitated by a sudden influx of cold gas ( as would happen , for example , in a gas - rich minor merger ) , it can proceed without catastrophic cooling in other galaxies within the cluster . repeated gas ejection outbursts are therefore possible without associated star formation . on the other hand , fr - i mode of feedback is quite different in that only the host galaxy is affected by the radio source . this means that gas must be supplied to the galaxy for another agn outburst to take place , and the outburst will therefore usually be accompanied by circumnuclear star formation . these galaxies will typically appear bluer . kaviraj et al . find that gas ejection is required to explain the colour evolution of early - type galaxies from the blue cloud to the red sequence . this could be provided by either of the two modes of feedback discussed here . however , it is difficult to envisage a scenario in which small - scale agn outbursts can explain the colours of the reddest observed galaxies . galaxies recently overrun by fr - ii radio sources show no correlation between the radio source age and @xmath129 colour , suggesting that the colours are only sensitive to agn feedback on timescales exceeding typical radio source lifetimes of @xmath193 myrs . we present an analysis of galaxy colours around fr - i and fr - ii type radio sources . we find that the less powerful fr - i radio sources can not affect the colours of galaxies near agn hosts ; while galaxies overrun by expanding fr - ii radio sources exhibit redder colours that are consistent with truncation of star formation following the passage of a radio source - driven bow - shock . we compare the fr - i and fr - ii samples with galaxies located in clusters that show no evidence of current agn activity , and find that these quiescent clusters are redder than expected for a secular evolution scenario . galaxy colours can be affected on timescales significantly exceeding the detectable agn lifetime , and rare powerful agn events thus play an important role in the colour evolution of local galaxies . we thank the referee , elaine sadler , for insightful , thorough and constructive comments that have undoubtedly improved the manuscript . sss thanks new college , oxford for a research fellowship and the bipac institute at oxford for support . sk acknowledges a research fellowship from the royal commission for the exhibition of 1851 , an imperial college junior research fellowship , a senior research fellowship form worcester college , oxford and support from the bipac institute . sss thanks anna scaife for assistance with the radio data , and dominic ford for creating pyxplot . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the us department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society and the higher education funding council for england . the sdss web site is http://www.sdss.org . the sdss is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , university of cambridge , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory and the university of washington . the nvss and first surveys were carried out using the national radio astronomy observatory vla . the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc .
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we investigate the effects of agn feedback on the colour evolution of galaxies found in local ( @xmath0 ) groups and clusters .
galaxies located within the lobes of powerful fanaroff - riley type ii ( edge - brightened ) sources show much redder colours than neighbouring galaxies that are not spatially coincident with the radio source .
by contrast , no similar effect is seen near fanaroff - riley type i ( core - dominated ) radio sources .
we show that these colours are consistent with fr - iis truncating star formation as the expanding bow shock overruns a galaxy .
we examine a sample of clusters with no detectable radio emission and show that galaxy colours in these clusters carry an imprint of past agn feedback .
agn activity in the low - redshift universe is predominantly driven by low - luminosity radio sources with short duty cycles .
our results show that , despite their rarity , feedback from powerful radio sources is an important driver of galaxy evolution even in the local volume .
galaxies : evolution galaxies : photometry galaxies : active
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