conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

unified approach to h optimization hankel approximation and balanced realization problems a unified approach is proposed here to solve three problems simultaneously namely h optimization hankel approximation and minimal balanced realization for scalar systems it is shown that all these problems can be reduced to the same standard form wherein merely the singular value decomposition svd of an easily constructed square matrix is required to establish the final solutions the formulation is solely in terms of the coefficients of a transfer function and the resulting algorithm can be performed in a very computationally efficient way

corrections to finite size scaling for quantum chains the authors compute the leading order correction to the energy gap at the critical point for the anisotropic ising and the z3 potts quantum chains periodic anti periodic and free boundary conditions are considered for the ising case the corrections are independent of the boundary conditions in the z3 case the correction is the same for anti periodic and free boundary conditions but is 2 5 smaller for the periodic case the authors have no explanation for this phenomenon

conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

one step extension approach to optimal hankel norm approximation and h sup infinity optimization problems a methodology is presented for hankel approximation and h sup infinity optimization problems that is based on a new formulation of a one step extension problem which is solved by the sarason interpolation theorem the parameterization of all optimal hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an hermitian matrix composed directly from the coefficients of a given transfer function matrix phi rather than starting with the state space realization of phi the authors use polynomial coefficients of phi as input data in terms of these data a natural basis is given for the finite dimensional sarason model space and all computations involve only manipulations with finite matrices

disorder solutions for ising and potts models with a field the exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard ising and potts models with a magnetic field are obtained different specialisations and extensions of these exact results are examined

the singular value decomposition of an infinite hankel matrix abstract let h be an infinite hankel matrix of known finite rank r a new algorithm for the numerical calculation of the singular values and vectors of h is presented the method proceeds by reduction to the singular value problem for an r r matrix this is achieved without solving for the poles of the symbol of h the resulting algorithm is of order r 3

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

an efficient algorithm on optimal hankel norm approximation for multivariable systems this note provides a novel methodology for hankel approximation and h sup infinity optimization problems based on a new formulation of the one step extension problem which is solved by the sarason interpolation theorem the present method does not require an initial balanced realization and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a hermitian matrix composed directly from the coefficients of the given transfer function matrix

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

correlation length in ising strips with free and fixed boundary conditions the correlation length of boundary spins in the ising model defined on strips of triangular lattice with free boundary conditions is determined with an efficient numerical procedure based on the star triangle transformation in the case of isotropic critical interactions the extrapolated amplitude of the correlation length is in excellent agreement with the value 2 pi eta sub predicted by conformal invariance an analytical formula for the amplitude in strips with anisotropic interactions is proposed fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor 1 2 the convergence of phenomenological renormalisation with free boundary conditions is studied

nonuniversal critical behaviour in planar ising models with extended radially symmetric defects using conformal mapping and finite size scaling the authors determine the local magnetic exponent near a defect characterized by couplings of the form j r j 1 a r in accordance with phenomenological considerations they find that it depends continuously on the amplitude a both for a defect in the interior and on the boundary

finite size scaling in strips antiperiodic boundary conditions the exact values of the universal amplitude a relating the correlation length xi approximately l a to the strip width l at the critical point of ising and 3 state potts models are obtained for the case of antiperiodic or twisted boundary conditions predictions are also made for the finite size scaling behaviour of the interfacial tension

stability and convergence properties of bergman kernel methods for numerical conformal mapping in this paper we study the stability and convergence properties of bergman kernel methods for the numerical conformal mapping of simply and doubly connected domains in particular by using certain wellknown results of carleman we establish a characterization of the level of instability in the methods in terms of the geometry of the domain under consideration we also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods when the basis set used contains functions that reflect the main singular behaviour of the conformal map

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

finite size scaling for directed bond percolation with and without cycles on a triangular lattice the authors apply finite size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic directed cyclic and undirected bond percolation on a triangular lattice the results are in good agreement with known estimates and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases as well as the distinction between locally directed and fully isotropic problems in addition the authors draw tentative comments on the extension to directed problems of the relation a pi eta where a is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point known to hold for various isotropic systems

universal functions in finite size scaling a numerical study in the ising and potts q 3 universality class the universality of the correlation length order parameter and susceptibility amplitudes is studied in the vicinity of the critical point of two dimensional models on strips with periodic boundary conditions for the ising and potts q 3 universality class the results confirm privman fisher conjecture 1984

gramian matrices and balanced model of generalized systems in this work the generalization of reachability and observability gramian matrices of control generalized systems have been studied the properties of these matrices to be solutions of lyapunov equations have been analyzed in addition an algorithm to obtain balanced generalized systems is given

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

accurate determination of the critical line of the square ising antiferromagnet in a field several points on the critical line of the simple quadratic antiferromagnetic ising model in a field are located with high precision this is accomplished by means of finite size scaling of the magnetic correlation length of n infinity strips the latter quantities were determined by means of the transfer matrix method for strip widths up to n 20 an expression possibly with the exact form is constructed which reproduces the critical line with an accuracy to the order of 10 10

h optimal observer based controller design for nonlinear multivariable systems abstract we propose a novel algorithm to synthesize an h optimal observer based controller for a nonlinear multivariable system based on the parametrization of the observer based controller a necessary and sufficient condition to achieve robust stability is developed moreover the nevanlinna pick interpolation theory is employed to test the solvability condition of the robust stabilization problem optimal robustness can be achieved via the technique of h optimization the exact solution of the model matching problem induced by the stability criterion is solved by the method of noniterative computation of the h norm

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

equilibrium crystal shape of hard squares with diagonal attractions the author exactly calculates the anisotropic interfacial tension of a square lattice gas model with nearest neighbour exclusion and next nearest neighbour attractions this is achieved by a method which introduces the shift operator into the standard transfer matrix argument from the calculated anisotropic interfacial tension the equilibrium crystal shape ecs is derived by the use of the wulff construction the ecs is a closed curve in the x y plane represented as cosh lambda x y kbt a3cosh lambda x y kbt a4 2 0 with a scale factor lambda and constants a3 a4 he argues that this shape is a universal one which appears as the ecss of a wide class of models

operator content of the blume capel quantum chain the authors calculated the conformal anomaly number of the blume capel quantum chain via finite size corrections to the ground state energy in addition they compute at the tricritical point the lowest excitations of the spectrum to be compared with the theoretical description in terms of the irreducible representations of the corresponding virasoro algebra

free energy amplitude in finite size scaling the baxter model the critical amplitude a0 of the free energy density of the baxter model is studied numerically on strips with widths l 2 12 as a function of the four spin interaction k4 it remains almost constant near to double the exactly known ising result a0 2a0ising pi 6 in the range of k4 values for which an extrapolation to infinite width is possible

the o 3 gauge transformation and 3 state vertex models the authors consider the o 3 gauge transformation for three state vertex models on lattices of coordination number three using an explicit mapping between o 3 and sl 2 they establish that there exist exactly six polynomials of the vertex weights which are fundamentally invariant under the o 3 transformation explicit expressions of these fundamental invariants are obtained in the case of symmetric vertex weights

a new algorithm for the minimal balanced realization of a transfer function matrix abstract a new algorithm is proposed here to obtain a minimal balanced realization directly from the transfer function matrix tfm this method which employs the singular value decomposition svd of an infinite block hankel matrix requires neither an initial minimal realization nor the solution of a lyapunov matrix equation the formulation is solely in terms of the coefficients of the transfer function matrix

minimal balanced realization of transfer function matrices using markov parameters a new relation between the controllability observability matrix and gramian for any stable realization is used to enable the development of an algorithm for obtaining a minimal balanced realization directly from the transfer function matrix the algorithm employs the markov parameters and a positive definite symmetric matrix which is composed solely from the coefficients of the least common denominator polynomial of the transfer function matrix

equality of directional critical exponents in multiparameter percolation models in multiparameter percolation models a directional critical exponent may be considered for each direction of approach in the parameter space to a point on the critical surface an elementary coupling argument shows that for each point on the critical surface the directional critical exponents are equal for all directions strictly in the positive orthant using the substitution method the set of directions with equal directional critical exponents can he extended beyond the orthant the anisotropic triangular lattice bond percolation model is used to illustrate the method

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

six vertex model with rotated boundary conditions we investigate the six vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes a model recently introduced by litvin and priezzhev auxiliary vertices are used to define an inhomogeneous system which leads to a one parameter family of commuting transfer matrices a product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system using an equation for commuting transfer matrices we calculate their eigenvalues finite size properties of the model are discussed from the viewpoint of the conformal field theory

balanced realization of stable transfer function matrices simple formulas are presented to compute the internally balanced minimal realization and the singular decomposition of the hankel operator of a given continuous time p m stable transfer function matrix e s d s the proposed formulas involve the schwarz numbers of d s and the singular eigenvalues eigenmatrices of a suitable finite matrix similar results are also obtained for a given discrete time transfer function matrix

on the symmetry of universal finite size scaling functions in anisotropic systems we postulate a symmetry of universal finite size scaling functions under a certain anisotropic scale transformation which connects the properties of a finite two dimensional 2d system at criticality with generalized aspect ratio 1 to a system with 1 the symmetry is formulated within a finite size scaling theory and expressions for several universal amplitude ratios are derived the predictions are confirmed within the exactly solvable weakly anisotropic 2d ising model and are checked within the strongly anisotropic 2d dipolar in plane ising model using monte carlo simulations

an exact result for the kagome lattice ising model with magnetic field an anisotropic ising model with three interaction parameters k1 k2 k3 and a magnetic field h formulated on the kagome lattice is solved exactly for an appropriate relation between ki and h when this relation is satisfied the system becomes equivalent to a free fermion model

exact solution of a triangular ising model in a nonzero magnetic field a closed form expression is obtained for the free energy per site of the ising model on the triangular lattice in a nonzero magnetic field and with two and three site interactions the solution is valid along a trajectory in the parameter space and is derived using a method of exact decimation a criterion determining the validity of the decimation method is also established

conformal invariance in incommensurate phases we study finite size corrections to the free energy of free fermion models on a torus with periodic twisted and fixed boundary conditions inside the critical striped incommensurate phase the free energy densityf n m on ann m square lattice with periodic or twisted boundary conditions scales asf n m f a s nm we derive exactly the finite size scaling fss amplitudesa s as a function of the aspect ratios m n these amplitudes are universal because they do not depend on details of the free fermion hamiltonian we establish an equivalence between the fss amplitudes of the free fermion model and the coulomb gas system with electric and magnetic defect lines the twist angle generates magnetic defect lines while electric defect lines are generated by competition between domain wall separation and system size the fss behavior of the free fermion model is consistent with predictions of the theory of conformal invariance with the conformal chargec l for instance the fss amplitude on an infinite cylinder with fixed boundary conditions is found to be one quarter of that with periodic boundary conditions finally we conjecture the exact form of the fss amplitudes for an interacting fermion model on a torus numerical calculations employing the bethe ansatz confirm our conjecture in the infinite cylinder limit

logarithmic corrections to finite size scaling in strips abstract the corrections to the finite size scaling behaviour of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width which occur when the hamiltonian contains a marginal operator are computed using conformal invariance they show a calculable universal logarthmic character for the four state potts model they agree with numerical data

universal amplitudes in finite size scaling generalisation to arbitrary dimensionality the relationship between the correlation length and critical exponents in finite width strips in two dimensions is generalised to cylindrical geometries of arbitrary dimensionality d for d not 2 these correspond however to curved spaces the result is verified for the spherical model

finite size scaling of the free energy levels in the ising and potts q 3 universality the finite size scaling of the free energy levels of privman and fisher 1984 is tested numerically on two dimensional lattice strips in the ising and q 3 potts universality the universal amplitudes and the free energy per site are obtained

an exact functional relation for the partition function of a 2d ising model with magnetic field an exact functional relation is established for the partition function of the ising model with magnetic field formulated on a kagome lattice of arbitrary size n an exact solution found in a previous work see ibid vol 21 p 31 1988 is a particular case of the result presented in this letter

duality of ordinary and extraordinary surface critical behaviour in the two dimensional potts model the dual relationship between semi infinite two dimensional q state potts models with free and fixed spin boundary conditions is used to study the surface critical behaviour at the bulk critical temperature with fixed boundary spins i e the extraordinary transition both the magnetisation and energy densities have singularities of the same form mod t tc mod 2 alpha as the bulk free energy at t tc both the spin spin and energy energy correlations decay as r 4 parallel to the surface

logarithmic corrections to gap scaling in random bond ising strips numerical results for the first gap of the lyapunov spectrum of the self dual random bond ising model on strips are analysed it is shown that finite width corrections can be fitted very well by an inverse logarithmic form predicted to hold when the hamiltonian contains a marginal operator

a simple algorithm on minimal balanced realization for transfer function matrices given a transfer function matrix it is shown that its minimal balanced realization can be obtained directly from the singular values and the singular vectors of a constant matrix constructed from the coefficients of the least common denominator polynomial the algorithm is based on the close relationship of controllability and the observability gramians to the singular value decomposition of the associated infinite block hankel matrix

universal ratio of magnetization moments in two dimensional ising models the authors calculate the universal critical point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic ising models on the square and on the triangular lattices periodic boundary conditions are used in accordance with the four fold and six fold rotational symmetries of the respective lattices these results which are obtained by means of an analysis of finite size data computed with a transfer matrix technique have an accuracy of the order of one millionth this analysis is also applied to rectangular systems with arbitrary aspect ratios

a uniqueness condition for gibbs measures with application to the 2 dimensional ising antiferromagnet a uniqueness condition for gibbs measures is given this condition is stated in terms of absence of a certain type of percolation involving two independent realisations this result can be applied in certain concrete situations by comparison with ordinary percolation in this way we prove that the ising antiferromagnet on a square lattice has a unique gibbs measure if 4 h 1 2ln pc 1 pc whereh denotes the external magnetic field the inverse temperature andpc the critical probability for site percolation on that lattice sincepc is larger than 1 2 this extends a result by dobrushin kolafa and shlosman whose proof was computer assisted

finite size analysis of the hard square lattice gas we investigate the hard square lattice gas model by means of transfer matrix calculations and a finite size scaling analysis using a minimal set of assumptions we find that the spectrum of correction to scaling exponents is consistent with that of the exactly solved ising model and that the critical exponents and correlation length amplitudes closely follow the relation predicted by conformal invariance assuming that these spectra are exactly identical and conformal invariance we determine the critical point the conformal anomaly and the temperature and magnetic exponents with numerical margins of 10 11 or less these results are in a perfect agreement with the exactly known ising universal parameters in two dimensions in order to obtain this degree of precision we included system sizes as large as feasible and used extended precision floating point arithmetic the latter resource provided a substantial improvement of the analysis despite the fact that it restricted the transfer matrix calculations to finite sizes of at most 34 lattice units

universal amplitude exponent relations for interfaces and conformal mappings the author considers the finite size fs scaling behaviour of the transverse correlation lengths xi sub n for an interface confined in a strip geometry exactly at a second order fluctuation dominated wetting transition he shows that the dimensionless amplitude ratios a sub n sigma l2 kbt xi sub n with sigma the surface stiffness coefficient are related to the short distance expansion critical exponent theta by a sub n pi 2n n theta 1 2 for all fluctuation dominated wetting transitions provided the binding potential v y in the capillary wave model is conformally mapped from the semi infinite plane he argues that the existence of this amplitude exponent relation reflects the conformal invariance of the capillary wave model lagrange density and lists a number of one point functions which can be shown to exhibit this invariance explicitly

approximation of conformal mappings of annular regions in this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal mapping f of an annular region a onto a circular annulus in particular we consider the case where f has an analytic extension in compl overline a and for this case we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform convergence we also estimate the rate of uniform convergence in the case where the annular region a has piecewise analytic boundary without cusps in both cases we also give the corresponding rates for the approximations to the conformal module of a

susceptibility of the kagome lattice ising model we explicitly calculate the zero field magnetic susceptibility of the anisotropic kagome lattice ising model on two different varieties of the parameter space one of them is the limith 0 of the solubility condition obtained in a previous paper by giacomini for the model with magnetic field the other one is the disorder variety of the model for which a dimensional reduction occurs these varieties do not contain any nontrivial critical behavior of the model a functional relation is also established which relates the zero field susceptibility for ferromagnetic and competing interactions

high dimensional lattice gases we investigate the critical behaviour of hard core lattice gases in four five and six dimensions by means of monte carlo simulations in order to suppress critical slowing down we use a geometrical cluster monte carlo algorithm in particular nearest neighbour exclusion lattice gases on simple hypercubic lattices are investigated these models undergo ising like ordering transitions where the majority of the lattice gas particles settle on one of two sublattices a finite size scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour the results agree with the renormalization predictions at and above the upper critical dimensionality in particular the predicted value of the binder cumulant is confirmed

finite size scaling amplitudes in a random tiling model we study finite size corrections to the free energy in a two dimensional random tiling model this model is equivalent to an asymmetric six vertex model and exhibits commensurate incommensurate phase transitions of the pokrovsky talapov type we calculate finite size scaling amplitudes of the free energy inside the incommensurate phase analytically and numerically by employing a transfer matrix method both periodic and free boundary conditions are considered these amplitudes are consistent with predictions of the theory of conformal invariance with the conformal charge c 1

finite size studies of the ashkin teller model the authors study the ashkin teller model in the time continuous hamiltonian version finite size scaling is used to calculate the magnetic gamma m electric gamma p and thermal alpha and nu critical exponents for several values of the coupling constant lambda the results confirm the believed extended scaling relations and suggest a conjecture relating the mass gap amplitudes and critical indices in the hamiltonian context

tests of hyperuniversality for self avoiding walks a universal combination of amplitudes for the end to end distance of n step self avoiding walks and for the number of n step self avoiding polygons is estimated numerically for various lattices in d 2 universality of this amplitude combination is confirmed to good accuracy while in d 3 the data are consistent with universality but the error limits are rather large for finite width square lattice strips the authors calculate universal critical point correlation amplitudes by fixed fugacity monte carlo simulations in addition these amplitudes are calculated analytically either exactly or approximately by cardy s conformal mapping approach numerical values agree well with the theoretical predictions

critical behaviour and logarithmic corrections of a quantum model with three spin interaction extended finite size scaling and series expansion methods are used to determine the critical properties of the one dimensional quantum ising model with three spin interaction by supposing the presence of logarithmic corrections the universality class of the model is investigated the results obtained by different methods support the conjecture that the model belongs to the same universality class as the four state potts model

a critical amplitude relation for one dimensional quantum transitions and determination of the exponent eta a new critical amplitude relation which holds for one dimensional quantum ground state transitions is presented the relation yields an estimate of the sound velocity which appears in the conformal field theory as an a priori unknown parameter with the use of this relation the exponent eta can be explicitly determined from only the scaling fit of the off critical energy gap the relation is confirmed in the transverse ising model the s 1 2 anisotropic xy model and the three state potts model

universal ising dynamics in two dimensions we explore several dominant eigenvalues of the spectra of markov matrices governing the dynamics of models in the universality class of the two dimensional ising model by means of a variational approximation we determine autocorrelation times of progressively rapid relaxation modes the approximation of one eigenstate associated with the slowest mode is employed in a variance reducing monte carlo method the resulting correlation times for which statistical errors exceed the systematic errors associated with the variational approximation are used for a finite size scaling analysis which corroborates universality of the dynamic critical exponent z for three distinct ising models on the square lattice tentative variational results for subdominant states strongly suggest that the amplitudes of the divergent time scales associated with different relaxation modes differ solely by metric factors setting a single non universal time scale for each model a by product of our analysis is a highly accurate confirmation of static universality

critical behaviour of a model with nnn interaction a one dimensional quantum model with next nearest neighbour interaction is studied under the long wavelength approximation the authors predict that this model belongs to the same universality class as the ising model this conclusion is checked with a numerical solution which is consistent with the prediction in both critical point and critical exponent

the non interacting hard square lattice gas ising universality the critical exponents of the non interacting hard square lattice gas model are determined by means of series analysis and finite size scaling the series analysis leads to results for the exponents that are consistent with the expected ising universal values a scaling analysis is performed on different types of numerical finite size data obtained by means of the transfer matrix method the results support ising universality a procedure based on conformal invariance reproduces the known ising temperature and magnetic exponents within 10 6 furthermore we estimate the critical density of the hard square model as rho c 0 367 743 000 5

pole type singularities and the numerical conformal mapping of doubly connected domains abstract let f be the function which maps conformally a given doubly connected domain onto a circular annulus and let h z f z f z 1 z in this paper we consider the problem of determining the main singularities of the function h in compl our purpose is to provide information regarding the location and nature of such singularities and to explain how this information can be used to improve the efficiency of certain expansion methods for numerical conformal mapping

spatial anisotropy and rotational invariance of critical hard squares we develop a new method to analyse solvable models defined on a square lattice rotated through an arbitrary angle phi with respect to the coordinate axes in the method we introduce auxiliary faces into a rotated system to relate it with an inhomogeneous one in the natural orientation phi 0 the inhomogeneous system is investigated by commuting a transfer matrix argument as an application of the method we consider interacting hard squares at the critical point we discuss relations between spatial anisotropy and rotational or conformal invariance of the model in calculations of finite size properties

finite size scaling of the quantum ising chain with periodic free and antiperiodic boundary conditions the authors give exact results for the energy spectrum of a chain of n ising spins in a transverse field with periodic free and antiperiodic boundary conditions the dependence of the energy gaps on boundary conditions is compatible with predictions of conformal invariance for correlation lengths in two dimensional strips the feasibility of calculating surface critical indices using phenomenological renormalisation with free boundary conditions and the convergence for large n are discussed