exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

families of graphs with w r g q functions that are nonanalytic at 1 q 0 denoting p g q as the chromatic polynomial for coloring an n vertex graph g with q colors and considering the limiting function w g q lim n to infty p g q 1 n a fundamental question in graph theory is the following is w r g q q 1 w g q analytic or not at the origin of the 1 q plane where the complex generalization of q is assumed this question is also relevant in statistical mechanics because w g q exp s 0 k b where s 0 is the ground state entropy of the q state potts antiferromagnet on the lattice graph g and the analyticity of w r g q at 1 q 0 is necessary for the large q series expansions of w r g q although w r is analytic at 1 q 0 for many g there are some g for which it is not for these w r has no large q series expansion it is important to understand the reason for this nonanalyticity here we give a general condition that determines whether or not a particular w r g q is analytic at 1 q 0 and explains the nonanalyticity where it occurs we also construct infinite families of graphs with w r functions that are non analytic at 1 q 0 and investigate the properties of these functions our results are consistent with the conjecture that a sufficient condition for w r g q to be analytic at 1 q 0 is that g is a regular lattice graph lambda this is known not to be a necessary condition

ground state entropy of potts antiferromagnets on homeomorphic families of strip graphs abstract we present exact calculations of the zero temperature partition function and ground state degeneracy per site w for the q state potts antiferromagnet on a variety of homeomorphic families of planar strip graphs g ch k1 k2 k m where k1 k2 and k describe the homeomorphic structure and m denotes the length of the strip several different ways of taking the total number of vertices to infinity by sending i m with k1 k2 and k fixed ii k1 and or k2 with m and k fixed and iii k with m and p k1 k2 fixed are studied and the respective loci of points b where w is nonanalytic in the complex q plane are determined the b s for limit i are comprised of arcs which do not enclose regions in the q plane and for many values of p and k include support for re q b for limits ii and iii is the unit circle q 1 1

absence of phase transition for antiferromagnetic potts models via the dobrushin uniqueness theorem we prove that theq state potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures including zero temperature wheneverq 2r we also prove slightly better bounds for several two dimensional lattices square lattice exponential decay forq 7 triangular lattice q 11 hexagonal lattice q 4 and kagome lattice q 6 the proofs are based on the dobrushin uniqueness theorem

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

ground state entropy in potts antiferromagnets abstract the q state potts antiferromagnet exhibits nonzero ground state entropy s0 g q 0 for sufficiently large q on a given n vertex lattice or graph g and its n limit g we present exact calculations of the zero temperature partition function z g q t 0 and w g q where s 0 k b ln w for this model on a number of families g these calculations have interesting connections with graph theory since z g q t 0 p g q where the chromatic polynomial p g q gives the number of ways of coloring the vertices of the graph g such that no adjacent vertices have the same color generalizing q from z to c we determine the accumulation set b of the zeros of p g q which constitute the continuous loci of points on which w is nonanalytic the potts antiferromagnet has a zero temperature critical point at the maximal value qc where b crosses the real q axis in particular exact solutions for w and b are given for infinitely long finite width strips of various lattices in addition to their intrinsic interest these yield insight into the approach to the 2d thermodynamic limit some corresponding results are presented for the exact finite temperature potts free energy on families of graphs finally we present rigorous upper and lower bounds on w for 2d lattices

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

ground state degeneracy of potts antiferromagnets homeomorphic classes with noncompact w boundaries abstract we present exact calculations of the zero temperature partition function z g q t 0 and ground state degeneracy w g q for the q state potts antiferromagnet on a number of families of graphs g for which generalizing q from z to c the boundary b of regions of analyticity of w in the complex q plane is noncompact passing through z 1 q 0 for these types of graphs since the reduced function w red q 1 w is nonanalytic at z 0 there is no large q taylor series expansion of w red the study of these graphs thus gives insight into the conditions for the validity of the large q expansions it is shown how such families of graphs can be generated from known families by homeomorphic expansion

ground state entropy of potts antiferromagnets bounds series and monte carlo measurements we report several results concerning w ensuremath lambda q mathrm exp s 0 k b the exponent of the ground state entropy of the potts antiferromagnet on a lattice ensuremath lambda first we improve our previous rigorous lower bound on w mathrm hc q for the honeycomb hc lattice and find that it is extremely accurate it agrees to the first 11 terms with the large q series for w mathrm hc q second we investigate the heteropolygonal archimedean 4 ensuremath cdot 8 2 lattice derive a rigorous lower bound on w 4 ensuremath cdot 8 2 q and calculate the large q series for this function to o y 12 where y 1 q ensuremath 1 remarkably these agree exactly to all 13 terms calculated we also report monte carlo measurements and find that these are very close to our lower bound and series third we study the effect of non nearest neighbor couplings focusing on the square lattice with next nearest neighbor bonds

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

on the herzog schonheim conjecture for uniform covers of groups let g be any group and a1g1 akgk k 1 be left cosets in g in 1974 herzog and schonheim conjectured that if a aigi i 1k is a partition of g then the finite indices n1 g g1 nk g gk cannot be pairwise distinct in this paper we show that if a covers all the elements of g the same number of times and g1 gk are subnormal subgroups of g not all equal to g then m max1 j k 1 i k ni nj is not less than the smallest prime divisor of n1 nk moreover min1 i klogni o mlog2m where the o constant is absolute

upper and lower bounds for the ground state entropy of antiferromagnetic potts models we derive rigorous upper and lower bounds for the ground state entropy of the q state potts antiferromagnet on the honeycomb and triangular lattices these bounds are quite restrictive especially for large q copyright ital 1997 ital the american physical society

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

ground state entropy of potts antiferromagnets and the approach to the two dimensional thermodynamic limit we study the ground state degeneracy per site exponent of the ground state entropy w lambda l x infty times l y q for the q state potts antiferromagnet on infinitely long strips with width l y of 2d lattices lambda with free and periodic boundary conditions in the y direction denoted fbc y and pbc y we show that the approach of w to its 2d thermodynamic limit as l y increases is quite rapid for moderate values of q and l y simeq 4 w lambda l x infty times l y q is within about 5 and cal o 10 3 of the 2d value w lambda l x infty times l y infty q for fbc y and pbc y respectively the approach of w to the 2d thermodynamic limit is proved to be monotonic non monotonic for fbc y pbc y it is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states

ground state degeneracy of potts antiferromagnets on two dimensional lattices approach using infinite cyclic strip graphs the q state potts antiferromagnet on a lattice lambda exhibits nonzero ground state entropy s0 kb ln w for sufficiently large q and hence is an exception to the third law of thermodynamics an outstanding challenge has been the calculation of w sq q on the square sq lattice we present here an exact calculation of w on an infinite length cyclic strip of the square lattice which embodies the expected analytic properties of w sq q similar results are given for the kagome lattice

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted