Lattice gas generalization of the hard hexagon model. III.q-Trinomial coefficients

In the first two papers in this series we considered an extension of the hard hexagon model to a solvable two-dimensional lattice gas with at most two particles per pair of adjacent sites, and we described the local densities in terms of elliptic theta functions. Here we present the mathematical theory behind our derivation of the local densities. Our work centers onq-analogs of trinomial coefficients.     

Rogers-Ramanujan identities in the hard hexagon model

The hard hexagon model in statistical mechanics is a special case of a solvable class of hard-square-type models, in which certain special diagonal interactions are added. The sublattice densities and order parameters of this class are obtained, and it is shown that many Rogers-Ramanujan-type identities naturally enter the working.Supported in part by the National Science Foundation under Grant No. PHY-79-06376A01.Part of this work was performed while the author was a visiting professor at the Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11794.     

Lattice gas generalization of the hard hexagon model. I. Star-triangle relation and local densities

In the solvable hard hexagon model there is at most one particle in every pair of adjacent sites, and the solution automatically leads to various mathematical identities, in particular to the Rogers-Ramanujan relations. These relations have been generalized by Gordon. Here we construct a solvable model with at most two particles per pair of adjacent sites, and find the solution involves the next of Gordon's relations. We conjecture the corresponding solution for a model with at mostn particles per pair of adjacent sites: this involves all Gordon's relations, as well as others that we will discuss in a subsequent paper.     

Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

The eight-vertex model is equivalent to a solid-on-solid (SOS) model, in which an integer heightl _i is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter . Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which =K/5 and the heights are restricted to the range 1l _i4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with =K/r (r an integer) and 1l _ir–1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponents, , are obtained.Supported by the Guggenheim Foundation and in part by the National Science Foundation, grant MCS 8201733.     

Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities

The restricted eight-vertex solid-on-solid (SOS) model is an exactly solvable class of two-dimensional lattice models. To each sitei of the lattice there is associated an integer heightl _i restricted to the range 1l _i r–1. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a parameter. In an earlier paper we considered the case=K/r. Here we generalize those considerations to the case=sK/r, s an integer relatively prime tor. We are again led to generalizations of the Rogers-Ramanujan identities.     

Modular properties of the hard hexagon model

The physical quantities (or powers thereof) in the hard-hexagon model that were computed exactly by Baxter are shown to be modular functions with respect to the number-theoretic group ₁[N]. This allows us to determine the analytic structure of, the partition function per site in the thermodynamic limit, and, the density, as functions of the activityz.     

Corner transfer matrices of the eight-vertex model. II. The Ising model case

In a previous paper certain corner transfer matrices were defined. It was conjectured that for the zero-field, eight-vertex model these matrices have a very simple eigenvalue spectrum. In this paper these conjectures are verified for the case when the eight-vertex model reduces to two independent and identical square-lattice Ising models. The Onsager-Yang expression for the magnetization follows immediately.     

Fermionic solution of the Andrews-Baxter-Forrester model. II. Proof of Melzer's polynomial identities

We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter, and Forrester, we prove polynomial identities for finitizations of the Virasoro characters as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers-Ramanujan-type identities for the unitary minimal Virasoro characters conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.Dedicated to the memory of Piet Kasteleyn.     

Free energy of the solvable chiral Potts model

Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent is 1–2/N.     

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.     

Corner transfer matrices of the chiral Potts model

We present some symmetry and factorization relations satisfied by the corner transfer matrices (CTMs) of the chiral Potts model. We show how the single-spin expectation values can be expressed in terms of the CTMs, and in terms of the related boost operator. Low-temperature calculations lead naturally to the variables that uniformize the Boltzmann weights of the model.     

Free energy of the chiral Potts model in the scaling region

We explicitly calculate the free energy of the general solvableN-state chiral Potts model in the scaling region, forT<T _c . We do this from both of the two available results for the free energy, and verify that they are mutually consistent. Ift=T _c –T, then we find that - _c /t has a Taylor expansion in powers oft ^2/N (together with higher-order non-scaling terms of ordert, ort logt).     

Interfacial tension of the chiral Potts model

We calculate the interfacial tension of theN-state chiral Potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions. Our result is valid for the general physical model (with positive Boltzmann weights) and at all subcritical temperatures. The interfacial tension has been calculated previously for the superintegrable chiral Potts model with skewed boundary conditions. UsingZ-invariance, Baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model. We show that this is indeed the case.     

Fermionic solution of the Andrews-Baxter-Forrester model. I. Unification of TBA and CTM methods

The problem of computing the one-dimensional configuration sums of the ABF model in regime III is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules. We thus obtain a newfermionic method to compute the local height probabilities of the model. Combined with the originalbosonic approach of Andrews, Baxter, and Forrester, we obtain a new proof of (some of) Melzer's polynomial identities. In the infinite limit these identities yield Rogers-Ramanujan type identities for the Virasoro characters _l,1 ^(r–l,r) (q) as conjectured by the Stony Brook group. As a result of our work the corner transfer matrix and thermodynamic Bethe Ansatz approaches to solvable lattice models are unified.     

Is the percolation transition of hard spheres a thermodynamic phase transition?

In hard-sphere systems, there is a fluid-solid transition, but no gas-liquid transition. In the fluid region, however, one can find a purely geometric percolation transition, which is studied in detail. The van der Waals model of hard spheres is treated. In this model, a uniform negative background potential is added. This modification does not change the structure, but induces a gas-liquid transition. In fact, percolation and the gas-liquid transition can be related to each other.     

Intersecting disks (and spheres) and statistical mechanics. II. the hard-disk system

A system of hard disks (diameter ) is studied by considering the corresponding exclusion disks (radius ). Thus, the results of previous papers on overlapping disks can be used for a geometrical analysis of the system. The concept of fluctuating free volume is compared with free volume theory. Finally, a series of computer experiments on hard disks is analyzed geometrically, especially with respect to the fluid-solid transition.     

Numerical results for the three-state critical Potts model on finite rectangular lattices

Partition functions for the three-state critical Potts model on finite square lattices and for a variety of boundary conditions are presented. The distribution of their zeros in the complex plane of the spectral variable is examined and is compared to the expected infinite-lattice result. The partition functions are then used to test the finite-size scaling predictions of conformal and modular invariance.     

Non-equilibrium phase transition of the fully frustrated square lattice Coulomb gas model

The non-equilibrium phase transitions of the fullyfrustrated (f = 1/2) square lattice Coulomb gas (CG) modeldriven by external electrical fields are studied in the frameworkof the short-time dynamic scaling approach. The criticaltemperature T_c, the static and dynamic critical exponents2β/ν, ν, and z are obtained for several smalldriving fields. The results show that T_c decreases with theincrease of electric field, and 2β/ν and z arestrongly dependent on the external electric field. Interestingly,contrary to the equilibrium case, in the presence of smallelectric field, the calculated exponent ν is close to that inpure 2D Ising model, which provides numerical evidence thatexternal electric field may change the universality class of thef = 1/2 CG system.     

The phase diagram of the Abelian lattice Higgs model. A review of rigorous results

We give a review (and some improvements) of the rigorous results on the phase structure of Abelian lattice Higgs models with gauge groups U(1) and . Emphasis is put on the relation between the Higgs mechanism (gauge-independent) and spontaneous symmetry breaking (gauge-dependent). We also discuss some nonperturbative effects due to Gribov copies in this context.This paper is based on an invited talk given by one of the authors (C.B.) at the conference on the Statistical Mechanics of Phase Transitions—Mathematical and Physical Aspects, Tebo, Czechoslovakia, September 1–6, 1986.