Hard Hexagon Research Articles

A solvable lattice model and related Rogers-Ramanujan type identities

An exactly solvable lattice model is presented, which is a natural generalization of Baxter's hard hexagon model to three spin states. The free energy and spin densities are calculated. Related Rogers-Ramanujan type identities are also studied.

Deviations from critical density in the generalised hard hexagon model

Ramanujan's elliptic function identities and others are used to obtain infinite product forms for the deviations rho - rho c from the critical density rho c=1/2(1-5-12/) in the four regimes of the generalised hard hexagon model. Product forms are also obtained for the singular part rho sing of the density confirming the simple scaling forms recently conjectured by Huse, (1983).

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 η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1⩽l i⩽4. 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 η=K/r (r an integer) and 1⩽l i⩽r−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 exponentsβ, α, $$\bar \alpha $$ are obtained.

Two-dimensional infinite repulsive systems: the Yang-Lee singularities are isolated branch points

The singularities of the pressure as a function of the activity z are examined by a certain limiting procedure applicable to the solved hard-hexagon model (Baxter 1980) and the tempered hard-square model (Fisher 1963). For hard hexagons the only singularities found in the finite complex z plane are two isolated branch points on the real z axis. For the tempered hard squares, the only singularities found are three isolated branch points on the real z axis. By the same procedure applied to the Ising model, the singularities of the energy as a function of temperature are found to be isolated branch points in the complex temperature plane. Comparison with numerical results shows that the dense distribution of singularities of finite systems disappears in the finite system limit, save for a few isolated points.

Hard squares with diagonal attractions

The authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next-nearest neighbour (diagonal) interactions on the special surface corresponding to regimes III and IV of the generalised hard hexagon model. The interfacial tension, correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row-to-row and corner transfer matrices. The associated critical exponents are found to be mu =v=5/4, beta =3/32 in regime III and mu '=v'=5/2, beta '=1/4 in regime IV. In particular, their results confirm the recent proposal by Huse that regime III is the first-order coexistence surface (separating the disordered fluid phase from the square ordered solid phases) and that the regime III/regime IV boundary is a line of tricritical points.

Ground-state properties of the antiferromagnetic potts model in an external field

The ground-state properties of the antiferromagnetic q-state Potts model in an external field are studied. At the upper critical field this model may be mapped onto a hard-core lattice gas with activity z = q −1. This allows us to get some exact results for the triangular lattice on which the corresponding hard hexagon problem has been recently solved by Baxter.

The inversion relation method for some two-dimensional exactly solved models in lattice statistics

The partition-functions-per-siteκ of several two-dimensional models (notably the eight-vertex, self-dual Potts and hard-hexagon models) can be easily obtained by using an inversion relation for local transfer matrices, together with symmetry and analyticity properties. This technique is discussed, the analyticity properties compared, and some equivalences (and nonequivalences) pointed out. In particular, the critical hard-hexagon model is found to have the sameκ as the self-dualq-state Potts model, withq=(3 + √5)/2 = 2.618 .... The Temperley-Lieb equivalence between the Potts and six-vertex models is found to fail in certain nonphysical antiferromagnetic cases.

Hard hexagons: interfacial tension and correlation length

Functional equations are derived for the eigenvalues of the row-to-row transfer matrix of the generalised hard hexagon model. These equations are exact for a lattice with N columns and are solved in the limit N to infinity . The partition function per site is rederived without the previous analyticity assumptions. The authors are also able to calculate the interfacial tension and the correlation length; the associated critical exponents are mu = nu = nu '=5/6 in agreement with the scaling relations.

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.

The hard-hexagon model and Rogers—Ramanujan type identities

In regime II of Baxter's solution of the hard-hexagon model [Baxter, R. J. (1980) J. Phys. A 13, L61-L70], he presents six conjectures identifying certain one-dimensional partition functions with infinite products. An outline of the proof of these conjectures is given here.

Duality relations and equivalences for models with O( N) and cubic symmetry

Formal expression for high-temperature series are derived for models with O( N) and cubic symmetry, with a special form of nearest neighbor interactions on the honeycomb lattice. By deriving low-temperature series for a class of generalized solid-on-solid and cubic models, a duality relation is established. Equivalences between cubic and SOS type models are also found. In the large- N limit, the series reduce to those of the hard hexagon model.

Phenomenological scaling approach to the triangular Ising antiferromagnet

We apply the phenomenological scaling approach of Nightingale to the triangular Ising model with nearest-neighbor interactions in the presence of a magnetic field. In order to do so, we calculate the largest eigenvalues of the transfer matrix of semi-infinite strips with widths 3, 6, 9,..., etc. The technique is first applied to two exactly soluble cases, the ferromagnetic and the hard hexagon model. Employing maximum strip widths of 9 and 15, respectively, in these two cases, we obtain results for the thermal exponent ${y}_{T}$ within 0.3% of exact values. Applied to the antiferromagnet with maximum strip width of 9, we obtain values of the exponent which vary with the applied field. This variation is only appreciable, however, near zero field where crossover to different critical behavior is anticipated. We formulate a criterion to determine the best values of the thermal exponent and obtain ${y}_{T}=1.2002$, in excellent agreement with the value 1.2 expected on the basis of symmetry.

Corner transfer matrices

Abstract For two-dimensional lattice models with interactions only between nearest (and diagonally nearest) neighbour spins, a well-known concept is the row-to-row transfer matrix. Less well-known is the “corner” transfer matrix (CTM). This has some very useful properties. If it is normalized so that its largest eigenvalue is unity, and the eigenvalues are arranged in numerically decreasing order, then each eigenvalue tends to a limit as the lattice becomes large. For those models which have been solved exactly (notably the Ising, eight-vertex and hard hexagon models), this limiting eigenvalue distribution is very simple, being basically that of a direct product of two-by-two matrices. From it the order parameter can easily be obtained. For all models one can write down formally exact matrix relations for the CTM, but the matrices are of infinite size. If one uses a representation in which the CTM is diagonal, and then truncates these relations to finite size, then one obtains a quite accurate approximation. The larger the size the greater the accuracy. I.G. Enting and I have thereby obtained comparatively long series expansions for the Ising model in a field, and for the hard squares model.

Vertex models for the hard-square and hard-hexagon gases, and critical parameters from the scaling transformation

In a general formulation of hard-core lattice gas models the hard-square and hard-hexagon models are expressed in terms of vertex models. It is shown that the hard-square gas is a 16-vertex model on the square lattice, and in one of its representations is equivalent to a lattice ramrod model first considered by Nagle (1968). A lattice transformation of the hard-hexagon model generates a special case of the 64-vertex model on the triangular lattice. In these two reductions clear differences between these two hard-core models emerge, and it is probably not surprising that they exhibit quite different critical behaviour. It is also shown that the block-site scaling transformation can be used to provide an alternative computational scheme for obtaining accurate numerical estimates of critical point parameters for these models. The method is illustrated by calculations performed on the hard-square gas.

Virial expansions for hard-core fluids

The authors discuss some of the difficulties involved in drawing conclusions about the asymptotic behaviour of the virial coefficients for a hard-core fluid when only the first few coefficients are known. Particular consideration is given to the recent work of Baram and Luban (1979) for hard discs and spheres, and that of Baxter (1980) for hard hexagons on the triangular lattice.

Series analysis for the three-state Potts model

Low-temperature series expansions for the zero-field partition function and order parameter of the three-state Potts model have been extended to u31. Estimates for alpha ' and beta show rather poor convergence but are consistent with the values alpha '=1/3, beta =1/9 which characterise the 'hard hexagons' lattice gas.

Hard hexagons: exact solution

The hard-hexagon model in lattice statistics (i.e. the triangular lattice gas with nearest-neighbour exclusion) has been solved exactly. It has a critical point when the activity z has the value 1/2(11+5 square root 5)=11.09017..., with exponents alpha =1/3, beta =1/9. More generally, a restricted class of square-lattice models with nearest-neighbour exclusion and non-zero diagonal interactions can be solved.

Entropy of hard hexagons

Metcalf and Yang (1978) have obtained the numerical estimate 0.3333 k for the entropy per site of hard hexagons. They therefore conjecture that the exact value is k/3. The authors have used the corner transfer matrix method to obtain a more accurate numerical estimate, namely 0.333 242 721 976 k, which contradicts the conjecture. The problem is a good example of the numerical accuracy of the method.

Phase transition of hard hexagons on a triangular lattice

Systems of hard hexagons on a triangular lattice are investigated. The orientation of the hexagons is kept fixed, while the size of the hexagons is varied. The existence of a phase transition is proved for all sizes by means of the Peierls'argument. The proof does not imply a phase transition in the continuous limit.

A Monte Carlo study of the thermodynamic properties of “hard hexagons” on the triangular lattice

A Monte Carlo method, developed originally to calculate thermodynamic properties in a grand canonical ensemble of lattice systems containing molecules which have finite mutual attractions, has been modified for studying lattice systems in which the intermolecular potential is an infinite repulsion. The grand ensemble formulation is reviewed briefly, and equations are given for a specific choice of transition probabilities defining a Markov process with a stationary distribution which is identical to the probability distribution of states in the grand ensemble. Results are presented for “hard hexagons” on the two-dimensional triangular lattice. For hexagons of a size that prevents simultaneous occupation of a pair of nearest-neighbor sites by two molecules, the results indicate a possible second-order phase transition for μ/ kT ≈ 2.3 at a density around 82% of the close-packed density. The existence, nature, and location of this transition have been more firmly established by Runnels and Combs, and by Gaunt, using other techniques. For larger hexagons excluding simultaneous second-neighbor occupancy, an apparent first-order transition occurs at μ f kT ≈ 1.68, with “fluid” and “solid” densities of about 69 % and 77 % of the close-packed density. These results agree, except for the effects of finite lattice size, with those extrapolated by Orban and Bellemans from exact calculations for semi-infinite lattices. When third neighbors are excluded also, the Monte Carlo results give no indication of a transition. However, Orban and Bellemans have estimated that the transition in this case occurs at μ/ kT ≈ 4.6, well above the value (about 3.0) where convergence of this particular Monte Carlo method becomes too slow for practical use.