Hard Hexagon Research Articles

An observation on the partition function zeros of the hard hexagon model

It is conjectured that part of the limiting locus of partition function zeros for the hard hexagon model is an algebraic curve generated by a simple and rational algebraic equation. This equation arises from the same mathematical mechanism as that which determines algebraic-invariant properties of self-dual systems. A remarkable approximation to the critical point of the three-state Potts model (triangular lattice) is mentioned in passing.

Exact results for the activity and isothermal compressibility of the hard-hexagon model

The Klein and Fricke theory of modular functions is used to derive closed-form algebraic expressions for the activity z( rho ) and isothermal compressibility kappa T( rho ) of the hard-hexagon lattice gas model in the disordered regime, where rho is the dimensionless number density. Similar results which are valid in the ordered regime are also presented.

Z2-11z-1 as an algebraic invariant for the hard-hexagon model

The limiting partition function per site Xi infinity (z) for the hard-hexagon model has two singular points at the roots of z2-11z-1=0. These two points appear to be algebraic invariants associated with algebraic functions defined on an infinite sequence of finite transfer matrices.

On the hard-hexagon model and the theory of modular functions

The mathematical properties of the exact solution of the hard-hexagon lattice gas model are investigated by using the Klein-Fricke theory of modular functions. In particular, it is shown that the order-parameter R and the reciprocal activity z' for the model can be expressed in terms of hauptmoduls that are associated with certain congruence subgroups of the full modular group Known modular equations are then used to prove that R (z') is an algebraic function of A connection is established between the singular points of this function and the geometrical properties of the icosahedron .

Deducing the lateral distribution of proteins in lipid bilayer membranes

We consider four models of the lateral distribution of proteins in lipid bilayer membranes and study the fraction of lipids which are adjacent to at least one protein (adjacent lipids) and how this quantity depends upon protein concentration. The models are (i) hard hexagons free to move from one lattice site to another; (ii) hard disks moving on a continuum; (iii) a mixture of two sizes of “nearly-hard” disks moving on a continuum; (iv) a modification of (ii). The hexagons or disks represent proteins, while unocupied lattice sites or the remainder of the continuum represents lipids. In (iii) large disks represent proteins and small disks represent lipids. In (iv) some of the continuum between pairs of disks, where packing defects might occur, is not occupied by lipids. We find that an analytical expression for the adjacent lipids (Hoffmann et al. 1981), which is in excellent agreement with the results of the Hexagon model (i), breaks down at a packing density of fA≊0.805, and we show by considering the hexagon pair correlation function, that this indicates the onset of random close packing, and that a transition to ordered close packing occurs at fA=0.866. We thus obtain an operational definition for a “random” distribution of hexagons: distributions of packing densities≲0.805. We show that the Disk model (ii) gives results for adjacent lipids that are greater than the Hexagon model and compare these results to the Two Disk model (iii) which gives a result substantially less than the Hexagon model (Mountain et al. 1986). We show that the Modified Disk model (iv) gives results in essential agreement with the Hexagon model except for fA≳0.77. Finally we discuss the general appearance of the “motion restricted” ESR spectrum and conclude that, of these four models, the Modified Disk or the Hexagon models best account for the data. We discuss why this is so with reference to the representation of a 3-dimensional membrane by a 2-dimensional plane.

Equation of state and isothermal compressibility for the hard hexagon model in the disordered regime

The equation of state for the hard hexagon model in the disordered regime is computed. The isothermal compressibility is given as a rational function of K and p. The hard hexagon model is a lattice gas on a triangular lattice with the rule that if a site is occupied, then the six neighbouring sites are necessarily vacant. Baxter (1980, 1981) has shown that this model is exactly solvable by the method of commuting transfer matrices (Baxter 1972, 1982). In the disordered regime, Baxter's exact results are

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 groupГ1[N]. This allows us to determine the analytic structure ofϰ, the partition function per site in the thermodynamic limit, andρ, the density, as functions of the activityz.

Scaling dimensions and conformal anomaly in anisotropic lattice spin models

The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(ay/ax) sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.

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.

Transfer-matrix inverison identities for exactly solvable lattice-spin models.

A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.

Structure factors associated with the melting of a (31) ordered phase on a centered-rectangular lattice gas: Effective scaling in a three-state chiral-clock-like model.

By computing the structure factor through the melting transition of a commensurate (3\ifmmode\times\else\texttimes\fi{}1) phase in a version of Baxter's generalized hard hexagon model, we simulate what might be observed in diffraction experiments (especially those using low-energy electrons) on chemisorption systems which have similar Landau-Ginzburg-Wilson Hamiltonians. In the commensurate and disordered phases we observe that the shift of the peak of the critical scattering from the commensurate position is roughly proportional to the inverse correlation length in the reduced temperature range 0.015--0.15. The proportionality constant appears to depend on the position of the transition on the phase boundary (i.e., on the chemical potential). We see no sign of an incommensurate floating phase between the disordered and commensurate phases. Effective critical exponents are consistent with those expected from the three-state Potts model. To the precision of this calculation (a few percent), the structure factor scales over approximately 10% of the Brillouin-zone area.

The emerging role of number theory in exactly solvable models in lattice statistical mechanics

We review the Baxter model, the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory. It is shown, for example, that various physical quantities in the hard hexagon model are all expressible in terms of modular functions. The use of Rogers-Ramanujan type identities in solvable models is also reviewed.

Continuous Hahn polynomials and the Heisenberg algebra

Continuous Hahn polynomials have surfaced in a number of somewhat obscure physical applications. For example, they have emerged in the description of two-photon processes in hydrogen, hard-hexagon statistical mechanical models, and Clebsch–Gordan expansions for unitary representations of the Lorentz group SO(3,1). In this paper it is shown that there is a simple and elegant way to construct these polynomials using the Heisenberg algebra.

Fusion of the eight vertex SOS model

A higher spin analogue is presented of the eight vertex-SOS correspondence in the sense of Andrews-Baxter-Forrester. The resulting hierarchy of solvable models contain the hard hexagon model and its recent multi-state generalizations.

Phase diagrams for H/Ni(111) based on model interactions: Effects of strong long-range attractions

We test the pairwise adatom-adatom interaction sets for H/Ni(111) computed by Muscat by using Monte Carlo techniques to determine the lattice gas phase diagrams they imply and then comparing with experiment. Although the correct ordered phase is reproduced and the maximum disordering temperature is approximately correct, there remain topological discrepancies with the experimental phase diagram. The origin of the main discrepancy is a relatively strong long-range attractive interaction. We describe the behavior this interaction causes and outline the circumstances under which similar phenomena might occur in other systems, developing insight through (re-)examination of the square-lattice Ising metamagnet problem. Finally we show how two previously studied interaction sets with relatively weak sixth-neighbor attractions lead to more accurate phase diagrams. One appendix collects some results on the metamagnet. Two others present transfer matrix scaling calculations of the critical properties of the disordering of two ordered states on a honeycomb lattice: hard hexagons (nearest neighbor exclusion) is Ising-like to high precision, while graphitic (2 × 2) is 4-state-Potts-like.

Lattice gas generalization of the hard hexagon model. II. The local densities as elliptic functions

In a previous paper 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. Here we use various mathematical identities (in particular Gordon's generalization of the Rogers-Ramanujan relations) to express the local densities in terms of elliptic functions. The critical behavior is then readily obtained.

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.

An exactly solvable 4-state IRF model

A new 4-state IRF model is constructed by solving the star-triangle equation. The model is a multi-state extension of Baxter's hard hexagon model and the 3-state model by the authors. This result strongly suggests that there exists a solvable hierarchy which corresponds to Gordon's generalization of the Rogers-Ramanujan identities.

Exactly Solvable IRF Models. I. A Three-State Model

Presented is an exactly solvable 3-state IRF model which generalizes the Baxter's hard hexagon model. The free energy is calculated. The model is related to a partition identity, the Gordon's generalization of the Rogers-Ramanujan identity.

Interfacial tension and correlation lengths for the three-state Potts model in two dimensions.

Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/${k}_{B}$T as T\ensuremath{\rightarrow}${T}_{c}$. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.