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Journal of Physics A Mathematical and General

Magnetic hard squares: exact solution

Paul A. Pearce

Published Nov 11, 1985• 21 citations
Publisher

Abstract

A three-state IRF model of a magnetic hard-square lattice gas with anisotropic interactions is solved exactly on special two-dimensional manifolds in the full five-dimensional thermodynamic space spanned by the (ferro- or antiferro-) magnetic interactions J, K, the (attractive or repulsive) lattice-gas interactions L, M and the activity z. The phase boundaries determined include multicritical lines and critical surfaces as well as first-order surfaces of three- and fivefold phase coexistence between the fluid and various (para, ferro, etc.) magnetic square-ordered solid phases. Analytic expressions are given for free energies, interfacial tensions, correlation lengths, sublattice densities and magnetisations. The critical behaviour of these quantities is studied and the associated critical exponents obtained.

DOI
10.1088/0305-4470/18/16/022

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A three-state IRF model of a magnetic hard-square lattice gas with anisotropic interactions is solved exactly on special two-dimensional manifolds in the full five-dimensional thermodynamic space spanned by the (ferro- or antiferro-) magnetic interactions J, K, the (attractive or repulsive) lattice-gas interactions L, M and the activity z. The phase boundaries determined include multicritical lines and critical surfaces as well as first-order surfaces of three- and fivefold phase coexistence between the fluid and various (para, ferro, etc.) magnetic square-ordered solid phases. Analytic expressions are given for free energies, interfacial tensions, correlation lengths, sublattice densities and magnetisations. The critical behaviour of these quantities is studied and the associated critical exponents obtained.
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.
The scaling behaviour of the magnetic hard-square lattice gas is examined in the vicinity of the multicritical line located by an exact solution. A simple scaling form is proposed for the singular part of the free energy and corresponding scaling forms are derived for the sublattice density difference, the magnetisation and the densities. These are all shown to be consistent with the exact results when restricted to the exact solution manifold. In the case of the densities this is achieved by proving appropriate elliptic function identities.
Baxter has solved a restricted class of square lattice gas models with nearest-neighbor exclusion (thus "hard squares") and next-nearest-neighbor interactions. Arguments are presented which demonstrate that the subspace spanned by his exact solution contains the line of tricritical points and the associated surface of first-order transitions of hard squares with attractive next-nearest-neighbor interactions. The tricritical exponents so identified confirm those obtained by Nienhuis for a dilute Ising model.
We present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.
The exact solution of an Ashkin-Teller model, with layer and spatial anisotropy and Z 2 *Z 2 symmetry, is presented. The model is related to the eight-vertex model by a sublattice or orbifold duality and it has the same free energy and magnetisation. The exact solution manifold is a three-dimensional submanifold of the full six-dimensional thermodynamic space. It admits a Z 3 symmetry relating three regimes each exhibiting different partial order. In the spatially isotropic case, these regimes are separated by three lines of continuously varying critical behaviour which coalesce at the four-state Potts critical point. The magnetisations and polarisation are derived using corner transfer matrices giving rise to the appearance of chiral and Virasoro characters. The associated critical exponents confirm the known values. In particular, one obtains the value x sigma =1/8 for the magnetic scaling dimension and identify the next-to-leading thermal field.
The scaling behaviour of the hard square lattice gas with diagonal interactions is examined on the basis of Baxter's (1980, 1981, 1982) recent exact solution. It is demonstrated that all the corrections to scaling found may be accounted for in terms of a single irrelevant scaling field that scales simply as a length. The lattice cut-off is proposed as the source of these corrections, although other sources cannot be ruled out. The melting of the 3*1 commensurate ordered phase in this model is examined in a more general context in which the line of Potts critical points found by Baxter appears to be a line of multicritical points on which the effective uniaxially chiral symmetry breaking field vanishes.