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Journal of Physics A Mathematical and General

Multicritical scaling in the magnetic hard-square lattice gas

Paul A. Pearce

Published Feb 1, 1987• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/20/2/030

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Physical review. B, Condensed matter•Sep 1, 1980•194 citations
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.
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 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.
The commuting row-to-row transfer matrices of magnetic hard squares on the multicritical T manifold are shown to satisfy a special functional equation called an inversion identity.For strip widths up to N=32, these equations are solved numerically for the transfer matrix eigenvalues. The central charge or conformal anomaly c=1 is obtained from 1/N 2 corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps. The magnetic scaling dimension is x m =1/8 and the sublattice density difference scaling dimension is x e =1/9(2-y), where y=2 lambda / pi and the interaction-dependent crossing parameter lambda varies between 0 and 2 pi /3. Scaling dimensions for further operators fall in the sequence x n =n 2 x e where n=1,2,3,4. The critical exponents also vary continuously along the multicritical line and are given by alpha =(14-9y)/(16-9y), beta m =9(2-y)/16(16-9y), beta e =1/2(16-9y), nu =9(2-y)/2(16-9y) and delta =15. These exponents are simply related to the exponents of the critical eight-vertex and Ashkin-Teller models, which also exhibit Z 2 *Z 2 symmetry, and the authors argue that these models lie in the same universality class.
The critical exponents of the non-interacting hard-square lattice gas model are determined by means of series analysis and finite-size scaling. The series analysis leads to results for the exponents that are consistent with the expected Ising universal values. A scaling analysis is performed on different types of numerical finite-size data obtained by means of the transfer-matrix method. The results support Ising universality. A procedure based on conformal invariance reproduces the known Ising temperature and magnetic exponents within 10 -6 . Furthermore we estimate the critical density of the hard-square model as rho c =0.367 743 000 (5).
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.
A theory of the scaling behavior near the qc=4{q}_{c}=4qc​=4 state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for qqq near qc{q}_{c}qc​. For the pure Potts model with q=qcq={q}_{c}q=qc​ we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At Tc{T}_{c}Tc​ we find ln(r)\mathrm{ln}(r)ln(r) corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with q=qcq={q}_{c}q=qc​ we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as TTT approaches the multicritical point from the first-order side. Results for q>qcq>{q}_{c}q>qc​ and q<qcq<{q}_{c}q<qc​ are also given.