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Modern Physics Letters B

GENERAL 8-VERTEX MODEL ON THE HONEYCOMB LATTICE: EQUIVALENCE WITH AN ISING MODEL

Ken‐Yu Lin, Fa Yueh Wu

Published Mar 10, 1990• 6 citations
Publisher

Abstract

It is shown that the general 8-vertex model on the honeycomb lattice is always reducible to an Ising model in a nonzero but generally complex magnetic field. In the most general case of the staggered 8-vertex model characterized by 16 independent vertex weights, the equivalent Ising model has three anisotropic interactions and a staggered magnetic field which assumes two different values on the two sublattices.

DOI
10.1142/s0217984990000398

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Physical Review Letters•Jun 12, 1989•38 citations
It is shown that the general 8-vertex model on the honeycomb lattice is always reducible to an Ising model in a nonzero but generally complex magnetic field. In the most general case of the staggered 8-vertex model characterized by 16 independent vertex weights, the equivalent Ising model has three anisotropic interactions and a staggered magnetic field which assumes two different values on the two sublattices.
An eight-vertex model with staggered (site-dependent) vertex weights is considered. The model is an extension of the usual one with translationally invariant weights and contains sixteen independent vertex weights. From its Ising representation it is seen that there are actually only eleven independent parameters. After discussing some general symmetry properties of this model, we consider in detail the soluble case of a free-fermion model. We find that the staggered free-fermion model may exhibit up to three phase transitions. Generally the specific heat has logarithmic singularities, expect in some special cases it has an exponent $\ensuremath{\alpha}=\frac{1}{2}$ and the system is frozen below a unique transition point. Conditions for these special cases are given.
A closed-form expression is proposed for the critical frontier, or the critical line, of the antiferromagnetic Ising model on the honeycomb lattice in a nonzero external magnetic field. We formulate the Ising model as an 8-vertex model and identify the critical frontier as a locus invariant under a generalized weak-graph transformation. In its simplest form the locus is an expression containing two unknown constants which we determine numerically. The resulting critical frontier lies very close to results of a finite-size analysis.