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Journal of Physics A Mathematical and General

Eight-vertex model on a ruby lattice

Ken‐Yu Lin

Published Nov 11, 1984• 3 citations
Publisher

Abstract

An eight-vertex model on a ruby lattice with staggered (site-dependent) vertex weights is considered. The special case of a free-fermion model is solved by the Pfaffian method. In general the special heat has logarithmic singularity at the critical temperature, except in some special cases where the system exhibits a first- or second-order phase transition.

DOI
10.1088/0305-4470/17/16/021

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An eight-vertex model on a ruby lattice with staggered (site-dependent) vertex weights is considered. The special case of a free-fermion model is solved by the Pfaffian method. In general the special heat has logarithmic singularity at the critical temperature, except in some special cases where the system exhibits a first- or second-order phase transition.
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.
We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered 8-vertex model. Using this equivalence we deduce the solution of the odd 8-vertex model when the weights satisfy a free-fermion condition. It is found that the freefermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of the free-fermion model with pure 2-spin interactions. Key words: Odd eight-vertex model, free-fermion model, exact solution. 1 2 1
The solution of the zero-field "eight-vertex" model is presented. This model includes the square lattice Ising, dimer, ice, FFF, and KDP models as special cases. It is found that in general the free energy has a branch-point singularity at a phase transition, with a continuously variable exponent.