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Chinese Journal of Physics

Ice-Rule Vertex Model on a 34-6 Lattice

Ken‐Yu Lin

Published Jul 1, 1985• 0 citations
OpenAlex

Abstract

An ice-rule vertex model on a 3-4-6 lattice with staggered (site-dependent) vertex weights is considered. The special case of a free-fermion model is solved exactly by the Pfaffian method. The system exhibits second-order phase transition and the specific heat diverges with an exponent 1/2.

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It is pointed out that the staggered ice-rule model contains a number of outstanding lattice statistical problems including the Ising model in a nonzero magnetic field. The most general Pfaffian solution of this staggered vertex model is studied in this paper. In special cases our solution reduces to that of two recently considered dimer models of phase transitions. It also leads to the exact solution of several vertex models as well as an exact isotherm of a general antiferroelectric model, all in the presence of both direct and staggered fields.
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 study the multicritical behavior of a class of two-dimensional ice-type vertex models on different lattices using renormalization-group theory. The models are classified by an integer m, with m=2 corresponding to the known square lattice case. For m>2, the specific-heat exponent is a=(m-2)/(m-1) with an upper critical dimensional confluent (lnt)1/2{)}^{1/2})1/2 divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.