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Journal of Physics A Mathematical and General

Local state probabilities for an infinite sequence of solvable lattice models

Atsuo Kuniba, Tetsu Yajima

Published Jan 21, 1988• 18 citations
Publisher

Abstract

The authors present a new infinite sequence of solvable lattice models. They contrast strikingly with the eight-vertex solid-on-solid models and admit extra degrees of freedom for the local fluctuation variables. The exact one-point functions are obtained. The result is neatly described in terms of theta-function identities. Using their modular invariance, critical behaviour is studied and exponents evaluated.

DOI
10.1088/0305-4470/21/2/029

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The authors present a new infinite sequence of solvable lattice models. They contrast strikingly with the eight-vertex solid-on-solid models and admit extra degrees of freedom for the local fluctuation variables. The exact one-point functions are obtained. The result is neatly described in terms of theta-function identities. Using their modular invariance, critical behaviour is studied and exponents evaluated.
In this Letter we report a new construction to obtain restricted solid-on-solid (RSOS) models out of loop models. The method is a generalization of ideas developed by Owczarek and Baxter, and by Pasquier. In particular we consider a solvable O(n) model and point out that some of the RSOS models thus obtained admit an off-critical extension. Among these models we find a spin-1 Ising model, which is solvable not only at the critical point, but also in a fieldlike deviation away from it. We calculate the critical exponent \ensuremath{\delta}=15 directly from the relation between the free energy and the field. This is the first determination of this exponent without the use of scaling relations.
Abstract Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.
An equivalence between generalized restricted solid-on-solid models, associated with sets of graphs, and multi-colour loop models is established. As an application the authors consider solvable loop models and, in this way, obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A L ,A L ) and related to the C 2 (1) vertex model, they give an off-critical extension, which breaks the Z 2 symmetry of the Dynkin diagrams.
Any planar set of intersecting straight lines forms a four-coordinated graph, or 'lattice', provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagome lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.