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Modern Physics Letters B

ON RELATION BETWEEN SYMMETRIC VERTEX MODEL AND ISING MODEL ON THE SQUARE LATTICE

Ladislav Šamaj, M. Kolesík

Published Jul 10, 1991• 5 citations
Publisher

Abstract

The symmetric version of the two-state vertex model on the square lattice is studied. It is shown that there exists a manifold in the vertex weights parameter space on which the model can be mapped onto the Ising model in a field. The choices of symmetric vertex weights for which the equivalence holds in the whole temperature range are briefly discussed.

DOI
10.1142/s0217984991001313

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Journal of the Physical Society of Japan•Mar 15, 1993•3 citations
The symmetric version of the two-state vertex model on the square lattice is studied. It is shown that there exists a manifold in the vertex weights parameter space on which the model can be mapped onto the Ising model in a field. The choices of symmetric vertex weights for which the equivalence holds in the whole temperature range are briefly discussed.
The mapping which produces a path between the p-state symmetric vertex model and the Ising model with spin [Formula: see text] on the honeycomb lattice is constructed. The mapping construction is based on a combination of decoration and gauge transformations. The formulation enables one to find the manifold in the vertex weights parameter space on which the mapping can be performed and to derive dual relations among the model parameters as well as the free energies of related symmetric vertex and Ising models. The extension of the mapping to arbitrary lattice coordination is outlined.
In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary n -state spin model and of a certain specific n -state vertex model coincide for finite lattice sizes. The equivalent vertex model has n 3 non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective R-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the R-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.
We consider the symmetric two-state 16-vertex model on the square lattice whose vertex weights are invariant under any permutation of adjacent edge states. The vertex-weight parameters are restricted to a critical manifold which is self-dual under the gauge transformation. The critical properties of the model are studied numerically with the Corner Transfer Matrix Renormalization Group method. Accuracy of the method is tested on two exactly solvable cases: the Ising model and a specific version of the Baxter eight-vertex model in a zero field that belong to different universality classes. Numerical results show that the two exactly solvable cases are connected by a line of critical points with the polarization as the order parameter. There are numerical indications that critical exponents vary continuously along this line in such a way that the weak universality hypothesis is violated.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.