CompletelyX-symmetricS-matrices corresponding to theta functions and models of statistical mechanics

✍ Scribed by D. V. Chudnovsky; G. V. Chudnovsky

• Publisher: Springer
• Year: 1981
• Tongue: English
• Weight: 318 KB
• Volume: 5
• Category: Article
• ISSN: 0377-9017
• DOI: 10.1007/bf00401826

✦ Synopsis

We consider general expressions of factorized S-matrices with Abelian symmetry expressed in terms of 0-functions. These expressions arise from representations of the Heisenberg group. New examples of factorized S-matrices lead to a large class of completely integrable models of statistical mechanics which generalize the XYZ-model of the eight-vertex model.

In this paper we consider the description of factorized S-matrices in the dimension 1 + 1 with Abelian internal symmetries. It is known that the complete knowledge of the S-matrix would enable us to find a canonical Hamiltonian structure and higher conservation laws

Here we analyze factorized S-matrices in the framework of the representation of the Heisenberg group. This leads to an expression of factorized S-matrices in terms of 0 functions. One should remember that, in non-trivial cases, solutions of factorization equations are always expressed in terms of Abelian functions [4], [6]. To find all factorized S-matrices with an Abelian group of symmetries, we substitute the condition of the law of addition by the condition of quasiperiodicity in the form of Weyl commutativity relations. Roughly speaking, we consider an S-matrix, which is non-trivial over complex torus V/L; this condition of nontriviality is replaced by the condition of triviality over V, and consistency conditions, connected with translations by elements of L. We present the corresponding general result from [6], [11] below, as a conjecture proved in important ca~es. These general expressions of S-matrices are then used to construct generalizations of the eight-vertex model [2] and the XYZ-model [12] for the case when spin variables assume values in an arbitrary finite Abelian group X.