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arXiv:hep-th/9211066 · Journal of Physics A Mathematical and General

Difference equations for the correlation functions of the eight-vertex model

M Jimbo, T Miwa, A Nakayashiki

Published May 7, 1993• 75 citations
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Abstract

The authors propose that the correlation functions of the inhomogeneous eight-vertex model in the anti-ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters. Solving the simplest difference equation they obtain the expression for the spontaneous staggered polarization conjectured by Baxter and Kelland (1974). They also discuss a related construction of vertex operators on the lattice.

DOI
10.1088/0305-4470/26/9/015

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The authors propose that the correlation functions of the inhomogeneous eight-vertex model in the anti-ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters. Solving the simplest difference equation they obtain the expression for the spontaneous staggered polarization conjectured by Baxter and Kelland (1974). They also discuss a related construction of vertex operators on the lattice.
A free field representation for the type III vertex operators and the corner transfer matrices of the eight-vertex model is proposed. The construction uses the vertex-face correspondence, which makes it possible to expresses correlation functions of the eight-vertex model in terms of correlation functions of the SOS model with a nonlocal insertion. This new nonlocal insertion admits of a free field representation in terms of Lukyanov's screening operator. The spectrum of the corner transfer matrix and the Baxter-Kelland formula for the average staggered polarization have been reproduced.
The basic properties of q-vertex operators are formulated in the context of the Andrews–Baxter–Forrester (ABF) series, as an example of face interaction models, the q-difference equations satisfied by their correlation functions are derived, and their connection with representation theory established. The q-difference equations of the Kashiwara–Miwa (KM) series are discussed as an example of edge interaction models. Next, the Ising model, the simplest special case of both ABF and KM series, is studied in more detail using the Jordan–Wigner fermions. In particular, all matrix elements of vertex operators are calculated.
Background of the problem The spin 1/21/21/2 XXZ model for Δ<−1\Delta <-1Δ<−1 The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The XXXXXXXXX limit q→−1q\rightarrow-1q→−1 Discussions List of formulas.
Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.