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

Michio Jimbo, Tetsuji Miwa, Atsushi Nakayashiki

Published Jan 1, 1993• 65 citations
OpenAlex

Abstract

We propose that the correlation functions of the inhomogeneous eightvertex model in the anti-ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters. Solving the simplest difference equation we obtain the expression for the spontaneous staggered polarization conjectured by Baxter and Kelland. We also discuss a related construction of vertex operators on the lattice. 0

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

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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.

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Algebraic analysis of solvable lattice models

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Background of the problem The spin $1/2$ XXZ model for $\Delta <-1$ 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 $XXX$ limit $q\rightarrow-1$ Discussions List of formulas.

American Mathematical Society eBooks•Jan 1, 1999•1 citation

Difference equations for correlation functions of Belavin's ℤn-symmetric model with boundary reflection

Yas-Hiro Quano

Belavin's n -symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K -matrices for n >2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A (1) n -1 -face model even for n ⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk n -symmetric model, up to a phase factor.

arXiv:hep-th/0003276•Journal of Physics A Mathematical and General•Nov 9, 2000•3 citations
We propose that the correlation functions of the inhomogeneous eightvertex model in the anti-ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters. Solving the simplest difference equation we obtain the expression for the spontaneous staggered polarization conjectured by Baxter and Kelland. We also discuss a related construction of vertex operators on the lattice. 0
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.
Belavin's n -symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K -matrices for n >2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A (1) n -1 -face model even for n ⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk n -symmetric model, up to a phase factor.