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

TL;DR

This paper derives difference equations for correlation functions in Belavin's $ ext{Z}_n$-symmetric elliptic model with boundary reflection, introducing non-diagonal $K$-matrices and calculating boundary spontaneous polarization.

Contribution

It introduces non-diagonal $K$-matrices satisfying the boundary Yang-Baxter equation and derives associated difference equations for boundary correlation functions.

Findings

Derived difference equations of the quantum Knizhnik-Zamolodchikov type for boundary models.

Obtained boundary spontaneous polarization from the simplest difference equations.

Expressed boundary polarization as the square of bulk polarization, up to a phase factor.

Abstract

Belavin's $\mathbb{Z}_n$-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 $n\geqq 2$. 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. The resulting quantity is the square of the spontaneous polarization for the bulk $\mathbb{Z}_n$-symmetric model, up to a phase factor.