Difference Equations in Spin Chains with a Boundary

TL;DR

This paper derives difference equations for semi-infinite boundary spin chains, enabling calculation of boundary magnetization and proposing a boundary S-matrix, extending integrable models with boundary conditions.

Contribution

It introduces difference equations for boundary spin chains, connecting vertex operator properties with boundary conditions, and extends the quantum Knizhnik-Zamolodchikov framework to systems with boundaries.

Findings

Derived difference equations for boundary spin chains.

Calculated boundary spontaneous magnetization.

Proposed and compared boundary S-matrix in sine-Gordon limit.

Abstract

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the 8-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary $S$-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal--Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.