Correlation Functions and the Boundary qKZ Equation in a Fractured XXZ Chain

TL;DR

This paper derives exact formulas for correlation functions in a split XXZ chain with boundary conditions, revealing a boundary qKZ equation of a different level and providing insights into boundary effects in quantum spin chains.

Contribution

It introduces a novel exact integral expression for correlation functions in a fractured XXZ chain with boundary magnetic fields, extending the vertex operator approach.

Findings

Exact expression for <vac|vac>' in a split XXZ chain.

Integral formula for <vac|O|vac>' with local operator O.

Correlation functions satisfy a boundary qKZ equation of a different level.

Abstract

We consider correlation functions of the form <vac|O|vac>', where |vac> is the vacuum eigenstate of an infinite antiferromagnetic XXZ chain, |vac>' is the vacuum eigenstate of an infinite XXZ chain which is split in two, and O is a local operator. The Hamiltonian of the split chain has no coupling between sites 1 and 0 and has a staggered magnetic field at these two sites; it arises from a tensor product of left and right boundary transfer matrices. We find a simple, exact expression for <vac|vac>' and an exact integral expression for general <vac|O|vac>' using the vertex operator approach. We compute the integral when O=\sigma^z_1 and find a conjectural expression that is analogous to the known formula for the XXZ spontaneous magnetisation and reduces to it when the magnetic field is zero. We show that correlation functions obey a boundary qKZ equation of a different level to the infinite XXZ chain with one boundary.