Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields

TL;DR

This paper derives correlation functions for open XXZ spin 1/2 chains with unparallel boundary magnetic fields using the quantum Separation of Variables framework, providing explicit formulas for finite and half-infinite chains.

Contribution

It extends the correlation function derivation to the more complex case of XXZ chains with arbitrary boundary fields at one end and a fixed field at the other, using the SoV method.

Findings

Complete spectrum characterized by homogeneous Baxter TQ-equation.

Explicit scalar product formulas for separate states.

Multiple integral formulas for correlation functions in finite and half-infinite chains.

Abstract

In this paper we continue our derivation of the correlation functions of open quantum spin 1/2 chains with unparallel magnetic fields on the edges; this time for the more involved case of the XXZ spin 1/2 chains. We develop our study in the framework of the quantum Separation of Variables (SoV), which gives us both the complete spectrum characterization and simple scalar product formulae for separate states, including transfer matrix eigenstates. Here, we leave the boundary magnetic field in the first site of the chain completely arbitrary, and we fix the boundary field in the last site $N$ of the chain to be a specific value along the $z$-direction. This is a natural first choice for the unparallel boundary magnetic fields. We prove that under these special boundary conditions, on the one side, we have a simple enough complete spectrum description in terms of homogeneous Baxter like $TQ$-equation. On the other side, we prove a simple enough description of the action of a basis of local operators on transfer matrix eigenstates as linear combinations of separate states. Thanks to these results, we achieve our main goal to derive correlation functions for a set of local operators both for the finite and half-infinite chains, with multiple integral formulae in this last case.