All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint

TL;DR

This paper develops a method to compute correlation functions in open XXX spin 1/2 chains with unparallel boundary magnetic fields using the quantum separation of variables, extending previous results to more general boundary conditions.

Contribution

It introduces a new approach to calculate correlation functions for chains with general boundary magnetic fields, including boundary-bulk decompositions and integral formulas.

Findings

Derived the ground state density in the thermodynamic limit.

Computed boundary-bulk decomposition of boundary states.

Obtained multiple integral formulas for correlation functions.

Abstract

In this first paper, we start the analysis of correlation functions of quantum spin chains with general integrable boundary conditions. We initiate these computations for the open XXX spin 1/2 quantum chains with some unparallel magnetic fields allowing for a spectrum characterization in terms of homogeneous Baxter like TQ-equations, in the framework of the quantum separation of variables (SoV). Previous SoV analysis leads to the formula for the scalar products of the so-called separate states. Here, we solve the remaining fundamental steps allowing for the computation of correlation functions. In particular, we rederive the ground state density in the thermodynamic limit thanks to SoV approach, we compute the so-called boundary-bulk decomposition of boundary separate states and the action of local operators on these separate states in the case of unparallel boundary magnetic fields. These findings allow us to derive multiple integral formulae for these correlation functions similar to those previously known for the open XXX quantum spin chain with parallel magnetic fields.