A Q-operator for the twisted XXX model

TL;DR

This paper introduces a new Q-operator construction for the XXX spin chain using quasi-periodic boundary conditions, revealing new Bethe ansatz equations and implications for correlation functions.

Contribution

It extends the representation theoretic construction of Baxter's Q-operators to the XXX model and highlights the necessity of quasi-periodic boundary conditions for convergence.

Findings

Derived a quantum Wronskian relation for the XXX model.

Identified two sets of Bethe ansatz equations related by the quantum Wronskian.

Found special solutions for chains up to 10 sites.

Abstract

Taking the isotropic limit in a recent representation theoretic construction of Baxter's Q-operators for the XXZ model with quasi-periodic boundary conditions we obtain new results for the XXX model. We show that quasi-periodic boundary conditions are needed to ensure convergence of the Q-operator construction and derive a quantum Wronskian relation which implies two different sets of Bethe ansatz equations, one above the other below the "equator" of total spin zero. We discuss the limit to periodic boundary conditions at the end and explain how this construction might be useful in the context of correlation functions on the infinite lattice. We also identify a special subclass of solutions to the quantum Wronskian for chains up to a length of 10 sites and possibly higher.