Q-operators for the open Heisenberg spin chain

TL;DR

This paper constructs Q-operators for the open Heisenberg spin chain with diagonal boundaries, demonstrating their polynomial nature, commutation with transfer matrices, and satisfaction of Baxter's TQ-equation, with explicit eigenvalues derived.

Contribution

It introduces a novel construction of Q-operators using reflection operators from the boundary Yang-Baxter equation for open spin chains.

Findings

Q-operators are polynomials in the spectral parameter.

Q-operators commute with the transfer matrix.

Eigenvalues are explicitly expressed in terms of Bethe roots.

Abstract

We construct Q-operators for the open spin-1/2 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang-Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.