A Q-operator for the quantum transfer matrix

TL;DR

This paper constructs a Q-operator for the quantum transfer matrix of the XXZ spin-chain, revealing new functional relations and simplifying the Bethe ansatz equations for numerical analysis at finite temperature.

Contribution

It introduces a novel Q-operator construction using quantum group representation theory and derives functional relations that improve numerical studies of the XXZ spin-chain.

Findings

Derived a new Q-operator for the quantum transfer matrix.

Established quadratic equations replacing Bethe ansatz equations under magnetic field.

Identified loop algebra symmetry at roots of unity for zero magnetic field.

Abstract

Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.