Low-temperature spectrum of the quantum transfer matrix of the XXZ chain in the massless regime

TL;DR

This paper rigorously analyzes the low-temperature spectrum of the quantum transfer matrix for the XXZ spin chain in the massless regime, confirming the emergence of conformal field theory behavior and providing mathematical validation for longstanding physics conjectures.

Contribution

It provides a rigorous mathematical analysis of the spectrum of the quantum transfer matrix for the XXZ chain, confirming the conformal spectrum emergence at low temperatures.

Findings

Identifies the dominant eigenvalue of the transfer matrix spectrum.

Shows divergence of correlation lengths aligns with conformal field theory predictions.

Establishes existence and uniqueness of solutions to the Bethe Ansatz equations in this context.

Abstract

The free energy per lattice site of a quantum spin chain in the thermodynamic limit is determined by a single `dominant' Eigenvalue of an associated quantum transfer matrix in the infinite Trotter number limit. For integrable quantum spin chains, related with solutions of the Yang-Baxter equation, an appropriate choice of the quantum transfer matrix enables to study its spectrum, e.g.\ by means of the algebraic Bethe Ansatz. In its turn, the knowledge of the full spectrum allows one to study its universality properties such as the appearance of a conformal spectrum in the low-temperature regime. More generally, accessing the full spectrum is a necessary step for deriving thermal form factor series representations of the correlation functions of local operators for the spin chain under consideration. These are statements that have been established by physicists on a heuristic level and that are calling for a rigorous mathematical justification. In this work we implement certain aspects of this programme with the example of the XXZ quantum spin chain in the antiferromagnetic massless regime and in the low-temperature limit. We rigorously establish the existence, uniqueness and characterise the form of the solutions to the non-linear integral equations that are equivalent to the Bethe Ansatz equations for the quantum transfer matrix of this model. This allows us to describe that part of the quantum transfer matrix spectrum that is related to the Bethe Ansatz and that does not collapse to zero in the infinite Trotter number limit. Within the considered part of the spectrum we rigorously identify the dominant Eigenvalue and show that those correlations lengths that diverge in the low-temperature limit are given, to the leading order, by the spectrum of the free Boson $c=1$ conformal field theory. This rigorously establishes a long-standing conjecture present in the physics literature.