Dressed energy of the XXZ chain in the complex plane

TL;DR

This paper analyzes the dressed energy function of the XXZ chain in the massless antiferromagnetic regime, extending it into the complex plane to understand solutions of the Bethe Ansatz equations at low temperatures.

Contribution

It provides a detailed description of the complex extension of the dressed energy and bounds its real part, advancing understanding of the quantum transfer matrix eigenvalues.

Findings

Describes the curve where the real part of the dressed energy vanishes.

Provides lower bounds for the real part of the dressed energy in certain complex regions.

Connects the complex analysis of the dressed energy to low-temperature eigenvalue solutions.

Abstract

We consider the dressed energy $\varepsilon$ of the XXZ chain in the massless antiferromagnetic parameter regime at $0 < \Delta < 1$ and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers it describes the energy of particle-hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve ${\rm Re}\, \varepsilon (\lambda) = 0$. We describe this curve and give lower bounds to the function ${\rm Re}\, \varepsilon$ in regions of the complex plane, where it is positive.