Dressed energy of the XXZ chain in the complex plane

Saskia Faulmann,Frank Göhmann,Karol K Kozlowski

doi:10.1007/s11005-021-01473-3

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 mathrm{Re}, varepsilon (lambda ) = 0. We describe this curve and give lower bounds to the function mathrm{Re}, varepsilon in regions of the complex plane, where it is positive.

Highlights

IntroductionThe XXZ chain [13,18,19,20,21] is an anisotropic deformation of the Heisenberg chain [2]
The XXZ chain [13,18,19,20,21] is an anisotropic deformation of the Heisenberg chain [2]. It is the prototypical example of a Yang–Baxter integrable model which is solvable by means of the algebraic Bethe Ansatz [14]
We have found a characterization of the full spectrum only in the massive antiferromagnetic regime ( > 1 and 0 < h < h, where h is a lower critical field) in the low-temperature limit [6]

Summary

Introduction

The XXZ chain [13,18,19,20,21] is an anisotropic deformation of the Heisenberg chain [2]. In our forthcoming work, we want to exclude the existence of strings in the low-temperature limit This will show that the Bethe roots of the dominant state, but the Bethe roots belonging to any Bethe eigenstate of the quantum transfer matrix come close to the curve Re ε(λ) = 0, when the temperature goes to zero. The latter will be a crucial input for the further investigation of the thermal form factor series of the two-point functions of the XXZ chain in the critical regime.

Properties of the kernel function

The general case of finite Q

Proof of Theorem 1

Proof of Theorem 2

Conclusions