Surface free energy of the open XXZ spin-1/2 chain

TL;DR

This paper derives an exact integral representation for the surface free energy of the open XXZ spin-1/2 chain with boundary fields, enabling analysis of boundary magnetization and thermodynamics.

Contribution

It connects the boundary free energy to the six-vertex model with reflecting ends and provides a method to analyze low-temperature boundary behavior.

Findings

Derived an integral equation for surface free energy.

Analyzed low-temperature boundary magnetization.

Numerically plotted boundary magnetization behavior.

Abstract

We study the boundary free energy of the XXZ spin-$\tf{1}{2}$ chain subject to diagonal boundary fields. We first show that the representation for its finite Trotter number approximant obtained by Bortz, Frahm and G\"{o}hmann is related to the partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya determinant representation for the latter quantity we are able to take the infinite Trotter number limit. This yields a representation for the surface free energy which involves the solution of the non-linear integral equation that governs the thermodynamics of the XXZ spin-1/2 chain subject to periodic boundary conditions. We show that this integral representation allows one to extract the low-$T$ asymptotic behavior of the boundary magnetization at finite external magnetic field on the one hand and numerically plot this function on the other hand.