The open XXZ-chain: Bosonisation, Bethe ansatz and logarithmic corrections

TL;DR

This paper analyzes the critical open XXZ spin-1/2 chain using bosonisation and Bethe ansatz, revealing cutoff-independent free energy contributions, temperature-dependent boundary susceptibility, and logarithmic corrections at special anisotropies.

Contribution

It provides explicit formulas for free energy and boundary susceptibility, including logarithmic corrections at specific anisotropies, verified by numerical and Bethe ansatz methods.

Findings

Agreement with Lukyanov's bulk free energy results

Explicit cutoff-independent boundary free energy terms

Logarithmic corrections at special anisotropies

Abstract

We calculate the bulk and boundary parts of the free energy for an open spin-1/2 XXZ-chain in the critical regime by bosonisation. We identify the cutoff independent contributions and determine their amplitudes by comparing with Bethe ansatz calculations at zero temperature T. For the bulk part of the free energy we find agreement with Lukyanov's result [Nucl.Phys.B 522, 533 (1998)]. In the boundary part we obtain a cutoff independent term which is linear in T and determines the temperature dependence of the boundary susceptibility in the attractive regime for $T\ll 1$. We further show that at particular anisotropies where contributions from irrelevant operators with different scaling dimensions cross, logarithmic corrections appear. We give explicit formulas for these terms at those anisotropies where they are most important. We verify our results by comparing with extensive numerical calculations based on a numerical solution of the T=0 Bethe ansatz equations, the finite temperature Bethe ansatz equations in the quantum-transfer matrix formalism, and the density-matrix renormalisation group applied to transfer matrices.