Asymmetric XXZ chain at the antiferromagnetic transition: Spectra and partition functions

TL;DR

This paper analytically solves the Bethe ansatz equations for the asymmetric XXZ chain near the antiferromagnetic boundary, revealing finite-size spectra, scaling behaviors, and explicit partition functions, with implications for phase transition analysis.

Contribution

It provides the first analytical derivation of finite-size spectra and partition functions of the asymmetric XXZ chain at critical boundaries using Bethe ansatz.

Findings

Energy gaps scale as N^{-1/2} near the phase boundary at zero vertical field.

Partition functions are explicitly obtained and relate to free fermion systems.

Identities between partition functions are derived from lattice symmetry.

Abstract

The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size $N$ as $N^{-1/2}$ and its amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. Similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as $N^{-2}$. The partition functions for this case are found to be that of a nonrelativistic free fermion system. From symmetry of the lattice model under $\pi /2$ rotation, several identities between the partition functions are found. The $N^{-1/2}$ scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky-Talapov transition with the space and time coordinates interchanged.