Root patterns and energy spectra of quantum integrable systems without $U(1)$ symmetry: antiperiodic $XXZ$ spin chain

TL;DR

This paper develops an analytic method to determine Bethe root patterns and energy spectra of the antiperiodic XXZ spin chain, a quantum integrable model without U(1) symmetry, aiding the study of its physical properties.

Contribution

It introduces a universal analytic approach to derive root patterns and physical properties for models lacking U(1) symmetry, exemplified by the antiperiodic XXZ spin chain.

Findings

Derived Bethe root patterns in the thermodynamic limit.

Calculated ground state energy and elementary excitations.

Provided a universal procedure for quantum integrable models.

Abstract

Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without $U(1)$ symmetry, their spectra are usually given by inhomogeneous $T-Q$ relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic $XXZ$ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived. The present method provides an universal procedure to compute physical properties of quantum integrable models in the thermodynamic limit.