Quantum spin chains of Temperley-Lieb type: periodic boundary conditions, spectral multiplicities and finite temperature

TL;DR

This paper analyzes the spectra of quantum spin chains of Temperley-Lieb type, establishing their relation to XXZ chains for different boundary conditions, and explores their thermodynamic properties at finite temperature.

Contribution

It introduces a Temperley-Lieb equivalence framework for spectral analysis of spin chains with open and periodic boundaries, including finite temperature and field considerations.

Findings

Spectra of open boundary models differ from XXZ only in eigenvalue multiplicities.

Periodic boundary spectra are derived from twisted XXZ chain spectra.

A formula for degeneracy of momentum eigenvalues is provided.

Abstract

We determine the spectra of a class of quantum spin chains of Temperley-Lieb type by utilizing the concept of Temperley-Lieb equivalence with the S=1/2 XXZ chain as a reference system. We consider open boundary conditions and in particular periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley-Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representation theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field.