Integrability of quantum chains: theory and applications to the spin-1/2 $XXZ$ chain

TL;DR

This paper reviews the theory of integrability in one-dimensional quantum systems, focusing on the spin-1/2 XXZ chain, and discusses finite-temperature properties using algebraic Bethe ansatz and transfer matrix methods.

Contribution

It provides a comprehensive overview of integrability concepts and applies them to derive thermodynamic properties of the XXZ chain at finite temperature.

Findings

Derived specific heat, magnetic susceptibility, and thermal conductivity of the XXZ chain.

Presented a lattice path integral and transfer matrix approach for finite-temperature analysis.

Reviewed key integrability concepts like the Yang-Baxter equation and algebraic Bethe ansatz.

Abstract

In this contribution we review the theory of integrability of quantum systems in one spatial dimension. We introduce the basic concepts such as the Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite extensively we present the treatment of integrable quantum systems at finite temperature on the basis of a lattice path integral formulation and a suitable transfer matrix approach (quantum transfer matrix). The general method is carried out for the seminal model of the spin-1/2 $XXZ$ chain for which thermodynamic properties like specific heat, magnetic susceptibility and the finite temperature Drude weight of the thermal conductivity are derived.