Statistical mechanics of integrable quantum spin systems

TL;DR

This paper provides an introduction to the equilibrium statistical mechanics of quantum spin chains, focusing on vertex models, transfer matrices, and deriving free energy representations for the XXZ model.

Contribution

It offers a minimal, accessible overview of quantum spin systems and their integrability, including graphical methods and integral representations for the free energy.

Findings

Derived an integral representation of the free energy for the XXZ model

Explained the graphical language of vertex models and transfer matrices

Provided background on equilibrium statistical mechanics of quantum spin chains

Abstract

This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School "Integrability in Atomic and Condensed Matter Physics" in the summer of 2018. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. The lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. The author was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. He was further asked to explain the graphical language of vertex models and to introduce the concepts of the Trotter decomposition and the quantum transfer matrix. This was basically the contents of the first four lectures presented at the school. In the remaining two lectures these notions were filled with life by deriving an integral representation of the free energy per lattice site for the Heisenberg-Ising chain (alias XXZ model) using techniques based on non-linear integral equations.