Statistical mechanics of integrable quantum spin systems

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.

Les Houches Lecture Notes 2018, 38 pages, v2: typos corrected, v3: minor changes, references and one figure added, version for publication in SciPost

References in corpus (8)

• Hidden Grassmann Structure in the XXZ Model II: Creation Operators
• Hidden Grassmann structure in the XXZ model
• Hidden Grassmann Structure in the XXZ Model III: Introducing Matsubara direction
• Thermal form-factor approach to dynamical correlation functions of integrable lattice models
• Temperature driven crossover phenomena in the correlation lengths of the one-dimensional t-J model
• Finite temperature correlation functions from discrete functional equations
• New exact results on density matrix for XXX spin chain
• Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach

Cited by in corpus (8)

• Entanglement spreading in non-equilibrium integrable systems
• Large time and long distance asymptotics of the thermal correlators of the impenetrable anyonic lattice gas
• Effective free-fermionic form factors and the XY spin chain
• On the Dynamics of Free-Fermionic Tau-Functions at Finite Temperature
• Analytical results for the low-temperature Drude weight of the XXZ spin chain
• Thermal form-factor expansion of the dynamical two-point functions of local operators in integrable quantum chains
• Thermodynamics of an Ising-like XXZ chain in a longitudinal magnetic field in the framework of the Quantum Transfer Matrix approach
• Integrable fishnet circuits and Brownian solitons