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arXiv:1909.09967 · SciPost Physics Lecture Notes

Statistical mechanics of integrable quantum spin systems

Frank Göhmann

Published Aug 17, 2020• 23 citations

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 responsibility for the selection of the material is partially with the organisers, Jean-Sebastien Caux, Nikolai Kitanine, Andreas Klümper and Robert Konik. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. My lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. I was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. I 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 I started filling these notions 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. Up to small corrections the following sections 1-6 display the six lectures almost literally. The only major change is that the example of the XXZ chain has been moved from section 5 to 2. During the school it was not really necessary to introduce the model, since other speakers had explained it before. But for these notes I thought it might be useful to introduce the main example rather early. I also supplemented each lecture with a comment section which contains additional references and material of the type that was discussed informally with the participants.

DOI: 10.21468/scipostphyslectnotes.16

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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 responsibility for the selection of the material is partially with the organisers, Jean-Sebastien Caux, Nikolai Kitanine, Andreas Klümper and Robert Konik. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. My lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. I was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. I 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 I started filling these notions 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. Up to small corrections the following sections 1-6 display the six lectures almost literally. The only major change is that the example of the XXZ chain has been moved from section 5 to 2. During the school it was not really necessary to introduce the model, since other speakers had explained it before. But for these notes I thought it might be useful to introduce the main example rather early. I also supplemented each lecture with a comment section which contains additional references and material of the type that was discussed informally with the participants.

These are lecture notes of a self-contained introduction to quantum integrability given at the Tenth Modave Summer School in Mathematical Physics, 2014, primarily aimed at PhD candidates and junior researchers in theoretical physics.We start by introducing spin chains and discussing the coordinate Bethe Ansatz (CBA) for a representative example, the Heisenberg XXZ model.We focus on the structure of the CBA and its main results, deferring a detailed treatment of the CBA in the general M-particle sector of the XXZ model to an appendix.Subsequently the transfer-matrix method is discussed for the six-vertex model, uncovering a relation between that model and the XXZ spin chain.Equipped with this background the quantum inverse-scattering method (QISM) and algebraic Bethe Ansatz (ABA) are treated.We emphasize the use of graphical notation for algebraic quantities and show how it can be used for computations.Finally we turn to quantum integrability in the context of theoretical high-energy physics.We discuss factorized scattering in two-dimensional QFT, and conclude with a qualitative introduction to one current research topic relating quantum integrability to theoretical high-energy physics: the Bethe/gauge correspondence.

These are lecture notes of an introduction to quantum integrability given at the Tenth Modave Summer School in Mathematical Physics, 2014, aimed at PhD candidates and junior researchers in theoretical physics. We introduce spin chains and discuss the coordinate Bethe ansatz (CBA) for a representative example: the Heisenberg XXZ model. The focus lies on the structure of the CBA and on its main results, deferring a detailed treatment of the CBA for the general

M

-particle sector of the XXZ model to an appendix. Subsequently the transfer-matrix method is discussed for the six-vertex model, uncovering a relation between that model and the XXZ spin chain. Equipped with this background the quantum inverse-scattering method (QISM) and algebraic Bethe ansatz (ABA) are treated. We emphasize the use of graphical notation for algebraic quantities as well as computations. Finally we turn to quantum integrability in the context of theoretical high-energy physics. We discuss factorized scattering in two-dimensional QFT, and conclude with a qualitative introduction to one current research topic relating quantum integrability to theoretical high-energy physics: the Bethe/gauge correspondence.

ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.

This is an extended and corrected version of lecture notes originally written for a one semester course at Leibniz University Hannover. The main aim of the notes is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites. Such systems can be used, for example, to model the materials in condensed matter physics. The notes provide the necessary background material to access recent literature in the field. Some of these recent results are also discussed. The contents are roughly as follows: (1) quick recap of essentials from functional analysis, (2) introduction to operator algebra, (3) algebraic quantum mechanics, (4) infinite systems (quasilocal algebra), (5) KMS and ground states, (6) Lieb-Robinson bounds, (7) algebraic quantum field theory, (8) superselection sectors of the toric code, (9) Haag-Ruelle scattering theory in spin systems, (10) applications to gapped phases. The level is aimed at students who have at least had some exposure to (functional) analysis and have a certain mathematical maturity.

Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.