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arXiv:2109.14280 · Journal of Physics A Mathematical and Theoretical

Introduction to classical and quantum integrability

Ana L. Retore

Published Mar 3, 2022• 27 citations
arXivHTMLPDF

Abstract

Abstract In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-matrix, solution of the quantum Yang–Baxter equation. The diagonalization of these charges is performed using the algebraic Bethe ansatz. At the end, the interpretation of the R-matrix as an S-matrix in a scattering process is also presented. These notes were written for the lectures delivered at the school ‘Integrability, Dualities and Deformations’, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

DOI
10.1088/1751-8121/ac5a8e

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Abstract In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-matrix, solution of the quantum Yang–Baxter equation. The diagonalization of these charges is performed using the algebraic Bethe ansatz. At the end, the interpretation of the R-matrix as an S-matrix in a scattering process is also presented. These notes were written for the lectures delivered at the school ‘Integrability, Dualities and Deformations’, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.
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 MMM-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.
Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents.
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.
This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of q−q-q−deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of SU^(2)qS{\hat U}(2)_qSU^(2)q​). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix τ(ł)τ(ł)τ(ł) are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit τ(ł)τ(ł)τ(ł) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].
The quantum inverse scattering method relates the Bethe Ansatz to the theory of classical completely integrable differential equations. These are sometimes called soliton equations. The modern way to solve them is called the classical inverse scattering method. In a sense this is a nonlinear generalization of the Fourier transform. In this Part the quantum inverse scattering method is expounded. The main statements of the classical inverse scattering method necessary for quantization are given in Chapter V where the Lax representation is introduced. The Hamiltonian structure of integrable models is also discussed along with the infinite number of integrals of motion. The most convenient method of analyzing the Hamiltonian structure relies on the classical r -matrix. Some concrete models will be considered. Chapter VI is devoted in particular to the quantum inverse scattering method. The R-matrix, which is the main object of this method, is introduced. The Yang-Baxter equation for the R-matrix is discussed. The main statements of the method are given and a number of examples are presented. The algebraic formulation of the Bethe Ansatz, one of the main achievements of the quantum inverse scattering method, is presented in Chapter VII. The notion of the determinant of the transition matrix in the quantum case is introduced in this chapter. (This is closely related to the concept of the antipode in quantum groups.)
Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.