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Acta Physica Slovaca Reviews and Tutorials

Introduction to Integrable Many-Body Systems II

Ladislav Šamaj

Published Apr 1, 2010• 5 citations
Publisher

Abstract

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.

DOI
10.2478/v10155-010-0002-2

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NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX

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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.
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.
Part I. Thermodynamics of Non-Interacting Systems and Ground States on Interacting Systems: 1. Free energy and correlation functions of XY models 2. Systems with delta-function potential 3. Isotropic Heisenberg model 4. XXZ model 5. XYZ and eight-vertex models 6. Hubbard model Part II. Finite Temperature Integral Equations for Un-Nested Systems: 7. Repulsive delta-function bosons 8. Thermodynamics of the XXX chain 9. Thermodynamics of the XXZ model 10. Thermodynamics of the XYZ model 11. Low-temperature thermodynamics Part III. Finite Temperature Integral Equations for Nested Systems: 12. S=1/2 fermions with repulsive potential in the continuum 13. S=1/2 fermions with an attractive potential 14. Thermodynamics of the Hubbard model Part IV. Quantum Transfer Matrix and Recent Developments: 15. Transfer matrix and correlation length 16. The spin 1/2 XXZ model in a magnetic field 17. The XYZ model with no magnetic field 18. Recent developments and related topics.
Introduction to integrable many-body systems I This is the first volume of a three-volume introductory course about integrable (exactly solvable) systems of interacting bodies. The aim of the course is to derive and analyze, on an elementary mathematical and physical level, the Bethe ansatz solutions, ground-state properties and the thermodynamics of integrable many-body systems in many domains of physics: Nonrelativistic one-dimensional continuum Fermi and Bose gases; One-dimensional quantum models of condensed matter physics like the Heisenberg, Hubbard and Kondo models; Relativistic models of the (1+1)-dimensional Quantum Field Theory like the Luttinger model, the sine-Gordon model and its fermionic analog the Thirring model; Two-dimensional classical models, especially the symmetric Coulomb gas. In the first part of this volume, we deal with nonrelativistic one-dimensional continuum Fermi and Bose quantum gases of spinless (identical) particles with specific types of pairwise interactions like the short-range δ-function and hard-core interactions, and the long-range 1/
This chapter reviews the various techniques developed for the study of the thermodynamics of integrable systems at the example of the simplest one, namely the spin-12 Heisenberg chain. As a matter of fact, the strict ground state is inaccessible due to the very fundamentals of thermodynamics. The minimization of the free energy functional yields what is known as the thermodynamical Bethe ansatz. Application of thermodynamical data of quantum spin chains is the microscopic modelling of magnetic systems such as spin-Peierls compounds and ladder systems. The microscopic interactions of a given substance are determined by a comparative analysis of the experimental results for the susceptibility and the theoretical data obtained for a model. Lowering the temperature, the systems enter a large regime with non-universal correlations and finally approach the quantum critical point at exactly zero temperature showing again universal yet nontrivial properties with divergent correlation lengths governed by conformal field theory.
The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.
NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX