Quantum Inverse Scattering Method. Selected Topics

Abstract

Four lectures given at Nankai Institute of Mathematics, Tianjin, China, 5--13 April 1991 present an elementary introduction into the quantum integrable models aimed for mathematical physicists and mathematicians. The stress is made on the algebraic aspects of the theory and the problem of determining the spectrum of quantum integrals of motion. The XXX magnetic chain is used as the basic example. Two lectures are devoted to a detailed exposition of the Functional Bethe Ansatz --- a new technique alternative to the Algebraic Bethe Ansatz --- and its relation to the separation of variable method. A possibility to extend FBA to the $SL(3)$ is discussed.

36 pages

Cited by in corpus (98)

• Families of quasi-local conservation laws and quantum spin transport
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Functional relations and Bethe Ansatz for the XXZ chain
• Renormalization of twist-three operators and integrable lattice models
• Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and twisted boundary conditions
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
• Separation of Variables in the open XXX chain
• Integrability and WKB solution of twist-three evolution equations
• Baxter's Q-operator for the homogeneous XXX spin chain
• Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
• Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof
• Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
• Separation of variables in the quantum integrable models related to the Yangian Y[sl(3)]
• Lectures on Yangian Symmetry
• Integrability of scattering amplitudes in N=4 SUSY
• Quantum Chaos, Delocalization, and Entanglement in Disordered Heisenberg Models
• Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
• Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism
• Quantum Sine(h)-Gordon Model and Classical Integrable Equations
• Separation of variables for the classical and quantum Neumann model
• Yangian Symmetry for Bi-Scalar Loop Amplitudes
• The open XXX spin chain in the SoV framework: scalar product of separate states
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• On-shell Diagrams, Graßmannians and Integrability for Form Factors
• Noncompact Heisenberg spin magnets from high-energy QCD: II. Quantization conditions and energy spectrum
• Long-range SL(2) Baxter equation in N=4 super-Yang-Mills theory
• On quantum separation of variables
• Quantum mechanics of null polygonal Wilson loops
• Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• The hexagon in the mirror: the three-point function in the SoV representation
• Integrability of two-loop dilatation operator in gauge theories
• Fixing the Quantum Three-Point Function
• Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame
• Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT
• R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain
• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• Towards Baxter equation in supersymmetric Yang-Mills theories
• Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices
• Slavnov and Gaudin-Korepin Formulas for Models without $\mathrm{U}(1)$ Symmetry: the Twisted XXX Chain
• Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors
• Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations
• On factorized overlaps: Algebraic Bethe Ansatz, twists, and Separation of Variables
• Problems with using separated variables for computing expectation values for higher ranks
• Universal R-matrix as integral operator
• Introduction to classical and quantum integrability
• Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation
• Quantum and Classical Integrable Systems
• Quasideterminant solutions of the generalized Heisenberg magnet model
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• Noncompact SL(2,R) spin chain
• On Complex Gamma-Function Integrals
• Modified Algebraic Bethe Ansatz: Twisted XXX Case
• A review of the AdS/CFT Quantum Spectral Curve
• Evolution equation for the higher-twist B-meson distribution amplitude
• Algebraic Bethe Ansatz approach to form factors and correlation functions of the cyclic eight-vertex solid-on-solid model
• A family of the Poisson brackets compatible with the Sklyanin bracket
• Unitarity of the SoV transform for the Toda chain
• Integrability in heavy quark effective theory
• On sl3 Knizhnik-Zamolodchikov equations and W3 null-vector equations
• Systematic classical continuum limits of integrable spin chains and emerging novel dualities
• A new representation for two- and three-point correlators of operators from sl(2) sector
• Bases of Bethe Vectors and Difference Equations with Regular Singular Points
• Scalar Products in Twisted XXX Spin Chain. Determinant Representation
• Dual conformal symmetry on the light-cone
• $\mathrm{SL}(2,\mathbb{C})$ Gustafson Integrals
• Gauss hypergeometric function and quadratic $R$-matrix algebras
• The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy
• Continuous and Discrete (Classical) Heisenberg Spin Chain Revised
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Baecklund transformations and Baxter's Q-operator
• Gauss hypergeometric function and quadratic R-matrix algebras
• Analytic Bethe Ansatz and Baxter equations for long-range psl(2|2) spin chain
• Non-Abelian $SU(3{)}_k$ anyons: inversion identities for higher rank face models
• Aspects of the inverse problem for the Toda chain
• A remark on the notion of independence of quantum integrals of motion in the thermodynamic limit
• A Collection of Exercises in Two-Dimensional Physics, Part 1
• Mellin$-$Barnes Transformation for Two-Loop Master-Diagram
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
• Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains
• Separation of Variables for a flux tube with an end
• Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables
• Separation of variables in the $A_2$ type Jack polynomials
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• Introduction to Yangian in Physics
• The higher order $q$-Dolan-Grady relations and quantum integrable systems
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Finite-size spectrum of the staggered six-vertex model with antidiagonal boundary conditions
• Quantum integrability: Lagrangian 1-form case
• Unitarity of the SoV Transform for $\mathrm{SL}(2,\mathbb{C})$ Spin Chains
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
• The molecular asymmetric rigid rotor Hamiltonian as an exactly solvable model
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• An Analytic Formula for the A_2 Jack Polynomials
• Momentum Evolution Numerics of an Impurity in a Quantum Quench
• On non-stationary Lamé equation from WZW model and spin-1/2 XYZ chain