Topics in Bethe ansatz

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.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.

arXiv:hep-th/9211111Nov 24, 1992179 citations

Introduction to the coordinate-space bethe ansatz and to the treatment of bethe ansatz equations

Ferenc Woynarovich

Springer eBooksApr 7, 200824 citations

Introduction to quantum integrability

Jules Lamers

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 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.

Modave Summer School in Mathematical PhysicsFeb 12, 20158 citations

Introduction to the Bethe Ansatz I

Michael Karabach, Gerhard H. Muller, Harvey Gould +1 more

Computers in PhysicsJan 1, 1997191 citations

A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics

Jules Lamers

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.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.

arXiv:1501.06805Jan 27, 20153 citations

SL(3)

is discussed.

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.