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 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.
Abstract The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.