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arXiv:hep-th/9404013 · International Journal of Modern Physics A

ALGEBRAIC ASPECTS OF THE BETHE ANSATZ

L. D. Faddeev

Published May 20, 1995• 164 citations
arXivHTMLPDF

Abstract

In this article an introduction to the algebraic aspects of the Bethe ansatz is given. The applications to the seminal spin 1/2 XXX model are discussed in detail and the generalization to higher spin as well as XXZ and the lattice sine-Gordon model are indicated. The origin of quantum groups and their appearance in CFT models are explained. The text can be considered as a guide to the research papers in this field.

DOI
10.1142/s0217751x95000905

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arXiv:solv-int/9810007•Oct 11, 1998•3 citations
In this article an introduction to the algebraic aspects of the Bethe ansatz is given. The applications to the seminal spin 1/2 XXX model are discussed in detail and the generalization to higher spin as well as XXZ and the lattice sine-Gordon model are indicated. The origin of quantum groups and their appearance in CFT models are explained. The text can be considered as a guide to the research papers in this field.
I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin sss, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.
The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when qqq is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact SU(2)qSU(2)_qSU(2)q​ invariance on the lattice. The algebra of the Yang-Baxter (YB) and SU(2)qSU(2)_qSU(2)q​ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the SU(2)qSU(2)_qSU(2)q​ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of SU(2)qSU(2)_qSU(2)q​. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(ppp) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of SU(2)qSU(2)_qSU(2)q​, at q=eiπ/pq=e^{iπ/p}q=eiπ/p, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J
This 1996 book is an introduction to integrability and conformal field theory in two dimensions using quantum groups. The book begins with a brief introduction to S-matrices, spin chains and vertex models as a prelude to the study of Yang–Baxter algebras and the Bethe ansatz. The basic ideas of integrable systems are then introduced, giving particular emphasis to vertex and face models. Special attention is given to explaining the underlying mathematical tools, including braid groups, knot invariants and towers of algebra. The book then goes on to give a detailed introduction to quantum groups as a prelude to chapters on integrable models, two-dimensional conformal field theories and superconformal field theories. The book contains many diagrams and exercises to illustrate key points in the text.
The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.