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Journal of Physics A Mathematical and General

Simple excitations in the nested Bethe-ansatz model

Junji Suzuki

Published Dec 21, 1988• 32 citations
Publisher

Abstract

The scaling dimensions of the operators in the nested Bethe ansatz model are studied. They explicitly show the structure of the underlying Lie algebras; they can be written in terms of the Cartan matrix. The author infers that they are the natural extension of those of SU(2), or equivalently of the Coulomb gas.
DOI
10.1088/0305-4470/21/24/003

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On the nested algebraic Bethe ansatz for spin chains with simple g\mathfrak{g}g-symmetry
We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with g\mathfrak{g}g-symmetry for any simple Lie algebra g\mathfrak{g}g. Starting the nesting process by removing a single simple root from g\mathfrak{g}g, we use the residual U(1)U(1)U(1) charge and the block Gauss decomposition of the RRR-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.
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.
The structure of Bethe vectors for generalised models associated with the rational and trigonometric R-matrix is investigated. The Bethe vectors in terms of two-component and multi-component models are described. Consequently, their structure in terms of local variables and operators is provided. This, as a consequence, proves the equivalence of coordinate and algebraic Bethe ansatzes for the Heisenberg spin chains. Hermitian conjugation of the elements of the monodromy matrix for the spin chains is studied.