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arXiv:math-ph/0004030

Nested Bethe Ansatz and Finite Dimensional Canonical Commutation Relations

Alex Kasman

Published Apr 24, 2000• 3 citations
arXivHTMLPDF

Abstract

Recent interest in discrete, classical integrable systems has focused on their connection to quantum integrable systems via the Bethe equations. In this note, solutions to the rational nested Bethe ansatz (RNBA) equations are constructed using the ``completed Calogero-Moser phase space'' of matrices which satisfy a finite dimensional analogue of the canonical commutation relationship. A key feature is the fact that the RNBA equations are derived only from this commutation relationship and some elementary linear algebra. The solutions constructed in this way inherit continuous and discrete symmetries from the CM phase space.

DOI
10.48550/arxiv.math-ph/0004030

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Recent interest in discrete, classical integrable systems has focused on their connection to quantum integrable systems via the Bethe equations. In this note, solutions to the rational nested Bethe ansatz (RNBA) equations are constructed using the ``completed Calogero-Moser phase space'' of matrices which satisfy a finite dimensional analogue of the canonical commutation relationship. A key feature is the fact that the RNBA equations are derived only from this commutation relationship and some elementary linear algebra. The solutions constructed in this way inherit continuous and discrete symmetries from the CM phase space.
A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's QQQ-operators, etc) with elements of classical non-linear integrable difference equations (τττ-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for Ak−1A_{k-1}Ak−1​-type models emerge as discrete time equations of motion for zeros of classical τττ-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed.
This chapter focuses on the diagonalization of a set of commuting scattering operators, or equivalently, transfer matrices, in models involving higher-rank symmetries. The discussion centers on modules more intricate than those of [N – M, M] type, ultimately considering arbitrary irreducible representations of the symmetric group. The core idea is to construct a suitable basis in the representation space that enables the reduction of a higher-dimensional diagonalization problem to a sequence of lower-dimensional ones that are already solved. This reduction is achieved through a recursive scheme in which each step lowers the matrix dimension, enabling its diagonalization–a procedure known as nested Bethe ansatz. The method is framed as a successive dimensional reduction, systematically building on solutions from simpler cases. Special attention is given to Yang’s connection formula and its role in enabling this recursive approach. The chapter extends the framework introduced in Chapter 3, developing a coordinate version of the nested Bethe ansatz and generalizing it to accommodate multiple levels of nesting. This recursive structure reveals deep algebraic insights and plays a pivotal role in the study of integrable systems with rich internal symmetries.
Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents.
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.
This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational GL(n)GL(n)GL(n)-invariant RRR-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.