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Institute of Physics eBooks

Unifying approaches in integrable systems: Quantum and statistical, ultralocal and non-ultralocal

Anjan Kundu

Published Nov 26, 2004• 3 citations
Publisher

Abstract

The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures.We restrict to quantum and statistical models belonging to trigonometric and rational classes with 2 × 2 Lax operators.The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation.Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.

DOI
10.1887/0750309598/b1385c6

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The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures.We restrict to quantum and statistical models belonging to trigonometric and rational classes with 2 × 2 Lax operators.The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation.Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.
The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures. We restrict to quantum and statistical models belonging to trigonometric and rational classes with (2 x 2)- Lax operators. The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation. Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.
Basic concepts of quantum integrable systems (QIS) are presented stressing on the unifying structures underlying such diverse models. Variety of ultralocal and nonultralocal models is shown to be described by a few basic relations defining novel algebraic entries. Such properties can generate and classify integrable models systematically and also help to solve exactly their eigenvalue problem in an almost model-independent way. The unifying thread stretches also beyond the QIS to establish its deep connections with statistical models, conformal field theory etc. as well as with abstract mathematical objects like quantum group, braided or quadratic algebra
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.
Formulating quantum integrability for nonultralocal models (NM) parallel to the familiar approach of inverse scattering method is a long standing problem. After reviewing our result regarding algebraic structures of ultralocal models, we look for the algebra underlying NM. We propose an universal equation represented by braided Yang-Baxter equation and able to derive all basic equations of the known models like WZWN model, nonabelian Toda chain, quantum mapping etc. As further useful application we discover new integrable quantum NM, e.g. mKdV model, anyonic model, Kundu-Eckhaus equation and derive SUSY models and reflection equation from the nonultralocal view point.
This chapter discusses a new type of vertex model will be constructed from the ancestor Lax operator using non-fundamental representations. It examines how these Lax operators can be generated in a systematic way confirming their integrability. The chapter focuses on the ultralocal systems due to their relative simplicity and examines a unifying scheme for generating quantum and statistical integrable models. It describes a new direction of study would be to investigate whether there could be any meaningful statistical model corresponding to the integrable non-ultralocal models. To describe an integrable system with such an involved structure, naturally one can no longer start from the Hamiltonian of the model as is customary in physics, since the Hamiltonian is merely one among many commuting conserved charges. Non-ultralocal models are mostly non-fundamental systems with the infinite dimensional representations defined in some Hilbert space.
An overview of the quantum integrable systems (QIS) is presented. Basic concepts of the theory are highlighted stressing on the unifying algebraic properties, which not only helps to generate systematically the representative Lax operators of different models, but also solves the related eigenvalue problem in an almost model independent way. Difference between the approaches in the integrable ultralocal and nonultralocal quantum models are explained and the interrelation between the QIS and other subjects are focussed on.