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arXiv:solv-int/9710018 · Indian journal of physics B and Proceedings of the Indian Association for the Cultivation of science B

Unifying structures in quantum integrable systems

Anjan Kundu

Published Oct 23, 1997• 0 citations
arXivHTMLPDF

Abstract

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

DOI
10.48550/arxiv.solv-int/9710018

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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 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.
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.
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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.