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arXiv:hep-th/9509029 · Birkhäuser Boston eBooks

Classical and Quantum Nonultralocal Systems on the Lattice

Michael Arsen'evich Semenov-Tian-Shansky, A. Sevostyanov

Published Jan 1, 1997• 22 citations
arXivHTMLPDF

Abstract

We classify nonultralocal Poisson brackets for 1-dimensional lattice systems and describe the corresponding regularizations of the Poisson bracket relations for the monodromy matrix. A nonultralocal quantum algebras on the lattices for these systems are constructed. For some class of such algebras an ultralocalization procedure is proposed. The technique of the modified Bethe-Anzatz for these algebras is developed and is applied to the nonlinear sigma model problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-2434-1_17

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We classify nonultralocal Poisson brackets for 1-dimensional lattice systems and describe the corresponding regularizations of the Poisson bracket relations for the monodromy matrix. A nonultralocal quantum algebras on the lattices for these systems are constructed. For some class of such algebras an ultralocalization procedure is proposed. The technique of the modified Bethe-Anzatz for these algebras is developed and is applied to the nonlinear sigma model problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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.
In the framework of the quantum inverse scattering method, we consider a problem of constructing local operators for one-dimensional quantum integrable models, especially for the lattice versions of the nonlinear Schrödinger and sine-Gordon models. We show that a certain class of local operators can be constructed from the matrix elements of the monodromy matrix in a simple way. They are closely related to the quantum projectors and have nice commutation relations with half of the matrix elements of the elementary monodromy matrix. The form factors of these operators can be calculated by using the standard algebraic Bethe ansatz techniques.