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arXiv:cond-mat/0111427 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum integrable system with two color components in two dimensions

Mu-Lin Yan

Published May 1, 2000• 0 citations
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Abstract

The Davey-Stewartson 1 (DS1) system [Proc. R. Soc. London, Ser. A 338, 101 (1974)] is an integrable model in two dimensions. A quantum DS1 system with two color components in two dimensions has been reformulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with two color components. The solutions to the two-dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For the latter problem with delta-function interactions and solution by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.

DOI
10.1103/physreve.61.4745

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The Davey-Stewartson 1 (DS1) system [Proc. R. Soc. London, Ser. A 338, 101 (1974)] is an integrable model in two dimensions. A quantum DS1 system with two color components in two dimensions has been reformulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with two color components. The solutions to the two-dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For the latter problem with delta-function interactions and solution by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.
A quantum N-body problem with 2-component in (2 + 1) dimensions deduced from an integrable model in (2 + 1) dimensions is investigated. The Davey–Stewartson 1 (DS1) system9 is an integrable model in two dimensions. A quantum DS1 system with 2 color components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 color components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimension. For the latter with δ-function interactions and being solved by the Bethe–Yang ansatz, we introduce symmetrical and anti-symmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solutions is discussed.
The quantum 2-component DS1 system in two dimensions has been reduced to two 2-component many-body problems both in one dimension by an analogue of the method of separation of variables. The solutions to 2 dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For latters with the delta\\deltadelta-function interactions and being solved by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.
The quantum 2-component DS1 system was reduced to two 1D many-body problems with δ−δ-δ−function interactions, which were solved by Bethe ansatz. Using the ansatz in ref.[1] and introducing symmetric and antisymmetric Young operators of the permutation group, we obtain the exact solutions for the system.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.