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arXiv:hep-th/9207007

A New Solution of the Supersymmetric TJ Model by Means of the Quantum Inverse Scattering Method

Fabian H. L. Eßler, V. E. Korepin

Published Jul 3, 1992• 11 citations
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Abstract

We construct the enveloping fundamental spin model of the t-J hamiltonian using the Quantum Inverse Scattering Method (QISM), and present all three possible Algebraic Bethe Ansätze. Two of the solutions have been previously obtained in the framework of Coordinate Space Bethe Ansatz by Sutherland and by Schlottmann and Lai, whereas the third solution is new. The formulation of the model in terms of the QISM enables us to derive explicit expressions for higher conservation laws.

DOI
10.48550/arxiv.hep-th/9207007

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We construct the enveloping fundamental spin model of the t-J hamiltonian using the Quantum Inverse Scattering Method (QISM), and present all three possible Algebraic Bethe Ansätze. Two of the solutions have been previously obtained in the framework of Coordinate Space Bethe Ansatz by Sutherland and by Schlottmann and Lai, whereas the third solution is new. The formulation of the model in terms of the QISM enables us to derive explicit expressions for higher conservation laws.
Introduction The quantum inverse scattering method (QSIM) appears as the quantized form of the classical inverse scattering method. It allows us to reproduce the results of the Bethe Ansatz and to move ahead. QISM is now a well developed branch of mathematical physics. In this chapter the fundamentals of QISM are given and illustrated by concrete examples. In section 1 the general scheme of QISM, which allows the calculation of commutation relations between elements of the transfer matrix (necessary to construct eigenfunctions of the Hamiltonian in Chapter VII) is presented, and the quantum R-matrix is introduced. As in the classical case, the existence of an R-matrix and trace identities ensures that a Lax representation for the model exists. Thus, there are infinitely many conservation laws. The Yang-Baxter equation, which is satisfied by the R-matrix, is discussed in section 2. Some important features of the R-matrix are also mentioned. The trace identities for the quantum nonlinear Schrödinger equation are proved in section 3. The general scheme of QISM is applied to the quantum sine-Gordon and Zhiber-Shabat-Mikhailov models in section 4. Spin models of quantum statistical physics are discussed in section 5. It is shown that a fundamental spin model can be constructed with the help of any given R-matrix. The connection between classical statistical models on a two-dimensional lattice and QISM is established in section 6. QISM is the generalization of the classical inverse scattering method.
Quantum integrability is established for the one-dimensional supersymmetric UUU model with boundary terms by means of the quantum inverse-scattering method. The boundary supersymmetric UUU chain is solved by using the coordinate-space Bethe-ansatz technique and Bethe-ansatz equations are derived. This provides us with a basis for computing the finite-size corrections to the low-lying energies in the system.