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Journal of Physics A Mathematical and General

GeneralizedSUq(1|2) Gaudin model

Jun-peng Cao, Bo-Yu Hou, Rui‐Hong Yue

Published Apr 27, 2001• 1 citation
Publisher

Abstract

In this paper we propose the Hamiltonians of the generalized SU q (1|2) Gaudin model corresponding to the periodic generalized t - J model. With the help of the well defined graded quantum determinant, we obtain the eigenstate and eigenvalues of the generating function and the Hamiltonians of the Gaudin model in the fermionic background in the framework of the graded quantum inverse scattering method. The Bethe ansatz equations are also obtained.

DOI
10.1088/0305-4470/34/18/305

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In this paper we propose the Hamiltonians of the generalized SU q (1|2) Gaudin model corresponding to the periodic generalized t - J model. With the help of the well defined graded quantum determinant, we obtain the eigenstate and eigenvalues of the generating function and the Hamiltonians of the Gaudin model in the fermionic background in the framework of the graded quantum inverse scattering method. The Bethe ansatz equations are also obtained.
We study the generalized supersymmetric $t\ensuremath{-}J$ model with boundaries in three different gradings: fermionic, fermionic, bosonic (FFB), bosonic, fermionic, fermionic (BFF), and fermionic, bosonic, fermionic (FBF). Starting from the trigonometric R matrix, and in the framework of the graded quantum inverse scattering method, we solve the eigenvalue problems for the supersymmetric $t\ensuremath{-}J$ model. Detailed calculations are presented to obtain the eigenvalues and Bethe ansatz equations of the supersymmetric $t\ensuremath{-}J$ model with boundaries in three different backgrounds.
In this paper we deal with the trigonometric Gaudin model, generalized using a nontrivial triangular reflection matrix (corresponding to non-periodic boundary conditions in the case of anisotropic XXZ Heisenberg spin-chain). In order to obtain the generating function of the Gaudin Hamiltonians with boundary terms we follow an approach based on Sklyanin’s derivation in the periodic case. Once we have the generating function, we obtain the corresponding Gaudin Hamiltonians with boundary terms by taking its residues at the poles. As the main result, we find the generic form of the Bethe vectors such that the off-shell action of the generating function becomes exceedingly compact and simple. In this way—by obtaining Bethe equations and the spectrum of the generating function—we fully implement the algebraic Bethe ansatz for the generalized trigonometric Gaudin model.
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.