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Journal of Physics A Mathematical and General

A new algebraic Bethe ansatz forgl(2,1) invariant vertex models

Markus P. Pfannmüller, Holger Frahm

Published Aug 7, 1997• 18 citations
Publisher

Abstract

The algebraic Bethe ansatz for the integrable vertex model constructed from the four-dimensional representation of the superalgebra gl(2,1) is calculated using a ferromagnetic reference state. This Bethe ansatz was known only for the three-dimensional representation leading to the supersymmetric t-J model. The necessary modification of the nested algebraic Bethe ansatz scheme and generalizations to related models are discussed.

DOI
10.1088/0305-4470/30/15/010

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The algebraic Bethe ansatz for the integrable vertex model constructed from the four-dimensional representation of the superalgebra gl(2,1) is calculated using a ferromagnetic reference state. This Bethe ansatz was known only for the three-dimensional representation leading to the supersymmetric t-J model. The necessary modification of the nested algebraic Bethe ansatz scheme and generalizations to related models are discussed.
On Bethe Ansatz for a Supersymmetric Vertex Model with Uq[osp(2∣2)(2)]\mathcal{U}_{\rm q}[{\rm osp}(2|2)^{(2)}]Uq​[osp(2∣2)(2)]
The Algebraic Bethe ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra Uq[osp(2∣2)(2)]\mathcal{U}_{q}[\mathrm{osp}(2|2)^{(2)}]Uq​[osp(2∣2)(2)] is presented in detail. The eigenvalues and eigenvectors of the row-to-row transfer matrix are calculated and the corresponding Bethe Ansatz equations are obtained and analyzed numerically.
In this work we investigate the supersymmetric t-J model in one dimension. The spectrum of the hamiltonian is obtained by means of the algebraic nested Bethe-ansatz method. Furthermore, we present a detailed analysis of the algebraic structure of the states. By combining the Bethe ansatz with the spl(2,1) supersymmetry of the t-J hamiltonian a complete set of eigenstates is constructed. The ground state and the elementary excitations of the model are investigated. We also introduce a new integrable vertex model, the anisotropic t-J model, which possesses splq(2, 1) "quantum supergroup" invariance.
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.
We review the recent results we have obtained in the framework of algebraic Bethe ansatz based on algebras and superalgebras of rank greater than 1 or on their quantum deformation. We present different expressions (explicit, recursive or using the current realization of the algebra) for the Bethe vectors. Then, we provide a general expression (as sum over partitions) for their scalar products. For some particular cases (in the case of gl(3) g l ( 3 ) or its quantum deformation, or of gl(2|1) g l ( 2 | 1 ) ), we provide determinant expressions for the scalar products. We also compute the form factors of the monodromy matrix entries, and give some general methods to relate them. A coproduct formula for Bethe vectors allows to get the form factors of composite models.
We present an algebraic Bethe ansatz for the supersymmetric U model for correlated electrons on the unrestricted 4L{4}^{\mathit{L}}4L-dimensional electronic Hilbert space ${\ensuremath{\bigotimes}}_{\mathit{L}}^{\mathit{n}=1}$C4{\mathit{C}}^{4}C4 (where L is the lattice length). The supersymmetry algebra of the model is the Lie superalgebra gl(2|1) and contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter U. The parameter U arises from the one-parameter family of inequivalent typical four-dimensional irreps of gl(2|1). Eigenstates of the model are determined by the algebraic Bethe ansatz on a one-dimensional periodic lattice. \textcopyright{} 1996 The American Physical Society.