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arXiv:cond-mat/9907344 · Journal of Physics A Mathematical and General

Integrable model of interactingXXand Fateev-Zamolodchikov chains

Francisco Castilho Alcaraz, R Z Bariev

Published Aug 19, 1999• 1 citation
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Abstract

We consider the exact solution of a model of correlated particles, which is presented as a system of interacting XX and Fateev-Zamolodchikov chains. This model can also be considered as a generalization of the multiband anisotropic t - J model in the case where we restrict the site occupations to at most two electrons. The exact solution is obtained for the eigenvalues and eigenvectors using the Bethe ansatz method.

DOI
10.1088/0305-4470/32/35/101

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We consider the exact solution of a model of correlated particles, which is presented as a system of interacting XX and Fateev-Zamolodchikov chains. This model can also be considered as a generalization of the multiband anisotropic t - J model in the case where we restrict the site occupations to at most two electrons. The exact solution is obtained for the eigenvalues and eigenvectors using the Bethe ansatz method.
Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz, the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian defines the algebraic properties of the matrices defining the amplitudes. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. The matrix product ansatz we propose allows an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t – J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with z -magnetic fields at the surface and arbitrary hard-core exclusion among the spins.
The Bethe ansatz in its several formulations is a common tool for the exact solution of one-dimensional quantum Hamiltonians. This ansatz asserts that several eigenfunctions of the Hamiltonians are given in terms of a sum of permutations of plane waves. We present results that induce us to expect that, alternatively, the eigenfunctions of all the exact integrable quantum chains can also be expressed by a matrix product ansatz. In this ansatz several components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. This ansatz allows an unified formulation of several exact integrable Hamiltonians. We show how to formulate this ansatz for a large family of quantum chains such as the anisotropic Heisenberg model, Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t – J model, Hubbard model, etc.
A new strongly correlated electron model is presented. This is formed by two types of sites: one where double occupancy is forbidden, as in the t - J model, and the other where double occupancy is allowed but vacancy is not allowed, as in an inverse t - J model. The Hamiltonian shows nearest and next-to-nearest neighbour interactions and it is solved by means of a modified algebraic nested Bethe ansatz. The number of sites where vacancy is not allowed, may be treated as a new parameter if the model is regarded as a t - J model with impurities. The ground and excited states are described in the thermodynamic limit.
Recently it was shown that the eigenfunctions for the the asymmetric exclusion problem and several of its generalizations as well as a huge family of quantum chains, like the anisotropic Heisenberg model, Fateev- Zamolodchikov model, Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc, can be expressed by a matrix product ansatz. Differently from the coordinate Bethe ansatz, where the eigenvalues and eigenvectors are plane wave combinations, in this ansatz the components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. In this work, we introduce a formulation of a matrix product ansatz for the six-vertex model with periodic boundary condition, which is the paradigmatic example of integrability in two dimensions. Remarkably, our studies of the six-vertex model are in agreement with the conjecture that all models exactly solved by the Bethe ansatz can also be solved by an appropriated matrix product ansatz.