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Physical review. B, Condensed matter

Algebraic Bethe ansatz for the supersymmetricUmodel

Katrina Hibberd, M. D. Gould, Jon Links

Published Sep 15, 1996• 18 citations
Publisher

Abstract

We present an algebraic Bethe ansatz for the supersymmetric U model for correlated electrons on the unrestricted ${4}^{\mathit{L}}$-dimensional electronic Hilbert space ${\ensuremath{\bigotimes}}_{\mathit{L}}^{\mathit{n}=1}$${\mathit{C}}^{4}$ (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.

DOI
10.1103/physrevb.54.8430

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A new lattice model is presented for correlated electrons on the unrestricted 4L{4}^{L}4L-dimensional electronic Hilbert space ${\ensuremath{\bigotimes}}_{n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{L}{\mathbf{C}}^{4}$ (where LLL is the lattice length). It is a supersymmetric generalization of the Hubbard model, but differs from the extended Hubbard model proposed by Essler, Korepin, and Schoutens. The supersymmetry algebra of the new model is superalgebra gl(2 $|$1). The model contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter UUU, and has its origin here in the one-parameter family of inequivalent typical 4-dimensional irreducible representations of gl(2 $|$1). On a one-dimensional lattice, the model is exactly solvable by the Bethe ansatz.
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.