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arXiv:cond-mat/9602093 · International Journal of Modern Physics B

INTEGRABLE SU (m|n) SUPERSYMMETRIC ELECTRONIC MODELS OF STRONG CORRELATIONS

JAMES T. LIU, D.F. WANG

Published Dec 15, 1996• 3 citations
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Abstract

We consider the SU (m|n) supersymmetric generalization of the one-dimensional SU (2|2) symmetric extended Hubbard model of 1/r2 interaction. This model, defined on a lattice, is part of a large family of integrable one-dimensional multicomponent systems with long range interactions. We study both uniform and nonuniform lattices and present the ground state wavefunctions and energy spectra of these models. In both cases the spectrum is highly degenerate, with the SU (m|n) supersymmetry explaining most of the degeneracies.

DOI
10.1142/s0217979296002002

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We consider the SU (m|n) supersymmetric generalization of the one-dimensional SU (2|2) symmetric extended Hubbard model of 1/r2 interaction. This model, defined on a lattice, is part of a large family of integrable one-dimensional multicomponent systems with long range interactions. We study both uniform and nonuniform lattices and present the ground state wavefunctions and energy spectra of these models. In both cases the spectrum is highly degenerate, with the SU (m|n) supersymmetry explaining most of the degeneracies.
A one-dimensional electron has both spin and charge degrees of freedom. In the limit of strong on-site repulsion, double occupation of a site can be neglected since it has a large energy cost. If the number N e of electrons is less than the number N of lattice sites, vacant sites appear and the system acquires both spin and charge degrees of freedom. The simplest model to incorporate these degrees of freedom is called the t – J model, which is discussed in this chapter. We begin by reviewing the SU(2,1) supersymmetry (SUSY) as an extension of the SU(2) symmetry in the spin chain. With the supersymmetry, the hopping and exchange in the t – J model can be treated in a unified way as the graded permutation. Then the ground state of the SUSY t – J model is derived in Section 6.2 by the coordinate representation of wave functions by using the fully polarized state as the reference. The static structure factor at zero temperature is completely determined by the wave function at the ground state. The spin and charge components are derived using the determinant and its generalizations in Section 6.3. In Section 6.4, we proceed to derive the spectrum of the SUSY t – J model. The elementary excitations consist of spinons, holons, and their antiparticles. It is proved in Section 6.5 that these excitations span the complete set. The argument relies on a generalization of Young diagrams, which are called ribbon diagrams. At the same time, the degeneracy of energy levels beyond the global SU(2) symmetry is ascribed to the presence of a Yangian supersymmetry Y ( sl 2|1 ).
We continue the study of the u(2|2)-supersymmetric extension of the Hubbard model in one dimension. We determine the excitation spectrum at zero temperature even in the sectors where the ground states are u(2|2)-descendants of Bethe states. The excitations include spinons, holons, electrons, localons (local electron pairs, moving coherently through the lattice) and their bound states. The spectra are found to be very different for repulsive and attractive on-site interaction. We also study the thermodynamics of the model.
The integrability of the one-dimensional long-range supersymmetric t-J model has previously been established for both open systems and those closed by periodic boundary conditions through explicit construction of its integrals of motion. Recently the system has been extended to include the effect of magnetic flux, which gives rise to a closed chain with twisted boundary conditions. While the t-J model with twisted boundary conditions has been solved for the ground state and full energy spectrum, proof of its integrability has so far been lacking. In this paper we extend the proof of integrability of the long-range supersymmetric t-J model and its SU(m|n) generalization to include the case of twisted boundary conditions.
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.