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Europhysics Letters (EPL)

Exact solution of a generalized Gross-Neveu-Thirring model

P.-A. Bares

Published Jul 1, 1999• 0 citations
Publisher

Abstract

We propose a model that describes approximately the strong-coupling limit of interacting electrons and holes in a one-dimensional quantum wire. An exact solution for this model at the SU ( n ) symmetric point of equal electron and hole masses and for a particular ratio of the forward- ( f ) to backward-scattering ( g ) is obtained by means of the Bethe ansatz. The model discussed is a hybrid between the chiral invariant Gross-Neveu model of quantum chromodynamics and the massive Thirring model. While the eigenstates exhibit a huge degeneracy in colour (spin) space as in the U → ∞ Hubbard model, the charge excitation spectrum comprizes excitons, bi-excitons and other electron-hole complexes (droplets) for not too repulsive interactions ( − 2/√3 ⩽ g ⩽ − 1). In the regime of strong repulsion ( g < − 2/√3), the bi-excitons and large droplets disappear from the spectrum.

DOI
10.1209/epl/i1999-00352-3

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A Hubbard-like model with SU(4) symmetry for electrons with twofold orbital degeneracy is studied extensively. Exact solution in one dimension is derived by means of Bethe ansatz, where the sites are supposed to be occupied by at most two electrons. The features of ground state and excited states for repulsive coupling are shown. For finite NNN number of electrons, the configurations of quantum numbers are given explicitly and the spectra of excitations are obtained by solving Bethe-ansatz equation numerically. For infinite N,N,N, the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.
A one-dimensional gas mixture consisting of bosons and fermions without spin interacting via a repulsive [Formula: see text]-function potential is considered. The model is integrable and soluble via two nested Bethe ansatz, if all particles are assumed to have equal masses and the interaction strength between the bosons and among the bosons and fermions is the same. The low energy excitation spectrum is a two-component Luttinger liquid and can be parametrized by a conformal field theory with conformal charges c = 1. In the low-energy limit, where the band curvature terms in the dispersion can be neglected, the linear dispersion of a Luttinger liquid is asymptotically exact. The spectral function, however, displays deviations from the Luttinger behavior for higher energy excitations. In the neighborhood of the single-particle (hole) energy, the spectral function is represented by an effective X-ray edge type model. Expressions of the critical exponents for the single-hole Green’s function are obtained using the Bethe ansatz solution in the limit of the bosonic gas. The results could be of relevance in the context of ultracold atoms confined to an elongated optical trap.
We consider an integrable model consisting of two one-dimensional parabolic bands of opposite mass (m=+or- 1/2 , respectively) separated by a gap, 2 Delta . The bands contain spinless fermions, one band corresponds to the conduction band and the other one to the valence band of a semiconductor or semimetal. The holes in the valence band and the spinless electrons in the conduction band are locally attracted via a delta -function potential. This model can be mapped onto the two-component Fermi gas with delta -function interaction, so that the two components label the bands: the chemical potential corresponds to a magnetic field, and Delta to the chemical potential. We use Gaudin and Yang's exact solution of the many-body problem to study the formation of exciton bands. The properties of the ground and metastable states, the excitation spectrum and the thermodynamics of the model are obtained. In the ground or metastable states the particles (electrons in the conduction band and holes in the valence band) are either paired in exciton bound states or unpaired. Their spectrum of elemental excitations is approximately parabolic. At finite T many-particle bound states (string solutions of the Bethe ansatz equations) can be populated; at low T these states are strongly delocalized and can be interpreted as electron-hole droplets. The low-T properties of the model are discussed.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.