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arXiv:cond-mat/0001297 · Physical review. B, Condensed matter

One-dimensional model for doubly degenerate electrons

You‐Quan Li, Shi-Jian Gu, Zu‐Jian Ying, Ulrich Eckern

Published Aug 1, 2000• 6 citations
arXivHTMLPDF

Abstract

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 $N$ 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 $N,$ the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.

DOI
10.1103/physrevb.62.4866

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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.
The Hubbard model for electrons with orbital degeneracy is shown to have an underlying SUd(4){\mathrm{SU}}_{d}(4)SUd​(4) symmetry of spin-orbital double. A hidden charge SUc(4){\mathrm{SU}}_{c}(4)SUc​(4) symmetry is exposed and an extended Lieb-Mattis transformation, which maps these two symmetries into each other, is given. On the basis of elementary degenerate perturbative theory, it is shown that the system with strong repulsive coupling is equivalent to a SO(6)\mathrm{SO}(6)SO(6) Heisenberg magnet at half-filling and a SU(4)\mathrm{SU}(4)SU(4) one at quarter-filling. The band is half-filled at all temperature for $\ensuremath{\mu}=3U/2.$ The features of ground-state and low-lying excitations in one dimension are indicated according to exact solutions.
We propose a novel characterization of the exact solution of the one-dimensional Hubbard model with unparallel boundary magnetic fields. The non-diagonal boundary terms break the U(1) symmetry of the system and the number of electrons with fixed spin is not conserved. Thus it is difficult to study the physical quantities in the thermodynamic limit based on the previously obtained inhomogeneous T−Q relation and Bethe ansatz equations. In order to solve this problem, we propose the t−W scheme. The basic idea is that the eigenvalues of the transfer matrix can be expressed by its zero-roots instead of the usual Bethe roots. This method allows us to study thermodynamic properties of the system such as the ground state, surface energy, charge and spin excitations. The t−W scheme is universal and can be applied to the systems either with or without U(1) symmetry.
The one-dimensional Heisenberg model in the presence of orbital degeneracy is studied at the SU(4)\mathrm{SU}(4)SU(4) symmetric point by means of Bethe ansatz. Following Sutherland's previous work on an equivalent model, we discuss the ground state and the low-lying excitations more extensively in connection to the spin systems with orbital degeneracy. We show explicitly that the ground state is a SU(4)\mathrm{SU}(4)SU(4) singlet. We study the degeneracies of the elementary excitations and the spectra of the generalized magnons consisting of these excitations. We also discuss the complex 2-strings in the context of the Bethe ansatz solutions.
The authors considers the one-dimensional SU(N)-invariant t-J model which consists of electrons with N spin components on a lattice with nearest-neighbour hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighbouring lattice sites. The model is integrable at the supersymmetric point t=J. The ground state Bethe ansatz equations are analysed and solved numerically for arbitrary band filling and several values of N. The ground state energy, the chemical potential and the spin susceptibility are obtained as a function of band filling. The elemental charge and spin excitations are derived for arbitrary N and band filling. The energy of the charge excitations vanishes at the Fermi surface. The Fermi velocity has a maximum as a function of band filling, vanishing for the empty and full bands. The spinwave velocity is inversely proportional to the susceptibility. For exactly one electron per site the charge fluctuations are suppressed and the Bethe ansatz equations map onto those of the SU(N)-invariant Heisenberg chain.
The thermodynamic Bethe-ansatz equations for the one-dimensional Hubbard model are derived for an attractive on-site interaction of the electrons, extending in this way Takahashi's result for the repulsive Hubbard chain. The results are discussed in several special limits, e.g., interaction much larger than the hopping, U\ensuremath{\rightarrow}0 case, and the high- and low-temperature limits. The dominating states at low temperatures consist of spin-paired electrons (resembling Cooper pairs) and bound states of such pairs. The excitation spectrum and its properties are discussed in the context of superconductivity and resonant valence bonds.
We consider the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N (spin and orbital degrees of freedom). This model is integrable and has several unusual properties at low temperatures. The Bethe-Ansatz equations at T=0 are analyzed in the thermodynamic limit in the absence of external fields. In the continuum limit, the effective interaction between the charge degrees of freedom corresponds to a potential of the form [sinh(ax)${]}^{\mathrm{\ensuremath{-}}2}$, where x is the distance between the particles involved and a is an inverse length scale. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty} and in the continuum limit, the charges reduce to a Bose gas interacting via a \ensuremath{\delta}-function potential. We further address here the properties of the charge degrees of freedom for a band filling close to one electron per site. The charge excitations obey Fermi statistics. We find a Mott metal-insulator transition at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​ of the Coulomb repulsion. Uc{\mathit{U}}_{\mathit{c}}Uc​ depends on N (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). A qualitative change in the charge-rapidity distribution is found at Uc{\mathit{U}}_{\mathit{c}}Uc​. The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​, diverges as U\ensuremath{\rightarrow}Uc{\mathit{U}}_{\mathit{c}}Uc​, and vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​.