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arXiv:cond-mat/0102211 · Journal of Mathematical Physics

Symmetries in the Hubbard model with n-fold orbital degeneracy

Zu-Jian Ying, You-Quan Li, Shi-Jian Gu

Published Oct 1, 2001• 8 citations

Abstract

The present paper studies the symmetries of the Hubbard model of electrons with generally n-fold orbital degeneracy. It is shown that SUd(2n) and SUc(2n) symmetries hold, respectively, for the model with completely repulsive or attractive on-site interaction and that with partly attractive interactions. An extended Lieb–Mattis transformation is given to map these two symmetries into each other. The subsymmetry SUd(e)(n)⊗SUd(o)(n) is found to be shared by the two models with arbitrary chemical potential μ. By assuming at most two electrons on each site it is found that SUd(2n)P and SUc(2n)P both exist in each kind of the two models and consequently lead to a larger symmetry SUd(2n)P×SUc(2n)P. Another underlying symmetry (SUc(e)(2)P×⋯×SUc(e)(2)P)⊗(SUc(o)(2)P×⋯×SUc(o)(2)P) is also revealed for the unified U model under the excluding. The symmetry is valid for the partially attractive model with chemical potential μ=−U.

DOI: 10.1063/1.1402956

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The present paper studies the symmetries of the Hubbard model of electrons with generally n-fold orbital degeneracy. It is shown that SUd(2n) and SUc(2n) symmetries hold, respectively, for the model with completely repulsive or attractive on-site interaction and that with partly attractive interactions. An extended Lieb–Mattis transformation is given to map these two symmetries into each other. The subsymmetry SUd(e)(n)⊗SUd(o)(n) is found to be shared by the two models with arbitrary chemical potential μ. By assuming at most two electrons on each site it is found that SUd(2n)P and SUc(2n)P both exist in each kind of the two models and consequently lead to a larger symmetry SUd(2n)P×SUc(2n)P. Another underlying symmetry (SUc(e)(2)P×⋯×SUc(e)(2)P)⊗(SUc(o)(2)P×⋯×SUc(o)(2)P) is also revealed for the unified U model under the excluding. The symmetry is valid for the partially attractive model with chemical potential μ=−U.

The Hubbard model for electrons with orbital degeneracy is shown to have an underlying

{\mathrm{SU}}_{d}(4)

symmetry of spin-orbital double. A hidden charge

{\mathrm{SU}}_{c}(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

\mathrm{SO}(6)

Heisenberg magnet at half-filling and a

\mathrm{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.

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.

Abstract Using the strong coupling diagram technique, we study the one-band repulsive Hubbard model on a two-dimensional square lattice in a wide range of chemical potentials μ . Infinite sequences of diagrams describing interactions of electrons with spin and charge fluctuations are taken into account. At low temperatures, regions of the negative electron compressibility are found for μ ≈ 0 and μ ≈ U , where U is the Hubbard repulsion. They are manifestations of non-rigid electron bands, which change drastically in the mentioned regions of chemical potentials due to the level crossing in the Hubbard atom.

In this paper we investigate the Hubbard model, with S-orbits on each site, for one-, two- and three-dimensional isotropic lattices with arbitrary values of transfer and on-site repulsion parameters. The method of diagonalization is based on equivalences due to Stevens for number conserving operators and a unitary transformation utilizing permutational symmetries of the problem. We obtained simple analytical formulas for single particle eigenstates and somewhat more involved expression for determinantal multi-electron states. Numerical calculations have been performed which demonstrate the boundaries of the lowest band for various filling factors using examples in one- and two-dimensions. The states within the band have also been identified and degeneracies evaluated numerically. These computations were based on a Monté Carlo technique in conjunction with particle conservation requirements. Agreement with some well-known limiting cases has been found.

We rigorously prove that an extended Hubbard model with attraction in two dimensions has an unconventional pairing ground state for any electron filling. The anisotropic spin-0 or anisotropic spin-1 pairing symmetry is realized, depending on a phase parameter characterizing the type of local attractive interactions. In both cases the ground state is unique. It is also shown that in a special case, where there are no electron-hopping terms, the ground state has Ising-type N\'eel order at half-filling, when on-site repulsion is furthermore added. Physical applications are mentioned.