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Journal of Physics Condensed Matter

Excitation gaps in the orbitally degenerate Hubbard model

Maria Emilia Amendola, Canio Noce

Published Aug 18, 2006• 5 citations
Publisher

Abstract

The excitation gaps for the two-orbital degenerate Hubbard model are investigated by applying a generalized version of Lieb's spin-reflection positivity. Combining the known exact results on the ground state, and making use of symmetry properties, we rigorously show that, at half-filling, the charge gaps are always larger than the spin-excitation gaps and properly defined orbital gaps.

DOI
10.1088/0953-8984/18/35/019

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The excitation gaps for the two-orbital degenerate Hubbard model are investigated by applying a generalized version of Lieb's spin-reflection positivity. Combining the known exact results on the ground state, and making use of symmetry properties, we rigorously show that, at half-filling, the charge gaps are always larger than the spin-excitation gaps and properly defined orbital gaps.
The relationship between charged and spin-excitation gaps of the half-filled Hubbard model, the symmetric periodic Anderson model, and the Kondo lattice model is considered for a general ddd-dimensional bipartite lattice. In a previous paper [G. S. Tian, Phys. Rev. B 58, 7612 (1998)], it was shown that the quasiparticle gaps of these models at half filling are always larger than their spin excitation gaps. In the present paper, we establish a theorem, which states that the charged gap is also greater than the corresponding spin gap in these models. This conclusion, which has been reached previously for one-dimensional systems based on variational and the density-matrix renormalization-group numerical calculations, is thus put on a rigorous and more general footing.
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.
The two-orbital degenerate Hubbard model proves to be a powerful tool in the investigation of several 3 d and 4 d transition metal oxides where orbital degeneracy is known to play a crucial role. We present here a finite-size cluster study of this model where an exact numerical diagonalization procedure is used, based on the implementation of the symmetries generated by the spin, the pairing and the orbital pseudospin operators. The technique is then applied to the solution of the model on a four-site ring, and an explicit comparison is presented between the behavior of the spin, charge and orbital gaps as the on-site Coulomb repulsion is varied.
In this paper, we study a revised definition c of the charged gap for the strongly correlated electron models on small clusters, proposed by Nishino, in a mathematically rigorous way. By applying a simplified version of Lieb's spin-reflection-positivity method, we show that this quantity is always positive for the half-filled Hubbard model, the periodic Anderson model and the Kondo lattice model. We also establish a model-dependent lower bound to the charged gap. Our results show explicitly the role played by electron repulsion in opening up a nonvanishing charged gap of a cluster.