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Journal of Physics A Mathematical and General

A rigorous study on the charged gap in strongly correlated electron clusters

Guang-Shan Tian, Jigang Wang

Published Jan 21, 2002• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/35/4/306

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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.
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.
In this paper, by applying a simplified version of Lieb's spin-reflection-positivity method, which was recently developed by one of us [G.S. Tian and J.G. Wang, J. Phys. A: Math. Gen. 35 (2002) 941], we investigate some general properties of the boson-fermion Hamiltonian, which has been widely used as a phenomenological model to describe the real-space pairing of electrons. On a mathematically rigorous basis, we prove that for either negative or positive coupling V, which represents the spontaneous decay and recombination process between boson and fermion in the model, the pairing energy of electrons is nonzero. Furthermore, we also show that the spin-excitation gap of the boson-fermion Hamiltonian is always larger than its charged gap, as predicted by the pre-paired electron theory.