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理论物理通讯：英文版

The Binding Energy, Spin-Excitation Gap, and Charged Gap in the Boson-Fermion Model

Yang, Kai-Hua, Tian, Guang-Shan, Han, Ru-Qi

Published Jan 1, 2003• 0 citations
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Abstract

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 boeon-fermion Hamiltonlan, 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 couping 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-palred electron theory.

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The Binding Energy, Spin-Excitation Gap, and Charged Gap in the Boson-Fermion Model

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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.

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Lieb’s spin-reflection-positivity method, Lieb lattice and all that

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In the present chapter we review some rigorous results derived with Lieb’s spin-reflectionpositivity method in a pedagogical manner. To begin with, we emphasize the important role played by the sign rule satisfied by the ground-state wave function of a quantum many-body system. For example, we study some localized spin systems, such as the antiferromagnetic Heisenberg model, in detail. Then we argue why such a rule is difficult to establish for the ground-state wave functions of itinerant electron models. By using the spin-reflection-positivity method, we show that this problem can be partially remedied. A weaker form of the sign rule is uncovered. It enables us to prove several inequalities satisfied by either an electron pairing correlation function or a magnetic correlation function of the Hubbard model on a bipartite lattice. These inequalities imply that, if a bipartite lattice has macroscopically unequal numbers of sublattice sites, then the Hubbard model at half-filling is either a supersolid or a ferrimagnet. Similar inequalities can also be proven for the periodic Anderson model and the Kondo lattice model. Finally, by applying a generalized version of the spin-reflection-positivity method, we further establish an inequality satisfied by the charged and spin gaps of these models. Based on them, we make some audacious conjectures, and then we summarize this chapter.

EMS Press eBooks•Jun 29, 2022•0 citations
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 boeon-fermion Hamiltonlan, 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 couping 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-palred electron theory.
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.
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.
In the present chapter we review some rigorous results derived with Lieb’s spin-reflectionpositivity method in a pedagogical manner. To begin with, we emphasize the important role played by the sign rule satisfied by the ground-state wave function of a quantum many-body system. For example, we study some localized spin systems, such as the antiferromagnetic Heisenberg model, in detail. Then we argue why such a rule is difficult to establish for the ground-state wave functions of itinerant electron models. By using the spin-reflection-positivity method, we show that this problem can be partially remedied. A weaker form of the sign rule is uncovered. It enables us to prove several inequalities satisfied by either an electron pairing correlation function or a magnetic correlation function of the Hubbard model on a bipartite lattice. These inequalities imply that, if a bipartite lattice has macroscopically unequal numbers of sublattice sites, then the Hubbard model at half-filling is either a supersolid or a ferrimagnet. Similar inequalities can also be proven for the periodic Anderson model and the Kondo lattice model. Finally, by applying a generalized version of the spin-reflection-positivity method, we further establish an inequality satisfied by the charged and spin gaps of these models. Based on them, we make some audacious conjectures, and then we summarize this chapter.