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International Journal of Modern Physics B

HARTREE–FOCK–BOGOLUBOV APPROXIMATION IN THE MODELS WITH GENERAL FOUR-FERMION INTERACTION

Nikolay N. Bogolubov, Andrey Vladimirovich Soldatov

Published Feb 28, 1996• 3 citations
Publisher

Abstract

An approach is proposed which allows one to investigate dynamical and thermodynamical properties of models with four-fermion interaction of general type. The approach combines ideas of the standard Bogolubov's approximating Hamiltonian method for the models with separable interaction with the method of Hartree–Fock approximation based on the ideas of self-consistency.

DOI
10.1142/s0217979296000246

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An approach is proposed which allows one to investigate dynamical and thermodynamical properties of models with four-fermion interaction of general type. The approach combines ideas of the standard Bogolubov's approximating Hamiltonian method for the models with separable interaction with the method of Hartree–Fock approximation based on the ideas of self-consistency.
We consider the Bogolubov–Hartree–Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose–Einstein condensate of fermion pairs. The latter can be described by means of the Gross–Pitaevskii energy functional.
We present numerical methods to solve the generalized Hartree–Fock theory for fermionic systems in lattices, both in thermal equilibrium and out of equilibrium. Specifically, we show how one can determine the covariance matrix corresponding to the fermionic Gaussian state that optimally approximates the quantum state of fermions. The methods apply to relatively large systems, because their complexity only scales quadratically with the number of lattice sites. Moreover, they are specially suited to describe inhomogeneous systems, as those typically found in recent experiments with atoms in optical lattices, at least in the weak interaction regime. As a benchmark, we have applied them to the two-dimensional Hubbard model on a 10 × 10 lattice with and without external confinement.
The students interested in the study of strongly correlated electron systems normally feel, in the beginning  of their studies, a great difficulty in understanding the complex correlations of these systems. We consider  that for these students it is very important to have texts that analyze pedagogically the theoretical research  methods of the subject. We show how to determine the thermodynamic properties of the Hubbard model  for the two-dimensional square lattice, utilizing the Hartree-Fock approximation. All the procedures and  peculiarities which lead to the determination of the thermodynamic properties and to the comprehension  of the meaning of the Hartree-Fock approximation applied to models of strongly correlated electron  systems are pedagogically presented in detail. We obtain the magnetization, the magnetic susceptibility,  the internal energy, the specific heat, and also a magnetic phase diagram of the studied system. Our  approach clearly shows that the use of the Hartree-Fock approximation, in this case, is very simple.
The hierarchy equations for the quantum thermal density matrices of an assembly of interacting particles in an external potential are derived in a compact form. This approach can complement and finally replace the phenomenological treatment of the thermodynamic properties of quantum gasses, liquids, and solids. We also derive a temperature Hartree-Fock approximation for fermions based on the symmetry properties of the exact hierarchy and finally we discuss the present approximation in relation to that of Kaplan and Argyres [T. A. Kaplan and P. N. Argyres, Ann. Phys. (N.Y.) 92, 1 (1975)].