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Construction of density operator for a general mean-field Hamiltonian and its application t
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Construction of density operator for a general
mean-field Hamiltonian and its application to the
models of correlated fermions
Jakub Je?drak, Jozef Spa lek
Marian Smoluchowski Institute of Physics, Jagiellonian University,
Reymonta 4, 30-059 Krako?w, Poland
Abstract
We analyze a class of the mean-field lattice-fermion Hamiltonians
and construct the corresponding grand-canonical density operator for
such a model system. New terms are introduced, which may be in-
terpreted as local fugacities, molecular fields, etc. The presence of
such terms is an unavoidable consequence of the consistent statistical
description. Although in some cases (e.g. the Hartree or the Hartree-
Fock type of approximations) the presented formalism is redundant,
in general (e.g. for a renormalized t-J model) it leads to nontrivial
modifications. The case of zero temperature is also briefly analyzed.
PACS: 05.30.-d, 71.10.Fd, 75.10.Jm.
1 Introduction
For most of realistic models of quantum many-particle systems an exact solu-
tion cannot be obtained. This is a typical situation in the condensed matter
or nuclear physics. As a consequence, a number of approximate methods
have been developed. Among them, widely used are various types of the
so-called mean-field approaches.
By mean-field (MF) description of the problem we understand making
use of the Hamiltonian which depends on some extra parameters, having the
meaning of expectation values of well-defined operators. Those parameters
are not a priori known and are to be determined. The Hartree-Fock method
is a good example of such approach.
Mean field methods often provide us with a valuable insight into the
physical properties of the system under consideration. They are capable
of describing many interesting phenomena, especially those involving phase
transitions, such as that from a normal metal to superconductor (in the form
of the Bardeen-Cooper-Schrief
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