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World Scientific series in 20th century physics

THE GENERALIZED MEAN FIELD EXPANSIONS FOR MANY FERMION SYSTEMS

Zhao-bing SU, Lu Yu, Guangzhao Zhou

Published May 1, 2009• 1 citation
Publisher

Abstract

The set of coupled, self-consistent equations for the order parameter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green's functions (CTPGF) is applied to systems with four-fermion interaction. The Hartree, the Hartree-Pock (HP) and the random phase approximations (RPA) for the order parameter and its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here can be applied to nuclear as well as other many fermion systems in both equilibrium and nonequilibrium states.

DOI
10.1142/9789814280389_0122

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The set of coupled, self-consistent equations for the order parameter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green's functions (CTPGF) is applied to systems with four-fermion interaction. The Hartree, the Hartree-Pock (HP) and the random phase approximations (RPA) for the order parameter and its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here can be applied to nuclear as well as other many fermion systems in both equilibrium and nonequilibrium states.
The set of coupled, self-consistent equations for the order parameter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green's functions (CTPGF) is applied to systems with four-fermion interaction. The Hartree, the Hartree-Pock (HP) and the random phase approximations (RPA) for the order parameter and its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here can be applied to nuclear as well as other many fermion systems in both equilibrium and nonequilibrium states.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.
In this article, we revisit the question of the validity of Hartree-Fock and random-phase approximations. We show that there is a connection between the two and while the RPA as it is known in much of the physics literature is of limited validity, there is a generalised sense in which the random phase approximation is of much wider applicability including to systems that do not possess Fermi surfaces. The main conclusion is that the Hartree-Fock approximation is a mean-field idea applied to the density operator, and the random-phase approximation is a mean-field idea applied to the number operator. The generalised RPA is used to compute single-particle properties such as momentum distribution and spectral functions. It is found that we have to go beyond the generalised RPA and include fluctuations in the momentum distribution in order to recover a nonzero imaginary part of the one-particle self energy, which is also explicitly computed, all this in any number of spatial dimensions and no bosonization is needed.
A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions d>2d>2d>2. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions d>1d>1d>1, in agreement with other recent field-theoretical treatments of clean Fermi liquids.