Exact Single-Particle Green Functions of Fermi Systems Without Using Bosonization or Are Hartree-Fock and Random Phase Approximations 'Controlled Approximations' ?

TL;DR

This paper critically examines the validity of Hartree-Fock and random-phase approximations, extending the applicability of RPA beyond Fermi surfaces and computing single-particle properties without bosonization.

Contribution

It demonstrates that generalized RPA can be applied to a wider class of systems and computes key single-particle properties without relying on bosonization or controlled approximations.

Findings

RPA's limited validity in traditional contexts.

Generalized RPA applies to systems without Fermi surfaces.

Including fluctuations in momentum distribution yields a nonzero imaginary self-energy.

Abstract

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.