Self-Consistent Random Phase Approximation from Projective Truncation Approximation Formalism

TL;DR

This paper derives a self-consistent RPA framework from the PTA formalism, applicable at arbitrary temperatures, and demonstrates its effectiveness on a 1D spinless fermion model, capturing key physical features.

Contribution

It introduces a general method to extend self-consistent RPA using the PTA formalism, unifying and expanding previous approaches.

Findings

Ground state energy and spectral functions agree with existing results.

Successfully captures Luttinger liquid features and spectral continuum.

Framework applicable to disordered ground states at finite temperature.

Abstract

We derive the self-consistent random phase approximations (sc-RPA) from the projective truncation approximation (PTA) for the equation of motion of two-time Green's function. The obtained sc-RPA applies to arbitrary temperature and recovers the Rowe's formalism at zero temperature. The PTA formalism not only rationalize Rowe's formula, but also provides a general framework to extend sc-RPA. We implement the sc-RPA calculation for the one-dimensional spinless fermion model in the parameter regime of disordered ground state, with the N-representability constraints enforced. The obtained ground state energy, correlation function, and density spectral function agree well with existing results. The features of the Luttinger liquid ground state and the continuum/bound state in the spectral function are well captured. We discuss several issues concerning the approximations made in RPAs, difficulties of RPA for symmetric state, and the static component problem of PTA.