Beyond Random Phase Approximation in electron-hole bilayer superfluidity

TL;DR

This paper extends the analysis of electron-hole bilayer superfluidity by including first-order corrections beyond RPA, showing that RPA remains a good approximation for screening and pairing across a range of densities.

Contribution

It derives the normal and anomalous polarization functions with first-order corrections, revealing their limited impact on screening and pairing in superfluid electron-hole bilayers.

Findings

Screening is negligible at low density due to normal-anomalous cancellation.

First-order corrections increase the normal-anomalous difference at high momenta.

RPA remains a valid approximation for superfluid screening up to high densities.

Abstract

We derive the normal and anomalous proper polarization functions and the screened Coulomb interactions in a two-dimensional superfluid electron-hole bilayer, including all first-order corrections beyond the Random Phase Approximation (RPA). This requires a modification of the perturbation method as first noted by Nozi\`eres and Schrieffer [1, 2]. We discuss the physical origin and magnitude of the first-order corrections in a superfluid system with long-range Coulomb interactions. Unlike conventional superconductivity, Migdal's theorem does not apply here, so exchange vertex corrections cannot be neglected. The screened electron-electron, hole-hole, and electron-hole interactions in the superfluid state are evaluated as functions of the carrier density. We find that at low density, the strong cancellations between the normal and anomalous components that make screening of the interactions negligible, apply not only within RPA but also with the first-order corrections included. As the density is increased, the normal-anomalous cancellation weakens and screening becomes increasingly significant. We find that the first-order corrections amplify the normal-anomalous difference but only at large momenta exchanged in the two-particle scattering, so their effect on the interactions remains modest. We conclude that the superfluid state RPA is an excellent approximation for the screening and for the effective electron-hole pairing in this superfluid system over the range of densities up to the maximum of the superfluid gap.