Beyond Random Phase Approximation in electron-hole bilayer superfluidity

TL;DR

This work develops a beyond-RPA framework for screening in a two-dimensional electron-hole bilayer superfluid by computing both normal and anomalous polarization functions to first order around a BCS-like ground state. It shows that Hartree/self-energy effects cancel due to the chosen mean-field ground state, leaving only exchange corrections, which dominate the beyond-RPA contributions. At low density, strong cancellations between normal and anomalous channels keep screening negligible, while at higher density the corrections grow with momentum transfer but still modestly affect the effective electron-hole interactions, preserving the reliability of RPA up to the maximum superfluid gap. The results establish that RPA is an excellent approximation for screening and pairing in this system across the physically relevant density range, with first-order corrections becoming more relevant only at large momenta and high density where the gap is already limited.

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 Nozières 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.

Figures (4)

• Figure 1: (a) Normal zero-order polarization function $\Pi_0^N$ (Eq. \ref{['P0N']}, blue solid line). Normal first-order polarization function $\Pi_1^N$ (red solid line) with components: $\Pi_{1,ee}^{N,EX}$, $\Pi_{1,hh}^{N,EX}$ (Eq. \ref{['P1eeNEX']}-\ref{['P1hhNEX']}, green dashed line), $\Pi_{1,eh}^{N,EX}$ (Eq. \ref{['P1ehNEX']}, magenta dashed line). (b) Anomalous zero-order polarization function $\Pi_0^A$ (Eq. \ref{['P0A']}, blue solid line). Anomalous first-order polarization function $\Pi_1^A$ (red solid line) with components: $\Pi_{1,ee}^{A,EX}$, $\Pi_{1,hh}^{A,EX}$ (Eq. \ref{['P1eeAEX']}, green dashed line), $\Pi_{1,eh}^{A,EX}$ (Eq. \ref{['P1ehAEX']}, magenta dashed line). Interparticle distance $r_0=10.0a_B^*$. All the curves are in unit of $\Pi_0^N(\mathbf{q})$.
• Figure 2: (a) Normal zero-order polarization function $\Pi_0^N$ (Eq. \ref{['P0N']}, blue solid line). Normal first-order polarization function $\Pi_1^N$ (red solid line) with its components as in Fig. \ref{['Fig_1']}. (b) Anomalous zero-order polarization function $\Pi_0^A$ (Eq. \ref{['P0A']}, blue solid line). Anomalous first-order polarization function $\Pi_1^A$ (red solid line) with its components as in Fig. \ref{['Fig_1']}. Interparticle distance $r_0=2.5a_B^*$. All the curves are in unit of $\Pi_0^N(\mathbf{q})$.
• Figure 3: The differences $(\Pi_0^N-\Pi_0^A)$ (dashed lines) and $\Pi_\textrm{tot}^N-\Pi_\textrm{tot}^A\equiv \Pi_0^N+\Pi_1^S-\Pi_0^A-\Pi_1^D$ (solid lines) in units of $\Pi_0^N(q)$. Red curves $r_0=2.5a_B^*$ and blue curves $r_0=10a_B^*$. All the curves are in unit of $\Pi_0^N(q)$.
• Figure 4: (a) Ratio of the electron-electron screened interaction to the bare Coulomb interaction: solid lines correspond to the zero-order (RPA) result, and dashed lines to the first-order screened interaction. The inset shows the ratio between the zero-order and first-order screened interactions. Results are shown for $r_0=10a_B^*$ (blue) and $r_0=2.5a_B^*$ (red). (b) Same as in panel (a), but for the electron-hole interaction.