Fermi Edge Singularities and Backscattering in a Weakly Interacting 1D Electron Gas

Abstract

The photon-absorption edge in a weakly interacting one-dimensional electron gas is studied, treating backscattering of conduction electrons from the core hole exactly. Close to threshold, there is a power-law singularity in the absorption, $I(ε) \propto ε^{-α}$, with $α= 3/8 + δ_+/π- δ_+^2/2π^2$ where $δ_+$ is the forward scattering phase shift of the core hole. In contrast to previous theories, $α$ is finite (and universal) in the limit of weak core hole potential. In the case of weak backscattering $U(2k_F)$, the exponent in the power-law dependence of absorption on energy crosses over to a value $α= δ_+/π- δ_+^2/2π^2$ above an energy scale $ε^* \sim [U(2k_F)]^{1/γ}$, where $γ$ is a dimensionless measure of the electron-electron interactions.