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

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, $I(\epsilon) \propto \epsilon^{-\alpha}$, with $\alpha = 3/8 + \delta_+/\pi - \delta_+^2/2\pi^2$ where $\delta_+$ is the forward scattering phase shift of the core hole. In contrast to previous theories, $\alpha$ 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 $\alpha = \delta_+/\pi - \delta_+^2/2\pi^2$ above an energy scale $\epsilon^* \sim [U(2k_F)]^{1/\gamma}$, where $\gamma$ is a dimensionless measure of the electron-electron interactions.