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, $I(\epsilon )\propto {\epsilon }^{-\alpha }$, 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 ${\epsilon }^{\ast }\sim [U(2k_F){]}^{1/\gamma }$, where $\gamma$ is a dimensionless measure of the electron-electron interactions.

8 pages + 1 postscript figure, preprint TPI-MINN-93/40-T