Exact Calculation of the Orthogonality Catastrophe in Metals

Abstract

We give exact calculation, using the linked-cluster theorem, of the overlap between the ground-state wave functions of a system of $N$ fermions in the absence and in the presence of a localized potential. It is found that the overlap vanishes as $N^{-\alpha }$, where $\alpha \left(\delta \right)$ contains both a nonperiodic and a $\pi$-periodic term in the phase shift $\delta \left({\epsilon }_{\mathrm{F}}\right)$ due to the potential. The present result reduces to that of Anderson and of Nozières and de Dominicis for small phase shifts.