The exponent in the orthogonality catastrophe for Fermi gases (Q329711)

Summary: We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in \(d\)-dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting \(N\)-Fermion systems. We interpret the decay exponent \(\gamma\) in terms of scattering theory and find \(\gamma=\pi^{-2}\|\arcsin|T_E/2|\|_{\mathrm {HS}}^2\), where \(T_E\) is the transition matrix at the Fermi energy \(E\). This exponent reduces to the one predicted by \textit{P. W. Anderson} [``Ground state of a magnetic impurity in a metal'', Phys. Rev. 164, No. 2, 352--359 (1967;: \url{doi:10.1103/PhysRev.164.352})] for the exact asymptotics in the special case of a repulsive point-like perturbation.