The exponent in the orthogonality catastrophe for Fermi gases

TL;DR

This paper rigorously quantifies how the ground-state overlap of two Fermi gases diminishes in the thermodynamic limit, linking the decay rate to scattering theory and confirming a longstanding prediction by Anderson.

Contribution

It provides a precise power-law bound on the ground-state overlap decay and expresses the decay exponent in terms of the transition matrix at the Fermi energy.

Findings

Decay exponent expressed as a Hilbert-Schmidt norm involving the transition matrix.

The decay exponent matches Anderson's prediction for point-like perturbations.

Provides a rigorous mathematical framework for the orthogonality catastrophe in Fermi gases.

Abstract

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 Schr\"odinger 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 $N$-particle systems. We interpret the decay exponent $\gamma$ in terms of scattering theory and find $\gamma = \pi^{-2}{\lVert\arcsin{\lvert T_E/2\rvert}\rVert}_{\mathrm{HS}}^2$, where $T_E$ is the transition matrix at the Fermi energy $E$. This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352-359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.