Anderson's orthogonality catastrophe

TL;DR

This paper establishes an upper bound on the ground-state overlap decay of two large fermionic systems differing by a potential, confirming Anderson's 1967 prediction of a power-law decay related to scattering cross sections.

Contribution

It provides a rigorous upper bound on the ground-state overlap decay in large fermionic systems, generalizing Anderson's original informal calculation.

Findings

Overlap vanishes in the thermodynamic limit

Decay follows a power-law with system size

Decay exponent relates to scattering cross section

Abstract

We give an upper bound on the modulus of the ground-state overlap of two non-interacting fermionic quantum systems with $N$ particles in a large but finite volume $L^d$ of $d$-dimensional Euclidean space. The underlying one-particle Hamiltonians of the two systems are standard Schr\"odinger operators that differ by a non-negative compactly supported scalar potential. In the thermodynamic limit, the bound exhibits an asymptotic power-law decay in the system size $L$, showing that the ground-state overlap vanishes for macroscopic systems. The decay exponent can be interpreted in terms of the total scattering cross section averaged over all incident directions. The result confirms and generalises P. W. Anderson's informal computation [Phys. Rev. Lett. 18, 1049--1051 (1967)].