Anderson's Orthogonality Catastrophe for One-dimensional Systems

TL;DR

This paper rigorously derives the asymptotic behavior of the Anderson integral in one-dimensional Fermi systems, relating the decay of ground state overlaps to the S-matrix, thus providing bounds on the orthogonality catastrophe exponent.

Contribution

It provides a rigorous derivation of the asymptotics of the Anderson integral and computes the coefficient in terms of the S-matrix for 1D Fermi systems.

Findings

Derived leading asymptotics of Anderson integral

Computed the coefficient in terms of the S-matrix

Established bounds on the orthogonality catastrophe exponent

Abstract

We derive rigorously the leading asymptotics of the so-called Anderson integral in the thermodynamic limit for one-dimensional, non-relativistic, spin-less Fermi systems. The coefficient, $\gamma$, of the leading term is computed in terms of the S-matrix. This implies a lower and an upper bound on the exponent in Anderson's orthogonality catastrophe, $\tilde CN^{-\tilde\gamma}\leq \mathcal{D}_N\leq CN^{-\gamma}$ pertaining to the overlap, $\mathcal{D}_N$, of ground states of non-interacting fermions.