Orthogonality Catastrophe Revisited

TL;DR

This paper revisits the orthogonality catastrophe in many-body systems, providing a simple theorem, a new derivation of the OC exponent, and a method to calculate XPS lineshapes, discussing conditions where OC fails.

Contribution

It offers a straightforward theorem on wavefunction overlaps, a compact derivation of the OC exponent, and a novel approach to compute XPS lineshapes from groundstate energy scaling.

Findings

Derived a simple theorem on wavefunction overlap and OC.

Provided a new derivation of the OC exponent in free electron gases.

Introduced a method to calculate XPS lineshapes from finite size scaling.

Abstract

We prove a simple theorem on the overlap of the wavefunctions of a manybody system with and without a single impurity and show how, and under which conditions, this leads to the ``Orthogonality Catastrophe'' (OC) described by Anderson. A compact new derivation of the known result for the OC exponent in a free electron gas, $\alpha=2 \sum_l(2l+1) (\delta_l(\epsilon_f)/\pi)^2 $ %$\alpha = 2\sum_l [2l+1] (\delta_l(\epsilon_f)/\pi)^2$, is given. We also introduce a simple way of calculating core level photoemission (XPS) lineshapes from the finite size scaling of groundstate energies, reproducing and extending results previously derived using boundary conformal field theory. Different conditions under which the OC fails are discussed, and simple physical predictions for XPS lineshapes made in each case.