Explicit proof of Anderson's orthogonality catastrophe for the one-dimensional Fermi polaron with attractive interaction

TL;DR

This paper analytically proves Anderson's orthogonality catastrophe for a one-dimensional Fermi polaron model, showing the quasi-particle residue decays algebraically with system size and confirming the relation of the decay exponent to the phase shift.

Contribution

It provides a fully analytical derivation of the orthogonality catastrophe in an integrable 1D Fermi polaron model using Bethe ansatz and Cauchy matrix properties.

Findings

Quasi-particle residue Z decays as W N^{- heta} in the thermodynamic limit.

The Anderson exponent θ equals 2δ_F^2/π^2, linking decay to phase shift.

The prefactor W is numerically obtained as a function of interaction.

Abstract

We provide a fully analytical derivation of Anderson's orthogonality catastrophe for the one dimensional Fermi polaron integrable model, describing a system of $N$ spin-up fermions, with fixed density $n=N/L$, interacting with a single spin-down fermion via an attractive contact potential. The proof combines the determinant representations of the norm of the many-body wave function and of its scalar product with the noninteracting ground state, obtained from the Bethe ansatz solution, with the special properties of Cauchy matrices. We derive the leading asymptotics of the two determinants in the thermodynamic limit and show that the quasi-particle residue $Z$ decays algebraically, $Z=W N^{-\theta}$. We confirm that the Anderson exponent $\theta$ is equal to $2\delta_F^2/\pi^2$, where $\delta_F$ is the Bethe-ansatz phase shift at the Fermi edge. The prefactor $W$ is obtained numerically as a function of the interaction parameter.