Exact calculation of spectral properties of a particle interacting with a one dimensional fermionic system

TL;DR

This paper uses Bethe ansatz analysis to exactly calculate the spectral properties of a particle interacting with a one-dimensional fermionic system, revealing how orthogonality catastrophe and phase shifts affect spectral exponents.

Contribution

It provides an exact calculation of spectral properties in a 1D fermionic system with equal masses, highlighting differences in quasiparticle weight exponents for finite and infinite mass cases.

Findings

Orthogonality catastrophe persists for infinite particle mass.

Exponents of quasiparticle weight depend on phase shifts at the Fermi surface.

Spectral function behavior is characterized for both repulsive and attractive interactions.

Abstract

Using the Bethe ansatz analysis as was reformulated by Edwards, we calculate the spectral properties of a particle interacting with a bath of fermions in one dimension for the case of equal particle-fermion masses. These are directly related to singularities apparent in optical experiments in one dimensional systems. The orthogonality catastrophe for the case of an infinite particle mass survives in the limit of equal masses. We find that the exponent $\beta$ of the quasiparticle weight, $Z\simeq N^{-\beta}$ is different for the two cases, and proportional to their respective phaseshifts at the Fermi surface; we present a simple physical argument for this difference. We also show that these exponents describe the low energy behavior of the spectral function, for repulsive as well as attractive interaction.