Dispersion relation and spectral function of an impurity in a one-dimensional quantum liquid

TL;DR

This paper investigates how an impurity's dispersion relation and spectral function behave in a one-dimensional quantum liquid, revealing effects analogous to the Kondo effect and potential quantum phase transitions.

Contribution

It introduces a detailed analysis of impurity dynamics in 1D quantum fluids, highlighting the impact of singular scattering processes and the analogy to the Kondo effect.

Findings

Dispersion relation is strongly affected near specific momenta due to singular scattering.

Analogies to the Kondo effect explain the impurity behavior.

Potential quantum phase transition indicated by a jump in the derivative of the dispersion relation.

Abstract

We consider the motion of an impurity particle in a general one-dimensional quantum fluid at zero temperature. The dispersion relation $\Omega(P)$ of the impurity is strongly affected by interactions with the fluid as the momentum approaches $\pm\pi\hbar n, \pm 3\pi\hbar n, ...$, where $n$ is the density. This behavior is caused by singular $\pm 2\pi\hbar n$ scattering processes and can be understood by analogy to the Kondo effect, both at strong and weak coupling, with the possibility of a quantum phase transition where $\Omega'(\pm \pi n)$ jumps to zero with increasing coupling. The low energy singularities in the impurity spectral function can be understood on the same footing.