Exact Friedel oscillations in the g=1/2 Luttinger liquid

TL;DR

This paper provides an exact analytical solution for Friedel oscillations caused by a single impurity in a g=1/2 Luttinger liquid, revealing detailed temperature and distance-dependent behaviors.

Contribution

It offers the first exact solution at the Toulouse point for arbitrary temperature and impurity coupling in the g=1/2 Luttinger liquid.

Findings

At zero temperature, charge density oscillations behave as ln x at small distances.

At large distances, oscillations decay as x^{-1/2}.

Charge density is expressed via hypergeometric functions.

Abstract

A single impurity in the 1D Luttinger model creates a local modification of the charge density analogous to the Friedel oscillations. In this paper, we present an exact solution of the case $g={1\over 2}$ (the equivalent of the Toulouse point) at any temperature $T$ and impurity coupling, expressing the charge density in terms of a hypergeometric function. We find in particular that at $T=0$, the oscillatory part of the density goes as $\ln x$ at small distance and $x^{-1/2}$ at large distance.