Friedel oscillations in Luttinger liquids

TL;DR

This paper analyzes how electron correlations in a one-dimensional Luttinger liquid affect Friedel oscillations caused by impurities, revealing a slower decay than in Fermi liquids and providing explicit decay laws for different interaction strengths.

Contribution

It provides a comprehensive calculation of Friedel oscillations in Luttinger liquids for arbitrary interactions and impurity strengths, extending previous Fermi liquid predictions.

Findings

Friedel oscillations decay as x^-g in Luttinger liquids with interaction constant g<1.

Decay of oscillations is slower than Fermi liquid predictions due to correlations.

Weak scatterers exhibit a crossover from a slower to the asymptotic decay law.

Abstract

We study the density disturbance of a correlated 1D electron liquid in the presence of a scatterer or a barrier. The 2k_F-periodic density profile away from the barrier (Friedel oscillation) is computed for arbitrary electron--electron interaction and arbitrary impurity strength. We find that in presence of correlations, the Friedel oscillation decays slower than predicted by Fermi liquid theory. In the case of a spinless Luttinger liquid characterized by an interaction constant g<1, the asymptotic decay of the Friedel oscillation is x^-g. For a weak scatterer, the decay is even slower at small-to-intermediate distances from the impurity, with a crossover to the asymptotic x^-g law.