Indirect forces between impurities in one-dimensional quantum liquids

TL;DR

This paper studies how two impurities interact indirectly in a one-dimensional quantum liquid, revealing oscillating attractive and repulsive forces influenced by electron interactions and impurity strength.

Contribution

It provides a detailed analysis of impurity interactions in a Luttinger liquid using the functional renormalization group, including effects of electron-electron interactions and impurity parameters.

Findings

Impurity interaction oscillates with distance, switching between attraction and repulsion.

Interaction amplitude decays as 1/r for U=0, with modifications for U>0.

Crossover from non-universal to universal decay behavior depending on impurity strength and interaction.

Abstract

We investigate the indirect interaction between two isolated impurities in a Luttinger liquid described by a microscopic lattice model. To treat the electron-electron interaction U the functional renormalization group method is used. For comparison we also study the U=0 case. We find that for a wide range of impurity parameters the impurity interaction V_{12} as a function of their separation r oscillates with decaying amplitude between being attractive and repulsive. For half-filling of the band and in a crossover regime between weak and strong impurities the interaction becomes purely attractive. For U=0 and independent of the impurity strength the amplitude of the interaction energy falls off as 1/r. For U>0 the decay for small separations and weak to intermediate impurities is governed by a U dependent exponent larger than -1, which crosses over to -1 for large r. The crossover scale depends on the impurity strength and U. We present simple pictures which explain our results in the limits of weak and strong impurities. We finally also consider attractive interactions U<0.