Functional Integral Bosonization for Impurity in Luttinger Liquid

TL;DR

This paper develops a functional integral approach to exactly analyze the electron Green function in a Luttinger liquid with an impurity, reproducing known results and providing new insights into local density of states behavior.

Contribution

It introduces an exact functional integral formalism for impurity effects in Luttinger liquids, extending previous bosonization methods.

Findings

Reproduces known suppression of local density of states at impurity

Derives analytic dependence of LDoS on distance from impurity

Shows crossover of LDoS behavior to pure Luttinger liquid

Abstract

We use a functional integral formalism developed earlier for the pure Luttinger liquid (LL) to find an exact representation for the electron Green function of the LL in the presence of a single backscattering impurity. This allows us to reproduce results (well known from the bosonization techniques) for the suppression of the electron local density of states (LDoS) at the position of the impurity and for the Friedel oscillations at finite temperature. In addition, we have extracted from the exact representation an analytic dependence of LDoS on the distance from the impurity and shown how it crosses over to that for the pure LL.