Impurity in a Luttinger liquid: a numerical study of the finite size energy spectrum and of the orthogonality catastrophe exponent

TL;DR

This paper numerically investigates how a single impurity affects a one-dimensional Luttinger liquid, confirming theoretical predictions on energy spectrum scaling and orthogonality catastrophe exponent using DMRG methods.

Contribution

It provides numerical validation of Kane and Fisher's theoretical predictions for impurity effects in Luttinger liquids, including energy spectrum scaling and orthogonality catastrophe exponent.

Findings

Confirmed theoretical finite size scaling predictions for energy spectrum.

Calculated orthogonality catastrophe exponent supporting existing theory.

Validated impurity effects in Luttinger liquids through numerical analysis.

Abstract

The behavior of a single impurity in a one-dimensional Luttinger liquid is numerically investigated by means of the density matrix renormalization group. By analyzing the finite size scaling behavior of the low energy spectrum, we confirm the theoretical prediction of Kane and Fisher [Phys. Rev. Lett. {\bf 68}, 1220 (1992)] both for attractive and repulsive interactions. Moreover, we calculate the exponent of the orthogonality catastrophe, which gives a further support to the above theoretical prediction.