Fermionic renormalization group methods for transport through inhomogeneous Luttinger liquids

TL;DR

This paper compares two fermionic renormalization group methods for analyzing electronic transport in inhomogeneous Luttinger liquids, highlighting their strengths, limitations, and the importance of channel coupling in capturing interaction effects.

Contribution

It provides a detailed comparison of a simple poor man's RG approach and a more comprehensive functional RG method for inhomogeneous Luttinger liquids.

Findings

Both methods can handle arbitrary inhomogeneity strength.

The functional RG better captures physics due to channel coupling.

Difficulties arise near the perfect chain fixed point in the poor man's approach.

Abstract

We compare two fermionic renormalization group methods which have been used to investigate the electronic transport properties of one-dimensional metals with two-particle interaction (Luttinger liquids) and local inhomogeneities. The first one is a poor man's method setup to resum ``leading-log'' divergences of the effective transmission at the Fermi momentum. Generically the resulting equations can be solved analytically. The second approach is based on the functional renormalization group method and leads to a set of differential equations which can only for certain setups and in limiting cases be solved analytically, while in general it must be integrated numerically. Both methods are claimed to be applicable for inhomogeneities of arbitrary strength and to capture effects of the two-particle interaction, such as interaction dependent exponents, up to leading order. We critically review this for the simplest case of a single impurity. While on first glance the poor man's approach seems to describe the crossover from the ``perfect'' to the ``open chain fixed point'' we collect evidence that difficulties may arise close to the ``perfect chain fixed point''. Due to a subtle relation between the scaling dimensions of the two fixed points this becomes apparent only in a detailed analysis. In the functional renormalization group method the coupling of the different scattering channels is kept which leads to a better description of the underlying physics.