Second order functional renormalization group approach to one-dimensional systems in real and momentum space

TL;DR

This paper develops a second order functional renormalization group method for one-dimensional spinless fermion systems, applicable in both real and momentum space, and demonstrates its effectiveness through Luttinger liquid physics problems.

Contribution

It introduces a second order FRG approach for 1D fermions in real and momentum space, improving accuracy over first order methods.

Findings

Accurately describes reflection at interfaces in Luttinger liquids

Reproduces power-law behavior in occupation numbers

Highlights strengths and limitations of the second order FRG scheme

Abstract

We devise a functional renormalization group treatment for a chain of interacting spinless fermions which is correct up to second order in the interaction strength. We treat both inhomogeneous systems in real-space as well as the translational invariant case in a k-space formalism. The strengths and shortcomings of the different schemes as well as technical details of their implementation are discussed. We use the method to study two proof-of-principle problems in the realm of Luttinger liquid physics, namely reflection at interfaces and power laws in the occupation number as a function of crystal momentum.