Renormalization flow of a weak extended backscattering Hamiltonian in a non-chiral Tomonaga-Luttinger liquid

TL;DR

This paper studies how extended backscattering interactions in a non-chiral Luttinger liquid evolve under renormalization, revealing different behaviors for repulsive and attractive interactions and providing insights relevant for nanoscale electronic systems.

Contribution

It introduces a momentum shell renormalization group analysis of extended backscattering in non-chiral Luttinger liquids, showing how the Hamiltonian's shape changes with interactions and temperature.

Findings

Repulsive interactions lead to delta-like scalar potentials upon renormalization.

Attractive interactions suppress the Hamiltonian amplitude, making the junction transparent.

The shape of the Hamiltonian varies with position, interaction strength, and renormalization stage.

Abstract

We consider a non-chiral Luttinger liquid in the presence of a backscattering Hamiltonian which has an extended range. Right/left moving fermions at a given location can thus be converted as left/right moving fermions at a different location, within a specific range. We perform a momentum shell renormalization group treatment which gives the evolution of the relative degrees of freedom of this Hamiltonian contribution under the renormalization flow, and we study a few realistic examples of this extended backscattering Hamiltonian. We find that, for repulsive Coulomb interaction in the Luttinger liquid, any such Hamiltonian contribution evolves into a delta-like scalar potential upon renormalization to a zero temperature cutoff. On the opposite, for attractive couplings, the amplitude of this kinetic Hamiltonian is suppressed, rendering the junction fully transparent. As the renormalization procedure may have to be stopped because of experimental constraints such as finite temperature, we predict the actual spatial shape of the kinetic Hamiltonian at different stages of the renormalization procedure, as a function of the position and the Luttinger interaction parameter, and show that it undergoes structural changes. This renormalized kinetic Hamiltonian has thus to be used as an input for the perturbative calculation of the current, for which we provide analytic expressions in imaginary time. We discuss the experimental relevance of this work by looking at one-dimensional systems consisting of carbon nanotubes or semiconductor nanowires.