One-dimensional conductance through an arbitrary potential

TL;DR

This paper presents a method to analyze one-dimensional conductance through arbitrary potentials by mapping the interacting Hamiltonian to a non-interacting one, allowing for conductance calculation in complex scenarios.

Contribution

It introduces a flow equation approach to transform the Tomonaga-Luttinger Hamiltonian with arbitrary potential into a non-interacting form, valid for small interactions and arbitrary potentials.

Findings

Reproduces Kane and Fisher's algebraic conductance behavior at low temperatures.

Validates the method for arbitrary external potentials.

Provides an alternative bosonization formula without Klein factors.

Abstract

The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary potential is mapped onto a non-interacting Fermi gas with renormalized potential. This is done by means of flow equations for Hamiltonians and is valid for small electron-electron interaction. This method also yields an alternative bosonization formula for the transformed field operator which makes no use of Klein factors. The two-terminal conductance can then be evaluated using the Landauer formula. We obtain similar results for infinite systems at finite temperature by identifying the flow parameter with the inverse squared temperature in the asymptotic regime. We recover the algebraic behavior of the conductance obtained by Kane and Fisher in the limit of low temperatures and weak electron-electron interaction, but our results remain valid for arbitrary external potential.