Landauer-B\"uttiker and Thouless conductance

TL;DR

This paper rigorously derives the Thouless conductance formula from quantum statistical mechanics, connecting it to the Landauer-B"uttiker approach through the concept of crystalline currents in periodic structures.

Contribution

It extends the Landauer-B"uttiker formula to periodic structures and establishes a rigorous link to the Thouless conductance formula from first principles.

Findings

Crystalline currents are closely related to Thouless currents.

Crystalline heat current is bounded by Thouless heat current.

Bound saturation occurs with reflectionless reservoir-sample coupling.

Abstract

In the independent electron approximation, the average (energy/charge/entropy) current flowing through a finite sample S connected to two electronic reservoirs can be computed by scattering theoretic arguments which lead to the famous Landauer-B\"uttiker formula. Another well known formula has been proposed by Thouless on the basis of a scaling argument. The Thouless formula relates the conductance of the sample to the width of the spectral bands of the infinite crystal obtained by periodic juxtaposition of S. In this spirit, we define Landauer-B\"uttiker crystalline currents by extending the Landauer-B\"uttiker formula to a setup where the sample S is replaced by a periodic structure whose unit cell is S. We argue that these crystalline currents are closely related to the Thouless currents. For example, the crystalline heat current is bounded above by the Thouless heat current, and this bound saturates iff the coupling between the reservoirs and the sample is reflectionless. Our analysis leads to a rigorous derivation of the Thouless formula from the first principles of quantum statistical mechanics.