The DMPK equation for mesoscopic quantum transport revisited

TL;DR

This paper revisits the DMPK equation for mesoscopic quantum transport, clarifying the proper mean free path and its implications for metallic and localized regimes, aligning theoretical models with microscopic transmission analysis.

Contribution

It corrects the understanding of the mean free path in the DMPK equation, reconciling it with microscopic transmission analysis and restoring the validity of conductance fluctuation predictions.

Findings

The Thouless localization length is a proper lower bound for macroscopic length scales.

The metallic regime's description aligns with Dorokhov's microscopic analysis.

Correcting the mean free path in the DMPK equation restores universal conductance fluctuation results.

Abstract

A recent Drude model description of the metallic regime and of a channel- averaged elastic mean free path (mfp), $\ell_0$, in an $N$-channel tight-binding wire identifies the Thouless localization length, $N\ell_0$, as a proper lower bound of macroscopic length scales ("mean free path") for the DMPK equation describing the localized regime of the wire. The mfp $\ell_0$ leads to a metallic regime which is consistent with Dorokhov's microscopic transmission analysis in terms of a nominal elastic mfp. On the other hand, the validity of Mello's derivation of universal conductance fluctuations in the metallic regime based on the DMPK equation is restored if the mfp $\ell'$, of order $N\ell_0$, in that equation is replaced by the correct mean free path $\ell_0$.