Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law

TL;DR

This paper rigorously connects microscopic disordered quantum wire models to random matrix theory, deriving conductance properties and proving Ohm's law across symmetry classes with novel insights into deviations from standard ensembles.

Contribution

It establishes a microscopic basis for RMT in quantum wires, derives a new ensemble close to but distinct from standard classes, and provides a simplified proof of Ohm's law.

Findings

Conductance matches ideal ensembles in  class

Deviation from standard ensemble in  class observed

Ohm's law proven for all symmetry classes

Abstract

We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire. In the literature, the standard description of such wires is via random matrix theory (RMT). Our objective is to firmly relate this theory to a microscopic model. We correct and extend previous work (arXiv:0912.1574) on the same topic. In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes \beta=1,2. In the \beta=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e.\ from TUE. In the \beta=1 class, we find a deviation from TOE. It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally relevant regime. For the ideal ensembles, we also prove Ohm's law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone. This proof bypasses the explicit but intricate solution methods that underlie most previous results.