Disordered Quantum Wires: Microscopic Origins of the DMPK Theory and Ohm’s Law

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. Our objective is to firmly relate this theory to a microscopic model. We correct and extend previous work (Bachmann and De Roeck in J. Stat. Phys. 139:541–564, 2010) 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 β=1,2. In the β=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e. from TUE. In the β=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 appropriate regime for metals. For the ideal ensembles, we also prove Ohm’s law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone in Phys. Rev. B 44:3559–3576, 1991. This proof bypasses the explicit but intricate solution methods that underlie most previous results.