Decohering d-dimensional quantum resistance

TL;DR

This paper extends the Landauer scattering approach to d-dimensional disordered resistors by incorporating decoherence through reservoir-like mechanisms, revealing that even minimal decoherence turns a sharp localization transition into a smooth crossover.

Contribution

It introduces a novel anisotropic Migdal-Kadanoff approach to analyze decoherence effects in higher-dimensional disordered conductors, generalizing from 1D to dD.

Findings

Small decoherence eliminates the localization transition, replacing it with a crossover.

Resistance moments of all orders become finite due to decoherence.

The anisotropic approximation simplifies the analytical treatment of the problem.

Abstract

The Landauer scattering approach to 4-probe resistance is revisited for the case of a d-dimensional disordered resistor in the presence of decoherence. Our treatment is based on an invariant-embedding equation for the evolution of the coherent reflection amplitude coefficient in the length of a 1-dimensional disordered conductor, where decoherence is introduced at par with the disorder through an outcoupling, or stochastic absorption, of the wave amplitude into side (transverse) channels, and its subsequent incoherent re-injection into the conductor. This is essentially in the spirit of B{\"u}ttiker's reservoir-induced decoherence. The resulting evolution equation for the probability density of the 4-probe resistance in the presence of decoherence is then generalised from the 1-dimensional to the d-dimensional case following an anisotropic Migdal-Kadanoff-type procedure and analysed. The anisotropy, namely that the disorder evolves in one arbitrarily chosen direction only, is the main approximation here that makes the analytical treatment possible. A qualitatively new result is that arbitrarily small decoherence reduces the localisation-delocalisation transition to a crossover making resistance moments of all orders finite.