Scattering Expansion for Localization in One Dimension

TL;DR

This paper introduces a perturbative method for analyzing localization in one-dimensional disordered systems, revealing how phase disorder influences localization length and providing analytical insights into quantum walk behavior.

Contribution

It develops a full-range phase disorder expansion method that links localization to phase information and derives a simplified distribution model for weak reflection scenarios.

Findings

Localization length can vary non-monotonically with phase disorder.

The joint distribution of transmission and reflection is explicitly derived.

Scaling theory applies for weak local reflection, depending on three parameters.

Abstract

We present a perturbative approach to disordered systems in one spatial dimension that accesses the full range of phase disorder and clarifies the connection between localization and phase information. We consider a long chain of identically disordered scatterers and expand in the reflection strength of any individual scatterer. As an example application, we show analytically that in a discrete-time quantum walk, the localization length can depend non-monotonically on the strength of phase disorder (whereas expanding in weak disorder yields monotonic decrease). More generally, we obtain to all orders in the expansion a particular non-separable form for the joint probability distribution of the transmission coefficient logarithm and reflection phase. Furthermore, we show that for weak local reflection strength, a version of the scaling theory of localization holds: the joint distribution is determined by just three parameters.