paper

Scattering Expansion for Localization in One Dimension: from Disordered Wires to Quantum Walks

arXiv:2211.13368 · doi:10.1103/PhysRevB.109.064201

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. We apply this expansion to several examples, including the Anderson model, a general class of periodic-on-average-random potentials, and a two-component discrete-time quantum walk, showing analytically in the latter case that 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.

25+19 pages, 9 figures, longer version of arXiv:2210.07999; minor corrections, published version

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