Transport and Localization in Quantum Walks on a Random Hierarchy of Barriers

TL;DR

This paper investigates how combining hierarchical and random barriers in a one-dimensional quantum walk can induce localization, revealing a transition from transport to localization with sparse randomness added to the system.

Contribution

It demonstrates that sparse randomness in a hierarchical barrier structure can cause localization, extending understanding of quantum walk transport phenomena.

Findings

Sparse randomness induces localization in hierarchical quantum walks.

A localization transition exists depending on barrier strength and randomness.

Transport ceases when randomness exceeds a critical threshold.

Abstract

We study transport within a spatially heterogeneous one-dimensional quantum walk with a combination of hierarchical and random barriers. Recent renormalization group calculations for a spatially disordered quantum walk with a regular hierarchy of barriers alone have shown a gradual decrease in transport but no localization for increasing (but finite) barrier sizes. In turn, it is well-known that extensive random disorder in the spatial barriers is sufficient to localize a quantum walk on the line. Here we show that adding only a sparse (sub-extensive) amount of randomness to a hierarchy of barriers is sufficient to induce localization such that transport ceases. Our numerical results suggest the existence of a localization transition for a combination of both, the strength of the regular barrier hierarchy at large enough randomness as well as the increasing randomness at sufficiently strong barriers in the hierarchy.