We discuss the model of a one-dimensional, discrete-time walk on a line with spatial heterogeneity in the form of a variable set of ultrametric barriers. Inspired by the homogeneous quantum walk on a line, we develop a formalism by which the classical ultrametric random walk as well as the quantum walk can be treated in parallel by using a “coined” walk with internal degrees of freedom. For the random walk, this amounts to a second-order Markov process with a stochastic coin, better known as an (anti-)persistent walk. When this coin varies spatially in the hierarchical manner of “ultradiffusion,” it reproduces the well-known results of that model. The exact analysis employed for obtaining the walk dimension $d_w$, based on the real-space renormalization group (RG), proceeds virtually identically for the corresponding quantum walk with a unitary coin. However, while the classical walk remains robustly diffusive ($d_w=\frac{1}{2}$) for a wide range of barrier heights, unitarity provides for a quantum walk dimension $d_w$ that varies continuously, for even the smallest amount of heterogeneity, from ballistic spreading ($d_w=1$) in the homogeneous limit to confinement ($d_w=\infty$) for diverging barriers. Yet for any $d_w<\infty$ the quantum ultra-walk never appears to localize.

• Received 2 December 2019
• Accepted 18 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023411

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Published by the American Physical Society