Persistent quantum walks: Dynamic phases and diverging timescales

Abstract

A discrete-time quantum walk is considered in which the step lengths are chosen to be either 1 or 2 with the additional feature that the walker is persistent with a probability p. This implies that with probability p, the walker repeats the step length taken in the previous step and is otherwise antipersistent. We estimate the probability P(x,t) that the walker is at x at time t and the first two moments. Asymptotically, (x2)tν for all p. For the extreme limits p=0 and 1, the walk is known to show ballistic behavior, i.e., ν=2. For both p=0+ and 1-p=0+ we find that the scaling behavior changes discontinuously with ν=1 for p=0+ and ν=1.5 for p→1. For 0