Persistent quantum walks: dynamic phases and diverging timescales

TL;DR

This paper investigates a quantum walk with variable step lengths and persistence, revealing four dynamic phases with diverging timescales and crossover behaviors near extreme persistence probabilities.

Contribution

It introduces a novel persistent quantum walk model with variable step lengths and identifies multiple phases and diverging timescales, expanding understanding of quantum walk dynamics.

Findings

Identifies four distinct phases characterized by different scaling exponents.

Discovers diverging timescales near extreme persistence probabilities.

Shows that suppressing antipersistence yields a uniform scaling behavior.

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, $\langle x^2 \rangle = t^\nu$ for all $p$. For the extreme limits $p=0$ and $1$, the walk is known to show ballistic behaviour, i.e., $\nu = 2$. As $p$ is varied from zero to 1, the system is found in four different phases characterised by the value of $\nu$: $\nu =2$ at $p=0$, $1 \leq \nu \leq 3/2$ for $0 < p < p_c$, $\nu = 3/2$ for $ p_c < p <1$ and $\nu = 2$ again at $p=1$. $p_c$ is found to be very close to $1/3$ numerically. Close to $p=0,1$, the scaling behaviour shows a crossover in time. Associated with this crossover, two diverging timescales varying as $1/p$ and $1/(1-p)$ close to $p=0$ and $p=1$ respectively are detected. Using a different scheme in which the antipersistence behaviour is suppressed, one gets $\nu= 3/2$ for the entire region $0 < p< 1$. Further, a measure of the entropy of entanglement is studied for both the schemes.