Persistent quantum walks: dynamic phases and diverging timescales

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.