Scaling and crossover behaviour in a truncated long range quantum walk

Abstract

We consider a discrete time quantum walker in one dimension, where at each step, the step length $\ell$ is chosen from a distribution $P(\ell )\propto {\ell }^{-\delta -1}$ with $\ell \le {\ell }_{max}$. We evaluate the probability $f(x,t)$ that the walker is at position $x$ at time $t$ and its first two moments. As expected, the disorder effectively localizes the walk even for large values of $\delta$. Asymptotically, $⟨x^2⟩\propto t^{3/2}$ and $⟨x⟩\propto t^{1/2}$ independent of $\delta$ and $\ell$, both finite. The scaled distribution $f(x,t)t^{1/2}$ plotted versus $x/t^{1/2}$ shows a data collapse for $x/t<\alpha (\delta ,{\ell }_{max})\sim \mathcal{O}(1)$ indicating the existence of a universal scaling function. The scaling function is shown to have a crossover behaviour at $\delta ={\delta }^{\ast }\approx 4.0$ beyond which the results are independent of ${\ell }_{max}$. We also calculate the von Neumann entropy of entanglement which gives a larger asymptotic value compared to the quantum walk with unique step length even for large $\delta$, with negligible dependence on the initial condition.

12 pages, 5 figures, version accepted in Physica A

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