Scaling and crossover behaviour in a truncated long range quantum walk

We consider a discrete time quantum walker in one dimension, where at each step, the step length $\mathrm{\ell }$ is chosen from a distribution $P(\ell) \propto \ell^{-\delta -1}$with$\ell \leq \ell_{max}$\mathrm{.}Weevaluatetheprobability$f(x,t)$thatthewalkerisatposition$x$attime$t$ and its first two moments. As expected, the disorder effectively localizes the walk even for large values of $\delta$. Asymptotically, $⟨x^2⟩\propto t^{3\mathrm{/}2}$ and $⟨x⟩\propto t^{1\mathrm{/}2}$ independent of $\delta$ and $\mathrm{\ell }$, both finite. The scaled distribution $f(x,t)t^{1\mathrm{/}2}$ plotted versus $x\mathrm{/}{t}^{1\mathrm{/}2}$ shows a data collapse for $x\mathrm{/}t<\alpha (\delta ,{\mathrm{\ell }}_{max})\sim \mathcal{O}(1)$ 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 ${\mathrm{\ell }}_{max}$. 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.