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 ℓ is chosen from a distribution P(ℓ)∝ℓ−δ−1 with ℓ≤ℓmax. 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 δ. Asymptotically, 〈x2〉∝t3∕2 and 〈x〉∝t1∕2 independent of δ and ℓ, both finite. The scaled distribution f(x,t)t1∕2 plotted versus x∕t1∕2 shows a data collapse for x∕t<α(δ,ℓmax)∼O(1) indicating the existence of a universal scaling function. The scaling function is shown to have a crossover behaviour at δ=δ∗≈4.0 beyond which the results are independent of ℓmax. 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 δ, with negligible dependence on the initial condition.