Localization and recurrence of a quantum walk in a periodic potential on a line

We present a numerical study of a model of quantum walk in a periodic potential on a line. We take the simple view that different potentials have different affects on the way in which the coin state of the walker is changed. For simplicity and definiteness, we assume that the walker＇s coin state is unaffected at sites without the potential, and rotated in an unbiased way according to the Hadamard matrix at sites with the potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks are studied numerically. It is found that, of the six cases, four cases display significant localization effect where the walker is confined in the neighborhood of the origin for a sufficiently long time. Associated with such a localization effect is the recurrence of the probability of the walker returning to the neighborhood of the origin.