High winding number of topological phase in periodic quantum walks

Abstract

We construct one-dimensional multi-period quantum walks to investigate the topological phases with high winding number. It is found that the maximum of winding number is determined by the number of shift operation in one-step evolution only under an appropriate time frame. For odd-period quantum walks, 0-energy edge states and π-energy edge states coexist equally on the boundary, but for even-period quantum walks, these two kinds of edge states may exist alone or together. The relations between the number of edge state and high winding number are discussed by considering coin parameter as a step function of position. The validity of the bulk-edge correspondence is confirmed. We also reveal that the occupation probability of the walker at the boundary exhibits dynamical oscillation due to the coexistence of 0-energy and π-energy edge states.