The bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice

Abstract

Suzuki's split-step quantum walk on the one-dimensional integer lattice can be naturally viewed as a chirally symmetric quantum walk. Given the unitary time-evolution of such a chirally symmetric quantum walk, we can separately introduce well-defined indices for the eigenvalues $\pm 1.$ The bulk-edge correspondence for Suzuki's split-step quantum walk is twofold. Firstly, we show that the multiplicities of the eigenvalues $\pm 1$ coincide with the absolute values of the associated indices. Note that this can be viewed as the symmetry protection of bound states, and that the indices we consider are robust in the sense that these depend only on the asymptotic behaviour of the parameters of the given model. Secondly, we show that that such bound states exhibit exponential decay at spatial infinity.