Edge states and topological analysis of discrete-time quantum walks in one-dimension

I will present recent work on the one-dimensional discrete-time quantum walk (DTQW), a quantum analog of the classical random walk. A DTQW, which is defined by a unitary operator, has motion that is controlled by its spin, which plays the role of a classical coin. These walks have attracted a great deal of interest, in part, because they can display topological insulating phases. In this talk, I will focus on the physics of the topologically protected edge modes and how the unitary controls their various properties, including their number and spin structure. I will present a transfer matrix approach that can capture these features. These results are in good agreement with numerical calculations.