Topological phases and delocalization of quantum walks in random environments

Abstract

We investigate one-dimensional (1D) discrete time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at non-zero energy in their dynamics. Contrary to the previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust for spatial disorder but not for temporal disorder.

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

“…Already the simplest one-dimensional (1D) * asboth.janos@wigner.mta.hu quantum walk with angle disorder presents such a case: rather than being completely localized, it spreads subdiffusively [22]. This feature was explained in Ref.…”

“…This feature was explained in Ref. [22] by mapping the effective Hamiltonian of the quantum walk to chiral symmetric quantum wires (also see Ref. [23]).…”

“…The simple quantum walk, as, e.g., in Ref. [22], has a single rotation operation per period. Its time step reads…”

“…In Sec. IV we apply this tool to treat uniform disorder in rotation angles, find Anderson localization or delocalization, depending on the parameters, and give a new perspective on the delocalization of the simple quantum walk [22]. In Sec.…”

“…This is a broad family of quantum walks with chiral and particle-hole symmetries, which contains as a special case the simple quantum walk in Ref. [22]. We use a cloning procedure to derive the real-space scattering matrix of the walk, which allows us to give simple and efficient formulas for the topological invariants as well as for the localization lengths.…”