1 September 2020 Tue 4 pm

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We  investigate  numerically  and  theoretically  the  effect  of  spatial  disorder  on  two-dimensional split-step discrete-time quantum walks with two internal “coin” states.  Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts.  We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization, but to a diffusive spread instead.  This is a delocalization, which happens because disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating  topological  phases.   We  base  this  explanation  on  the  relationship  of  this  general  quantum walk to a simpler case more studied in the literature, and for which disorder-induced delocalization of a topological origin has been observed.  We review topological delocalization for the simpler quantum  walk,  using  time-evolution  of  the  wavefunctions  and  level  spacing  statistics.   We  apply scattering theory to two-dimensional quantum walks, and thus calculate the topological invariants of  disordered  quantum  walks,  substantiating  the  topological  interpretation  of  the  delocalization, and finding signatures of the delocalization in the finite-size scaling of transmission.  We show signatures of criticality in the wavefunctions of the walk eigenstates. Our results showcase how theoretical ideas and numerical tools from solid-state physics can help us understand spatially random quantum walks.