Topologically-protected delocalization in 1d quantum walks

Disordered systems in low dimension, which are generically Anderson-localized, can be protected from complete localization by being topologically nontrivial. This occurs, for instance, in the integer quantum Hall effect. Moving beyond static systems, we show that delocalization of eigenstates is a topologically-protected feature of single-particle, discrete-time evolutions (e.g., in quantum walks and Floquet systems) in one spatial dimension. Using spectral flow and flux insertion — appropriately modified for unitary operators instead of Hamiltonians — we argue that any topologically nontrivial time evolution operator must have an extensive number of delocalized states with non-vanishing density throughout the quasienergy spectrum.