  Anderson localization in two-dimensional topological insulators takes place
via the so-called levitation and pair annihilation process. As disorder is
increased, extended bulk states carrying opposite topological invariants move
towards each other in energy, reducing the size of the topological gap,
eventually meeting and localizing. This results in a topologically trivial
Anderson insulator. Here, we introduce the anomalous levitation and pair
annihilation, a process unique to periodically-driven, or Floquet systems. Due
to the periodicity of the quasienergy spectrum, we find it is possible for the
topological gap to increase as a function of disorder strength. Thus, after all
bulk states have localized, the system remains topologically nontrivial,
forming an anomalous Floquet Anderson insulator (AFAI) phase. We show a
concrete example for this process, adding disorder via onsite potential "kicks"
to a Chern insulator model. By changing the period between kicks, we can tune
which type of (conventional or anomalous) levitation-and-annihilation occurs in
the system. We expect our results to be applicable to generic Floquet
topological systems and to provide an accessible way to realize AFAIs
experimentally, without the need for multi-step driving schemes.
