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Periodically driven quantum many-body systems support anomalous topologicalphases of matter, which cannot be realized by static systems. In many cases,these anomalous phases can be many-body localized, which implies that they arestable and do not heat up as a result of the driving. What types of anomaloustopological phenomena can be stabilized in driven systems, and in particular,can an anomalous phase exhibiting non-Abelian anyons be stabilized? We addressthis question using an exactly solvable, stroboscopically driven 2D Kitaev spinmodel, in which anisotropic exchange couplings are boosted at consecutive timeintervals. The model shows a rich phase diagram which contains anomaloustopological phases. We characterize these phases using weak and strongscattering-matrix invariants defined for the fermionic degrees of freedom. Ofparticular importance is an anomalous phase whose zero flux sector exhibitsfermionic bands with zero Chern numbers, while a vortex binds a pair ofMajorana modes, which as we show support non-Abelian braiding statistics. Wefurther show that upon adding disorder, the zero flux sector of the modelbecomes localized. However, the model does not remain localized for a finitedensity of vortices. Hybridization of Majorana modes bound to vortices form"vortex bands", which delocalize by either forming Chern bands or a thermalmetal phase. We conclude that while the model cannot be many-body localized, itmay still exhibit long thermalization times, owing to the necessity to create afinite density of vortices for delocalization to occur.

Topology and localization of a periodically driven Kitaev model