Superconductors with anomalous Floquet higher-order topology

We develop a general theory for two-dimensional (2D) anomalous Floquet higher-order topological superconductors (AFHOTSCs), which are dynamical Majorana-carrying phases of matter with no static counterpart. Despite the triviality of its bulk Floquet bands, an AFHOTSC generically features the simultaneous presence of corner-localized Majorana modes at both zero and $\ensuremath{\pi}/T$ quasienergies, a phenomenon beyond the scope of any static topological band theory. We show that the key to AFHOTSCs is their unavoidable singular behavior in the phase spectrum of the bulk time-evolution operator. By mapping such evolution-phase singularities to the stroboscopic boundary signatures, we classify 2D AFHOTSCs that are protected by a rotation group symmetry in symmetry class $D$. We further extract a higher-order topological index for unambiguously predicting the presence of Floquet corner Majorana modes, which we confirm numerically. Our theory serves as a milestone towards a dynamical topological theory for Floquet superconducting systems.