Floquet first-and higher-order topological superconductors: A review

The current trend in quantum technologies is deeply rooted in quantum computation, with Majorana fermions and their topological protection believed to play a central role. In this context, topological superconductors (TSCs) offer a promising arena for investigating Majorana modes in the context of topological quantum computation. In dimension d > 1, the spatial localization profile of the topological Majorana modes can be tuned in higher-order TSC extending beyond the realm of conventional first-order topological superconductor (FOTSC). In this review, we systematically survey the zoo of topological superconductivity, ranging from FOTSCs to third-order topological superconductors (TOTSCs), wherein the spatial dimensionality of Majorana zero modes (MZMs) is progressively reduced. We show that periodic driving protocols enable the Floquet engineering of all the above phases, even when the corresponding undriven systems are topologically trivial. We further investigate various driving schemes and the interplay among spinorbit coupling, magnetic fields, magnetic impurities, and symmetry-breaking mass terms to design Floquet higher-order topological superconducting (FHOTSC) phases, which are characterized by appropriate topological invariants. We extensively discuss the development of invariants that are able to distinguish between regular MZMs and anomalous Majorana π-modes, appearing at two distinct energies, where the latter does not exhibit any static analog.