Topological singularities and the general classification of Floquet–Bloch systems

Recent works have demonstrated that the Floquet-Bloch bands of periodically-driven systems feature a richer topological structure than their non-driven counterparts.The additional structure in the driven case arises from the periodicity of quasienergy, the energy-like quantity that defines the spectrum of a periodically-driven system.Here we develop a new paradigm for the topological classification of Floquet-Bloch bands, based on the time-dependent spectrum of the driven system's evolution operator throughout one driving period.Specifically, we show that this spectrum may host topologically-protected degeneracies at intermediate times, which control the topology of the Floquet bands of the full driving cycle.This approach provides a natural framework for incorporating the role of symmetries, enabling a unified and complete classification of Floquet-Bloch bands and yielding new insight into the topological features that distinguish driven and non-driven systems.After the discovery 1, 2 and explanation 3-7 of the quantized Hall effects, topology gained new importance as a mechanism for generating extremely robust quantum mechanical phenomena.The realization that the Bloch bands of solid state systems could possess non-trivial topological characteristics led to the prediction 8-10 and experimental discovery 11,12 of whole new classes of materials 13, 14-the topological insulators and superconductors-which host a variety of remarkable and potentially useful phenomena.On a theoretical level, a complete topological classification 15,16 of such systems has been developed, predicting a number of new phases.However, finding materials that realize these phases remains a very challenging task, with no known examples for many topological classes.Motivated by the great successes and open challenges in the arena of topological matter, many authors have begun to explore the possibilities for realizing topological phenomena in driven quantum systems .Time-dependent driving offers the opportunity to control a material's properties in a variety of new ways, potentially opening new routes for studying topological phenomena in solid state 44, atomic 22, 45, 46, and optical systems 47,48.Intriguingly, driven systems may host an even richer array of topological phenomena than their non-driven counterparts.To date several examples of topological phenomena which can only be realized in driven systems have been found 19,22,29,49,50, such as the existence of robust chiral edge states in two dimensional (2D) systems whose Floquet bands have trivial Chern indices 49, and pairs of non-degenerate Majorana end modes with protected quasienergy splittings in one-dimensional (1D) systems 22.This indicates that periodically driven systems feature additional topological structure beyond that found in non-driven systems.However, a unifying principle for understanding and classifying these new phenomena remains lacking.In this work we answer the question: under what conditions does the evolution of a driven system become topologically distinct from that of a non-driven system?In doing so we develop a powerful and general framework that can be used to understand the topology of periodically driven systems.In the analysis of periodically driven systems, the Floquet operator, denoted U T , ( ) plays a central role as the stroboscopic evolution operator that propagates the system forward in time through each complete driving period, T. The spectrum of the Floquet operator, given by U T e ,plays an analogous role to the spectrum of the Hamiltonian in a non-driven system, with real-valued energies replaced by periodically-