Anomalous Floquet Phases. A resonance phenomena

Floquet topological phases emerge when systems are periodically driven out-of-equilibrium. They gained attention due to their external control, which allows to simulate a wide variety of static systems by just tuning the external field in the high frequency regime. However, it was soon clear that their relevance goes beyond that, as for lower frequencies, anomalous phases without a static counterpart are present and the bulk-to-boundary correspondence can fail. In this work we discuss the important role of resonances in Floquet phases. For that, we present a method to find analytical solutions when the frequency of the drive matches the band gap, extending the well-known high frequency analysis of Floquet systems. With this formalism, we show that the topology of Floquet phases with resonances can be accurately captured in analytical terms. We also find a bulk-to-boundary correspondence between the number of edge states in finite systems and a set of topological invariants in different frames of reference, which crucially do not explicitly involve the micromotion. To illustrate our results, we periodically drive a SSH chain and a $\pi$-flux lattice, showing that our findings remain valid in various two-band systems and in different dimensions. In addition, we notice that the competition between rotating and counter-rotating terms must be carefully treated when the undriven system is a semi-metal. To conclude, we discuss the implications to experimental setups, including the direct detection of anomalous topological phases and the measurement of their invariants.