▎ 摘　　要

We study the Floquet phase diagram of two-dimensional Dirac materials such as graphene and the onedimensional (1D) spin-1/2 XY model in a transverse field in the presence of periodic time-varying terms in their Hamiltonians in the lowdrive frequency (omega) regimewhere standard 1/omega perturbative expansions fail. For graphene, such periodic time-dependent terms are generated via the application of external radiation of amplitude A(0) and time period T = 2 pi/omega, while for the 1D XY model, they result from a two-rate drive protocol with a time-dependent magnetic field and nearest-neighbor couplings between the spins. Using the adiabatic-impulse method, whose predictions agree almost exactly with the corresponding numerical results in the low-frequency regime, we provide several semianalytic criteria for the occurrence of changes in the topology of the phase bands (eigenstates of the evolution operator U) of such systems. For irradiated graphene, we point out the role of the symmetries of the instantaneous Hamiltonian H(t) and the evolution operator U behind such topology changes. Our analysis reveals that at low frequencies, topology changes of irradiated graphene phase bands may also happen at t = T/3 and 2T/3 (apart from t = T) showing the necessity of analyzing the phase bands of the system for obtaining its phase diagrams. We chart out the phase diagrams at t = T/3, 2T/3, and T, where such topology changes occur, as a function of A(0) and T using exact numerics, and compare them with the prediction of the adiabatic-impulse method. We show that several characteristics of these phase diagrams can be analytically understood from results obtained using the adiabatic-impulse method and point out the crucial contribution of the high-symmetry points in the graphene Brillouin zone to these diagrams. We study the modes that can appear at the edges of a finite-width strip of graphene and show that the change in the number of such modes agrees with the change in the Chern number of bulk graphene as we go across a phase band crossing. Finally, we study the 1D XY model with a two-rate driving protocol. After studying the symmetries of the system, we use the adiabatic-impulse method and exact numerics to study its phase band crossing which occurs at t = T/2 and k = pi/2. We also study the end modes generated by such a drive and show that there can be anomalous modes whose Floquet eigenvalues are not equal to +/- 1. We suggest experiments to test our theory.