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We study the Floquet phase diagram of two-dimensional Dirac materials such asgraphene and the one-dimensional (1D) spin-1/2 $XY$ model in a transverse fieldin the presence of periodic time-varying terms in their Hamiltonians in the lowdrive frequency ($\omega$) regime where standard $1/\omega$ perturbativeexpansions fail. For graphene, such periodic time dependent terms are generatedvia 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 driveprotocol with time-dependent magnetic field and nearest-neighbor couplingsbetween the spins. Using the adiabatic-impulse method, we provide severalsemi-analytic criteria for the occurrence of changes in the topology of thephase bands of such systems. For irradiated graphene, we point out the role ofthe symmetries of $H(t)$ and $U$ behind such topology changes. Our analysisreveals that at low frequencies, phase band topology changes may also happen at$t= T/3, 2T/3$ (apart from $t=T$). We chart out the phase diagrams at $t=T/3,2T/3,\, {\rm and }\, T$ as a function of $A_0$ and $T$ using exact numerics,and compare them with the prediction of the adiabatic-impulse method. We showthat several characteristics of these phase diagrams can be analyticallyunderstood from results obtained using the adiabatic-impulse method and pointout the crucial contribution of the high-symmetry points in the grapheneBrillouin zone to these diagrams. Finally we study the 1D $XY$ model with atwo-rate driving protocol using the adiabatic-impulse method and exact numericsrevealing a phase band crossing at $t=T/2$ and $k=\pi/2$. We also study theanomalous end modes generated by such a drive. We suggest experiments to testour theory.

Low-frequency phase diagram of irradiated graphene and periodically driven spin-1/2 $XY$ chain