  We study the Floquet phase diagram of two-dimensional Dirac materials such as
graphene and the one-dimensional (1D) spin-1/2 $XY$ model in a transverse field
in the presence of periodic time-varying terms in their Hamiltonians in the low
drive frequency ($\omega$) regime where 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 time-dependent magnetic field and nearest-neighbor couplings
between the spins. Using the adiabatic-impulse method, we provide several
semi-analytic criteria for the occurrence of changes in the topology of the
phase bands of such systems. For irradiated graphene, we point out the role of
the symmetries of $H(t)$ and $U$ behind such topology changes. Our analysis
reveals 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 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. Finally we study the 1D $XY$ model with a
two-rate driving protocol using the adiabatic-impulse method and exact numerics
revealing a phase band crossing at $t=T/2$ and $k=\pi/2$. We also study the
anomalous end modes generated by such a drive. We suggest experiments to test
our theory.
