  We construct a systematic high-frequency expansion for periodically driven
quantum systems based on the Brillouin-Wigner (BW) perturbation theory, which
generates an effective Hamiltonian on the projected zero-photon subspace in the
Floquet theory, reproducing the quasienergies and eigenstates of the original
Floquet Hamiltonian up to desired order in $1/\omega$, with $\omega$ being the
frequency of the drive. The advantage of the BW method is that it is not only
efficient in deriving higher-order terms, but even enables us to write down the
whole infinite series expansion, as compared to the van Vleck degenerate
perturbation theory. The expansion is also free from a spurious dependence on
the driving phase, which has been an obstacle in the Floquet-Magnus expansion.
We apply the BW expansion to various models of noninteracting electrons driven
by circularly polarized light. As the amplitude of the light is increased, the
system undergoes a series of Floquet topological-to-topological phase
transitions, whose phase boundary in the high-frequency regime is well
explained by the BW expansion. As the frequency is lowered, the high-frequency
expansion breaks down at some point due to band touching with nonzero-photon
sectors, where we find numerically even more intricate and richer Floquet
topological phases spring out. We have then analyzed, with the Floquet
dynamical mean-field theory, the effects of electron-electron interaction and
energy dissipation. We have specifically revealed that phase transitions from
Floquet-topological to Mott insulators emerge, where the phase boundaries can
again be captured with the high-frequency expansion.
