  Floquet-Magnus (FM) expansion theory is a powerful tool in periodically
driven (Floquet) systems under high-frequency drives. In closed systems, it
dictates that their stroboscopic dynamics under a time-periodic Hamiltonian is
well captured by the FM expansion, which gives a static effective Hamiltonian.
On the other hand, in dissipative systems driven by a time-periodic
Liouvillian, it remains an important and nontrivial problem whether the FM
expansion gives a static Liouvillian describing continuous-time Markovian
dynamics, which we refer to as the Liouvillianity of the FM expansion. We
answer this question for generic systems with local interactions. We find that,
while noninteracting systems can either break or preserve Liouvillianity of the
FM expansion, generic few-body and many-body interacting systems break it under
any finite drive, which is essentially caused by propagation of interactions
via higher order terms of the FM expansion. Liouvillianity breaking implies
that Markovian dissipative Floquet systems in the high-frequency regimes do not
have static (Markovian) counterparts, giving a signature of emergent
non-Markovianity. Our theory provides a useful insight for questing unique
phenomena in dissipative Floquet systems.
