  Fractional Chern insulators are the proposed phases of matter mimicking the
physics of fractional quantum Hall states on a lattice without an overall
magnetic field. The notion of Floquet fractional Chern insulators refers to the
potential possibilities to generate the underlying topological bandstructure by
means of Floquet engineering. In these schemes, a highly controllable and
strongly interacting system is periodically driven by an external force at a
frequency such that double tunneling events during one forcing period become
important and contribute to shaping the required effective energy bands. We
show that in the described circumstances it is necessary to take into account
also third order processes combining two tunneling events with interactions.
Referring to the obtained contributions as micromotion-induced interactions, we
find that those interactions tend to have a negative impact on the stability of
of fractional Chern insulating phases and discuss implications for future
experiments.
