  We introduce the magnonic Floquet Hofstadter butterfly in the two-dimensional
insulating honeycomb ferromagnet. We show that when the insulating honeycomb
ferromagnet is irradiated by an oscillating space- and time-dependent electric
field, the hopping magnetic dipole moment (i.e. magnon quasiparticles)
accumulate the Aharonov-Casher phase. In the case of only space-dependent
electric field, we realize the magnonic Hofstadter spectrum with similar
fractal structure as graphene subject to a perpendicular magnetic field, but
with no spin degeneracy due to broken time-reversal symmetry by the
ferromagnetic order. In addition, the magnonic Dirac points and Landau levels
occur at finite energy as expected in a bosonic system. Remarkably, this
discrepancy does not affect the topological invariant of the system.
Consequently, the magnonic Chern number assumes odd values and the magnon Hall
conductance gets quantized by odd integers. In the case of both space- and
time-dependent electric field, the theoretical framework is studied by the
Floquet formalism. We show that the magnonic Floquet Hofstadter spectrum
emerges entirely from the oscillating space- and time-dependent electric field,
which is in stark contrast to electronic Floquet Hofstadter spectrum, where
irradiation by circularly polarized light and a perpendicular magnetic field
are applied independently. We study the deformation of the fractal structure at
different laser frequencies and amplitudes, and analyze the topological phase
transitions associated with gap openings in the magnonic Floquet Hofstadter
butterfly.
