  The Floquet spectra of a class of driven SU(2) systems have been shown to
display butterfly patterns with multifractal properties. The implication of
such critical spectral behavior for the Floquet eigenstate statistics is
studied in this work. Following the methodologies for understanding the fractal
behavior of energy eigenstates of time-independent systems on the Anderson
transition point, we analyze the distribution profile, the mean value, and the
variance of the logarithm of the inverse participation ratio of the Floquet
eigenstates associated with multifractal Floquet spectra. The results show that
the Floquet eigenstates also display fractal behavior, but with features
markedly different from those in time-independent Anderson-transition models.
This motivated us to propose a new type of random unitary matrix ensemble,
called "power-law random banded unitary matrix" ensemble, to illuminate the
Floquet eigenstate statistics of critical driven systems. The results based on
the proposed random matrix model are consistent with those obtained from our
dynamical examples with or without time-reversal symmetry.
