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The Floquet spectra of a class of driven SU(2) systems have been shown todisplay butterfly patterns with multifractal properties. The implication ofsuch critical spectral behavior for the Floquet eigenstate statistics isstudied in this work. Following the methodologies for understanding the fractalbehavior of energy eigenstates of time-independent systems on the Andersontransition point, we analyze the distribution profile, the mean value, and thevariance of the logarithm of the inverse participation ratio of the Floqueteigenstates associated with multifractal Floquet spectra. The results show thatthe Floquet eigenstates also display fractal behavior, but with featuresmarkedly 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 theFloquet eigenstate statistics of critical driven systems. The results based onthe proposed random matrix model are consistent with those obtained from ourdynamical examples with or without time-reversal symmetry.

Generating a Fractal Butterfly Floquet Spectrum in a Class of Driven SU(2) Systems: Eigenstate Statistics