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

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.