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We present generic conditions for phase band crossings for a class of periodically driven integrable systems represented by free fermionic models subjected to arbitrary periodic drive protocols characterized by a frequency ?D. These models provide a representation for the Ising and XY models in d=1, the Kitaev model in d=2, several kinds of superconductors, and Dirac fermions in graphene and atop topological insulator surfaces. Our results demonstrate that the presence of a critical point/region in the system Hamiltonian (which is traversed at a finite rate during the dynamics) may change the conditions for phase band crossings that occur at the critical modes. We also show that for d>1, phase band crossings leave their imprint on the equal-time off-diagonal fermionic correlation functions of these models; the Fourier transforms of such correlation functions, Fk? 0(?0), have maxima and minima at specific frequencies which can be directly related to ?D and the time at which the phase bands cross at k =k0. We discuss the significance of our results in the contexts of generic Hamiltonians with N>2 phase bands and the underlying symmetry of the driven Hamiltonian. [ABSTRACT FROM AUTHOR]

We study periodically driven closed quantum systems where two parameters of the system Hamiltonian are varied periodically in time with frequencies ω1 and ω2=rω1. We show that such drives may be used to tune towards dynamics-induced freezing where the wave function of the state of the system after a drive cycle at time T=2π/ω1 has almost perfect overlap with the initial state. We locate regions in the (ω1,r) plane where the freezing is near exact for a class of integrable models and a specific nonintegrable model. The integrable models that we study encompass Ising and XY models in d=1, Kitaev model in d=2, and Dirac fermions in graphene and atop a topological insulator surface, whereas the nonintegrable model studied involves the experimentally realized one-dimensional tilted Bose-Hubbard model in an optical lattice. In addition, we compute the relevant correlation functions of such driven systems and describe their characteristics in the region of the (ω1,r) plane where the freezing is near exact. We supplement our numerical analysis with semianalytic results for integrable driven systems within adiabatic-impulse approximation and discuss experiments which may test our theory. [ABSTRACT FROM AUTHOR]