Signatures and conditions for phase band crossings in periodically driven integrable systems

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 omega(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, F-(k) over right arrow0 (omega(0)), have maxima and minima at specific frequencies which can be directly related to omega(D) and the time at which the phase bands cross at (k) over right arrow = (k) over right arrow (0). We discuss the significance of our results in the contexts of generic Hamiltonians withN > 2 phase bands and the underlying symmetry of the driven Hamiltonian