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We present generic conditions for phase band crossings for a class ofperiodically driven integrable systems represented by free fermionic modelssubjected to arbitrary periodic drive protocols characterized by a frequency$\omega_D$. These models provide a representation for the Ising and $XY$ modelsin $d=1$, the Kitaev model in $d=2$, several kinds of superconductors, andDirac fermions in graphene and atop topological insulator surfaces. Our resultsdemonstrate that the presence of a critical point/region in the systemHamiltonian (which is traversed at a finite rate during the dynamics) maychange the conditions for phase band crossings that occur at the criticalmodes. We also show that for $d>1$, phase band crossings leave their imprint onthe equal-time off-diagonal fermionic correlation functions of these models;the Fourier transforms of such correlation functions, $F_{\vec k_0}(\omega_0)$, have maxima and minima at specific frequencies which can bedirectly related to $\omega_D$ and the time at which the phase bands cross at$\vec k = \vec k_0$. We discuss the significance of our results in the contextsof generic Hamiltonians with $N>2$ phase bands and the underlying symmetry ofthe driven Hamiltonian.

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