The last decade has witnessed tremendous experimental progress in the study of "Floquet media," crystalline materials whose properties are altered by time-periodic parametric forcing. Theoretical advancements, however, have so far been achieved through discrete and approximate models. Understanding these materials from their underlying, first-principle PDE models, however, remains an open problem.

Specifically, semi-metals such as graphene are known to transform into insulators under periodic driving. While traditionally this phenomenon is modeled by a spectral gap, in PDE models no such gaps are conjectured to form. How do we reconcile these seemingly contradictory statements? We show that the driven Schrödinger equation possesses an “effective gap” – a novel and physically relevant relaxation of a spectral gap. Adopting a broader perspective, we study the influence of time-periodic forcing on a general band structure. A spectrally-local notion of stability is formulated and proven, using methods from periodic homogenization theory.