Title: Floquet Hamiltonians - Spectrum and Dynamics


Abstract: The last decade has witnessed tremendous experimental progress in the study of "Floquet media," crystalline materials whose properties are changed by applying a time-periodic parametric forcing. The theory of Floquet media has so far been restricted to discrete models, which are often heuristic and approximate. Understanding these materials from their underlying PDE models, such as the Schrödinger equation, remains an open problem. 

Specifically, semi-metals such as graphene are known to transform into "Floquet Insulators" under such 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 prove the existence of an “effective gap” – a novel and physically-relevant notion which generalizes a (proper) spectral gap. Adopting a broader perspective, we then study the influence of time-periodic forcing on general band structures. A spectrally-local notion of near-invariance is formulated and proven, using methods from homogenization theory.