EFFECTIVE GAPS IN CONTINUOUS FLOQUET HAMILTONIANS

We consider two-dimensional Schrodinger equations with honeycomb potentials and slow time-periodic forcing of the form iψt(t, x) “H^\varepsilon (t)ψ “(H⁰ + 2iεA(εt) ¨ ∇) ψ, H⁰ “´\Delta +V (x). The unforced Hamiltonian, H⁰, is known to generically have Dirac (conical) points in its band spectrum. The evolution under H^\varepsilon (t) of band-limited Dirac wave-packets (spectrally localized near the Dirac point) is well-approximated on large time scales (t À ε^´2`) by an effective time-periodic Dirac equation with a gap in its quasi-energy spectrum. This quasi-energy gap is typical of many reduced models of time-periodic (Floquet) materials and plays a role in conclusions drawn about the full system: conduction vs. insulation, topological vs. nontopological bands. Much is unknown about the nature of the quasi-energy spectrum of original time-periodic Schr odinger equations, and it is believed that no such quasi-energy gap occurs. In this paper, we explain how to transfer quasi-energy gap information about the effective Dirac dynamics to conclusions about the full Schrodinger dynamics. We introduce the notion of an effective quasi-energy gap and establish its existence in the Schrodinger model. In the current setting, an effective quasi-energy gap is an interval of quasi-energies which does not support modes with large spectral projection onto band-limited Dirac wave-packets. The notion of effective quasi-energy gap is a physically relevant relaxation of the strict notion of quasi-energy spectral gap; if a system is tuned to drive or measure at momenta and energies near the Dirac point of H⁰, then the resulting modes in the effective quasi-energy gap will only be weakly excited and detected.