2021/05/03 by Sagiv, Amir, Weinstein, Michael I. · 2 citations
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2105.00958
We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: iψt (t,x) = Hε(t)ψ=(H0+2iε A (ε t) ⋅ ∇ )ψ, H0=-Δ+V (x) . The unforced Hamiltonian, H0, is known to generically have Dirac (conical) points in its band spectrum. The evolution under Hε(t) of \it band limited Dirac wave-packets (spectrally localized near the Dirac point) is well-approximated on large time scales (t\lesssim 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. non-topological bands. Much is unknown about nature of the quasi-energy spectrum of original time-periodic Schroedinger equation, 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 Schroedinger dynamics. We introduce the notion of an \it effective quasi-energy gap, and establish its existence in the Schroedinger 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.