Effective gaps in continuous Floquet Hamiltonians

TL;DR

This paper demonstrates the existence of an effective quasi-energy gap in the Floquet Hamiltonian of a 2D Schrödinger system with honeycomb potentials, linking effective Dirac dynamics to full system spectral properties.

Contribution

It introduces the concept of an effective quasi-energy gap and proves its existence in the Schrödinger model with time-periodic forcing, connecting reduced Dirac models to the full dynamics.

Findings

Effective quasi-energy gap exists in the Schrödinger model.

Modes near the Dirac point are weakly excited within this gap.

The effective gap informs about the full system's spectral behavior.

Abstract

We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: $$i\psi_t (t,x) = H^\varepsilon(t)\psi=\left(H^0+2i\varepsilon A (\varepsilon t) \cdot \nabla \right)\psi,\quad H^0=-\Delta +V (x) .$$ The unforced Hamiltonian, $H^0$, is known to generically have Dirac (conical) points in its band spectrum. The evolution under $H^\varepsilon(t)$ of {\it band limited Dirac wave-packets} (spectrally localized near the Dirac point) is well-approximated on large time scales ($t\lesssim \varepsilon^{-2+}$) 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 $H^0$, then the resulting modes in the effective quasi-energy gap will only be weakly excited and detected.