Floquet formulation of the dynamical Berry-phase approach to nonlinear optics in solids (I. Alliati)

We present a Floquet scheme for the ab-initio calculation of nonlinear optical properties in extended systems. This entails a reformulation of the real-time approach based on the dynamical Berry-phase polarisation [Attaccalite & Grüning, PRB 88, 1–9 (2013)] and retains the advantage of being non-perturbative in the electric field. The proposed method applies to periodically-driven Hamiltonians and makes use of this symmetry to turn a time-dependent problem into a self-consistent time-independent eigenvalue problem. We implemented this Floquet scheme at the independent particle level and compared it with the real-time approach. Our reformulation reproduces real-time-calculated 2nd and 3rd order susceptibilities for a number of bulk and two-dimensional materials, while reducing the associated computational cost by one or two orders of magnitude. An extension of this formalism to include local fields and excitonic effects is also discussed, alongside the instabilities it introduces into the scheme and some preliminary results. Finally, a method to extract Floquet states from the time-dependent solution of the real-time approach is used to identify the issues appearing in the proposed Floquet approach upon inclusion of many-body effects.