Periodicity of the time-dependent Kohn-Sham equation and the Floquet theorem

TL;DR

This paper investigates the applicability of the Floquet theorem within time-dependent density functional theory, revealing that the effective Kohn-Sham potential is generally not periodic unless certain adiabatic conditions are met.

Contribution

It analyzes conditions under which the Floquet theorem can be applied in density functional theory, highlighting limitations and specific scenarios for its validity.

Findings

The exact Kohn-Sham potential is not unconditionally periodic.

Multiple Floquet states lead to non-periodic Hartree-exchange-correlation potentials.

The Floquet theorem applies only under weak, adiabatic external fields.

Abstract

The Floquet theorem allows to reformulate periodic time-dependent problems such as the interaction of a many-body system with a laser field in terms of time-independent, field-dressed states, also known as Floquet states. If this was possible for density functional theory as well, one could reduce in such cases time-dependent density functional theory to a time-independent Floquet density functional theory. We analyze under which conditions the Floquet theorem is applicable in a density-functional framework. By employing numerical {\em ab initio} solutions of the interacting time-dependent Schr\"odinger equation with time-periodic external potentials we show that the exact effective potential in the corresponding Kohn-Sham equation is {\em not} unconditionally periodic. Whenever several Floquet states in the interacting system are involved in a physical process the corresponding Hartree-exchange-correlation potential is not periodic with the external frequency only. Using an analytically solvable example we demonstrate that, in general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be restored by choosing a different initial state. Only if the external periodic potential is sufficiently weak such that the initial state of the interacting system evolves adiabatically to a single, field-dressed state, the resulting Kohn-Sham system admits the application of the Floquet theorem.