Time dependent, external perturbations of Hubbard clusters with explicit many electron interactions and their implications

We examine the effects of harmonic and other time dependent external perturbations on a many electron system
using exact diagonalization and Lanczos method for Hubbard clusters in our search for time crystal and other behaviors such as
precursors to superconducting states.
It is possible to open time dependent breathing modes of the Hubbard Hamiltonian by exposing the cluster to electromagnetic radiation.
The analysis based on exact solutions of the Heisenberg equation demonstrates that it is possible to reach highly populated states with localized pairs of electrons having opposite spins (related to possible superconductivity). Since the system is principally time dependent in the pre-thermalization regime, it can be treated as an example of a time crystal.