Heating effects in Floquet engineered system are detrimental to the control
of physical properties. In this work, we show that the heating of periodically
driven strongly correlated systems can be suppressed by multi-color driving,
i.e., by applying auxiliary excitations which interfere with the absorption
processes from the main drive. We focus on the Mott insulating single-band
Hubbard model and study the effects of multi-color driving with nonequilibrium
dynamical mean-field theory. The main excitation is a periodic electric field
with frequency Ω smaller than the Mott gap, while for the auxiliary
excitations, we consider additional electric fields and/or hopping modulations
with a higher harmonic of Ω. To suppress the 3-photon absorption of the
main excitation, which is a parity-odd process, we consider auxiliary
electric-field excitations and a combination of electric-field excitations and
hopping modulations. On the other hand, to suppress the 2-photon absorption,
which is a parity-even process, we consider hopping modulations. The conditions
for an efficient suppression of heating are well captured by the Floquet
effective Hamiltonian derived with the high-frequency expansion in a rotating
frame. As an application, we focus on the exchange couplings of the spins
(pseudo-spins) in the repulsive (attractive) model, and demonstrate that the
suppression of heating allows to realize and clearly observe a significant
Floquet-induced change of the low energy physics.Comment: 15 pages, 15 figure