  We study the effects of a periodically varying electric field on the Hubbard
model at half-filling on a triangular lattice. The electric field is
incorporated through the phase of the nearest-neighbor hopping amplitude via
the Peierls prescription. When the on-site interaction $U$ is much larger than
the hopping, the effective Hamiltonian $H_{eff}$ describing the spin sector can
be found using a Floquet perturbation theory. To third order in the hopping,
$H_{eff}$ is found to have the form of a Heisenberg antiferromagnet with three
different nearest-neighbor couplings $(J_\alpha,J_\beta,J_\gamma)$ on bonds
lying along the different directions. Remarkably, when the periodic driving
does not have time-reversal symmetry (TRS), $H_{eff}$ is also found to have a
chiral three-spin interaction in each triangle, with the coefficient $C$ of the
interaction having opposite signs on up- and down-pointing triangles. Thus
periodic driving which breaks TRS can simulate the effect of a perpendicular
magnetic flux which is known to generate such a chiral term in the spin sector,
even though our model does not have a magnetic flux. The four parameters
$(J_\alpha,J_\beta,J_\gamma,C)$ depend on the amplitude, frequency and
direction of the oscillating electric field. We then study the spin model as a
function of these parameters using exact diagonalization and find a rich phase
diagram of the ground state with seven different phases consisting of two kinds
of ordered phases (colinear and coplanar) and disordered phases. Thus periodic
driving of the Hubbard model on the triangular lattice can lead to an effective
spin model whose couplings can be tuned over a range of values thereby
producing a variety of interesting phases.
