  Ground state properties of the spin$-1/2$ Falicov-Kimball model on a
triangular lattice in the presence of uniform external magnetic field are
explored. Both the orbital and the Zeeman field-induced effects are taken into
account and in each unit cell only rational flux fractions are considered.
Numerical results, obtained with the help of Monte Carlo simulation algorithm,
reveal that the ground state properties strongly depend on the onsite Coulomb
correlation between itinerant and localized electrons, orbital magnetic field
as well as the Zeeman splitting. Strikingly, for the on-site Coulomb
correlation $U/t \approx 1$, the Zeeman splitting produces a phase transition
from paramagnetic metal/insulator to ferromagnetic insulator/metal transition
in the itinerant electron subsystem accompanied by the phase segregation to the
bounded/regular phase in the localized electrons subsystem. For the onsite
Coulomb correlation $U/t \approx 5$, although no metal to insulator transition
is observed but a magnetic phase transition from paramagnetic phase to
ferromagnetic phase in the itinerant electron subsystem is observed with the
Zeeman splitting. These results are applicable to the layered systems e.g.
cobaltates, rare earth and transition metal dichalcogenides, $GdI_{2}$,
$NaTiO_{2}$, $NaVO_{2}$ and $Be_{x}Zn_{1-x}O$ etc. It has been also proposed
that the results can be realized in the optical lattices with mixtures of light
atoms and heavy atoms using the cold atomic techniques.
