  Quantum simulators based on cold atomic gases can provide an ideal platform
to study the microscopic mechanisms behind intriguing properties of solid
materials and further explore novel exotic phenomena inaccessible by chemical
synthesis. Here we propose and theoretically analyze a coherently coupled
binary mixture of Fermi atoms in a triangular optical lattice as a promising
realization of synthetic frustrated antiferromagnets. We perform a cluster
mean-field plus scaling analysis to show that the ground state exhibits several
nontrivial magnetic phases and a novel spin reorientation transition caused by
the quantum order-by-disorder mechanism. Moreover, we find from Monte Carlo
simulations that thermal fluctuations induce an unexpected coexistence of
Berezinskii-Kosterlitz-Thouless physics and long-range order in different
correlators. These predictions, besides being relevant to present and future
experiments on triangular antiferromagnetic materials, can be tested in the
laboratory with the combination of the currently available techniques for cold
atoms.
