  We demonstrate that a triangular optical lattice of two atomic species,
bosonic or fermionic, can be employed to generate a variety of novel spin-1/2
Hamiltonians. These include effective three-spin interactions resulting from
the possibility of atoms tunneling along two different paths. Such interactions
can be employed to simulate particular one or two dimensional physical systems
with ground states that possess a rich structure and undergo a variety of
quantum phase transitions. In addition, tunneling can be activated by employing
Raman transitions, thus creating an effective Hamiltonian that does not
preserve the number of atoms of each species. In the presence of external
electromagnetic fields, resulting in complex tunneling couplings, we obtain
effective Hamiltonians that break chiral symmetry. The ground states of these
Hamiltonians can be used for the physical implementation of geometrical or
topological objects.
