In this letter, the superfluid-Mott-insulator phase transition of two-species cold bosonic atoms in an optical lattices is studied. The Hamiltonian of this model is diagonalized by means of Bogliubov transformations and by the inversion symmetry of the optical lattice, the energy spectrum of this system is obtained. From the energy gap of the excitation spectrum, the quantum phase transition condition is obtained and it is determined by the competition between the interatomic repulsions and the tunnel coupling. It is found that there exists an ordinary fluid phase when taking the zero wave-vector limit.
We demonstrate that a triangular optical lattice of two atomic species, bosonic or fermionic, can be employed to generate a variety of 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.