Quantum Antiferromagnents and Emergent Orders on Spatial Anisotropic Triangular Lattices

Schwinger boson representation and large $N$ expansion technique is applied to the quantum antiferromagnetic Heisenberg model on triangular lattices with spatial anisotropic nearest-neighbor and next-nearest-neighbor coupling. In the large N limit, we found several degenerate ground states with different magnetic ordering on sub-lattices,where the non-zero bonds form honeycomb lattice or dice lattice. Large $\kappa\left(=\frac{n_{b}}{N}\right)$ expansion is used to lift the degeneracy and to obtain the phase diagram. Possible applications to recent discovered compound $LiZnMo_{3}O_{8}$ are discussed.