Squaring the Triangle: Insulating Ground State of $Na_{0.5} CoO_{2}$

We demonstrate that at a filling of $n=1.5$, an hexatic insulating state obtains in the extended Hubbard model on a triangular lattice. Composed of two tetragonal sublattices with fillings of $n=1$ and $n=2$, the insulating state is charge ordered and possesses an antiferromagnetic superlattice with dimension $a\times\sqrt{3}$. Two distinct energy scales arise in our model, a charge gap for the insulator and the effective exchange interaction in the antiferromagnet. Our model is capable of explaining the sign change of the Hall-coefficient as a function of temperature as well as the persistence of antiferromagnetism above the insulating state.