Charge and spin order on the triangular lattice -- Na$_x$CoO$_2$ at $x=0.5$

The nature of electronic states due to strong correlation and geometric frustration on the triangular lattice is investigated in connection to the unconventional insulating state of Na$_x$CoO$_2$ at $x=0.5$. We study an extended Hubbard model using a spatially unrestricted Gutzwiller approximation. We find a new class of charge and spin ordered states at $x=1/3$ and $x=0.5$ where antiferromagnetic (AF) frustration is alleviated via weak charge inhomogeneity. At $x=0.5$, we show that the $\sqrt{3}a\times2a$ off-plane Na dopant order induces weak $\sqrt{3}a\times1a$ charge order in the Co layer. The symmetry breaking enables successive $\sqrt{3}a\times1a$ AF and $2a\times2a$ charge/spin ordering transitions at low temperatures. The Fermi surface is truncated by the $2a\times2a$ hexagonal zone boundary into small electron and hole pockets. We study the phase structure and compare to recent experiments.