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

The nature of charge and spin order of strongly correlated triangular lattice fermions is investigated in connection to the unconventional insulating state of Na$_x$CoO$_2$ at $x=0.5$. We study an extended Hubbard ($t$-$U$-$V$) model of the electron doped Co $a_{1g}$ band 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 the system alleviates antiferromagnetic (AF) frustration via charge inhomogeneity. We show that the $\sqrt{3}a\times2a$ off-plane Na dopant order at $x=0.5$ plays an important but subtle role. It induces weak $\sqrt{3}a\times1a$ charge order in the Co layer without gapping the Fermi surface and allows successive $\sqrt{3}a\times1a$ AF and $2a\times2a$ charge/spin ordering transitions at low temperatures. The nesting with the $2a\times2a$ hexagonal zone boundary gaps out almost the entire Fermi surface at $x=0.5$. We study the phase structure and compare to the findings of recent experiments.