Textured electronic states of the triangular lattice Hubbard model and Na$_x$CoO$_2$ near $x=1/3$

The interplay between geometric frustration and strong correlation is studied in the triangular lattice Hubbard model near electron doping $x=1/3$, in connection to the sodium cobaltates Na$_x$CoO$_2$. We found a mechanism of alleviated magnetic frustration via charge and spin inhomogeneity. At $x=1/3$, the uniform paramagnetic ground state for $U < U_{c1}$ transforms into a $\sqrt{3}\times\sqrt{3}$ spin-charge textured insulating state for $U > U_{c2}$ with antiferromagnetic order on the underlying unfrustrated honeycomb lattice. The transition region, $U_{c1} < U < U_{c2}$, shows several textured semi-metallic states with both collinear and noncollinear magnetic order. We obtain the phase diagram and show that the strongly correlated phases near $x=1/3$ corresponds to doping the ``1/3 state'' with excess carriers forming electron or hole Fermi surface pockets, and compare to experimental findings. We thus propose that the cobaltates near $x=1/3$ are in proximity to such ``hidden'' textured phases with spin and charge order and the enhanced electronic fluctuations can mediate the superconducting pairing interaction.