Textured electronic states of the triangular-lattice Hubbard model and ${\mathrm{Na}}_x{\mathrm{CoO}}_2$

We show that geometric frustration and strong correlation in the triangular-lattice Hubbard model lead a rich phase structure of $\sqrt{3}\times \sqrt{3}$ spin-charge textured electronic states over a wide region of electron doping $0\le x\le 0.40$. In addition to the ${120}^{\circ }$ Néel ordered insulator at half filling, we found a spin charge ordered insulator at $x=1/3$ with collinear antiferromagnetic (AF) order on the underlying unfrustrated honeycomb lattice. Separating the two insulating phases is a Lifshitz transition between a noncollinear AF ordered metal and one with coexisting charge order. We obtain the phase diagram and the evolution of the Fermi surface (FS). Remarkably, the correlated ground states near $x=1/3$ emerge as doping the “1/3 AF insulator” by excess carriers, leading to electron and hole FS pockets with important implications for the cobaltate superconducting state.

Figure 1

Phase diagram of the triangular-lattice Hubbard model. $(U_{\mathrm{te}},x_{\mathrm{te}})$ denotes the tetra-critical point (red circle) where the first-order transition line (black), two second-order transition lines, and the Lifshitz transition (dashed line) meet. Note the different horizontal scale for $x\le 0.2$.

Figure 2

Magnetic ordered insulating states at large $U$. (a) ${120}^{\circ }$ noncollinear Néel order at $x=0$ and $U=2W$. (b) Unfrustrated AF order on the underlying honeycomb lattice with charge order at $x=1/3$ and $U=3W$. Solid circles indicate the charge density.

Figure 3

FS of electronic textured phases at $U=3W$ and doping (a) $x=0.28$, (b) $x=0.32$, (c) $x=0.36$, (d) $x=0.45$.

Figure 4

(a) Schematic phase diagram at $x=1/3$ as a function of $U$. The evolution of the charge density, magnitude, and orientation of the spin density on the three sublattices sketched in (a) are shown quantitatively in (b), (c), and (d), respectively.

Figure 5

Band dispersion (left panel) and FS topology at $x=1/3$: (a) CSCO-AFM at $U=1.97W$, (b) NSCO-AFM at $U=2.08W$, (c) C-FRM at $U=2.18W$, and (d) CSCO-AFI at $U=6.7W$. The single-particle gap in the CSCO-AFI phase is shown in (e) as a function of $U/W$.