Hubbard Model on the Triangular Lattice: Spiral Order and Spin Liquid

We investigate the half-filled Hubbard model on an isotropic triangular lattice with the variational cluster approximation. By decreasing the on-site repulsion $U$ (or equivalently increasing pressure) we go from a phase with long-range, three-sublattice, spiral magnetic order, to a nonmagnetic Mott insulating phase—a spin liquid—and then, for $U\lesssim 6.7t$, to a metallic phase. Clusters of sizes 3, 6, and 15 with open boundary conditions are used in these calculations, and an extrapolation to infinite size is argued to lead to a disordered phase at $U=8t$, but to a spiral order at $U\gtrsim 12$.

Figure 1

clusters used in our study. The 6-site and 15-site clusters tile the lattice only when paired with identical, inverted clusters. Superlattice basis vectors are shown.

Figure 2

Scaled Potthoff functional $\Omega$ as a function of Weiss field $h$ for various values of $U$ (3-site cluster). The local minima are indicated by arrows.

Figure 3

Left panel: $U$ dependence of the spiral order parameter for $L=3$, 6, and 15. Right panel: Spiral order parameter as a function of scaling parameter $Q$, for various $U$’s. The $U=8$ curve is a guide to the eye only. Left inset: Néel order parameter as a function of $U$ on a square lattice (12 sites). Right inset: The same as a function of $Q$.

Figure 4

Scaled Weiss field as a function of $Q$ (solid lines) and $1-1/L$ (dashed lines) for various values of $U$. The data are obtained for 3-, 6-, and 15-sites triangular clusters. Top left panel: square-lattice results at $U=16$ for the Néel Weiss field, with $L=2$, 4, 8, 10, 12, and 16 sites.

Figure 5

Bottom panel (B): density of states at $U=8$ (see text for details). Inset: the Mott gap as a function of $Q$. Top panel (A): $U$ dependence of the infinite-size extrapolated Mott gap.