Noncollinear and noncoplanar magnetic order in the extended Hubbard model on anisotropic triangular lattice

Motivated by the importance of noncollinear and noncoplanar magnetic phases in determining various electrical properties in magnets, we investigate the magnetic phase diagram of the extended Hubbard model on an anisotropic triangular lattice. We map out the ground-state phase diagram within a mean-field scheme that treats collinear, noncollinear, and noncoplanar phases on equal footing. In addition to the standard ferromagnet and ${120}^{\circ }$ antiferromagnet states, we find the four-sublattice flux, the 3Q noncoplanar, and the noncollinear charge-ordered states to be stable at specific values of filling fraction $n$. Inclusion of a nearest-neighbor Coulomb repulsion leads to intriguing spin-charge-ordered phases. The most notable of these are the collinear and noncollinear magnetic states at $n=2/3$, which occur together with a pinball-liquid-like charge order. Our results demonstrate that the elementary single-orbital extended Hubbard model on a triangular lattice hosts unconventional spin-charge ordered phases, which are similar to those reported in more complex and material-specific electronic Hamiltonians.

Figure 1

(a) A schematic view of the anisotropic triangular lattice with hopping parameters $t$ and $t^{\prime }$ shown via solid and dashed lines, respectively. (b)–(g) Building blocks of various long-range ordered magnetic phases. The cone angle $\theta$ connects the flux state to the antiferromagnetic state via the 3Q state.

Figure 2

$n\text{−}U$ phase diagrams obtained within the grand-canonical approach for (a) $t^{\prime }/t=0$, (b) $t^{\prime }/t=0.5$, (c) $t^{\prime }/t=0.7$, and (d) $t^{\prime }/t=1$. PS denotes phase separation, P. spiral (C. spiral) refers to planar (conical) spiral phase, and 3Q and NC-CO are the four-sublattice and six-sublattice ordered phases found at $n=1/2$ and $n=2/3$, respectively.

Figure 3

$t^{\prime }\text{−}U$ phase diagrams obtained within the canonical ensemble approach at commensurate values of average electronic fillings: (a) $n=1/2$, (b) $n=2/3$, (c) $n=1$, and (d) $n=3/2$. The notation for different phases is same as that in Fig. 2.

Figure 4

(a) Change in the variational parameters corresponding to the ground states for $n=1/2,n=1$, and $n=3/2$ as a function of $t^{\prime }$. The parameters are the spiral wave vector $q_x=q_y$ for $n=1$, and the polar angle $\theta$ describing the $2\times 2$ block-periodic states for $n=1/2,3/2$. For $n=3/2$ the ground state is FM for $t^{\prime }/t>0.2$. The electronic density of states for various ground states for different values of $t^{\prime }$ are as follows: (b) the NC-CO state at $n=2/3$ and $t^{\prime }=1$; (c) the flux and planar spiral states at $n=0.5$ and $n=0.7$, respectively, for $t^{\prime }=0$; (d) the stripe and ${120}^{\circ }$ states at $n=0.62$ and $n=0.5$, respectively, for $t^{\prime }/t=0.5$; (e) the flux and the staggered AFM states for $t^{\prime }/t=0.7$; and (f) the noncoplanar 3Q and conical spiral states for $t^{\prime }/t=1$.

Figure 5

Results on the effect of Coulomb repulsion at $n=1/2$. The value of (a) the local charge densities and (b) the local magnetic moments at two inequivalent sites as a function of $V$. The inset in (d) shows the change in variational angle $\theta$ which connects the flux state $\left(\theta =\pi /2\right)$ to the 3Q state $\left(\theta \sim 0.31\pi \right)$. Density of states of the spin-charge ordered phases at different values of $V$ for (c) $t^{\prime }=0$ and (d) $t^{\prime }=1$. These results are obtained for $U=8$ and $V^{\prime }=0$.

Figure 6

A schematic view of the spin-charge ordered phases that appear for finite intersite Coulomb interaction for filling fraction: (a)–(c) $n=1/2,t^{\prime }=0$ and (d)–(g) $n=2/3,t^{\prime }=t$. The sizes of the circles represent the local charge density. The higher-density sites are shown as open circles for clarity.

Figure 7

Results on the effect of Coulomb repulsion at $n=2/3$. The value of (a) the local charge densities and (b) the local magnetic moments of three inequivalent sites as a function of $V$ for $V^{\prime }=0$. The inset in (b) shows the change in variational angle $\mathrm{\Theta }$ which connects the FM state $\left(\mathrm{\Theta }=0\right)$ to the ${120}^{\circ }$ state $\left(\mathrm{\Theta }=\pi /2\right)$. (c)-(d) Density of states of the spin-charge ordered phases at different values of $V$. These results are obtained for $U=8$ and $V^{\prime }=0$.

Figure 8

The value of (a) the local charge densities and (b) the local magnetic moments at two inequivalent sites as a function of $V$ for $n=1/2$. The value of (c) the local charge densities and (d) the local magnetic moments at three inequivalent sites as a function of $V$ for $n=2/3$. These results are obtained for $U=8$ and $V^{\prime }=V$.