Figure 1

Three competing periodic $2\times 2$ spin textures in the triangular KL model at $3/4$ filling. The (a) $1q$, (b) $2q$, and (c) $3q$ phases are named according to their number of reciprocal lattice vectors. The $3q$ phase maximizes chirality $\chi ={\mathbf{S}}_i\times {\mathbf{S}}_j·{\mathbf{S}}_k=\pm 4/3^{3/2}$ averaged over triangular plaquettes $\left[ijk\right]$ and yields a quantum Hall effect at 1/4 and $3/4$ fillings. The $1q$ and $2q$ phases break rotational symmetry.Reuse & Permissions

Figure 2

Phase diagram of the triangular KL model with $J=0.2$ and $\mu =1.947$, corresponding to filling fraction $\sim 3/4$. (a) At low temperatures the preferred $3q$ phase is identified by $\langle \chi \rangle \ne 0$. System size $N={100}^2$ (circles) is sufficient to stabilize the $3q$ phase, but $N={60}^2$ is not. (b) Spin-spin correlation functions $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ for three nearest-neighbor orientations. Three first-order phase transitions are apparent: $3q\to 2q$ at $T=0.0010$, $2q\to 1q$ at $T=0.0017$, and $1q\to$ paramagnet at $T=0.0029$.Reuse & Permissions

Figure 3

Phase ordering in the $200\times 200$ triangular KL model after a quench from $T=\infty$ to $T=0$. The color gradient, ranging from red to blue, is the local chirality. Langevin time is in units of $\tau =1.28\times {10}^4$. (a) $J=3$, $\mu =-3.2$ ($\sim 1/4$ filling). Domain coarsening with strong anisotropy is observed. ${\mathbb{Z}}_2$ vortices (white dots) rapidly annihilate each other. (b) $J=0.2$, $\mu =1.947$ ($\sim 3/4$ filling). The system is trapped in a complex metastable state. (c) $J=0.2$, $\mu =1.947$, $B_z=8\pi /\sqrt{3N}$ ($\sim 3/4$ filling). An external field breaks chiral symmetry, and the system rapidly evolves to the $3q$ ground state. A ${\mathbb{Z}}_2$ vortex is identified by the winding of a Burger's circuit (green) in $SO\left(3\right)$ space, a filled projective sphere in the axis-angle representation. A ${\mathbb{Z}}_2$ vortex is enlarged in the third panel.Reuse & Permissions