Chirality and ${\mathbb{Z}}_2$ vortices in a Heisenberg spin model on the kagome lattice

The phase diagram of the classical $J_1–J_2$ model on the kagome lattice is investigated by using extensive Monte Carlo simulations. In a realistic range of parameters, this model has a low-temperature chiral-ordered phase without long-range spin order. We show that the critical transition marking the destruction of the chiral order is preempted by the first-order proliferation of ${\mathbb{Z}}_2$ point defects. The core energy of these vortices appears to vanish when approaching the $T=0$ phase boundary, where both ${\mathbb{Z}}_2$ defects and gapless magnons contribute to disordering the system at very low temperatures. This situation might be typical of a large class of frustrated magnets. Possible relevance for real materials is also discussed.

Figure 1

(Color online) 12-sublattice Néel order on the kagome lattice. Top: The 12-site magnetic unit cell. The different sublattices are indicated by different numbers. Bottom: The order parameter in spin space and its mirror image. The two degenerate ground states only differ by their scalar chirality, namely, the triple product of three spins on a (shaded) triangle.

Figure 2

(Color online) Transition temperature $T^{\ast }$ from chiral order to full disorder versus the coupling ratio $J_2/\left|J_1\right|$. For $0\le J_2/\left|J_1\right|\le 1/3$, the ground state is ferromagnetic. For $J_2/\left|J_1\right|>1/3$, the ground state is the 12-sublattice Néel ordered phase described in the text and abbreviated here as “cuboc.” Dots: Results of Monte Carlo simulations (size effects are much smaller than the size of the symbols). The solid line is a guide to the eye.

Figure 3

(Color online) Top: Temperature dependence of the staggered chirality $⟨\left|\mathcal{C}\right|⟩$ and the specific heat for $J_2/\left|J_1\right|=0.38$ and $L=64$. Temperatures are measured relatively to $T^{\ast }$, the temperature where $C_v$ is maximum. Bottom: spatially averaged spin-spin correlation length ${\xi }_{\text{spin}}$ and the vortex density versus temperature.

Figure 4

(Color online) Evolution of the vortex density $⟨n_v⟩$ around the transition temperature $T^{\ast }$ as a function of $J_2/\left|J_1\right|$.