Clustering of Topological Charges in a Kagome Classical Spin Liquid

Fractionalization is a ubiquitous phenomenon in topological states of matter. In this work, we study the collective behavior of fractionalized topological charges and their instabilities, through the $J_1\text{−}{J}_2\text{−}{J}_3$ Ising model on a kagome lattice. This model can be mapped onto a Hamiltonian of interacting topological charges under the constraint of Gauss’ law. We find that the recombination of topological charges gives rise to a yet unexplored classical spin liquid. This spin liquid is characterized by an extensive residual entropy, as well as the formation of hexamers of same-sign topological charges. The emergence of hexamers is reflected by a half-moon signal in the magnetic structure factor, which provides a signature of this new spin liquid in elastic neutron-scattering experiments. To study this phase, a worm algorithm has been developed which does not require the usual divergence-free condition.

Figure 1

(a) Red, blue, and green lines are, respectively, the first-, second-, and third-neighbor interactions on the kagome lattice. Couplings across a hexagon (dashed line) are not considered. The unit vectors ${\mathit{a}}_{i=1,2}$ are represented by orange arrows. (b) All possible charge states on a triangle with ${\sigma }^z=+1(-1)$ marked by red (blue) dots. (c)–(e) The ground state configurations for (c) $J<-1$, (d) $-1<J<0$, (e) $J>\frac{1}{3}$, whose magnetic unit cell is represented by a black rhombus. (f) One of the ground state configurations for $0<J<\frac{1}{3}$. The dashed circles denote hexamers. The region enclosed by the bold line is one of the same-charge clusters, and pink (light blue) circles denote the internal (boundary) sites of the cluster. (g) A hexamer CSL state on the Husimi cactus. Even if there are no hexagonal loops, the Husimi cactus allows for each triangle to neighbor exactly two triangles of the same charge, as is the case on average on the kagome lattice. Accordingly, it is a good approximation to compute the thermodynamics of the hexamer CSL [Fig. 2].

Figure 2

(a) Finite-temperature phase diagram. Circles and crosses are, respectively, phase transitions and crossovers. Black stars denote the exact phase boundaries at $T=0$. (b)–(e) Temperature dependence of (b) $C$ and (c) $S$ at $J=-0.1$, and that of (d) $C$ and (e) $S$ at $J=0.1$. Insets of (b) and (d) are the magnetic susceptibilities $\chi$. Red lines are results of Husimi-cactus calculations, and orange, green, and blue dots are, respectively, the results of Monte Carlo simulations with $L=32$, 64, and 84. $S$ is computed for $L=64$. (f) Temperature dependence of the density of hexamers ${\rho }_{\text{hex}}$ for $J=0.1$ for $L=64$. (g) Autocorrelation, $R_{\text{spin}}(t)\equiv \sum _i{\sigma }_i^z(0){\sigma }_i^z(t)/N_{\text{site}}$, for $J=0.1$, $T=0.03$. Red, green, and blue dots are for the single spin flip, the loop-update algorithm, and the new worm algorithm, respectively.

Figure 3

Magnetic structure factors for $(J,T)=$ (a) $(-0.1,0.5)$, (b) $(-0.1,0.1)$, (c) (0.1,0.5), and (d) (0.1,0.1). (e) The contribution from isolated hexamers. It confirms that the half-moon patterns observed in (d) are signatures of the hexamer formation. Additional features, such as the scattering at Brillouin-zone corners, are artifacts due to the fact that spins outside of the hexamers are not included in the computation of the structure factor in panel (e). Black lines denote Brillouin zones.