Chiral broken symmetry descendants of the kagome lattice chiral spin liquid

The breaking of chiral and time-reversal symmetries provides a pathway to exotic quantum phenomena and topological phases. Recent work has extensively explored the resulting emergence of chiral charge orders and chiral spin liquids (CSLs) on the kagome lattice. Such CSLs are closely tied to bosonic fractional quantum Hall states with anyonic quasiparticles; however, their connection to nearby ordered states has remained a mystery. Here, we use spin-wave theory, parton Gutzwiller wave functions, and exact diagonalization to show that two distinct magnetic orders with uniform scalar chirality—the XYZ umbrella state and the octahedral spin crystal–emerge as competing orders in close proximity to the CSL. In this letter, we highlight the intimate link between a topologically ordered liquid and broken symmetry states with nontrivial real-space topology.

Figure 1

Kagome lattice. (a) The chiral three-spin interactions and further neighbor two-spin couplings in the model $H_{\mathrm{spin}}$. (b) Proposed phase diagram of the model $H_{\mathrm{spin}}$ as we tune $J_3$ and the strength of quantum fluctuations via $1/S$ where $S$ is the spin length. For spin $S=\frac{1}{2}$, the extensively degenerate classical point evolves into an emergent chiral spin liquid, bounded by noncoplanar magnetic orders. The light blue region indicates where spin-wave fluctuations can destabilize the noncoplanar orders, hinting at a quantum spin liquid. (c) The classical limit XYZ ordering pattern, indicating spins pointing along $x,y,z$. The XYZ umbrella order has the same symmetries as the classical XYZ order but has an opening angle between the three spins on the triangle which lies between the orthogonal XYZ triad and coplanar ${120}^{\circ }$ order. (d) Octahedral magnetic order with quadrupled unit cell and spins pointing along $(x,\overline{x}), (y,\overline{y}), (z,\overline{z})$, where $\overline{x}\equiv -x$.

Figure 2

Parton theory phase diagram. Mean-field phase diagram of the $S=\frac{1}{2}$ parton theory of Eq. (2) as we vary $J_3/J_{\chi }$. For $J_3>0.092J_{\chi }$, we find a phase transition from the mean-field chiral spin liquid (CSL) into the octahedral state, while for $J_3<-0.077J_{\chi }$, we find an instability into the XYZ umbrella order, a state with the same symmetries as the XYZ state. The top line depicts the total Chern number of the half-filled parton bands in various phases as we tune $J_3/J_{\chi }$.

Figure 3

Gutzwiller wave function study. (a) Variational parameters used in our kagome wave function ansatz include a second-neighbor hopping of strength $\gamma$, fluxes $\mathrm{\Theta },\mathrm{\Phi }$ through elementary and large triangular plaquettes (shaded), a local Jastrow factor which suppresses the total $S^z$ on bow ties (excluding the central site), and a Weiss field ${\mathcal{B}}_{\mathrm{oct}}$ which induces octahedral order; field directions are depicted on the sites. (b) Variational energy vs ${\mathcal{B}}_{\mathrm{oct}}$ for an $8\times 8$ lattice for different $J_3$, showing a stable chiral spin liquid (CSL) state for $J_3=0$ and an instability to octahedral order for $J_3\gtrsim 0.02$. The inset shows the order parameter ${\mathcal{M}}_{\mathrm{oct}}$ which becomes nonzero in the ordered phase.

Figure 4

Exact diagonalization results. (a) Gaps to the lowest-lying $S_T^z=0,...7$ states for the 36-site cluster. (b) The induced $\langle S^{\alpha }\rangle$ for the 36-site cluster with a Zeeman field $h_{\mathrm{XYZ}}$ applied to a single triangle (shaded) at $J_3=-0.5$, and (c) with a $h_{\mathrm{oct}}$ applied to a single triangle (shaded) at $J_3=0.1$. The radius of the points are proportional to the overlap with the expected XYZ or octahedral ordering direction at each site. (d) $\langle S^{\alpha }\rangle$ on adjacent sites taken anticlockwise on any given single up triangle for the $24Rh$-site cluster in a uniform $h_{\mathrm{XYZ}}$ field with $J_3=-1.4$. (e) $\langle S^{\alpha }\rangle$ on adjacent sites on a single triangle for the 24-site cluster in a uniform $h_{\mathrm{oct}}$ field with $J_3=0.15$.