Spin Liquid Landscapes in the Kagome Lattice: A Variational Monte Carlo Study of the Chiral Heisenberg Model and Experimental Signatures

Abstract

Chiral spin liquids, which break time-reversal symmetry, are of great interest due to their topological properties and fractionalized excitations (anyons). In this work, we investigate chiral spin liquids (CSL) on the kagome lattice arising from the competition between the third-nearest-neighbor Heisenberg interaction across hexagons ($J_d$) and a staggered scalar spin chirality term ($J_χ$). Using variational Monte Carlo methods, we map out the phase diagram and identify various gapped and gapless CSL phases, each characterized by a distinct flux pattern. Notably, the interplay between $J_d$ and $J_χ$ induces a tricritical point, which we analyze using Landau-Ginzburg theory. Additionally, we identify potential signatures of these CSLs-including distinctive spin-spin correlations, anomalies in the static spin structure factor, longitudinal thermal conductivity, and magentoelectric effects-which offer practical guidance for their future experimental detection.

Figures (3)

• Figure 1: (a) The flux pattern of Eq. \ref{['eq:J1JdJcMeanHamiltonian']} on the kagome lattice. The $U(1)$ gauge fluxes are $\Phi_{\vartriangle} = \theta + \phi$ and $\Phi_{\triangledown} = \theta - \phi$ through the triangles and $\Phi_{\hexagon} = -2\theta$ through the hexagons. The red dotted line indicates the enlarged unit cell of the type-1 flux pattern. The fermionic spinon hopping amplitude is $|t_{ij}| = 1$ for all nearest-neighbor pairs. (b) The variational phase diagram of the $J_{1}-J_{d}-J_{\chi}$ model for a $N=3 \times 12 \times 12$ site cluster, constructed by comparing energies calculated with VMC for competing Gutzwiller-projected states with the flux parameters $\theta$ and $\phi$ optimized at each point in parameter space. Within this framework, four stable phases appear: the Dirac spin liquid (DSL), the Dirac chiral spin liquid (Dirac CSL), the gapped chiral spin liquid (gapped CSL), and the staggered chiral spin liquid (staggered CSL). The radius of the blue circles encodes the magnitude of the staggered flux $\phi$, while the intensity of the gray shading is proportional to $\theta$, where $\Phi_{\hexagon} = \pi - 2\theta$. In the magenta region $\phi = \pi/2$, $\theta = 0$, and $\Phi_{\hexagon} = 0$. The flux patterns and point group symmetries of each phase are detailed in Table \ref{['tab:tab']}. Solid lines represent second-order phase transition boundaries, while dotted lines indicate first-order transitions. Turning on $J_{\chi}$ immediately leads to time-reversal symmetry breaking and drives a phase transition from the DSL to the Dirac-CSL.
• Figure 2: (Top) Spin-spin correlation of (a) Dirac spin liquid ($\Phi_\vartriangle = 0$, $\Phi_\triangledown=0$, $\Phi_{\hexagon}=\pi$), (b) gapped chiral spin liquid ($\Phi_\vartriangle = \pi/8$, $\Phi_\triangledown = \pi/8$, $\Phi_{\hexagon}=3\pi/4$), and (c) staggered chiral spin liquid ($\Phi_\vartriangle=\pi/2$, $\Phi_\triangledown=-\pi/2$, $\Phi_{\hexagon}=0$), where ($\Phi_\vartriangle$, $\Phi_\triangledown$, $\Phi_{\hexagon}$) denote the fluxes through the up and down triangles and hexagons of the kagome lattice, respectively. We plot variational Monte Carlo (VMC) and mean-field results for each case. (Bottom) Corresponding static spin structure factor. The dotted and solid lines represent the 1st and the extended Brillouin zone, respectively. The structure factor is normalized over the extended Brillouin zone, $\sum_{\bm{q}\in \text{EBZ}}S(\bm{q}) = 1$.
• Figure 3: Diagrams illustrating the reduction of the grey magnetic point group symmetry $\{1,\mathcal{T}\} \otimes \mathsf{D_{6h}}$ in spin liquids with different flux patterns, indicated by red arrows on the edges of the kagome lattice. Panel (a) shows the full grey point group, while panels (b)–(d) correspond to symmetry reductions induced by finite fluxes: $\theta\neq 0$ in (b) and (d), and a staggered flux ($\phi\neq 0$) in (c) and (d). The stereograms inside the hexagons represent the magnetic point group associated with each flux configuration. Black symbols denote elements of the unitary subgroup, whereas red symbols indicate elements of the antiunitary subgroup, combining point-group symmetry operations with time-reversal symmetry $\mathcal{T}$.