Non-Abelian chiral spin liquid in a spin-$1$ antiferromagnet on the square lattice

RESUMO

Non-Abelian chiral spin liquids (CSLs) host non-Abelian anyon excitations and are promising platforms for topological quantum computation. However, realizing non-Abelian CSLs in short-range interacting spin systems remains elusive. Here, we study a spin-$1$ square-lattice model with the first- and second-neighbor bilinear ($J_1,J_2$) and biquadratic interactions ($K_1,K_2$), as well as the three-spin scalar chiral coupling $J_{\chi}$. Inspired by the evidence of the Moore-Read state in exact diagonalization and infinite Projected Entangled Pair States calculations, we fix $J_1=1.0$, $J_2/J_1=0.623$, $K_1/J_1=-0.176$, and obtain a quantum phase diagram for $0 \leq K_2/J_1 \leq 0.4$ and $0.3 \leq J_{\chi}/J_1 \leq 0.6$ by using the density matrix renormalization group (DMRG) calculations. We identify a stripe antiferromagnetic phase, a Néel antiferromagnetic phase, a magnetically ordered chiral spin state phase, and a non-Abelian CSL phase emerging near the boundaries of the ordered phases. The unbiased DMRG results provide strong evidence for the Moore-Read state, including the three topological sectors, the quantized spin Chern number $C=1$, and the level counting of low-lying entanglement spectra which is consistent with the chiral SU(2)$_2$ conformal field theory. Our results may shed new light on searching for non-Abelian CSLs in other higher-spin ($S > 1/2$) systems with short-range couplings.