Chiral spin liquid in a $\mathbb{Z}_3$ Kitaev model

We study a $\mathbb{Z}_3$ Kitaev model on the honeycomb lattice with nearest neighbor interactions. Based on matrix product state simulations and symmetry considerations, we find evidence that, with ferromagnetic isotropic couplings, the model realizes a chiral spin liquid, characterized by a possible $\mathrm{U}(1)_{12}$ chiral topological order. This is supported by simulations on both cylinder and strip geometries. On infinitely long cylinders with various widths, scaling analysis of entanglement entropy and maximal correlation length suggests that the model has a gapped 2D bulk. The topological entanglement entropy is extracted and found to be in agreement with the $\mathrm{U}(1)_{12}$ topological order. On infinitely long strips with moderate widths, we find the model is critical with a central charge consistent with the chiral edge theory of the $\mathrm{U}(1)_{12}$ topological phase. We conclude by discussing several open questions.