Projective symmetry group classification of ${\mathbb{Z}}_3$ parafermion spin liquids on a honeycomb lattice

To study exotic excitations described by parafermions in the possible spin liquid states of $\mathrm{SU}\left(n\right)$ spin systems, we introduce a parafermion parton approach. The $\mathrm{SU}\left(n\right)$ spin operators can be represented by clock and shift matrices, which are shown to be the polynomials of parafermion operators in the parafermion representation. We find that $\mathrm{SU}\left(n\right)$ spins can be decomposed into $n$ parafermion matrices of degree one. In this decomposition, the spin has a ${\{⨂\mathrm{SU}\left(n\right)\}}^{n-1}$ gauge symmetry. As an application, we study the one-dimensional three-state clock model and generalized Kitaev model by a mean-field theory; both of them have been proved to be related to parafermion excitations. We find that with the symmetries of translations, sixfold rotation, and combination of parity and time reversal, there are 9 types and 102 solutions for two-dimensional ${\mathbb{Z}}_3$ parafermion spin liquids on the honeycomb lattice. By contrast, there are 9 types and 36 solutions if both parity and time-reversal symmetries are present. Our results provide a route for the systematic search of types of spin liquids with exotic anyon excitations.

Figure 1

Space transformations of the honeycomb lattice. The three bond types are colored differently.