Projective symmetry group classification of $Z_3$ parafermion spin liquids on a honeycomb lattice

TL;DR

This paper introduces a parafermion parton approach to classify $Z_3$ parafermion spin liquids on a honeycomb lattice, revealing numerous symmetric solutions and providing a systematic method to discover exotic spin liquids with anyonic excitations.

Contribution

It develops a novel parafermion parton framework and projective symmetry group classification for $Z_3$ spin liquids, expanding the understanding of their symmetry properties and solution space.

Findings

Identified 9 types and 102 solutions with certain symmetries.

Found 9 types and 36 solutions under combined parity and time-reversal symmetry.

Provided a systematic classification method for exotic spin liquids with parafermion excitations.

Abstract

To study exotic excitations described by parafermions in the possible spin liquid states of SU($n$) spin systems, we introduce a parafermion parton approach. The SU($n$) 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 SU($n$) spins can be decomposed into $n$ parafermion matrices of degree one. In this decomposition, the spin has a $\{\bigotimes{\rm SU}(n)\}^{n-1}$ 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, $6$-fold rotation and combination of parity and time reversal, there are $9$ types and $102$ solutions for two-dimensional $Z_3$ parafermion spin liquids on the honeycomb lattice. On the contrast, there are $9$ types and $36$ solutions if both parity and time-reversal symmetries are present. Our results provide a novel route for the systematic search of new types of spin liquids with exotic anyon excitations.