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A projective symmetry group (PSG) has been regarded as a classificationtheory of spin liquids. However, it does not include a symmetry-protectedtopological order of fermionic spinon excitations, and thus the classificationof gapped spin liquids is incomplete. We demonstrate the classification beyondPSG by utilizing the Kitaev model on the squareoctagon lattice, where twogapped spin liquids are distinguished by a topological $Z_2$ invariant. This$Z_2$ invariant can be defined solely by the time-reversal and translationsymmetries on condition that the time-reversal symmetry is implementedprojectively. Thus, it is a hidden class of topological Kitaev spin liquidswith helical edge states, which has been ignored for a long time. This suggeststhat there exists an unknown classification scheme of gapped spin liquidsbeyond PSG.

Topological $Z_2$ invariant in Kitaev spin liquids: Classification of gapped spin liquids beyond projective symmetry group