Entanglement spectrum as a diagnostic of chirality of topological spin liquids: Analysis of SU(3) projected entangled pair states

We address the key question of representation of chiral topological quantum states in $2+1$ dimensions (i.e., with nonzero chiral central charge) by projected entangled pair states (PEPS). A noted result (due to Wahl et al. [Phys. Rev. Lett. 111, 236805 (2013)], and Dubail and Read [Phys. Rev. B 92, 205307 (2015)] says that this is possible for noninteracting fermions, but the answer is as yet unknown for interacting systems. Characteristic counting of degeneracies of low-lying states in the entanglement spectrum (ES) at fixed transverse momentum of bipartitioned long cylinders (``Li-Haldane counting'') provides often-used supporting evidence for chirality. However, nonchiral PEPS (with zero chiral central charge), yet with strong breaking of time-reversal and reflection symmetries, with invariance under the product of these two operations (i.e., ``apparently'' chiral states), are known whose low-lying ES exhibits the same Li-Haldane counting as a chiral state in certain topological sectors [Kure\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. B 99, 045116 (2019); Arildsen et al., Phys. Rev. B 108, 245150 (2023)]. In this work, we identify a distinct indicator and hallmark of chirality in the ES of PEPS with global $\mathrm{SU}(3)$ symmetry: the splittings of conjugate irreps. We prove that in the ES of the chiral states conjugate irreps are exactly degenerate because the operators that would split them [related to the cubic Casimir invariant of $\mathrm{SU}(3)$] are forbidden. By contrast, in the ES of nonchiral states, conjugate splittings are demonstrably nonvanishing. Such a diagnostic provides an unambiguous and powerful tool to distinguish chiral and nonchiral topological states in $2+1$ dimensions via their entanglement spectra.