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(2+1)-D chiral topological phases are often identified by studying low-lyingentanglement spectra (ES) of their wavefunctions on long cylinders of finitecircumference. For chiral topological states that possess global$\mathrm{SU}(3)$ symmetry, we can now understand, as shown in this work, thenature of the topological phase from the study of the splittings ofdegeneracies in the finite-size ES, at a given momentum, solely from theperspective of conformal field theory (CFT). This is a finer diagnostic thanLi-Haldane "state-counting", extending the approach of PRB 106, 035138 (2022)by two of the authors. We contrast ES of such chiral topological states withthose of a non-chiral PEPS (Kure\v{c}i\'c, Sterdyniak, and Schuch [PRB 99,045116 (2019)]) also possessing $\mathrm{SU}(3)$ symmetry. That latter PEPS hasthe same discrete symmetry as the chiral PEPS: strong breaking of separatetime-reversal and reflection symmetries, with invariance under the product ofthese two operations. However, the full analysis of the topological sectors ofthe ES of the latter PEPS in prior work [arXiv:2207.03246] shows lack ofchirality, as would be manifested, e.g., by a vanishing chiral central charge.In the present work, we identify a distinct indicator and hallmark of chiralityin the ES: the splittings of conjugate irreps. We prove that in the ES of thechiral states conjugate irreps are exactly degenerate, because the operators[related to the cubic Casimir invariant of $\mathrm{SU}(3)$] that would splitthem are forbidden. By contrast, in the ES of non-chiral states, conjugatesplittings are demonstrably non-vanishing. Such a diagnostic significantlysimplifies identification of non-chirality in low-energy finite-size ES for$\mathrm{SU}(3)$-symmetric topological states.

Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS