Chiral topological order in tensor networks is typically identified through a characteristic counting of multiplets in the entanglement spectrum. This, however, is challenging: Those entanglement spectra can only be computed on cylinders of relatively small circumference, and there are non-chiral phases which exhibit precisely the same counting as chiral phases unless one probes them on large enough cylinders. In this work, the authors devise a novel way to identify chiral phases, and to disambiguate them from non-chiral phases which display the same counting: As they prove, non-chiral phases exhibit a splitting of conjugate multiplets in the entanglement spectrum which is forbidden for chiral phases. This splitting, or the absence thereof, can already be probed for small system sizes, and thus provides a very sensitive novel probe for chirality in Projected Entangled Pair State simulations of entangled quantum matter.

If you would like to know more, take a look at the abstract below, or at the published paper in Physical Review B [Phys. Rev. B 110, 235147 (2024)] or the open-access version on arXiv.

Abstract
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 [Kurečić 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 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 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.

This work has received support through the ERC grant SEQUAM and the Austrian Science Fund FWF (grant DOIs 10.55776/P36305 and 10.55776/COE1). Calculations were partly performed on the Vienna Scientific Cluster (VSC).