Entanglement spectra of nonchiral topological (2+1)-dimensional phases with strong time-reversal symmetry breaking, Li-Haldane state counting, and PEPS

The Li-Haldane correspondence [Phys. Rev. Lett. 101, 010504 (2008)] is often used to help identify wave functions of ( 2 + 1)-dimensional chiral topological phases (i.e., with nonzero chiral central charge) by studying low-lying entanglement spectra (ES) on long cylinders of finite circumference. Here we consider such ES of states [in fact, of wave functions of certain projected entangled pair states (PEPSs)] that are not chiral (i.e., having zero chiral central charge) but which strongly break time-reversal as well as reflection symmetry, while preserving their product, thus exhibiting the same symmetry properties as a chiral topological state. This leads to ES which have branches of both right- and left-moving chiralities, but with vastly different velocities. For circumferences much smaller than the inverse entanglement gap scale, the low-lying ES appear chiral in some topological sectors, and precisely follow the Li-Haldane state counting of a truly chiral phase. On its face, this could lead one to mistakenly identify the phase as chiral. However, by considering the ES in all possible sectors, one can observe distinct differences from a chiral phase. We explore this phenomenon in the setting of an SU(3) spin liquid PEPS studied by Kure.ci ' c et al. [Phys. Rev. B 99, 045116 (2019)], where the topologically trivial sector appeared to have the characteristic Li-Haldane counting of a chiral SU( 3)-level-one [SU(3)(1)] conformal field theory (CFT). In fact, the PEPS has D(Z(3)) topological order, with nine sectors. We compute the ES in minimally entangled states corresponding to all these sectors, which map to the nine anyon types of doubled SU(3)-level-one Chern-Simons topological field theory. The state countings of the ES in all sectors are exactly consistent with our expectation: specifically, the ES contain representations of global SU(3) symmetry from the tensor products of the (lowest-lying) multiplet of primary states of a "high-velocity" chiral SU(3)(1) CFT with the full content of a "low-velocity" chiral SU(3)(1) CFT sector, a nonchiral structure beyond that observable in the topologically trivial sector of the ES.