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

The Li-Haldane correspondence [Li, Haldane, PRL 101, 010504 (2008)] is often used to help identify wave functions of (2+1)-D chiral topological phases, 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 [PEPS]) that are not chiral, but which strongly break time-reversal as well as reflection symmetry. 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 corresponding inverse entanglement gap scale, the low-lying ES appear chiral in some topological sectors, and precisely follow the Li-Haldane state counting of a corresponding 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 the chiral phase. We explore this phenomenon in the setting of an SU(3) spin liquid PEPS [Kurečić, Vanderstraeten, Schuch, PRB 99, 045116 (2019)]. Potential implications on the so-far unresolved question concerning interacting chiral PEPS wave functions will be discussed.