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The Li-Haldane correspondence [PRL 101, 010504 (2008)] is often used to helpidentify wave functions of (2+1)-D chiral topological phases (i.e., withnon-zero 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, certain Projected Entangled Pair States (PEPS)] that are not chiral(i.e., having zero chiral central charge), but which strongly breaktime-reversal as well as reflection symmetry, while preserving their product,the same symmetry as a chiral state. This leads to ES with branches of bothright- and left-moving chiralities, but with vastly different velocities. Forcircumferences much smaller than the inverse entanglement gap scale, thelow-lying ES appear chiral in some topological sectors, and precisely followthe Li-Haldane state counting of a truly chiral phase. This could lead one tomisidentify the phase as chiral. However, considering the ES in all sectors,one can observe distinct differences from a chiral phase. We explore this in an$SU(3)$ spin liquid PEPS studied by Kure\v{c}i\'c, et al. [PRB 99, 045116(2019)], where the topologically trivial sector has the state counting of achiral $SU(3)$-level-one [$SU(3)_1$] Conformal Field Theory (CFT). In fact, thePEPS has $D(\mathbb{Z}_3)$ topological order, with 9 sectors. We compute the ESin minimally entangled states corresponding to these sectors, which map to the9 anyon types of doubled $SU(3)_1$ Chern-Simons Topological Field Theory. Thestate countings of the ES coincide with our expectation: the ES contain irrepsof global $SU(3)$ symmetry from the tensor products of the (lowest-lying) irrepof primary states of a "high-velocity" chiral $SU(3)_1$ CFT with the fullcontent of a "low-velocity" chiral $SU(3)_1$ CFT sector, a non-chiral structurebeyond that observable in the topologically trivial sector of the ES.

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