Understanding the Chiral Topological Behavior of Certain PEPS Models Through Detection of Conserved Quantities from the Conformal Boundary State Description of the Real Space Entanglement Spectrum

Degeneracies in momentum of low-lying levels of a (2+1)D chiral topological state's entanglement spectrum (ES), numerically computed across a finite real-space cut, serve as a fingerprint of the topological nature of the bulk state. Using the conformal boundary state description, we examined the splittings of these degeneracies in systems with global SU(2) symmetry. The splittings arise from higher-order conservation laws of the conformal field theory (CFT) in the sense of a generalized Gibbs ensemble (GGE), strongly limited in number by global SU(2) symmetry. The ability to explain the splittings solely in the context of CFT via a GGE serves as a finer diagnostic (than the degeneracies alone) of the chiral topological behavior of the bulk. The capability of interacting PEPS tensor networks to describe chiral topological states is not understood. We analyzed earlier PEPS numerical studies with ES degeneracies characteristic of SU(2) Chern-Simons theory at levels one and two. We find that splittings in the low-lying ES are well characterized by a number of higher-order conservation laws. Thus, at the considered system sizes, the PEPS states appear chiral also under our more sensitive diagnostic. Notably, for level two, fermionic conservation laws of fractional spin are required.