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\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} 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(Z3) 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.
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 [Kure\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} 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.
Projected entangled pair states (PEPS) provide exact representations for many nonchiral topologically ordered states whereas their range of applicability to interacting chiral topological phases remains largely unsettled. In this context, the symmetries of the local PEPS tensors are crucial for determining the characteristic topological features of the state. In this paper we examine the constraints that arise when different symmetries are imposed simultaneously on the local tensor such as internal
SU(2), point group, and translation symmetry. We show how the interplay of these symmetries manifests in the entanglement spectrum which is the main diagnostic tool for chiral topological order. We apply our results to a spin liquid PEPS introduced previously as a chiral generalization of the resonating valence bond state. Our findings explain the discrepancies observed between the entanglement spectrum of this state and the expected edge spectrum described by a chiral conformal field theory. Finally, in a certain parameter region where this PEPS possesses an additional
U(1) symmetry we are able to resolve these discrepancies and obtain an entanglement spectrum with the expected state countings and conformal weight.
Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-
21 quantum spin liquids with
Z2 gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)
1), as is the case for the $\ensuremath{\nu}$=
21 (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.
Abstract Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the Z 8 classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.
Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.