Chiral gapped states are universally non-topological

TL;DR

This work develops an operator-based bulk/edge framework for chiral gapped 2+1D states, showing that the entanglement Hamiltonian decomposes into bulk and boundary pieces whose boundary content maps to edge CFT data. By regulating sharp corners as holes, it derives universal corner contributions to entanglement entropy, a corner vector fixed-point equation, and modular-commutator expressions tied to the edge chiral central charge $c_{-}$, all captured by a corner conformal ruler. A logical framework of corner axioms is formulated, introducing the corner central charge $rak{c}_{ ext{tot}}$ and its stationarity, with $(rak{c}_{ ext{tot}})_{ ext{min}}>0$ signaling edge ungappability and a nonzero correlation length. Numerics on a lattice $p+ip$ superconductor corroborate the corner-entropy predictions, the modular-commutator relation, and demonstrate error reduction via a Hamiltonian-reconstruction gradient descent toward a zero-correlation fixed point. Overall, the paper argues that universal corner geometry reveals a conformal structure in the bulk entanglement that is not captured by TQFT alone and provides a practical diagnostic for ungappable boundaries.

Abstract

We propose an operator generalization of the Li-Haldane conjecture regarding the entanglement Hamiltonian of a disk in a 2+1D chiral gapped groundstate. The logic applies to regions with sharp corners, from which we derive several universal properties regarding corner entanglement. These universal properties follow from a set of locally-checkable conditions on the wavefunction. We also define a quantity $(\mathfrak{c}_{\text{tot}})_{\text{min}}$ that reflects the robustness of corner entanglement contributions, and show that it provides an obstruction to a gapped boundary. One reward from our analysis is that we can construct a local gapped Hamiltonian within the same chiral gapped phase from a given wavefunction; we conjecture that it is closer to the low-energy renormalization group fixed point than the original parent Hamiltonian. Our analysis of corner entanglement reveals the emergence of a universal conformal geometry encoded in the entanglement structure of bulk regions of chiral gapped states that is not visible in topological field theory.

Figures (39)

• Figure 1: Modular commutator on a "complete" and "incomplete" disk in (a) and (b) respectively.
• Figure 2: Partition of a disk into four regions.
• Figure 3: A region $A$ with a sharp corner (dashed line), regulated by a hole. We will show that it is useful to envision a boundary CFT groundstate living on the boundary of the hole (blue line).
• Figure 4: A0 and A1 partitions. The linear size of the regions are larger than the correlation length $\xi$.
• Figure 5: Implications from A0 and A1 conditions.

Theorems & Definitions (3)

• Theorem 3: Vector fixed-point equation
• Theorem 4
• proof