Topological phases of matter are characterized by invariants that resist local perturbation. Three apparently different diagnostic tools have been used to detect these phases: the feature spectrum (projecting the Hamiltonian onto subspaces defined by quantum observables like spin), the entanglement spectrum (eigenvalues of the reduced density matrix for a spatial partition), and the Wilson loop spectrum (the holonomy of the Berry connection around the Brillouin zone). Each has its own mathematical framework, physical motivation, and community of practitioners.
Hung, Ong, and Lin (arXiv:2603.13128) prove that for non-interacting fermionic systems, the three are equivalent. Feature spectrum topology, entanglement properties, and Wilson loop spectra are different views of the same underlying structure. The feature spectrum encodes entanglement between quantum observable sectors (spin-up versus spin-down, orbital angular momentum components), and its spectral flow is identical to Wilson loop winding — both manifest what the authors call feature-energy complementarity.
The recursive construction makes this precise: applying projection operators to feature spectra subsectors generates a nested hierarchy, and the topology at each level maps exactly onto the entanglement spectrum at the corresponding partition and the Wilson loop at the corresponding cycle. Topological boundary modes can appear in the feature spectrum even when the energy spectrum is fully gapped — the feature spectrum sees topology that the energy spectrum misses.
This refines the bulk-boundary correspondence. The conventional statement — gapless boundary modes signal nontrivial bulk topology — is a special case of a broader principle where the “boundary” can be in feature space rather than real space. The topology isn't bound to edges; it lives in the complementarity between any two conjugate descriptions of the same system.