Nested Feature Spectrum Topology: Tripartite Topological Equivalence of Feature, Entanglement, and Wilson Loop Spectrum

• URL: http://arxiv.org/abs/2603.13128v1
• Date: Fri, 13 Mar 2026 16:19:26 GMT
• Title: Nested Feature Spectrum Topology: Tripartite Topological Equivalence of Feature, Entanglement, and Wilson Loop Spectrum
• Abstract summary: We show that the feature spectrum encodes the entanglement between sectors of the quantum observable.<n>We prove that spectral flow in the entanglement spectrum and the Wilson loop winding in the feature spectrum are equivalent manifestations of the feature-energy complementarity.
• Score: 0.8870188183999853
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Topological phases of matter are traditionally characterized through symmetry-based classifications. In cases of symmetry breaking, the projected spectrum - obtained by projecting the ground state onto the eigenstates of a pertinent quantum observable, such as spin or orbital angular momentum - provides a clear method for classifying topological phases. This approach underpins well-known frameworks such as spin-resolved topology and feature spectrum topology. Here we introduce nested feature spectrum topology, in which projection operators are applied recursively to subsectors of the feature spectrum, generating a hierarchy of feature spectra. We uncover a fundamental tripartite equivalence among the topology of feature, the entanglement, and the Wilson loop spectra in non-interacting fermionic systems. This equivalence reveals that the feature spectrum encodes the entanglement between sectors of the quantum observable, such as the spin-up and spin-down states in spin-resolved topology. We further prove that spectral flow in the entanglement spectrum and the Wilson loop winding in the feature spectrum are equivalent manifestations of the feature-energy complementarity: the appearance of gapless spectral flow in either energy or projected spectra on the boundary. This complementarity refines the conventional bulk-boundary correspondence by demonstrating that topological boundary modes may persist in the feature spectrum even when energy spectra are gapped. Our results provide a deeper understanding and solid foundation for the origin of band topology in the feature spectrum.

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