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

TL;DR

This paper introduces nested feature spectrum topology, revealing a tripartite equivalence among feature, entanglement, and Wilson loop spectra, which deepens understanding of topological phases and boundary modes in non-interacting fermionic systems.

Contribution

It establishes a fundamental tripartite equivalence among feature, entanglement, and Wilson loop spectra, and introduces nested feature spectrum topology with hierarchical projections.

Findings

Uncovered tripartite equivalence among feature, entanglement, and Wilson loop spectra.

Proved spectral flow and Wilson loop winding are equivalent manifestations of feature-energy complementarity.

Showed topological boundary modes can exist in the feature spectrum even when energy spectra are gapped.

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.