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Communications Physics

Feature spectrum topology and bulk boundary correspondence in energy and projective operator spectra

Baokai Wang, Yi-Chun Hung, Xiaoting Zhou, Tze Tzen Ong, Hsin Lin

Published Jul 30, 2026• 0 citations
Publisher

Abstract

Abstract Bulk-boundary correspondence, a key result of ground-state topology, predicts gapless boundary states at interfaces between distinct phases. Motivated by the limitations of conventional band topology in describing boundary states under weak symmetry-breaking perturbations, we propose feature spectrum topology, a generalized framework based on the spectrum of projected operators of the form $$F=P\widehat{O}P$$ F = P O ̂ P , where P projects onto a manifold of interest, and $$\widehat{O}$$ O ̂ is an operator representing physical observables. We rigorously establish a bulk-boundary correspondence in this projected spectrum, showing that the topological invariant guarantees gapless boundary states appearing in either the energy spectrum or the projected operator spectrum. This feature-energy complementarity uncovers boundary phenomena that are overlooked by traditional energy-based approaches. Our framework provides a unifying perspective for characterizing topological features tied to observables, offering new insights into systems that extend beyond the conventional paradigm. We demonstrate the application and discuss possible experimental observations of our theory in topological materials.

DOI
10.1038/s42005-026-02705-5

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Abstract Bulk-boundary correspondence, a key result of ground-state topology, predicts gapless boundary states at interfaces between distinct phases. Motivated by the limitations of conventional band topology in describing boundary states under weak symmetry-breaking perturbations, we propose feature spectrum topology, a generalized framework based on the spectrum of projected operators of the form F=PO^PF=P\widehat{O}PF=POP F = P O ̂ P , where P projects onto a manifold of interest, and O^\widehat{O}O O ̂ is an operator representing physical observables. We rigorously establish a bulk-boundary correspondence in this projected spectrum, showing that the topological invariant guarantees gapless boundary states appearing in either the energy spectrum or the projected operator spectrum. This feature-energy complementarity uncovers boundary phenomena that are overlooked by traditional energy-based approaches. Our framework provides a unifying perspective for characterizing topological features tied to observables, offering new insights into systems that extend beyond the conventional paradigm. We demonstrate the application and discuss possible experimental observations of our theory in topological materials.
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.
Topology is a fundamental aspect of quantum physics, and it has led to key breakthroughs and results in various fields of quantum materials. In condensed matters, this has culminated in the recent discovery of symmetry-protected topological phases. However, symmetry-based topological characterizations rely heavily on symmetry analysis and are incapable of detecting the topological phases in systems where the symmetry is broken, thus missing a large portion of interesting topological physics. Here, we propose a new approach to understanding the topological nature of quantum materials, which we call feature spectrum topology. In this framework, the ground-state is separated into different partitions by the eigenspectrum of a feature, a particular chosen internal quantum degree of freedom, such as spin or pseudo-spin, and the topological properties are determined by analysis of these ground-state partitions. We show that bulk-boundary correspondence guarantees gapless spectral flows in either one of the energy or feature spectrum. Most importantly, such 'feature-energy duality' of gapless spectral flows serves as a fundamental manifestation of a topological phase, thereby paving a new way towards topological characterizations beyond symmetry considerations. Our development reveals the topological nature of a quantum ground state hidden outside symmetry-based characterizations, hence, providing a platform for a more refined search of unconventional topological materials.
In this study, we discuss a type of bulk-boundary correspondence, which holds for topological insulators and superconductors when the parity-time ( P T ) and/or parity-particle-hole ( P C ) symmetry are present. In these systems, even when the bulk topology is nontrivial, the edge spectrum is generally gapped, and thus the conventional bulk-boundary correspondence does not hold. We find that, instead of the edge spectrum, the single-particle entanglement spectrum becomes gapless when the bulk topology is nontrivial: i.e., the holds in P T - and/or P C -symmetric topological insulators and superconductors. After showing the correspondence using K -theoretic approach, we provide concrete models for each symmetry class up to three dimensions where nontrivial topology because of P T and/or P C is expected. An implication of our results is that, when the bulk topology under P T and/or P C symmetry is nontrivial, the noninteracting many-body entanglement spectrum is multiply degenerate in one dimension and is gapless in two or higher dimensions. Published by the American Physical Society 2024
We establish a quantitative bulk-boundary correspondence in interacting topological insulators by relating many-body topological invariants to characteristic degeneracy structures in the entanglement spectrum. Focusing on generalized Su-Schrieffer-Heeger chains with higher winding number, we construct a gauge-invariant many-body winding invariant based on Pancharatnam geometric phases that remains well defined in the presence of interactions. We show that this invariant uniquely determines the low-lying entanglement-spectrum degeneracy, which exhibits a universal 4ν4^ν4ν scaling with the winding number ννν, providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.
The discovery of topological phases has ushered in a new era of condensed matter physics and revealed a variety of natural and artificial materials. They obey the bulk-boundary correspondence (BBC), which guarantees the emergence of boundary states with nonzero topological invariants in the bulk. Widespread attention has been paid to extending topological phases to nonlinear and non-Hermitian systems. However, the BBC and topological invariants of non-Hermitian nonlinear systems remain largely unexplored. Here, we establish a complete BBC and topological characterization of the topological phases in a class of non-Hermitian nonlinear-eigenvalue systems by introducing an auxiliary system. We restore the BBC broken by non-Hermiticity via employing the generalized Brillouin zone on the auxiliary system. Remarkably, we discover that the interplay between non-Hermiticity and nonlinearity creates an exotic complex-band topological phase that coexists with the real-band topological phase. Our results enrich the family of nonlinear topological phases and lay a foundation for exploring novel topological physics in metamaterial systems.
Abstract The topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.