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.
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.
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=POP F = P O ̂ P , where P projects onto a manifold of interest, and 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.
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.
The entanglement spectrum is a useful tool to study topological phases of matter, and contains valuable information about the ground state of the system. Here, we study its properties for free non-Hermitian systems for both point-gapped and line-gapped phases. While the entanglement spectrum only retains part of the topological information in the former case, it is very similar to Hermitian systems in the latter. In particular, it not only mimics the topological edge modes, but also contains all the information about the polarization, even in systems that are not topological. Furthermore, we show that the Wilson loop is equivalent to the many-body polarization and that it reproduces the phase diagram for the system with open boundaries, despite being computed for a periodic system.
Topological insulators lie outside the standard Landau paradigm: Instead of being characterized by local order parameters and spontaneous symmetry breaking, their defining properties are encoded in the global structure of their quantum states and manifested as robust boundary states. Crystalline higher-order topological insulators can extend this principle by hosting localized corner modes in two and higher dimensions. Although these boundary modes can be detected spectroscopically through the local density of states, determining the underlying bulk topology, for example through nested Wilson loops, remains an outstanding experimental challenge. Here we use up to 728 qubits in a superconducting analogue-digital quantum simulator to realize and probe a hierarchy of one-, two-, and three-dimensional topological insulators governed by effective tight-binding Hamiltonians. By reconstructing complex propagators from real-space measurements, we obtain complementary boundary and bulk characterizations: local densities of states reveal localized boundary modes, whereas experimentally reconstructed band projectors yield bulk topological invariants, culminating in a three-dimensional octupole moment extracted through third nested Wilson loops. These results establish a route to realizing, characterizing, and manipulating topologically protected states in programmable three-dimensional quantum matter.
This paper explores the rigorous formulation of entanglement spectra within the framework of C*-algebras, proposing their utility as robust topological invariants for classifying quantum phases of matter. Entanglement spectra, derived from the eigenvalues of reduced density matrices, have emerged as powerful diagnostic tools for characterizing topological order, often revealing hidden patterns beyond conventional symmetry-breaking paradigms. By embedding quantum systems and their subsystem decompositions into the abstract mathematical structure of C*-algebras, we provide a generalized and mathematically rigorous approach to define and analyze these spectra. This C*-algebraic perspective allows for a unified treatment of finite and infinite quantum systems, including those arising in quantum field theory, and naturally incorporates the algebraic notion of locality and quantum states. We demonstrate how specific features of the entanglement spectrum, such as degeneracy patterns or gaps, can be viewed as invariants under C*-algebraic deformations that preserve the underlying topological order. This framework suggests a deep connection between the algebraic properties of observables, the structure of entanglement, and the classification of topological phases, potentially linking to established mathematical tools like K-theory. The work establishes a theoretical foundation for leveraging entanglement spectra as fundamental topological indicators within a highly abstract yet physically relevant algebraic setting.
Using bulk gapless topological superconductors in both 1D and 2D as free fermion model examples, we demonstrate the power of subsystem correlation spectrum (the spectrum of correlation matrix), or equivalently the entanglement spectrum for the case of free fermions, in characterizing the topology of the nontrivial ground state. For the systems considered, we show that signatures of the low-energy spectrum, including both the edge modes and the bulk modes, appear in the correlation spectrum, albeit with different behaviors. This work generalizes the 2D Li-Haldane entanglement spectrum characterization of topological edge states to 2D topological systems with gapless bulk.