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

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.

Figures (4)

• Figure 1: The tripartite equivalence between the Wilson loop spectrum, spatially-resolved entanglement spectrum, and spatially-resolved feature spectrum has been extensively studied PhysRevLett.107.036601J.Phys.A:Math.Gen.36.L205PhysRevB.91.085119. Previous research has elucidated the connections between the Wilson loop spectrum and both the spatially-resolved entanglement spectrum and the spatially-resolved feature spectrum. In this work, we highlight the equivalence between the feature and the entanglement spectra and extend the tripartite equivalence to sectors within the feature spectrum.
• Figure 2: (A) The schematic feature spectrum with feature $\hat{O} = \hat{S}_z$ of a spin-Chern insulator when spin-$U(1)$ symmetry is broken, leading to two sectors determined by the sign of their dispersion in the feature spectrum. (B) The schematic of the corresponding feature energy complementarity, where the gapless spectral flows (magenta line) simultaneously appear in the feature spectrum when the energy edge states are gapped.
• Figure 3: (A) A schematic illustration of the entanglement spectrum from $(\mathbf{\hat{C}_A})_{ij}$ and the feature spectrum from $(\mathbf{\hat{F}_A})_{ij}$. In both spectra, the $1$-sector represents the occupied subspace of partition $A$ ($\mathcal{H}_{O_A,\text{(occ.)}}$). The $0$-sector in the former corresponds to unoccupied states in $A$ ($\mathcal{H}_{O_A,\text{(unocc.)}}$), while the one in the latter corresponds to occupied states in complementary partitions $A^{(c)}$ ($\mathcal{H}_{O_{A^{(c)}},\text{(occ.)}}$). (B) Under symmetry-breaking perturbations, even though boundary states become gapped in the energy spectrum, (C) the entanglement spectrum and (D) the feature spectrum still retain gapless spectral flows. The flows marked by ①–③(③') track the evolution of boundary states. The color map shows the texture of the chosen feature, with darker (brighter) colors corresponding to occupied (unoccupied) states.
• Figure 4: (A) Schematic illustration of the nested feature spectrum derived from $(\mathbf{\tilde{F}_{O_A,O^{'}\alpha}})_{ij}$. The 1 (0) sector indicates the intersection of the occupied subspace associated with feature $\hat{O}'$ ($\mathcal{H}_{O^{'}\alpha}^{(\text{occ.})}$) and the 1 (0) sector of the feature spectrum defined by $\hat{O}$. These intersections define the subspaces $\mathcal{H}_{O_A}^{(O^{'}\alpha)}$ and $\mathcal{H}_{O_{A^{(c)}}}^{(O^{'}_\alpha)}$ for the 1- and 0-sectors, respectively. (B) Feature-resolved spatial entanglement spectrum $\mathbf{\tilde{C}_{\hat{P}_A,O^{'}\alpha}}(\vec{k}\parallel,r_\perp)$ for the $+$ and $-$ sectors of the feature spectrum defined by $s_z\tau_x\sigma_0$ in a high-pseudospin-Chern insulator. Here, ${s_i}$, ${\tau_i}$, and ${\sigma_i}$ denote the spin, layer, and orbital degrees of freedom, respectively (Supplemental Materials Section S1 for model details SM).