Yi-Chun Hung

Yi-Chun Hung

A physics PhD student at NEU

Band topology in projected spectrum

Symmetry protected topological (SPT) phases establishes a robust framework to efficiently characterize ground state topology through internal (Chiu et al., 2016) or crystalline symmetries (Bradlyn et al., 2017). However, the topological characterization in symmetry-breaking phases is comparatively less well-understood than that in SPT phases. Previous approach focusing on quantum spin Hall insulators (QSHIs) without the spin-U(1) symmetry or time-reversal symmetry (TRS) utlizes the spectrum of the projected operator:

$$ \mathcal{S}_{\mathbf{\hat{p}}}=P\mathbf{\hat{S}}\cdot\mathbf{\hat{n}}P, $$

where \(P\) is the projection operator of the manifold of interest and \(\mathbf{\hat{S}}\cdot\mathbf{\hat{n}}\) is the spin operator (pointing to \(\mathbf{\hat{n}}\)-direction) of interest (Prodan, 2009), leading to the spin-resolved topology (Lin et al., 2024). In this framework, the ground state topology is characterized by the band topology of the spectrum of \(\mathcal{S}_{\mathbf{\hat{p}}}\), which remains well-defined even when the spin-U(1) symmetry and TRS are broken. The corresponding bulk-boundary correspondence is also numerically demonstrated in previous work (Li et al., 2012). However, a rigorous proof and extensions to general cases beyond QSHIs remain to be elucidated.

To address this long-standing issue, we generalize this scheme to consider a general translationally-invariant operator \(\hat{O}\) that characterize relevant features of the system. By following the idea proposed in Prodan, 2009, the proposed framework focuses on the spectrum of:

$$ \mathcal{O}_{\hat{P}}=P\hat{O}P. $$

Most importantly, we provide a rigorous mathematical proof of the corresponding bulk-boundary correspondence in the projected spectrum along with an experimental method to measure it [1]. This idea is then demonstrated in mirror Chern insulators without mirror symmetry and tailored lattice models [1], as well as the topological orbital Hall effect in group IV materials [2]. Furthermore, we proof a tripartite equivalence between the projected, entanglement, and the Wilson loop spectra [3]. This reveal the fundamental relation between band topology and entanglement beyond SPT phases, where the corresponding entanglement entropy can be probed indirectly via optical methods.

In addition to band topology and their characterization, by using the framework of projected spectrum, we extend the topological lower bound of quantum weight \(K\) (Onishi and Fu, 2024) beyond the SPT phases [4], leading to

$$ K + K_c \geq \sum_\alpha |C_\alpha|. $$

Here, \(K_c\) is a quantum geometric correction arising from symmetry-breaking perturbations and \(C_\alpha\) is the Chern number of the projected spectrum. We also provide an examples in which the conventional topological lower bound fails but our proposed lower bound remain valid. These results demonstrate the fundamental relation between band topology and quantum geometry beyond SPT phases.

  • [1] Baokai Wang*, Yi-Chun Hung*, Xiaoting Zhou, Tzen Ong, and Hsin Lin. Feature Spectrm Topology. arXiv:2310.14832. The proof is provided in our later revision presented to the referees in the peer review section, which will be published soon.

  • [2] Baokai Wang*, Yi-Chun Hung*, Hsin Lin, Sheng Li, Rui-Hua He, and Arun Bansil. Topological characteristics and bulk-boundary correspondence in the orbital Hall effect. Phys. Rev. B 111, 195102 (2025).

  • [3] Yi-Chun Hung*, Tzen Ong*, and Hsin Lin. Tripartite Topological Equivalence of Feature, Entanglement, and Wilson Loop Spectrum from Nested Feature Spectrum Topology. arXiv:2603.13128.

  • [4] Yi-Chun Hung*, Yugo Onishi*, Hsin Lin, Liang Fu, and Arun Bansil. Extending Topological bound on Quantum Weight Beyond Symmetry-Protected Topological Phase. arXiv:2603.13041.