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:

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

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 \(\mathcal{P}_{\mathbf{\hat{S}}\cdot\mathbf{\hat{n}}}\), 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 \(\hat{O}\) by following the idea proposed in Prodan, 2009, focusing on the spectrum of:

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

Most importantly, we provide a rigorous mathematical proof of the corresponding bulk-boundary correspondence 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].

  • [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).