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$\mathbb{Z}_2$ topological invariant in three-dimensional PT- and PC-symmetric class CI band structures
by Ken Shiozaki
Submission summary
| Authors (as registered SciPost users): | Ken Shiozaki |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2509.19825v3 (pdf) |
| Date submitted: | Jan. 5, 2026, 3:33 a.m. |
| Submitted by: | Ken Shiozaki |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
English writing using ChatGPT (GPT-5)
Abstract
We construct a previously missing $\mathbb{Z}_2$ topological invariant for three-dimensional band structures in symmetry class CI defined by parity-time (PT) and parity-particle-hole (PC) symmetries. PT symmetry allows one to define a real Berry connection and, based on the $η$-invariant, a spin-Chern--Simons (spin-CS) action. We show that PC symmetry quantizes the spin-CS action to $\{0,2π\}$ with $4π$ periodicity, thereby yielding a well-defined $\mathbb{Z}_2$ invariant. This invariant is additive under direct sums of isolated band structures, reduces to a known $\mathbb{Z}_2$ index when a global Takagi factorization exists, and in general depends on the choice of spin structure. Finally, we demonstrate lattice models in which this newly introduced $\mathbb{Z}_2$ invariant distinguishes topological phases that cannot be detected by the previously known topological indices.