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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 |
| Specialties: |
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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.
Current status:
Reports on this Submission
Report
While most part of the report has been well responded, I am still not convinced by the author's answer to my question about the dependence of the 3D topological invariant on the spin structure.
My understanding is that the K group has a canonical decomposition along dimensions, and therefore the 3D Z2 component is canonically separated from the lower dimensional components. The group addition corresponds to the direct sum of the Hamiltonians. Thus, there should exist a 3D Z2 topological invariant that can tell whether the 3D topological component is trivial or nontrivial.
Now, the spin CS term proposed by the author depends on the spin structure in the presence of nontrivial Stiefel-Whitney classes in lower dimensions. This contradicts my understanding stated above.
I think there are two possibilities: 1) there exists a physically prefered spin structure; 2) the correct spin strucure depends on the Stiefel-Whitney classes in order to determine whether the 3D component is nontrivial.
In my previous report, I proposed to fix this issue by considering the direct sum of the concerned q with q_{0,1} with global Takagi factorization. Here, q_{0,1} has trivial and nontrivial 3D topology, respectively.
In conclusion, the author has successfully formulated 3D topological invariants in the presence of nontrivial Stiefel-Whitney classes, where the previous invariant using global Takagi factorization fails. However, the spin dependence remains an unresolved issue. This point should be highlighted and may be clarified in future work.
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