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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/2509.19825v2 (pdf) |
| Date submitted: | Sept. 30, 2025, 5:27 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 $\eta$-invariant, a spin-Chern--Simons (spin-CS) action. We show that PC symmetry quantizes the spin-CS action to $\{0,2\pi\}$ with $4\pi$ 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.
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In this work, the author points out that in general the Takagi factorization gives rise to an emergent principal O(N)-bundle, whose topological class can be characterized by its Stiefel-Whitney classes w1 and w2, and the global Takagi factor Q(k) exists if and only if the principal O(N)-bundle is trivial, namely w1=w2=0. In addition, w1 and w2 are just the Stiefel-Whiteney classes as lower dimensional topological invariants protected by PT symmetry. Thus, a global Takagi factor Q(k) exists if and only if lower-dimensional topological invariants w1 and w2 are all trivial.
Based on these observations, the author formulated a general 3D topological invariant, namely the spin Chern Simons term of the principal O(N)-bundle with the canonical connection (Eq. 13) from the Takagi factorization. The definition of the spin Chern Simons term requires a presumed spin structure, and the term depends on the spin structure in the presence of nontrivial w1 and w2.
While this is certainly an important contribution to this field, I have a concern about this spin-structure dependency. Physically q has trivial or nontrivial 3D topological invariant should be independent of the spin structure \sigma, since \sigma is just a convention. I elaborate this point from two aspects as follows.
First, when the lower dimensional topological invariants are trivial, there exists q_0(k) and q_1(k) with trivial and nontrivial 3D topological invariants, respectively, which can be determined by global Takagi factors. Then, the 3D Z_2 invariant \nu(q_1\oplus q, \sigma) should be 0 if q is nontrivial and should be 1 if q is trivial. This specifies a physically preferred spin structure.
Second, the splitting form of the K group in Eq. 16 implies that there exists a 3D Z2 invariant which is independent of the lower dimensional topological invariants. But now, the spin Chern-Simons term depends on lower dimensional topological invariants from its dependence on the spin structure.
Below, some additional comments are listed for the author’s consideration.
1. I wonder whether w_2(P_q) of the emergent O(N)-bundle is the same as the second Stiefel Whitney class of the real valence bands from PT symmetry.
2. For 3-torus, the eight spin structures correspond to the eight possible periodic/anti-periodic boundary conditions, which may be introduced in the manuscript for readers who are not familiar with the abstract concept of spin structure.
3. In the paragraph below Eq. 11, the gauge transformation of A misses the transposition of the first S.
4. In Eq. 13, Hermitian conjugation is missed for the first Q.
5. The connection in Eq. 13 and that in Eq. 15 are related by a unitary gauge transformation, which may be explicitly stated.
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