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$\mathbb{Z}_2$ topological invariant in three-dimensional PT- and PC-symmetric class CI band structures
by Ken Shiozaki
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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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Author: Ken Shiozaki on 2026-01-05 [id 6198]
(in reply to Report 1 on 2025-10-21)We thank the referee for the careful reading of our manuscript and for the constructive comments. Below we address each point in turn and describe the corresponding changes made in the revised version.
**Referee’s comment: **
>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.
**Our reply: **
We thank the referee for the comments.
In the presence of translational symmetry, the trivial phase is uniquely represented by the constant matrix q = 1_N (and those adiabatically connected to it). This representative has trivial lower-dimensional invariants w1 = w2 = 0, and therefore its triviality is free from any spin-structure dependence.
Correspondingly, nontrivial phases, which are not adiabatically connected to a constant matrix, are detected by the set of topological invariants w1, w2, and \nu. The detectability of nontrivial phases, namely whether a phase can or cannot be connected to a trivial one, is independent of the choice of spin structure, even though the value of \nu(q,\sigma) may depend on \sigma.
We have added the following paragraph:
We note that the apparent spin-structure dependence does not lead to any inconsistency. The physical distinction between trivial and nontrivial phases remains spin-structure independent. In translationally invariant systems, the trivial phase is uniquely represented by the constant matrix $q = 1_n$ (and those adiabatically connected to it), for which all lower-dimensional invariants vanish and no spin-structure dependence arises. Accordingly, the distinction between trivial and nontrivial phases, namely, whether a given phase can or cannot be connected to a trivial one, is independent of the choice of spin structure, even though the value of $\nu(q,\sigma)$ may depend on $\sigma$.
**Referee's comment: **
>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.
**Our reply: **
Yes. This is because the valence frame (12) obeys the same O(N) gauge transformation as the Takagi factors Q^\alpha. We have added the following sentence:
“Note that the SW class (10) coincides with that of the real occupied bands protected by PT symmetry, since the occupied-band frame (12) obeys the same gauge transformation $S^\alpha\beta$ as the Takagi factorization.”
**Referee's comment: **
>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.
**Our reply: **
We have added the following sentence just after the first appearance of the spin structure:
“A spin structure corresponds to a choice of periodic or anti-periodic boundary conditions for fermions along each nontrivial one-cycle. For instance, on the three-torus $T^3$, there are eight distinct spin structures corresponding to periodic or anti-periodic boundary conditions along the three non-contractible loops.”
**Referee's comment: **
>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.
**Our reply: **
>Thank you for the careful reading. We have corrected them.
**Referee's comment: **
>5. The connection in Eq. 13 and that in Eq. 15 are related by a unitary gauge transformation, which may be explicitly stated.
**Our reply:**
We have added the following sentence:
Actually, the real Berry connection (13) is obtained from $A^{\rm U}_\bk$ by the gauge transformation $A^{\alpha}_\bk = (Q_\bk^\alpha)^\top (A^{\rm U}_\bk + d) (Q_\bk^\alpha)^*$.