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Pfaffian invariant identifies magnetic obstructed atomic insulators

by Isidora Araya Day, Anastasiia Varentcova, Daniel Varjas, Anton R. Akhmerov

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Submission summary

Authors (as registered SciPost users): Anton Akhmerov · Isidora Araya Day · Daniel Varjas
Submission information
Preprint Link:  (pdf)
Code repository:
Date accepted: 2023-07-31
Date submitted: 2023-06-05 13:02
Submitted by: Araya Day, Isidora
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We derive a $\mathbb{Z}_4$ topological invariant that extends beyond symmetry eigenvalues and Wilson loops and classifies two-dimensional insulators with a $C_4 \mathcal{T}$ symmetry. To formulate this invariant, we consider an irreducible Brillouin zone and constrain the spectrum of the open Wilson lines that compose its boundary. We fix the gauge ambiguity of the Wilson lines by using the Pfaffian at high symmetry momenta. As a result, we distinguish the four $C_4 \mathcal{T}$-protected atomic insulators, each of which is adiabatically connected to a different atomic limit. We establish the correspondence between the invariant and the obstructed phases by constructing both the atomic limit Hamiltonians and a $C_4 \mathcal{T}$-symmetric model that interpolates between them. The phase diagram shows that $C_4 \mathcal{T}$ insulators allow $\pm 1$ and $2$ changes of the invariant, where the latter is overlooked by symmetry indicators.

Author comments upon resubmission

Resubmission letter

We thank the referees for their feedback and for their overall positive evaluation. Below we list the detailed response to the referee inquiries, followed by the list of changes in the manuscript.

Response to referee 1

  1. The term mod 4 in Eq. (8) needs additional explanation. The statement in the text "The invariant [...] is well-defined modulo $4$ due to the gauge-fixing procedure" is somewhat cryptic.

    We have clarified the sentence by changing it to "The invariant is defined modulo $4$ because each dressed Wilson line determinant is well-defined modulo $2 \pi$."

  2. It would be helpful to explain the origin of the term $mod \; 2 \pi$ in Eq. (2) as this term is normally absent in Stokes' theorem.

    We have added the following statement "Since the Berry connection integral may change by multiples of $2\pi$ upon singular gauge transformations, while the Berry flux is fully gauge-invariant, Eq.(2) only holds modulo $2\pi$."

  3. Below Eq. (13) it is stated "we choose the upper right quadrant of the Brillouin zone as the IBZ for simplicity". Is this really allowed when evaluating Eq. (8)? The lower half of the IBZ highlighted in Fig. 2 is rotated into the upper right quadrant by $C_4 \mathcal{T}$. Since $C_4 \mathcal{T}$ involves time-reversal, it flips the sign of Berry curvature. Hence the Berry curvature integral over the upper right quadrant will in general be different from the integral over the IBZ of Fig. 2.

    The sign of the Berry curvature indeed flips upon the action of $C_4 \mathcal{T}$, however Equation (8) does not depend on the choice of the IBZ. Specifically, the IBZ in Fig. 2 deforms into the upper right quadrant of the BZ by a smooth deformation that changes both the Berry flux and the Wilson line continuously. This means that in the numerical implementation we consider the Wilson loop $M \to X \to M$ and twice the Wilson line $M \to Y \to \Gamma$. Since Eq. (8) only takes integer values, its value cannot change upon smooth deformations of the IBZ. Therefore, using either IBZ gives the same invariant, which we have confirmed numerically. We only use the right upper quadrant to simplify the integration over a square grid in the Brillouin zone. To clarify this we have added a sentence "Choosing a different IBZ does not change the invariant because all the possible IBZ are smoothly connected to one another, but Eq.(8) only takes integer values."

  4. Could there be a sign error in the second term of Eq. (4)? Using $\text{log} \; \text{det} = \text{tr} \; \text{log}$ results in $\int \text{tr} \mathcal{A}$ , not $-\int \text{tr} \mathcal{A}$ as would be required from Eq. (2).

    We thank the referee for finding a typo in our manuscript. In the previous version, Eq. (4) was consistent with Figure 2, but the Wilson loop's path went against the convention of using counter-clockwise paths in Stokes' theorem. In the current version, we have reversed the path of Figure 2, and changed the sign in Eq. (4), so these agree with Eq.(2-3). Since the inconsistency was only present in the text and not in the code, our results remain the same.

  5. The invariants $\delta$ and $\nu$ appear in Fig. 1 without explanation, and much earlier than their definition in the text.

    We have modified paragraphs 2 and 3 to introduce $\delta$ and $\nu$ before Figure 1.

  6. I do not agree with the statement "...because the phases within each pair are equivalent up to a fractional lattice vector translation, the symmetry indicators only provide an incomplete topological classification". For instance, take inversion symmetry in 1D. The two atomic insulator phases (with Wannier centers at the 1a or 1b Wyckoff positions) are equivalent up to a half lattice vector translation, yet they are fully resolved by symmetry indicators.

    We have modified the sentence to "However, even the full set of symmetry indicators only provides an incomplete topological classification: phases $\nu=0$ and $\nu=2$ have identical representation content at every high-symmetry momentum."

Response to referee 2

We thank the referee for the interesting suggestions about altermagnetism and the experimental relevance of our work. We believe that addressing these questions would require a separate investigation, and therefore we leave them to future work. We have addressed the referee's suggestion in the concluding remarks of the new version.

  1. “On the other hand, because the phases within each pair are equivalent ...” As pointed out in another review report, this sentence needs some further clarification. Fractional lattice vector translation itself doesn’t seem to provide a sufficient condition for the symmetry indicators to be identical. I believe that it would need to involve more details about the space group symmetry, e.g., location of the rotation center, to fully clarify this statement here''

    We have modified the sentence to "However, even the full set of symmetry indicators only provides an incomplete topological classification: phases $\nu=0$ and $\nu=2$ have identical representation content at every high-symmetry momentum."

List of changes

- Fixed a minus sign typo in Eqs. 4 and 8
- Fixed the orientation of the IBZ in Fig. 2
- Introduced $\nu$ and $\delta$ before Fig. 1
- Clarified the origin of mod 4 in Eq. 8
- Clarified the IBZ used for numerical integration
- Complemented the conclusion with altermagnets

We also attach a PDF file with the changes highlighted:

Published as SciPost Phys. 15, 114 (2023)

Reports on this Submission

Report 2 by Kai Sun on 2023-6-14 (Invited Report)


All concerns raised previously have been fully addressed by the authors, and I am delighted to recommend the publication of this manuscript.

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Report 1 by Frank Schindler on 2023-6-5 (Invited Report)


The authors have addressed my previous concerns in a satisfactory manner and I am happy to recommend publication.

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  • clarity: -
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