SciPost Submission Page
Isotropic 3D topological phases with broken time reversal symmetry
by Helene Spring, Anton R. Akhmerov, Daniel Varjas
Submission summary
| Authors (as registered SciPost users): | Anton R. Akhmerov · Daniel Varjas |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202505_00069v2 (pdf) |
| Code repository: | https://doi.org/10.5281/zenodo.8212020 |
| Date accepted: | Feb. 3, 2026 |
| Date submitted: | Jan. 15, 2026, 11:27 a.m. |
| Submitted by: | Daniel Varjas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Axial vectors, such as current or magnetization, are commonly used order parameters in time-reversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a three-dimensional medium with average isotropy and inversion symmetry where time-reversal symmetry is systematically broken. We devise a model of an amorphous material with scalar time-reversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the bonds. The presence of only average spatial symmetries---continuous rotation and inversion---is sufficient to protect a topological phase, yielding a statistical topological insulator. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant for the effective continuum model, which guarantees gapless surface spectrum on any surface with an odd number of Dirac nodes, analogous to crystalline mirror Chern insulators. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant, which guarantees gapless surface spectrum on any surface with several overlapping Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a three-dimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We hereby resubmit our manuscript titled "Isotropic 3D topological phases with broken time reversal symmetry". Despite the overall positive evaluation, both Referees asked for major revision. In response to the Referees' comments, we revised our manuscript, improving its logical flow, expanding it to make it more self-contained and reproducible, and updating the list of references. We believe that our updated manuscript meets the standards of SciPost Physics.
Sincerely:
The Authors
List of changes
We have significantly revised the manuscript, focusing on making the narrative flow easier to follow, the arguments more self-contained, expanding references to literature, and improving the reproducibility of the results. A list of the major changes is below, and minor changes are mentioned in the replies to the Referees.
- We have restructured the manuscript by removing the "Results" section, and making its subsections into sections. We have also changed the order of sections to improve the logical flow of the presentation. The relabeling in effect is 2.1.1 -> 2.1, 2.1.2 -> 2.3, 2.1.3 -> 2.2, 2.2 -> 3, 3 -> 4.
- We significantly expanded the discussion of the symmetry constraints both on the continuum and tight-binding models in order to make the manuscript more self-contained.
- We added details of the calculations that would make them reproducible without relying on the code and software packages that we used to carry out these calculaion.
- We added more detailed plots about the phase transitions in both the single and double models.
- We explained the connection to the concept of toroidal moments, and added relevant references.
- We expanded the discussion to clarify our interpretation of the results in the single and doubled cases.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
I understand the apparent paradox from the C=2 case stems from the fact that in disordered systems topological protection in transport is not equivalent to the topological stability of the bulk invariants. The authors have made an effort to explain the limitations of their approach and expanded their numerics to further support their statements, including a computation of the invariant of the doubled model.
I believe the authors response and new manuscript are satisfactory and I recommend publication.
Recommendation
Publish (meets expectations and criteria for this Journal)