SciPost Submission Page
Two-dimensional higher-order Weyl semimetals
by Lizhou Liu, Qing-Feng Sun, and Ying-Tao Zhang
Submission summary
| Authors (as registered SciPost users): | Yingtao Zhang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202507_00076v2 (pdf) |
| Date submitted: | Dec. 5, 2025, 8:24 a.m. |
| Submitted by: | Yingtao Zhang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a theoretical scheme to realize two-dimensional higher-order Weyl semimetals using a trilayer topological film coupled with a $d$-wave altermagnet. Our results show that the trilayer topological film exhibits two-dimensional Weyl semimetal characteristics with helical edge states. Notably, the Weyl points are located at four high-symmetry points in the Brillouin zone, and the topology of symmetric subspaces governs the formation of these Weyl points and edge states. Upon introducing a $d$-wave altermagnet oriented along the $z$-direction, gaps open in the helical edge states while preserving two Weyl points, leading to the realization of two-dimensional higher-order Weyl semimetals hosting topological corner states. The nonzero winding number in the subspace along the high-symmetry line serves as a topological invariant characterizing these corner states, and the other subspace Hamiltonian confirms the existence of the Weyl points. Finally, a topological phase diagram provides a complete topological description of the system.
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
Dear Editor,
We thank you for forwarding the referee’s report on our manuscript “Two-dimensional higher-order Weyl semimetals” (scipost_202507_00076v1). We appreciate the referee’s detailed and constructive comments and have revised the manuscript accordingly.
The referee provided an overall positive and encouraging assessment. He/She rated the validity and significance of the work as high, the originality as top, and the clarity, formatting, and grammar as high or excellent. The report emphasizes the novelty of realizing a two-dimensional gapless phase in which bulk Weyl points coexist with higher-order corner states, and the usefulness of the algebraic subspace decomposition for defining analytical Chern and winding numbers supported by numerical band structures and nanoflake spectra.
In addition, the referee also offered several suggestions to further strengthen the manuscript, calling for clarification and extension at both the conceptual level and in terms of experimental relevance and numerical robustness. In the revised manuscript, we have implemented targeted changes along these lines: we clarify the scope of our symmetry-guided minimal model and the role of the algebraic symmetry M, provide a sharper characterization of the Weyl nodes and phase diagram through explicit low-energy and subspace analyses, and add an edge-theory and disorder study that underpins the origin and robustness of the corner states. We also expand the discussion of higher-order topology in two dimensions, finite-size effects, possible experimental probes, and numerical convergence. All new or modified text is highlighted in blue in the revised version.
We hereby resubmit the revised manuscript together with a detailed point-by-point response to the referee’s comments. We believe that the concerns raised in the report have been satisfactorily addressed and that the main results are now presented in a clearer and more comprehensive manner. We hope that the revised version will be suitable for publication in SciPost Physics.
Yours sincerely,
Lizhou Liu, Qing-Feng Sun, and Ying-Tao Zhang
We thank you for forwarding the referee’s report on our manuscript “Two-dimensional higher-order Weyl semimetals” (scipost_202507_00076v1). We appreciate the referee’s detailed and constructive comments and have revised the manuscript accordingly.
The referee provided an overall positive and encouraging assessment. He/She rated the validity and significance of the work as high, the originality as top, and the clarity, formatting, and grammar as high or excellent. The report emphasizes the novelty of realizing a two-dimensional gapless phase in which bulk Weyl points coexist with higher-order corner states, and the usefulness of the algebraic subspace decomposition for defining analytical Chern and winding numbers supported by numerical band structures and nanoflake spectra.
In addition, the referee also offered several suggestions to further strengthen the manuscript, calling for clarification and extension at both the conceptual level and in terms of experimental relevance and numerical robustness. In the revised manuscript, we have implemented targeted changes along these lines: we clarify the scope of our symmetry-guided minimal model and the role of the algebraic symmetry M, provide a sharper characterization of the Weyl nodes and phase diagram through explicit low-energy and subspace analyses, and add an edge-theory and disorder study that underpins the origin and robustness of the corner states. We also expand the discussion of higher-order topology in two dimensions, finite-size effects, possible experimental probes, and numerical convergence. All new or modified text is highlighted in blue in the revised version.
We hereby resubmit the revised manuscript together with a detailed point-by-point response to the referee’s comments. We believe that the concerns raised in the report have been satisfactorily addressed and that the main results are now presented in a clearer and more comprehensive manner. We hope that the revised version will be suitable for publication in SciPost Physics.
Yours sincerely,
Lizhou Liu, Qing-Feng Sun, and Ying-Tao Zhang
List of changes
List of Modifications
1) We have added a new Fig. 4, and the original Fig. 4 has been renumbered as Fig. 5.
2) We have added a new reference,
[77] R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13, 3398 (1976).
3) We have clarified the definition of 2D higher-order topology by stating that gapped edges and in-gap corner states characterize a second-order phase (see page 1, column 1, paragraph 1, lines 7–13.)
4) We have clarified the physical meaning and realizability of the algebraic symmetry M in symmetric trilayer vdW structures (see page 3, column 2, last paragraph, last lines 1-6 and page 4, column 1, lines 1-4).
5) We have demonstrated the linear nature of all Weyl points by adding an explicit low-energy expansion around the four high-symmetry momenta (see page 4, column 1, paragraph 1, last lines 4-14).
6) We have added explicit statements confirming the numerical convergence of the Chern numbers and the size-independence of the corner modes (see page 4, column 2, paragraph 2, last lines 3-6; page 5, column 1, paragraph 1, last lines 1-4).
7) We have added a clarification on finite-size hybridization, noting that the corner-state splitting is exponentially small for the system sizes used (see page 5, column 1, paragraph 2, last lines 3-8).
8) We have added a full edge theory derivation that analytically explains the origin of the corner zero modes and shows how alternating Dirac masses arise on the four edges (see page 6-7, Sec. III C Edge theory).
9) We have added a subsection on robustness against disorder, showing that magnetic disorder preserves the corner modes while scalar disorder destroys them (see page 7, Sec. III D Robustness against disorder).
10) We have explained why the gap-closing conditions, and thus the phase diagram, remain independent of J for altermagnetism (see page 8, column 2, paragraph 2, last lines 1-7).
11) We have expanded the discussion of experimental probes and tunability of the corner modes (see page 9, column 1, paragraph 1, lines 10-15).
12) We added a concise materials motivation in the Conclusions to highlight that MnTe/Bi₂(Se,Te)₃/MnTe and MnSe/Bi₂Se₃ heterostructures exhibit proximity induced exchange coupling and related topological phases, supporting the feasibility of the ingredients used in our model (see page 9, column 2, paragraph 1). And, We added two new references associated with this materials motivation:
[80] N. Pournaghavi et al., Phys. Rev. B 103, 195308 (2021).
[81] A. V. Matetskiy et al., Appl. Phys. Lett. 107, 091604 (2015).
1) We have added a new Fig. 4, and the original Fig. 4 has been renumbered as Fig. 5.
2) We have added a new reference,
[77] R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13, 3398 (1976).
3) We have clarified the definition of 2D higher-order topology by stating that gapped edges and in-gap corner states characterize a second-order phase (see page 1, column 1, paragraph 1, lines 7–13.)
4) We have clarified the physical meaning and realizability of the algebraic symmetry M in symmetric trilayer vdW structures (see page 3, column 2, last paragraph, last lines 1-6 and page 4, column 1, lines 1-4).
5) We have demonstrated the linear nature of all Weyl points by adding an explicit low-energy expansion around the four high-symmetry momenta (see page 4, column 1, paragraph 1, last lines 4-14).
6) We have added explicit statements confirming the numerical convergence of the Chern numbers and the size-independence of the corner modes (see page 4, column 2, paragraph 2, last lines 3-6; page 5, column 1, paragraph 1, last lines 1-4).
7) We have added a clarification on finite-size hybridization, noting that the corner-state splitting is exponentially small for the system sizes used (see page 5, column 1, paragraph 2, last lines 3-8).
8) We have added a full edge theory derivation that analytically explains the origin of the corner zero modes and shows how alternating Dirac masses arise on the four edges (see page 6-7, Sec. III C Edge theory).
9) We have added a subsection on robustness against disorder, showing that magnetic disorder preserves the corner modes while scalar disorder destroys them (see page 7, Sec. III D Robustness against disorder).
10) We have explained why the gap-closing conditions, and thus the phase diagram, remain independent of J for altermagnetism (see page 8, column 2, paragraph 2, last lines 1-7).
11) We have expanded the discussion of experimental probes and tunability of the corner modes (see page 9, column 1, paragraph 1, lines 10-15).
12) We added a concise materials motivation in the Conclusions to highlight that MnTe/Bi₂(Se,Te)₃/MnTe and MnSe/Bi₂Se₃ heterostructures exhibit proximity induced exchange coupling and related topological phases, supporting the feasibility of the ingredients used in our model (see page 9, column 2, paragraph 1). And, We added two new references associated with this materials motivation:
[80] N. Pournaghavi et al., Phys. Rev. B 103, 195308 (2021).
[81] A. V. Matetskiy et al., Appl. Phys. Lett. 107, 091604 (2015).
Current status:
In refereeing