SciPost Submission Page
Real-space approach for the Euler class and fragile topology in quasicrystals and amorphous lattices
by Dexin Li, Citian Wang, Huaqing Huang
Submission summary
| Authors (as registered SciPost users): | Huaqing Huang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202405_00006v2 (pdf) |
| Date submitted: | 2024-07-22 20:05 |
| Submitted by: | Huang, Huaqing |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
We propose a real-space formalism of the topological Euler class, which characterizes the fragile topology of two-dimensional systems with real wave functions. This real-space description is characterized by local Euler markers whose macroscopic average coincides with the Euler number, and it applies equally well to periodic and open boundary conditions for both crystals and noncrystalline systems. We validate this by diagnosing topological phase transitions in clean and disordered crystalline systems with the reality endowed by the space-time inversion symmetry $\mathcal{I}_{ST}$. Furthermore, we demonstrated the topological Euler phases in quasicrystals and even in amorphous lattices lacking any spatial symmetries. Our work not only provides a local characterization of the fragile topology but also significantly extends its territory beyond $\mathcal{I}_{ST}$-symmetric crystalline materials.
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
Thank you for handling our manuscript, and we thank the referees for their helpful comments, which we found useful and constructive, and have helped us to further clarify several technical points in the paper. We have responded to all the comments and suggestions by the referees, and accordingly revised the manuscript. Attached below please find our revised manuscript, responses to all comments and suggestions by the referee, and a summary of the changes made. We trust that the revised manuscript can now be accepted for publication in SciPost Physics.
Sincerely,
Huaqing Huang
/On behalf of all Authors/
List of changes
According to the referees’ suggestions, we have made the following revisions (Note: all the line numbers mentioned in the following revision list are given according to the revised manuscript):
1. Added a citation in line 60.
2. Added the following paragraph “To generalize a formula ... in appendix C” in line 96.
3. Added the following sentence “Formally, Eq. (6) share a similar expression to the real-space Chern number except for the substituting from Tr to Pfocc.” in line 110.
4. Added the following sentence “where |r> denotes the basis to construct the external space indexed by the Wannier cell r.” in line 114.
5. Added the following sentence “(See details in appendix H.1)” in line 152.
6. Revised the following sentence “but this difference can be diminished ...(see appendix ??)” to “but this difference can be diminished by increasing lattice size (see appendix I.4)” in line 194.
7. Added the following sentence “Each atom is assigned with a random displacement following the Gaussian distribution with standard deviation σ = 0.2.” in the caption of Fig. 3.
8. We used another distinct fraktur font “r” to denote all the Wannier cell of the specific basis |r>.
9. Added the following paragraph “To be more specific, the explicit expression of Slater-Koster ... is the unit direction vector.” in line 519.
10. Added the following sentence “The amplitude d of atomic displacements are determined by Gaussian distributions with standard deviation σ = 0.2.” in line 549.
11. Added the following paragraphs “Since the real-space Euler number ... functional optimization method. ” in line 571.
12. Added the following paragraphs “In such functional optimization process ... without the initialization requirement.” in line 584.
13. Added the following paragraph “Another issue to be clarified... the continuity of these markers.” in line 596.
14. Added the following sentences “q-|e| with ... in this case” in the caption of Fig S2.
15. Added the figure S5 and the section I.4 in line 661.
16. Added the figure S8 and the section I.7 in line 717.
17. Revised Figure 1 and Figure S3.