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Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
by Lucien Jezequel, Pierre Delplace
Submission summary
| Authors (as registered SciPost users): | Lucien Jezequel |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202404_00026v1 (pdf) |
| Date submitted: | 2024-04-18 16:59 |
| Submitted by: | Jezequel, Lucien |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a theory, that we call the mode-shell correspondence, which relates the topological zero-modes localised in phase space to a shell invariant defined on the surface forming a shell enclosing these zero-modes. We show that the mode-shell formalism provides a general framework unifying important results of topological physics, such as the bulk-edge correspondence, higher-order topological insulators, but also the Atiyah-Singer and the Callias index theories. In this paper, we discuss the already rich phenomenology of chiral symmetric Hamiltonians where the topological quantity is the chiral number of zero-dimensionial zero-energy modes. We explain how, in a lot of cases, the shell-invariant has a semi-classical limit expressed as a generalised winding number on the shell, which makes it accessible to analytical computations.
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 thank the reviewers for their careful reading and critical assessments of our manuscript. We address their concerns point by point. We have also incorporated many changes in the revised manuscript to take into account their many remarks. In particular, we have fully revised the derivation of our main formula in appendix F. We hope this new version is now suitable for publication in SciPost Physics.
List of changes
Changes are highlighted in green in the new manuscript.
We added “phase space” to the title of the article to emphasize its importance.
We completely revised the derivation of our main formula in Appendix F. We chose a different phase space orientation convention which changed some signs in the expressions of the article.
We moved some remarks in section 3.2 to the appendix.
We modified the parameter $\epsilon$ of the figure in section 3.2 to emphasize that the important separation is in wavenumber rather than in position
We added a figure in section 3.3.
In section 3.4, we swapped the order of presentation of the two examples.
We added clarifications in the text to respond to the comments of the referres.