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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
  • Condensed Matter Physics - Theory
Approach: Theoretical


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

Dear editor,
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.

Current status:
Awaiting resubmission

Reports on this Submission

Anonymous Report 2 on 2024-5-20 (Contributed Report)


The manuscript provides a unified description of invariants in phase space.


The updated version clarifies the questions raised by the referees in the first round, however I believe the updated manuscript does not completely address the question about the HOTI by the 2nd referee.

Specifically, I believe the referee was asking about the *intrinsic* higher order TI, as defined and described in and references therein. These phases do have bulk-edge correspondence, but may not have a gapped boundary. The authors' response appropriately indicates that the mode-shell correspondence unlikely applies to these phases, but the manuscript does not clarify this and does not credit this line of research.

Requested changes

I would like to ask the authors to:

- Clearly state that mode-shell correspondence, as described in the manuscript does not clearly apply to higher order TIs.
- Cite the relevant literature, including for example


Ask for minor revision

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2024-5-6 (Invited Report)


The revised manuscript answers all my concerns that I have raised in the first round. As I have outline in my first report, I do believe hat semiclassical mode-shell correspondence in phase-space allows to obtain a unified view on different topological invariant, by having both access to momentum and position on equal footing. I do recommend that the paper is published in the present form. More information, why I deem that the manuscript fulfills the acceptance criteria you find in my first report.


Publish (easily meets expectations and criteria for this Journal; among top 50%)

  • validity: top
  • significance: high
  • originality: top
  • clarity: high
  • formatting: excellent
  • grammar: -

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