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Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants
by Lucien Jezequel, Pierre Delplace
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Lucien Jezequel |
| Submission information | |
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| Preprint Link: | scipost_202504_00006v1 (pdf) |
| Code repository: | https://github.com/ljezeq/Code-Mode-shell-correspondence |
| Date accepted: | May 19, 2025 |
| Date submitted: | April 3, 2025, 6:34 p.m. |
| Submitted by: | Jezequel, Lucien |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The mode-shell correspondence relates the number $I_M$ of gapless modes in phase space to a topological \textit{shell invariant} $I_S$ defined on a close surface -- the shell -- surrounding those modes, namely $I_M=I_S$. In part I, we introduced the mode-shell correspondence for zero-modes of chiral symmetric Hamiltonians (class AIII). In this part II, we extend the correspondence to arbitrary dimension and to both symmetry classes A and AIII. This allows us to include, in particular, $1D$-unidirectional edge modes of Chern insulators, massless $2D$-Dirac and $3D$-Weyl cones, within the same formalism. We provide an expression of $I_M$ that only depends on the dimension of the dispersion relation of the gapless mode, and does not require a translation invariance. Then, we show that the topology of the shell (a circle, a sphere, a torus), that must account for the spreading of the gapless mode in phase space, yields specific expressions of the shell index. Semi-classical expressions of those shell indices are also derived and reduce to either Chern or winding numbers depending on the parity of the mode's dimension. In that way, the mode-shell correspondence provides a unified and systematic topological description of both bulk and boundary gapless modes in any dimension, and in particular includes the bulk-boundary correspondence. We illustrate the generality of the theory by analyzing several models of semimetals and insulators, both on lattices and in the continuum, and also discuss weak and higher-order topological phases within this framework. Although this paper is a continuation of Part I, the content remains sufficiently independent to be mostly read separately.
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Author comments upon resubmission
List of changes
-Remark added to contextualise equation (1)
-Explanation of the $\Gamma$ parameter added near eq (5)
-Clarification of the robustness of modes separated in wavenumber to disorder at page 13
-Code uploaded for Figures 8 and 10 to GitHub repository
-Numerical treatment of the disordered QWZ model in Appendix C with code put available
Published as SciPost Phys. 18, 193 (2025)
Reports on this Submission
Report #1 by Isidora Araya Day (Referee 1) on 2025-4-30 (Invited Report)
Report
After reviewing the code, I noticed that Fig. 10 uses the analytical expression for the Wigner transform. I believe it is more accurate to describe it this way, rather than 'brute force calculation', as done in text or caption of the figure. I would like to ask the authors to clarify how the Wigner-Weyl transform is computed.
Aside from this minor remark, I am happy to recommend publication.
Recommendation
Ask for minor revision
Author: Lucien Jezequel on 2025-05-02 [id 5441]
(in reply to Report 1 by Isidora Araya Day on 2025-04-30)We thank the referee for their valuable feedback. Regarding Fig. 10, the model's simplicity allows for an exact analytical computation of the Wigner-Weyl transform, which we used for the plot. At the same time for the QWZ model in Fig. 8, we employed explicit numerical computation of the Wigner-Weyl transform for the zero modes. Following the referee's suggestion, we will modify "Numerical computation of" to "Characterization of" in Fig. 10's caption to better reflect that these results involve less numeric computations.