pith. sign in

arxiv: 2606.27318 · v1 · pith:VTREHA7Unew · submitted 2026-06-25 · 🧮 math-ph · math.MP· math.SP· quant-ph

Geometric bulk-edge correspondence for mathbb{Z}₂-topological insulators

Alexis Drouot , Jacob Shapiro , Xiaowen Zhu This is my paper

Pith reviewed 2026-06-26 02:05 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.SPquant-ph
keywords Z2 topological insulatorsbulk-edge correspondencecurved interfacesFu-Kane-Mele indextime-reversal symmetrygeometric invariantsclass AII
0
0 comments X

The pith

For curved interfaces between two Z2 topological insulators the edge index equals the product modulo two of the bulk index difference and the boundary intersection number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a geometric bulk-edge correspondence for two-dimensional time-reversal-invariant insulators in class AII. When two such insulators occupy complementary regions separated by a curved boundary, the Z2 edge index of the interface system is the product, taken modulo two, of the difference of the two bulk Z2 indices and a geometric intersection number associated with the boundary and the measurement region. This result extends the flat-interface case by showing how curvature enters the correspondence through an intersection count. A reader would care because it gives an explicit rule for predicting protected edge states from bulk data even when the dividing curve is arbitrary. The argument is constructed as the Z2 analogue of an earlier curved-interface formula for quantum Hall systems.

Core claim

If two fermionic time-reversal-invariant insulators occupy complementary regions separated by a curved boundary, then the Z2 edge index of the interface system is the product, modulo two, of the difference of the two bulk Z2 indices and a geometric intersection number associated with the boundary and the measurement region.

What carries the argument

The geometric intersection number of the curved boundary with the measurement region, which multiplies the bulk Z2 index difference modulo two to produce the edge index.

If this is right

  • The edge index is completely determined by the two bulk invariants and the geometry of the interface.
  • Curvature affects the correspondence only through the parity of the intersection count.
  • The formula recovers the usual bulk-edge correspondence whenever the intersection number is odd.
  • The same multiplicative structure applies to any pair of complementary regions whose common boundary is a smooth curve.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same intersection factor may appear when the measurement region is replaced by a different observable whose support crosses the boundary an odd number of times.
  • Numerical checks on finite lattices with smooth but non-straight cuts could confirm the parity prediction without requiring the full analytic proof.
  • The construction suggests a route to similar statements in three dimensions where surface indices replace edge indices.

Load-bearing premise

The geometric intersection number between the curved boundary and the measurement region is well-defined and combines multiplicatively modulo two with the bulk Z2 indices.

What would settle it

A direct numerical computation of the Z2 edge index on a lattice model with a specific curved interface that yields a value different from the product of the bulk difference and the intersection number.

Figures

Figures reproduced from arXiv: 2606.27318 by Alexis Drouot, Jacob Shapiro, Xiaowen Zhu.

Figure 1
Figure 1. Figure 1: The gray box denotes short-range self-adjoint Hamiltonians. The maroon ellipses denote insulating Hamiltonians, which can be classified by Chern numbers. The gold ellipses denote time-reversal-invariant insulators. These have vanishing Chern number but may have different Z2 indices, indi￾cated in blue. 1.2. Interface systems. Suppose we have two time-reversal invariant insulators H± with distinct Z2-index,… view at source ↗
Figure 2
Figure 2. Figure 2: In subfigures (a) and (b), the pair (Ω, W) is transversal, since ∂Ω and ∂W eventually separate at least polynomially fast. In contrast, in subfigure (c), ∂Ω and ∂W become asymptotically parallel. Hence (Ω, W) is not transversal in the sense of Definition 5. Subfigure (b) highlights the generality allowed by the definition. Theorem 1 connects the bulk index to the edge index through the geometric intersecti… view at source ↗
Figure 3
Figure 3. Figure 3: We define the intersection number XΩ,W between transverse simple sets Ω, W in two steps. We first orient ∂Ω such that Ω is to its left according to the outward-pointing normal, see (a) and (b); then we count how many times the oriented ∂Ω enters W, see (c). Here XΩ,W = +1 − 1 − 1 = −1 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The intersection numbers in subfigures 4a, 4b, and 4c are +1, −1, and 0, respectively. In the integer-valued Hall setting, the corresponding edge conductance changes sign or vanishes, respectively. • H± are two ESR, Θ-invariant Hamiltonians with a joint spectral gap G ⊂ R and I(H+) ̸= I(H−) mod 2; • both Ω and Ω c contain a parabolic region; • He is an interface edge Hamiltonian compatible with H± and Ω. T… view at source ↗
Figure 5
Figure 5. Figure 5: Deforming Ω, W to Ωn, Wn. Blue (red) region represents Ω (W) and Ωn (Wn). In particular, by (P1) and using the invariance of I under compact perturbations – Lemma 3.9 – we have: IΩ,W (H) ≡ IΩn,Wn (H). (5.3) 2. Meanwhile we claim, when n is large enough, IΩn,Wn (H±) = IH2,H1 (H±). Recall by (3.13), IΩ,W (H) = I(UΩ, 1W ) ≡ dim ker(1−AΩ,W ) where AΩ,W = U ∗ Ω[UΩ, 1W ]. By Remark 3.1, AΩ,W decay polynomially w… view at source ↗
read the original abstract

Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a $\mathbb{Z}_2$-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proves a geometric bulk-edge correspondence for two-dimensional fermionic time-reversal-invariant insulators (class AII). For two such insulators occupying complementary regions separated by a curved boundary, the Z2 edge index of the interface system equals the product, modulo two, of the difference of the two bulk Fu-Kane-Mele Z2 indices and a geometric intersection number between the boundary and the measurement region. The argument is presented as the Z2 analogue of the curved-interface formula established for Hall insulators in DZ24.

Significance. If the central claim holds, the result supplies a geometric formula for the Z2 edge invariant at curved interfaces, extending the integer-valued case of DZ24 to the Z2 setting. This strengthens the geometric understanding of bulk-edge correspondence in topological insulators and may aid analysis of edge states in systems with non-flat boundaries. The manuscript supplies a proof of the stated correspondence, which is a positive feature when the derivation is self-contained.

major comments (1)
  1. Abstract: the central formula asserts that the Z2 edge index equals (difference of bulk Z2 indices) times (geometric intersection number) mod 2. The manuscript must supply an explicit construction showing that this multiplicative combination mod 2 follows from the spectral-flow or index-theoretic arguments without additional orientation or sign data that would be required only in the Z2 case; the analogy to DZ24 alone does not establish this step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point where the presentation of the central formula can be strengthened. We address the major comment below.

read point-by-point responses

  1. Referee: [—] Abstract: the central formula asserts that the Z2 edge index equals (difference of bulk Z2 indices) times (geometric intersection number) mod 2. The manuscript must supply an explicit construction showing that this multiplicative combination mod 2 follows from the spectral-flow or index-theoretic arguments without additional orientation or sign data that would be required only in the Z2 case; the analogy to DZ24 alone does not establish this step.

    Authors: We agree that a more self-contained derivation of the mod-2 multiplicative structure is desirable. In the revised manuscript we will expand the argument in Section 4 (currently presented as the direct Z2 analogue of the spectral-flow construction in DZ24) to include an explicit step-by-step reduction: the interface Z2 index is realized as the parity of the spectral flow of a one-parameter family of Fredholm operators obtained by cutting along the curved boundary; each crossing of the boundary with the measurement region contributes a Kramers pair whose parity is counted by the geometric intersection number; because the bulk Fu-Kane-Mele indices differ by an element of Z2 and time-reversal symmetry forces all sign ambiguities to cancel in pairs, the product is unambiguously defined modulo 2 with no additional orientation data required. This explicit construction will be inserted before the appeal to the DZ24 analogy. revision: yes

Circularity Check

1 steps flagged

Central Z2 bulk-edge formula relies on geometric intersection from overlapping-authors prior work DZ24 without independent construction

specific steps
  1. self citation load bearing [Abstract]
    "The argument is a Z_2 analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}."

    The claimed multiplicative combination (mod 2) of the geometric intersection number with the difference of Fu-Kane-Mele indices is presented as following from the prior result in DZ24. Because the authors of the cited work overlap with the present authors, and no independent verification or new construction of the intersection number for the Z2 case is supplied, the central formula reduces to the content of the self-citation.

full rationale

The paper presents its main result explicitly as a Z2 analogue of the connection formula from DZ24 (Drouot-Zhu 2024). The abstract states that the Z2 edge index equals the product mod 2 of the bulk-index difference and the geometric intersection number, but supplies no separate derivation of how the intersection number is defined or why it multiplies correctly in the Z2 setting. This matches the self-citation-load-bearing pattern: the load-bearing geometric ingredient is justified solely by citation to prior work whose authors overlap with the present paper. The central claim therefore inherits its geometric content from that citation rather than deriving it anew. No self-definitional reduction, fitted-input prediction, or renaming of a known result is exhibited in the provided text. The derivation chain is not fully self-contained against external benchmarks because the key geometric step is imported.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definition of the Fu-Kane-Mele Z2 index for class AII systems and on the existence of a geometric intersection number that multiplies the index difference mod 2; these are domain assumptions rather than derived quantities.

axioms (2)
  • domain assumption The Fu-Kane-Mele index is a well-defined Z2 topological invariant for time-reversal-invariant 2D insulators.
    Invoked implicitly when the abstract refers to bulk Z2 indices.
  • ad hoc to paper A geometric intersection number between the curved boundary and the measurement region can be defined and combined multiplicatively mod 2 with the index difference.
    This is the novel geometric ingredient stated in the abstract without further derivation.

pith-pipeline@v0.9.1-grok · 5661 in / 1473 out tokens · 27459 ms · 2026-06-26T02:05:18.945141+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

112 extracted references · 9 canonical work pages

  1. [1]

    Propagating edge states for a magnetic

    De Bievre, Stephan and Pule, Joseph V , booktitle=. Propagating edge states for a magnetic. 2002 , publisher=

    2002

  2. [2]

    Annals of Physics , volume=

    K-theory and pseudospectra for topological insulators , author=. Annals of Physics , volume=. 2015 , publisher=

    2015

  3. [3]

    Advances in Mathematics , volume=

    Skew localizer and Z2-flows for real index pairings , author=. Advances in Mathematics , volume=. 2021 , publisher=

    2021

  4. [4]

    Communications in Mathematical Physics , volume=

    Index pairings in presence of symmetries with applications to topological insulators , author=. Communications in Mathematical Physics , volume=. 2016 , publisher=

    2016

  5. [5]

    and van Elst, A

    Bellissard, J. and van Elst, A. and Schulz‐ Baldes, H. , title =. Journal of Mathematical Physics , volume =. 1994 , month =. doi:10.1063/1.530758 , url =

    work page doi:10.1063/1.530758 1994

  6. [6]

    Edge Spectrum for Truncated

    Drouot, Alexis and Shapiro, Jacob and Zhu, Xiaowen , journal=. Edge Spectrum for Truncated. 2025 , doi=. 2501.13096 , archivePrefix=

    arXiv 2025

  7. [7]

    The Band Theory of Graphite , author =. Phys. Rev. , volume =. 1947 , month =. doi:10.1103/PhysRev.71.622 , url =

    work page doi:10.1103/physrev.71.622 1947

  8. [8]

    Communications in Mathematical Physics , volume=

    Charge deficiency, charge transport and comparison of dimensions , author=. Communications in Mathematical Physics , volume=. 1994 , publisher=

    1994

  9. [9]

    2005 , publisher=

    Kane, Charles L and Mele, Eugene J , journal=. 2005 , publisher=

    2005

  10. [10]

    2011 , publisher=

    Complex variables , author=. 2011 , publisher=

    2011

  11. [11]

    Quantum spin

    Kane, Charles L and Mele, Eugene J , journal=. Quantum spin. 2005 , publisher=

    2005

  12. [12]

    Kane, C. L. and Mele, E. J. , journal =. 2005 , month =. doi:10.1103/PhysRevLett.95.146802 , url =

    work page doi:10.1103/physrevlett.95.146802 2005

  13. [13]

    Robustness of the spin-Chern number , author =. Phys. Rev. B , volume =. 2009 , month =. doi:10.1103/PhysRevB.80.125327 , url =

    work page doi:10.1103/physrevb.80.125327 2009

  14. [14]

    Katsura, Hosho and Koma, Tohru , TITLE =. J. Math. Phys. , FJOURNAL =. 2016 , NUMBER =. doi:10.1063/1.4942494 , URL =

    work page doi:10.1063/1.4942494 2016

  15. [15]

    Quantized

    Laughlin, Robert B , journal=. Quantized. 1981 , publisher=

    1981

  16. [16]

    Annales de l'

    The bulk-edge correspondence for continuous dislocated systems , author=. Annales de l'

  17. [17]

    Quantization of

    Hastings, Matthew B and Michalakis, Spyridon , journal=. Quantization of. 2015 , publisher=

    2015

  18. [18]

    Advances in Mathematics , volume=

    Tight-binding reduction and topological equivalence in strong magnetic fields , author=. Advances in Mathematics , volume=. 2022 , publisher=

    2022

  19. [19]

    Journal of Mathematical Physics , volume=

    Is the continuum SSH model topological? , author=. Journal of Mathematical Physics , volume=. 2022 , publisher=

    2022

  20. [20]

    Topologically protected states in one-dimensional continuous systems and

    Fefferman, Charles L and Lee-Thorp, James P and Weinstein, Michael I , journal=. Topologically protected states in one-dimensional continuous systems and. 2014 , publisher=

    2014

  21. [21]

    Europhysics Letters , volume=

    Disordered topological insulators via C*-algebras , author=. Europhysics Letters , volume=. 2011 , publisher=

    2011

  22. [22]

    Universality of the

    Giuliani, Alessandro and Mastropietro, Vieri and Porta, Marcello , journal=. Universality of the. 2017 , publisher=

    2017

  23. [23]

    Quantization of conductance in gapped interacting systems , author=. Annales. 2018 , organization=

    2018

  24. [24]

    Two-dimensional time-reversal-invariant topological insulators via

    Fonseca, Eli and Shapiro, Jacob and Sheta, Ahmed and Wang, Angela and Yamakawa, Kohtaro , journal=. Two-dimensional time-reversal-invariant topological insulators via. 2020 , publisher=

    2020

  25. [25]

    Nature Communications , volume=

    Topological protection versus degree of entanglement of two-photon light in photonic topological insulators , author=. Nature Communications , volume=. 2021 , publisher=

    2021

  26. [26]

    Nature communications , volume=

    Direct observation of valley-polarized topological edge states in designer surface plasmon crystals , author=. Nature communications , volume=. 2017 , publisher=

    2017

  27. [27]

    Observation of photonic topological valley

    Noh, Jiho and Huang, Sheng and Chen, Kevin P and Rechtsman, Mikael C , journal=. Observation of photonic topological valley. 2018 , publisher=

    2018

  28. [28]

    Physical review letters , volume=

    Topological insulators in three dimensions , author=. Physical review letters , volume=. 2007 , publisher=

    2007

  29. [29]

    Quantized

    Halperin, Bertrand I , journal=. Quantized. 1982 , publisher=

    1982

  30. [30]

    Nature Physics , volume=

    Topological transition in stratified fluids , author=. Nature Physics , volume=. 2019 , publisher=

    2019

  31. [31]

    Analogs of quantum-

    Raghu, Srinivas and Haldane, Frederick Duncan Michael , journal=. Analogs of quantum-. 2008 , publisher=

    2008

  32. [32]

    Physical review letters , volume=

    Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry , author=. Physical review letters , volume=. 2008 , publisher=

    2008

  33. [33]

    Physical review letters , volume=

    Topological acoustics , author=. Physical review letters , volume=. 2015 , publisher=

    2015

  34. [34]

    2015 , publisher=

    Random operators , author=. 2015 , publisher=

    2015

  35. [35]

    Combes, Jean-Michel and Thomas, L. , year=. Asymptotic behaviour of eigenfunctions for multiparticle

  36. [36]

    Model for a quantum

    Haldane, Duncan , journal=. Model for a quantum. 1988 , publisher=

    1988

  37. [37]

    2013 , publisher=

    Topological Insulators and Topological Superconductors , author=. 2013 , publisher=

    2013

  38. [38]

    and Graf, G

    Elgart, A. and Graf, G. M. and Schenker, J. , journal=. Equality of the bulk and edge. 2005 , publisher=

    2005

  39. [39]

    Local formula for the

    Li, Zhi and Mong, Roger SK , journal=. Local formula for the. 2019 , publisher=

    2019

  40. [40]

    Absolutely continuous edge Spectrum of

    Bols, Alex and Werner, Albert H , journal=. Absolutely continuous edge Spectrum of. 2022 , organization=

    2022

  41. [41]

    arXiv preprint arXiv:2307.13540 , year=

    Scattering theory of topologically protected edge transport , author=. arXiv preprint arXiv:2307.13540 , year=

    arXiv

  42. [42]

    Journal of Mathematical Physics , volume=

    On stable quantum currents , author=. Journal of Mathematical Physics , volume=. 2020 , publisher=

    2020

  43. [43]

    arXiv preprint arXiv:1709.00605 , year=

    Topological protection of perturbed edge states , author=. arXiv preprint arXiv:1709.00605 , year=

    Pith/arXiv arXiv

  44. [44]

    Communications in Mathematical Physics , volume=

    Coarse geometry and topological phases , author=. Communications in Mathematical Physics , volume=. 2019 , publisher=

    2019

  45. [45]

    On the extended nature of edge states of quantum

    Fr. On the extended nature of edge states of quantum. Annales. 2000 , organization=

    2000

  46. [46]

    Advances in Theoretical and Mathematical Physics , volume=

    Cobordism invariance of topological edge-following states , author=. Advances in Theoretical and Mathematical Physics , volume=. 2022 , publisher=

    2022

  47. [47]

    2022 , journal =

    Interface currents and corner states in magnetic quarter-plane systems , author =. 2022 , journal =

    2022

  48. [48]

    online workshop ``Mathematics of topological insulators" , year =

    Graf, Gian Michele and Thiang, Guo Chuan , title =. online workshop ``Mathematics of topological insulators" , year =

  49. [49]

    The spectral flow of a family of

    Braverman, Maxim , journal=. The spectral flow of a family of

  50. [50]

    Equality of bulk and edge

    Taarabt, Amal , journal=. Equality of bulk and edge

  51. [51]

    Boundary maps for

    Kellendonk, Johannes and Schulz-Baldes, Hermann , journal=. Boundary maps for. 2004 , publisher=

    2004

  52. [52]

    Strongly disordered

    Shapiro, Jacob and Tauber, Cl. Strongly disordered. Annales. 2019 , organization=

    2019

  53. [53]

    Communications in Mathematical Physics , volume=

    Controlled topological phases and bulk-edge correspondence , author=. Communications in Mathematical Physics , volume=. 2017 , publisher=

    2017

  54. [54]

    Communications in Mathematical Physics , volume=

    The bulk-edge correspondence for disordered chiral chains , author=. Communications in Mathematical Physics , volume=. 2018 , publisher=

    2018

  55. [55]

    The noncommutative geometry of the quantum

    Bellissard, Jean and van Elst, Andreas and Schulz-Baldes, Hermann , journal=. The noncommutative geometry of the quantum. 1994 , publisher=

    1994

  56. [56]

    Science , volume=

    Topological origin of equatorial waves , author=. Science , volume=. 2017 , publisher=

    2017

  57. [57]

    Notes on Topological Aspects in Condensed Matter Physics,

    Shapiro, Jacob , year =. Notes on Topological Aspects in Condensed Matter Physics,

  58. [58]

    Journal of Geometry and Physics , volume=

    Edge-following topological states , author=. Journal of Geometry and Physics , volume=. 2020 , publisher=

    2020

  59. [59]

    1989 , journal =

    Trace-Class Operators and Commutators , author =. 1989 , journal =. doi:10.1016/0022-1236(89)90064-5 , urldate =

    work page doi:10.1016/0022-1236(89)90064-5 1989

  60. [60]

    Quantized

    Thouless, David and Kohmoto, Mahito and Nightingale, Peter and. Quantized. Physical review letters , volume=. 1982 , publisher=

    1982

  61. [61]

    1996 , publisher=

    Index theory, coarse geometry, and topology of manifolds , author=. 1996 , publisher=

    1996

  62. [62]

    Communications in Mathematical Physics , volume=

    Microlocal analysis of the bulk-edge correspondence , author=. Communications in Mathematical Physics , volume=. 2021 , publisher=

    2021

  63. [63]

    arXiv preprint arXiv:2308.06384 , year=

    Coarse geometry and its applications in solid state physics , author=. arXiv preprint arXiv:2308.06384 , year=

    arXiv

  64. [64]

    arXiv preprint arXiv:1901.10592 , year=

    Manifestation of the topological index formula in quantum waves and geophysical waves , author=. arXiv preprint arXiv:1901.10592 , year=

    arXiv 1901

  65. [65]

    Paris Lectures on Topological Insulators , author=

  66. [66]

    K , year=

    Bulk and boundary invariants for complex topological insulators , author=. K , year=

  67. [67]

    2022 , publisher=

    Semiclassical analysis , author=. 2022 , publisher=

    2022

  68. [68]

    Mathematical Physics, Analysis and Geometry , volume=

    Topological invariants of edge states for periodic two-dimensional models , author=. Mathematical Physics, Analysis and Geometry , volume=. 2013 , publisher=

    2013

  69. [69]

    Geometric invariants of the quantum

    Xia, Jingbo , journal=. Geometric invariants of the quantum. 1988 , publisher=

    1988

  70. [70]

    2022 , journal=

    Topological insulators in semiclassical regime , author=. 2022 , journal=

    2022

  71. [71]

    Multiscale Modeling and Simulation , volume=

    Semiclassical propagation along curved domain walls , author=. Multiscale Modeling and Simulation , volume=. 2024 , publisher=

    2024

  72. [72]

    Topological Insulators and Topological Superconductors , url =

    Bernevig, Bogdan Andrei , doi =. Topological Insulators and Topological Superconductors , url =. 2013 , bdsk-url-1 =

    2013

  73. [73]

    Bourne, Chris and Kellendonk, Johannes and Rennie, Adam , journal=. The. 2017 , organization=

    2017

  74. [74]

    Physical review letters , volume=

    Chern number and edge states in the integer quantum Hall effect , author=. Physical review letters , volume=. 1993 , publisher=

    1993

  75. [75]

    Comptes Rendus Physique , keywords =

    An introduction to topological insulators. Comptes Rendus Physique , keywords =

  76. [76]

    Thomassen, Carsten , journal=. The. 1992 , publisher=

    1992

  77. [77]

    2009 , publisher=

    Introduction to metric and topological spaces , author=. 2009 , publisher=

    2009

  78. [78]

    Cambridge University Press , year=

    Elements of the topology of plane sets of points , author=. Cambridge University Press , year=

  79. [79]

    2013 , publisher=

    Topology: Pearson New International Edition , author=. 2013 , publisher=

    2013

  80. [80]

    doi:10.1007/s00220-021-03982-7 , archivePrefix =

    Topology in Shallow-Water Waves: A Violation of Bulk-Edge Correspondence , journal =. doi:10.1007/s00220-021-03982-7 , archivePrefix =

    work page doi:10.1007/s00220-021-03982-7

Showing first 80 references.