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arxiv: 2312.04396 · v2 · submitted 2023-12-07 · ❄️ cond-mat.mes-hall · physics.class-ph· physics.optics

Canonical scattering problem in topological metamaterials: Valley-Hall modes through a bend

Theo Torres , C\'edric Bellis , R\'egis Cottereau , Antonin Coutant This is my paper

Pith reviewed 2026-05-24 04:44 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.class-phphysics.optics
keywords Valley Hall modestopological metamaterialsscatteringbackscatteringtight-binding modeltransfer matrixgraphene ribbonsvalley conservation
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The pith

Valley Hall modes transmit near-maximally through sharp bends even without valley index conservation.

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

The paper tests whether conservation of the valley index accounts for the high transmission observed for Valley Hall modes at bends in reciprocal topological systems. It models graphene ribbons with an interface via a tight-binding approximation and applies a transfer-matrix method to calculate reflection and transmission at bends that either preserve or break valley index. The computations show transmission stays close to maximal across all bend configurations, with no evident link to valley conservation. This directly challenges a standard explanation and supplies a baseline case for metamaterial design.

Core claim

In a tight-binding model of graphene ribbons with an interface, Valley Hall modes exhibit transmission coefficients close to maximal when encountering sharp bends, including those that do not conserve the valley index; consequently no correlation exists between valley conservation and transmission quality.

What carries the argument

Transfer-matrix computation of reflection and transmission coefficients on a tight-binding model of graphene ribbons with an interface, applied to Valley Hall modes at bends.

If this is right

  • Transmission remains close to maximal for Valley Hall modes in every bend configuration examined.
  • Valley index conservation shows no correlation with transmission quality.
  • The results hold for both valley-conserving and valley-nonconserving bends in the model.
  • The computed coefficients provide a reference standard for Valley Hall metamaterial design.

Where Pith is reading between the lines

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

  • High transmission may arise from interface properties rather than valley symmetry alone.
  • Design rules for bends in these metamaterials could relax the requirement to preserve valley index.
  • The same scattering analysis might apply to other index-protected modes in reciprocal systems.

Load-bearing premise

The chosen tight-binding model together with its transfer-matrix implementation faithfully reproduces the scattering of Valley Hall modes at sharp bends without large discretization artifacts.

What would settle it

A measurement or exact computation that finds reflection coefficients substantially above zero for Valley Hall modes at a valley-nonconserving bend would contradict the near-maximal transmission result.

Figures

Figures reproduced from arXiv: 2312.04396 by Antonin Coutant, C\'edric Bellis, R\'egis Cottereau, Theo Torres.

Figure 1
Figure 1. Figure 1: FIG. 1: Graphene structure and its band structure. Panel a) depicts the graphene bulk structure, its unit [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Graphene structure with symmetry breaking and illustration of the valley Hall effect with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Band structure and edge modes of graphene ribbons with zigzag edges. Panel (a) shows a schematic [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Band structure and edge modes of graphene ribbons with zigzag edges and onsite potential [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Band structure and interface modes of graphene ribbons with a bridge interface. Panel (a) shows [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Band structure and interface modes of graphene ribbons with a zigzag interface. Panel (a) shows [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Illustration of the supercells of the graphene ribbons with zigzag edges and a bridge interface [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Illustration of the various supercells in a ribbon configuration with a bridge interface and a bend of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Different geometrical configurations combining the two possible interfaces and the two possible [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Scattering properties for a particular choice of [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Mode reconstruction for the case of a zigzag interface and a bend with angle [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Mode reconstruction for the case of a zigzag interface and a bend with angle 2 [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Mode reconstruction for the cas of bridge interface and a bend with angle [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Mode reconstruction for the case of a bridge interface and a bend with angle 2 [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Influence of the ribbon width [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Influence of the onsite potential [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Schematic of the Rice-Mele chain with an interface located at an intercell hopping and its [PITH_FULL_IMAGE:figures/full_fig_p036_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Schematic of the infinite Rice-Mele chain and its reduction to a semi-infinite chain. [PITH_FULL_IMAGE:figures/full_fig_p037_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Comparison between the analytical expression for the energy of the interface states in infinite [PITH_FULL_IMAGE:figures/full_fig_p039_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Scattering in a straight ribbon with flip of the onsite potential. Panel a) shows a particular [PITH_FULL_IMAGE:figures/full_fig_p040_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Comparison between interface and bulk modes scattering in a sharp bend at angle 2 [PITH_FULL_IMAGE:figures/full_fig_p041_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Graphene ribbon with armchair edges (Panel a) and its band structure (Panel b). The band [PITH_FULL_IMAGE:figures/full_fig_p045_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Graphene ribbon with armchair edges and onsite potential (Panel a) and its band structure (Panel [PITH_FULL_IMAGE:figures/full_fig_p046_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24: Armchair ribbon with an interface (Panel a) and its band structure (Panel b). The blue and ref [PITH_FULL_IMAGE:figures/full_fig_p047_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: Reflection and transmission spectrum for the armchair ribbon with an interface with [PITH_FULL_IMAGE:figures/full_fig_p047_25.png] view at source ↗
read the original abstract

We study the amount of backscattering of Valley Hall modes in a classical topological insulator. In reciprocal systems, the conservation of the valley index has been argued to be at the root of the high-transmission of Valley Hall modes, observed in many experimental realisations. Here, we reconsider this hypothesis by quantitatively analysing the canonical scattering problem of interface Valley Hall modes impinging on sharp bends which may or may not conserve the valley index. We consider a tight binding model of graphene ribbons with an interface and compute the reflection and transmission coefficients using a transfer matrix formalism. We find that, in all configurations considered, the transmission of Valley Hall modes is close to being maximal, even in cases where the valley index is not conserved. Hence there appears to be no correlation between valley conservation and good transmission. Our results serve as a reference case for the design of Valley Hall type metamaterial.

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

2 major / 2 minor

Summary. The manuscript studies backscattering of Valley Hall modes at sharp bends in a tight-binding model of graphene ribbons with an interface. Using a transfer-matrix formalism, it computes reflection and transmission for bend geometries that do and do not conserve the valley index. The central claim is that transmission remains close to maximal in all cases examined, implying no correlation between valley conservation and high transmission; the results are positioned as a reference case for metamaterial design.

Significance. If the numerical results are free of discretization or truncation artifacts, the work supplies a clean, parameter-free benchmark that directly tests and challenges the hypothesis that valley-index conservation is the dominant mechanism for suppressed backscattering in Valley-Hall systems. This could usefully inform both theoretical modeling and experimental design in topological metamaterials.

major comments (2)
  1. [Methods (transfer-matrix formalism)] Transfer-matrix implementation (methods section): the paper must demonstrate convergence of the reported transmission coefficients with respect to both ribbon width and the number of modes retained in the basis. Without such checks, the near-unity transmission found for non-valley-conserving bends could arise from truncation that artificially limits coupling to additional scattering channels at the lattice-scale corner.
  2. [Results (bend scattering calculations)] Results for non-conserving configurations: the claim that transmission is insensitive to valley conservation rests on the specific tight-binding ribbon model; the manuscript should quantify how transmission changes when the interface termination or ribbon width is varied, to confirm that the observed high transmission is not an artifact of the finite-width discretization.
minor comments (2)
  1. Figure captions should explicitly state the numerical values of transmission and reflection for each geometry rather than relying solely on visual inspection.
  2. [Introduction] The introduction would benefit from a brief statement of the precise valley-conservation hypothesis being tested, with a citation to the key prior works that advanced it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the work's potential significance as a benchmark. We address each major comment below and will revise the manuscript to incorporate the requested checks and additional data.

read point-by-point responses

  1. Referee: [Methods (transfer-matrix formalism)] Transfer-matrix implementation (methods section): the paper must demonstrate convergence of the reported transmission coefficients with respect to both ribbon width and the number of modes retained in the basis. Without such checks, the near-unity transmission found for non-valley-conserving bends could arise from truncation that artificially limits coupling to additional scattering channels at the lattice-scale corner.

    Authors: We agree that explicit convergence checks are necessary to rule out possible truncation artifacts in the transfer-matrix calculations. In the revised manuscript we will add an appendix (or dedicated subsection in Methods) containing convergence data. This will include transmission versus ribbon width (extending to at least twice the widths used in the main figures) at fixed mode count, and transmission versus number of retained modes at the largest ribbon width. These plots will confirm that the reported near-unity values have saturated and are insensitive to further increases in basis size or width. revision: yes

  2. Referee: [Results (bend scattering calculations)] Results for non-conserving configurations: the claim that transmission is insensitive to valley conservation rests on the specific tight-binding ribbon model; the manuscript should quantify how transmission changes when the interface termination or ribbon width is varied, to confirm that the observed high transmission is not an artifact of the finite-width discretization.

    Authors: We accept that robustness with respect to interface termination and ribbon width should be demonstrated explicitly. The revised manuscript will include additional calculations for alternative interface terminations (different edge configurations at the bend) and for a broader range of ribbon widths. These new results will be presented in the Results section (or a supplementary figure) to show that the high transmission persists across these variations, thereby supporting that the finding is not an artifact of the particular finite-width discretization originally employed. revision: yes

Circularity Check

0 steps flagged

Numerical scattering computation on fixed lattice model yields transmission results independent of valley conservation.

full rationale

The paper's central result is obtained by direct numerical solution of a linear scattering problem: a tight-binding Hamiltonian on graphene ribbons is discretized, a transfer-matrix method is applied to compute reflection/transmission coefficients for Valley-Hall interface modes at bends, and the output coefficients are reported. No parameter is fitted to the target transmission data, no self-referential definition equates the output to an input, and no self-citation chain is invoked to justify the model or the conclusion. The computation is therefore self-contained; its outputs are not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes a standard tight-binding graphene model and transfer-matrix scattering formalism without introducing new free parameters or entities; the central claim therefore rests on domain-standard assumptions rather than paper-specific inventions.

axioms (1)
  • domain assumption The tight-binding Hamiltonian on a graphene lattice with an interface supports Valley Hall modes whose scattering can be computed via transfer matrix.
    Standard modeling choice in condensed-matter and metamaterial literature for valley-Hall systems.

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Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    As for the infinite lattice, we now introduce an onsite potential to open a gap around the energy E = 0

    Graphene ribbons with onsite potential The graphene ribbon described above does not exhibit any band gap. As for the infinite lattice, we now introduce an onsite potential to open a gap around the energy E = 0. We consider a graphene ribbon with zigzag edges, with the addition of an onsite potential + u on the A sites and −u on the B sites of the lattice....

    work page

  2. [2]

    The interface will always be considered in the middle of the ribbon hence there are two possible configurations for the interface : i) a bridge interface or ii) a zigzag interface

    Graphene ribbons with an interface We now consider graphene ribbons with the addition of an onsite potential and in the presence of an interface. The interface will always be considered in the middle of the ribbon hence there are two possible configurations for the interface : i) a bridge interface or ii) a zigzag interface. The ribbons are made of superc...

    work page

  3. [3]

    This can most easily be seen from Eqs

    Conserved current Due to the fact that the Hamiltonian of an isolated supercell of the ribbon is self-adjoint, there exists a conserved current in the ribbon. This can most easily be seen from Eqs. (3.5) and (3.6). By multiplying Eq. (3.5) by B† n and Eq. (3.6) by A† n, we get B† nAn+1 + B† n(UB − E1N)Bn = −B† n 1N + J T An (3.10) A† n (1N + J) Bn = A† n ...

    work page

  4. [4]

    Writing the eigenvalues as λj = eikj, we see that if |λj| = 1 the mode is propagative and if |λj| ̸ = 1, the mode is evanescent

    Eigenmodes The modes of the ribbon are given by the eigenvectors of the transfer matrix, M ψj = λjψj, (3.17) with j = 1 , ..., N. Writing the eigenvalues as λj = eikj, we see that if |λj| = 1 the mode is propagative and if |λj| ̸ = 1, the mode is evanescent. For propagating modes, kj is real and it corresponds to the Bloch wavenumber. The wavevector assoc...

    work page

  5. [5]

    We now need to connect those modes on each side of the bend

    Connection across a bend In the previous section, we have constructed the propagating and evanescent eigenmodes and their associated eigenvalues in a ribbon configuration. We now need to connect those modes on each side of the bend. To do so, we will construct the transfer matrix of the bend cell. 18 FIG. 8: Illustration of the various supercells in a rib...

    work page

  6. [6]

    We can then define a scattering problem by imposing specific boundary conditions

    Reflection and transmission across a bend Using the modal decomposition constructed from the transfer matrix of the ribbons, Ψ n outside the bend cell can be decomposed as a linear superposition of the ribbons eigenmodes. We can then define a scattering problem by imposing specific boundary conditions. A general scattering solution can be written as Ψn = ...

    work page

  7. [7]

    Spectrum The reflection and transmission coefficients are computed over the energy range where there exists a single pair of propagating modes on each side of the bend and are presented in Fig. 10. The main observation from this example is the fact that in all configurations, the reflection coefficient is relatively small and therefore the transmission is...

    work page

  8. [8]

    (3.36) & (3.37) as well as the reflection an transmission matrices, we can reconstruct the field in the ribbons of both sides of the bend as well as within the bend

    Visualisation of scattering solutions Using Eqs. (3.36) & (3.37) as well as the reflection an transmission matrices, we can reconstruct the field in the ribbons of both sides of the bend as well as within the bend. This allows to visualise the scattering solutions which we present below for a fixed value of the energy for all possible configurations. The ...

    work page 2030

  9. [9]

    2013 Photonic topological insulators

    Khanikaev AB, Hossein Mousavi S, Tse WK, Kargarian M, MacDonald AH, Shvets G. 2013 Photonic topological insulators. Nature materials 12, 233–239

    work page 2013

  10. [10]

    2014 Topological photonics.Nature photonics 8, 821–829

    Lu L, Joannopoulos JD, Soljaˇ ci´ c M. 2014 Topological photonics.Nature photonics 8, 821–829

    work page 2014

  11. [11]

    2019 Topological photonics

    Ozawa T, Price HM, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman MC, Schuster D, Simon J, Zilberberg O et al.. 2019 Topological photonics. Reviews of Modern Physics 91, 015006

    work page 2019

  12. [12]

    2019 Topological phases in acoustic and mechanical systems

    Ma G, Xiao M, Chan CT. 2019 Topological phases in acoustic and mechanical systems. Nature Reviews Physics 1, 281–294

    work page 2019

  13. [13]

    2020 Topological wave insulators: a review

    Zangeneh-Nejad F, Al` u A, Fleury R. 2020 Topological wave insulators: a review. Comptes Rendus. Physique 21, 467–499

    work page 2020

  14. [14]

    2021 Design of topological elastic waveguides

    Miniaci M, Pal R. 2021 Design of topological elastic waveguides. Journal of Applied Physics 130

    work page 2021

  15. [15]

    2022 Topological sound in two dimensions

    Yves S, Ni X, Al` u A. 2022 Topological sound in two dimensions. Annals of the New York Academy of Sciences 1517, 63–77

    work page 2022

  16. [16]

    1992 Quantum Hall Effect

    Stone M. 1992 Quantum Hall Effect . World Scientific

    work page 1992

  17. [17]

    2011 The quantum spin Hall effect

    Maciejko J, Hughes TL, Zhang SC. 2011 The quantum spin Hall effect. Annu. Rev. Condens. Matter Phys. 2, 31–53

    work page 2011

  18. [18]

    2005 Quantum spin Hall effect in graphene

    Kane CL, Mele EJ. 2005 Quantum spin Hall effect in graphene. Physical review letters 95, 226801

    work page 2005

  19. [19]

    2015 Scheme for achieving a topological photonic crystal by using dielectric material

    Wu LH, Hu X. 2015 Scheme for achieving a topological photonic crystal by using dielectric material. Physical review letters 114, 223901

    work page 2015

  20. [20]

    2016 Acoustic topological insulator and robust one-way sound transport

    He C, Ni X, Ge H, Sun XC, Chen YB, Lu MH, Liu XP, Chen YF. 2016 Acoustic topological insulator and robust one-way sound transport. Nature physics 12, 1124–1129

    work page 2016

  21. [21]

    2017 Pseudospins and topological effects of phonons in a kekul´ e lattice

    Liu Y, Lian CS, Li Y, Xu Y, Duan W. 2017 Pseudospins and topological effects of phonons in a kekul´ e lattice. Physical review letters 119, 255901

    work page 2017

  22. [22]

    2018 Experimental Observation of Topologically Protected Helical Edge Modes in Patterned Elastic Plates

    Miniaci M, Pal RK, Morvan B, Ruzzene M. 2018 Experimental Observation of Topologically Protected Helical Edge Modes in Patterned Elastic Plates. Phys. Rev. X 8, 031074. (10.1103/PhysRevX.8.031074)

    work page doi:10.1103/physrevx.8.031074 2018

  23. [23]

    2016 Pseudo-time-reversal symmetry and topological edge states in two-dimensional acoustic crystals

    Mei J, Chen Z, Wu Y. 2016 Pseudo-time-reversal symmetry and topological edge states in two-dimensional acoustic crystals. Scientific reports 6, 32752

    work page 2016

  24. [24]

    2017 Crystalline metamaterials for topological properties at subwavelength scales

    Yves S, Fleury R, Berthelot T, Fink M, Lemoult F, Lerosey G. 2017 Crystalline metamaterials for topological properties at subwavelength scales. Nature communications 8, 16023

    work page 2017

  25. [25]

    2018 Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials

    Yang Y, Xu YF, Xu T, Wang HX, Jiang JH, Hu X, Hang ZH. 2018 Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials. Physical review letters 120, 217401

    work page 2018

  26. [26]

    2017 Observation of topological valley transport of sound in sonic crystals

    Lu J, Qiu C, Ye L, Fan X, Ke M, Zhang F, Liu Z. 2017 Observation of topological valley transport of sound in sonic crystals. Nature Physics 13, 369–374

    work page 2017

  27. [27]

    2017 Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect

    Pal RK, Ruzzene M. 2017 Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect. New Journal of Physics 19, 025001

    work page 2017

  28. [28]

    2018 Valley topological phases in bilayer sonic crystals

    Lu J, Qiu C, Deng W, Huang X, Li F, Zhang F, Chen S, Liu Z. 2018 Valley topological phases in bilayer sonic crystals. Physical review letters 120, 116802

    work page 2018

  29. [29]

    2020 Acoustic graphene network loaded with Helmholtz resonators: a first-principle modeling, Dirac cones, edge and interface waves

    Zheng LY, Achilleos V, Chen ZG, Richoux O, Theocharis G, Wu Y, Mei J, Felix S, Tournat V, Pagneux V. 2020 Acoustic graphene network loaded with Helmholtz resonators: a first-principle modeling, Dirac cones, edge and interface waves. New Journal of Physics 22, 013029

    work page 2020

  30. [30]

    2017 Observation of topological valley modes in an elastic hexagonal lattice

    Vila J, Pal RK, Ruzzene M. 2017 Observation of topological valley modes in an elastic hexagonal lattice. Physical Review B 96, 134307

    work page 2017

  31. [31]

    2018 Tunable acoustic valley–hall edge states in reconfigurable phononic elastic waveg- uides

    Liu TW, Semperlotti F. 2018 Tunable acoustic valley–hall edge states in reconfigurable phononic elastic waveg- uides. Physical Review Applied 9, 014001. 33

    work page 2018

  32. [32]

    2018 Achieving acoustic topological valley-Hall states by modulating the subwavelength honeycomb lattice

    Zhang Z, Cheng Y, Liu X. 2018 Achieving acoustic topological valley-Hall states by modulating the subwavelength honeycomb lattice. Scientific Reports 8, 16784

    work page 2018

  33. [33]

    2020 Dispersion tuning and route reconfiguration of acoustic waves in valley topological phononic crystals

    Tian Z, Shen C, Li J, Reit E, Bachman H, Socolar JE, Cummer SA, Jun Huang T. 2020 Dispersion tuning and route reconfiguration of acoustic waves in valley topological phononic crystals. Nature communications 11, 762

    work page 2020

  34. [34]

    2019 Programmable elastic valley Hall insulator with tunable interface propagation routes

    Zhang Q, Chen Y, Zhang K, Hu G. 2019 Programmable elastic valley Hall insulator with tunable interface propagation routes. Extreme Mechanics Letters 28, 76–80

    work page 2019

  35. [35]

    2022 Gigahertz topological valley Hall effect in nanoelectromechanical phononic crystals

    Zhang Q, Lee D, Zheng L, Ma X, Meyer SI, He L, Ye H, Gong Z, Zhen B, Lai K et al.. 2022 Gigahertz topological valley Hall effect in nanoelectromechanical phononic crystals. Nature Electronics 5, 157–163

    work page 2022

  36. [36]

    2022 Localized waves in elastic plates with perturbed honeycomb arrays of constraints

    Haslinger S, Frecentese S, Carta G. 2022 Localized waves in elastic plates with perturbed honeycomb arrays of constraints. Philosophical Transactions of the Royal Society A 380, 20210404

    work page 2022

  37. [37]

    2019 Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides

    Orazbayev B, Fleury R. 2019 Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides. Nanophotonics 8, 1433–1441

    work page 2019

  38. [38]

    2019 A comparison study between acoustic topological states based on valley Hall and quantum spin Hall effects

    Deng Y, Lu M, Jing Y. 2019 A comparison study between acoustic topological states based on valley Hall and quantum spin Hall effects. The Journal of the Acoustical Society of America 146, 721–728

    work page 2019

  39. [39]

    2021 Direct quantification of topological protection in symmetry-protected photonic edge states at telecom wavelengths

    Arora S, Bauer T, Barczyk R, Verhagen E, Kuipers L. 2021 Direct quantification of topological protection in symmetry-protected photonic edge states at telecom wavelengths. Light: Science & Applications 10, 9

    work page 2021

  40. [40]

    2023 Quantifying topological protection in valley photonic crystals using triangular resonators

    L´ evˆ eque G, Pennec Y, Szriftgiser P, Amo A, Mart´ ınez A. 2023 Quantifying topological protection in valley photonic crystals using triangular resonators. arXiv preprint arXiv:2301.10565

    work page arXiv 2023

  41. [41]

    2023 Observation of strong backscattering in valley-Hall photonic topological interface modes

    Rosiek CA, Arregui G, Vladimirova A, Albrechtsen M, Vosoughi Lahijani B, Christiansen RE, Stobbe S. 2023 Observation of strong backscattering in valley-Hall photonic topological interface modes. Nature Photonics pp. 1–7

    work page 2023

  42. [42]

    2023 Reciprocal topological photonic crystals allow backscattering.Nature Photonics 17, 383–384

    Rechtsman MC. 2023 Reciprocal topological photonic crystals allow backscattering.Nature Photonics 17, 383–384

    work page 2023

  43. [43]

    2020 Analytical methods for perfect wedge diffraction: A review

    Nethercote MA, Assier RC, Abrahams ID. 2020 Analytical methods for perfect wedge diffraction: A review. Wave Motion 93, 102479

    work page 2020

  44. [44]

    2020 Observations of symmetry-induced topological mode steering in a reconfigurable elastic plate

    Tang K, Makwana M, Craster RV, Sebbah P. 2020 Observations of symmetry-induced topological mode steering in a reconfigurable elastic plate. Physical Review B 102, 214103

    work page 2020

  45. [45]

    2018 Designing multidirectional energy splitters and topological valley supernetworks

    Makwana MP, Craster RV. 2018 Designing multidirectional energy splitters and topological valley supernetworks. Physical Review B 98, 235125

    work page 2018

  46. [46]

    2020 Manipulating topological valley modes in plasmonic metasurfaces

    Proctor M, Huidobro PA, Maier SA, Craster RV, Makwana MP. 2020 Manipulating topological valley modes in plasmonic metasurfaces. Nanophotonics 9, 657–665

    work page 2020

  47. [47]

    2017 Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals

    Vanel AL, Schnitzer O, Craster RV. 2017 Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals. Europhysics Letters 119, 64002

    work page 2017

  48. [48]

    2018 A study of topological effects in 1D and 2D mechanical lattices

    Chen H, Nassar H, Huang G. 2018 A study of topological effects in 1D and 2D mechanical lattices. Journal of the Mechanics and Physics of Solids 117, 22–36

    work page 2018

  49. [49]

    2021a Acoustic Su-Schrieffer- Heeger lattice: Direct mapping of acoustic waveguides to the Su-Schrieffer-Heeger model

    Coutant A, Sivadon A, Zheng L, Achilleos V, Richoux O, Theocharis G, Pagneux V. 2021a Acoustic Su-Schrieffer- Heeger lattice: Direct mapping of acoustic waveguides to the Su-Schrieffer-Heeger model. Physical Review B 103, 224309

    work page

  50. [50]

    2021b Topological two-dimensional Su–Schrieffer– Heeger analog acoustic networks: Total reflection at corners and corner induced modes

    Coutant A, Achilleos V, Richoux O, Theocharis G, Pagneux V. 2021b Topological two-dimensional Su–Schrieffer– Heeger analog acoustic networks: Total reflection at corners and corner induced modes. Journal of Applied Physics 129

    work page

  51. [51]

    2023 Topologically invisible defects in chiral mirror lattices

    Coutant A, Zheng LY, Achilleos V, Richoux O, Theocharis G, Pagneux V. 2023 Topologically invisible defects in chiral mirror lattices. arXiv preprint arXiv:2303.02029

    work page arXiv 2023

  52. [52]

    2020 Emulating tightly bound electrons in crystalline solids using mechanical waves

    Ram´ ırez-Ram´ ırez F, Flores-Olmedo E, B´ aez G, Sadurn´ ı E, M´ endez-S´ anchez R. 2020 Emulating tightly bound electrons in crystalline solids using mechanical waves. Scientific Reports 10, 10229

    work page 2020

  53. [53]

    2022 Hermitian and non-Hermitian topological edge states in one-dimensional perturbative elastic metamaterials

    Fan H, Gao H, An S, Gu Z, Liang S, Zheng Y, Liu T. 2022 Hermitian and non-Hermitian topological edge states in one-dimensional perturbative elastic metamaterials. Mechanical Systems and Signal Processing 169, 108774. 34

    work page 2022

  54. [54]

    2008 Wave propagation: from electrons to photonic crystals and left-handed materials

    Markos P, Soukoulis CM. 2008 Wave propagation: from electrons to photonic crystals and left-handed materials . Princeton University Press

    work page 2008

  55. [55]

    2022 Theory of edge states in graphene-like systems

    Lado J, Fern´ andez-Rossier J. 2022 Theory of edge states in graphene-like systems. arXiv preprint arXiv:2210.07568

    work page arXiv 2022

  56. [56]

    2010 Electronic states of graphene nanoribbons and analytical solutions

    Wakabayashi K, Sasaki Ki, Nakanishi T, Enoki T. 2010 Electronic states of graphene nanoribbons and analytical solutions. Science and technology of advanced materials 11, 054504

    work page 2010

  57. [57]

    1979 Solitons in polyacetylene

    Su WP, Schrieffer JR, Heeger AJ. 1979 Solitons in polyacetylene. Physical review letters 42, 1698

    work page 1979

  58. [58]

    2016 A short course on topological insulators.Lecture notes in physics 919, 166

    Asb´ oth JK, Oroszl´ any L, P´ alyi A. 2016 A short course on topological insulators.Lecture notes in physics 919, 166

    work page 2016

  59. [59]

    2016 Of bulk and boundaries: Generalized transfer matrices for tight-binding models

    Dwivedi V, Chua V. 2016 Of bulk and boundaries: Generalized transfer matrices for tight-binding models. Physical Review B 93, 134304

    work page 2016

  60. [60]

    However, the associated Rice-Mele chain obeys the mirror symmetry

    Note1. However, the associated Rice-Mele chain obeys the mirror symmetry. Hence it is the eigenmodes of the Rice-Mele and not directly of the graphene ribbon which display symmetric and anti-symmetric behaviours

    work page

  61. [61]

    2011 Zak phase and the existence of edge states in graphene

    Delplace P, Ullmo D, Montambaux G. 2011 Zak phase and the existence of edge states in graphene. Physical Review B 84, 195452. 35 Appendix A: Berry curvature of a two level system Here we present the details to compute the Berry curvature of the graphene lattice with onsite potential [50]. The Bloch Hamiltonian (2.5) is a 2 × 2 matrix which can be decompos...

    work page 2011

  62. [62]

    We consider the general Rice-Mele chain with intracell hopping h1, intercell hopping h2 and onsite potential u

    Spectrum in the fully dimerized limit One can gain valuable insight by considering the fully dimerized limit, that is the case where the intracell hopping vanishes. We consider the general Rice-Mele chain with intracell hopping h1, intercell hopping h2 and onsite potential u. A schematic of the chain is shown in Fig. 17. To connect the Rice-Mele chain to ...

    work page

  63. [63]

    We derive here analytically the expression for the energy as well as for the eigenvectors of interface states in a ribbon with infinite transverse width

    Interface states of the periodic chain Another instructive limit is the one of an infinite Rice-Mele chain, which corresponds to a ribbon with infinite width. We derive here analytically the expression for the energy as well as for the eigenvectors of interface states in a ribbon with infinite transverse width. The Rice-Mele chain with an interface is sho...

    work page

  64. [64]

    III A was done directly from the specific ribbon configuration and its decomposition in tilted supercells

    Transfer matrix for arbitrary ribbons The construction of the transfer matrix in Sec. III A was done directly from the specific ribbon configuration and its decomposition in tilted supercells. In particular, we used the fact the supercell only contained sites which were connected to adjacent supercells. Here, we present a more abstract discussion allowing...

    work page

  65. [65]

    As an illustration we now apply this formalism to the case of graphene ribbons with armchair edges

    Application to ribbons with armchair edges We have now constructed the transfer matrix for general ribbons with arbitrary supercell. As an illustration we now apply this formalism to the case of graphene ribbons with armchair edges. a. Graphene ribbons As for the case of the zigzag ribbon, we begin by briefly reviewing the band properties of the armchair ...

    work page