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arxiv: 2508.03026 · v1 · submitted 2025-08-05 · ⚛️ physics.optics

Free Extension of Topological States via Double-zero-index Media

Rui Dong , Changhui Shen , Changqing Xu , Yun Lai , Ce Shang This is my paper

Pith reviewed 2026-05-19 01:23 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords topological statesdouble-zero-index mediaphotonic latticesSu-Schrieffer-Heeger modelbulk-edge correspondencetopological photonicsmetamaterials
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The pith

Double-zero-index media expand topological states by acting as effective points that reshape interfaces.

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

The paper proposes using double-zero-index media to overcome the spatial confinement of topological states in photonic systems. These media occupy finite space yet behave optically like points with zero size, which changes the geometry of the interfaces where topological states live. This breaks the usual bulk-edge correspondence and allows the states to extend uniformly over larger areas. A sympathetic reader would care because it could make robust topological devices more practical by removing geometric limits on their size and shape. The approach is demonstrated in a two-dimensional photonic Su-Schrieffer-Heeger lattice using simulations and experiments.

Core claim

Double-zero-index media, although finite in extent, are optically equivalent to infinitesimal points. When placed in a topological photonic lattice, they alter the effective geometry of the topological interfaces, breaking conventional bulk-edge correspondence and enabling the free spatial expansion of uniform topological states beyond their native interfaces.

What carries the argument

Double-zero-index media that occupy finite space but function optically as points, thereby modifying interface geometry and violating bulk-edge correspondence.

If this is right

  • Topological states can be expanded spatially while preserving their robustness against disorder.
  • This offers a way to design larger-scale topological photonic devices without geometric constraints.
  • The method extends to general wave systems including acoustic metamaterials.
  • Uniform topological states become possible over expanded regions rather than being limited to narrow interfaces.

Where Pith is reading between the lines

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

  • This approach might enable topological features to be incorporated into devices with complex or arbitrary shapes.
  • Similar techniques could apply to other topological systems in electronics or mechanics.
  • Future work could test if this extension preserves all topological invariants under various perturbations.

Load-bearing premise

That double-zero-index media placed in finite regions are optically equivalent to points of zero size, which changes the effective geometry of the topological interface.

What would settle it

A simulation or microwave experiment showing that topological states remain confined to the original interface without extension after introducing the double-zero-index media, or that the optical behavior deviates from that of an infinitesimal point.

Figures

Figures reproduced from arXiv: 2508.03026 by Ce Shang, Changhui Shen, Changqing Xu, Rui Dong, Yun Lai.

Figure 1
Figure 1. Figure 1: FIG. 1. Double-zero-index media and the reshaping of topo [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A topological edge state extended by the DZIM layer. (a) A PC with different unit cell selections, labeled A and B. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A topological corner state extended by DZIM layer. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Experimental realization of spatially extended topo [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Topological states, known for their robustness against disorder, offer promising avenues for disorder-resistant devices. However, their intrinsic spatial confinement at interfaces imposes geometric constraints that limit the scalability of topological functionalities. Here, we propose a strategy to overcome this limitation by using double-zero-index media to expand topological interfaces. Although occupying finite space, these media are optically equivalent to infinitesimal points, effectively altering the geometry of topological interfaces and breaking conventional bulk-edge correspondence. This strategy enables the spatial expansion of uniform topological states beyond their native interface, offering new possibilities for topological photonic devices. We have verified this behavior through numerical simulations and microwave experiments in a two-dimensional photonic Su-Schrieffer-Heeger lattice. Our findings offer a universal framework to overcome the inherent dimensional limitations of topological states, with implications extending to general wave systems such as acoustic metamaterials.

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 / 2 minor

Summary. The manuscript proposes using double-zero-index media to spatially expand uniform topological edge states in a 2D photonic Su-Schrieffer-Heeger lattice beyond their native interfaces. Despite occupying finite space, these media are asserted to be optically equivalent to infinitesimal points, thereby reshaping the effective geometry of the topological interfaces and breaking conventional bulk-edge correspondence. The strategy is supported by numerical simulations and microwave experiments demonstrating the expanded mode profiles.

Significance. If the central mechanism holds, the work provides a route to overcome the inherent spatial confinement of topological states, enabling more scalable and flexible topological photonic devices while preserving robustness. The combination of simulations and experiments is a clear strength, and the framework's claimed generality to other wave systems (e.g., acoustics) adds value. Explicit verification of the topological invariant under the proposed modification would further solidify the contribution.

major comments (1)
  1. [§3] §3 (Theoretical Model and Effective Medium Description): The claim that finite double-zero-index regions (ε = μ = 0) are optically equivalent to points and thereby break bulk-edge correspondence while preserving the topological invariant is load-bearing for the central result. In the discrete lattice setting, this equivalence rests on an unverified long-wavelength assumption; local scattering or modified dispersion at the lattice-constant scale could alter the Zak-phase winding. An explicit check (e.g., computation of the invariant before/after insertion or a homogenization-limit analysis) is required to confirm that the observed spatial expansion arises from geometric decoupling rather than other effects.
minor comments (2)
  1. [Experimental Results] The experimental section would benefit from additional details on the precise realization of the double-zero-index inclusions (e.g., metamaterial unit-cell parameters and operating-frequency confirmation) to allow independent reproduction.
  2. [Figures 3 and 5] Figure captions for the simulated and measured field profiles should explicitly indicate the spatial scale of the expanded region relative to the native interface for clearer comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will revise the manuscript to incorporate the requested verification of the topological invariant.

read point-by-point responses

  1. Referee: [§3] §3 (Theoretical Model and Effective Medium Description): The claim that finite double-zero-index regions (ε = μ = 0) are optically equivalent to points and thereby break bulk-edge correspondence while preserving the topological invariant is load-bearing for the central result. In the discrete lattice setting, this equivalence rests on an unverified long-wavelength assumption; local scattering or modified dispersion at the lattice-constant scale could alter the Zak-phase winding. An explicit check (e.g., computation of the invariant before/after insertion or a homogenization-limit analysis) is required to confirm that the observed spatial expansion arises from geometric decoupling rather than other effects.

    Authors: We agree that an explicit verification strengthens the central claim. Although the observed mode expansion in our simulations and experiments is consistent with preservation of the topological character, we will add a direct computation of the Zak phase (or equivalent winding number) for the lattice both before and after insertion of the finite double-zero-index regions. This analysis will be performed in the long-wavelength limit via effective-medium homogenization and also through explicit band-structure calculations on the discrete lattice to address possible local scattering or dispersion modifications at the lattice scale. The revised manuscript will include these results to confirm that the spatial expansion originates from geometric decoupling rather than unintended changes to the invariant. revision: yes

Circularity Check

0 steps flagged

Minor self-citation present but central derivation remains independent and externally verified

full rationale

The paper grounds its core strategy in established topological band theory for the SSH lattice and standard effective-medium properties of zero-index media. The equivalence of finite double-zero-index regions to infinitesimal points is introduced as a modeling premise and then checked via independent numerical simulations plus microwave experiments rather than being fitted to the target topological extension result. No equation reduces by construction to a prior fitted parameter, no uniqueness theorem is imported solely from the authors' own prior work, and no ansatz is smuggled via self-citation. The single minor self-citation (likely to zero-index literature) is not load-bearing for the claimed geometric alteration or breaking of bulk-edge correspondence. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the domain assumption of optical equivalence for double-zero-index media and standard topological band theory; no free parameters or new invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Double-zero-index media are optically equivalent to infinitesimal points while occupying finite space
    This equivalence is invoked to alter the geometry of topological interfaces without physical resizing.

pith-pipeline@v0.9.0 · 5670 in / 1113 out tokens · 31976 ms · 2026-05-19T01:23:15.962888+00:00 · methodology

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett.49, 405 (1982)

    work page 1982

  2. [2]

    M. Z. Hasan and C. L. Kane, Rev. Mod. Phys.82, 3045 (2010)

    work page 2010

  3. [3]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys.83, 1057 (2011)

    work page 2011

  4. [4]

    M. G. Silveirinha, Phys. Rev. X9, 011037 (2019)

    work page 2019

  5. [5]

    H. Jia, J. Hu, R.-Y. Zhang, Y. Xiao, D. Wang, M. Wang, S. Ma, X. Ouyang, Y. Zhu, and C. T. Chan, Phys. Rev. Lett.134, 206603 (2025)

    work page 2025

  6. [6]

    Chen, Y.-H

    Z.-X. Chen, Y.-H. Zhang, X.-C. Sun, R.-Y. Zhang, J.-S. Tang, X. Yang, X.-F. Zhu, and Y.-Q. Lu, Phys. Rev. Lett.134, 136601 (2025)

    work page 2025

  7. [7]

    Wang, R.-Y

    M. Wang, R.-Y. Zhang, C. Zhang, H. Xue, H. Jia, J. Hu, D. Wang, T. Jiang, and C. T. Chan, Sci. Adv.11, eadq9285 (2025)

    work page 2025

  8. [8]

    Fu, R.-Y

    T. Fu, R.-Y. Zhang, S. Jia, C. T. Chan, and S. Wang, Phys. Rev. Lett.132, 233801 (2024)

    work page 2024

  9. [9]

    M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Sza- meit, Nature496, 196 (2013)

    work page 2013

  10. [10]

    Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Nature461, 772 (2009)

    work page 2009

  11. [11]

    A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater.12, 233 (2013)

    work page 2013

  12. [12]

    Wu and X

    L.-H. Wu and X. Hu, Phys. Rev. Lett.114, 223901 (2015)

    work page 2015

  13. [13]

    M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, Nat. Nanotechnol.14, 31 (2019)

    work page 2019

  14. [14]

    Dong, X.-D

    J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, Nat. Mater.16, 298 (2017)

    work page 2017

  15. [15]

    B. Yan, B. Liao, F. Shi, X. Xi, Y. Cao, K. Xiang, Y. Meng, L. Yang, Z. Zhu, J. Chen, X.-D. Chen, G.- G. Liu, B. Zhang, and Z. Gao, Phys. Rev. Lett.134, 033803 (2025)

    work page 2025

  16. [16]

    J. Kang, R. Wei, Q. Zhang, and G. Dong, Adv. Phys. Res.2, 2200053 (2023)

    work page 2023

  17. [17]

    A. B. Khanikaev and A. Al` u, Nat. Commun.15, 931 (2024)

    work page 2024

  18. [18]

    Harari, M

    G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, Science359, eaar4003 (2018)

    work page 2018

  19. [19]

    M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Science359, eaar4005 (2018)

    work page 2018

  20. [20]

    Dikopoltsev, T

    A. Dikopoltsev, T. H. Harder, E. Lustig, O. A. Egorov, J. Beierlein, A. Wolf, Y. Lumer, M. Emmerling, C. Schneider, S. H¨ ofling,et al., Science373, 1514 (2021)

    work page 2021

  21. [21]

    Zhang, X

    W. Zhang, X. Xie, H. Hao, J. Dang, S. Xiao, S. Shi, H. Ni, Z. Niu, C. Wang, K. Jin,et al., Light Sci. Appl. 9, 109 (2020)

    work page 2020

  22. [22]

    Chen, Z.-X

    X.-D. Chen, Z.-X. Gao, X. Cui, H.-C. Mo, W.-J. Chen, R.-Y. Zhang, C. T. Chan, and J.-W. Dong, Phys. Rev. Lett.133, 133802 (2024)

    work page 2024

  23. [23]

    G.-G. Liu, S. Mandal, X. Xi, Q. Wang, C. Devescovi, A. Morales-P´ erez, Z. Wang, L. Yang, R. Banerjee, Y. Long, Y. Meng, P. Zhou, Z. Gao, Y. Chong, A. Garc´ ıa- Etxarri, M. G. Vergniory, and B. Zhang, Science387, 162 (2025)

    work page 2025

  24. [24]

    Banerjee, S

    R. Banerjee, S. Mandal, Y. Y. Terh, S. Lin, G.-G. Liu, B. Zhang, and Y. D. Chong, Phys. Rev. Lett.133, 233804 (2024)

    work page 2024

  25. [25]

    Y. Liu, S. Leung, F.-F. Li, Z.-K. Lin, X. Tao, Y. Poo, and J.-H. Jiang, Nature589, 381 (2021)

    work page 2021

  26. [26]

    Landi, J

    M. Landi, J. Zhao, W. E. Prather, Y. Wu, and L. Zhang, Phys. Rev. Lett.120, 114301 (2018)

    work page 2018

  27. [27]

    Zhang, Y

    L. Zhang, Y. Yang, Z.-K. Lin, P. Qin, Q. Chen, F. Gao, E. Li, J.-H. Jiang, B. Zhang, and H. Chen, Adv. Sci.7, 1902724 (2020)

    work page 2020

  28. [28]

    Jiang, C

    T. Jiang, C. Zhang, R.-Y. Zhang, Y. Yu, Z. Guan, Z. Wei, Z. Wang, X. Cheng, and C. Chan, Nat. Commun.15, 10863 (2024)

    work page 2024

  29. [29]

    C. Xu, H. Chu, J. Luo, Z. H. Hang, Y. Wu, and Y. Lai, Phys. Rev. Lett.127, 123902 (2021). [30]See Supplementary Materials at link for further informa- tion on the equivalence of a point and a square of DZIM, the original 0D corner states and 1D edge states between the topologically trivial and nontrivial PC, the derivation of effective parameter method ba...

    work page 2021

  30. [30]

    Liberal and N

    I. Liberal and N. Engheta, Nat. Photonics11, 149 (2017)

    work page 2017

  31. [31]

    Al` u, M

    A. Al` u, M. G. Silveirinha, A. Salandrino, and N. En- gheta, Phys. Rev. B75, 155410 (2007)

    work page 2007

  32. [32]

    Moitra, Y

    P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, Nat. Photonics7, 791 (2013)

    work page 2013

  33. [33]

    Y. Li, S. Kita, P. Mu˜ noz, O. Reshef, D. I. Vulis, M. Yin, M. Lonˇ car, and E. Mazur, Nat. Photonics9, 738 (2015)

    work page 2015

  34. [34]

    Huang, Y

    X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, Nat. Mater.10, 582 (2011)

    work page 2011

  35. [35]

    J. Luo, Z. H. Hang, C. T. Chan, and Y. Lai, Laser & Photonics Rev.9, 523 (2015)

    work page 2015

  36. [36]

    Fleury and A

    R. Fleury and A. Al` u, Phys. Rev. Lett.111, 055501 (2013). 6

    work page 2013

  37. [37]

    Edwards, A

    B. Edwards, A. Al` u, M. E. Young, M. Silveirinha, and N. Engheta, Phys. Rev. Lett.100, 033903 (2008)

    work page 2008

  38. [38]

    C. Xu, G. Ma, Z.-G. Chen, J. Luo, J. Shi, Y. Lai, and Y. Wu, Phys. Rev. Lett.124, 074501 (2020)

    work page 2020

  39. [39]

    C. Xu, J. Shi, X. Liu, Z.-G. Chen, Y. Wu, and Y. Lai, Phys. Rev. Lett.135, 046902 (2025)

    work page 2025

  40. [40]

    Liberal, A

    I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, Science355, 1058 (2017)

    work page 2017

  41. [41]

    Liberal and N

    I. Liberal and N. Engheta, Sci. Adv.2, e1600987 (2016)

    work page 2016

  42. [42]

    J. Hao, W. Yan, and M. Qiu, Appl. Phys. Lett.96, 101109 (2010)

    work page 2010

  43. [43]

    H. Li, Z. Zhou, Y. He, W. Sun, Y. Li, I. Liberal, and N. Engheta, Nat. Commun.13, 3568 (2022)

    work page 2022

  44. [44]

    H. Li, P. Fu, Z. Zhou, W. Sun, Y. Li, J. Wu, and Q. Dai, Sci. Adv.8, eabq6198 (2022)

    work page 2022

  45. [45]

    Zhang, X

    R.-Y. Zhang, X. Cui, Y.-S. Zeng, J. Chen, W. Liu, M. Wang, D. Wang, Z.-Q. Zhang, N. Wang, G.-B. Wu, and C. T. Chan, Nature641, 1142 (2025)

    work page 2025

  46. [46]

    Zhang, B.-Y

    X. Zhang, B.-Y. Xie, H.-F. Wang, X. Xu, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Nat. Commun.10, 5331 (2019)

    work page 2019

  47. [47]

    Z.-G. Chen, C. Xu, R. Al Jahdali, J. Mei, and Y. Wu, Phys. Rev. B100, 075120 (2019)

    work page 2019

  48. [48]

    Zhang, H.-X

    X. Zhang, H.-X. Wang, Z.-K. Lin, Y. Tian, B. Xie, M.- H. Lu, Y.-F. Chen, and J.-H. Jiang, Nat. Phys.15, 582 (2019)

    work page 2019