pith. sign in

arxiv: 2604.15983 · v1 · submitted 2026-04-17 · ❄️ cond-mat.mes-hall

Observation of ring states in a delicate topological insulator

Caroline Tornow , Julia Rupprecht , Pascal Engeler , Ute Drechsler , Kukka-Emilia Huhtinen , Chiara Devescovi , Sebastian D. Huber This is my paper

Pith reviewed 2026-05-10 08:07 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords delicate topologytopological insulatorring statesphononic metamaterialimpurity bound statesin-gap statesmulticellularitybound states
0
0 comments X

The pith

Strong impurities create frequency-pinned ring states that diagnose delicate topological insulators.

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

The paper shows that ring states induced by strong impurities serve as a spectroscopic signature for delicate topological insulators. Delicate topology differs from standard forms because its multicellular character can be erased by couplings to other orbitals, even far from the gap, so surface-state probes often fail to confirm it. The authors realize the phase in a phononic metamaterial, tune impurity strength, and use orbital-resolved readout to detect in-gap bound states that lock to fixed frequency in the strong-impurity limit while forming ring-shaped profiles around the impurity site. These ring states remain after an added orbital removes multicellularity, demonstrating that the signature captures complex multiband features. A reader would conclude this supplies a practical local probe for topologies that conventional low-energy methods miss.

Core claim

By realizing a delicate topological insulator in a phononic metamaterial and introducing strong local impurities with orbital-resolved readout, the authors observe ring states: in-gap bound states whose frequencies remain pinned in the strong-impurity limit and whose real-space profiles form a pronounced ring around the impurity. These states persist after multicellularity is removed by a weakly hybridizing additional orbital. The results establish impurity-induced ring states as probes of delicate topological phases and of complex multiband physics more generally.

What carries the argument

Ring states: in-gap bound states that remain frequency-pinned under strong impurity perturbation while forming ring-like spatial profiles around the impurity.

If this is right

  • Ring states remain a valid diagnostic even after multicellularity is eliminated, widening their use beyond the strict delicate invariant.
  • The impurity-tuning and orbital-readout method works in metamaterial platforms that host delicate or related multiband topologies.
  • The approach supplies a local spectroscopic tool that does not require direct access to boundary modes or global invariants.
  • The persistence of the states implies the signature is tied to broader band-structure features associated with delicate phases.

Where Pith is reading between the lines

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

  • Similar impurity experiments could be performed in electronic or photonic crystals to check whether ring states appear in other realizations of delicate topology.
  • If the frequency pinning and ring formation depend mainly on the presence of a gap plus certain orbital characters, the method might extend to any gapped multiband system regardless of delicate character.
  • Controlled impurities in fabricated devices could read out topological information locally without relying on surface transport measurements.

Load-bearing premise

The phononic metamaterial accurately reproduces the delicate topological insulator model and the ring states specifically indicate its topological features rather than arising from lattice artifacts or generic impurity effects.

What would settle it

If the bound-state frequencies shift continuously with increasing impurity strength or the spatial profiles fail to form clear rings in orbital-resolved maps, the identification of these states as indicators of delicate topology would be falsified.

Figures

Figures reproduced from arXiv: 2604.15983 by Caroline Tornow, Chiara Devescovi, Julia Rupprecht, Kukka-Emilia Huhtinen, Pascal Engeler, Sebastian D. Huber, Ute Drechsler.

Figure 2
Figure 2. Figure 2: Dispersion along high-symmetry lines. (a) Mea [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental observation of impurity-induced ring states. (a) Measured eigenfrequencies of impurity-induced states as [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Three band model. (a) Energy eigenvalues along [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Topological insulators are typically characterized by particularly stable properties, such as global invariants, and can be identified by probing their robust surface states. A recently discovered novel form of band topology, delicate topology, challenges this paradigm: its defining property, multicellularity, can be removed by introducing a coupling to local orbitals anywhere in the spectrum, even far above the relevant band gap. This makes it hard to diagnose delicate topology with conventional probes that access only low-energy degrees of freedom. Here, we introduce strong local impurities as a spectroscopic probe of a delicate topological insulator which we realize in a phononic metamaterial. By tuning the impurity strength and performing orbital-resolved readout, we observe recently proposed indicators of topology: ring states, in-gap bound states whose frequencies remain pinned in the strong-impurity limit while their real-space profiles form a pronounced ring around the impurity site. We find that these ring states persist even when the multicellularity in our system is removed by a weakly hybridizing additional orbital. Our results establish impurity-induced ring states as probes of complex multiband physics, including delicate topological phases.

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 paper claims to experimentally observe ring states—in-gap bound states that remain frequency-pinned in the strong-impurity limit and exhibit ring-like real-space profiles around the impurity—in a phononic metamaterial realizing a delicate topological insulator. By tuning impurity strength and performing orbital-resolved readout, these states are reported to persist even after multicellularity is removed via a weakly hybridizing additional orbital, leading to the broadened conclusion that such states serve as probes of complex multiband physics including delicate topological phases.

Significance. If the central observations hold, this work would establish impurity-induced ring states as a practical spectroscopic tool for accessing delicate topology and related multiband phenomena where conventional surface-state probes are ineffective due to removable multicellularity. The metamaterial platform's tunability and direct visualization capability represent a clear experimental strength, potentially enabling broader studies of complex band topologies.

major comments (2)
  1. [Abstract and multicellularity removal section] Abstract and the section on multicellularity removal: The persistence of ring states after introducing the additional orbital (which removes multicellularity) is presented as extending the applicability, but this directly weakens the specificity to delicate topology. Without explicit controls, such as measurements in a trivial multiband system or theoretical modeling demonstrating that the delicate invariant is required for the pinned ring profiles, the states could arise from generic multiband impurity scattering or metamaterial details rather than the delicate band structure.
  2. [Model realization section] Model realization section: The assertion that the phononic metamaterial faithfully emulates the delicate topological insulator requires stronger quantitative support, including direct band-structure comparisons with error bars to the theoretical model and confirmation that finite-size or orbital-hybridization effects do not independently produce the observed pinned states.
minor comments (2)
  1. [Figure captions] Figure captions should explicitly state the impurity strength values, orbital components shown, and any normalization used in the real-space profiles to improve interpretability.
  2. [Notation and definitions] The term 'ring states' and the criterion for 'pinned' frequencies should be defined with reference to specific quantitative thresholds or equations in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the scope of our claims. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract and multicellularity removal section] Abstract and the section on multicellularity removal: The persistence of ring states after introducing the additional orbital (which removes multicellularity) is presented as extending the applicability, but this directly weakens the specificity to delicate topology. Without explicit controls, such as measurements in a trivial multiband system or theoretical modeling demonstrating that the delicate invariant is required for the pinned ring profiles, the states could arise from generic multiband impurity scattering or metamaterial details rather than the delicate band structure.

    Authors: We agree that the persistence of the ring states after multicellularity removal broadens their applicability to complex multiband systems and that this requires careful framing to avoid overstating specificity to delicate topology alone. In the revised manuscript, we will add a dedicated theoretical subsection that compares the impurity-induced states in our delicate topological model to an otherwise similar trivial multiband model (with the same number of bands and comparable gap structure but trivial invariants). This modeling will show that the frequency pinning and ring-like spatial profiles in the strong-impurity limit are tied to the delicate band topology and do not appear in the trivial case. We will also revise the abstract and discussion to emphasize that the ring states serve as probes of delicate topology while remaining robust to the removal of multicellularity, rather than claiming they are unique to it. revision: yes

  2. Referee: [Model realization section] Model realization section: The assertion that the phononic metamaterial faithfully emulates the delicate topological insulator requires stronger quantitative support, including direct band-structure comparisons with error bars to the theoretical model and confirmation that finite-size or orbital-hybridization effects do not independently produce the observed pinned states.

    Authors: We will strengthen the model realization section with quantitative band-structure comparisons. Specifically, we will include plots of the measured dispersion (extracted from Fourier transforms of the experimental fields) overlaid with the theoretical tight-binding bands, together with error bars obtained from repeated measurements across multiple samples and excitation conditions. We will also add finite-size scaling analysis and additional simulations with the weak hybridizing orbital turned off, demonstrating that neither finite-size effects nor the orbital hybridization alone produce in-gap pinned states with ring profiles. These additions will be supported by both experimental data and numerical results. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental observation without reductive derivation

full rationale

This paper is an experimental report of impurity-induced ring states observed in a phononic metamaterial. No derivation chain, first-principles calculation, or fitted parameter is presented that reduces any central claim to its own inputs by construction. The abstract and text explicitly state the persistence of ring states after multicellularity removal via an additional orbital, which broadens rather than circularly narrows the interpretation to 'probes of complex multiband physics, including delicate topological phases.' Any cited prior proposals for ring states as indicators are treated as external benchmarks, not self-referential definitions. The result is self-contained against the reported measurements.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work is an experimental observation and introduces no new mathematical axioms, free parameters, or invented entities; delicate topology is referenced as a recently discovered concept from prior literature.

pith-pipeline@v0.9.0 · 5513 in / 1200 out tokens · 41724 ms · 2026-05-10T08:07:43.158436+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

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    M. Z. Hasan and C. L. Kane,Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010), URL

    work page 2010

  2. [2]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang,Topological insulators and su- perconductors, Rev. Mod. Phys.83, 1057 (2011), URL

    work page 2011

  3. [3]

    K¨ onig, S

    M. K¨ onig, S. Wiedmann, C. Br¨ une, A. Roth, H. Buhman, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang,Quantum Spin Hall Insulator State in HgTe Quantum Wells, Sci- ence318, 766 (2007), URL

    work page 2007

  4. [4]

    Roushan, J

    P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani,Topological surface states protected from backscattering by chiral spin texture, Nature460, 1106 (2009), URL

    work page 2009

  5. [5]

    F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Sch¨ afer, and R. Claessen, Bismuthene on a SiC substrate: A candidate for a high- temperature quantum spin Hall material, Science357, 287 (2017), URL

    work page 2017

  6. [6]

    J. L. Collins, A. Tadich, W. Wu, L. C. Gomes, J. N. B. Rodrigues, C. Liu, J. Hellerstedt, H. Ryu, S. Tang, S.-K. Mo, et al.,Electric-field-tuned topological phase transition in ultrathin Na3Bi, Nature564, 390 (2018), URL

    work page 2018

  7. [7]

    K. v. Klitzing, G. Dorda, and M. Pepper,New Method for High-Accuracy Determination of the Fine-Structure Con- stant Based on Quantized Hall Resistance, Phys. Rev. Lett.45, 494 (1980), URL

    work page 1980

  8. [8]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang,Quan- tum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science314, 1757 (2006), URL

    work page 2006

  9. [9]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. Vergniory, Z. Wang, C. Felser, M. Aroyo, and B. A. Bernevig,Topological quantum chemistry, Nature537, 298 (2017), URL

    work page 2017

  10. [10]

    Queiroz, R

    R. Queiroz, R. Ilan, Z. Song, B. A. Bernevig, and A. Stern,Ring states in topological materials, arXiv:2406.03529 (2024), URL

    work page arXiv 2024

  11. [11]

    Nelson, T

    A. Nelson, T. Neupert, T. Bzduˇ sek, and A. Alexandrad- 6 inata,Multicellularity of Delicate Topological Insulators, Phys. Rev. Lett.126, 216404 (2021), URL

    work page 2021

  12. [12]

    Nelson, T

    A. Nelson, T. Neupert, A. Alexandradinata, and T. Bzduˇ sek,Delicate topology protected by rotation sym- metry: Crystalline Hopf insulators and beyond, Phys. Rev. B106, 075124 (2022), URL

    work page 2022

  13. [13]

    Chen, Y.-P

    Y.-C. Chen, Y.-P. Lin, and Y.-J. Kao,Chern dartboard insulator: sub-Brillouin zone topology and skyrmion mul- tipoles, Commun. Phys.7, 32 (2024), URL

    work page 2024

  14. [14]

    H. C. Po, H. Watanabe, and A. Vishwanath,Fragile Topology and Wannier Obstructions, Phys. Rev. Lett. 121, 126402 (2018), URL

    work page 2018

  15. [15]

    Z.-D. Song, L. Elcoro, and B. A. Bernevig,Twisted bulk- boundary correspondence of fragile topology, Science367, 794 (2020), URL

    work page 2020

  16. [16]

    Bouhon, A

    A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager,Wil- son loop approach to fragile topology of split elementary band representations and topological crystalline insula- tors with time-reversal symmetry, Phys. Rev. B100, 195135 (2019), URL

    work page 2019

  17. [17]

    V. Peri, Z. Song, M. Serra-Garcia, P. Engeler, R. Queiroz, X. Huang, W. Deng, Z. Liu, B. A. Bernevig, and S. D. Huber,Experimental characterization of spectral flow be- tween fragile bands, Science367, 797 (2020), URL

    work page 2020

  18. [18]

    Wilczek and A

    F. Wilczek and A. Zee,Appearance of Gauge Structure in Simple Dynamical Systems, Phys. Rev. Lett.52, 2111 (1984), URL

    work page 1984

  19. [19]

    Zak,Band Center—A Conserved Quantity in Solids, Phys

    J. Zak,Band Center—A Conserved Quantity in Solids, Phys. Rev. Lett.48, 359 (1982), URL

    work page 1982

  20. [20]

    Coh and D

    S. Coh and D. Vanderbilt,Electric Polarization in a Chern Insulator, Phys. Rev. Lett.102, 107603 (2009), URL

    work page 2009

  21. [21]

    Cheng, S

    Z. Cheng, S. Yue, Y. Long, W. Xie, Z. Yu, H. T. Teo, Y. X. Zhao, H. Xue, and B. Zhang,Observation of return- ing Thouless pumping, Nature Comm.16, 9669 (2025), URL

    work page 2025

  22. [22]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger,Solitons in Polyacetylene, Phys. Rev. Lett.42, 1698 (1979), URL

    work page 1979

  23. [23]

    Misawa and Y

    T. Misawa and Y. Yamaji,Zeros of Green functions in topological insulators, Phys. Rev. Res.4, 023177 (2022), URL

    work page 2022

  24. [24]

    S.-S. Diop, L. Fritz, M. Vojta, and S. Rachel,Impurity bound states as detectors of topological band structures revisited, Phys. Rev. B101, 245132 (2020), URL

    work page 2020

  25. [25]

    Pangburn, A

    E. Pangburn, A. Banerjee, C. P´ epin, and C. Bena, Impurity-induced Mott ring states and Mott zeros ring states in the Hubbard operator formalism, Phys. Rev. B 112, 125157 (2025), URL

    work page 2025

  26. [26]

    Liu and T

    Q. Liu and T. Ma,Classical spins in topological insula- tors, Phys. Rev. B80, 115216 (2009), URL

    work page 2009

  27. [27]

    W.-Y. Shan, J. Lu, H.-Z. Lu, and S.-Q. Shen,Vacancy- induced bound states in topological insulators, Phys. Rev. B84, 035307 (2011), URL

    work page 2011

  28. [28]

    Naselli, V

    G. Naselli, V. K¨ onye, S. K. Das, G. G. N. Angilella, A. Isaeva, J. van den Brink, and C. Fulga,Nontrivial gapless electronic states at the stacking faults of weak topological insulators, Phys. Rev. B106, 094105 (2022), URL

    work page 2022

  29. [29]

    Michel, A

    S. Michel, A. F¨ unfhaus, R. Quade, R. Valent´ ı, and M. Potthoff,Bound states and local topological phase di- agram of classical impurity spins coupled to a Chern in- sulator, Phys. Rev. B109, 155116 (2024), URL

    work page 2024

  30. [30]

    A. J. Mains, J.-X. Zhong, Y. Jing, and B. Roy,Ordinary lattice defects as probes of topology, arXiv:2511.10646 (2025), URL

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  31. [31]

    K. H. Matlack, M. Serra-Garcia, A. Palermo, S. D. Hu- ber, and C. Daraio,Designing perturbative metamaterials from discrete models, Nature Mat.17, 323 (2018), URL

    work page 2018

  32. [32]

    Serra-Garcia, V

    M. Serra-Garcia, V. Peri, R. S¨ usstrunk, O. R. Bilal, T. Larsen, L. Villanueva, and S. D. Huber,Observation of a phononic quadrupole insulator, Nature555, 342 (2018), URL

    work page 2018

  33. [33]

    6.4,https: //www.comsol.com, Stockholm, Sweden

    COMSOL AB,COMSOL Multiphysics®v. 6.4,https: //www.comsol.com, Stockholm, Sweden

    work page

  34. [34]

    atomic Wannier function

    E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz,Boundary-obstructed topological phases, Phys. Rev. Res.3, 013239 (2021), URL. 1 Supplemental Material: Observation of ring states in a delicate topological insulator CONTENTS SI. Wilson loop phase and hybrid Wannier-center position 2 A. Quantization on the mirror-invariant lines in the two-band mod...

    work page 2021