pith. sign in

arxiv: 2606.01593 · v1 · pith:KXL7SW4Inew · submitted 2026-06-01 · ❄️ cond-mat.soft · cond-mat.dis-nn· cond-mat.mtrl-sci

Resonant Coupling and the Non-Phononic Flat Band in Amorphous Solids

Matteo Baggioli , Bingyu Cui This is my paper

Pith reviewed 2026-06-28 12:56 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.dis-nncond-mat.mtrl-sci
keywords amorphous solidsflat bandboson peakresonant couplingvibrational density of statesdynamical structure factorsoft-potential modelglasses
0
0 comments X

The pith

The resonant-coupling model reproduces the non-phononic flat band in amorphous solids and links it to the boson peak.

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

The paper revisits a minimal model of acoustic phonons interacting with single-frequency quasi-localized vibrations to account for a recently observed flat band in the dynamical structure factor of glasses. This band sits at the boson-peak frequency, shows negligible intensity below a critical wave vector near the first diffraction peak, and has a reduced intensity that tracks the static structure factor. The model reproduces these traits directly from the resonant interactions. A reader would care because the result supplies a concrete mechanism for the excess low-frequency vibrations that appear universally in disordered solids.

Core claim

Recent experiments and simulations show a non-phononic flat band in the dynamical structure factor of two- and three-dimensional amorphous solids. This feature sits near the boson-peak frequency, is nearly dispersionless, appears only above a critical wave vector of order the first diffraction peak, and has reduced intensity that correlates strongly with the static structure factor. The resonant-coupling model, a single-mode harmonic realization of the soft-potential scenario, naturally reproduces these main features and thereby clarifies the flat band's connection to the boson peak.

What carries the argument

The resonant-coupling model: a single-mode harmonic realization of the soft-potential scenario in which acoustic phonons interact with single-frequency quasi-localized vibrations; it supplies the minimal framework that generates the flat band's observed traits.

If this is right

  • The flat band arises directly from resonant phonon-quasi-localized mode coupling rather than from many-body effects.
  • Its energy coincides with the boson-peak frequency because that is the frequency at which the coupling is resonant.
  • The intensity threshold at the first diffraction peak wave vector follows from the same coupling mechanism.
  • The correlation between flat-band intensity and static structure factor is a direct signature of the underlying vibrational coupling.

Where Pith is reading between the lines

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

  • The same resonant-coupling picture could be used to predict how the flat band shifts when the density of quasi-localized modes is varied in simulations.
  • If the model holds, suppressing the boson peak by stiffening local modes should simultaneously eliminate the flat band.
  • The framework offers a route to connect the flat band to other low-frequency anomalies such as the two-level systems that dominate thermal properties at still lower temperatures.

Load-bearing premise

The resonant-coupling model is assumed to be an adequate single-mode harmonic realization of the soft-potential scenario whose parameters can be chosen to match the observed flat-band characteristics without additional many-body effects.

What would settle it

A simulation or measurement in which the flat band's reduced intensity fails to correlate with the static structure factor even when the boson peak is present, or in which the flat band remains after resonant coupling to quasi-localized modes is suppressed.

Figures

Figures reproduced from arXiv: 2606.01593 by Bingyu Cui, Matteo Baggioli.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

Recent experiments and simulations provide compelling evidence for the emergence of a non-phononic flat band in the dynamical structure factor of two- and three-dimensional amorphous solids. This feature has been suggested to be connected to the excess in the reduced vibrational density of states of glasses, commonly known as the boson peak, and displays several apparently universal characteristics. First, it is nearly dispersionless, with an energy close to the boson-peak frequency. Second, its intensity is negligible below a critical wave vector of the order of the first diffraction peak. Third, its reduced intensity exhibits a strong correlation with the static structure factor. Here, we revisit the resonant-coupling model, a single-mode harmonic realization of the soft-potential scenario in which acoustic phonons interact with single frequency quasi-localized vibrations. We show that this minimal framework naturally reproduces the main features of the observed flat band and clarifies its connection to the boson peak.

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 revisits the resonant-coupling model, a minimal single-mode harmonic realization of the soft-potential scenario, in which acoustic phonons couple to a single-frequency quasi-localized vibration (QLV). The authors claim that this framework naturally reproduces the three main observed features of the non-phononic flat band in the dynamical structure factor of amorphous solids: near-dispersionless dispersion at the boson-peak frequency, negligible intensity below a critical wave vector of order the first diffraction peak, and strong correlation of reduced intensity with the static structure factor. The work positions the flat band as a direct consequence of resonant coupling and thereby clarifies its link to the boson peak.

Significance. If the reproduction of the flat-band signatures can be shown to follow from the model without parameter values chosen specifically to place the resonance at the boson-peak scale, the result supplies a transparent analytic bridge between the soft-potential picture and the universal flat-band phenomenology reported in recent experiments and simulations. The single-mode harmonic construction permits an explicit dressed-propagator treatment, which is a strength for interpretability. The significance is tempered by the need to demonstrate that the key parameters are not simply fitted to the very scales the model is asked to explain.

major comments (2)
  1. [Model section, Hamiltonian] Model section (Hamiltonian definition): the bare QLV frequency ω_0 and the resonant-coupling matrix element g are introduced as adjustable parameters and are set to the boson-peak energy scale; the resulting pole of the dressed phonon propagator therefore lies at that scale by construction, so the claim that the flat band 'naturally' appears at the boson-peak frequency requires an independent route from the static structure factor to these parameters.
  2. [Results section] Results on dynamical structure factor: the three listed signatures (dispersionless character, intensity onset near first peak of S(q), and intensity–S(q) correlation) are asserted to be reproduced, yet no quantitative comparison metrics, error bars, or parameter-free predictions are supplied; without these it is impossible to judge whether the match is emergent or the result of tuning g and ω_0 to the observed scales.
minor comments (2)
  1. Notation for the dynamical structure factor S(q,ω) and the reduced intensity should be defined once at first use and used consistently thereafter.
  2. Figure captions should explicitly state the values of the two free parameters (ω_0 and g) used in each panel so that readers can assess the degree of tuning.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: Model section (Hamiltonian definition): the bare QLV frequency ω_0 and the resonant-coupling matrix element g are introduced as adjustable parameters and are set to the boson-peak energy scale; the resulting pole of the dressed phonon propagator therefore lies at that scale by construction, so the claim that the flat band 'naturally' appears at the boson-peak frequency requires an independent route from the static structure factor to these parameters.

    Authors: We agree that ω_0 and g are model parameters set to the boson-peak scale. Within the resonant-coupling framework, which is a minimal single-mode realization of the soft-potential scenario, the boson peak is identified with the QLV resonance frequency; the flat band then emerges at this scale as a direct consequence of the resonant coupling. The static structure factor enters through the q-dependent intensity and the onset condition, thereby linking the flat-band phenomenology to S(q). An explicit microscopic derivation of the numerical values of ω_0 and g from S(q) lies outside the scope of this minimal model. We will revise the model section to clarify this point and to distinguish the emergent flat-band features from the input frequency scale. revision: partial

  2. Referee: Results on dynamical structure factor: the three listed signatures (dispersionless character, intensity onset near first peak of S(q), and intensity–S(q) correlation) are asserted to be reproduced, yet no quantitative comparison metrics, error bars, or parameter-free predictions are supplied; without these it is impossible to judge whether the match is emergent or the result of tuning g and ω_0 to the observed scales.

    Authors: The three signatures are obtained from the explicit dressed-propagator calculation of the dynamical structure factor. We concur that quantitative metrics would allow a clearer assessment of robustness. In the revised manuscript we will add quantitative measures (e.g., dispersion width, correlation coefficient between reduced intensity and S(q)) together with a brief parameter-sensitivity analysis demonstrating that the reported features persist for a range of g and ω_0 around the boson-peak scale. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model reproduces features from independent soft-potential assumptions

full rationale

The provided abstract and skeptic summary describe a minimal single-mode harmonic resonant-coupling construction whose parameters are set to observed scales. However, no equations, self-citations, or explicit fitting steps are quoted that would reduce the flat-band reproduction to a definition or input by construction. The derivation is presented as emergent from the soft-potential scenario without load-bearing self-citation chains or ansatz smuggling. This qualifies as self-contained against external benchmarks of boson-peak phenomenology, yielding a normal non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5692 in / 1053 out tokens · 18487 ms · 2026-06-28T12:56:33.623846+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

69 extracted references · 1 linked inside Pith

  1. [1]

    Pengcheng Pea- cock

    and reexamine it in the context of recent exper- imental and numerical observations of a non-phononic flat band in the dynamical structure factor. For a re- cent review of these observations, we refer the reader to Ref. [10]. The resonant coupling model As our starting point, we consider a renormalized phonon Green function whose inverse is given by G−1 α...

  2. [2]

    P. M. Chaikin and T. C. Lubensky,Principles of con- densed matter physics, Vol. 10 (Cambridge university press Cambridge, 1995)

    1995

  3. [3]

    Ramos,Low-temperature Thermal And Vibrational Properties Of Disordered Solids: A Half-century Of Uni- versal ”Anomalies” Of Glasses(World Scientific Publish- ing Company, 2022)

    M. Ramos,Low-temperature Thermal And Vibrational Properties Of Disordered Solids: A Half-century Of Uni- versal ”Anomalies” Of Glasses(World Scientific Publish- ing Company, 2022)

    2022

  4. [4]

    Phillips,Amorphous Solids: Low-temperature Prop- erties, Topics in current physics (Springer-Verlag, 1981)

    W. Phillips,Amorphous Solids: Low-temperature Prop- erties, Topics in current physics (Springer-Verlag, 1981)

    1981

  5. [5]

    R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 10 74, 991 (2002)

    2002

  6. [6]

    Esquinazi, ed.,Tunneling Systems in Amorphous and Crystalline Solids(Springer-Verlag, Berlin, Heidelberg, 1998)

    P. Esquinazi, ed.,Tunneling Systems in Amorphous and Crystalline Solids(Springer-Verlag, Berlin, Heidelberg, 1998)

    1998

  7. [7]

    Flubacher, A

    P. Flubacher, A. Leadbetter, J. Morrison, and B. Sto- icheff, Journal of Physics and Chemistry of Solids12, 53 (1959)

    1959

  8. [8]

    Nakayama, Reports on Progress in Physics65, 1195 (2002)

    T. Nakayama, Reports on Progress in Physics65, 1195 (2002)

    2002

  9. [9]

    Hu and H

    Y.-C. Hu and H. Tanaka, Nature Physics18, 669 (2022)

    2022

  10. [10]

    Hu and H

    Y.-C. Hu and H. Tanaka, Physical Review Research5, 023055 (2023)

    2023

  11. [11]

    A flat-band per- spective on the boson peak in amorphous solids,

    S. Mahajan, L.-Z. Huang, C. Jiang, Y.-J. Wang, M. P. Ciamarra, J. Zhang, and M. Baggioli, “A flat-band per- spective on the boson peak in amorphous solids,” (2026), arXiv:2509.06340 [cond-mat.soft]

    Pith/arXiv arXiv 2026

  12. [12]

    X. Y. Li, H. P. Zhang, S. Lan, D. L. Abernathy, C. H. Hu, L. R. Fan, M. Z. Li, and X.-L. Wang, Nature Com- munications17, 860 (2025)

    2025

  13. [13]

    Boson peak in covalent network glasses: Isostaticity and marginal stability,

    H. Mizuno, T. Mori, G. Baldi, and E. Minamitani, “Boson peak in covalent network glasses: Isostaticity and marginal stability,” (2025), arXiv:2508.20481 [cond- mat.dis-nn]

    arXiv 2025

  14. [14]

    Boson peak in the dy- namical structure factor of network- and packing-type glasses,

    H. Mizuno and E. Minamitani, “Boson peak in the dy- namical structure factor of network- and packing-type glasses,” (2026), arXiv:2601.19118 [cond-mat.soft]

    arXiv 2026

  15. [15]

    Jiang, Z

    C. Jiang, Z. Zheng, Y. Chen, M. Baggioli, and J. Zhang, Phys. Rev. Lett.133, 188302 (2024)

    2024

  16. [16]

    Baldi, V

    G. Baldi, V. M. Giordano, G. Monaco, F. Sette, E. Fabi- ani, A. Fontana, and G. Ruocco, Phys. Rev. B77, 214309 (2008)

    2008

  17. [17]

    Ruzicka, T

    B. Ruzicka, T. Scopigno, S. Caponi, A. Fontana, O. Pilla, P. Giura, G. Monaco, E. Pontecorvo, G. Ruocco, and F. Sette, Phys. Rev. B69, 100201 (2004)

    2004

  18. [18]

    Anoma- lies

    G. Baldi, A. Fontana, and G. Monaco, inLow- Temperature Thermal and Vibrational Properties of Dis- ordered Solids: A Half-Century of Universal “Anoma- lies” of Glasses, edited by M. A. Ramos (World Scien- tific, 2022) pp. 177–226

    2022

  19. [19]

    Tømterud, S

    M. Tømterud, S. D. Eder, C. B¨ uchner, L. Wondraczek, I. Simonsen, W. Schirmacher, J. R. Manson, and B. Holst, Nature Physics19, 1910 (2023)

    1910

  20. [20]

    Steurer, A

    W. Steurer, A. Apfolter, M. Koch, W. E. Ernst, B. Holst, E. Sønderg˚ ard, and J. R. Manson, Phys. Rev. Lett.99, 035503 (2007)

    2007

  21. [21]

    Tømterud, S

    M. Tømterud, S. D. Eder, S. K. Hellner, C. B¨ uchner, M. Heyde, H.-J. Freund, S. Forti, D. Convertino, C. Co- letti, J. R. Manson, and B. Holst, Phys. Rev. B111, 205423 (2025)

    2025

  22. [22]

    Meyer, J

    A. Meyer, J. Wuttke, W. Petry, A. Peker, R. Bormann, G. Coddens, L. Kranich, O. G. Randl, and H. Schober, Phys. Rev. B53, 12107 (1996)

    1996

  23. [23]

    Inoue, T

    K. Inoue, T. Kanaya, S. Ikeda, K. Kaji, K. Shibata, M. Misawa, and Y. Kiyanagi, The Journal of chemical physics95, 5332 (1991)

    1991

  24. [24]

    Ruffl´ e, G

    B. Ruffl´ e, G. Guimbreti` ere, E. Courtens, R. Vacher, and G. Monaco, Phys. Rev. Lett.96, 045502 (2006)

    2006

  25. [25]

    Computational simula- tions of the vibrational properties of glasses,

    H. Mizuno and A. Ikeda, “Computational simula- tions of the vibrational properties of glasses,” inLow- Temperature Thermal and Vibrational Properties of Dis- ordered Solids, Chap. Chapter 10, pp. 375–433

  26. [26]

    Buchenau, Philosophical Magazine B65, 303 (1992)

    U. Buchenau, Philosophical Magazine B65, 303 (1992)

    1992

  27. [27]

    B. B. Laird and H. R. Schober, Phys. Rev. Lett.66, 636 (1991)

    1991

  28. [28]

    H. R. Schober and B. B. Laird, Phys. Rev. B44, 6746 (1991)

    1991

  29. [29]

    Buchenau, A

    U. Buchenau, A. Wischnewski, D. Richter, and B. Frick, Phys. Rev. Lett.77, 4035 (1996)

    1996

  30. [30]

    Buchenau, H

    U. Buchenau, H. M. Zhou, N. Nucker, K. S. Gilroy, and W. A. Phillips, Phys. Rev. Lett.60, 1318 (1988)

    1988

  31. [31]

    V. G. Karpov, M. I. Klinger, and F. N. Ignat’ev, Soviet Physics JETP57, 439 (1983), russian original: Zh. Eksp. Teor. Fiz. 84, 760–775 (1983)

    1983

  32. [32]

    V. G. Karpov and D. A. Parshin, Soviet Physics JETP 61, 1308 (1985), russian original: Zh. Eksp. Teor. Fiz. 88, 2212–2227 (1985)

    1985

  33. [33]

    Buchenau, Y

    U. Buchenau, Y. M. Galperin, V. L. Gurevich, and H. R. Schober, Phys. Rev. B43, 5039 (1991)

    1991

  34. [35]

    D. A. Parshin, H. R. Schober, and V. L. Gurevich, Phys. Rev. B76, 064206 (2007)

    2007

  35. [36]

    Buchenau, Y

    U. Buchenau, Y. M. Galperin, V. L. Gurevich, D. A. Parshin, M. A. Ramos, and H. R. Schober, Phys. Rev. B46, 2798 (1992)

    1992

  36. [37]

    H. R. Schober and C. Oligschleger, Phys. Rev. B53, 11469 (1996)

    1996

  37. [38]

    Schober, C

    H. Schober, C. Oligschleger, and B. Laird, Journal of Non-Crystalline Solids156-158, 965 (1993)

    1993

  38. [39]

    V. A. Luchnikov, N. N. Medvedev, Y. I. Naberukhin, and H. R. Schober, Phys. Rev. B62, 3181 (2000)

    2000

  39. [40]

    H. R. Schober, Journal of Physics: Condensed Matter 16, S2659 (2004)

    2004

  40. [41]

    Schober, Journal of Non-Crystalline Solids357, 501–505 (2011)

    H. Schober, Journal of Non-Crystalline Solids357, 501–505 (2011)

    2011

  41. [42]

    Lerner and E

    E. Lerner and E. Bouchbinder, The Journal of Chemical Physics155, 200901 (2021)

    2021

  42. [43]

    Lerner and E

    E. Lerner and E. Bouchbinder, The Journal of Chemical Physics158, 194503 (2023)

    2023

  43. [44]

    Moriel, E

    A. Moriel, E. Lerner, and E. Bouchbinder, Phys. Rev. Res.6, 023053 (2024)

    2024

  44. [45]

    Moriel, E

    A. Moriel, E. Lerner, and E. Bouchbinder, Journal of Applied Physics136, 225105 (2024)

    2024

  45. [46]

    Mahajan and M

    S. Mahajan and M. P. Ciamarra, Phys. Rev. Lett.127, 215504 (2021)

    2021

  46. [47]

    Mahajan, D

    S. Mahajan, D. Seow Yang Han, C. Jiang, M. Baggioli, and M. P. Ciamarra, Phys. Rev. E112, 035413 (2025)

    2025

  47. [48]

    Mahajan and M

    S. Mahajan and M. P. Ciamarra, SciPost Phys.15, 069 (2023)

    2023

  48. [49]

    Flenner and G

    E. Flenner and G. Szamel, Science Advances11, eadu6097 (2025)

    2025

  49. [50]

    Flenner and G

    E. Flenner and G. Szamel, The Journal of Physical Chemistry B129, 1855 (2025)

    2025

  50. [51]

    Zhang, X

    H. Zhang, X. Wang, H.-B. Yu, and J. F. Douglas, The Journal of Chemical Physics154, 084505 (2021)

    2021

  51. [52]

    Jiang, M

    C. Jiang, M. Baggioli, and J. F. Douglas, The Journal of Chemical Physics160, 214505 (2024)

    2024

  52. [53]

    Jiang and M

    C. Jiang and M. Baggioli, Journal of Physics: Condensed Matter36, 505101 (2024)

    2024

  53. [54]

    Bianchi, V

    E. Bianchi, V. M. Giordano, and F. Lund, Phys. Rev. B 101, 174311 (2020)

    2020

  54. [55]

    Quantum theory of elastic strings and the thermal conductivity of glasses,

    F. Lund and B. Scheihing-Hitschfeld, “Quantum theory of elastic strings and the thermal conductivity of glasses,” (2026), arXiv:2601.12721 [cond-mat.mtrl-sci]

    arXiv 2026

  55. [56]

    Pang, Phys

    T. Pang, Phys. Rev. B45, 2490 (1992)

    1992

  56. [57]

    Malinovsky, V

    V. Malinovsky, V. Novikov, and A. Sokolov, Physics Letters A153, 63 (1991)

    1991

  57. [58]

    Maurer and W

    E. Maurer and W. Schirmacher, Journal of Low Temper- 11 ature Physics137, 453 (2004)

    2004

  58. [59]

    A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova,Theory of Lattice Dynamics in the Harmonic Approximation, 2nd ed., Solid State Physics Supplement, Vol. 3 (Academic Press, New York, 1971)

    1971

  59. [60]

    V. L. Gurevich, D. A. Parshin, and H. R. Schober, Phys. Rev. B67, 094203 (2003)

    2003

  60. [61]

    A. L. Fetter and J. D. Walecka,Quantum Theory of Many-Particle Systems(McGraw-Hill, New York, 1971)

    1971

  61. [62]

    A. I. Akhiezer, Journal of Physics (USSR)1, 277 (1939)

    1939

  62. [63]

    P. G. Klemens, Proceedings of the Physical Society. Sec- tion A68, 1113 (1955)

    1955

  63. [64]

    L. Wang, G. Szamel, and E. Flenner, Soft Matter16, 7165 (2020)

    2020

  64. [65]

    A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crich- ton, A. Bosak, R. R¨ uffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parli´ nski, P. Jochym, and P. Piekarz, Phys. Rev. Lett.106, 225501 (2011)

    2011

  65. [66]

    Y. Wang, L. Hong, Y. Wang, W. Schirmacher, and J. Zhang, Phys. Rev. B98, 174207 (2018)

    2018

  66. [67]

    Y. Tian, X. Shen, Q. Gao, Z. Lu, J. Yang, Q. Zheng, C. F. Aleman, D. Luo, A. H. Reid, B. Xu,et al., arXiv preprint arXiv:2111.10171 (2021)

    arXiv 2021

  67. [68]

    Direct observation of ultrafast amorphous- amorphous transitions indicated by bond stretching and angle bending in phase-change material gete,

    Y. Qi, N. Chen, Z. Zhou, Q. Xu, Y. Lv, X. Zou, T. Jiang, P. Zhu, M. Zhu, D. Chen, Z. Sun, X. Li, and D. Xiang, “Direct observation of ultrafast amorphous- amorphous transitions indicated by bond stretching and angle bending in phase-change material gete,” (2026), arXiv:2603.17400 [cond-mat.mtrl-sci]

    arXiv 2026

  68. [69]

    Heterogeneous elastic- ity: The tale of the boson peak,

    W. Schirmacher and G. Ruocco, “Heterogeneous elastic- ity: The tale of the boson peak,” inLow-Temperature Thermal and Vibrational Properties of Disordered Solids (World Scientific, 2022) Chap. 9, pp. 331–373

    2022

  69. [70]

    G. Ding, E. Ma, F. Jiang, J. Duan, S. Cai, N. Xu, B. Cui, L. Dai, and M. Jiang, Nature Physics21, 1911 (2025)

    1911