Simulations of soft-sphere packings in spatial dimensions d=3d=3–77 show that, for compressions up to roughly one jamming spacing above the transition, the vibrational density of states (DOS) collapses onto a universal non-Debye master form D(ω)≈Aω2/(ω2+B Δϕ/ϕJ)D(\omega)\approx A\omega^2/(\omega^2+B\,\Delta\phi/\phi_J), consistent with infinite-dimensional mean-field theories of marginal stability (effective medium theory and the perceptron). Charbonneau, Corwin, Parisi, Poncet and Zamponi (Phys. Rev. Lett. 117, 045503 (2016)), who established this, left three linked issues open: the origin of the systematic deviations from the master form at high compression (Δϕ/ϕJ≳1\Delta\phi/\phi_J\gtrsim1) — a breakdown of the assumptions behind the mean-field comparison, or a distinct zero-temperature phase transition; the true value and scaling of the Debye threshold ω0\omega_0 in finite dimensions — does it vanish identically at finite compression, as in infinite-dimensional models, or only scale weakly with Δϕ\Delta\phi; and the precise geometric or structural mechanism generating quasilocalized low-frequency modes in low dimensions, and whether these modes can be derived from (perturbations of) mean-field solutions. The problem is to construct the quantitative finite-dimensional theory of the vibrational spectrum that resolves these three issues, delimiting exactly where the infinite-dimensional marginal-stability description of amorphous solids is valid in finite dd and what finite-dimensional physics replaces it where it fails.

Background

In a crystal, Debye theory predicts that the vibrational density of states (DOS) — the distribution D(ω)D(\omega) over the frequencies ω\omega of the normal modes — grows as D(ω)∼ωd−1D(\omega)\sim\omega^{d-1} at low frequency in dd spatial dimensions, because the only low-energy excitations are acoustic phonons, i.e. sound waves. Amorphous solids (glasses) instead exhibit an excess of low-frequency modes over the Debye prediction, the so-called boson peak, and in addition host spatially localized soft modes that govern low-temperature thermodynamics, sound attenuation, and the plastic rearrangements underlying yielding.

The theoretically cleanest model of an amorphous solid is a disordered packing of NN soft, purely repulsive frictionless spheres at temperature T=0T=0, interacting through a one-sided harmonic contact potential. Below a critical packing fraction ϕJ\phi_J such a packing is mechanically unstable; this loss of rigidity is the jamming transition. For packing fraction ϕ=ϕJ+Δϕ\phi=\phi_J+\Delta\phi with Δϕ>0\Delta\phi>0, the packing is a marginally coordinated solid — marginally stable, i.e. poised at the edge of mechanical instability, with the minimum number of interparticle contacts needed to sustain rigidity — whose pressure grows as P∝Δϕ1/2P\propto\Delta\phi^{1/2}. The vibrational normal modes are the eigenvectors of the Hessian matrix — the matrix of second derivatives of the potential energy at a mechanically stable configuration — with squared frequencies equal to the eigenvalues. Particles bearing no force ('rattlers') are conventionally excluded from the analysis. The spatial extent of a mode is measured by its inverse participation ratio: a mode extended over the whole system has a value of order 1/N1/N, while a mode concentrated on a few particles has a value of order one.

Two mean-field descriptions — effective medium theory, a self-consistent theory of a disordered spring network (DeGiuli, Laversanne-Finot, Düring, Lerner and Wyart, 2014), and the exact solution of the perceptron, a constraint-satisfaction model in the same universality class as spheres near jamming (Franz, Parisi, Sevelev, Urbani and Zamponi, 2017) — predict that a marginally stable amorphous solid away from jamming obeys

D(ω)∼{ωd−1ω≪ω0ω2/ω∗2ω0≪ω≪ω∗constantω≫ω∗D(\omega)\sim\begin{cases}\omega^{d-1} & \omega\ll\omega_0\\ \omega^2/\omega_*^2 & \omega_0\ll\omega\ll\omega_*\\ \text{constant} & \omega\gg\omega_*\end{cases}

where ω∗\omega_* is a characteristic frequency that vanishes at jamming (scaling as ω∗∝Δϕ1/2\omega_*\propto\Delta\phi^{1/2}) and ω0\omega_0 is the Debye threshold separating the lowest-frequency Debye regime from the anomalous non-Debye regime; ωmax⁡\omega_{\max} denotes the highest vibrational frequency, i.e. the top of the spectrum. In d=3d=3 the Debye and anomalous regimes both scale as ω2\omega^2 and differ only in prefactor, the anomalous one (1/ω∗21/\omega_*^2) being much larger. Crucially, in infinite-dimensional marginally stable models ω0\omega_0 vanishes identically over a finite range of compressions around jamming — the non-Debye law is then gapless, and this gaplessness is a direct signature of marginal stability.

Charbonneau, Corwin, Parisi, Poncet and Zamponi (2016) tested this picture in packings of N=8192N=8192 harmonic spheres in d=3d=3–77. For Δϕ/ϕJ≲1\Delta\phi/\phi_J\lesssim1 the spectra collapse after rescaling ω\omega by ω∗(Δϕ)\omega_*(\Delta\phi) and are described by the perceptron-inspired form D(ω,Δϕ)≈A ω2/(ω2+B Δϕ/ϕJ)D(\omega,\Delta\phi)\approx A\,\omega^2/(\omega^2+B\,\Delta\phi/\phi_J) with nearly dimension-independent fit parameters A≈0.4A\approx0.4 and B≈1B\approx1; the anomalous ω2\omega^2 regime is present in all dimensions down to the lowest accessible frequencies, so ω0\omega_0 lies below the observable window. Their analysis of mode localization (inverse participation ratios) showed that low-frequency modes become more extended with increasing dd and with decreasing compression, although delocalization is never complete — quasilocalized modes (QLMs) are a low-dimensional feature whose precise geometric origin was, and remains, an active problem. In parallel, simulations of quenched structural glasses had established that their lowest-frequency non-phononic modes are quasilocalized on cores of a few tens of particles and follow a universal law D(ω)∼ω4D(\omega)\sim\omega^4 independent of preparation protocol and dimension (Lerner, Düring and Bouchbinder, 2016); such modes are absent from the completely delocalized mean-field spectrum. The unresolved tension between the mean-field master form and the finite-dimensional mode structure is the subject of this problem.

Problem Statement

Determine whether, where, and why the infinite-dimensional mean-field description of the vibrational density of states of jammed amorphous solids fails in finite spatial dimensions, by resolving the three linked issues left open by Charbonneau et al. (2016):

(i) The nature and origin of the systematic deviations of the DOS from the low-compression master form at large compressions, Δϕ/ϕJ≳1\Delta\phi/\phi_J\gtrsim1: establish whether the deviations reflect a breakdown of the assumptions underlying the mean-field comparison (mixing of the crossover scales ω0∼ω∗∼ωmax⁡\omega_0\sim\omega_*\sim\omega_{\max}, changes in contact topology, or the incorporation into the force network of particles that were rattlers at jamming) or the existence of a distinct zero-temperature phase transition at high compression, and give a decisive discrimination between these explanations.

(ii) The true value and scaling behavior of the Debye threshold ω0\omega_0 in finite dd: determine whether ω0\omega_0 vanishes identically at any finite compression, as in infinite-dimensional marginally stable models, or follows a weak scaling with Δϕ\Delta\phi that has merely lain below the accessible frequency windows, and characterize the crossover (including its dimension and compression dependence) at which the anomalous non-Debye regime gives way to the low-frequency phonon/quasilocalized-mode structure.

(iii) The precise geometric or structural mechanism that generates quasilocalized low-frequency modes in low dimensions — modes concentrated on cores of a few tens of particles whose weight is strongly suppressed with increasing dd — including whether they can be obtained, perturbatively or otherwise, from the mean-field solutions, or instead require an intrinsically non-perturbative finite-dimensional mechanism, and what that mechanism is.

These three issues are physically linked — they concern the same finite-dimensional spectra and the same mean-field baseline — and serve a single overarching objective: a quantitative finite-dimensional account of the vibrational spectrum of jammed amorphous solids that reduces to the mean-field master form wherever that form is valid and identifies the finite-dimensional physics that replaces it where it fails.

References

  1. Patrick Charbonneau, Eric I. Corwin, Giorgio Parisi, Alexis Poncet, Francesco Zamponi, 'Universal Non-Debye Scaling in the Density of States of Amorphous Solids', Phys. Rev. Lett. 117, 045503 (2016). DOI: 10.1103/PhysRevLett.117.045503, https://doi.org/10.1103/PhysRevLett.117.045503 (also arXiv:1512.09100, https://arxiv.org/abs/1512.09100)
  2. Eric DeGiuli, Adrien Laversanne-Finot, Gustavo Düring, Edan Lerner, Matthieu Wyart, 'Effects of coordination and pressure on sound attenuation, boson peak and elasticity in amorphous solids', Soft Matter 10, 5628 (2014). DOI: 10.1039/c4sm00561a, https://doi.org/10.1039/c4sm00561a
  3. S. Franz, G. Parisi, P. Sevelev, P. Urbani, F. Zamponi, 'Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems', SciPost Phys. 2, 019 (2017). DOI: 10.21468/SciPostPhys.2.3.019, https://doi.org/10.21468/SciPostPhys.2.3.019
  4. Edan Lerner, Gustavo Düring, Eran Bouchbinder, 'Statistics and Properties of Low-Frequency Vibrational Modes in Structural Glasses', Phys. Rev. Lett. 117, 035501 (2016). DOI: 10.1103/PhysRevLett.117.035501, https://doi.org/10.1103/PhysRevLett.117.035501
  5. Giulio Biroli, Pierfrancesco Urbani, 'Breakdown of elasticity in amorphous solids', Nature Physics 12, 1130 (2016). DOI: 10.1038/nphys3845, https://doi.org/10.1038/nphys3845
  6. Hideyuki Mizuno, Hayato Shiba, Atsushi Ikeda, 'Continuum limit of the vibrational properties of amorphous solids', Proc. Natl. Acad. Sci. USA 114, E9767 (2017). DOI: 10.1073/pnas.1709015114, https://doi.org/10.1073/pnas.1709015114
  7. Edan Lerner, Eran Bouchbinder, 'Effect of instantaneous and continuous quenches on the density of vibrational modes in model glasses', Phys. Rev. E 96, 020104(R) (2017). DOI: 10.1103/PhysRevE.96.020104, https://doi.org/10.1103/PhysRevE.96.020104
  8. Geert Kapteijns, Eran Bouchbinder, Edan Lerner, 'Universal Nonphononic Density of States in 2D, 3D, and 4D Glasses', Phys. Rev. Lett. 121, 055501 (2018). DOI: 10.1103/PhysRevLett.121.055501, https://doi.org/10.1103/PhysRevLett.121.055501
  9. Harukuni Ikeda, 'Universal non-mean-field scaling in the density of states of amorphous solids', Phys. Rev. E 99, 050901(R) (2019). DOI: 10.1103/PhysRevE.99.050901, https://doi.org/10.1103/PhysRevE.99.050901
  10. Harukuni Ikeda, 'Note: Effect of localization on mean-field density of state near jamming', arXiv:1808.01635 (2018, unpublished note). arXiv: 1808.01635, https://arxiv.org/abs/1808.01635
  11. Masanari Shimada, Hideyuki Mizuno, Ludovic Berthier, Atsushi Ikeda, 'Low-frequency vibrations of jammed packings in large spatial dimensions', Phys. Rev. E 101, 052906 (2020). DOI: 10.1103/PhysRevE.101.052906, https://doi.org/10.1103/PhysRevE.101.052906
  12. Edan Lerner, 'Finite-size effects in the nonphononic density of states in computer glasses', Phys. Rev. E 101, 032120 (2020). DOI: 10.1103/PhysRevE.101.032120, https://doi.org/10.1103/PhysRevE.101.032120
  13. Francesco Arceri, Eric I. Corwin, 'Vibrational Properties of Hard and Soft Spheres Are Unified at Jamming', Phys. Rev. Lett. 124, 238002 (2020). DOI: 10.1103/PhysRevLett.124.238002, https://doi.org/10.1103/PhysRevLett.124.238002
  14. Corrado Rainone, Pierfrancesco Urbani, Francesco Zamponi, Edan Lerner, Eran Bouchbinder, 'Mean-field model of interacting quasilocalized excitations in glasses', SciPost Phys. Core 4, 008 (2021). DOI: 10.21468/SciPostPhysCore.4.2.008, https://doi.org/10.21468/SciPostPhysCore.4.2.008
  15. Edan Lerner, Eran Bouchbinder, 'Low-energy quasilocalized excitations in structural glasses', J. Chem. Phys. 155, 200901 (2021). DOI: 10.1063/5.0069477, https://doi.org/10.1063/5.0069477
  16. James D. Sartor, Sean A. Ridout, Eric I. Corwin, 'Mean-Field Predictions of Scaling Prefactors Match Low-Dimensional Jammed Packings', Phys. Rev. Lett. 126, 048001 (2021). DOI: 10.1103/PhysRevLett.126.048001, https://doi.org/10.1103/PhysRevLett.126.048001
  17. Patrick Charbonneau, Eric I. Corwin, R. Cameron Dennis, Rafael Díaz Hernández Rojas, Harukuni Ikeda, Giorgio Parisi, Federico Ricci-Tersenghi, 'Finite-size effects in the microscopic critical properties of jammed configurations: A comprehensive study of the effects of different types of disorder', Phys. Rev. E 104, 014102 (2021). DOI: 10.1103/PhysRevE.104.014102, https://doi.org/10.1103/PhysRevE.104.014102
  18. Masanari Shimada, Hideyuki Mizuno, Atsushi Ikeda, 'Novel elastic instability of amorphous solids in finite spatial dimensions', Soft Matter 17, 346 (2021). DOI: 10.1039/d0sm01583k, https://doi.org/10.1039/d0sm01583k
  19. Harukuni Ikeda, Masanari Shimada, 'Vibrational density of states of jammed packing at high dimensions: Mean-field theory', Phys. Rev. E 106, 024904 (2022). DOI: 10.1103/PhysRevE.106.024904, https://doi.org/10.1103/PhysRevE.106.024904
  20. Lijin Wang, Licun Fu, Yunhuan Nie, 'Density of states below the first sound mode in 3D glasses', J. Chem. Phys. 157, 074502 (2022). DOI: 10.1063/5.0102081, https://doi.org/10.1063/5.0102081
  21. Lijin Wang, Ding Xu, Shiyun Zhang, Yunhuan Nie, Hua Tong, Ning Xu, 'Low-frequency non-phononic vibrations of amorphous solids', Rep. Prog. Phys. 88, 104601 (2025). DOI: 10.1088/1361-6633/ae0e34, https://doi.org/10.1088/1361-6633/ae0e34
  22. Yusuke Hara, Ryosuke Matsuoka, Hiroyuki Ebata, Daisuke Mizuno, Atsushi Ikeda, 'A link between anomalous viscous loss and the boson peak in soft jammed solids', Nature Physics 21, 262 (2025). DOI: 10.1038/s41567-024-02722-7, https://doi.org/10.1038/s41567-024-02722-7
  23. Hideyuki Mizuno, Kuniyasu Saitoh, Yusuke Hara, Atsushi Ikeda, 'Unified study of viscoelasticity and sound damping in hard and soft amorphous solids', Communications Physics 8, 23 (2025). DOI: 10.1038/s42005-025-01933-5, https://doi.org/10.1038/s42005-025-01933-5