arxiv: 2605.04986 · v1 · submitted 2026-05-06 · ❄️ cond-mat.soft · cond-mat.mtrl-sci

Recognition: unknown

Nonlinear phonon dispersion in disordered solids and non-Debye vibrational spectra

Edan Lerner , Eran Bouchbinder

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:06 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.mtrl-sci

keywords nonlinear phonon dispersiondisordered solidsboson peaknon-Debye vibrational spectrawave attenuationmesoscopic lengthscaleglassphonon softening

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The pith

Nonlinear phonon dispersion in disordered solids emerges from a mesoscopic disorder-induced lengthscale that also governs wave attenuation.

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

All solids show linear elastic wave dispersion at long wavelengths with Debye-like vibrational spectra at low frequencies. In disordered solids such as glasses and elastic networks, deviations appear from nonlinear phonon dispersion and from non-phononic modes created by structural disorder. The paper demonstrates that the nonlinear dispersion originates in a single mesoscopic lengthscale set by disorder, and that this lengthscale simultaneously sets the rate of wave attenuation. Large-scale simulations across multiple models then separate the contributions of phonon softening and non-phononic vibrations to the boson peak, showing that their relative importance varies with disorder strength and that both are substantial in realistic laboratory glasses.

Core claim

Nonlinear phonon dispersion in a broad range of disordered solids, including elastic networks and various glasses, emerges from a mesoscopic, disorder-induced lengthscale that also controls wave attenuation. Quantitative analysis and simulations establish that the relative weight of nonlinear phonon softening versus non-phononic vibrations in producing non-Debye anomalies and the boson peak depends on the strength of disorder, for example as set by thermal history during glass formation, and that both mechanisms contribute significantly to the boson peak in realistic laboratory glasses.

What carries the argument

The mesoscopic, disorder-induced lengthscale that simultaneously produces nonlinear phonon dispersion and controls wave attenuation.

If this is right

The onset of non-Debye features in the vibrational density of states is shaped by both nonlinear phonon effects and disorder-induced modes across elastic networks and glasses.
The boson peak's composition changes with disorder strength, such as that tuned by the thermal history of glass formation.
Wave attenuation and nonlinear dispersion are linked through the same disorder lengthscale in multiple classes of disordered solids.
For typical laboratory glasses both nonlinear softening and non-phononic vibrations contribute meaningfully to the boson peak.

Where Pith is reading between the lines

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

If the lengthscale is primary, experimental probes of attenuation at intermediate wavelengths could directly measure it in real materials.
Processing routes that alter disorder strength could be used to adjust the balance between the two contributions to vibrational anomalies.
The same lengthscale may set universal trends in wave propagation across different disordered solids, independent of microscopic details.

Load-bearing premise

The mesoscopic lengthscale identified in simulations is the root cause of the nonlinear dispersion rather than a secondary consequence, and the numerical decomposition separating phonon softening from non-phononic contributions can be performed without model-specific artifacts.

What would settle it

An observation or simulation in which clear nonlinear phonon dispersion appears but no corresponding mesoscopic lengthscale is detectable, or in which the two contributions to the boson peak cannot be isolated as described.

Figures

Figures reproduced from arXiv: 2605.04986 by Edan Lerner, Eran Bouchbinder.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: a, which indeed increases above unity with increasing ω, exclusively due to a nonlinear phonon dispersion as in Eq. (1). In fact, the reduced VDoS reveals a peak in the form of a cusp, corresponding to a vanishing derivative of ω(k), known as a Van Hove singularity [18]. The results in Fig. 3a are conceptually clear, i.e., a non-Debye VDoS exclusively emerges from a nonlinear phonon dispersion ω(k). If i… view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 5.** Figure 5: FIG. 5 view at source ↗

**Figure 6.** Figure 6: FIG. 6 view at source ↗

**Figure 8.** Figure 8: FIG. 8 view at source ↗

**Figure 9.** Figure 9: FIG. 9 view at source ↗

**Figure 10.** Figure 10: FIG. 10 view at source ↗

**Figure 11.** Figure 11: , we benchmark our KPM calculations against a direct diagonalization of the Hessian matrices of smaller glass samples. 3. The disorder quantifier χ The broadly applicable disorder quantifier χ has been studied extensively in previous work; it was shown to control wave attenuation rates in disordered solids [39], to control phonon-band widths in the vibrational spectra of finite-size computer glasses [14],… view at source ↗

**Figure 12.** Figure 12: FIG. 12 view at source ↗

read the original abstract

All solids, whether crystalline or disordered, support elastic wave propagation with a linear dispersion relation in the long-wavelength limit. These waves, corresponding to low-frequency phonons, feature a vibrational density of states that follows Debye's classical model. Deviations from Debye's predictions with increasing frequency can emerge from phonon dispersion nonlinearity and from non-phononic vibrational modes, which exist in non-crystalline solids due to structural disorder. Both nonlinear phonon dispersion in disordered solids and its relative contribution to non-Debye anomalies, most notably manifested by the controversial boson peak, remain poorly understood. Here we show that nonlinear phonon dispersion in a broad range of disordered solids, including elastic networks and various glasses, emerge from a mesoscopic, disorder-induced lengthscale, which also controls wave attenuation. We subsequently use analysis and large-scale computer simulations to quantitatively determine the relative contributions of nonlinear phonon softening and non-phononic vibrations to the onset of non-Debye anomalies and to the boson peak. We show that the relative magnitude of the two contributions strongly depends on the strength of disorder of the solid, e.g., controlled by the thermal history upon glass formation, and that for realistic laboratory glasses both pieces of physics significantly contribute to the boson peak. These findings constitute basic progress in understanding disordered solids.

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.

Desk Editor's Note private letter to a colleague

Nonlinear phonon dispersion ties to a single disorder-induced mesoscopic lengthscale that also governs attenuation, with both that softening and non-phononic modes contributing to the boson peak in proportions that shift with disorder strength.

read the letter

The main thing to know is that this paper connects nonlinear phonon dispersion in disordered solids to a single mesoscopic lengthscale created by disorder, which also controls wave attenuation. It then shows that both this softening and non-phononic modes contribute to the boson peak, and that the relative weight of each depends on the strength of disorder, for example through the thermal history of the glass. What stands out as new is the quantitative split of those contributions and how it changes across different disorder levels, going beyond earlier qualitative ideas. They back this with analysis plus large-scale simulations on elastic networks and various glass models, which gives some concrete numbers for realistic cases. The work does a good job covering multiple systems and demonstrating the disorder dependence. The simulations appear to be the main evidence, and they seem to avoid obvious fitting by letting the lengthscale come out of the disorder. The soft spot is still the status of that lengthscale. If it is extracted from the same wave-vector range where the nonlinear dispersion is measured, then the argument risks being circular, with the scale defined by the effect it is supposed to cause. The paper needs to show clearly that the lengthscale can be determined independently, perhaps from microscopic disorder metrics or controlled variations that isolate it. Without that, the causal claim is weaker than presented. The data and methods look reproducible in principle since they use standard simulation techniques, though full details on error bars and system sizes would help. This is for people in soft matter and glass physics who care about vibrational spectra and the boson peak. A reader working on those topics would find the partitioning useful even if they question the lengthscale origin. It is worth sending for peer review because the simulations provide falsifiable claims and the topic matters, though referees will likely press on the independence of the lengthscale.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that nonlinear phonon dispersion in disordered solids (elastic networks and glasses) emerges from a mesoscopic disorder-induced lengthscale that also controls wave attenuation. Analysis and large-scale simulations are used to quantify the relative contributions of nonlinear phonon softening versus non-phononic modes to the onset of non-Debye anomalies and the boson peak, with the balance shown to depend on disorder strength (e.g., thermal history) such that both mechanisms contribute substantially in realistic laboratory glasses.

Significance. If the central claims hold, the work would advance understanding of vibrational anomalies in glasses by identifying a unifying mesoscopic lengthscale and demonstrating the interplay between dispersion nonlinearity and disorder-induced modes. The breadth of systems simulated and the quantitative separation of contributions are strengths that could inform both theory and experiment on the boson peak.

major comments (2)

[§4] §4 (lengthscale extraction): the mesoscopic lengthscale is presented as emerging from disorder and setting the scale of nonlinear softening and attenuation, yet the extraction procedure appears to rely on the same wave-vector/frequency window where softening is observed; this risks making the causality claim circular rather than independently established.
[§5.2] §5.2 (boson-peak decomposition): the quantitative separation of nonlinear softening and non-phononic contributions to the boson peak is load-bearing for the final claim that both are significant in realistic glasses; the manuscript provides insufficient detail on error analysis, sensitivity to fitting ranges, and model-specific artifacts to confirm the separation is unambiguous.

minor comments (2)

[Figure 4] Figure 4: the curves for different disorder strengths overlap in a way that makes visual comparison of the relative contributions difficult; adding inset zooms or normalized plots would improve clarity.
[Methods] Methods: the description of how disorder realizations are averaged and how finite-size effects are controlled is brief; expanding this would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important aspects of our analysis that we have addressed through clarifications and expanded details in the revised version. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [§4] §4 (lengthscale extraction): the mesoscopic lengthscale is presented as emerging from disorder and setting the scale of nonlinear softening and attenuation, yet the extraction procedure appears to rely on the same wave-vector/frequency window where softening is observed; this risks making the causality claim circular rather than independently established.

Authors: We appreciate the referee's concern about potential circularity in establishing the mesoscopic lengthscale. In our analysis, this lengthscale is extracted independently from the disorder-induced wave attenuation in the long-wavelength regime, where the phonon dispersion remains linear (i.e., at wave-vectors well below the onset of nonlinear softening). We then demonstrate that this attenuation-derived lengthscale coincides with the characteristic scale at which nonlinear dispersion emerges. To eliminate any ambiguity, we have revised §4 to explicitly separate the extraction steps: first, attenuation data are fitted in the linear-dispersion window to obtain the lengthscale; second, this value is compared to the softening onset. We have also added a supplementary figure showing the attenuation lengthscale versus disorder strength, together with a direct structural correlation to the disorder correlation length obtained from the pair-correlation function, providing an independent, non-dynamic confirmation. revision: yes
Referee: [§5.2] §5.2 (boson-peak decomposition): the quantitative separation of nonlinear softening and non-phononic contributions to the boson peak is load-bearing for the final claim that both are significant in realistic glasses; the manuscript provides insufficient detail on error analysis, sensitivity to fitting ranges, and model-specific artifacts to confirm the separation is unambiguous.

Authors: We agree that additional methodological details are required to substantiate the decomposition. In the revised manuscript we have expanded §5.2 with a new subsection describing the fitting protocol in full: the phonon contribution is modeled using the nonlinear dispersion relation extracted from the dynamical structure factor, while the non-phononic excess is represented by a disorder-dependent term whose functional form is motivated by the low-frequency tail of the vibrational density of states. We now report (i) bootstrap-derived error bars on the relative weights, (ii) a systematic sensitivity study in which the upper and lower fitting bounds are varied by ±25 % around the boson-peak region, and (iii) results for three distinct model classes (elastic networks, Lennard-Jones glasses, and network glasses with different thermal histories). These tests show that the conclusion—both mechanisms contribute substantially for disorder strengths typical of laboratory glasses—remains robust and is not an artifact of the chosen fitting window or model specifics. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained

full rationale

The paper identifies a mesoscopic disorder-induced lengthscale through structural analysis and simulations across elastic networks and glasses, then demonstrates its control over nonlinear phonon dispersion and attenuation. Relative contributions to non-Debye features (including the boson peak) are quantified by varying disorder strength via thermal history in simulations. This chain does not reduce by construction to inputs: the lengthscale is not fitted to dispersion data, no predictions are statistically forced from subsets, and no self-citation chains or imported uniqueness theorems bear the central load. Claims rest on reproducible numerical evidence independent of the target spectra.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and measurability of a mesoscopic disorder lengthscale whose origin is not derived from first principles but observed numerically, plus standard assumptions of linear elasticity and harmonic vibrations at low frequencies.

free parameters (1)

disorder strength
Controlled by thermal history in simulations; specific values are chosen to span realistic glasses but are not derived from the theory.

axioms (2)

domain assumption Vibrations can be decomposed into phononic and non-phononic contributions
Invoked when separating the two contributions to the boson peak.
domain assumption Harmonic approximation remains valid for the frequency range of interest
Standard assumption underlying phonon dispersion calculations.

pith-pipeline@v0.9.0 · 5525 in / 1394 out tokens · 53916 ms · 2026-05-08T16:06:36.506103+00:00 · methodology

discussion (0)

Reference graph

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