arxiv: 2605.09436 · v1 · submitted 2026-05-10 · ❄️ cond-mat.dis-nn

Recognition: 2 theorem links

· Lean Theorem

Microscopic origin of Boson peak in amorphous solids

Cunyuan Jiang

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:28 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords Boson peakamorphous solidsvibrational density of statescoordination number fluctuationsspring constant fluctuationsdynamical matrixdisordered networks

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The pith

Fluctuation of coordination numbers alone produces the Boson peak in amorphous solids.

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

The paper proposes a simple model based on a network of nodes connected by springs to explain the Boson peak, an excess of low-frequency vibrational modes in glasses. It splits disorder into two independent sources: variations in spring strengths and variations in the number of springs meeting at each node. Calculations on the scalar dynamical matrix show that only the variations in node coordination create the characteristic hump in the density of states at low frequencies. Variations in spring strength mainly damp the modes and have little effect on the low-frequency spectrum. This reduces the problem of anomalous vibrations in amorphous solids to a question of network topology.

Core claim

We proposed a non-analytic model to explain the microscopic origin of the anomalous vibrational density of states (DOS), the Boson peak (BP), in amorphous solids based on the scalar dynamical matrix of a network with springs and nodes. We argue that disorder can be classified into two factors: fluctuation of spring strength and fluctuation of coordination numbers (the number of springs connected to a node). The results suggest that BP originates solely from fluctuation of coordination numbers, while the fluctuation of spring strength only contributes to the effect of damping and has very limited effect on low frequency DOS. This work converts complexity into simplicity and provides a direct

What carries the argument

Scalar dynamical matrix of a spring-node network that isolates fluctuations in spring strengths from fluctuations in coordination numbers to determine their separate effects on the vibrational density of states.

If this is right

The position and intensity of the Boson peak are set by the variance in the number of connections per node.
Disorder in bond strengths produces damping and broadening but does not generate excess modes at low frequency.
The low-frequency vibrational spectrum depends primarily on topological disorder rather than on energetic disorder.
Model systems can vary coordination statistics independently to tune the Boson peak without changing spring constants.

Where Pith is reading between the lines

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

Materials could have their Boson peak adjusted by changing local coordination distributions while keeping bond stiffness fixed.
The result points to fluctuations around the isostatic coordination as a possible universal driver of anomalous vibrations.
The scalar separation of disorders may be tested by extending the model to vector displacements in three dimensions.
Simulations of amorphous solids can focus computational resources on accurate sampling of coordination number variance.

Load-bearing premise

The scalar dynamical matrix on a spring-node network with two independent disorder types fully captures the vibrational physics of real three-dimensional amorphous solids.

What would settle it

Calculate the density of states for a network in which every node has exactly the same number of springs but spring constants are randomized; the absence of a Boson peak would support the claim that coordination fluctuations are required.

Figures

Figures reproduced from arXiv: 2605.09436 by Cunyuan Jiang.

**Figure 2.** Figure 2: FIG. 2: The vibrational density of states (DOS) reduced by Debye’s law [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

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

The paper uses a scalar spring-node model to claim the Boson peak arises solely from coordination-number fluctuations, with spring-strength disorder adding only damping.

read the letter

The main thing to know is that this work separates disorder into two independent types on a scalar dynamical matrix and concludes the excess low-frequency density of states comes only from coordination fluctuations. Spring-strength variations are said to contribute little beyond damping. They introduce a non-analytic model to reach this separation and present it as a direct answer to the microscopic origin of the Boson peak. That classification and the explicit downplaying of spring disorder look new relative to the usual mixed-disorder treatments in the literature. The approach has the virtue of turning a messy problem into something simpler and more falsifiable in a network setting. If the numerics hold, it gives theorists a cleaner handle on which kind of disorder actually drives the anomaly. The soft spot is the scalar approximation itself. Real amorphous solids are three-dimensional with vector displacements, so longitudinal and transverse modes couple, and coordination disorder typically creates local strains and bond-length correlations that the model treats as independent. Those effects could easily shift or broaden the low-frequency excess DOS in ways the scalar case misses. Without explicit checks against vector Hessians or atomistic simulations of real glasses, the “solely” claim stays provisional. The abstract gives no error analysis or sensitivity tests, so the full derivations will need close scrutiny. This is for people who already work with effective network models of glasses and vibrational anomalies. They can use the separation as a starting point even if they later add vector or correlation terms. It is worth sending to referees because the claim is specific, the model is reproducible in principle, and the field needs concrete proposals that can be tested or extended rather than more simulations without a clear mechanism.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a non-analytic model based on the scalar dynamical matrix of a spring-node network to explain the microscopic origin of the Boson peak (BP) in amorphous solids. Disorder is classified into two independent factors: fluctuations of spring strength and fluctuations of coordination numbers. The central claim is that the BP originates solely from coordination-number fluctuations, while spring-strength fluctuations contribute only to damping and have very limited effect on the low-frequency density of states (DOS).

Significance. If the result holds, the work offers a simplified microscopic explanation for the BP by attributing it to a single dominant source of disorder in a minimal network model, converting complexity into simplicity. This could provide a direct answer to a long-standing puzzle if the separation of effects is robust. The model is presented as explanatory rather than tautological, with no free parameters highlighted in the abstract.

major comments (2)

[Model definition and abstract] The scalar dynamical matrix on a spring-node network with independently imposed fluctuations (as described in the model) may not capture the vectorial nature of vibrations in real 3D amorphous solids. Real systems require a vectorial Hessian whose eigenvalues include both longitudinal and transverse modes; coordination disorder also induces local strain fields and topology-bond length correlations absent when the two disorder types are imposed independently. This assumption is load-bearing for the claim that BP originates solely from coordination fluctuations.
[Results on DOS (likely §4 or equivalent)] The conclusion that spring-strength fluctuations have very limited effect on low-frequency DOS (while coordination fluctuations are the sole origin) is derived entirely within the scalar model. Without explicit comparison to vectorial 3D simulations or checks for how correlations between disorder types alter the excess DOS, the separation into 'sole origin' and 'limited effect' does not necessarily survive in more realistic settings.

minor comments (2)

[Abstract] The abstract uses 'We proposed' in past tense; consistent present tense ('We propose') is preferable for describing the work.
[Methods/Model] Clarify how the non-analytic model is constructed and solved (e.g., any specific equations for the dynamical matrix or averaging procedure) to allow reproducibility.

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, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Model definition and abstract] The scalar dynamical matrix on a spring-node network with independently imposed fluctuations (as described in the model) may not capture the vectorial nature of vibrations in real 3D amorphous solids. Real systems require a vectorial Hessian whose eigenvalues include both longitudinal and transverse modes; coordination disorder also induces local strain fields and topology-bond length correlations absent when the two disorder types are imposed independently. This assumption is load-bearing for the claim that BP originates solely from coordination fluctuations.

Authors: We appreciate the referee pointing out the distinction between scalar and vectorial descriptions. Our scalar dynamical matrix is intentionally adopted as a minimal model that enables independent control and isolation of the two disorder types, which is essential for demonstrating their separate contributions to the DOS. This approach follows a long tradition in the field where scalar models have been shown to reproduce the essential low-frequency anomalies, including the Boson peak. We will revise the manuscript to expand the discussion of model limitations, explicitly note the absence of local strain fields and bond-length correlations, and reference prior vectorial studies that support the dominant role of coordination disorder. revision: partial
Referee: [Results on DOS (likely §4 or equivalent)] The conclusion that spring-strength fluctuations have very limited effect on low-frequency DOS (while coordination fluctuations are the sole origin) is derived entirely within the scalar model. Without explicit comparison to vectorial 3D simulations or checks for how correlations between disorder types alter the excess DOS, the separation into 'sole origin' and 'limited effect' does not necessarily survive in more realistic settings.

Authors: We agree that the quantitative separation is demonstrated within the scalar framework. In the revised version we will add a dedicated paragraph clarifying the scope of the conclusions and discussing consistency with existing vectorial simulations in the literature, where coordination fluctuations similarly produce excess low-frequency modes. Performing new, fully independent vectorial 3D simulations with decoupled disorder types would require substantial additional work outside the present scope, but the minimal-model insight remains useful for guiding such extensions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained within model assumptions

full rationale

The paper introduces a scalar dynamical matrix model on a spring-node network and partitions disorder into two independent types (spring-strength fluctuations and coordination-number fluctuations). It then computes the vibrational DOS within this model and reports that the boson peak appears only for the coordination disorder. No load-bearing step reduces to a fitted parameter renamed as a prediction, no self-citation chain is invoked to justify uniqueness, and no ansatz is smuggled via prior work. The separation of effects is an output of the explicit model calculations rather than a definitional identity. The modeling choice (scalar 2-D network) is an assumption about physics, not a circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model assumes a scalar dynamical matrix suffices and that disorder factors cleanly into two independent fluctuations; no free parameters or invented entities are named in the abstract.

axioms (2)

domain assumption Vibrational modes of amorphous solids are adequately described by the scalar dynamical matrix of a spring-node network.
Invoked in the abstract as the basis for the entire calculation.
domain assumption Disorder can be partitioned into independent fluctuations of spring strength and coordination number.
Stated as the classification that allows the separation of effects.

pith-pipeline@v0.9.0 · 5399 in / 1186 out tokens · 29587 ms · 2026-05-12T02:28:50.372617+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

BP originates solely from fluctuation of coordination numbers, while the fluctuation of spring strength only contributes to the effect of damping
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

model the amorphous solid as a network of springs and nodes... scalar dynamical matrix of size N×N

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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