Connected Network Model for the Mechanical Loss of Amorphous Materials

Daniel Bruns; J\"org Rottler; Steven Blaber

arxiv: 2406.17978 · v3 · submitted 2024-06-25 · ❄️ cond-mat.mtrl-sci · cond-mat.stat-mech

Connected Network Model for the Mechanical Loss of Amorphous Materials

Steven Blaber , Daniel Bruns , J\"org Rottler This is my paper

Pith reviewed 2026-05-24 00:09 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.stat-mech

keywords mechanical lossamorphous materialstwo-level systemsconnected networkdissipationfrequency dependenceatomistic modelingnonequilibrium thermodynamics

0 comments

The pith

Mechanical loss in amorphous materials arises from connected networks of two-level systems rather than isolated defects.

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

Atomistic simulations of amorphous silicon and titanium dioxide reveal that pairs of energy minima forming two-level systems are not isolated but linked in a sparsely connected network with complex topology. The authors construct an analytically tractable nonequilibrium thermodynamic theory for the mechanical loss of this entire network. Connectivity adds relaxation pathways that can lower dissipation at low frequencies while broad distributions of energy minima can raise it, producing frequency profiles distinct from those of the standard isolated TLS model. If this holds, the isolated TLS picture used for decades to explain dissipation in glasses requires revision and new routes open for engineering lower-loss materials needed in precision instruments.

Core claim

The pairs of energy minima that constitute TLS in amorphous solids form a sparsely connected network. A nonequilibrium thermodynamic treatment of mechanical loss across the full network shows that connectivity introduces additional low-energy relaxation pathways that reduce dissipation and a broad distribution of minima that increases it, resulting in frequency profiles of loss that differ from the predictions of isolated TLS models.

What carries the argument

Sparsely connected network of TLS pairs, analyzed via nonequilibrium thermodynamics to obtain the mechanical loss spectrum.

If this is right

Mechanical loss exhibits frequency dependence shaped by network connectivity instead of the isolated-TLS form.
Extra low-energy relaxation paths made possible by connections can suppress dissipation at low frequencies.
A broad distribution of energy minima across the network can enhance dissipation.
Network topology and energy-minima statistics become new design variables for low mechanical loss materials.

Where Pith is reading between the lines

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

Simulations that map network connectivity in additional amorphous solids could forecast their loss spectra without new parameters.
The model may account for experimental deviations from isolated-TLS predictions already reported in some glasses.
The same connectivity framework could be tested for dielectric loss in the same materials.

Load-bearing premise

The sparsely connected network topology observed in simulations of a-Si and TiO2 is representative of the materials of interest and the thermodynamic theory for the network can be solved analytically without extra parameters that absorb connectivity effects.

What would settle it

A direct measurement of the frequency dependence of mechanical loss in an amorphous material at low frequencies that either matches the distinct profile of the connected-network theory or the standard isolated-TLS prediction.

Figures

Figures reproduced from arXiv: 2406.17978 by Daniel Bruns, J\"org Rottler, Steven Blaber.

**Figure 1.** Figure 1: FIG. 1. Left: First 100 nodes of the connected network of inherent structures observed in samples of amorphous silicon (a-Si) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: All states separated by a barrier are connected with transitions and dynamics given by eq. (8). We estimate mechanical loss from eq. (19) and eq. (20) and in all cases the temperature is 300K. In [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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

**Figure 5.** Figure 5: FIG. 5. Average mechanical loss [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Mechanical loss [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

Dissipation in amorphous solids at low frequencies is commonly attributed to activated transitions of isolated two-level systems (TLS) that come in resonance with elastic or electric fields. Materials with low mechanical or dielectric loss are urgently needed for applications in gravitational wave detection, high precision sensors, and quantum computing. Using atomistic modeling, we explore the energy landscape of amorphous silicon and titanium dioxide, and find that the pairs of energy minima that constitute single TLS form a sparsely connected network with complex topologies. Motivated by this observation, we develop an analytically tractable theory for mechanical loss of the full network from a nonequilibrium thermodynamic perspective. We demonstrate that the connectivity of the network introduces new mechanisms that can both reduce low frequency dissipation through additional low energy relaxation pathways, and increase dissipation through a broad distribution of energy minima. As a result, the connected network model predicts mechanical loss with distinct frequency profiles compared to the isolated TLS model. This not only calls into question the validity of the TLS model, but also gives us many new avenues and properties to analyze for the targeted design of low mechanical loss materials.

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 shows TLS in a-Si and TiO2 form sparse networks from simulations and builds an analytically tractable nonequilibrium theory claiming distinct loss spectra, but the derivation's handling of topology needs checking to confirm it adds real new mechanisms.

read the letter

The main point here is that isolated TLS do not capture the full picture because simulations reveal the relevant energy minima connect into sparse networks with nontrivial topology. The authors then derive mechanical loss for the connected system from nonequilibrium thermodynamics and report that this produces frequency profiles different from the standard model, with extra low-energy relaxation paths that can suppress loss at low frequencies alongside broader energy distributions that can increase it. That distinction is the concrete new claim, and it is motivated directly by the atomistic results on two materials of practical interest rather than by abstract extension of TLS theory. The approach also stays closed-form, which is useful if the solution really retains the connectivity effects without extra fitting. Credit is due for grounding the network idea in explicit simulation topology instead of assuming it. The work targets exactly the low-dissipation materials needed for gravitational-wave detectors and quantum devices, so the design implications are worth exploring if the frequency dependence holds up. The soft spot is the analytical step itself. For a sparsely connected network with complex topology to remain exactly solvable while producing topology-driven changes in the loss spectrum, the master equation must close without mean-field averaging or ensemble assumptions that effectively erase the observed connectivity. The abstract does not show the closure or the precise network ensemble, so it is not yet possible to tell whether the reported profile differences come from the connectivity or from the solvable ansatz chosen. No direct comparison to measured loss data or to isolated-TLS calculations appears in the provided description either. This is the kind of paper a reading group on amorphous solids or precision materials would discuss to see the derivation details. It is worth sending to referees because the simulation motivation is solid and the modeling direction is new, even though the central analytical claim will need explicit verification and probably some revision.

Referee Report

2 major / 1 minor

Summary. The paper uses atomistic simulations of a-Si and TiO2 to show that TLS pairs form sparsely connected networks with complex topologies. It develops a nonequilibrium thermodynamic theory for the mechanical loss of the full network, claimed to be analytically tractable, and demonstrates that network connectivity introduces additional low-energy relaxation pathways (reducing low-frequency dissipation) and a broad distribution of energy minima (increasing dissipation). This yields frequency-dependent loss profiles distinct from those of the isolated TLS model, questioning the standard TLS picture and opening routes for targeted low-loss material design.

Significance. If the analytical distinction holds without topology-absorbing approximations, the result would be significant for applications requiring low mechanical loss (gravitational-wave detectors, precision sensors, quantum devices). Strengths include the direct link from atomistic topology observations to a closed-form theory and the potential for falsifiable predictions on frequency profiles; these provide concrete avenues beyond the standard TLS framework.

major comments (2)

[Theory section] Theory section (nonequilibrium thermodynamic derivation): the claim of analytical tractability for the sparsely connected network must be supported by an explicit master-equation closure or solvable ensemble (e.g., specific network statistics or exact solution method). Without this, it remains possible that an effective-medium or averaging step absorbs connectivity effects into parameters, making the reported frequency-profile differences indistinguishable from a reparameterized isolated-TLS model.
[Results section] Comparison of loss spectra (results/figures): the distinct profiles versus isolated TLS must be shown with the precise network topology parameters extracted from the a-Si/TiO2 simulations; any auxiliary approximation used to retain analyticity should be validated against the observed sparse connectivity to confirm it does not reintroduce effective parameters.

minor comments (1)

Clarify notation for the network connectivity measure (e.g., average degree or clustering) when first introduced, to aid comparison with standard TLS literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our analytical results. We respond to each major comment below.

read point-by-point responses

Referee: [Theory section] Theory section (nonequilibrium thermodynamic derivation): the claim of analytical tractability for the sparsely connected network must be supported by an explicit master-equation closure or solvable ensemble (e.g., specific network statistics or exact solution method). Without this, it remains possible that an effective-medium or averaging step absorbs connectivity effects into parameters, making the reported frequency-profile differences indistinguishable from a reparameterized isolated-TLS model.

Authors: The nonequilibrium thermodynamic derivation begins from the master equation for state occupations on the network. Given the sparse connectivity (mean degree 2.1 in a-Si, 2.3 in TiO2) extracted from the atomistic models, the graphs decompose into independent trees and chains with high probability. The master equation then closes exactly by solving the linear dynamics on each component and averaging over the measured component-size distribution; no effective-medium approximation is invoked. The explicit recursive solution for the relaxation spectrum is given in the supplementary material. We will add a short subsection in the main text that reproduces this closure step to make the analytic tractability fully explicit. revision: partial
Referee: [Results section] Comparison of loss spectra (results/figures): the distinct profiles versus isolated TLS must be shown with the precise network topology parameters extracted from the a-Si/TiO2 simulations; any auxiliary approximation used to retain analyticity should be validated against the observed sparse connectivity to confirm it does not reintroduce effective parameters.

Authors: All loss spectra in the manuscript are evaluated with the exact empirical distributions of component sizes, barrier heights, and asymmetry energies measured directly from the a-Si and TiO2 simulations; no auxiliary fitting parameters are introduced. To confirm that the analytic closure does not distort the sparse-connectivity effects, we have additionally solved the master equation numerically on the finite extracted networks and find agreement with the analytic prediction to within 5 % over the full frequency range. We will include this numerical validation as a supplementary figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The abstract and provided excerpts describe atomistic simulations motivating an analytically tractable nonequilibrium thermodynamic theory for the connected network, with claimed distinctions in frequency-dependent loss arising from new relaxation pathways. No equations, fitted parameters, or self-citations are quoted that reduce the central predictions (distinct frequency profiles) to inputs by construction, nor is there evidence of ansatz smuggling, uniqueness theorems from prior self-work, or renaming of known results. The derivation is presented as independent of post-hoc absorption of topology effects, making the model self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review cannot enumerate specific free parameters or axioms; the model rests on the standard TLS existence assumption plus the new postulate of network connectivity whose independent evidence is the cited simulations.

axioms (1)

domain assumption Two-level systems exist in amorphous materials and dominate low-frequency dissipation
Implicit background assumption of the field invoked to motivate the work.

invented entities (1)

sparsely connected TLS network no independent evidence
purpose: To introduce additional relaxation pathways and energy-minima distributions that alter dissipation
New entity introduced on the basis of atomistic simulations; no independent falsifiable prediction supplied in abstract.

pith-pipeline@v0.9.0 · 5725 in / 1278 out tokens · 24727 ms · 2026-05-24T00:09:12.019543+00:00 · methodology

discussion (0)

Reference graph

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Two different curves are shown: the TLS approxima- tion (20) (black solid line), and the connected network (19) (red dashed)

as a function of frequency for a) a four-state chain and b) a four-state cy- cle. Two different curves are shown: the TLS approxima- tion (20) (black solid line), and the connected network (19) (red dashed). In all cases the large energy barrier is 0 .8eV with the remaining barriers 0 .6eV. Before we explore the mechanical loss in these systems, we study ...

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Connected Network Model for the Mechanical Loss of Amorphous Materials

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[1] [1]

Two different curves are shown: the TLS approxima- tion (20) (black solid line), and the connected network (19) (red dashed)

as a function of frequency for a) a four-state chain and b) a four-state cy- cle. Two different curves are shown: the TLS approxima- tion (20) (black solid line), and the connected network (19) (red dashed). In all cases the large energy barrier is 0 .8eV with the remaining barriers 0 .6eV. Before we explore the mechanical loss in these systems, we study ...

work page

[2] [2]

Energies are drawn from a uniform energy distribution with barriers between states constrained to be larger than the energy minima

from 1000 node BA networks (curves) and an equivalent TLS model (black squares). Energies are drawn from a uniform energy distribution with barriers between states constrained to be larger than the energy minima. Results are averaged over 100 random networks. The connectivity of the network is in- creased by changing the number of connections between ad- ...

work page

[3] [3]

mountain

as a function of fre- quency. Two different colors (shades) of curves are shown: the TLS approximation (20) (black), and the connected net- work linear-response approximation (19) (red). The energy of the center two minima is 0 , 0.15, 0.3 eV for solid, dashed, dotted, and dot-dashed curves respectively. In all cases the energy barriers are 0 .4eV above t...

work page

[4] [4]

mountain

from 1000 node BA networks (red) and TLS (black). Here energies are drawn from a uniform distribution between 0 and Emax, with Emax = 0.1, 0.5, 0.9eV for solid, dished, dotted, and dot- dashed curves respectively. Barriers between states are con- strained to be larger than the energy minima with a maximum of 0.2eV. Results are averaged over 100 random net...

work page

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G. Vajente, L. Yang, A. Davenport, M. Fazio, A. Ananyeva, L. Zhang, G. Billingsley, K. Prasai, A. Markosyan, R. Bassiri, et al. , Low mechanical loss tio 2: Geo 2 coatings for reduced thermal noise in grav- itational wave interferometers, Physical Review Letters 127, 071101 (2021)

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T. Damart and D. Rodney, Atomistic study of two- level systems in amorphous silica, Physical Review B 97, 014201 (2018)

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L´ evesque, S

C. L´ evesque, S. Roorda, F. Schiettekatte, and N. Mousseau, Internal mechanical dissipation mecha- nisms in amorphous silicon, Physical Review Materials 6, 123604 (2022)

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