# Mean-field model of interacting quasilocalized excitations in glasses

### Submission summary

 As Contributors: Edan Lerner Preprint link: scipost_202102_00013v1 Date accepted: 2021-03-30 Date submitted: 2021-02-09 14:22 Submitted by: Lerner, Edan Submitted to: SciPost Physics Core Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

Structural glasses feature quasilocalized excitations whose frequencies $\omega$ follow a universal density of states ${\cal D}(\omega)\!\sim\!\omega^4$. Yet, the underlying physics behind this universality is not yet fully understood. Here we study a mean-field model of quasilocalized excitations in glasses, viewed as groups of particles embedded inside an elastic medium and described collectively as anharmonic oscillators. The oscillators, whose harmonic stiffness is taken from a rather featureless probability distribution (of upper cutoff $\kappa_0$) in the absence of interactions, interact among themselves through random couplings (characterized by strength $J$) and with the surrounding elastic medium (an interaction characterized by a constant force $h$). We first show that the model gives rise to a gapless density of states ${\cal D}(\omega)\!=\!A_{\rm g}\,\omega^4$ for a broad range of model parameters, expressed in terms of the strength of stabilizing anharmonicity, which plays a decisive role in the model. Then --- using scaling theory and numerical simulations --- we provide a complete understanding of the non-universal prefactor $A_{\rm g}(h,J,\kappa_0)$, of the oscillators' interaction-induced mean square displacement and of an emerging characteristic frequency, all in terms of properly identified dimensionless quantities. In particular, we show that $A_{\rm g}(h,J,\kappa_0)$ is a non-monotonic function of $J$ for a fixed $h$, varying predominantly exponentially with $-(\kappa_0 h^{2/3}\!/J^2)$ in the weak interactions (small $J$) regime --- reminiscent of recent observations in computer glasses --- and predominantly decays as a power-law for larger $J$, in a regime where $h$ plays no role. We discuss the physical interpretation of the model and its possible relations to available observations in structural glasses, along with delineating some future research directions.

Published as SciPost Phys. Core 4, 008 (2021)

### Submission & Refereeing History

Submission scipost_202102_00013v1 on 9 February 2021

## Reports on this Submission

### Anonymous Report 2 on 2021-3-15 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202102_00013v1, delivered 2021-03-15, doi: 10.21468/SciPost.Report.2701

### Strengths

This paper presents a mean field analysis of the low frequency of a density of states for a simple Hamiltonian designed to model local anharmonic vibrations that are pairwise coupled. The analysis recovers a quartic power law in frequency, similar to that observed in simulations of more realistic model glasses. Numerical calculations establish the dependence of the absolute magnitude of the density of states on the coupling strength J. This is a valuable contribution to the study of the novel low frequency mechanical properties of glassy solids.

### Weaknesses

The coupling between these localized anharmonicities seemed poorly rationalized, given its significance. I was left to wonder about the more obvious coupling between each individual anharmonicity and the surrounding elastic medium. I can imagine that this coupling may not contribute significantly at low frequencies but some discussion on this point would be helpful.

### Report

I recommend the paper be accepted for publication once the authors have had a chance to consider the comment above.

• validity: high
• significance: high
• originality: good
• clarity: high
• formatting: good
• grammar: good

### Anonymous Report 1 on 2021-2-18 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202102_00013v1, delivered 2021-02-18, doi: 10.21468/SciPost.Report.2669

### Report

In this paper the properties of a model introduced by Kühn and Horstmann and Gurevich et al. describing local defects in glasses are investigated. The model consists of a set or random anharmonic oscillators, coupled linearly to each other in the presence of an external force h. The randomness is given in terms of spatially fluctuating harmonic stiffness coefficients and couplings. The distribution density of the stiffness coefficients is assumed to be uniform and extend to zero. The couplings are assumed to fluctuate according to a Gaussian of variance $\sim J^2$.

By means of the combination of a scaling analysis and numerical simulations the authors establish the following results:

- There is always a low-frequency density of states (DOS) proportional to the fourth power of frequency
- For weak interaction strength $J$ the crossover to a linear behavior of the DOS occurs at a frequency $w_x$, which is proportional to $h^3/2$, for larger $J$, $w_x$ becomes independent of $h$ and is related to the effective mean shift of the oscillator stiffness due to the anharmonicity.
- These findings lead to a strong non-monotonic dependence of the prefactor of the $w^4$ DOS on the interaction strength $J$: in the weak-interaction regime it increases exponentially and then crosses over to an inverse-power law.

The presentation, the results and the discussion are highly interesting and important for the glass community, in which the role of quasilocalized oscillaters has been widely debated recently.

The paper would benefit from proof-reading by a native-English speaking person.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -