Mean-field model of interacting quasilocalized excitations in glasses

Abstract

Structural glasses feature quasilocalized excitations whose frequencies \omegaω follow a universal density of states {D}(\omega)\!\sim\!\omega^4D(ω)∼ω4. Yet, the underlying physics behind this universality is not 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_0κ0) in the absence of interactions, interact among themselves through random couplings (characterized by a strength JJ) and with the surrounding elastic medium (an interaction characterized by a constant force hh). We first show that the model gives rise to a gapless density of states {D}(\omega)\!=\!A_{g}\,\omega^4D(ω)=Agω4 for a broad range of model parameters, expressed in terms of the strength of the oscillators’ 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_{g}(h,J,\kappa_0)Ag(h,J,κ0), 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_{g}(h,J,\kappa_0)Ag(h,J,κ0) is a non-monotonic function of JJ for a fixed hh, varying predominantly exponentially with -(\kappa_0 h^{2/3}\!/J^2)−(κ0h2/3/J2) in the weak interactions (small JJ) regime — reminiscent of recent observations in computer glasses — and predominantly decays as a power-law for larger JJ, in a regime where hh 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.

Marginal stability of soft anharmonic mean field spin glasses

“…The model and its phase diagram -We consider the KHGPS model recently introduced in [29,30]. This model is an evolution of a model introduced by Gurevich Parshin and Schöber [31] and whose mean field limit has been considered by Kühn and Hortsmann in [32].…”

Effects of intraspecific cooperative interactions in large ecosystems

“…In [60] it has been shown that at small external field the density of states is purely quadratic D(ω) ∼ ω 2 as correctly predicted by mean-field models, and the corresponding modes are delocalized. On the other hand, upon increasing the external field, a quartic pseudogap D(ω) ∼ ω 4 takes place associated with the appearance of localized modes.…”

“…In a recent paper [60], a generalization of the Kühn and Horstmann (KH) model combined with the Gurevich, Parshin and Schober (GPS) three-dimensional lattice version in the presence of random interactions and a constant external field has been proposed in order to study zerotemperature vibrational modes of a specifically designed disordered system. The model is formulated as a collection of anharmonic oscillators subject to a random distribution p(κ) for the stiffnesses κ, which are taken uniform in the interval [κ min , κ max ] for positive or zero values.…”

Low energy excitations of mean-field glasses

“…The internal free-energies of TAP states and their Complexity are obtained by eqs. (17) and they read…”

“…It appears that deeper states in the landscape, corresponding to better optimized glasses, have less and less the low energy excitations, reflecting in smaller and smaller values of A 4 , and correspondingly, the excitations are more and more localized [12,13]. This spectrum of localized modes was first rationalized through phenomenological theories [14,15], while new predictions have recently enriched the picture [11][12][13][16][17][18][19]. In addition to typical ungapped minima, found by usual minimization protocols, it has been noticed in [20] that in some model glasses gapped mimima can be found through the use of smart minimization protocols that include particle swap [21,22].…”