Continuum limit of the vibrational properties of amorphous solids

Abstract

The low-frequency vibrational and low-temperature thermal properties of amorphous solids are markedly different from those of crystalline solids. This situation is counter-intuitive because any solid material is expected to behave as a homogeneous elastic body in the continuum limit, in which vibrational modes are phonons following the Debye law. A number of phenomenological explanations have been proposed, which assume elastic heterogeneities, soft localized vibrations, and so on. Recently, the microscopic mean-field theories have been developed to predict the universal non-Debye scaling law. Considering these theoretical arguments, it is absolutely necessary to directly observe the nature of the low-frequency vibrations of amorphous solids and determine the laws that such vibrations obey. Here, we perform an extremely large-scale vibrational mode analysis of a model amorphous solid. We find that the scaling law predicted by the mean-field theory is violated at low frequency, and in the continuum limit, the vibrational modes converge to a mixture of phonon modes following the Debye law and soft localized modes following another universal non-Debye scaling law.

Universal disorder-induced broadening of phonon bands: from disordered lattices to glasses

“…On the other hand, the onset of glassy modes is expected to follow L −2/5 in 2D (with possible logarithmic corrections, see discussion in [45]), i.e.it is larger than the crossover frequency ω † . This means that in 2D, above some system size, we expect harmonic QLGMs to be unobservable altogether, and no coexistence regime with phonons to exist, as indeed reported in [18].…”

“…We presented extensive computer simulation data that suggest that QLGMs loose their quasilocalized nature if they occur within frequency intervals occupied by phonon bands, due to hybridizations and mixing with those phonons. This, in turn, implies that a coexistence frequency window of phonons and harmonic QLGMs opens at intermediate system sizes, explaining the observations of [18,19] that report coexistence of these two types of low-frequency excitations, distinguished by their participation ratio. Our results further indicate that the said coexistence frequency window vanishes in the thermodynamic limit, seriously questioning the claim in [18,19] that coexistence persists in the continuum (i.e.…”

“…We therefore calculated the histogram of the vibrational frequencies, and fitted a Gaussian curve to each peak that was cleanly distinguishable, as demonstrated in figures B1(b) and (c). For our 3D glasses, we followed the approach of [18] and calculated histograms of vibrational frequencies after filtering phonons from non-phononic modes by discarding of modes with participation ratios e<0.03. The standard deviation of the Gaussian fits are reported as the phonon band widths.…”

“…It has been recently discovered [13] that non-phononic vibrational modes populate the low-frequency tails of the vibrational density of states of structural glasses, whose frequencies are distributed according to a universal gapless ω 4 law, independently of microscopic details. It is now broadly accepted that these soft non-phononic excitations are spatially quasilocalized [13,14,17], i.e.they feature a disordered core characterized by a localization length of a few atomic sizes in linear dimension and are accompanied by either a power-law decay (unlike Andersonlocalized modes that appear at the high-frequency end of the phononic spectrum and decay exponentially in space [37]), or an extended, wave-like background that spans the entire system [18,38,39]. Furthermore, these soft non-phononic excitations are known to emerge from the presence of frustration-induced internal stresses in the glass [16].…”

“…Efforts to observe these glassy excitations of a given frequency ω have been hampered for a long time because their unique quasilocalized nature is destroyed due to hybridization and mixing with extended phonons that share similar frequencies [14,25]. Consequently, quasilocalized glassy excitations can be observed in gaps/holes in the phononic spectrum that exist for ω<ω † (L), explaining the recent observations of [18,19]. Finally, as ω † (L)→0 in the L  ¥ limit, quasilocalized glassy excitations cannot be directly observed in the thermodynamic limit.…”