Origin, protocol dependence, and thermal-transport consequences of the low-frequency nonphononic vibrational spectrum of structural glasses

Simulations of small glass-forming systems showed that the character of the low-frequency vibrational spectrum is controlled by glass preparation history: instantaneous quenches from high parental temperature (the temperature of the equilibrium liquid before the quench) retain Debye-like extended modes below the lowest transverse resonant frequency, while quenches from parental temperatures near the dynamical transition (where the liquid's relaxation time diverges) suppress those extended modes and expose a quasi-localized population with $D(\omega)\sim\omega^{4}$. The originating work (Paoluzzi, Angelani, Parisi and Ruocco, Phys. Rev. Research 2, 043248 (2020)) left open the mechanism behind this protocol dependence, the spatial structure of the modes at larger sizes, and their role in macroscopic thermal and transport properties. These questions remain open in the current literature: there is still no accepted microscopic theory of the nonphononic spectrum, the universality of the $\omega^{4}$ law is under active controversy between competing frameworks, and the thermal and transport consequences of these modes are only partially addressed.

Background

A structural glass is an amorphous solid produced by rapidly cooling a liquid: it lacks the periodic lattice of a crystal, and its low-temperature properties differ from crystalline solids in characteristic ways. In the harmonic approximation one diagonalizes the Hessian (also called the dynamical matrix: the matrix of second derivatives of the potential energy evaluated at a mechanically stable configuration, an 'inherent structure' reached by energy minimization) to obtain normal modes and the vibrational density of states (VDoS) $D(\omega)$. In crystals, low-frequency modes are phonons — spatially extended sound-wave-like modes that arise, by Goldstone's theorem, as the generic soft excitations accompanying the breaking of translational symmetry — giving the Debye law $D(\omega)\sim\omega^{d-1}$ in $d$ dimensions ($\omega^{2}$ in 3D). Glasses additionally host nonphononic excitations: soft, quasi-localized modes (QLEs) that involve only a few tens of particles. The soft potential model (a phenomenological framework treating these modes as anharmonic soft directions in the energy landscape) and mean-field marginal-stability arguments (which derive the spectrum from the assumption that the glass sits at the threshold of mechanical stability) predict for these a universal $D(\omega)\sim\omega^{4}$ law, independent of dimension; the same excitations are implicated in two-level systems (quantum tunneling degrees of freedom dominating below about 1 K) and in the boson peak (the excess of the glass VDoS over the Debye prediction at terahertz frequencies).

Because extended phonon modes dominate the counted VDoS, isolating the nonphononic population is difficult. One strategy exploits small systems: below the lowest resonant transverse mode, at frequency $\omega_{0}=v_{T}\,2\pi/L$ (with $v_{T}$ the transverse sound velocity and $L$ the box side), there are no phonon resonances, so — the idea of this strategy goes — any modes appearing there can be counted as nonphononic. Paoluzzi, Angelani, Parisi and Ruocco tested this idea in a three-dimensional 50:50 binary soft-sphere mixture ($N=12^{3}$ particles at number density $\rho=1$ in reduced units), using three observables computed in the harmonic approximation: the VDoS $D(\omega)$, its cumulative $F(\omega)$ (which avoids binning since it only requires the sorted eigenvalues), and the dynamical structure factor $S(q,\omega)$ (the frequency- and wavelength-resolved density response); mode localization was quantified by the inverse participation ratio $R=\sum_{i}|e_{i}(\omega)|^{4}/\left[\sum_{i}|e_{i}(\omega)|^{2}\right]^{2}$, where $e_{i}(\omega)$ are the components of the mode eigenvector ($R$ of order $1/N$ for extended modes, $R\to1$ for a mode concentrated on a single particle). They found that for instantaneous quenches from high parental temperature (the temperature of the equilibrium liquid before the quench; here $T/T_{d}=2.38$, with $T_{d}\approx0.185$ in reduced units the dynamical, mode-coupling-like transition temperature at which the liquid's relaxation time diverges and dynamical heterogeneity grows), extended modes still populate the spectrum below $\omega_{0}$ and Debye's law survives, consistent with a plateau of $S(q,\omega)$ as $\omega\to0$. For quenches from parental temperatures close to $T_{d}$ — the protocol of their earlier work, which used the parental temperature as a control parameter in place of the pinning fraction of a preceding random-pinning study (in which a randomly chosen subset of liquid particles is frozen in place) — extended modes below $\omega_{0}$ are suppressed and only soft localized modes remain, with $D(\omega)\sim\omega^{4}$ and $F(\omega)\sim\omega^{5}$. Their conclusion: annealing protocols, not small system size, are the decisive ingredient, so small systems are an efficient tool for the VDoS (diagonalization cost grows as $N^{3}$), while larger systems are needed for the spatial properties of the modes.

These findings sit inside a broader controversy. A series of works reported the $\omega^{4}$ law to be universal across dimensions, glass-forming classes, and preparation protocols, including ultra-stable glasses equilibrated deep in the supercooled regime by swap Monte Carlo (a method that exchanges particle radii to accelerate equilibration), and an analysis extracting the nonphononic contribution from experimental data of diverse glasses. In opposition, heterogeneous-elasticity theory (HET) and its generalization (GHET) attribute non-Debye modes to spatial fluctuations of the elastic constants and predict a nonuniversal exponent set by the statistics of frozen-in local stresses, which depends on the pair potential and on preparation. A 2021 perspective review of quasi-localized excitations concluded that, unlike phonons, no fundamental (Goldstone-like) theoretical understanding of the generic existence of QLEs exists, and that current models cannot yet be related quantitatively to the properties of the parent liquid. It is against this background — an established empirical phenomenon with a contested theoretical interpretation — that the originating paper's open questions are posed.

Problem Statement

The problem is to replace the current descriptive picture with a predictive, mechanism-level account of the low-frequency vibrational spectrum of structural glasses and its observable consequences. Its components are scientifically linked facets of one objective — connecting glass preparation history to the low-frequency spectrum, and the spectrum to measurable properties:

(1) Mechanism and protocol dependence (the central contested core). Identify the microscopic mechanism by which the parental temperature of the instantaneous quench — from well above the dynamical transition temperature $T_{d}$ down to temperatures close to $T_{d}$, where dynamical heterogeneities proliferate, together with the associated degree of glass stability — controls the low-frequency spectrum: why instantaneous quenches from high parental temperature retain extended nonphononic modes below the lowest transverse resonant frequency $\omega_{0}$, while quenches from parental temperatures near the dynamical transition temperature suppress them; and whether the resulting quasi-localized population obeys a universal $D(\omega)\sim\omega^{4}$ law or an exponent governed by preparation- and potential-dependent quantities such as the statistics of frozen-in local stresses. A resolution must adjudicate or reconcile the competing frameworks (universality scenarios versus heterogeneous-elasticity-type theories predicting nonuniversal exponents) against controlled evidence.

(2) Spatial structure. Characterize the spatial structure, localization core sizes, correlation lengths, and propagation and attenuation characteristics of the extended and quasi-localized modes — including the finite-size scaling of the spatial mode profiles — at system sizes large enough to separate the two populations.

(3) Consequences for thermal and transport properties. Quantify the contribution of the low-frequency extended and quasi-localized modes to macroscopic observables of structural glasses, specifically the low-temperature specific heat, the thermal conductivity, and sound attenuation, and thereby connect the low-frequency spectrum to measurable macroscopic properties.

References

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