Glassy solids exhibit a wide variety of generic thermomechanical properties, ranging from universal anomalous specific heat at cryogenic temperatures to nonlinear plastic yielding and failure under external driving forces, which qualitatively differ from their crystalline counterparts. For a long time, it has been believed that many of these properties are intimately related to nonphononic, low-energy quasilocalized excitations (QLEs) in glasses. Indeed, recent computer simulations have conclusively revealed that the self-organization of glasses during vitrification upon cooling from a melt leads to the emergence of such QLEs. In this Perspective, we review developments over the past three decades toward understanding the emergence of QLEs in structural glasses and the degree of universality in their statistical and structural properties. We discuss the challenges and difficulties that hindered progress in achieving these goals and review the frameworks put forward to overcome them. We conclude with an outlook on future research directions and open questions.

Structural glasses are formed by cooling liquids quickly enough so that they avoid crystallization. As liquids are supercooled below their melting temperature, their viscosity increases dramatically by several orders of magnitude; at some point in this process, the viscosity becomes so large that the supercooled liquid falls out of equilibrium and is deemed to become a solid. This vitrification occurs at a temperature known as the glass transition temperature Tg, which is operationally defined as the temperature at which the liquid’s viscosity reaches 1013P.1,2 The nature of the glass transition remains a highly debated topic under intense investigation.1–4 

Glassy solids below their respective glass-transition temperature feature several intriguing properties associated with their disordered nature, absent in their crystalline counterparts that feature long-range order. The experimental work of Zeller and Pohl5 revealed the anomalous thermal conductivity and specific heat of glasses below 10K. It is now well established that the specific heat of glasses grows from zero temperature approximately as T and that the thermal conductivity of glasses grows from zero temperature approximately as T2 for a very large variety of glasses6,7 instead of the phonon-mediated T3 scaling predicted by Debye for both observables.8 These anomalies were addressed in the early 1970s by phenomenological tunneling models.9,10 These models postulate the existence of localized excitations—known as Two-Level Systems (TLSs)—which are small, localized groups of particles that can tunnel between two mechanically stable configurations.

Nonphononic low-energy excitations find another widely studied manifestation in glassy solids, known as the boson peak (BP). The BP is observed when plotting the vibrational density of states (VDoS) of a glass, D(ω), normalized by Debye’s phononic VDoS (in three dimensions) DD(ω)=ADω2, where ω denotes the angular frequency and AD is a known frequency-independent prefactor.8 The reduced VDoS D(ω)/DD(ω) of glasses generically exceeds unity at low frequencies, indicating the existence of excess low-frequency vibrational modes on top of the low-frequency phononic excitations. Furthermore, the reduced VDoS of glasses generically features a peak, typically at a frequency ωBP in the THz range. It is commonly believed that this BP is more pronounced in glassy states that feature a greater degree of structural disorder.11–15 Despite decades of investigations, there is no consensus regarding the origin of the boson peak.11,14–24

The structural disorder of glasses manifests itself in their mechanical response to external forces as well. Different from ordered crystalline solids, the micro-scale disorder of glasses leads to correlated, non-affine motions of the constituent particles in response to external forces,25–27 even in the elastic/reversible regime of small deformation. At larger deformation levels, plastic/irreversible processes become abundant, taking the form of localized immobile rearrangements of a few tens of particles, yet again in sharp contrast to mobile dislocations in crystalline solids. These rearrangements—coined “shear transformations” by Argon28,29 [occurring at regions that were later on coined “shear transformations zones” (STZs) by Falk and Langer30]—have been the subject of extensive theoretical, computational, and experimental investigations.28–42 Under a broad set of circumstances,43–46 predominantly at low temperatures/high strain-rates, STZs are activated in a collective/correlative manner, resulting in plastic strain localization in the form of shear bands.32,47–51

The generic nature of the aforementioned phenomena suggests that they share a common, universal origin, presumably associated with disorder-induced emergent excitations that differ from, and coexist with, phonons at low frequencies. Indeed, over the past few years, it has been established that structural glasses quenched from a melt generically host a population of soft, quasilocalized nonphononic excitations, similar to that depicted in Fig. 1(a). These soft excitations feature a disordered core of linear size ξg, typically of the order of ten particle diameters (with some exceptions to be discussed below). As shown in Fig. 1(b), particle dispalcements at distance r away from a quasilocalized excitation’s (QLE’s) core, decay as r−(đ−1) in đ spatial dimensions. As such, they echo the continuum-level elastic fields of Eshelby-like inclusions,54 e.g., the far-field displacement response to local dipolar perturbations.52,55 Under certain conditions—to be discussed at length in what follows—QLEs assume the form of harmonic vibrational modes whose frequencies ω follow a universal ∼ω4 distribution, apparently independent of spatial dimension,53 glass formation history,55,56 or microscopic details.57–59 As described in detail below, this universal form of the nonphononic VDoS had been predicted theoretically since the late 1980s60,61 and has been more firmly established using computer simulations in recent years.

FIG. 1.

(a) A quasilocalized excitation (QLE) observed in a computer glass in two-dimensions (see Ref. 52 for details). The circle delineates the spatial extent of the QLE’s core of linear size ξg. (b) The spatial decay of QLEs’ amplitude, as observed in computer glasses in various spatial dimensions (see the legend and Ref. 53 for details). Agreement with the continuum elastic response to a local dipolar perturbation ∼r−(đ−1), in đ spatial dimensions, is typically observed above rξg ≈ 10 particle diameters. This Perspective reviews past and recent advances in understanding the properties of QLEs and the degree of universality of their emergent statistics in structural glasses.

FIG. 1.

(a) A quasilocalized excitation (QLE) observed in a computer glass in two-dimensions (see Ref. 52 for details). The circle delineates the spatial extent of the QLE’s core of linear size ξg. (b) The spatial decay of QLEs’ amplitude, as observed in computer glasses in various spatial dimensions (see the legend and Ref. 53 for details). Agreement with the continuum elastic response to a local dipolar perturbation ∼r−(đ−1), in đ spatial dimensions, is typically observed above rξg ≈ 10 particle diameters. This Perspective reviews past and recent advances in understanding the properties of QLEs and the degree of universality of their emergent statistics in structural glasses.

Close modal

In this Perspective, we review past and recent developments in understanding the universal emergence of soft, quasilocalized excitations in structural glasses and their connection to glass properties. While we build on extensive effort and progress accumulated in the literature over the last few decades, we are also quite strongly biased toward our own recent work and our understanding of the history of this scientific field. Consequently, this Perspective is necessarily somewhat subjective and is not meant to provide a comprehensive and technical review of this topic. This Perspective is structured as follows: in Sec. II, we review early theoretical, numerical, and experimental developments in understanding the emergence of QLEs in structural glasses. In Sec. III, we explain how phononic modes dwelling in the low-frequency tail of the vibrational spectrum of structural glasses tend to hybridize with QLEs and as such suppress the latter’s realization as distinct harmonic vibrational modes. We further trace out the set of conditions in which QLEs can be cleanly and directly observed in the vibrational spectrum of finite-size computer glasses. In Sec. IV, we review recent progress in resolving the degree of universality of the statistical properties of QLEs and discuss what affects the characteristic length and frequency scales associated with them. In addition, we review recently proposed mean-field models of QLEs and report recent progress in developing methods that allow to define non-hybridized QLEs by incorporating anharmonicities of the potential energy landscape. Finally, in Sec. V, we discuss experimental evidence for QLEs, and in Sec. VI, we briefly discuss some open questions and future research directions.

In this section, we provide a concise chronological perspective on what we view as the key observations, as well as the accompanying evolution of concepts, regarding the generic existence of QLEs in structural glasses, along with the roles they play in determining glass properties. As stressed above, we note that in the framework of such a perspective, we cannot possibly offer an exhaustive account of the huge literature on this topic.

To the best of our knowledge, the first suggestion that structural glasses embed a population of low-energy localized excitations was put forward in 1962 by Rosenstock.62 Rosenstock argued that “non-elastic” (i.e., non-Debye or non-wave-like) soft localized excitations—emanating from weakly bounded groups of atoms in the glass structure—should be expected to emerge in disordered solids. This claim was based on earlier observations63,64 of discrepancies between the measured specific heat of glasses and that expected from elastic moduli measurements and Debye’s theory.8 Later inelastic cold neutron scattering experiments by Leadbetter and Litchinsky65 suggested the existence of resonant modes associated with particular defects in the structure of Vitreous Germania. Indeed, some authors used the term “resonant modes” interchangeably with QLEs (cf. Ref. 66) similar to resonant modes associated with self-interstitials in metals.67 

More well known are the works by Phillips9 and by Anderson, Halperin, and Varma,10 who independently formulated phenomenological models that address the anomalous temperature-dependence of the thermal conductivity and specific heat of structural glasses at very low temperatures, as revealed earlier by the experimental work of Zeller and Pohl.5 A key assumption in these models is the existence of localized excitations—the “Two-Level-Systems” (TLSs)—envisioned as small groups of atoms or molecules that can tunnel between two mechanically stable states, typically at temperatures of ≈1K and below. In Ref. 68, Phillips proposed that the anomalous thermodynamic and transport properties of glasses reflect the behavior of intrinsic low-frequency vibrational modes of the structure—a proposition that nicely corresponds to the subject of this Perspective. In Sec. IV J, we discuss in more detail possible connections between tunneling TLSs—more precisely double-well potentials—and QLEs.

In the late 1970s, the fundamental mechanism of plastic deformation in externally driven structural glasses was studied by Spaepen31 and Argon.28 Further reinforced by observations from mechanical experiments on bubble-rafts29 and computer simulations,69 these studies argued that plastic flow in amorphous solids proceeds via immobile, localized shear-like rearrangements of a few tens of particles. Subsequent computer simulations70–72 defined various structural defects in computer glasses and studied their statistical properties and the level of correlations between those defects and plastic-flow events.

Between the early 1980s and the early 1990s, a series of papers by Klinger, Karpov, Ignatiev, Galperin, Il’in, Buchenau, Gurevich, Schober, and others60,61,73–79 proposed that glasses generically host local regions in which the stiffness associated with atomic motion is anomalously small. These frameworks were collectively termed the “Soft Potential Model” (SPM). The SPM assumes that localized groups of particles can be envisioned as non-interacting anharmonic oscillators, each of which is described by a smooth, random potential energy function U(q) that admits a Taylor expansion in the form U(q)=n=1anqn/n! [for convenience, we set U(0) = 0]. Here, the coefficients {an} follow a regular (i.e., featuring no zeros or singularities) joint distribution function p({an}). Assuming then that U(q) attains a minimum at q = q0, one can obtain the following quartic expansion:

U(s)U0+12b2s2+13!b3s3+14!b4s4+Os5,
(1)

where sqq0 and U0U(q0). Transforming from p({an}) to p({bn}) and integrating over all of the coefficients but b2, one can show that p(b2) ∼ |b2| for b2 → 0.80 

In view of Eq. (1), TLSs naturally emerge as double-well potentials in the framework of the SPM. Invoking then quantum tunneling (relevant at very low temperatures) and using p(b2) ∼ |b2| for b2 → 0, the SPM offers various predictions (not discussed here) for tunneling TLSs.76 More directly relevant for our discussion here is that p(b2) ∼ |b2|, together with ωb20, implies a density of vibrational modes that grows from zero frequency as ω3. As pointed out in Ref. 80, further demanding that the minimum at q0 is a global minimum of U(q),80 which for the quartic expansion of Eq. (1) is guaranteed for b323b2b4, the SPM predicts that the contribution of the local soft potentials to the density of vibrational modes grows from zero frequency as ω4.60,61,76 In Sec. IV G, we further discuss the condition b323b2b4—which may be viewed as a stability bound—as well as numerical evidence supporting it and its implications.

While the simplicity of the SPM picture is appealing, it leaves some key questions unanswered. For example, the SPM does not explain what determines the degree of localization of soft excitations, nor does it fully describe the physical factors that control their number density. In addition, the SPM appears to lack a physical description of the minimal set of conditions necessary for the model’s key predictions to hold.

In the mid-2000s Gurevich, Parshin, and Schober (GPS)—who were part of the group of authors that previously formulated the SPM—put forward a phenomenological theory for the VDoS of glassy solids that goes beyond the SPM.22–24 The theory envisions a glass as a collection of interacting anharmonic oscillators—as opposed to the non-interacting SPM picture—that are meant to represent mesoscopic material elements. According to the theory, anharmonic oscillators a distance r from each other interact bilinearly, with an interaction strength that decays with distance r as rđ (in đ spatial dimensions, consistently with the spatial decay ∼r−(đ−1) of a QLE’s displacement field), mimicking elastic dipole–dipole interactions (here “dipole” stands for a “force dipole;” sometimes, these are also termed “displacement quadrupoles”). The physical picture according to which a glass is represented by interacting anharmonic oscillators as described by GPS bears some similarities to earlier propositions by Grannan, Randeria, and Sethna81,82 and by Kühn and Horstmann (KH).83 

The GPS approach involves two steps: the first step considers the effect of interactions on soft oscillators, which are a priori assumed to exist: the stiffnesses associated with soft oscillators are reduced due to interactions with stiff oscillators. This softening creates a “traffic” of oscillators’ stiffnesses toward zero stiffness. This interaction-induced softening leads to the destabilization of some of the soft oscillators: they assume a negative stiffness and—together with the local anharmonicity—become double-well potentials. Since the system must evolve toward a mechanically stable state, those destabilized oscillators restore stability by assuming a shifted equilibrium position that corresponds to the minimum of one of the two potential wells formed. This reconstruction of soft oscillators leads to a stiffness distribution that is flat near zero stiffness, resulting in a generic D(ω)ω reconstructed VDoS, independent of the initial distribution of oscillator-stiffnesses (as long as it has no hard gap; see further discussion in Sec. IV E).

In the second step of the GPS theory, the interactions between the reconstructed soft oscillators are considered. Since those oscillators have assumed shifted equilibrium positions, they exert random static forces on each other. These may be viewed as frustration-induced internal stresses, which generically exist in glasses. In the presence of local anharmonicities, these static forces lead to a further stabilization of the softest reconstructed oscillators, resulting in a universal VDoS D(ω)ω4 for frequencies ω smaller than a characteristic crossover frequency ωb,22–24 above which the first reconstructed D(ω)ω persists. These predictions imply that the reduced VDoS D(ω)/DD(ω) features a boson peak in the vicinity of ωb.

In addition to the phenomenological theory outlined above, GPS put forward a lattice model in three dimensions and studied it numerically in order to validate their theoretical predictions. The lattice model assumes that each lattice site is occupied by an anharmonic oscillator with a stiffness κi ≥ 0 drawn from a gapless parent distribution g0(κ) ∼ κβ and is described by the Hamiltonian

H GPS=12iκixi2+i<jJij(rij)xixj+A4!ixi4.
(2)

Here, xi denotes the (scalar) coordinate of the ith oscillator, and Jij(rij)=Jgij/rij3 is a space-dependent random variable representing the elastic coupling between the ith and jth oscillators, where gij ∈ [−1/2, 1/2] is a uniformly distributed random variable, J is an interaction-strength parameter, and rij is the distance between the oscillators. In Ref. 22, GPS report on numerical simulations of this lattice model, verifying that it features a VDoS D(ω)ω4 as ω → 0, independent of the exponent β that characterizes the initial gapless distribution g0(κ) of oscillator stiffnesses. In addition, at higher frequencies, the model’s VDoS was shown to follow a ∼ω scaling, resulting in a boson peak in the reduced VDoS as predicted by the phenomenological theory. Several additional predictions from the phenomenological theory were verified by numerical simulations of the lattice model in Refs. 22 and 24. The GPS phenomenological theory and lattice model were critically discussed in Refs. 84 and 85.

In the early 1990s, inspired by neutron scattering experiments on vitreous silica,88 Schober and co-workers turned to atomistic simulations,66,86,87,89–93 with the aim of testing and possibly validating the SPM predictions. To the best of our knowledge, the first robust numerical observations of low-frequency quasilocalized vibrations in simple computer glasses were put forward in 1991 by Laird and Schober66,86 (however, see also Ref. 94). In these works, monodisperse soft-sphere computer glasses with N = 500 and N = 1024 particles were studied, and the (quasi-) localization of the lowest-frequency vibrational modes was established by measuring those modes’ participation ratio

e(Ψ)iΨiΨi2NiΨiΨi2,
(3)

where Ψi denotes the đ-dimensional vector of Cartesian components pertaining to the ith particle of a vibrational mode Ψ. The participation ratio e(Ψ) of a given mode Ψ (i.e., a displacement vector field defined on each particle in the system) is a quantifier of the degree of localization of that mode. If a mode is localized on a compact core of Nc particles, one expects eNc/N, whereas if it is extended, then e ∼ 1. In Fig. 2(a), we show an example of a scatter plot of the participation ratio of vibrational modes of a simple computer glass vs their frequency, adapted from the original work of Schober and Laird.86 Those authors argued, based on their numerical results, that the low-frequency quasilocalized modes are centered typically on ∼20 particles. Later, in Ref. 90, it was shown that localization of low-frequency vibrational modes also occurs in a computer model of Selenium, which includes directional bonds, reinforcing that QLEs generically emerge in glassy solids.

FIG. 2.

(a) Early numerical evidence for the existence of low-energy QLEs in computer glasses by Schober and Laird;86 shown is a scatter-plot of the participation ratio of vibrational modes vs their frequency, calculated in a single soft-sphere computer glass of N = 500 particles in three dimensions. Adapted with permission from H. R. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991). Copyright 1991 American Physical Society. (b) Early numerical observations by Schober and Oligschleger87 that are consistent with the universal ∼ω4 VDoS of QLEs. Plotted is the reduced low-frequency VDoS D(ω)/ω2 of the same computer-glass model87 but with N = 5488 particles, which appears to increase superlinearly from zero frequency. Adapted with permission from H. R. Schober and C. Oligschleger, Phys. Rev. B 53, 11469 (1996). Copyright 1996 American Physical Society.

FIG. 2.

(a) Early numerical evidence for the existence of low-energy QLEs in computer glasses by Schober and Laird;86 shown is a scatter-plot of the participation ratio of vibrational modes vs their frequency, calculated in a single soft-sphere computer glass of N = 500 particles in three dimensions. Adapted with permission from H. R. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991). Copyright 1991 American Physical Society. (b) Early numerical observations by Schober and Oligschleger87 that are consistent with the universal ∼ω4 VDoS of QLEs. Plotted is the reduced low-frequency VDoS D(ω)/ω2 of the same computer-glass model87 but with N = 5488 particles, which appears to increase superlinearly from zero frequency. Adapted with permission from H. R. Schober and C. Oligschleger, Phys. Rev. B 53, 11469 (1996). Copyright 1996 American Physical Society.

Close modal

The statistical samples accessible in Refs. 66 and 86 were insufficient in order to robustly validate the predicted ω4 scaling of the VDoS. Later work87 employed the same computer-glass model but larger glass samples (up to N = 5488). These larger computer-glass samples allowed the authors to establish that the reduced nonphononic VDoS grows at least as ω3 and possibly stronger [see numerical data of Ref. 87 in Fig. 2(b)], compatible with the predicted ω4 scaling of the SPM for quasilocalized vibrations’ VDoS.

In that same work,87 the generic occurrence of hybridizations between low-frequency extended, plane-wave (phononic) modes and low-frequency quasilocalized vibrations is discussed. These hybridizations were further elaborated upon in Ref. 92. Since phonons are always present in the low-frequency spectrum of solids due to the breakdown of global continuous symmetries (Goldstone’s theorem), their hybridization with QLEs in the harmonic spectrum of glasses has for a long time hindered progress in revealing the statistical and structural properties of QLEs and in understanding the low-frequency spectrum of structural glasses using computational tools. In Sec. III, we describe in more detail the hybridization of phononic modes and QLEs in the low-frequency spectrum of structural glasses.

Within the harmonic approximation, one generically expects different classes of excitations to strongly hybridize and mix if those different excitations interact and share similar frequencies. As described early on by several authors87,92,95 and more recently by others,96–101 low-energy QLEs and plane-wave-like phononic excitations are no exception to this rule. In Fig. 3, we show an example of a low-frequency vibrational mode in a computer glass in two dimensions (2D), which is comprised of a QLE hybridized with a plane-wave (phononic) mode.

FIG. 3.

An example of a QLE (marked by the green circle) that is strongly hybridized with the lowest-frequency phononic vibrational mode in a 2D computer glass (see Ref. 52 for model details).

FIG. 3.

An example of a QLE (marked by the green circle) that is strongly hybridized with the lowest-frequency phononic vibrational mode in a 2D computer glass (see Ref. 52 for model details).

Close modal

The hybridization of QLEs and phonons in the harmonic spectrum of glasses affects these modes’ localization properties. In particular, the participation ratio of hybridized phonon-QLE excitations can assume any value between Nc/N and 1, as demonstrated in Fig. 4(a). The data presented therein were obtained in a polydisperse soft-sphere computer glass of N = 64 000 particles in 3D (see Ref. 102 for details). It is apparent from these data that QLEs can be realized as harmonic vibrations below the first phonon band and between phonon bands, which appear as peaks with eO(1). In addition, it appears that phonon bands “burn” holes in the low-e range corresponding to the participation ratio of this system’s quasilocalized modes. In other words, although QLEs certainly exist at frequencies that match phonon bands’ frequencies (as explicitly demonstrated in Ref. 103), their clean realization as quasilocalized harmonic vibrations is largely destroyed by hybridizations with phonons if their frequencies lie in the close vicinity of phonon frequencies. It is crucial to stress, in this context and more generally, that QLEs manifest the existence of soft glassy structures embedded inside a glass, and as such, their ontological status is independent of whether they can be realized as quasilocalized normal (harmonic) modes or not. Hybridizations with phonons, however, can have serious implications for one’s ability to detect QLEs.

FIG. 4.

(a) Scatter-plot of the participation ratio e [cf. Eq. (3)] of vibrational modes of a 3D computer glass vs their frequency ω. Frequencies are expressed in terms of ω0cs/a0, where cs is the shear wave-speed, a0 = (V/N)1/3 is an interparticle distance (with V denoting the system’s volume), and here N = 64 000. The ellipse engulfs those quasilocalized modes that escaped hybridizations (see text for further discussion). (b) The VDoS of the same glasses analyzed in panel (a). Discrete phonon bands, which are apparent, acquire a finite width Δω due to the glass’s mechanical disorder.102,103 The vertical lines in both panels mark the crossover frequency scale ω [cf. Eq. (5)], above which phonon bandwidths become comparable to the gaps between them, such that QLEs can no longer be cleanly realized as harmonic vibrations. Note that the lower envelope of the presented VDoS, in fact, follows an ω4 scaling (not marked on the figure, but see Fig. 1 in Ref. 102, where it is marked).

FIG. 4.

(a) Scatter-plot of the participation ratio e [cf. Eq. (3)] of vibrational modes of a 3D computer glass vs their frequency ω. Frequencies are expressed in terms of ω0cs/a0, where cs is the shear wave-speed, a0 = (V/N)1/3 is an interparticle distance (with V denoting the system’s volume), and here N = 64 000. The ellipse engulfs those quasilocalized modes that escaped hybridizations (see text for further discussion). (b) The VDoS of the same glasses analyzed in panel (a). Discrete phonon bands, which are apparent, acquire a finite width Δω due to the glass’s mechanical disorder.102,103 The vertical lines in both panels mark the crossover frequency scale ω [cf. Eq. (5)], above which phonon bandwidths become comparable to the gaps between them, such that QLEs can no longer be cleanly realized as harmonic vibrations. Note that the lower envelope of the presented VDoS, in fact, follows an ω4 scaling (not marked on the figure, but see Fig. 1 in Ref. 102, where it is marked).

Close modal

Also apparent from the data of Fig. 4 is that above a system-size- and mechanical-disorder-dependent frequency scale denoted by ω(L, χ), QLEs can no longer be realized as harmonic vibrations due to phonon-hybridizations. Here, LN1/đ denotes the linear size of the glass, and χ is a measure of mechanical disorder that is intimately related to the relative spatial fluctuations of the shear modulus field.102 The frequency scale ω(L, χ) is understood as follows: the mechanical disorder intrinsic to structural glasses lifts the degeneracy of low-frequency phononic excitations that share the same wavelength. As a result, sets of low-frequency, iso-wavelength phonons form discrete bands with finite widths Δω, as illustrated in Fig. 4(b). In Refs. 102 and 103, it was shown that

ΔωχωnzN,
(4)

where nz is the degeneracy level of the zth phonon band in an ideally isotropic homogeneous elastic medium.102,103 This relation implies that Δω increases with increasing frequency. In Refs. 102 and 103, it was shown that at the crossover frequency

ωL,χχL2đ+2,
(5)

Δω becomes comparable to the gaps between consecutive phonon-bands. Consequently, above ω, phononic excitations are no longer clustered into discrete bands but are instead distributed quasi-continuously over the frequency axis. Therefore, QLEs with frequencies ω > ω exclusively hybridize with phonons, as shown in Fig. 4.

We finally note that in 2D, ωL−1/2, whereas the typical frequency of the softest quasilocalized mode in a finite-size glassy sample, ωmin, is of order L−2/5 (with logarithmic corrections, see Refs. 53 and 57). This implies that in 2D, ω < ωmin, and therefore, 2D computer glasses of sizes of a few thousand particles and above typically do not feature many non-hybridized quasilocalized vibrations, as indeed observed numerically in several works.58,105

Since the numerical work of Schober in the 1990s, many theoretical, computational, and experimental investigations of the vibrational spectra of structural glasses were put forward; some examples include Refs. 11, 14, 17, 21, and 106112 and see additional references therein. It was not until 2011, however, that the first numerical evidence of the universal ω4 nonphononic VDoS of quasilocalized excitations was indirectly revealed by Karmakar et al.,113 who studied the statistics of the lowest-frequency vibrational modes per computer glass. A few years later, in 2015, simulations by Baity-Jesi et al.96 of the 3D Heisenberg spin glass revealed a quartic VDoS of QLEs. This was accomplished by applying a fluctuating external field that penalizes Goldstone modes that emerge due to the rotational invariance of the Heisenberg spin glass’s Hamiltonian (analogous to phonons in structural glasses), hence overcoming the aforementioned hybridization issues and exposing the universal statistics of QLEs in a spin glass.

A more direct route to observe the form of the nonphononic spectrum of structural glasses was taken in 2016.57 The main idea of this work was that in finite systems of linear size L, the (finite) lowest-frequency phonons and QLEs follow a different scaling with L. Consequently, it was shown that the system size L of computer glasses can be carefully tuned such that it is small enough so as to push phononic excitations to higher frequencies, cleanly exposing the VDoS of QLEs without hybridizations with phonons,92,93,114 yet appreciably larger than QLEs’ core size ξg. Therefore, it turned out that small systems can, in fact, be beneficial in this context. Following this idea, extensive ensembles of glassy samples were generated such that the nonphononic ∼ω4 VDoS of QLEs could be directly and robustly observed. In the very same work, the ∼r−2 spatial decay of QLEs (in 3D) away from their respective cores was shown—consistent with previous observations,36,115 and see also Fig. 1(b). Furthermore, the core size ξg ≈ 10a0 was estimated, and the scaling e ∼ 1/N of the participation ratio of QLEs was established. Finally, that work established that the softest QLEs per glassy sample follow Weibullian statistics, suggesting that QLEs are largely uncorrelated.

Subsequent work showed that the ω4 scaling of the nonphononic VDoS of structural glasses is robust to changes in spatial dimension53,116 and to extreme supercooling55,56 (made possible by applying the swap Monte Carlo algorithm117–119 to the glass-forming model by Ninarello, Berthier, and Coslovich104). More recently, the robustness of the ω4 VDoS to changes in the interaction potential was established.59,120 Some of these efforts to establish the universality of the nonphononic VDoS are presented in Fig. 5.

FIG. 5.

Numerical evidence supporting the universality of the ω4 nonphononic VDoS of structural glasses. (a) Low-frequency spectra of (from left to right) soft spheres interacting via the Hertz law, the Stillinger–Weber network glass-former, a triatomic OTP-like molecular glass former, a model polymeric glass, and a model CuZr bulk metallic glass. The vertical dashed lines mark the lowest phonon frequency, whereas the continuous lines correspond to the universal ω4 scaling. Details about the models and methods can be found in Ref. 59. (b) Low-frequency spectra of soft-sphere computer glasses in 2D, 3D, and 4D (the spectra are shifted for visual clarity) (see Ref. 53 for details). (c) Low-frequency spectra of a polydisperse soft-sphere glass (see Ref. 102 for details) that can be subjected to extremely deep supercooling using the swap Monte Carlo algorithm.104 Here, Tp is the equilibrium parent temperature from which glassy configurations were instantaneously quenched to zero temperature.

FIG. 5.

Numerical evidence supporting the universality of the ω4 nonphononic VDoS of structural glasses. (a) Low-frequency spectra of (from left to right) soft spheres interacting via the Hertz law, the Stillinger–Weber network glass-former, a triatomic OTP-like molecular glass former, a model polymeric glass, and a model CuZr bulk metallic glass. The vertical dashed lines mark the lowest phonon frequency, whereas the continuous lines correspond to the universal ω4 scaling. Details about the models and methods can be found in Ref. 59. (b) Low-frequency spectra of soft-sphere computer glasses in 2D, 3D, and 4D (the spectra are shifted for visual clarity) (see Ref. 53 for details). (c) Low-frequency spectra of a polydisperse soft-sphere glass (see Ref. 102 for details) that can be subjected to extremely deep supercooling using the swap Monte Carlo algorithm.104 Here, Tp is the equilibrium parent temperature from which glassy configurations were instantaneously quenched to zero temperature.

Close modal

As demonstrated in Subsection IV A, QLEs appear to generically exist in glasses formed by quenching a liquid and to follow a universal VDoS. What are the salient, non-universal properties of QLEs, apart from the universal ∼ω4 form of their distribution over frequency? As mentioned in the Introduction, QLEs are characterized by a core of linear size ξg, typically on the order of ten interparticle distances53,55,57 (see, e.g., Fig. 1). Another important attribute of QLEs is their number density, as encapsulated in the non-universal prefactor Ag of the nonphononic VDoS, where the latter is written for small frequencies as

D(ω)=Agω4.
(6)

The non-universal prefactor Ag has dimensions of [frequency]−5; its physical significance was discussed at length in Refs. 52, 55, 121, and 122. It encompasses information both about the number density N of QLEs and about their characteristic frequency ωg,52,55 in complete analogy to the corresponding prefactor AD (see the Introduction) in Debye’s VDoS of phonons. It is important to note that unlike AD, at present there exist no first principles predictions for Ag; rather, it is extracted by fitting the VDoS of QLEs (once properly isolated) to the ω4 law. In Refs. 52 and 55, it was suggested that the prefactor can be meaningfully decomposed into a product of the form Ag=Nωg5, since changes in Ag may stem both from the stiffening or softening of QLEs (i.e., variations in their characteristic frequency ωg as discussed in Refs. 55 and 123) and from QLEs’ depletion or proliferation. If both ωg and Ag can be measured independently, then the QLE number density N=Agωg5 can be estimated, and its dependence on the formation history of a glass and on other factors can be studied.

How do the length ξg, characteristic frequency ωg, and nonphononic VDoS prefactor Ag depend on the formation history of a glass? Here, we discuss this question in the context of the “parent temperature” simulational protocol. This protocol amounts to equilibrating a computer liquid at some parent temperature Tp and following it by an instantaneous quench to zero temperature to form a glassy solid. We note that at high Tp’s, this protocol generates rather unrealistically unstable glasses compared to laboratory liquids, which cannot be driven through their respective glass transition temperatures at comparable cooling rates. Nevertheless, this protocol is useful as an investigative tool, as it allows us to probe the full variety of glassy structures accessible to a single glass-forming liquid model.

In Fig. 6(a), we illustrate the effect of Tp on QLEs’ properties: upon deeper supercooling of glasses’ ancestral liquid configurations, the core size of QLEs decreases, in parallel to their depletion, as discussed in Refs. 55, 56, and 124 (note that TLSs have also been shown to undergo depletion upon deep supercooling125). Direct numerical evidence for the decrease in QLEs’ core size is shown in Fig. 6(b); these data were adapted from Ref. 55, where in addition to directly probing the core size of QLEs [brown diamonds in Fig. 6(b)], it was further compared to (i) the length obtained via 2πcs/ωg (green squares), where cs is the shear wave-speed and ωg was estimated as the typical frequency associated with the response of the glass to local dipolar forces,52,55,126 and to (ii) the length obtained by analyzing the spatial response to the same local force dipoles, where the distance from the imposed dipoles in which the onset of the expected ∼r−(đ−1) continuum-elastic scaling is observed was estimated126–128 (yellow circles). The agreement between all of these lengths supports the relation ξgcs/ωg and indicates that ξg can be accurately estimated via responses to local dipolar forces. The latter thus emerge as important physical quantities for probing QLEs’ properties.52,55,126

FIG. 6.

(a) Illustration of the Tp dependence of QLEs’ properties (see text for discussion). (b) The lengthscale ξg that characterizes the core size of QLEs decreases upon deeper supercooling of glasses’ ancestral liquid configuration (see text and Ref. 55 for further details) [and compare to panel (a)]. (c) The number density N of QLEs follows a Boltzmann-like dependence on the parent temperature Tp (see discussion in the text and in Ref. 55).

FIG. 6.

(a) Illustration of the Tp dependence of QLEs’ properties (see text for discussion). (b) The lengthscale ξg that characterizes the core size of QLEs decreases upon deeper supercooling of glasses’ ancestral liquid configuration (see text and Ref. 55 for further details) [and compare to panel (a)]. (c) The number density N of QLEs follows a Boltzmann-like dependence on the parent temperature Tp (see discussion in the text and in Ref. 55).

Close modal

Finally, with an estimation of the characteristic frequency ωg of QLEs and the prefactors Ag of their VDoS at hand, the number density of QLEs can be estimated as NAgωg5; N is plotted against the inverse-parent-temperature in Fig. 6(c), revealing a Boltzmann-like dependence,55 

N(Tp)expE QLETp,
(7)

below a crossover parent temperature Tx (discussed e.g., in Ref. 129), with EQLE representing the formation energy of a QLE. Equation (7) suggests that Tp plays the role of a nonequilibrium thermodynamic temperature that carries memory of the equilibrium state at which a glass falls out of equilibrium, in particular, of the configurational (as opposed to vibrational) degrees of freedom of the liquid, deep into the nonequilibrium glassy state.55 

The existence of a nonequilibrium temperature in glasses, sometimes termed the fictive/effective/configurational temperature,130–135 and its relation to the number of QLEs (additional discussions regarding the number density of QLEs can be found in Refs. 136 and 137) strongly echo the nonequilibrium thermodynamic Shear-Transformation-Zones (STZs) theory of glassy deformation.50,138,139 This theory is based on a two-temperature nonequilibrium thermodynamic framework, where the number of STZs—the “flow defects” in a glass—follows a Boltzmann-like relation with the effective temperature as in Eq. (7), which satisfies its own field equation.50,138,139 Consequently, if QLEs can be identified with STZs (or at least if they are strongly correlated with them, as shown in Subsections IV H and IV I) and if Tp can be identified with the effective temperature, then Eq. (7) and Fig. 6(c) provide interesting support to one of the main predictions of the STZ theory of glassy deformation.

The STZ theory provides various predictions regarding the elasto-plastic deformation of glasses in a wide variety of physical situations.50 In particular, the strong depletion of STZs with decreasing effective temperature, consistent with the Boltzmann-like relation, has been predicted to give rise to a ductile-to-brittle transition in the fracture toughness of glasses140,141 (a possibly related phenomenology emerged in models of athermal quasistatic deformation of glasses, where stress–strain curves—but not the fracture toughness—have been considered43,44). This prediction has been recently supported by experiments on the fracture toughness of bulk metallic glasses, where the effective temperature has been carefully controlled and varied.142 In addition to establishing a connection between individual QLEs and STZs, as discussed in Subsections IV H and IV I, large-scale computer simulations can shed light on the collective effect of AgN (once the former is properly nondimensionalized) on the nonlinear and dissipative mechanics of glasses. Indeed, very recently, it has been shown that the variation of the fracture toughness of various computer glasses with thermal history and the underlying interparticle interaction potential is largely controlled by the dimensionless Ag, proving strong support to the central role played by QLEs in the physics of glasses.122 

In Subsection IV B, we have seen that while the quartic law of the nonphononic VDoS appears to universally hold, the number density and structural and energetic properties of QLEs’ can be affected by the formation history of structural glasses. Recent work122,126 has demonstrated that the nature of interparticle potentials may affect QLE-properties as well. Here, we concisely review the results of Ref. 126 in which the effect of variations of a simple, pairwise interparticle potential on the properties of QLEs was studied.

The main results of Ref. 126 relevant to the current discussion are summarized in Fig. 7. Panel (a) shows the tunable pairwise interaction potential; the interaction range, denoted by rc, is the key control parameter that affects QLE-properties, along with other mechanical and elastic observables. Panels (b) and (c) show the dependence of the (linear) core size ξg of QLEs and the (dimensionless) prefactor Agω05 of the nonphononic VDoS, respectively, on the cutoff rc (recall that ω0cs/a0, where cs is the shear wave-speed and a0 is a typical interparticle distance). The length ξg varies by over a factor of two, which exceeds the Tp-induced variation of ξg as shown in Fig. 6(b). The dimensionless prefactor Agω05 varies by over two decades; in Ref. 126, it is further demonstrated that the variability of Agω05 stems mostly from the stiffening of QLEs (i.e., the increase in ωg or decrease in ξg) with reducing rc rather than from their depletion (with the exception of the rc = 1.1 systems, see further discussions in Refs. 126 and 144).

FIG. 7.

Effect of interparticle potential on QLE-properties (data from Ref. 126). (a) The sticky-spheres pairwise potential φSS(r) put forward in Ref. 143. The interaction-range cutoff rc serves as the key control parameter affecting QLEs’ properties (note that r0 denotes the location of the minimum of the potential). (b) The lengthscale ξg that characterizes the core size of QLEs decreases with decreasing rc by roughly a factor of two. (c) The dimensionless nonphononic VDoS prefactor Agω05 varies by over two decades under changes of rc. (d) The ratio of dilatant-to-shear strain associated with QLEs (see text for exact definitions) grows significantly with decreasing rc.

FIG. 7.

Effect of interparticle potential on QLE-properties (data from Ref. 126). (a) The sticky-spheres pairwise potential φSS(r) put forward in Ref. 143. The interaction-range cutoff rc serves as the key control parameter affecting QLEs’ properties (note that r0 denotes the location of the minimum of the potential). (b) The lengthscale ξg that characterizes the core size of QLEs decreases with decreasing rc by roughly a factor of two. (c) The dimensionless nonphononic VDoS prefactor Agω05 varies by over two decades under changes of rc. (d) The ratio of dilatant-to-shear strain associated with QLEs (see text for exact definitions) grows significantly with decreasing rc.

Close modal

In the same work,126 a quantifier of the geometry of QLEs was put forward, with the aim of assessing the ratio of shear vs dilatant strain that QLEs feature. The quantifier is constructed as follows: for a QLE given by a normalized displacement field π, we define the tensor

Fπ2Uϵxπ,
(8)

where U(x) is the potential energy, which depends on coordinates x and ϵ is the strain tensor.145 Next, Fπ is decomposed into its deviatoric and dilatational contributions as Fπ=Fπ iso+Fπ dev, where Fπ isoITr(Fπ)/đ (I is the identity tensor) and Fπ devFπFπ iso. Fπ dev is then diagonalized and its eigenvalue with the largest absolute magnitude λmax is recorded. The ratio of dilatational to shear strain of the QLE π is finally defined as

dilationshear=Tr(Fπ)/đλmax.
(9)

The behavior of the dilation-to-shear ratio calculated over a few thousand QLEs observed in computer glasses of different rc’s is presented in Fig. 7(d); the color bars cover the second and third quartiles of the dilation-to-shear ratio, and the middle horizontal line represents the mean ratio. Interestingly, reducing rc leads to the development of a much larger dilatational component of the strain fields associated with QLEs.

We note that the same computer glasses whose QLEs feature large dilation-to-shear strain ratios were also shown to have relatively small Poisson’s ratios126 and to fail in a brittle fashion under uniaxial loading.122 A continuum analog of the geometric quantifier of QLEs discussed here was introduced and compared to the microscopic quantifier described above in Ref. 54. Similar approaches toward quantifying the geometry of plastic instabilities in computer glasses were discussed in Refs. 136, 146, and 147. We finally note that while the introduction of strong attractive interactions may affect QLEs’ properties—as shown in Fig. 7—they do not necessarily do so, as discussed at length in Ref. 126.

The unjamming transition is an elasto-mechanical instability that occurs in gently compressed disordered packings of soft spheres upon reducing their pressure p toward zero.106,148–150 Ikeda and co-workers58,121 have recently studied using computer simulations how QLEs’ statistical and structural properties in harmonic soft-sphere packings are affected by the proximity of those packings to the unjamming transition.

The key microscopic observable in the context of the unjamming transition is the coordination difference δZZZc to the so-called Maxwell threshold Zc = 2đ, where Z denotes the mean number of contacts per particle in a packing. For many canonical soft-sphere models near the unjamming point, p/KδZ2 (where K is the bulk modulus).106,148–150 In Ref. 121, it was shown that ξgδZ−1/2 using numerical simulations, consistent with previous observations of diverging lengthscales near the unjamming point.107,127,151–153 In Refs. 58 and 121, it was claimed based on scaling arguments and demonstrated numerically that the dimensionless prefactor of the quartic nonphononic VDoS near unjamming follows Agω05δZ3/2.

Interestingly, it was shown in Ref. 121 that the product Ne [with e denoting the participation ratio, cf. Eq. (3)]—which represents the effective volume of QLEs’ core—scales as 1/δZξg2 instead of the naïve expectation ξgđ in đ dimensions. This result is consistent with the 1/δZ scaling of the sum-of-squares of the displacement response to local dipole forces in disordered networks of relaxed Hookean springs, as spelled out in Ref. 127. According to a recent replica calculation of the overlap correlation function,154 this scaling between the effective volume of a QLE and its core length ξg stems from the pre-asymptotic spatial decay ∼r−(đ−2)/2 of QLE displacements at distances r < ξg.

Several efforts to understand the disorder-induced properties of the low-frequency spectra of structural glasses based on mean-field models have been put forward in previous literature; some notable examples include Fluctuating Elasticity Theory (FET),14,84,156 effective medium theory,21,111 the perceptron model,112 and the mean-field theories for hard-sphere glasses158–159 and jammed packings.160 Some of these models, e.g., Refs. 21, 112, and 161, predict that the nonphononic VDoS follows an ω2 scaling with frequency, independent of spatial dimension. FET predicts a dimension-dependent, Rayleigh-like scaling ∼ωđ+1 for excess modes that are spatially extended (i.e., FET does not predict QLEs).84 

Recently, a mean-field model for QLEs in structural glasses was put forward.155,162 The model is a generalization of the 3D model for anharmonic interacting oscillators by Gurevich, Parshin, and Schober (GPS)22–24 discussed at length in Sec. II B, which is also somewhat reminiscent of the earlier model by Kühn and Horstmann (KH);83 as such, it was termed the KHGPS model in Refs. 155 and 162.

The KHGPS mean-field model, defined through the following Hamiltonian:

H KHGPS=12iκixi2+i<jJijxixj+A4!ixi4hixi,
(10)

can be formally obtained from Eq. (2) by taking the space-dependent interaction coefficients Jij(rij) to be space independent (corresponding to taking the infinite-dimensional limit, đ, of the dipole–dipole elastic interaction rđ) and Gaussian, i.i.d. random variables of variance J2/N; here, J represents the interaction strength, and N is the number of interacting oscillators described by the scalar coordinates xi. In addition, the anharmonic oscillators are assumed to linearly interact with a constant field h, which breaks the xi → −xi symmetry of the Hamiltonian, a feature missing in Eq. (2). Finally, the oscillator stiffnesses κi are assumed to be drawn from a uniform parent distribution over the interval [κmin, κ0], where κmin < κ0 may be finite, which is yet another deviation from the GPS model that considered only gapless parent distributions for κi.

The KHGPS model defined in Eq. (10) is similar to the KH model considered in Ref. 83, with the notable difference that in the latter, the stiffnesses κi were not taken to be random variables. Formally, in terms of the formulation above, the model in Ref. 83 corresponds to κmin = κ0 = 1. While the VDoS has not been studied in Ref. 83, it was shown in Ref. 155 to give rise to an ω2 VDoS as ω → 0 and to delocalized modes [i.e., O(N) oscillators feature sizable displacements xi at minima of the Hamiltonian that populate the ω2 regime]. The KHGPS model defined in Eq. (10) also bears some similarity to the soft-spin version of the Sherrington–Kirkpatrick model,163 although the latter has not been previously shown to be related to soft vibrational excitations. The KHGPS model has been recently analyzed in Refs. 155 and 162 and the main results are briefly reviewed next.

In Ref. 155, it was rigorously shown that if the parent stiffnesses distribution is gapped, i.e., if κmin > 0, then for a fixed h and sufficiently small interaction strength J, the VDoS corresponding to HKHGPS in Eq. (10) is also gapped. This phase is replica-symmetric (RS) and hence is denoted as the RS phase in Fig. 8(a). With increasing J, there exists a critical line Jc(h) in the hJ plane on which a gapless VDoS emerges. Interestingly, for small h, the gapless VDoS is populated by delocalized modes that follow a quadratic behavior ∼ω2, similar to previous mean-field models.21,112,161 On the other hand, there exists a special point on the critical Jc(h) line above which (i.e., for large enough h) the gapless VDoS is populated by localized modes that follow a quartic behavior ∼ω4. This result shows that in contrast to previous belief, mean-field models can, in fact, feature localized modes with an ω4 VDoS, similar to direct observations in finite-dimensional computer glasses.53 

FIG. 8.

(a) The hJ phase diagram of the KHGPS model, defined in Eq. (10), with κmin = 0.1 and κ0 = 1. The critical transition line Jc(h) separates the RS phase from the glassy (RSB) phase (see text and Ref. 155 for additional details). On the dotted line, one has D(ω)ω2, while on the solid line, one has D(ω)ω4. (b) D(ω) upon approaching the ω2-transition line [dotted line in panel (a)]. (c) D(ω) upon approaching the ω4-transition line [solid line in panel (a)].

FIG. 8.

(a) The hJ phase diagram of the KHGPS model, defined in Eq. (10), with κmin = 0.1 and κ0 = 1. The critical transition line Jc(h) separates the RS phase from the glassy (RSB) phase (see text and Ref. 155 for additional details). On the dotted line, one has D(ω)ω2, while on the solid line, one has D(ω)ω4. (b) D(ω) upon approaching the ω2-transition line [dotted line in panel (a)]. (c) D(ω) upon approaching the ω4-transition line [solid line in panel (a)].

Close modal

Finally, for J > Jc(h), the model features a replica-symmetry breaking (RSB); hence, this regime is denoted as the glassy (RSB) phase in Fig. 8(a). This phase has not been analyzed in Ref. 155. The results described above are visually presented in Fig. 8, where the hJ phase diagram is shown in panel (a), the VDoS upon approaching the critical Jc(h) line (from the RS phase) in the ω2 regime is shown in panel (b), and the VDoS upon approaching the critical Jc(h) line (from the RS phase) in the ω4 regime is shown in panel (c).

When κmin is reduced toward zero, the Jc(h) line is pushed toward smaller values of J, and the ω4 regime increases; for κmin = 0, the KHGPS model is in the RSB phase for all J values.155 While—as stated above—this glassy (RSB) regime has not been analyzed in Ref. 155, it has been rather thoroughly studied numerically and through a scaling theory in Ref. 162. First, it was shown that the model in this regime gives rise to a gapless VDoS D(ω)=Agω4 for a broad range of model parameters, as demonstrated in Fig. 9(a). This result shows that the ω4 scaling of the VDoS in the KHGPS model persists deep inside the glassy (RSB) regime. In addition, a complete understanding of the non-universal prefactor Ag(h,J,κ0) has been developed; recall that here κmin = 0 and that the anharmonicity amplitude A is fixed; hence, the model is fully characterized by the parameters h, J, and κ0.

FIG. 9.

(a) D(ω) of the KHGPS model in the glassy (RSB) phase, with κmin = 0, calculated numerically in systems of N = 16 000 oscillators, for h = 0.01 and various values of the parameters J and κ0 as indicated by the legend. The generic emergence of D(ω)ω4 is demonstrated (see the 4:1 triangle and note the double-logarithmic axes). (b) Numerical validation of the theoretical prediction for the predominantly exponential variation of Ag(h,J,κ0) with −κ0h2/3J−2 (see text and Ref. 162 for details). (c) Numerical validation of the theoretical prediction for the predominantly power-law decay of Ag(h,J,κ0) with J when the latter is sufficiently larger than the crossover interaction strength Jx(h,κ0) (see text and Ref. 162 for details). The weak interactions regime, discussed in panel (b), is also presented such that the non-monotonic variation of Ag(h,J,κ0) with J is evident.

FIG. 9.

(a) D(ω) of the KHGPS model in the glassy (RSB) phase, with κmin = 0, calculated numerically in systems of N = 16 000 oscillators, for h = 0.01 and various values of the parameters J and κ0 as indicated by the legend. The generic emergence of D(ω)ω4 is demonstrated (see the 4:1 triangle and note the double-logarithmic axes). (b) Numerical validation of the theoretical prediction for the predominantly exponential variation of Ag(h,J,κ0) with −κ0h2/3J−2 (see text and Ref. 162 for details). (c) Numerical validation of the theoretical prediction for the predominantly power-law decay of Ag(h,J,κ0) with J when the latter is sufficiently larger than the crossover interaction strength Jx(h,κ0) (see text and Ref. 162 for details). The weak interactions regime, discussed in panel (b), is also presented such that the non-monotonic variation of Ag(h,J,κ0) with J is evident.

Close modal

It was theoretically predicted that in the weak interactions regime, i.e., for J smaller than a crossover level Jx(h,κ0), the prefactor Ag(h,J,κ0) satisfies logκ01/2h2/3JAgκ0h2/3J2. The validity of this predominantly exponential variation of Ag(h,J,κ0) with −κ0h2/3J−2 is numerically demonstrated in Fig. 9(b). Interestingly, this result is reminiscent of the predominantly exponential variation of the number density N of QLEs with −1/Tp in computer glasses, shown in Fig. 6(c) [see also Eq. (7)]. This similarity is suggestive, calling for a better understanding of the possible relations between the KHGPS model parameters h, J, and κ0 and the parent temperature Tp that characterizes the liquid state at which the glass falls out of equilibrium during a quench.

Furthermore, it has been predicted in Ref. 162 that for J larger than a crossover interaction strength Jx(h,κ0), Ag predominantly decays with J as a power-law, as is numerically demonstrated in Fig. 9(c). In addition, the analysis in Ref. 162 has revealed the existence of a characteristic frequency ωx scale in the KHGPS model, which is reminiscent of the crossover frequency ωb mentioned in Sec. II B in the context of the reconstruction picture, and studied its properties. The importance of this frequency scale has been further highlighted in a very recent work,164 where additional intriguing features of the KHGPS model are revealed using long time gradient descent dynamics in the glassy (RSB) phase (based on a dynamical mean field theory) and replica method calculations. All in all, the formulation of the KHGPS model and its revealed properties as of now show, contrary to previous belief, that mean-field models can share similar properties with finite-dimensional computer glasses in relation to QLEs, most notably modes localization and the ∼ω4 VDoS.

At the same time, the KHGPS model raises various questions and challenges. One class of questions concerns the relations between the model and finite-dimensional, realistic glasses. In particular, it remains a challenge to understand the relations between the model’s inputs, such as the parent stiffnesses distribution and the parameters J and h, and the self-organizational processes taking place during glass formation when a liquid is quenched. Establishing such connections, qualitative or even quantitative, appears essential for clarifying what mean-field models such as the KHGPS one may teach us about the physics of glasses at the fundamental level. Another class of challenges, more on the mathematical physics side, concerns the rigorous analysis of the glassy (RSB) phase, which has been so far mainly analyzed numerically and through a scaling theory (but see Ref. 164). Finally, exploring whether such mean-field models can teach us something deep about glassy dynamics—not just statistical properties of glassy structures—appears to be an interesting direction for future investigation.

In Sec. III, we described how finite-size scaling of QLEs and phononic excitations may be utilized to overcome their hybridization.57 Back in 1996, Schober and Oligschleger already proposed a demixing/dehybridization procedure to disentangle extended phonons and QLEs.87 In recent years, several other computational frameworks for overcoming phonon-QLE hybridizations have been put forward.96,98,100,101 In this subsection, we describe one of these frameworks, which enables to robustly single out QLEs, regardless of whether or not phononic excitations with similar frequencies exist in the glass.

Consider a zero-temperature glass, namely, a configuration x0 that constitutes a (local) minimum of the glass’s potential energy U(x). Consider next the energy variation δUU(x) − U(x0) that results from displacing the particles a distance s in the configuration-space direction prescribed by a given putative displacement field z, namely,

δU(s)12b2s2+13!b3s3,
(11)

where

b2(z)2Uxxx0:zz,
(12)
b3(z)3Uxxxx0:zzz.
(13)

Here, :, : · represent double and triple contractions, respectively, and we assume that z is normalized such that z · z = 1. Notice that the absence of a first-order term in Eq. (11) stems from the mechanical equilibrium condition Uxx0=0 that applies to any minimum x0 of U(x). The cubic expansion as appearing in Eq. (11) implies that an energy barrier of height

B(z)=2[b2(z)]33[b3(z)]2
(14)

exists at a displacement of s = −2b2/b3 away from the minimum at s = 0 of the potential energy U.

Important to the present discussion is that the barrier B(z) is fully expressed in terms of the configuration-space directionz; all of its dependence on the details of the minimum x0 and on the potential energy topography in the vicinity of x0 are embodied in the tensors 2Uxx and 3Uxxx, evaluated at x0. Consequently, one can ask which configuration-space direction z should be chosen such that the energy barrier on the way toward an adjacent, nearby minimum of the potential energy is small.

Finding the aforementioned directions that minimize B(z) is equivalent to finding fields π that solve the equation

2Uxxπ=b2b33Uxxx:ππ,
(15)

obtained by requiring Bzπ=0. In Refs. 97, 100, 115, and 165, it was established that solutions π are QLEs; they can be obtained either by solving Eq. (15) via an iterative scheme spelled out in Ref. 97 or by performing a numerical minimization of the barrier B(z) with respect to the direction z.166 Solutions π that pertain to minima of B(z) are both low-energy—due to the numerator b23 of B(z)—and highly localized—due to the denominator b32 of B(z), which was shown in Refs. 97 and 165 to be larger for more localized modes. In addition, they feature the same properties as those described above for QLEs: their frequency distribution follows the same universal ω4 law [see Fig. 10(a)], where new results that have not been published earlier are presented, and their spatial structure features the same disordered core and algebraic far-field decay of QLEs.97 Importantly, in Ref. 100, it was shown that the energies associated with QLEs obtained via the nonlinear framework described here are comparable to the energies of non-hybridized QLEs obtained via harmonic analyses.

FIG. 10.

(a) Cumulative distribution function (CDF) of the sample-to-sample statistics of minimal-frequency QLEs (per sample), measured in ≈10 000 computer glasses in 2D using the nonlinear framework described here (see Ref. 52 for details about the model and Appendix B in Ref. 100 for details about the calculation of QLEs). The striped vertical lines mark the two lowest-phonon frequencies. (b) Participation ratio e of vibrational modes measured in the same 2D computer glasses analyzed in (a), scatter-plotted against frequency ω. These data demonstrate the robustness of the nonlinear framework against phonon-hybridizations of QLEs seen in harmonic analyses. The results presented in the figure have not been published elsewhere.

FIG. 10.

(a) Cumulative distribution function (CDF) of the sample-to-sample statistics of minimal-frequency QLEs (per sample), measured in ≈10 000 computer glasses in 2D using the nonlinear framework described here (see Ref. 52 for details about the model and Appendix B in Ref. 100 for details about the calculation of QLEs). The striped vertical lines mark the two lowest-phonon frequencies. (b) Participation ratio e of vibrational modes measured in the same 2D computer glasses analyzed in (a), scatter-plotted against frequency ω. These data demonstrate the robustness of the nonlinear framework against phonon-hybridizations of QLEs seen in harmonic analyses. The results presented in the figure have not been published elsewhere.

Close modal

One key advantage of the nonlinear-QLE framework is its robustness against hybridization, primarily with phonons,97 but also between QLEs that are spatially adjacent and close in frequency.100 These properties allow us to establish the universal quartic law for QLE-frequencies in 2D, in system sizes for which hybridizations with phonons largely obscure QLEs within the harmonic analysis, as demonstrated in Fig. 10. Some useful generalizations of the nonlinear-QLE framework reviewed here are discussed in Refs. 97, 100, and 167.

QLEs obtained via the nonlinear framework discussed in Subsection IV F feature some interesting properties different from their harmonic counterparts. Consider again a soft potential given by a quartic expansion of the potential energy along a QLE π [and recall Eq. (1)], namely,

δU(s)12b2s2+13!b3s3+14!b4s4,
(16)

where we have introduced the quartic coefficient associated with π,

b44Uxxxx::ππππ,
(17)

and the notation :: stands for a contraction over four fields. The nonlinear framework for QLEs π and the accompanying definitions of the coefficients b2, b3, and b4 [Eqs. (12), (13), and (17), respectively] offer a concrete realization of the local soft-potentials as postulated within the Soft Potential Model (SPM) discussed at length in Sec. II A.

In Fig. 11, we scatter-plot b32 vs the product b2b4 measured for QLEs obtained via the nonlinear framework applied to glasses quenched from equilibrium liquid states at parent temperatures as seen in the legend (see Ref. 100 for further details about the model and calculation). Also marked by the continuous line is the stability bound b32=3b2b4 as spelled out in the SPM. In essence, this bound assumes that if soft potentials have a double-well form, the system will typically reside in the lower-energy well. Indeed, lower-Tp ensembles—comprised of glasses with enhanced mechanical stability—seem to better satisfy this bound.

FIG. 11.

The square of the cubic expansion coefficient b32 associated with QLEs is scatter-plotted against the product b2b4 of the quadratic and quartic expansion coefficients [cf. Eq. (16)], all calculated using the nonlinear-QLE framework.97,100,165 The continuous line represents the stability bound b323b2b4 as spelled out in the SPM (cf. Sec. II A), whereas the dashed line marks an empirical lower-bound that still lacks a theoretical explanation. Here, the percentages of unstable soft-potentials with b32>3b2b4 are 6.6%, 4.6%, and 0.5% for Tp = 1.3, 0.5, and 0.35, respectively.

FIG. 11.

The square of the cubic expansion coefficient b32 associated with QLEs is scatter-plotted against the product b2b4 of the quadratic and quartic expansion coefficients [cf. Eq. (16)], all calculated using the nonlinear-QLE framework.97,100,165 The continuous line represents the stability bound b323b2b4 as spelled out in the SPM (cf. Sec. II A), whereas the dashed line marks an empirical lower-bound that still lacks a theoretical explanation. Here, the percentages of unstable soft-potentials with b32>3b2b4 are 6.6%, 4.6%, and 0.5% for Tp = 1.3, 0.5, and 0.35, respectively.

Close modal

Not predicted by the SPM is the curious observation that a lower bound b320.4b2b4—marked by the dashed line in Fig. 11—appears to exist. In Ref. 100, it was shown that QLEs calculated via other definitions (e.g., QLEs that assume the form of harmonic vibrations) do not satisfy any lower bound—their associated cubic coefficient b3 can be arbitrarily small. There is currently no theoretical explanation for the emergence of this lower bound.

In addition, since b4 is largely independent of b2ω2,57,100 the lower and upper bounds on b32 imply a tight correlation between a QLE’s stiffness b2 and its associated cubic coefficient b3; in particular, we deduce that

|b3|b2ω.
(18)

Consider next a quartic soft potential in the form of Eq. (16), and let us assume that b32>(8/3)b2b4, i.e., that the quartic expansion corresponds to a double-well potential. In such cases and using the scaling relation Eq. (18) spelled out above, we expect the barrier B between the two potential wells [cf. Eq. (14)] to follow77,100

Bb22ω4.
(19)

Incorporating next the universal quartic law D(ω)ω4, we can predict the distribution p(B) of energy barriers B (in the B → 0 limit) using p(B)dBD(ω)dωω4dω and Eq. (19), arriving at

p(B)B1/4.
(20)

This prediction assumes that the frequency distribution of QLEs with b32>(8/3)b2b4 follows the universal quartic law. Future research should validate or refute this prediction, which we expect to hold for any structural glass quenched from a melt. If validated, it would be interesting to subsequently explore the implications of Eq. (20) on various low-temperature activated processes in glasses, such as plastic deformation and structural relaxation.

QLEs π obtained from the nonlinear framework discussed in Subsection IV F are natural candidates to serve as the “Shear-Transformation-Zones” (STZs) envisioned by Falk and Langer in the late 1990s.30 In addition to their robustness against hybridization with phononic excitations, the stiffness b2=2Uxx:ππ associated with QLEs π follows a compact and physically transparent equation of motion with respect to the imposed deformations in the athermal, quasistatic limit;115,165 keeping the most singular contribution (see below), it reads

db2dϵb3Fπb2,
(21)

where Fπ and b3 were defined above in Eqs. (8) and (13), respectively, and ϵ is the imposed-deformation tensor.

If one considers an imposed-deformation tensor ϵ(γ) parameterized by a strain parameter γ, Eq. (21) would imply that near a plastic instability at a critical strain γc, one expects b2γcγ,34 as indeed verified for a variety of system sizes in Fig. 12 (outlined symbols). We note that the strain interval γcγ, over which this scaling holds, is much larger than the corresponding quantity observed for deformation-induced destabilizing harmonic vibrational modes (pale symbols in Fig. 12) due to phonon-hybridizations.115 In addition, in Ref. 167, it was shown that QLEs that are activated in plastic instabilities can be detected using the nonlinear framework discussed above at strains of order 5% away from plastic instabilities, establishing that STZs are a priori encoded in a glass’s microstructure, at odds with the claims of Ref. 168. We finally note again that the prefactor Ag of the nonphononic VDoS was shown in Ref. 122 to control the fracture toughness of computer glasses subjected to athermal, quasistatic external loading, further reinforcing the role of QLEs as the carriers of plastic flow in structural glasses.

FIG. 12.

The strain-induced softening of the stiffness b2 of QLEs near plastic instabilities (occurring at γc) observed in 2D computer glasses under athermal, quasistatic simple shear.115 The outlined symbols represent QLEs measured using the nonlinear framework, whereas the pale symbols represent the stiffness of the lowest harmonic vibrational mode in the glass.

FIG. 12.

The strain-induced softening of the stiffness b2 of QLEs near plastic instabilities (occurring at γc) observed in 2D computer glasses under athermal, quasistatic simple shear.115 The outlined symbols represent QLEs measured using the nonlinear framework, whereas the pale symbols represent the stiffness of the lowest harmonic vibrational mode in the glass.

Close modal

In Subsection IV B, the effect of QLEs—in particular of their number—on the nonlinear and dissipative properties of glasses has been briefly discussed. QLEs also have significant effects on the statistical and spatial properties of physical observables defined with respect to glassy inherent structures [local minima of U(x)] in the low-temperature limit. To see this, let us consider a general physical observable O(x) that depends on the coordinates x (strictly speaking, the components of the vector x represent here the deviations of the system’s degrees of freedom from a local minimum of its potential energy U) under constant volume. The thermal average of O is given by OT=Z(T)1O(x)expU(x)kBTdx, where Z(T)=expU(x)kBTdx is the partition function and kB is Boltzmann’s constant.

Performing then an expansion of OT to leading order in T and taking its derivative with respect to T, one obtains169 

112kBdOTdTT=02Oxx:M1OxM1U:M1,
(22)

where M2Uxx is the dynamical (Hessian) matrix (that defines the eigenvalue equation MΨ=ω2Ψ, where Ψ is the eigenmode), U3Uxxx is a third-order anharmonicity tensor, and all derivatives are evaluated at a minimum of U(x).

The structure of Eq. (22) immediately provides some physical insight. First, it is observed that anharmonicity contributes to any observable O featuring O/x0, which is typical for local observables (i.e., observables defined at the interaction scale). Second, the anharmonic contribution is expected to dominate the harmonic one in many situations and to be controlled by QLEs, implying strong spatial localization/heterogeneity and anomalous statistics. To see this, first note that M1ω2, i.e., M1 is dominated by soft excitations. The question then is whether low-frequency phonons and QLEs, both featuring ω → 0, make markedly different contributions to the anharmonic term in Eq. (22). The point is that typically for local observables, spatial gradients (e.g., as appearing in U and O/x) are vanishingly small for low-frequency phonons (which are extended objects, featuring long wavelengths) but finite for QLEs. Consequently, the anharmonic term in Eq. (22) is expected to be dominated by QLEs and feature anomalously large values due to extremely soft QLEs, ω → 0.

To demonstrate this in the context of a fundamental physical observable, let us consider O=εα,169,170 where ɛα is the interaction energy of a pair of particles denoted by α, such that the total potential energy is given as a sum over pairwise interactions, U = ∑αɛα. In light of Eq. (22), we then consider 12kBcαdεαTdTT=0, where cα can be viewed as the classical (non-quantum) local heat capacity such that ∑αcα is the global specific heat in the classical limit. The global specific heat, taking into account quantum effects, is discussed in Sec. V in the context of experimental evidence for QLEs. For O=εα, O/x is nothing but the internal force fα, which is generically finite in glasses due to glassy frustration (note that mechanical equilibrium implies that the sum of fα per particle vanishes, not the individual contributions169). Consequently, we expect the local heat capacity cα to feature anomalous statistics due to QLEs and their ω4 VDoS and to feature strong spatial localization, i.e., to attain anomalously large values in spatial locations where soft QLEs reside.

The anomalous statistics of cα, i.e., the fat-tailed (power-law) nature of the probability distribution function (PDF) p(cα), and its quantitative relation to the universal VDoS ω4 of QLEs have been demonstrated and discussed in Ref. 169. In Fig. 13, we present a spatial map of cα for a 2D computer glass composed of 10 000 particles, where each interacting pair of particles α is represented by a line whose thickness stands for the magnitude of cα, with red (black) representing negative (positive) values. It is observed that the spatial distribution of cα is highly heterogeneous, featuring strong localization in positions where soft QLEs reside; note in this context the visual resemblance between Figs. 6(a) and 13. Other physical observables, e.g., the local thermal expansion coefficient, are expected to feature similar anomalous statistics and spatial heterogeneity due to QLEs.

FIG. 13.

A map of the local heat capacity cα in a 2D computer glass169 (see text for definitions and discussion and Ref. 52 for model details). Each interacting pair of particles α is represented by a line whose thickness stands for the magnitude of cα, with red (black) representing negative (positive) values. Note the resemblance between this figure and Fig. 6(a), both revealing the existence of QLEs.

FIG. 13.

A map of the local heat capacity cα in a 2D computer glass169 (see text for definitions and discussion and Ref. 52 for model details). Each interacting pair of particles α is represented by a line whose thickness stands for the magnitude of cα, with red (black) representing negative (positive) values. Note the resemblance between this figure and Fig. 6(a), both revealing the existence of QLEs.

Close modal

The strong spatial heterogeneity of physical observables such as cα implies heterogeneous dynamics, e.g., structural relaxation under finite temperatures T or irreversible rearrangements under the application of external forces. Indeed, it has been shown that locations of anomalously high cα values are susceptible to local irreversible rearrangements when the glass is sheared externally,169 further strengthening the relation between QLEs and STZs, discussed in Subsection IV H. Furthermore, it has been shown that QLEs are, in fact, anisotropic objects that feature orientation-dependent coupling to driving forces,170 a property that is important for their effect on plastic deformation. Finally, we note that strictly athermal quantities (i.e., physical observables that are defined in inherent structures without involving any thermal averaging) are also strongly affected by QLEs. For example, it has been recently shown that fluctuations in the athermal shear modulus—that are relevant for sound attenuation at low temperatures14—feature anomalous statistics due to QLEs and their universal ω4 VDoS.171 

Within the Soft Potential Model (SPM) and the reconstruction picture, which were concisely reviewed in Sec. II, soft QLEs and tunneling two-level systems (TLSs) are intimately related. Indeed, in Ref. 79, it was argued that experimental observations support a common basis for the universal properties of glasses, both in the extremely low temperatures regime—where quantum tunneling dominates—and slightly above it. Further experimental support for this argument was provided in Refs. 172 and 173, which concluded that tunneling TLSs and QLEs that give rise to the increase in the reduced heat capacity C/T3 at T ≈ 10 K [cf. Fig. 15(a)] share a common origin.

In terms of micromechanics, tunneling TLSs are a subset of double-well potentials whose quantum energy splitting is relevant for a given temperature. The energy splitting is determined by the asymmetry of the double-well potential and the associated tunneling amplitude, which, in turn, depends on the properties of the barrier (height and width) that separates the two potential wells and requires the solution of a one-dimensional Schrödinger equation. It is then interesting to ask whether QLEs are related to tunneling TLSs, as argued in the framework of the SPM. Some discussion of this question appears in the recent literature.174 Here, we would like to pose a different, yet potentially related, question; in particular, we aim at addressing the possible relations between QLEs and double-well potentials,175 independently of whether the latter belong to the subset of relevant tunneling TLSs or not. Even more specifically, we aim at quantifying the degree of similarity between QLEs and double-well potentials, which are intrinsically nonlinear micromechanical objects.

To this aim, consider then the one-dimensional sketch of a double-well potential presented in the inset of Fig. 14, whose minima occur at the positions x0 and x1, and denote by Δx the displacement between them, i.e., Δxx1x0. The corresponding quantity in a glass is the displacement vector Δxx1x0, where x0 and x1 are the particle position vectors in two adjacent mechanically stable states. Consequently, a double-well potential can be characterized by its normalized displacement vector Δ̂xΔx/|Δx|, which represents a direction in the multi-dimensional configurational space of a glass. The normalized displacement vector π of a QLE corresponding to one of the potential wells, say, the one at x0, is also a direction in the multi-dimensional configurational space of a glass. Consequently, one measure of similarity between QLEs and double-well potentials is the degree of overlap between these two directions, i.e., |Δ̂xπ|.

FIG. 14.

Probability distribution function (PDF) of the overlap between normalized displacement Δ̂x(x1x0)/|x1x0| separating pairs x0 and x1 of adjacent potential energy landscape minima and the QLE π used to detect those pairs. The inset schematically illustrates the notations employed (see text for more details). The results presented in the figure have not been published elsewhere.

FIG. 14.

Probability distribution function (PDF) of the overlap between normalized displacement Δ̂x(x1x0)/|x1x0| separating pairs x0 and x1 of adjacent potential energy landscape minima and the QLE π used to detect those pairs. The inset schematically illustrates the notations employed (see text for more details). The results presented in the figure have not been published elsewhere.

Close modal

To quantify the latter, we employ a close variant of the polydisperse, soft-spheres model put forward in Ref. 104, which can be subjected to extreme supercooling via the swap Monte Carlo method. Further details about the model can be found in Ref. 176. We studied 40 000 glasses of N = 16 000 particles that were instantaneously quenched to T = 0 from supercooled states equilibrated at Tp = 0.4; as a reference, the onset (crossover) temperature of this system was estimated at Tx0.66.129 

In each glassy sample, we pick up one of the softest QLEs π using the nonlinear framework described above (initial conditions for detecting QLEs were chosen as described in Appendix B of Ref. 126). Once we have a soft QLE π at hand, starting from the initial state x0, we displace particles a distance s along π and follow this displacement by an energy minimization. If a new mechanically stable state x1 is detected, the overlap |Δ̂xπ| is computed. In the search for the second stable state x1 in each glass, we varied s with small increments between (0, a0], where a0 is a typical interparticle distance.

In Fig. 14, we show the probability distribution function (PDF) of |Δ̂xπ|. It is observed to be sharply peaked near unity, being predominantly supported over the relatively small range |Δ̂xπ|>0.8. These results indicate that nearby minima in the potential energy landscape of a glass can be found by following QLEs and that a high degree of overlap between the normalized displacements of QLEs and of double-well potentials (quantified by Δ̂x) exists. The latter suggest that QLEs and double-well potentials are closely related, at least as far as Δ̂x and π are concerned. Finally, we note that (i) the mean displacement amplitude between x0 and x1 is found to satisfy ⟨|Δx|⟩/a0 ≈ 0.5, and (ii) the fraction of QLEs that yielded a second mechanically stable state x1 by following their respective configurational directions (as described above) is ≈14%; preliminary data (not shown) indicate that these fractions can substantially increase in systems with strong attractive forces, whose potential-energy-landscapes are highly fragmented.177 

While we are not able to provide an exhaustive review of the literature relevant to understanding the emergence of low-energy QLEs in structural glasses in its entirety, we do concisely discuss here some additional, relevant recent efforts.

Several authors178–181 put forward theoretical frameworks that propose links between the statistical properties of various observables in a glass and QLEs. For example, in Ref. 178, an attempt to relate the distribution p(x) ∼ xθ of local strain instability thresholds x to the nonphononic VDoS of QLEs was presented based on a picture of interacting soft anharmonic oscillators that bears similarities to the KHGPS model discussed in Sec. IV E. The proposed theory, accompanied by supporting numerical simulations, predicts that D(ω)ω3+4θ for θ < 1/4 and D(ω)ω4 for θ > 1/4. Similar relations between the ω4 law of QLEs and (i) the statistics of “finite-đ fluctuations” attributed to vibrational-frequencies179 and (ii) the distribution of pairwise bond-stiffnesses180,181 were proposed. However, these works provided no clear physical motivation for the form of the assumed input distributions. Furthermore, these predictions potentially disagree with a recent numerical demonstration128 that observations of D(ω)ωβ with β < 4 suffer from finite-size effects, even for computer glasses instantaneously quenched from high temperature liquid states.

A different route was taken in Ref. 182, where a generic field-theoretic model of a quenched glass was put forward, featuring a nonphononic VDoS of excess modes that grows from zero frequency as ∼ω3. The model assumes an initial random stress field that is then driven by overdamped dynamics to a mechanical-equilibrium inherent structure; this quench procedure gives rise to various emergent properties, such as long-range spatial correlations in the resulting stress field, as discussed previously.183,184 Moreover, the work in Ref. 182 also offers an interesting attempt to relate parameters of mean-field models to the quench that generates a glass, albeit not in the context of the ω4 law of QLEs (as mentioned above).

Other authors put forward simplified, atomistic models attempting to explain the origin of QLEs and their statistical properties. In Ref. 185, an atomistic model of an effectively one-dimensional, weighted spring-network with additional bonds—that pushes the system away from isostaticity—was put forward. It was established numerically that the model features a ∼ω4 scaling of the VDoS in an intermediate frequency regime. In Ref. 186, it was shown that mechanically frustrated and positionally disordered local structures— termed “minimal complexes” — constitute a minimal model of QLEs. It was further demonstrated that ensembles of marginally stable minimal complexes feature a ∼ω4 VDoS.

Finally, we highlight a few recent studies of QLE-properties in generic computer-glass models. In Ref. 187, a contribution ∼ω3 to the nonphononic VDoS was predicted (but not observed) based on a combination of numerical simulations of soft-sphere packings and an analysis of compression- and shear-induced plastic instabilities. In Refs. 188 and 189, it was claimed that extended “phonon-like” modes always exist below the lowest phonon frequency for high-Tp glasses and that lowering Tp leads to the attenuation of the Debye spectrum in favor of the non-Debye one—claims that stand in stark contradiction with the results reviewed here.

In Ref. 190, it was shown that the ω4 nonphononic VDoS holds at finite temperatures deep in the glass phase. In Ref. 191, it was argued based on numerical simulations that the low-frequency nonphononic VDoS is crucially sensitive to the stress-ensemble considered. In particular, it was shown that the sample-to-sample minimal vibrational frequency of 2D glasses from which residual macroscopic stress was removed grows as ωmin5 instead of the expected ωmin4 for as-cast computer glasses.57 Even more recently, a numerical study of low-frequency vibrations in 2D computer glasses was presented;193 in that work, it was claimed that D(ω)ω3 at frequencies below the lowest phonon frequency; however, finite-size and glass-formation-protocol effects were not tested for (see also Fig. 10). In the same work, an analysis of the intermediate-frequency vibrations revealed an excess ∼ω2 nonphononic VDoS, in agreement with previous predictions21,112 and some simulations.116,192

The search for QLEs, and the pressing need to elucidate their statistical and mechanical properties, was strongly motivated by various experimental observations in glasses, as explained in detail in the Introduction (cf. Sec. I). Moreover, it is clear that QLEs significantly affect various glass properties and glassy phenomena. At the same time, up until now, we did not discuss direct and quantitative experimental evidence for the existence of QLEs and their properties but rather mostly focused on the study of computer glasses.

A natural physical quantity that may provide a route for obtaining direct and quantitative experimental evidence for the VDoS of QLEs is the specific heat C(T), in particular, its T dependence in the low temperature limit. Other physical quantities, such as the sound absorption coefficient (cf. the recent review in Ref. 193), can also be considered. The reason for the direct relevance of the specific heat is that it is given as C(T)=kBD(ω)x2ex(1ex)2dω, where kBx2ex(1ex)2 is the single harmonic (normal) mode contribution, with xωkBT. Consequently, a contribution D(ω)ωδ to the VDoS translates into a contribution C(T) ∼ Tδ+1 to the specific heat.8 

The latter relations imply that the experimentally observed linear T dependence of C(T) in the limit of extremely low temperatures (below 1 K; see the Introduction) is associated with a constant VDoS D(ω). It is important to note that a constant VDoS cannot be read off or extracted from the ∼ω4 VDoS presented above and is not discussed in this Perspective. Yet, this approximately linear T dependence of C(T), which is attributed to tunneling TLSs, is commonly associated with a constant TLSs VDoS (i.e., δ = 0).10,76 Consequently, the tunneling TLSs contribution to the specific heat can be denoted as CTLST. Debye’s VDoS of low-frequency phonons takes the form D(ω)=ADω2 (in 3D, i.e., here δ = 2). Consequently, the low-frequency phonons contribution to the specific heat can be denoted as CDeb(T) = CDT3, which is the famous Debye’s prediction.8 Taken together, the combined contributions of TLSs and phonons to C(T) take the form CTLST + CDT3.

If one assumes that there exist no other excitations in the low-frequency limit, ω → 0, then C(T) ≃ CTLST + CDT3 is expected to accurately describe low temperature calorimetric measurements. In fact, since CDAD and since AD depends on the linear elastic coefficients of a glass (or the elastic wave-speeds), CD can be independently obtained from elastic measurements. Consequently, C(T) ≃ CTLST + CDT3 should both accurately describe calorimetric measurements of C(T), and the resulting coefficient CD should be consistent with the outcome of independent elastic measurements. However, it has been shown that fitting the measured specific heat to C(T) ≃ CTLST + CDT3 leads to systematic differences between the calorimetric and elastic estimates of CD (where the former exceeds the latter),194 clearly indicating that C(T) ≃ CTLST + CDT3 is incomplete, i.e., that additional low-frequency excitations exist.

The VDoS of QLEs, D(ω)=Agω4, corresponds to δ = 4 and implies a contribution to the specific heat of the form CsmT5 (here “sm” corresponds to “soft modes,” and we use Csm instead of the more natural CQLE in order to adhere to the notation of Ref. 194). Consequently, the low-temperature dependence of the specific heat of glasses is expected to take the form

C(T)CTLST+CDT3+CsmT5.
(23)

In Fig. 15(a), we reproduce low-temperature measurements of C(T) of the (B2O3)84(Na2O)16 glass,194 where a fit to Eq. (23) is superimposed (solid line). In the figure, which is adapted from Ref. 194, Debye’s contribution CDeb(T) is subtracted from C(T) and its scaling with T [i.e., CDeb(T) ∝ T3] is used to normalize the outcome. Consequently, in this reduced representation of the specific heat, the tunneling TLSs contribution corresponds to a T−2 dependence (left dashed line) and the QLE contribution to a T2 dependence (right dashed line) (see Ref. 194 for the details of the fitting procedure and the selection of the temperature range to be fitted).

FIG. 15.

Analysis of the measured low-temperature specific heat of glasses in relation to Eq. (23). Note that the experimental data are for the constant pressure specific heat Cp(T), and following previous authors (e.g., Refs. 196–196), we assume that it follows the same temperature dependence of the constant volume specific heat of Eq. (23). (a) The reduced specific heat (Cp(T) − CDeb(T))/T3 of the (B2O3)84(Na2O)16 glass is plotted against T on a double-logarithmic scale.194 The solid and dashed lines correspond to a fit to Eq. (23) (see text and Ref. 194 for details). The intersection of the dashed lines is denoted by Tmin. (b) The extracted ratio CTLS/Csm for 12 different glasses [obtained from the fitting procedure presented in (a); see Eq. (23), text, and Ref. 194 for details] is plotted against the experimentally measuredTmin. The solid line corresponds to CTLS/CsmTmin4. Both panels are adapted with permission from M. A. Ramos, Philos. Mag. 84, 1313 (2004). Copyright 2004 Taylor and Francis.

FIG. 15.

Analysis of the measured low-temperature specific heat of glasses in relation to Eq. (23). Note that the experimental data are for the constant pressure specific heat Cp(T), and following previous authors (e.g., Refs. 196–196), we assume that it follows the same temperature dependence of the constant volume specific heat of Eq. (23). (a) The reduced specific heat (Cp(T) − CDeb(T))/T3 of the (B2O3)84(Na2O)16 glass is plotted against T on a double-logarithmic scale.194 The solid and dashed lines correspond to a fit to Eq. (23) (see text and Ref. 194 for details). The intersection of the dashed lines is denoted by Tmin. (b) The extracted ratio CTLS/Csm for 12 different glasses [obtained from the fitting procedure presented in (a); see Eq. (23), text, and Ref. 194 for details] is plotted against the experimentally measuredTmin. The solid line corresponds to CTLS/CsmTmin4. Both panels are adapted with permission from M. A. Ramos, Philos. Mag. 84, 1313 (2004). Copyright 2004 Taylor and Francis.

Close modal

The fitting procedure shown in Fig. 15(a) closely follows the original procedure of Buchenau et al.76 (cf. Figs. 2 and 3 therein). Repeating it for many glasses (cf. Table 1 in Ref. 194) has shown that calorimetric and elastic estimates of CD agree with each other within experimental error, eliminating the above-mentioned systematic deviations between the two independent estimates. Moreover, Eq. (23) suggests a self-consistency constraint on the analysis of the experimental data; it predicts that the minimum of the reduced specific heat (C(T) − CDeb(T))/T3, occurring at Tmin [cf. the intersection of the two dashed lines in Fig. 15(a)], satisfies CTLS/CsmTmin4. In Fig. 15(b), we show the ratio CTLS/Csm for 12 different glasses [obtained from the fitting procedure presented in Fig. 15(a)] against the experimentally measuredTmin, revealing favorable agreement with the CTLS/CsmTmin4 prediction.194 All in all, the analysis of the low-temperature specific heat of glasses appears to offer a rather direct experimental evidence for the existence of QLEs and their universal nonphononic VDoS ∼ω4.

We conclude this Perspective by briefly delineating several open questions and future research directions in relation to QLEs in structural glasses.

The existence of low-frequency phonons, i.e., of soft extended excitations, is understood in general theoretical terms to emerge from the breakdown of global continuous symmetries, following Goldstone’s theorem. As of now, there is no comparable theoretical understanding of the generic existence of QLEs, i.e., of soft (quasi-) localized nonphononic excitations, and of their statistical and mechanical properties. Developing such a fundamental understanding is a challenge for future work.

Several theoretical frameworks, such as the GPS and KHGPS models discussed in Secs. II B and IV E respectively, a priori assume the existence of localized excitations and aim at offering some understanding of their VDoS. Such frameworks appear to be quite successful in rationalizing the emergence of a gapless VDoS that increases from zero frequency as ω4. These models also highlight some generic physical ingredients involved, such as long-range elastic interactions, anharmonicity, and frustration-induced internal stresses. As these models focus on studying the minima of some effective Hamiltonians, it is tempting to interpret the latter as representing the liquid state from which a glass is formed and the minimization process to mimic the self-organizational processes taking place during vitrification in realistic finite dimensions. Yet, it is currently unclear how to substantiate such an interpretation and how to quantitatively support it. Likewise, it is currently unclear how to relate the parameters appearing in such models to physical properties of the ancestral liquid state and/or the nonequilibrium glassy state. Addressing these challenges seems important for developing a minimal model of QLEs.

QLEs populate the low-frequency tail of excess modes/excitations in glasses, i.e., soft excitations that are not phononic in nature and that do not follow Debye’s VDoS. It is now established, as discussed in this Perspective, that QLEs feature a characteristic frequency (or equivalently an energy) scale ωg.52,55 It would be interesting to clarify whether and how ωg might be related to other, previously identified frequency/energy scales in glasses. In particular, in the context of excess modes/excitations in glasses, a most well-characterized quantity is the boson peak frequency ωBP. Consequently, a challenge for future work would be to elucidate the relation between ωBP and the characteristic frequency ωg of QLEs (the possible existence of such relations has been raised in Ref. 55).

Direct experimental evidence for the existence of QLEs and their universal properties has been discussed in Sec. V, focusing on low-temperature measurements of the specific heat of glasses. A combined theoretical and experimental challenge for future work is identifying additional, measurable low-temperature observables that can cleanly reveal QLEs. On the experimental side, it is also desirable to probe the physics of QLEs not just through their effect on low-temperature observables but also through the identification of the glassy structure that underlie them, which, in turn, requires the development of experimental techniques featuring enhanced spatial, and possibly temporal, resolution.

Finally, future research should further elucidate the roles played by QLEs in various glass properties and dynamics. The latter include properties such as the fracture toughness122 and viscoelastic response functions, as well as the physics of plastic deformation. Another potentially interesting line of investigation would be the exploration of the roles played by QLEs in structural relaxation in equilibrium supercooled liquids177,197,198 and in out-of-equilibrium aging dynamics,89,199 with the hypothesis that relaxation/aging occurs at locations where QLEs reside and that the long-range elastic interactions between QLEs is of importance.

We warmly thank Karina González-López, Geert Kapteijns, David Richard, Corrado Rainone, Gustavo Düring, Francesco Zamponi, Pierfrancesco Urbani, Avraham Moriel, Talya Vaknin, and Robert Pater for their invaluable contributions to some the works reviewed here. We are also grateful to David Richard and Karina González-López for their help with the graphics. We thank Omar Benzine, Zhiwen Pan, and Lothar Wondraczek for calling our attention to the work of Ramos, discussed in Sec. V. We are grateful to Herbert Schober, Ulrich Buchenau, Geert Kapteijns, Karina González-López, and Avraham Moriel for reading an earlier version of the manuscript and for providing useful and constructive comments. E.L. acknowledges support from the NWO (Vidi Grant No. 680-47-554/3259). E.B. acknowledges support from the Ben May Center for Chemical Theory and Computation and the Harold Perlman Family.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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