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# Growing timescales and lengthscales characterizing vibrations of amorphous solids

Contributed by Giorgio Parisi, May 23, 2016 (sent for review February 2, 2016; reviewed by Kunimasa Miyazaki and Grzegorz Szamel)

## Significance

Amorphous solids constitute most of solid matter but remain poorly understood. The recent solution of the mean-field hard-sphere glass former provides, however, deep insights into their material properties. In particular, this solution predicts a Gardner transition below which the energy landscape of glasses becomes fractal and the solid is marginally stable. Here we provide, to our knowledge, the first direct evidence for the relevance of a Gardner transition in physical systems. This result thus opens the way toward a unified understanding of the low-temperature anomalies of amorphous solids.

## Abstract

Low-temperature properties of crystalline solids can be understood using harmonic perturbations around a perfect lattice, as in Debye’s theory. Low-temperature properties of amorphous solids, however, strongly depart from such descriptions, displaying enhanced transport, activated slow dynamics across energy barriers, excess vibrational modes with respect to Debye’s theory (i.e., a boson peak), and complex irreversible responses to small mechanical deformations. These experimental observations indirectly suggest that the dynamics of amorphous solids becomes anomalous at low temperatures. Here, we present direct numerical evidence that vibrations change nature at a well-defined location deep inside the glass phase of a simple glass former. We provide a real-space description of this transition and of the rapidly growing time- and lengthscales that accompany it. Our results provide the seed for a universal understanding of low-temperature glass anomalies within the theoretical framework of the recently discovered Gardner phase transition.

Understanding the nature of the glass transition, which describes the gradual transformation of a viscous liquid into an amorphous solid, remains an open challenge in condensed matter physics (1, 2). As a result, the glass phase itself is not well understood either. The main challenge is to connect the localized, or “caged,” dynamics that characterizes the glass transition to the low-temperature anomalies that distinguish amorphous solids from their crystalline counterparts (3⇓⇓⇓–7). Recent theoretical advances, building on the random first-order transition approach (8), have led to an exact mathematical description of both the glass transition and the amorphous phases of hard spheres in the mean-field limit of infinite-dimensional space (9). A surprising outcome has been the discovery of a novel phase transition inside the amorphous phase, separating the localized states produced at the glass transition from their inherent structures. This Gardner transition (10), which marks the emergence of a fractal hierarchy of marginally stable glass states, can be viewed as a glass transition deep within a glass, at which vibrational motion dramatically slows down and becomes spatially correlated (11). Although these theoretical findings promise to explain and unify the emergence of low-temperature anomalies in amorphous solids, the gap remains wide between mean-field calculations (9, 11) and experimental work. Here, we provide direct numerical evidence that vibrational motion in a simple 3D glass-former becomes anomalous at a well-defined location inside the glass phase. In particular, we report the rapid growth of a relaxation time related to cooperative vibrations, a nontrivial change in the probability distribution function of a global order parameter, and the rapid growth of a correlation length. We also relate these findings to observed anomalies in low-temperature laboratory glasses. These results provide key support for a universal understanding of the anomalies of glassy materials, as resulting from the diverging length- and timescales associated with the criticality of the Gardner transition.

## Preparation of Glass States

Experimentally, glasses are obtained by a slow thermal or compression annealing, the rate of which determines the location of the glass transition (1, 2). We find that a detailed numerical analysis of the Gardner transition requires the preparation of extremely well-relaxed glasses (corresponding to structural relaxation timescales challenging to simulate) to study vibrational motion inside the glass without interference from particle diffusion. We thus combine a very simple glass-forming model––a polydisperse mixture of hard spheres––to an efficient Monte Carlo scheme to obtain equilibrium configurations at unprecedentedly high densities, i.e., deep in the supercooled regime. The optimized swap Monte Carlo algorithm (12), which combines standard local Monte Carlo moves with attempts at exchanging pairs of particle diameters, indeed enhances thermalization by several orders of magnitude. Configurations contain either *N* = 1,000 or *N* = 8,000 (results in Figs. 1–3 are for *N* = 1,000; results in Fig. 4 are for *N* = 8,000) hard spheres with equal unit mass *m* and diameters independently drawn from a probability distribution *SI Appendix*.

We mimic slow annealing in two steps (Fig. 1). First, we produce equilibrated liquid configurations at various densities *ρ* is the number density, *β* is the inverse temperature, and *P* is the system pressure, is described by*k*th moment of *SI Appendix*). We have not analyzed the compression of equilibrium configurations with

Second, we use these liquid configurations as starting points for standard molecular dynamics simulations during which the system is compressed out of equilibrium up to various target *SI Appendix* for a discussion on the *C* weakly depends on

Our numerical protocol is analogous to varying the cooling rate––and thus the glass transition temperature––of thermal glasses, and then further annealing the resulting amorphous solid. Each value of *β*-relaxation processes) are fully separated (2). For sufficiently large *β*-relaxation processes (4).

## Growing Timescales

A central observable to characterize glass dynamics is the mean-squared displacement (MSD) of particles from position *t* starts after waiting time *φ*. The MSD plateau height at long times quantifies the average cage size (*SI Appendix*). Because some of the smaller particles manage to leave their cages, the sum in Eq. **4** is here restricted to the larger half of the particle size distribution (*SI Appendix*). When *φ* is not too large, *A*). When the glass is compressed beyond a certain

To determine the timescale associated with this slowdown, we estimate the distance between independent pairs of configurations by first compressing two independent copies, *A* and *B*, from the same initial state at *φ*, and then measuring their relative distance*B*. The two copies share the same positions of particles at *φ* increases, the corresponding dynamics becomes slower (Fig. 2*C*). In other words, as *φ* grows particles take longer to explore a smaller region of space. In a crystal, by contrast, *τ*, can be extracted from the decay of *t*, whose logarithmic form, *τ* dramatically increases (Fig. 2*D*), which provides direct evidence of a marked cross-over characterizing the evolution of the glass upon compression.

## Global Fluctuations of the Order Parameter

This sharp dynamical cross-over corresponds to a loss of ergodicity inside the glass, i.e., time and ensemble averages yield different results. To better characterize this cross-over, we define a timescale *SI Appendix*], and the corresponding order parameters

The evolution of the probability distribution functions, *A* and *B* for a range of densities across *φ* in this regime (Fig. 3*B*) suggests that states are then pushed further apart in phase space, which is consistent with theoretical predictions (11). When compressing a system across *A*). Repeated compressions from a same initial state at *A*). These results are essentially consistent with theoretical predictions (9, 11), which suggest that for

To quantify these fluctuations we measure the variance *SI Appendix* and ref. 11) of *C* and *D*). The global susceptibility *C*). Whereas *D*). This reflects the roughly symmetric shape of

## Growing Correlation Length

The rapid growth of *ξ*. Its measurement requires spatial resolution of the fluctuations of *i* we define *μ*. Even for the larger system size considered, measuring *A*). Fitting the results to an empirical form that takes into account the periodic boundary conditions in a system of linear size *L*,*a* and *b* are fitting parameters, nonetheless confirms that *ξ* grows rapidly with *φ* and becomes of the order of the simulation box at *B*). Note that although probed using a dynamical observable, the spatial correlations captured by

## Experimental Consequences

The system analyzed in this work is a canonical model for colloidal suspensions and granular media. Hence, experiments along the lines presented here could be performed to investigate more closely vibrational dynamics in colloidal and granular glasses, using a series of compressions to extract *T* instead of density. Let us therefore rephrase our findings from this viewpoint. As the system is cooled, the supercooled liquid dynamics is arrested at the laboratory glass transition temperature *SI Appendix* for a discussion of the phase diagram as a function of *T*).

Around *β*-relaxation dynamics inside the glass thus becomes highly cooperative (24, 25) and ages (26). The fragmentation of phase space below

A key prediction is that the aforementioned anomalies appear simultaneously around a

## Conclusion

Since its prediction in the mean-field limit, the Gardner transition has been regarded as a key ingredient to understand the physical properties of amorphous solids. Understanding the role of finite-dimensional fluctuations is a difficult theoretical problem (37). Our work shows that clear signs of an apparent critical behavior can be observed in three dimensions, at least in a finite-size system, which shows that the correlation length becomes at least comparable to the system size as *φ* approaches

## Acknowledgments

P.C. acknowledges support from the Alfred P. Sloan Foundation and National Science Foundation (NSF DMR-1055586). B.S. acknowledges the support by Ministerio de Economía y Competitividad (MINECO) (Spain) through Research Contract FIS2012-35719-C02. This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement 654971, as well as from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 306845. This work was granted access to the High-Performance Computing (HPC) resources of Mésocentre de Calcul-Université de Recherche Paris Sciences et Lettres (MesoPSL) financed by the Region Ile de France and the project Equip@Meso (Reference ANR-10-EQPX-29-01) of the program Investissements d’Avenir supervised by the Agence Nationale pour la Recherche. This project has received funding from the ERC under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement 694925).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: jinyuliang{at}gmail.com, giorgio.parisi{at}roma1.infn.it, or beatriz.seoane.bartolome{at}lpt.ens.fr.

Author contributions: L.B., P.C., Y.J., G.P., B.S., and F.Z. designed research, performed research, analyzed data, and wrote the paper.

Reviewers: K.M., Nagoya University; and G.S., Colorado State University.

The authors declare no conflict of interest.

Data deposition: Data relevant to this work have been archived and can be accessed at doi.org/10.7924/G8QN64NT.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1607730113/-/DCSupplemental.

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