Universal spectrum of normal modes in low-temperature glasses

Abstract

We report an analytical study of the vibrational spectrum of the simplest model of jamming, the soft perceptron. We identify two distinct classes of soft modes. The first kind of modes are related to isostaticity and appear only in the close vicinity of the jamming transition. The second kind of modes instead are present everywhere in the glass phase and are related to the hierarchical structure of the potential energy landscape. Our results highlight the universality of the spectrum of normal modes in disordered systems, and open the way toward a detailed analytical understanding of the vibrational spectrum of low-temperature glasses.glasses | jamming | normal modes | boson peak L ow-energy excitations in disordered glassy systems have received a great deal of attention because of their multiple interesting features and their importance for thermodynamic and transport properties of low-temperature glasses. Much debate has been concentrated around the deviation of the spectrum from the Debye law for solids, due to an excess of low-energy excitations, known as the "boson peak" (1).The vibrational spectrum of glasses is a natural problem of random matrix theory. In fact, the Hessian of a disordered system is a random matrix due to the random position of particles in the sample. The distribution of the particles induces nontrivial correlations between the matrix elements. Many attempted to explain the observed spectrum of eigenvalues by replacing the true statistical ensemble with some simpler ones, in which correlations are neglected or treated in approximate ways (2-11). However, most of these models are not microscopically grounded, thus making it difficult to assess which of the proposed mechanisms are the most relevant and understand their interplay.In this work we will focus on two ways of inducing a boson peak in random matrix models. First, it has been suggested that the boson peak is due to the vicinity to the jamming transition where glasses are isostatic (12, 13). Isostaticity means that the number of degrees of freedom is exactly equal to the number of interactions. Isostaticity implies marginal mechanical stability (MMS): cutting one particle contact induces an unstable soft mode that allows particles to slide without paying any energy cost (14,15). From this hypothesis, scaling laws have been derived that characterize the spectrum as a function of the distance from an isostatic point (11,12,16). Second, it has been proposed that low-temperature glasses have a complex energy landscape with a hierarchical distribution of energy minima and barriers (17). Minima are marginally stable (18) and display anomalous soft modes (11,19) related to the lowest energy barriers (20-22). We will denote this second kind of marginality as landscape marginal stability (LMS).Both mechanisms described above are highly universal. LMS is a generic property of mean-field strongly disordered models (18). MMS holds for a broad class of simple random matrix models (6,10,11,16) and for realistic glass models (12,23,24) at the iso...

Breakdown of elasticity in amorphous solids

“…This new Gardner transition is completely disconnected from the jamming point.) This re-entrant behaviour is in agreement with very recent studies on the spectrum of harmonic vibrations of elastic sphere glasses 22,32 . In summary, we have shown that, by cooling, amorphous solids undergo the same transition towards a marginal glass state as found for hard spheres 12 .…”

“…In summary, we have shown that, by cooling, amorphous solids undergo the same transition towards a marginal glass state as found for hard spheres 12 . The marginality of this new solid phase is related to the presence of long-range correlations and soft modes 10,22 . We can now turn to our main concern, which is the change in the elastic properties of the solid approaching the Gardner transition.…”

“…Within mean-field theory, the results found at jamming are a consequence of a more general marginal stability: the appearance of multiple competing particle arrangements is accompanied by the emergence of long-range power-law correlations and soft modes. These are related to the Goldstone modes of the Gardner transition and are present not just at jamming-that is, at infinite pressure-but in the entire high-pressure regime beyond the critical Gardner point 10,22 . Here we show that also models of structural glasses exhibit this transition when decreasing the temperature, and that this drastically affects their elastic behaviour.…”

Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions

“…The exact solution for the vibrational modes around jamming presents a vast excess of low-frequency modes compared with the Debye model for solids (a so-called Boson peak) (75). The low-frequency limit of the density of states does not vanish as in the Debye model but tends to a constant (89,93,94,95,96).…”

“…The physical origin of this effect has no fully satisfying explanation. The universality of the phenomenon is also thought to apply far beyond structural glasses (124,75). Second, renormalization group analysis of these effects have yet to be fully developed (86).…”