Disordered Crystals Reveal Soft Quasilocalized Glassy Excitations

Structural glasses formed by quenching a melt are known to host a population of low-energy quasilocalized (nonphononic) excitations whose frequencies $\omega$ follow a universal $\sim {\omega }^4$ distribution as $\omega \to 0$, independently of the glass formation history, the interparticle interaction potential, or spatial dimension. Here, we show that the universal quartic law of nonphononic excitations also holds in disordered crystals featuring finite long-range order, which is absent in their glassy counterparts. We thus establish that the degree of universality of the quartic law extends beyond structural glasses quenched from a melt. We further find that disordered crystals, whose level of disorder can be continuously controlled, host many more quasilocalized excitations than expected based on their degree of mechanical disorder—quantified by the relative fluctuations of the shear modulus—as compared to structural glasses featuring a similar degree of mechanical disorder. Our results are related to glasslike anomalies experimentally observed in disordered crystals. More broadly, they constitute an important step toward tracing the essential ingredients necessary for the emergence of universal nonphononic excitations in disordered solids.

Figure 1

Dimensionless quantifiers of mechanical disorder across the amorphization transition: the main panel shows the relative sample-to-sample fluctuations of the shear modulus $\chi$; see definition in main text. These data were measured over about 700 independent realizations of systems of $N=108000$ particles at fixed vanishing pressure. The vertical line marks an estimation of the transition point at ${\delta }_c\approx 0.49$. The black squares mark the disordered-crystal ensembles studied below, namely $\delta =0.38$, 0.40, 0.42, 0.44, all of which are located below the amorphization transition. Inset: the ratio $G_{\mathrm{na}}/G$ vs $\delta$; see text for discussion.

Figure 2

(a)–(d) Pair correlation functions $g(r)$ between the “large” particle species (see Ref. [23] for details), measured for ensembles of disordered solids of $N=108000$ particles created with different $\delta$ values, as indicated in the legends. $r$ is normalized by the interparticle distance $a_0\equiv (V/N{)}^{1/3}$, where $V$ is the solid’s volume. We indeed find that all the way up to the amorphization transition at ${\delta }_c\approx 0.49$, long-range order persists.

Figure 3

(a) The vibrational density of states of disordered crystals, for different values of the parameter $\delta$ (as indicated in the legend), plotted against rescaled frequency $\omega /{\omega }_0$, where ${\omega }_0\equiv c_s/a_0$ with $c_s$ being the shear wave speed. The continuous lines correspond to fits to the $\sim {\omega }^4$ law, from which the prefactors $A_g$ are extracted. Inset: the spatial decay of (two randomly selected) soft quasilocalized excitations $\mathit{\pi }$ (obtained by the nonlinear framework presented in [23]) follows the expected continuumlike $\sim r^{-2}$ scaling in three dimensions; the colors match the $\delta$ values in the legend. (b) Dimensionless prefactors $A_g{\omega }_0^5$ vs $\delta$ on a log-lin scale. (c) The products $N\overline{e}$—representing the mean size of the disordered cores of QLEs ($e$ is the participation ratio, defined in the text)—plotted vs $\delta$. The vertical bars cover the 2nd and third quartiles.

Figure 4

Relations between quantifiers of mechanical disorder. (a) $G_{\mathrm{na}}/G$ vs $\chi$ for disordered crystals and structural glasses, as indicated by the legend. The two datasets appear to approximately form a single function; see text for further discussion. (b) The dimensionless nonphononic VDOS prefactor $A_g{\omega }_0^5$ vs $\chi$ for disordered crystals and structural glasses [the same symbols as in panel (a)]. In sharp contrast to the results of panel (a), we find that the two classes of disordered solids follow very different curves. In particular, the disordered crystals appear to have anomalously low $\chi$ values relative to the number of QLEs they host.