Detecting low-energy quasilocalized excitations in computer glasses

Soft, quasilocalized excitations (QLEs) are known to generically emerge in a broad class of disordered solids and to govern many facets of the physics of glasses, from wave attenuation to plastic instabilities. In view of this key role of QLEs, shedding light upon several open questions in glass physics depends on the availability of computational tools that allow one to study QLEs' statistical mechanics. The latter is a formidable task since harmonic analyses are typically contaminated by hybridizations of QLEs with phononic excitations at low frequencies, obscuring a clear picture of QLEs' abundance, typical frequencies, and other important micromechanical properties. Here we present an efficient algorithm to detect the field of quasilocalized excitations in structural computer glasses. The algorithm introduced takes a computer-glass sample as input and outputs a library of QLEs embedded in that sample. We demonstrate the power of the algorithm by reporting the spectrum of glassy excitations in two-dimensional computer glasses featuring a huge range of mechanical stability, which is inaccessible using conventional harmonic analyses due to phonon hybridizations. Future applications are discussed.

Figure 1

Different realizations of the same nonlinear mode extracted from the cubic (a), quartic (b), and PHM (c) cost function. Mode energies are ${\kappa }_{\mathrm{cubic}}=0.428, {\kappa }_{\mathrm{quartic}}=0.347$, and ${\kappa }_{\mathrm{PHM}}=0.357$, respectively.

Figure 2

Sketch of stage 1 of our algorithm: (1) we probe glassy heterogeneities on a scale $\xi$, (2) we pinch two particles with a force $\mathit{f}$ at the center of each block (here rendered with different colors), and (3) we map $\mathit{f}$ onto a solution $\mathit{\pi }$ of a given nonlinear cost function $\mathcal{C}$.

Figure 3

First stage of our detection algorithm applied on a very stable (a) and poorly annealed (b) glass prepared at $T_{\mathrm{p}}=0.2$ and 0.7, respectively. Different colors correspond to different dipole forces used to start the mapping procedure, such as shown in Fig. 2. (c) Occurrence number (red) and density of states (black) as a function of the mode frequency $\omega$ for a glass prepared at $T_{\mathrm{p}}=0.7$. (d) Example of the halo effect in a large mildly annealed glass composed of $N=102400$. (e) Example of the halo effect when approaching a plastic instability. The black curve is the mode frequency as a function of the strain. Insets show our algorithm at different strains marked by colored squares. (f) Mode radial distribution function $g_{\pi }\left(r\right)$ for different mode $\pi$ with frequency ${\omega }_{\pi }$. (g) Halo radius ${\xi }_H$ as a function of ${\omega }_{\pi }$ for two different box lengths $L$. ${\omega }_0\equiv c_s/a_0$ with $c_s$ and $a_0$ denoting the shear-wave speed $c_s=\sqrt{\mu /\rho }$ and typical interparticle distance, respectively. The black line indicates the scaling ${\xi }_H\sim 1/{\omega }_{\pi }$. (h) Sketch of the cost function landscape $\mathcal{C}\left(\mathit{z}\right)$ approaching the limit ${\omega }_{\pi }\to 0$ as illustrated in (e). Different colors correspond to different frequencies ${\omega }_{\pi }$ of one soft excitation with ${\omega }_1<{\omega }_2<{\omega }_3$. Each star represents a distinct solution associated to one minimum of $\mathcal{C}\left(\mathit{z}\right)$. The size of the halo created by the lowest excitation is indicated by horizontal double arrows. All modes are extracted using ${\mathcal{C}}_{\mathrm{PHM}}$.

Figure 4

Example of our detection algorithm on a glassy configuration at the onset of a plastic instability with critical mode ${\mathit{\pi }}_c$ (${\omega }_{{\pi }_c}/{\omega }_0\simeq 0.02$) rendered in red. Each blob of particles colored in cyan corresponds to the core of a distinct mode.

Figure 5

(a) Comparison between the harmonic and PHM vDOS in mildly annealed 3D glasses composed of $N=2000$ with $\xi =5$. The arrow indicates the first shear wave. The inset shows eight modes detected in a sample composed of $N=16000$ with $\xi =10$. (b) Effect of partitioning length $\xi$ on the PHM spectrum. (c) PHM spectrum for different cost function with $\xi =5$. The system is a 3D polydisperse glass prepared at $T_{\mathrm{p}}=0.6$.

Figure 6

Location of PHM excitations extracted from our biasing algorithm with different coarse-graining length $\xi$. PHMs that populate the quartic scaling (${\omega }_{\pi }/{\omega }_0<0.2$) are rendered as red displacement fields. Stiffer excitations are rendered as cyan blobs.

Figure 7

Low- (left) and high-frequency (right) cubic modes extracted in a mildly annealed 2D polydisperse glass prepared at $T_{\mathrm{p}}=0.6$. Modes have been shifted to the center of the simulation box for the sake of visibility.

Figure 8

(a) Comparison between the full harmonic (solid lines) and PHM (empty circles) vDOS for 2D glasses with $N=4096$ quenched from parent temperatures $T_{\mathrm{p}}$ as indicated by the legend. PHM spectra are computed using $\xi =5$. The dashed vertical lines indicate the shift of the average mode frequency ${\omega }_{\pi }$ between our most stable glassy configurations (blue) and hyperquenched glasses (red). (b) Rescaled PHM vDOS for various parent temperature $T_{\mathrm{p}}$ with $\xi =5$. The horizontal lines indicate $A_{\mathrm{g}}$. (c) Rescaled square of the mode amplitude decay ${\left|\pi \right|}^2$ by $r^2$. Vertical dashed lines indicate the reduced core size ${\xi }_{\pi }$ when decreasing $T_{\mathrm{p}}$.

Figure 9

(a) Comparison between the PHM vDOS extracted with $\xi =10$ (empty circles) and $\xi =5$ (solid lines) for different parent temperatures. (b) The prefactors $A_{\mathrm{g}}$ plotted vs the parent temperature $T_{\mathrm{p}}$ for various $\xi$. (c) The normalized characteristic frequency $\langle {\omega }_{\pi }\rangle /\langle {\omega }_{\infty }\rangle$ of quasilocalized modes for various $\xi$. Filled black circles are the normalized characteristic frequency of dipole responses taken from Ref. [45].

Figure 10

(a) The prefactors $A_{\mathrm{g}}$ plotted vs the parent temperature $T_{\mathrm{p}}$ for both 2D and 3D glasses with $\xi =5$. $A_{\mathrm{g}}$ is normalized by the high-temperature value $A_{\infty }=A_{\mathrm{g}}(T_{\mathrm{p}}=1)$. (b) The normalized characteristic frequency $\langle {\omega }_{\pi }\rangle /\langle {\omega }_{\infty }\rangle$ of quasilocalized modes for $\xi =3$. The orange data represent the characteristic frequency scale of dipole responses, taken from Ref. [45]. (c) The density $\mathcal{N}$ of QLEs, plotted as a function of $1/T_{\mathrm{p}}$. Solid lines are linear regressions. (d) Core size ${\xi }_{\pi }$ vs $T_{\mathrm{p}}$.

Figure 11

Average time $t_{\mathrm{algo}}$ to build a catalog of QLEs in a glass composed of $N$ particles with a block length $\xi =20$. The solid line indicates the scaling $t_{\mathrm{algo}}\sim N^{2.4}$.

Figure 12

Comparison between the 2D harmonic vDOS (solid black line) and the PHM spectra obtained from different cost functions: Cubic (purple), quartic (gray), and PHM (orange). The ensemble employed corresponds to poorly annealed glasses prepared at $T_{\mathrm{p}}=1$ with $N=4096$.

Figure 13

Normalized cumulative distribution of the nonphonic vDOS for $\xi =20$ and different system sizes. Frequencies are scales by $\sqrt{lnN}$ to account for the log-correction present in 2D solids [51]. Solid lines are linear regressions, and the inset shows the corresponding exponents for the asymptotic scaling of the vDOS $D\left(\omega \right)\sim {\omega }^{\beta }$. Solid and dashed black lines indicate $\beta =4$ and $\beta =3.5$, respectively.

Figure 14

(a) Rescaled PHM vDOS for various parent temperature $T_{\mathrm{p}}$ with $\xi =5$. The horizontal lines indicate $A_{\mathrm{g}}$. (b) Rescaled square of the mode amplitude decay ${\left|\pi \right|}^2$ by $r^4$. Vertical dashed lines indicate the reduced core size ${\xi }_{\pi }$ when decreasing $T_{\mathrm{p}}$.