Theory for the density of interacting quasilocalized modes in amorphous solids

Quasilocalized modes appear in the vibrational spectrum of amorphous solids at low frequency. Though never formalized, these modes are believed to have a close relationship with other important local excitations, including shear transformations and two-level systems. We provide a theory for their frequency density, $D_L\left(\omega \right)\sim {\omega }^{\alpha }$, that establishes this link for systems at zero temperature under quasistatic loading. It predicts two regimes depending on the density of shear transformations $P\left(x\right)\sim x^{\theta }$ (with $x$ the additional stress needed to trigger a shear transformation). If $\theta >1/4$, then $\alpha =4$ and a finite fraction of quasilocalized modes form shear transformations, whose amplitudes vanish at low frequencies. If $\theta <1/4$, then $\alpha =3+4\theta$ and all quasilocalized modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically.

Figure 1

Illustration of the flow of block parameters in the $\left(\lambda ,\kappa \right)$ plane, under increasing shear stress. Examples of trajectories are shown on the left. The shape of the associated potentials $u$ are shown on the right. The blue line is an example of a passive mode, for which $\mathrm{\Phi }<0$. The red line is an example of a shear transformation, for which $\mathrm{\Phi }>0$. The black line corresponds to the marginal case for which $\mathrm{\Phi }=0$. The dashed green line marks the appearance of the second minimum in $u$, and thus the locations of reinsertion after a failure.

Figure 2

[(a) and (b)] Stress-strain curves averaged over 1000 realizations for different $N$ (see legend). [(c) and (d)] Finite-size scaling of the mean strain increment between plastic events $\langle {\gamma }_{min}\rangle$ at three representative applied strains to determine $\theta$ (see legend). [(e) and (f)] Finite-size scaling of the mean lowest frequency is used at three representative strains $\gamma$ to determine $\alpha$ (see legend). [(g) and (h)] Density of quasilocalized modes $D_L\left(\omega \right)$ at low frequencies $\omega$: (g) 1000 realizations at $N=16000$ and (h) 5000 realizations at $N=32000$. Modes with a participation ratio above the threshold (g) $e_c=0.125$ and (h) $e_c=0.0625$ have been removed following the procedure of Ref. [30]. Insets of (g) and (h) show participation ratios $e$ and their thresholds $e_c$ (solid black lines). The green markers on the axes of (g) and (h) indicate the fitting range of $\omega$.

Figure 3

[(a) and (b)] Exponent $\theta$ extracted by finite-size scaling of $\langle {\gamma }_{min}\rangle$ at different $\gamma$. The employed system sizes are reported in the legends. [(c) and (d)] Green line: Prediction of $\alpha$ based on measured $\theta$. Blue line: $\alpha$ extracted by finite-size scaling (FSS) of $\langle {\omega }_{min}\rangle$ at different $\gamma$ using system size range $N=2000,4000,8000,16000$ and $N=4000,8000,16000,32000$. Red line: $\alpha$ obtained from a direct fit $D_L\left(\omega \right)$ at $N=16000$ and $N=32000$. Our prediction for $\alpha$ is indicated using a solid green line. The error bars for $\theta$ and $\alpha$ correspond to 95% confidence intervals for the coefficient estimates obtained by linear fit on a log-log scale.

Figure 4

Sketch of the protocol to detect the instability. The cyan curve, the blue curve, and the red curve correspond to shear steps $\delta \gamma$, $\delta \gamma /10$, $\delta \gamma /100$, respectively. The dashed curves represent the trial steps. If at some point $Q_2>max\left(Q_1,Q_3,Q_c\right)$ the trial steps are reversed and the strain step is reduced by a factor 10.

Figure 5

The left column shows results obtained after a rapid quench of a system of $N=16000$ particles and the right column after a slow quench for $N=32000$. [(a) and (b)] The comparison between $D_L\left(\omega \right)$ and $D_L\left({\omega }_{min}\right)$ at three representative $\gamma =0$, 0.02, 0.09 (red, yellow, blue). The black markers on the axes indicate the fitting range of $D_L\left({\omega }_{min}\right)$. The green markers on the axes indicate the same the fitting range used in the main text [cf. Figs. 2 and 2]. [(c) and (d)] The distribution of $P\left({\gamma }_{min}\right)$ at the same three representative $\gamma =0$, 0.02, 0.09 (red, yellow, blue).

Figure 6

The left column shows results obtained after a rapid quench for $N=16000$ using 1000 realizations and the right column results obtained after a slow quench for $N=32000$ using 5000 realizations. [(a) and (b)] ${\theta }^{\prime }$ is obtained by fitting $P\left({\gamma }_{min}\right)$ at different $\gamma$. [(c) and (d)] Measured values of ${\alpha }^{\prime }$ (red line) are in good agreement with the theoretical prediction ${\alpha }^{\prime }=4{\theta }^{\prime }+3$ (blue line).

Figure 7

$\alpha$ measured at different lower bounds of fitting ranges after (a) a rapid quench and (b) a slow quench. The results that are reported in the main text (Figs. 3 and 3) correspond to the red lines.

Figure 8

(a) $\theta$ obtained from finite-size scaling analysis in steady state. (b) The density of states of quasilocalized vibrational modes $D_L\left(\omega \right)$ at low frequencies $\omega$ for $N=16000$ after modes with a participation ratio above the threshold $e_c$ have been removed. The green markers on the axes indicate the same fitting range that is used in the main text (cf. Fig. 2).