Bond-space operator disentangles quasilocalized and phononic modes in structural glasses

The origin of several emergent mechanical and dynamical properties of structural glasses is often attributed to populations of localized structural instabilities, coined quasilocalized modes (QLMs). Under a restricted set of circumstances, glassy QLMs can be revealed by analyzing computer glasses' vibrational spectra in the harmonic approximation. However, this analysis has limitations due to system-size effects and hybridization processes with low-energy phononic excitations (plane waves) that are omnipresent in elastic solids. Here we overcome these limitations by exploring the spectrum of a linear operator defined on the space of particle interactions (bonds) in a disordered material. We find that this bond-force-response operator offers a different interpretation of QLMs in glasses and cleanly recovers some of their important statistical and structural features. The analysis presented here reveals the dependence of the number density (per frequency) and spatial extent of QLMs on material preparation protocol (annealing). Finally, we discuss future research directions and possible extensions of this work.

Figure 1

A single soft mode in a two-dimensional (2D) computer glass with $N=4096$ as revealed by (a) the Hessian matrix $\mathcal{M}$ and by (b) the bond-force-response operator $\mathcal{A}$, see text for details. The clear phonon hybridization of the Hessian mode of panel (a) is entirely suppressed in the corresponding $\mathcal{A}$ mode of panel (b), cleanly revealing a quasilocalized mode. (c) Harmonic spectrum $D\left(\omega \right)$ vs frequency $\omega$ for computer glasses in three dimensions (3D), featuring the universal asymptotic $\sim {\omega }^4$ scaling. The vertical line marks the expected lowest phonon frequency ${\omega }_{\text{ph}}=2\pi c_s/L$ with $c_s$ denoting the shear wave speed and $L$ denoting the linear size of the glass. All frequencies are divided by ${\omega }_0=c_s/a_0$, where $a_0\sim {\rho }^{-1/d}$ and $\rho$ is the density of the system (with particle masses set to unity). (d) Distribution of eigenvalues $P\left(\lambda \right)$ of $\mathcal{A}$ calculated for the same glasses as (c). All eigenvalues are divided by ${\lambda }_0={\omega }_0^{-2}$ for sake of comparison with the spectrum of the Hessian. The observed $P\left(\lambda \right)\sim {\lambda }^{-7/2}$ scaling at large $\lambda$ echoes the $\sim {\omega }^4$ scaling of the nonphononic spectrum of (c), as we explain in what follows. The vertical line marks the $\lambda$ value corresponding to the expected lowest phonon frequency in the system, ${\lambda }_{\text{ph}}\sim {\omega }_{\text{ph}}^{-2}$

Figure 2

Bond operator formalism. For an example 2D IPL system: (a) Coordinate-space unit dipole on $\alpha , |{\mathrm{\Xi }}^{\left(\alpha \right)}\rangle$ and displacement response $|\delta R^{\left(\alpha \right)}\rangle$. (b) Contact network around $\alpha$. (c) Contact network around $\beta$. Inset: ${\mathcal{A}}_{\alpha \beta }$ measures the change in length of bond $\beta$ in response to an applied dipolar force on $\alpha$.

Figure 3

Spectrum of bond operator for varying material preparation protocols. (a) Distribution of $\mathcal{A}$ eigenvalues for four different $T_{\mathrm{p}}$. Lighter colors (bottom) represent deeper annealing. The scale triangles depict the predicted high-$\lambda P\left(\lambda \right)\sim {\lambda }^{-7/2}$ power law and mid-$\lambda P\left(\lambda \right)\sim {\lambda }^{-2}$ scaling respectively. (b) Participation ratio of particle force modes multiplied by $N$ as a function of $\lambda$ for varying $T_{\mathrm{p}}$, color scale as in (a). We use the plateau values of $Ne$ in the high-$\lambda$ regime to extract QLM length scales $\xi$ from $\mathcal{A}$ modes.

Figure 4

Eigenmodes of $\mathcal{A}$ and their correspondence to PHMs. (a) High-$\lambda$ bond force mode $|f\rangle$ obtained by diagonalizing $\mathcal{A}$. Red-colored bonds represent extension, and blue-colored bonds represent compression. Line opacity and thickness are adjusted to reflect the magnitude of extension or compression, and we color only the top $10\%$ (in magnitude) of bonds. (b) Particle force mode $|F\rangle$ (blue) corresponding to the bond force mode in (a) and related pseudoharmonic force mode $|F_{\mathrm{PH}}\rangle$ (orange). (c) Particle displacement mode $|D\rangle$ (green) corresponding to the particle force mode in (b) and related pseudoharmonic displacement mode $|D_{\mathrm{PH}}\rangle$ (purple). (d) Spatial decay profiles of polarization vector magnitudes as a function of distance from disordered core. Colored points for each type of mode are the same as in (a)–(c). (e) Convergence of particle displacement and particle force modes to their corresponding PHMs. The results shown in (d) and (e) are for our ensemble of 3D IPL glasses, whereas (a)–(c) are presented in 2D for ease of visualization.

Figure 5

Effect of thermal annealing on $A_{\mathrm{g}}$ and $\xi$. Prefactor ratios $R_{A_{\mathrm{g}}}$ extracted from $P\left(\lambda \right)$ distributions in Fig. 3 as a function of parent temperature (open blue circles), overlaid on data reproduced from Ref. [24] (black connected dots). The inset shows the QLM length scale ratio $R_{\xi }$ extracted from the participation plateaus in Fig. 3 as a function of parent temperature (open red circles), also overlaid on data from Ref. [24] (black connected dots).

Figure 6

(a) Participation ratio as a function of $\mathcal{A}$ eigenvalue $\lambda$ for particle force modes (blue, bottom) and particle displacement modes (green, top). Markers represent the participations and eigenvalues associated with the sample $|D\rangle$ and $|F\rangle$ modes in (b)–(g), which are arranged in the same order. (b)–(d) Particle displacement modes corresponding to the profile in (a), arranged from low to high $\lambda$ from left to right. (e)–(g) Particle force modes corresponding to the profile in (a), arranged from low to high $\lambda$ from left to right.

Figure 7

$P\left(\lambda \right)$ and $D\left(\omega \right)$ distributions for ensembles of harmonic sphere packings. Spectra for varying system sizes are represented with different shapes and colors of marker as shown. (a) Full $P\left(\lambda \right)$ spectrum, where the scale triangle represents $P\left(\lambda \right)\sim {\lambda }^{-7/2}$ scaling. (b) The low-frequency $D\left(\omega \right)$ is shown to emphasize the presence of phonons. The scale triangle represents $D\left(\omega \right)\sim {\omega }^4$ scaling. Dashed vertical lines depict the lowest phonon frequency given by ${\omega }_{\text{ph}}=2\pi c_s/L$ where corresponding peaks are visible.