Internally Stressed and Positionally Disordered Minimal Complexes Yield Glasslike Nonphononic Excitations

Glasses, unlike their crystalline counterparts, exhibit low-frequency nonphononic excitations whose frequencies $\omega$ follow a universal $\mathcal{D}(\omega )\sim {\omega }^4$ density of states. The process of glass formation generates positional disorder intertwined with mechanical frustration, posing fundamental challenges in understanding the origins of glassy nonphononic excitations. Here we suggest that minimal complexes—mechanically frustrated and positionally disordered local structures—embody the minimal physical ingredients needed to generate glasslike excitations. We investigate the individual effects of mechanical frustration and positional disorder on the vibrational spectrum of isolated minimal complexes, and demonstrate that ensembles of marginally stable minimal complexes yield $\mathcal{D}(\omega )\sim {\omega }^4$. Furthermore, glasslike excitations emerge by embedding a single minimal complex within a perfect lattice. Consequently, minimal complexes offer a conceptual framework to understand glasslike excitations from first principles, as well as a practical computational method for introducing them into solids.

Figure 1

Examples of minimal complexes in (a) $đ=1$, (b) 2, and (c) 3. In (a) forces are one-dimensional, and each particle has two interactions. Force balance is satisfied by choosing a constant force of any magnitude. In (b), bonds are colored according to a force-balanced internally stressed state—red bonds are repulsive, and blue bonds are attractive (this could be reversed by a negative multiplication scaling factor). In (c) a similar coloring scheme is used.

Figure 2

Visualization of the eigenmodes $\mathit{\psi }$’s and eigenvalues $\lambda$’s of $\widehat{\mathcal{H}}$ and $\widehat{\mathcal{F}}$. The three translational and rotational zero modes of $\widehat{\mathcal{H}}$ and $\widehat{\mathcal{F}}$ are shown in red, green, and blue, respectively (left). The fourfold degeneracy of the ${\lambda }_{\widehat{\mathcal{H}}}=2$ shear mode band is lifted in the presence internal stresses $\widehat{\mathcal{F}}$ to a threefold degenerate band, and a single mode. The threefold degenerate band is characterized by shearing of repulsive (red) interactions, while the single mode is obtained by shearing attractive (blue) interactions. While dilation is associated with ${\lambda }_{\widehat{\mathcal{H}}}=4$, it is an additional zero mode for $\widehat{\mathcal{F}}$ as no bond is sheared.

Figure 3

Stability phase diagram of $\widehat{\mathcal{M}}$ as a function of the positional disorder $\delta$ and the internal stress $\epsilon$. Each point in the phase diagram corresponds to the lowest non-zero eigenvalue of ${10}^5$ realizations within the $(\delta ,\epsilon )$ ensemble. Blue regions signify stable ensembles (positive semidefinite $\widehat{\mathcal{M}}$’s), while white regions indicate $(\delta ,\epsilon )$ ensembles with at least a single unstable minimal complex (ensembles on the stability boundary exhibit vanishingly small lowest non-zero eigenvalues). Solid and dashed red lines correspond to theoretical predictions. The inset depicts a representative destabilizing positional perturbation for the $\epsilon >0$ stability boundary. Markers correspond to the $(\delta ,\epsilon )$ ensembles examined in Fig. 4.

Figure 4

Density of states $\mathcal{D}(\omega )$ of four different $(\delta ,\epsilon )$ ensembles [using ${10}^7$ realizations for each $(\delta ,\epsilon )$ combination], as denoted in Fig. 3. Marginal ensembles’ low-frequency spectrum (green-yellow triangles) follow a power-law distribution close to the glassy $\mathcal{D}(\omega )\sim {\omega }^4$ [6, 7, 8, 9, 10, 11]. The nonmarginal ensemble (orange square) does not exhibit such power-law scaling, emphasizing the importance of mechanical marginality as a coupling mechanism.

Figure 5

(a) An example of a glassy mode from an inverse-power-law glass ($N=80^2$, details in Ref. [62]). (b) An example of the emerging glasslike mode $\mathit{\psi }$ from a minimal complex embedded within a lattice ($N=80^2$), obtained with $\delta =0.1$ and $\epsilon =1.44$ (enlarging the system increases ${\epsilon }_c$). The obtained quadrupolar structure is reminiscent of the one observed in (a). (c) Decay of the magnitude $|\mathit{\psi }|\equiv \sqrt{\mathit{\psi }·\mathit{\psi }}$, as a function of the distance $r$ away from the core, scaling as $r^{-1}$, similar to glassy modes [8, 62]. (d) The first 10 eigenvalues of $\mathcal{M}$ of the system. The mode presented in (b) is marked in blue, shown to exist below the first phononic band.