Emergent Percolation Length and Localization in Random Elastic Networks

We study, theoretically and numerically, a minimal model for phonons in a disordered system. For sufficient disorder, the vibrational modes of this classical system can become Anderson localized, yet this problem has received significantly less attention than its electronic counterpart. We find rich behavior in the localization properties of the phonons as a function of the density, frequency, and spatial dimension. We use a percolation analysis to argue for a Debye spectrum at low frequencies for dimensions higher than one, and for a localization-delocalization transition (at a critical frequency) above two dimensions. We show that in contrast to the behavior in electronic systems, the transition exists for arbitrarily large disorder, albeit with an exponentially small critical frequency. The structure of the modes reflects a divergent percolation length that arises from the disorder in the springs without being explicitly present in the definition of our model. Within the percolation approach, we calculate the speed of sound of the delocalized modes (phonons), which we corroborate with numerics. We find the critical frequency of the localization transition at a given density and find good agreement of these predictions with numerical results using a recursive Green-function method that was adapted for this problem. The connection of our results to recent experiments on amorphous solids is discussed.

Popular Summary

How mechanical vibrations propagate (or sound waves travel) in a solid with a perfectly crystalline structure is well understood. What happens to the propagation of such vibrational modes when some structural disorder is present in the solid? The understanding is actually still incomplete, despite decades of both theoretical and experimental effort. It is known that in the presence of structural disorder, vibrational modes above a certain threshold frequency in a solid can stop being propagating modes and become localized, a phenomenon akin to the disorder-induced localization of electronic wave functions in semiconductors predicted by P. W. Anderson in 1957. But, limited understanding, especially in terms of analytical results, exists of the properties of localized and propagating vibrational modes, of the nature of the transition between these two types, and of the dependence of the transition on the dimensionality of the solid and the degree of the structural disorder. In this theoretical paper, we present a minimal model for describing vibrations in disorder solids, a new analytical approach, and report a number of novel analytical results.

Our model describes a disordered solid in a minimal way as a random network of masses connected by elastic springs whose spring constants are not uniform, but are given by a statistical distribution. Being simple, it lends itself to analytical treatment. Another original aspect of our work is the use of a percolation theory approach, which effectively zooms the original mass-spring network out into a coarser network with percolating paths. Remarkably, this coarser network has a characteristic length scale. This “emergent percolation length” sets the length scale below which continuum theories of elastic waves in a solid break down. Based on this approach, we have been able to provide both analytical arguments for the existence of a critical frequency separating the localized and propagating modes in dimensions above two, and also an expression for the speed of sound of the propagating vibrational modes as a function of the strength of the disorder.

Our study provides an analytically tractable model for localization of vibrational modes in disordered solids and introduces into this context the novel concept of the emergent percolation length. Our work should be relevant to the study of vibrations in amorphous solids such as colloidal glasses and to the understanding of thermal conductance in disordered solids.

Figure 1

Illustration of our calculational scheme. (a) Start with a network of identical masses placed randomly, connected by springs with broadly distributed spring constants—their value exponential in the Euclidean distance between each pair of masses. Only springs with a strength exceeding a threshold $K_{\mathrm{opt}}$, defined in the main text, are drawn as thin lines. (b) A percolating network of connected strong springs is obtained, yielding a meshwork of pathways, with characteristic length scale $l^{*}$ [61]. (c) The disordered network is coarse grained to an effective ordered elastic network, with masses $m^{*}$, spring constants $K_{\mathrm{opt}}$, and length scale of order $l^{*}$. For system size $L\gg l^{*}$ and dimensions $d>1$, the low-lying modes follow continuum elasticity, leading to a Debye spectrum.

Figure 2

The speed of sound of the low-frequency modes as a function of the density parameter $\epsilon$, for a two-dimensional system. A low value of $\epsilon$ corresponds to a highly disordered, sparse network, where the distribution of spring constants is very broad. The energy scale $U=1$ [see Eq. (1)], and $\xi =1$. The straight line shows the prediction at high densities, with no fitting parameters; see Appendix . The other line shows the prediction of Eq. (3) of an exponentially suppressed speed at low densities, as predicted from percolation theory, taking the prefactor as a fitting parameter. The points are numerically determined from finite systems, as described in the main text.

Figure 3

Phase diagram showing the localized and delocalized modes’ dependence on density ($\epsilon$) and frequency in two and three dimensions. As predicted in Eq. (4), there are both localized and delocalized modes at any density, and at low densities, the critical frequency ${\omega }^{*}$ below which modes are delocalized depends on $\epsilon$, according to Eq. (4). The only fitting parameter in each line is the prefactor. Modes are labeled as localized or delocalized according to the criterion in Appendix . White squares indicate parameters where numerical results did not converge, as can occur for delocalized or exceptionally weakly localized states.

Figure 4

The lowest-frequency mode is shown for a single matrix with $N=500000$, $\epsilon =0.1$, in two dimensions. The amplitudes are binned along one of the sides of the square in which the points are randomly chosen, and the graph shows the sum of amplitudes in each bin. While the eigenmode itself is not a simple plane wave, after coarse graining in this way, one obtains a cosine profile, precisely as would be obtained in an ordered system.

Figure 5

Part of the data used to produce the phase diagram in Fig. 3. For $\epsilon =0.097$ in $d=2$, we show ${\Lambda }_w(w)$. The phase boundary between the localized (sloping down) and delocalized (sloping up) energies is clear.