Localization of soft modes at the depinning transition

We characterize the soft modes of the dynamical matrix at the depinning transition, and compare the matrix with the properties of the Anderson model (and long-range generalizations). The density of states at the edge of the spectrum displays a universal linear tail, different from the Lifshitz tails. The eigenvectors are instead very similar in the two matrix ensembles. We focus on the ground state (soft mode), which represents the epicenter of avalanche instabilities. We expect it to be localized in all finite dimensions, and make a clear connection between its localization length and the Larkin length of the depinning model. In the fully connected model, we show that the weak-strong pinning transition coincides with a peculiar localization transition of the ground state.

Figure 1

Left: Phase diagram for disordered elastic system within the Larkin approach. When $D<D_{\mathrm{uc}}=4$, pinning is always strong. Varying the disorder strength $\sigma$ results in a crossover between individual (${\ell }_c\sim 1$) and collective (${\ell }_c\gg 1$) pinning regimes. When $D>D_{\mathrm{uc}}$, ${\ell }_c$ diverges, and the crossover disappears. In the fully connected model $D=\infty$, there is a transition between weak and strong pinning phases. We believe that such a transition does not exist for any finite $D$. Right: Velocity-force characteristics: for pure systems ($\sigma =0$), $v\propto f$; for weak pinning ($\sigma <{\sigma }_c$), the velocity is nonzero for any $f>0$; for strong pinning ($\sigma >{\sigma }_c$), there is a critical force $f_c>0$, below which $v=0$.

Figure 2

An illustration of the disordered elastic model in the $d=1$ case. Each point $j$ has its own disorder force $F_j\left(u_j\right)=-{\partial }_{u_j}{V}_j\left(u_j\right)$ as a function of $u_j$. We depict one of them, and indicate the parameters characterizing its correlation length $r_f$ and its magnitude $\sigma$; see Eq. (4).

Figure 3

Rescaled averaged structure factor of the critical configuration, as a function of disorder, for $\sigma <1$. Dashed lines indicate two regime of roughness: at small length scales we find the Larkin regime $S_q\sim q^{-4}$, and at large length scales $S_q\sim q^{-(1+2\zeta )}$ with $\zeta \approx 1.25$ the random-manifold exponent at depinning. The crossover length corresponds to the Larkin length ${\ell }_c\sim {\sigma }^{-2/3}$. Inset: Raw data.

Figure 4

(a) Probability distribution of the stability matrix diagonal elements as a function of the disorder strength $\sigma$. (b) The covariance of diagonal elements $C\left(\right|i-j\left|\right):={\overline{M_{ii}{M}_{jj}}}^c.$

Figure 5

(a) Filled markers: Density of states (DOS) of dynamical matrix of marginally metastable configurations (before an avalanche of size $S>1$), in a disordered potential whose values at $u_j=1,2,\cdots$ are normally (circles) or uniformly (triangles) distributed, with variance ${\sigma }^2=0.1^2$ for both. The thick dashed curves around $\lambda =0$ show the contribution of the ground state ${\lambda }_0$ to the DOS. The black line is a linear fit $\rho \left(\lambda \right)=5\lambda$. Empty markers: DOS of dynamical matrices with diagonal elements randomly shuffled. Solid (dashed) red curve: 1d Anderson model with uniform (Gaussian) diagonal elements, with the same mean (0.345) and standard deviation (0.225) of the dynamical matrix diagonal elements. (b) Distribution of rescaled eigenvalues $\widehat{\lambda }=\lambda r_f/\sigma$ of the dynamical matrix, with different $\sigma =1,0.5,0.2,0.1,0.05$. The solid curves are guides to the eyes. The straight line is a linear fit $\rho \left(\widehat{\lambda }\right)=\widehat{\lambda }/20.$ The histograms are binned in log scale, resolving the peak at $\lambda =0$.

Figure 6

(a) Samples of dynamical matrix ground states of 1d short-range depinning model of $L=1024$, with $\sigma =0.002,0.02,0.2$, centered around its maximum. (b) Disorder average of the localization length ${\xi }_{\text{loc}}$ [Eq. (24)] as a function of $\sigma$, of the ground state (“edge”) and of the 100 states in the middle of the spectrum (“bulk”).

Figure 7

(a) Numerical measure of the stretched-exponential decay of the ground state of the dynamical matrix, of size $L=2^{12}$, and of different disorder strengths. They compare well to the ansatz Eq. (25). (b): The same measure for the short-range, 1d Anderson model. The matrices have sizes $L=2^{13}$. The diagonal disorders are drawn from a uniform distribution in $[0,W_d]$. The “depinning” data are those with $\sigma =0.0002$, which matches numerically to $W_d=0.0026$ with Eq. (20). The comparison shows a discrepancy in the stretched-exponential prefactor between Anderson and dynamical matrix ensembles.

Figure 8

(a) Illustration of Eq. (28) in the weak pinning phase; only one solution of $v_j$ exists for each ${\alpha }_j$. (b) Illustration of Eq. (28) and Eq. (29) in the strong pinning phase. (c), (d) The cumulative distribution of $v_j$ Eq. (30) in the weak and strong pinning phases. In (d), the dash-dotted curve corresponds to the unstable solution of Eq. (28) that is discarded; see main text.

Figure 9

Main plot: Numerical measure of the critical force $f_c$ in the fully connected model, with the quadratic disorder Eq. (41), and system sizes $2^7,2^9,2^{11},2^{13}$ (from top to bottom). The theoretical prediction of $f_c$ is given by Eq. (42) in the $\sigma >1$ phase, and vanishes when $\sigma \le 1$. Insets: The ground state of the pinned configuration, with $\sigma =0.7$ (left) and $\sigma =1.5$ (right), and with $L=2^{13}.$ See also Fig. 11.

Figure 10

The cumulative distribution function of ${\alpha }_j$ and of $v_j$ $[P_{<}\left(\alpha \right)$ and $P_{<}\left(v\right)$, respectively], in two representative samples of $L=2^{13}$, with $\sigma =0.9<1$ (a) and $\sigma =1.1>1$ (b), respectively. For both value of $\sigma$, the distribution of $\alpha$ agrees with the uniform distribution in the interval $(-1,1)$. When $\sigma =0.9<1$ the distribution of $v_j$ is continuous, while for $\sigma =1.1>1$, it displays a jump.

Figure 11

The coefficients of the ground state, in decreasing order. The analytical curve is obtained by Eqs. (31), (33) by assuming that ${\alpha }_j=jU/L$ is equidistributed. In the $\sigma <1$ phase, it is expected to represent the whole ground state. In the $\sigma >1$, it is expected to cover only the continuous part of the coefficients.

Figure 12

Localization of the ground state in the 1d long-range Anderson model, with $\alpha =0.25$. The diagonal disorder $W_j$ is independent and uniformly distributed in $[0,W_d]$, where $W_d$ is indicated beside each curve using the same color. The inverse participation ratio exponent $q_2$, Eq. (47), is averaged over ${10}^2\sim {10}^3$ realizations for each system size $L=2^8,\cdots ,2^{24}$.

Figure 13

(a) A sample of the disorder potential $W_j$ (in blue, lower curves, with $W_d=3.0$) and the ground state of the corresponding long-range Anderson model, with $\alpha =0.25$. The system size is $L=2^{16}$. The insets are zoom-in, linear scale plots of the disorder and the ground state near the maximal peak (indicated by black dashed lines in the main plot). (b) Ground states of $\alpha =0.25$ long-range Anderson model with $W_d=3.0$ in different system sizes, with the maximum displaced to the origin. The wave-function decay is compared to the power law $r^{-\alpha -1}$ argued in Ref. [44]. The value $q_2$, Eq. (47), of each wave function is noted above it with the same color.

Figure 14

Numerical measure of the decay of ground state of the 1d long-range Anderson model with $\alpha =1$, compared to the prediction Eq. (C11). The diagonal elements are uniformly distributed in $[0,W_d]$. Close to the maximum $j_{max}$, we observe a small deviation from the $r^{1/2}$ power law. A similar behavior is also observed around the maximum of a Brownian bridge (red curve).