Finite-time scaling at the Anderson transition for vibrations in solids

A model in which a three-dimensional elastic medium is represented by a network of identical masses connected by springs of random strengths and allowed to vibrate only along a selected axis of the reference frame exhibits an Anderson localization transition. To study this transition, we assume that the dynamical matrix of the network is given by a product of a sparse random matrix with real, independent, Gaussian-distributed nonzero entries and its transpose. A finite-time scaling analysis of the system's response to an initial excitation allows us to estimate the critical parameters of the localization transition. The critical exponent is found to be $\nu =1.57\pm 0.02$, in agreement with previous studies of the Anderson transition belonging to the three-dimensional orthogonal universality class.

Figure 1

(a) Average square of the penetration depth $\langle R^2(\omega ,t)\rangle$ for a set of frequencies $(\omega =4.5$, 4.75, 5, 5.25, 5.5, 5.75, 6, and 6.25 from top to bottom) around the localization transition of a 3D system. The solid and dashed straight lines illustrate $\langle R^2\rangle \propto t$ for $\omega <{\omega }_c\simeq 5.5$ and $\langle R^2\rangle \propto t^{2/3}$ expected at the mobility edge $\omega ={\omega }_c$, respectively. (b) The same as (a) for a 2D system.

Figure 2

Best fits to numerical results for two widths of the critical region: $\mathrm{\Delta }=0.5$ (a) and 1 (b). Different colors correspond to different times $t$: 100 (black), 200 (red), 400 (green), 800 (blue), 1600 (orange), 3200 (purple), 6400 (cyan), 12 800 (magenta), and 25 600 (brown). The fits were performed using Eqs. (12, 13, 14) with $m_1=2,n_1=1$, and $m_2=n_2=0$ (a) and $m_1=3,n_1=2$, and $m_2=n_2=0$ (b). The best-fit values of the mobility edge ${\omega }_c$ are shown by vertical dashed lines. The best-fit values of the critical exponent $\nu$ are written in the plots.

Figure 3

Best-fit values of the mobility edge ${\omega }_c$ as functions of the minimum time $t_{\min }$ used in the fit procedure and for two different widths of the critical region: $\mathrm{\Delta }=0.5$ (a) and 1 (b). Error bars show the standard errors of the best-fit values, the horizontal dashed lines show the values $\langle {\omega }_c\rangle$ of ${\omega }_c$ averaged over all $t_{\min }$, and the gray shaded regions show the errors of $\langle {\omega }_c\rangle$.

Figure 4

Best-fit values of the critical exponent $\nu$ as functions of the minimum time $t_{\min }$ used in the fit procedure and for two different widths of the critical region: $\mathrm{\Delta }=0.5$ (a) and 1 (b). Error bars show the standard errors of the best-fit values, the horizontal dashed lines show the values $\langle \nu \rangle$ of $\nu$ averaged over all $t_{\min }$, and the gray shaded regions show the errors of $\langle \nu \rangle$.

Figure 5

Data of Fig. 2 represented as a function of a single parameter $L/\xi$. All data points (symbols) fall on a single master curve following from Eq. (14) (black solid line) justifying the hypothesis of single-parameter scaling.