Scaling and Localization Lengths of a Topologically Disordered System

We consider a noninteracting disordered system designed to model particle diffusion, relaxation in glasses, and impurity bands of semiconductors. Disorder originates in the random spatial distribution of sites. We find strong numerical evidence that this model displays the same universal behavior as the standard Anderson model. We use finite-size scaling to find the localization length as a function of energy and density, including localized states away from the delocalization transition. Results at many energies all fit onto the same universal scaling curve.

Figure 1

Scaling function for data with $w\ge 25$, including one correction to scaling $\eta$ with $y=2.7$. The thin solid line is the fit. In (a), the dashed line shows the asymptote $\alpha (x)=\nu {sinh}^{-1}(x/2)$, showing good agreement in the strongly localized regime (bottom right). In (b), the relevant function $h_0(x)$ minus the asymptotic $\alpha (x)$ is plotted. Dashed lines show the three Gaussians from the fit of $\delta$. Statistical errors in $\Lambda$ are smaller than point sizes. There are 5829 data points and 893 fit parameters—8 for the Gaussians, 1 for $y$, 2 for $h_1$, and 441 each for $\varphi (\rho ,E)$ and $\eta (\rho ,E)$. The normalized ${\chi }^2$ statistic per degree of freedom $d$ is 7.5. 70% of the data points are within error of the fit. [Inset in (a)] A wire of width $w$ is constructed by sequentially adding slices (shown at right) to an existing wire. Each slice has length $L_0\ge L_c$ and has randomly distributed sites of the chosen density. [Inset in (b)] Density of states for three site densities, calculated with $1500$ sites and 100 realizations of disorder. Curves offset for clarity. At high density, the DOS becomes asymmetric.

Figure 2

(a) Correlation lengths $\xi (\rho ,E)$ from the fit of Fig. 1, for each of the 13 densities which produced enough data to be studied, offset for clarity. Solid circles mark localized states, and crosses mark extended states. Solid and dashed lines are guides to the eye. As expected, the correlation length increases smoothly as the mobility edge is approached from either side. This figure shows the choice of energies for study, which are focused on the area of interest for the critical density, near ${\rho }^{1/3}=0.24$. The mobility edge is asymmetric just as is the density of states, with the upper mobility edge closer to $E=0$ than the lower mobility edge. The $(\rho ,E)$ with low density of states (lower left) are excluded. (b) Deviations in fitted values of the scaling variable $\varphi (\rho ,E)$ from 92 different fits with and without corrections to scaling and with the smallest value of $w$ taken to be 22, 25, or 28 (in units of $a_B^{*}$). Localized states with $\varphi <-0.05$ are determined within 10% by the scaling fits. See 30.