Critical Parameters from a Generalized Multifractal Analysis at the Anderson Transition

We propose a generalization of multifractal analysis that is applicable to the critical regime of the Anderson localization-delocalization transition. The approach reveals that the behavior of the probability distribution of wave function amplitudes is sufficient to characterize the transition. In combination with finite-size scaling, this formalism permits the critical parameters to be estimated without the need for conductance or other transport measurements. Applying this method to high-precision data for wave function statistics obtained by exact diagonalization of the three-dimensional Anderson model, we estimate the critical exponent $\nu =1.58\pm 0.03$.

Figure 1

Evolution of the wave function intensity distributions $\mathcal{P}(\tilde{\alpha };W,L)$ as a function of disorder $W$ across the Anderson transition, at fixed $\lambda =0.1$ for two system sizes $L$. Each distribution has been computed with ${10}^4$ wave functions. The data points (●) and solid lines on the bottom plane mark the trajectories of the maximum ${\tilde{\alpha }}_m$. For clarity, distributions are shown at $W=15$, 16.6, and 18.0 only.

Figure 2

Two-parameter finite-size scaling result for ${\tilde{\alpha }}_m(\lambda )$ at energy $E=0$ when fitted according to Eq. (2) indicated by the shaded surface. Data (●, ○) and fit parameters are as described in the second row and caption of Table . The black lines on the surface highlight the different values of $\lambda$ [symbol ○ highlights $\lambda =0.1$]. The estimated ${\alpha }_0=4.09$ corresponds to the extrapolation of ${\tilde{\alpha }}_m(\lambda )$ as $\lambda \to 0$.

Figure 3

Plot of ${\tilde{\tau }}_2$ (top) and ${\tilde{\alpha }}_m$ (bottom) at $E=0$ and $\lambda =0.1$ as a function of disorder at various system sizes $L\in [20,100]$. The error bars denote standard deviations obtained from averaging over disorder. The lines are plotted according to Eqs. (1, 2), respectively, for the fit parameters as in the first and fourth rows of Table  and with irrelevant exponents $y=-1.8\pm 0.4$ and $y=-1.65\pm 0.12$. The vertical dashed lines show the estimated $W_c$ and its confidence interval is indicated by the grey region. The insets show the FSS functions in an enlarged view of the critical region.