Absence of Anderson localization in certain random lattices

We report on the transition between an Anderson localized regime and a conductive regime in a one-dimensional microwave scattering system with correlated disorder. We show experimentally that when long-range correlations are introduced, in the form of a power-law spectral density with power larger than 2, the localization length becomes much bigger than the sample size and the transmission peaks typical of an Anderson localized system merge into a pass band. As other forms of long-range correlations are known to have the opposite effect, i.e., to enhance localization, our results show that care is needed when discussing the effects of correlations, as different kinds of long-range correlations can give rise to very different behavior.

Figure 1

(a) Schematic of the scattering potential: a set of slabs with refractive index $n=1.594$ with different thicknesses are positioned at 2.5-cm intervals in the waveguide. The sequence of thicknesses $d^{\prime }$ is obtained using Eq. (2) to discretize the random sequence $d$ [Eq. (1)]. (b) The sequence $d$ is designed to be random but also to have long-range correlations, in the form of a power-law spectral density $S\left(k\right)\propto k^{-\alpha }$. The black line shows an example for $\alpha =2$. The discretization process leads to a sequence that has the same power-law spectral density, albeit noisier (red line).

Figure 2

Typical transmission spectra for different values of $\alpha$. For low values of $\alpha$ the peaks corresponding to single Anderson localized modes are clearly visible, but as $\alpha$ increases the modes become wider and start to merge into each other, forming a smooth transmission band for large values of $\alpha$.

Figure 3

Ratio of the mode separation $\mathrm{\Delta }\omega$ to the mode width $\delta \omega$ as a function of $\alpha$. Performing a multipeak fit to the transmission spectra we obtained the average spacing between the peaks and their average width. The error bars represent the standard deviation of the ratio $\mathrm{\Delta }\omega /\delta \omega$ over the different realization of the disorder. The Thouless criterion for Anderson localization states that a 1D system can be considered localized if this ratio is larger than 1 (dashed red line), i.e., if the (localized) modes are spectrally separated. Once the ratio becomes smaller than 1 we cannot talk about separated modes anymore and the system becomes conductive again.

Figure 4

Values of the localization length $\xi$ as a function of $\alpha$. The error bars represent the fit uncertainties for $\mu$ and $b$ propagated to $\xi$. The part of the exponential decay of $T\left(L\right)$ that cannot be accounted for by absorption gives us an estimate of the localization length $\xi$. For low values of $\alpha$ the measured localization length is significantly shorter than the total sample thickness (shown by the red dashed line), but when $\alpha \gtrsim 2$, there is a crossver and the system stops being localized.

Figure 5

Example of a multipeak fit for a transmission spectrum at $\alpha =0.5$. The black dots are the experimental data points, colored thin lines are the fitted peaks, and the thick red line is their sum (which has to be compared to the experimental data).

Figure 6

Fits of $T\left(L\right)$ (black points) and $T\left(L\right)+R\left(L\right)$ (blue points) to obtain, respectively, $b$ and $\mu$, for the various vales of $\alpha$.