Contrary to the theoretical predictions of one-parameter scaling theory that all waves in two-dimensional disordered materials are localized, Anderson localization is observed only for sufficiently high frequencies in an isotropically jammed two-dimensional disordered granular packing of photoelastic disks. More specifically, we have performed an experiment in analyzing the level statistics of normal mode vibrations. We find that the level-distance distribution obeys Gaussian-orthogonal-ensemble (GOE) statistics of the Wigner-Dyson classification in the low-frequency (boson-peak) and intermediate-frequency regimes, whereas in the high-frequency regime Poisson statistics are observed. This means that at low and intermediate frequencies we have delocalized modes, and only at very high frequencies do localized modes exist. Evaluating the system-size dependence of the delocalization-localization crossover frequency, we obtain evidence of a true transition with a mobility edge at 80% of the Debye frequency and a value of the critical correlation-length exponent $\nu \sim$ 1.66 that is similar to that of the three-dimensional electronic Anderson model. We argue that for force-constant disorder one-parameter scaling might not be applicable.

中文翻译：

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二维粒状材料的料位统计和安德森离域

与一维定标理论的理论预测相反，二维无序材料中的所有波都被局域化，仅在各向同性阻塞的二维二维弹性无序粒状填料中，只有足够高的频率才能观察到安德森局域化。更具体地说，我们进行了一项分析正常模式振动的水平统计的实验。我们发现，在低频（玻色子峰）和中频模式中，水平距离分布服从Wigner-Dyson分类的高斯-正交系（GOE）统计，而在高频模式中，泊松统计是观测到的。这意味着在低频和中频处，我们有离域模式，只有在非常高的频率下，才存在局部模式。$\nu 〜$1.66与三维电子安德森模型的相似。我们认为，对于力恒定性疾病，一参数缩放可能不适用。