Universal mechanism for Anderson and weak localization
Universal mechanism for Anderson and weak localization
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DOI：
10.1073/pnas.1120432109
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发表时间：
2012-09-11
影响因子：
11.1
通讯作者：
Mayboroda, Svitlana
中科院分区：
文献类型：
--
作者：
Filoche, Marcel;Mayboroda, Svitlana
驻波的局部化发生在各种各样的振动系统中，无论是机械的、声学的、光学的还是量子的。它是由不均匀介质、复杂几何形状或猝灭无序的存在引起的。其中最引人注目和著名的表现是安德森局域化，负责例如无序合金中的金属-绝缘体转变。然而，尽管有大量的相关文献，一个清晰和统一的本地化图片仍然有待发现，以及它的许多表现形式之间的确切关系。在本文中，我们证明了这两个安德森和弱局域化起源于相同的普遍机制，作用于任何类型的振动，在任何维度，并为任何域形状。这种机制将系统划分为弱耦合的子区域。这些子区域的边界对应于一个隐藏的景观，从波算子和系统几何形状之间的相互作用出现的山谷。地形沿着山谷的高度决定了各次区域之间的耦合强度。景观及其对本地化的影响可以通过解决一个特殊的边界问题来严格确定。该理论允许人们预测的本地化属性，限制区域，并估计通过一个几何对象的属性的振动本征模式的能量。特别是，安德森本地化可以理解为一个特殊的情况下，弱本地化在一个非常粗糙的景观。
Localization of stationary waves occurs in a large variety of vibrating systems, whether mechanical, acoustical, optical, or quantum. It is induced by the presence of an inhomogeneous medium, a complex geometry, or a quenched disorder. One of its most striking and famous manifestations is Anderson localization, responsible for instance for the metal-insulator transition in disordered alloys. Yet, despite an enormous body of related literature, a clear and unified picture of localization is still to be found, as well as the exact relationship between its many manifestations. In this paper, we demonstrate that both Anderson and weak localizations originate from the same universal mechanism, acting on any type of vibration, in any dimension, and for any domain shape. This mechanism partitions the system into weakly coupled subregions. The boundaries of these subregions correspond to the valleys of a hidden landscape that emerges from the interplay between the wave operator and the system geometry. The height of the landscape along its valleys determines the strength of the coupling between the subregions. The landscape and its impact on localization can be determined rigorously by solving one special boundary problem. This theory allows one to predict the localization properties, the confining regions, and to estimate the energy of the vibrational eigenmodes through the properties of one geometrical object. In particular, Anderson localization can be understood as a special case of weak localization in a very rough landscape.