Ammari Habib, Davies Bryn, Hiltunen Erik Orvehed
Department of Mathematics, ETH Zürich, Zürich, Switzerland.
Department of Mathematics, Imperial College London, London, UK.
Commun Math Phys. 2024;405(1):1. doi: 10.1007/s00220-023-04880-w. Epub 2024 Jan 14.
In this paper, we use recent breakthroughs in the study of coupled subwavelength resonator systems to reveal new insight into the mechanisms responsible for the fundamental features of Anderson localization. The occurrence of strong localization in random media has proved difficult to understand, particularly in physically derived multi-dimensional models and systems with long-range interactions. We show here that the scattering of time-harmonic waves by high-contrast resonators with randomly chosen material parameters reproduces the characteristic features of Anderson localization. In particular, we show that the hybridization of subwavelength resonant modes is responsible for both the repulsion of energy levels as well as the widely observed phase transition, at which point eigenmode symmetries swap and very strong localization is possible. We derive results from first principles, using asymptotic expansions in terms of the material contrast parameter and obtain a characterization of the localized modes in terms of generalized capacitance matrices. This model captures the long-range interactions of the wave-scattering system and provides a concise framework to explain the exotic phenomena that are observed.
在本文中，我们利用耦合亚波长谐振器系统研究中的最新突破，揭示了对安德森局域化基本特征背后机制的新见解。事实证明，随机介质中强局域化的出现难以理解，尤其是在物理推导的多维模型以及具有长程相互作用的系统中。我们在此表明，具有随机选择的材料参数的高对比度谐振器对时谐波的散射再现了安德森局域化的特征。特别是，我们表明亚波长谐振模式的杂化既导致了能级的排斥，也导致了广泛观察到的相变，在该相变点本征模式对称性互换，并且可能出现非常强的局域化。我们从第一性原理出发，利用材料对比度参数的渐近展开式，并根据广义电容矩阵获得了局域模式的特征描述。该模型捕捉了波散射系统的长程相互作用，并提供了一个简洁的框架来解释所观察到的奇异现象。
