Research on the Anderson metal-insulator transport transition and otherphenomena in disordered systems

无序系统中Anderson金属-绝缘体输运转变及其他现象的研究

• 批准号: 0200710
• 负责人: Abel Klein
• 金额: $ 19.2万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200710&HistoricalAwards=false
• 关键词: Research; Anderson; metal; insulator; transport

The subject of this research proposal is the Anderson metal-insulator transport transition and other phenomena in disordered systems. A new approach to the Anderson metal-insulator transition based on transport instead of spectral properties will be investigated. In addition, several related topics will be investigated. Constructive criteria for localization in random media will be developed; an application is planned for the Landau Hamiltonian with a random potential. Local Poisson statistics for the strong insulator spectrum of Anderson-type Hamiltonians in the continuum will be studied. The spectrum of the Anderson model on the Bethe lattice will be studied. Localization at low disorder in one or two dimensions will be investigated.Fortysome years have passed since P. Anderson's seminal article on localization of electrons in random media, but our mathematical understanding of the metal-insulator transition is still very unsatisfactory. In three or more dimensions a transition is believed to occur from an insulator regime, characterized by localized states, to a very different metallic regime characterized by extended states. The energy at which this metal insulator transition occurs is called the mobility edge. The standard mathematical interpretation of this picture is that there should be a transition in the spectrum of the random SchrAdinger from pure point spectrum (localized states) to absolutely continuous spectrum (extended states). But up to now there are no mathematical results on the existence of continuous spectrum and a metal-insulator transition (except for the special case of the Anderson model on the Bethe lattice). A new approach to the Anderson metal-insulator transition is proposed based on transport instead of spectral properties. It is motivated by the fact that the intuitive physical notion of localization has a dynamical interpretation: an initially localized wave packet should remain localized under time evolution, and delocalization may be interpreted as nontrivial transport. The main goal of this proposal is to show the existence of such a transport transition.

本研究计划的主题是无序系统中的安德森金属-绝缘体输运跃迁和其他现象。一种新的方法，安德森金属-绝缘体转变的基础上的传输，而不是光谱性质将被调查。 此外，还将研究几个相关的主题。在随机介质中的本地化建设性的标准将被开发;计划的应用程序的朗道哈密顿与随机潜在的。研究了连续介质中强绝缘子谱的局部泊松统计。 研究了Bethe格点上安德森模型的谱。从P.安德森关于电子在随机介质中局域化的开创性文章发表至今已经过去了40多年，但我们对金属-绝缘体转变的数学理解仍然很不令人满意。 在三个或更多的维度中，过渡被认为是从以局域态为特征的绝缘体状态到以扩展态为特征的非常不同的金属状态。这种金属绝缘体转变发生的能量称为迁移率边缘。对这幅图的标准数学解释是，随机薛定谔的光谱应该有一个从纯点光谱（局域态）到绝对连续光谱（扩展态）的过渡。但迄今为止，除了Bethe晶格上的安德森模型的特殊情况外，还没有关于连续谱和金属-绝缘体相变存在性的数学结果。 提出了一种基于输运而不是光谱性质研究安德森金属-绝缘体相变的新方法。它的动机是这样一个事实，即直观的物理概念的本地化有一个动力学的解释：一个最初本地化的波包应该保持本地化下的时间演化，和离域可以被解释为非平凡的运输。 本提案的主要目标是表明这种运输过渡的存在。