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.

这项研究建议的主题是安德森金属 - 绝缘体运输过渡和无序系统中的其他现象。将研究一种基于运输而不是光谱特性的安德森金属绝缘体过渡的新方法。 此外，还将研究几个相关主题。将开发在随机媒体中定位的建设性标准；计划为具有随机潜力的Landau Hamiltonian申请。将研究Continuum Anderson型哈密顿量强的绝缘谱的当地泊松统计数据。 将研究安德森（Anderson）模型在伯特（Bethe）格子上的光谱。自P.安德森（P. 在三个或多个维度中，据信从以局部状态为特征的绝缘体制度出现过渡到以扩展状态为特征的非常不同的金属状态。这种金属绝缘子过渡的能量称为迁移率边缘。这张图片的标准数学解释是，随机施拉丁格从纯点光谱（局部状态）到绝对连续的光谱（扩展状态）中应该存在过渡。但是到目前为止，还没有关于连续频谱和金属 - 绝缘体过渡的数学结果（除了伯特（Bethe）晶格上的安德森模型的特殊情况除外）。 基于运输而不是光谱特性提出了一种针对安德森金属绝缘体过渡的新方法。它的动机是，直观的定位概念具有动态解释：最初的局部波数据包应在时间演化下保持定位，并且可以将定位解释为非平凡的运输。 该提案的主要目标是表明这种运输过渡的存在。