Supersymmetry approach to disordered quantum systems

小型电子结构制造技术的快速发展使我们现在可以研究几纳米尺寸的系统（纳米器件）。在这些尺度上，电子的波动性质变得很重要，因此它们的行为被量子力学所描述。许多纳米器件的另一个基本特征是不可避免地存在缺陷或无序。因此，统计描述在这种情况下是最合适的。包含不同类型无序的量子系统的统计性质是无序量子系统理论的主题。它最初用于研究固体的电子特性，后来被应用到其他物理领域，在这些领域中，干涉效应和无序之间的相互作用是必不可少的。例如，光在无序介质中的传播或超冷原子气体的物理学。无序量子系统理论中最有力的工具之一是场理论方法。该方法起源于高能物理，在现代凝聚态理论中起着越来越重要的作用。特别是，它在无序量子系统中的应用是非常成功的。尽管如此，该领域的一些基本问题仍难以得到严格的理解。这类问题的核心之一是安德森定位现象。1958年，p·w·安德森（P. W. anderson）推测，由于相消干涉效应，受随机势作用的电子的扩散传播可以被完全抑制。尽管有很长的历史和许多重要的见解，但仍然缺乏一个严格的D bbbb1维的安德森局域化理论。这个问题引起了物理学家和数学家的极大兴趣。事实上，这将是剑桥艾萨克牛顿研究所即将开展的为期六个月的项目“安德森定位的数学和物理：50年后”的主题。该项目的目标是开发新的场理论技术，以解决量子无序系统领域的各种问题。将这些技术应用于特定的系统，例如多维安德森模型，将大大提高我们对量子无序系统在局域和临界状态下的性质的理解。

The rapid development of fabrication technology for small electronic structures allows us now tostudy systems with dimensions of a few nanometres (nanodevices). At these scales the wave nature of electronsbecomes important and so their behaviour is described by quantum mechanics. Another essential feature of many nanodevices is an inevitable presence of imperfections or disorder. Therefore the statistical description is most appropriate in this situation. Statistical properties of quantum systems containing different kind of disorder isthe subject of the theory of disordered quantum systems. Developed initially to study electronic properties of solids, it was applied later to other areas of physics, in which interplay between interference effects and disorder is essential. Among them, for instance, are light propagation in disordered media or physics of ultra cold atomic gases.One of the most powerful tools in the theory of disordered quantum systems is the field-theoretical approach.Originated in high-energy physics, the approach plays an increasingly important role in the modern condensed matter theory. In particular, its application to disordered quantum systems was extremely successful. In spite of this, a number of fundamental problems in the field resisted rigorous understanding.One of the central problems of that kind is the Anderson localization phenomenon. In 1958, P. W. Andersonconjectured that the diffusive propagation of an electron subject to a random potential can be completelysuppressed due to the destructive interference effects. Despite its long history and many important insights obtained, a rigorous theory of Anderson localization in dimensions D>1 is still lacking. The issue attracts a lot of interest from physicists and mathematicians. Indeed, it will be the subject of a forthcoming six-month programme ``Mathematics and Physics of Anderson localization: 50 Years After'' at the Isaac Newton Institutein Cambridge.The goal of this project is to develop novel field-theoretical techniques allowing to solve various problems in the field of quantum disordered systems. Application of those techniques to particular systems, such as, for example, the multidimensional Anderson model, should improve significantly our understanding of the properties of quantum disordered systems, both in the localized and the critical regimes.