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能够研究尺寸为几纳米的系统(纳米器件)。在这些尺度上，电子的波动性质变得很重要，因此它们的行为由量子力学来描述。许多纳米设备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。含有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。它最初是为了研究固体的电子性质而开发的，后来被应用到其他物理学领域，在这些领域中，干扰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。无序量子系统理论中最强大的工具之一是场论方法。该方法起源于高能物理，在现代凝聚态理论中发挥着越来越重要的作用。特别是，它在无序量子系统中的应用非常成功。尽管如此，该领域的一些基本问题仍未能得到严格的理解，其中一个核心问题就是安德森本土化现象。1958年，P·W·Anderson猜想，由于相消干涉效应，受无规势作用的电子的扩散传播可以被完全抑制。尽管它有很长的历史和许多重要的见解，但在D&gt；1维上安德森局部化的严谨理论仍然缺乏。这个问题引起了物理学家和数学家的极大兴趣。事实上，这将是即将在剑桥的艾萨克·牛顿研究所举办的为期六个月的项目《安德森本地化的数学和物理：50年后》的主题。该项目的目标是开发新的场论技术，使其能够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Anderson模型，应该会极大地提高我们对量子无序系统的性质的理解，无论是在定域区还是临界区。

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.