Topological states of quantum matter

电子在固体中的行为受量子力学的特殊定律支配。在量子世界中，粒子通常表现为波，而波现象，如光，获得了粒子般的属性。能带理论是20世纪世纪量子物理学的一大成就：它解释了为什么某些晶体固体（即金属）导电，而另一些则是电绝缘体。能带理论也是我们理解半导体的基础，而半导体又是过去50年来取得的大部分技术进步的基础。直到最近，人们还普遍认为能带理论及其应用已经被完全理解了。2005年，一群理论家发现了能带理论的一个全新方面，这是一个巨大的惊喜，它植根于从数学领域被称为拓扑学的思想。这些开创性的理论工作和随后的实验研究确立了一类被称为"拓扑绝缘体"的全新固体的存在。这些材料有绝缘体，但值得注意的是，它们的表面导电。这种不寻常的行为形成了许多提议的实际应用的基础。 该提议涉及拓扑绝缘体以及相关类别的拓扑超导体材料中的新量子现象的理论探索。例如，所提出的研究的一个目标是分析和分类各种类型的无序的后果，即总是存在于固体中，对上述导电表面状态。 另一个目标是发展对电子-电子相互作用效应的理解。除了有助于解决潜在的基本知识挑战外，拟议研究的目的是开发可能导致这些非凡材料在量子计算，信息存储和处理等领域实际应用的想法。

Behavior of electrons in solids is governed by the peculiar laws of quantum mechanics. In the quantum world particles often behave as waves, and wave phenomena, such as light, acquire particle-like attributes. Among the triumphs of the 20th century quantum physics is the band theory: it explains, among other things, why certain crystalline solids (i.e. metals) conduct electricity while others are electrical insulators. Band theory also forms the basis for our understanding of semiconductors which in turn underlie most of the technological progress achieved over the past 50 years. Until very recently it was widely believed that the band theory and its applications were fully and completely understood. It came as a great surprise when in 2005 a group of theorists uncovered a completely new aspect of the band theory rooted in the ideas adopted from the area of mathematics known as topology. These pioneering theoretical works and the subsequent experimental studies established the existence of a brand new class of solids known as `topological insulators'. These materials have insulating bulk but, remarkably, their surfaces conduct electricity. This unusual behavior forms a basis for numerous proposed practical applications. This proposal is concerned with theoretical exploration of novel quantum phenomena in topological insulators as well as in the related class of materials known as topological superconductors. For example, one goal of the proposed research is to analyze and classify the consequences of various types of disorder, that is invariably present in solids, on the above mentioned conducting surface states. Another goal is to develop understanding of the effect of electron-electron interactions. Besides contributing to resolving the underlying fundamental intellectual challenges the aim of the proposed research is to develop ideas that could lead to practical applications of these remarkable materials in areas such as quantum computation, information storage and processing.