Correlations and phase transitions in Dirac systems

许多固体材料的电绝缘或导电行为，如铜或铝，可以很好地用量子力学模型来解释，该模型几乎完全忽略了组成电子之间的排斥相互作用。然而，也有许多已知的材料在这种简单的图像中失败了，包括电子之间的相互作用变得至关重要。在所谓的狄拉克材料中尤其如此，这种材料的电子结构类似于石墨烯；这是一种真正的二维单层碳层，有望给当今的电子产品带来革命性的变化。这些狄拉克固体减少了可以与外部探测器反应的电子数量，这减少了通常的电子-电子斥力的“屏蔽”，使它们更容易受到电子本身的反馈效应的影响。这些特性不一定对它们的导电、磁性和其他性质有害，但它们确实需要一种新的理论方法来研究这种类型的材料。*我建议从理论上研究电子间相互作用和杂质相互作用对狄拉克体系二维和三维电子性质的新影响。这个提议建立在我自己和其他研究小组最近进行的一些研究的基础上，这些研究表明，相互作用的狄拉克系统可能会显示出有趣的新的有序低温相，这些相与当时所谓的向列相有一些相似之处，否则在物理上完全不同，液晶。其他的可能性也很多；最重要的是，人们长期寻找的介于正常金属和有序绝缘体之间的相也被认为是可能的。确定众多电子相中最终将实现哪一相的物理机制是拟议研究的主要目标之一。要实现这一目标，需要提高对相互作用量子系统研究的主要理论工具--所谓的重整化群--及其所允许的各种可能性的数学理解。人们希望，这种加深的理解也可能导致在其他形式上类似的问题上取得进展，例如，发生在高能物理中的问题。

The electrically insulating or conducting behavior of many solid materials, such as cooper or aluminum, may be well understood by a quantum mechanical model which almost entirely neglects the repulsive interaction between the constituent electrons. However, there are also many known materials where such a simple picture fails, and the inclusion of interactions between electrons becomes paramount. This is particularly true in the so-called Dirac materials, which have their electronic structure similar to graphene; a truly two-dimensional single layer of carbon which holds a promise to revolutionize today's electronics. These Dirac solids have their number of electrons that can react to external probes reduced, which diminishes the usual "screening" of the electron-electron repulsion, and makes them more vulnerable to its feedback effects on the electrons themselves. These features are not necessarily detrimental to their conducting, magnetic, and other properties, but they do necessitate a new theoretical approach to this type of materials. ****** I propose a theoretical study of the qualitatively new effects of the interactions between electrons and of the impurities on the electronic properties of the Dirac systems in both two and three dimensions. The proposal builds on a number of recent studies performed by my own and by other research groups which indicate that interacting Dirac systems may show interesting new ordered low-temperature phases which share some similarity with then so-called nematic phases of, otherwise physically entirely different, liquid crystals. Other possibilities abound; most importantly, a long-sought phase that is by its physical characteristics somewhere in between a normal metal and an ordered insulator is also thought to be possible. A formulation of the physical mechanism that decides which of the multitude of the electronic phases is ultimately the one to be realized is one of the principal goals of the proposed research. Achieving this goal requires improved mathematical understanding of the main theoretical tool for the studies of interacting quantum systems, the so-called renormalization group, and the various possibilities that it allows. The hope is that such a deepened understanding might also lead to advances on other formally similar problems, which, for example, occur in high-energy physics.