Correlations and phase transitions in Dirac systems

许多固体材料，如铜或铝的电绝缘或导电行为，可以用量子力学模型很好地理解，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，这样一个简单的图片失败，电子之间的相互作用的包含变得至关重要。在所谓的狄拉克材料中尤其如此，它的电子结构与石墨烯相似；一种真正的二维单层碳有望彻底改变当今的电子产品。这些狄拉克固体可以对外部探针产生反应的电子数量减少了，这减少了通常的电子-电子排斥的“筛选”，使它们更容易受到对电子本身的反馈效应的影响。这些特性并不一定会对它们的导电性、磁性和其他特性造成损害，但它们确实需要一种新的理论方法来研究这类材料。我提出了一个理论研究电子和杂质之间的相互作用对狄拉克系统在二维和三维电子性质的定性新影响。这个建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。其他的可能性还有很多；最重要的是，一种长期寻找的相位，其物理特性介于普通金属和有序绝缘体之间，也被认为是可能的。决定众多电子相中哪一个最终要实现的物理机制的公式是拟议研究的主要目标之一。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的数学理解，即所谓的重整化群，以及它所允许的各种可能性。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，例如高能物理学中的问题。

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.