Correlations and phase transitions in Dirac systems

许多固体材料的电绝缘或导电行为，如库珀或铝，可以很好地理解量子力学模型，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，并包括电子之间的相互作用变得至关重要。这在所谓的狄拉克材料中尤其如此，狄拉克材料的电子结构类似于石墨烯;一种真正的二维单层碳，有望彻底改变当今的电子学。这些狄拉克固体减少了可以对外部探针起反应的电子数量，这减少了通常的电子-电子排斥的“屏蔽”，使它们更容易受到电子本身的反馈效应的影响。这些特征不一定会损害它们的导电，磁性和其他特性，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的定性新的影响电子之间的相互作用和杂质的狄拉克系统的电子性质在二维和三维。 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，这些相与当时所谓的液晶相有一些相似之处，否则物理上完全不同，液晶。其他可能性比比皆是;最重要的是，长期以来一直在寻找的相也被认为是可能的，该相的物理特性介于正常金属和有序绝缘体之间。制定的物理机制，决定了众多的电子阶段中的哪一个是最终要实现的是一个拟议的研究的主要目标之一。 要实现这一目标，需要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--有更好的数学理解。希望这种深入的理解也可能导致其他形式上类似的问题的进展，例如，发生在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.