Correlations and phase transitions in Dirac systems

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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.