Interactions and symmetries of electronic systems with quadratic band touching

In this essay we study two-dimensional electronic systems that exhibit quadratic band touching (QBT) in their low energy dispersion. A lattice realization is shown to introduce the emergence of a QBT in the framework of the tight binding approximation. We then make a continuum limit, which allows for a more general approach on the level of low energy Hamiltonians. The basic concepts and the applied methods are introduced via a simplified problem, a system of spinless fermions. We use a renormalization group (RG) approach to examine the effects of short-range electron-electron interactions. Once it is found that the system is unstable toward ordered phases, we use the mean field approximation to investigate how the phases compete energetically. We then introduce the spin degree of freedom of the electrons and follow analogous calculations using the RG approach. A number of ordered phases are identified, and the flow equations of the susceptibilities are calculated. We find a symmetry of the action that is larger than what we start with and emerges under some plausible conditions.