Novel behaviors in fermion systems with point-like Fermi surfaces and singular interactions

Modern condensed matter physics owes much of its success to Fermi liquid theory. In recent times, however, the study of strongly correlated particle systems is opening up all-new territory in many-body physics. A peculiar class of such systems, that of nodal fermions, is characterized by the quasiparticle density of states vanishing at the point-like Fermi surface and long-range inter-particle interactions surviving unscreened. In the first part, I discuss fermion pairing in one such system, 2D graphene. By solving the gap equation for the excitonic order parameter, I obtain values of critical interaction strength for a variety of power-law interactions and densities of states. Furthermore, these results are then used to compute the respective free energies and analyze possible phase transitions. In the second part, I discuss the effects of lattice disorder and impurities on the density of states in nodal fermion systems. These results and predictions will be of interest to a broad range of physical problems involving nodal fermions, and can be tested in a variety of experimental directions.