Fermi Liquid Theory of a Fermi Ring*

We study the effect of electron-electron interactions in the electronic properties of a biased graphene bilayer. This system is a semiconductor with conduction and valence bands characterized by an unusual ``mexican-hat'' dispersion. We focus on the metallic regime where the chemical potential lies in the ``mexican-hat'' in the conduction band, leading to a topologically non-trivial Fermi surface in the shape of a ring. We show that due to the unusual topology of the Fermi surface electron-electron interactions are greatly enhanced. We discuss the possibility of an instability towards a ferromagnetic phase due to this enhancement. We compute the electronic polarization function in the random phase approximation and show that, while at low energies the system behaves as a Fermi liquid (albeit with peculiar Friedel oscillations), at high frequencies it shows a highly anomalous response when compare to ordinary metals.