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 show that the ferromagnetic instability can occur provided a low density of carriers. 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 compared to ordinary metals.