Pomeranchuk and Topological Fermi Surface Instabilities from Central Interactions

We address at the mean field level the emergence of a Pomeranchuk instability in a uniform Fermi liquid with \emph{central} particle-particle interactions. We find that Pomeranchuk instabilities with all symmetries except $l=1$ can take place if the interaction is repulsive and has a finite range $r_{0}$ of the order of the inter-particle distance. We demonstrate this by solving the mean field equations analytically for an explicit model interaction, as well as numerical results for more realistic potentials. We find in addition to the Pomeranchuk instability other, subtler phase transitions in which the Fermi surface changes topology without rotational symmetry-breaking. We argue that such interaction-driven topological transitions may be as generic to such systems as the Pomeranchuk instability.