RG flow of the coupling functions around the Fermi liquid phase

In this work, we propose a new functional renormalization group (RG) method for the fermi liquid phase. Instead of the coupling constants, we study the flow of coupling functions parametrized by the angle of Fermi momenta. Particularly, we consider the nearly forward scattering and nearly backward scattering respectively. Different from the forward scattering which is exactly marginal, the nearly forward scattering can induce RG flow towards the Fermi liquid. The RG flow is traced back to an unstable fixed point which describes the transition between the Fermi liquid and the Ising nematic phase. The critical theory has been studied before as a non-Fermi liquid. For the first time, we are able to give a RG formulation for the Pomeranchuk instability which has been studied for a long time. Besides, by studying the nearly backward scattering, we find a new fixed point separating Fermi liquid and superconductor at 2+epsilon spatial dimension. It is described by a critical theory distinct from the Ising nematic transition. Therefore, this generalized RG method focusing on the flow of coupling functions opens a new route towards Fermi liquid and its transitions to other phases.