Fermi liquid has infinite number of zero modes on the fermi surface. In this talk, I will present the complete low energy dynamics of a fermi liquid which is scale invariant and corresponds to a fixed point from the perspective of the renormalization group (RG). We especially consider the non-forward scatterings with small momentum transfer in addition to the forward scatterings in the effective field theory. All the scattering processes are encoded in the quartic coupling function which flows under RG as the energy scale is lowered. This shows how the fermi liquid is emerged and how the low energy effective theory becomes a scale invariant description of a RG fixed point. Moreover, we have also found an unstable fixed point which represents the transition out of fermi liquid. It is also known as the Pomeranchuk instability where the fermi surface is destroyed. This is the first time that such an instability is uncovered in the framework of RG. The RG flow for BCS instability induced by nearly backward scattering is also studied. Our work provide a more careful and complete study for the low energy physics of the fermi liquid. It opens a new route to the system with extended zero energy phase space characterized by an intrinsic scale by using the concept and technique of RG.

# Renormalization group for metals

Host: Hae-Young Kee