Fermi liquid near a q=0 charge quantum critical point

We analyze the quasiparticle interaction function (the fully dressed and antisymmetrized interaction between fermions) for a two-dimensional Fermi liquid at zero temperature close to a q=0 charge quantum critical point (QCP) in the s-wave channel (the one leading to phase separation). By the Ward identities, this vertex function must be related to quasiparticle residue Z, which can be obtained independently from the fermionic self-energy. We show that to satisfy these Ward identities, one needs to go beyond the standard diagrammatic formulation of Fermi-liquid theory and include a series of additional contributions to the vertex function. These contributions are not present in a conventional Fermi liquid, but do emerge near a QCP, where the effective 4-fermion interaction is mediated by a soft dynamical boson. We demonstrate explicitly that including these terms restores the Ward identity. Our analysis is built on previous studies of the vertex function near an antiferromag netic QCP [Phys. Rev. B 89, 045108 (2014)1098-012110.1103/PhysRevB.89.045108] and a d-wave charge-nematic QCP [Phys. Rev. B 81, 045110 (2010)1098-012110.1103/PhysRevB.81.045110]. We show th at for s-wave charge QCP the analysis is more straightforward and allows one to obtain the full quasiparticle interaction function (the Landau function) near a QCP. We show that all partial components of this function (Landau parameters) diverge near a QCP, in the same way as the effective mass m∗, except for the s-wave charge component, which approaches -1. Consequently, the susceptibilities in all channels, except for the critical one, remain finite at a QCP, as they should.