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 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 antiferromagnetic QCP [Phys. Rev. B 89, 045108 (2014)] and a d-wave charge-nematic QCP [Phys. Rev. B 81, 045110 (2010)]. We show that 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.