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)] and a d-wave charge-nematic QCP [Phys. Rev. B 81 , 045110 (2010)]. 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 & lowast; , 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.