Quasiparticle interaction function in a two-dimensional Fermi liquid near an antiferromagnetic critical point

We present the expression for the quasiparticle vertex function Gamma(omega)(K-F, P-F) (proportional to the Landau interaction function) in a 2D Fermi liquid (FL) near an instability towards antiferromagnetism. This function is relevant in many ways in the context of metallic quantum criticality. Previous studies have found that near a quantum critical point, the system enters into a regime in which the fermionic self-energy is large near hot spots on the Fermi surface [points on the Fermi surface connected by the antiferromagnetic ordering vector q(pi) = (pi, pi)] and has much stronger dependence on frequency than on momentum. We show that in this regime, which we termed a critical FL, the conventional random-phase-approximation- (RPA) type approach breaks down, and to properly calculate the vertex function one has to sum up an infinite series of terms which were explicitly excluded in the conventional treatment. Besides, we show that, to properly describe the spin component of Gamma(omega)(K-F, P-F) even in an ordinary FL, one has to add Aslamazov-Larkin (AL) terms to the RPA vertex. We show that the total Gamma(omega)(K-F, P-F) is larger in a critical FL than in an ordinary FL, roughly by an extra power of magnetic correlation length xi, which diverges at the quantum critical point. However, the enhancement of Gamma(omega)(K-F, P-F) is highly nonuniform: It holds only when, for one of the two momentum variables, the distance from a hot spot along the Fermi surface is much larger than for the other one. This fact renders our case different from quantum criticality at small momentum, where the enhancement of Gamma(omega)(K-F, P-F) was found to be homogeneous. We show that the charge and spin components of the total vertex function satisfy the universal relations following from the Ward identities related to the conservation of the particle number and the total spin. We show that in a critical FL, the Ward identity involves Gamma(omega)(K-F, P-F) taken between particles on the FS. We find that the charge and spin components of Gamma(omega)(K-F, P-F) are identical to leading order in the magnetic correlation length. We use our results for Gamma(omega)(K-F, P-F) and for the quasiparticle residue to derive the Landau parameters F-c(l=0) = F-s(l=0), the density of states, and the uniform (q = 0) charge and spin susceptibilities chi(l=0)(c) = chi(l=0)(s). We show that the density of states N-F diverges as log xi; however, F-c,s(l=0) also diverge as log xi, such that the total chi((l=0))(c,s) proportional to N-F/(1 + F-c(l=0)) remain finite at xi = infinity. We show that at weak coupling these susceptibilities are parametrically smaller than for free fermions.