Quasiparticle interaction function in a 2D 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 function) in a 2D Fermi liquid (FL) near a $T=0$ instability towards antiferromagnetism. Previous studies have found that near an instability, the system enters into a critical FL regime, in which the fermionic self-energy is large near hot spots (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 to properly calculate the vertex function in this regime 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 include Aslamazov-Larkin terms. 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 $\xi$. However, the enhancement of $\Gamma ^{\omega }(K_{F},P_{F})$ is highly non-uniform: 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. 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 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 derive the Landau parameters, the density of states $N_F$, and the uniform ($q=0$) charge and spin susceptibilities $\chi ^{l=0}_{c} = \chi^{l=0}_s$. We show that the susceptibilities remain finite at $\xi = \infty$ despite that $N_F$ diverges as $\log \xi$.