Conductivity close to antiferromagnetic criticality

We study the conductivity of a 3D disordered metal close to the antiferromagnetic instability within the framework of the spin-fermion model using the diagrammatic technique. We calculate the interaction correction $\delta\sigma(\omega,T)$ to the conductivity, assuming that the latter is dominated by the disorder scattering, and the interaction is weak. Although the fermionic scattering rate shows critical behaviour on the entire Fermi surface, the interaction correction is dominated by the processes near the hot spots, narrow regions of the Fermi-surface corresponding to the strongest spin-fermion scattering. Exactly at the critical point $\delta\sigma\propto[max(\omega, T)]^{3/2}$. At sufficiently large frequencies $\omega$ the conductivity is independent of the temperature, and $\delta\sigma\propto(\tau^{-1}-i\omega)^{-2}$, $\tau$ being the elastic scattering time. In a certain intermediate frequency range $\delta\sigma(\omega)\propto i\omega(\tau^{-1}-i\omega)^{-2}$.