First-Matsubara-frequency rule in a Fermi liquid. Part II: Optical conductivity and comparison to experiment

Motivated by recent optical measurements on a number of strongly correlated electron systems, we revisit the dependence of the conductivity of a Fermi liquid, \sigma(\Omega,T), on the frequency \Omega and temperature T. Using the Kubo formalism and taking full account of vertex corrections, we show that the Fermi liquid form Re\sigma^{-1}(\Omega,T)\propto \Omega^2+4\pi^2T^2 holds under very general conditions, namely in any dimensionality above one, for a Fermi surface of an arbitrary shape (but away from nesting and van Hove singularities), and to any order in the electron-electron interaction. We also show that the scaling form of Re\sigma^{-1}(\Omega,T) is determined by the analytic properties of the conductivity along the Matsubara axis. If a system contains not only itinerant electrons but also localized degrees of freedom which scatter electrons elastically, e.g., magnetic moments or resonant levels, the scaling form changes to Re\sigma^{-1}(\Omega,T)\propto \Omega^2+b\pi^2T^2, with 1\leq b<\infty. For purely elastic scattering, b =1. Our analysis implies that the value of b\approx 1, reported for URu_2Si_2 and some rare-earth based doped Mott insulators, indicates that the optical conductivity in these materials is controlled by an elastic scattering mechanism, whereas the values of b\approx 2.3 and b\approx 5.6, reported for underdoped cuprates and organics, correspondingly, imply that both elastic and inelastic mechanisms contribute to the optical conductivity.