First-Matsubara-frequency rule in a Fermi liquid. 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) proportional to Omega(2) + 4 pi T-2(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) proportional to Omega(2) + b pi T-2(2), with 1 <= b < infinity. For purely elastic scattering, b = 1. Our analysis implies that the value of b approximate to 1, reported for URu2Si2 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 approximate to 2.3 and 5.6, reported for underdoped cuprates and organics, correspondingly, imply that both elastic and inelastic mechanisms contribute to the optical conductivity.