Electrical conductivity of Fermi liquids. I. Many-body effect on the Drude weight

On the basis of the Fermi liquid theory, we investigate the many-body effect on the Drude weight. In a lattice system the Drude weight D is modified hy electron-electron interaction, while it is not renormalized in a Galilean invariant system. This is explained by showing that the effective mass m' for D proportional to n/m' is defined through the current, not velocity, of quasiparticle. It is shown that the inequality D > 0 is required for the stability against the uniform shift of the Fermi surface. The result of perturbation theory applied for the Hubbard model indicates that D as a function of the density n is qualitatively modified around half filling n similar to 1 by Umklapp processes.