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The question of conductivity is revisited. Using the total momentum shiftoperator to construct the perturbed many-body Hamiltonian and ground state wavefunction the second derivative of the ground state energy with respect to theperturbing field is expressed in terms of the one and two-body momentumdensities. The distinction between the adiabatic and envelope functionderivatives, hence that between the Drude and superfluid weights can beintroduced in a straightforward manner. It is shown that a discontinuity in themomentum density leads to a contribution to the Drude weight, but not thesuperfluid weight, however a $\delta$-function contribution in the two-bodymomentum density (such as in the BCS wave-funtion) contributes to bothquantities. The connection between the discontinuity in the momentum densityand localization is also demonstrated.

Drude and Superfluid Weights in Extended Systems: the Role of Discontinuities and $\delta$-peaks in the One and Two-Body Momentum Densities