Polarization fluctuations in insulators and metals: new and old theories merge

The ground-state fluctuation of polarization P is finite in insulators and divergent in metals, owing to the SWM sum rule [I. Souza, T. Wilkens, and R. M. Martin, Phys. Rev. B 62, 1666 (2000)]. This is a virtue of periodic (i.e., transverse) boundary conditions. I show that within any other boundary conditions the P fluctuation is finite even in metals, and a generalized sum rule applies. The boundary-condition dependence is a pure correlation effect, not present at the independent-particle level. In the longitudinal case inverted triangle x P = -rho, and one equivalently addresses charge fluctuations: the generalized sum rule reduces then to a well-known result of the many-body theory.