Optical sum increase due to electron undressing

The total optical sum is a constant, independent of many-body interactions, of impurity scattering and of temperature. For a single band in a metal, such a sum rule is no longer independent of the interactions or temperature, when the dispersion and/or finite bandwidth is accounted for. We adopt such a model, with electrons coupled to a single Einstein oscillator of frequency omega(E) and study the optical spectral weight. The optical sum depends on both the strength of the coupling and on the characteristic phonon frequency, omega(E). A hardening of omega(E) leads to electron undressing and translates into a decrease in the electron kinetic energy and an increase in the total optical sum. Our model produces a temperature dependence in this sum as seen in recent experiments.