Optical sum increase due to electron undressing
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For a system with a fixed number of electrons, 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 $\omega_{E}$, due, for example, to a phase transition, leads to
electron undressing and translates into a decrease in the electron kinetic
energy and an increase in the total optical sum, as observed in recent
experiments in the cuprate superconductors.