Many-Electron Effects in the Optical Conductivity of Simple Metals by Kubo Formula

Abstract

Starting from the Kubo formula for the optical conductivity, we review, reformulate, and generalize our previous one-electron theory of the optical absorption of solid and liquid alkali metals to include many-body effects due to electron-electron Coulomb interactions. The optical matrix element $〈{\psi }_k,\left|\overrightarrow{\nabla }\right|{\psi }_k〉$, which was previously calculated in terms of an "optical pseudopotential," is rederived from a second-order scattering of an electron (quasiparticle in the many-body theory) by the Coulomb field of the ions and the applied photon field. The result is then represented by Feynman graphs similar in the lowest order to the "bremsstrahlung" of quantum electrodynamics, and shown, accordingly, to give rise to four categories of many-electron effects, viz., screening effects (usually incorporated in the one-electron approximation), electron self-energy effects, electron-photon and electron-ion vertex corrections, and final-state interactions. The changes of the one-electron result due to self-energy and to vertex corrections counteract each other in sodium and potassium, leading to no more than 10% net change; accordingly, the only appreciable enhancement comes from final-state interactions involving virtual exchange of plasmons, considered previously by Mahan.