Corrections to Migdal's theorem for spectral functions: a cumulant treatment of the time-dependent Green's function.

Gunnarsson O., Meden V., Schönhammer K.

The electron spectral function is calculated for a model including electron-phonon coupling to Einstein phonons. The spectrum is studied as a function of the electronic bandwidth and the energy epsilon k of the level from which the electron is removed. A cumulant expansion is used for the time-dependent Green's function, and the second- and fourth-order cumulants are studied. This approach is demonstrated to give accurate results for an exactly solvable two-level model with two electronic levels coupling to local phonons. For a one-band, infinite, three-dimensional model the cumulant expansion gives one satellite in the large-bandwidth limit. As the bandwidth is reduced, the spectrum calculated with the fourth-order cumulant develops multiple satellites, if epsilon k is close to the Fermi energy EF, and as the bandwidth becomes small, results similar to the two-level model are obtained. If epsilon k is more than a phonon energy below EF, the spectrum instead shows a very broad peak, due to the decay of the hole into a hole closer to EF and a phonon. If the spin degeneracy of the electrons is taken into account, the broadening due to the decay of a hole into a hole closer to EF and an electron-hole pair becomes important, even if epsilon k is closer to EF than the phonon energy. The validity of Migdal's theorem for A3C60 (A=K, Rb) is discussed. The intersubband electron-phonon coupling is appreciable for A3C60, and it may be argued that the effective bandwidth is large. It is shown that Migdal's theorem is, nevertheless, not valid for A3C60.

Physical Review B, 50 10462-73, 1994.

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