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The electron-phonon interaction causes thermal and zero-point motion shiftsof electron quasiparticle (QP) energies $\epsilon_k(T)$. Other consequences ofinteractions, visible in angle-resolved photoemission spectroscopy (ARPES)experiments, are broadening of QP peaks and appearance of sidebands, containedin the electron spectral function  $A(k,\omega)=-{\Im m}G_R(k,\omega) /\pi$, where $G_R$ is the retarded Green'sfunction. Electronic structure codes (e.g. using density-functional theory) arenow available that compute the shifts and start to address broadening andsidebands. Here we consider MgO and LiF, and determine their nonadiabaticMigdal self energy. The spectral function obtained from the Dyson equationmakes errors in the weight and energy of the QP peak and the position andweight of the phonon-induced sidebands. Only one phonon satellite appears, withan unphysically large energy difference (larger than the highest phonon energy)with respect to the QP peak. By contrast, the spectral function from a cumulanttreatment of the same self energy is physically better, giving a quite accurateQP energy and several satellites approximately spaced by the LO phonon energy.In particular, the positions of the QP peak and first satellite agree closelywith those found for the Fr\"ohlich Hamiltonian by Mishchenko $\textit{et al.}$(2000) using diagrammatic Monte Carlo. We provide a detailed comparison betweenthe first-principles MgO and LiF results and those of the Fr\"ohlichHamiltonian. Such an analysis applies widely to materials with infra-red activephonons. We also compare the retarded and time-ordered cumulant treatments:they are equivalent for the Fr\"ohlich Hamiltonian, and only slightly differ infirst-principles electron-phonon results for wide-band gap materials.

physical review. b./physical review. b

Quasiparticles and phonon satellites in spectral functions of  semiconductors and insulators: Cumulants applied to full first principles  theory and Fr\"ohlich polaron