Carrier doping by the electric field effect has emerged recently as an ideal route for monitoring many-body physics in two-dimensional (2D) materials where the Fermi level is tuned in a way that -indirectly-the strength of the interactions can also be scanned 1, 2 . The possibility of systematic doping in combination with high resolution photoemission has allowed to uncover a genuinely many-body electron spectrum in single-layer MoS 2 transition metal dichalcogenide, resolving three clear quasi-particle states, where only one state should be expected if the electron-phonon interaction vanished 3 . Our analysis combines first-principles and consistent complex plane analytic approaches and brings into light the presence and the physical origin of two gaps and the three quasi-particle bands which are unambiguously present in the photoemission spectrum. One of these states, though being strongly interacting with the accompanying virtual phonon cloud, presents a notably long lifetime which is an appealing property when trying to understand and take advantage of many-body interactions to modulate the transport properties 4-7 .The effective velocity and the lifetime of electron states close to the Fermi level determine most of the transport properties of metals, and the interactions with collective excitactions e.g. phonons, magnons or plasmons, are responsible for modifying or renormalizing these properties 8 .Specifically, phonons are the low energy excitations that more strongly couple to electron states in normal metals 9 . The interaction of electrons and phonons has a many-body character primarily because the Pauli exclusion principle prohibits the scattering to occupied states and because quantum mechanics allows the virtual excitation of phonons even in the absence of available energy for low energy electrons. All this physics is already contained in the most drastically simplified Einstein model, where one single optical phonon mode with energy ω 0 interacts with a single electron band with a parabolic dispersion in absence of coupling (Fig. 1a) 10 . The many-body coupling divides the spectrum in two regions, below and above ω 0 . On the one hand, for energies below ω 0 , elec-1 arXiv:1905.05168v1 [cond-mat.mes-hall]

Ab initio calculations of the phonon-induced band structure renormalization are currently based on the perturbative Allen-Heine theory and its many-body generalizations. These approaches are unsuitable to describe materials where electrons form localized polarons. Here, we develop a selfconsistent, many-body Green's function theory of band structure renormalization that incorporates localization and self-trapping. We show that the present approach reduces to the Allen-Heine theory in the weak-coupling limit, and to total energy calculations of self-trapped polarons in the strongcoupling limit. To demonstrate this methodology, we reproduce the path-integral results of Feynman and diagrammatic Monte Carlo calculations for the Fröhlich model at all couplings, and we calculate the zero point renormalization of the band gap of an ionic insulator including polaronic effects.

We present a comprehensive first-principles analysis of the non-adiabatic effects due to the electron-phonon interaction on the vibrational spectrum of the electron-doped monolayer MoS 2 . Deep changes in the Fermi surface upon doping cause the linewidth broadening of the normal modes governing the spin-conserving intervalley electronic scattering, which become unstable with the population of all the spin-split conduction valleys. We find that the non-adiabatic spectral effects modify dramatically the adiabatic dispersion of the longwavelength optical phonon modes, responsible for intra-valley scattering, as soon as inequivalent valleys get populated. These results are illustrated by means of a simple analytical model. Finally, we explain the emergence of an intricate dynamical structure for the strongly interacting out-of-plane polarized A 1 optical vibrational mode spectrum by means of a multiple-phonon quasi-particle picture defined in the full complex frequency plane, showing that this intriguing spectral structure originates from the splitting of the original adiabatic branch induced by the electron-phonon coupling. arXiv:1911.00311v2 [cond-mat.mtrl-sci]