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In polar semiconductors and insulators, the Fr\"ohlich interaction betweenelectrons and long-wavelength longitudinal optical phonons induces a many-bodyrenormalization of the carrier effective masses and the appearence ofcharacteristic phonon sidebands in the spectral function, commonly dubbed'polaron satellites'. The simplest model that captures these effects is theFr\"ohlich model, whereby electrons in a parabolic band interact with adispersionless longitudinal optical phonon. The Fr\"ohlich model has beenemployed in a number of seminal papers, from early perturbation-theoryapproaches to modern diagrammatic Monte Carlo calculations. One limitation ofthis model is that it focuses on undoped systems, thus ignoring carrierscreening and Pauli blocking effects that are present in real experiments ondoped samples. To overcome this limitation, we here extend the Fr\"ohlich modelto the case of doped systems, and we provide exact solutions for the electronspectral function, mass enhancement, and polaron satellites. We perform theanalysis using two approaches, namely Dyson's equation with the Fan-Migdalself-energy, and the second-order cumulant expansion. We find that these twoapproaches provide qualitatively different results. In particular, the Dyson'sapproach yields better quasiparticle masses and worse satellites, while thecumulant approach provides better satellite structures, at the price of worsequasiparticle masses. Both approaches yield an anomalous enhancement of theelectron effective mass at finite doping levels, which in turn leads to abreakdown of the quasiparticle picture in a significant portion of the phasediagram.

Many-body Green's function approaches to the doped Fr\"ohlich solid: Exact solutions and anomalous mass enhancement