Competing Coulomb and electron-phonon interactions in NbS$_2$

The interplay of Coulomb and electron-phonon interactions with thermal and quantum fluctuations facilitates rich phase diagrams in two-dimensional electron systems. Layered transition metal dichalcogenides hosting charge, excitonic, spin and superconducting order form an epitomic material class in this respect. Theoretical studies of materials like NbS$_2$ have focused on the electron-phonon coupling whereas the Coulomb interaction, particularly strong in the monolayer limit, remained essentially untouched. Here, we analyze the interplay of short- and long-range Coulomb as well as electron-phonon interactions in NbS$_2$ monolayers. The combination of these interactions causes electronic correlations that are fundamentally different to what would be expected from the interaction terms separately. The fully interacting electronic spectral function resembles the non-interacting band structure but with appreciable broadening. An unexpected coexistence of strong charge and spin fluctuations puts NbS$_2$ close to spin and charge order, suggesting monolayer NbS$_2$ as a platform for atomic scale engineering of electronic quantum phases.