We study the interplay between electron-electron interaction and electron-phonon coupling in a two-orbital Hubbard model [1] which can be seen as an idealized description of transition-metal oxides, molecular crystals like the fullerides [2] , and twisted bilayer systems [3].We demonstrate that the e-ph interaction coexists with the Mott localization driven by the Hubbard repulsion U, but it competes with the Hund's exchange coupling. For a half-filled system, this interplay leads to two spectacularly different Mott insulators, a standard high-spin Mott insulator with frozen phonons which is stable when the Hund's coupling prevails, and a low-spin insulator favored by phonons, where the characteristic features of Mott insulators and bipolarons coexist. The two phases are separated by a sharp boundary along which an intriguing intermediate solution emerges as a kind of compromise between the two insulators. We discuss the relevance of these results to different materials and their stability to doping.[1] A. Scazzola, A. Amaricci, and M. Capone, Phys. Rev. B 107, 085131 (2023)[2] Y. Nomura, S. Sakai, M. Capone and R. Arita, Science Advances 1 (7), e1500568 (2015)[3] A. Blason and M. Fabrizio, Phys. Rev. B 106, 235112 (2022)
