Ferromagnetism and superconductivity in twisted double bilayer graphene

We present a theory of competing ferromagnetic and superconducting orders in twisted double bilayer graphene (TDBG). In our theory, ferromagnetism is induced by Coulomb repulsion, while superconductivity with intervalley equal-spin pairing can be mediated by electron-acoustic phonon interactions. We calculate the transition temperatures for ferromagnetism and superconductivity as a function of moir\'e band filling factor, and find that superconducting domes can appear on both the electron and hole sides of the ferromagnetic insulator at half filling. We show that the ferromagnetic insulating gap has a dome shape dependence on the layer potential difference, which provides an explanation to the experimental observation that the ferromagnetic insulator only develops over a finite range of external displacement field. We also verify the stability of the half-filled ferromagnetic insulator against two types of collective excitations, i.e., spin magnons and valley magnons.