Swapping exchange and spin-orbit induced correlated phases in proximitized Bernal bilayer graphene

Ex-so-tic van der Waals heterostructures take advantage of the electrically tunable layer polarization to swap proximity exchange and spin-orbit coupling in the electronically active region. Perhaps the simplest example is Bernal bilayer graphene (BBG) encapsulated by a layered magnet from one side and a strong spin-orbit material from the other. Taking WS$_2$/BBG/Cr$_2$Ge$_2$Te$_6$ as a representative ex-so-tronic device, we employ realistic \emph{ab initio}-inspired Hamiltonians and effective electron-electron interactions to investigate the emergence of correlated phases within the random phase approximation. We find that for a given doping level, exchange and spin-orbit coupling induced Stoner and intervalley coherence instabilities can be swapped, allowing to explore the full spectrum of correlated phases within a single device.