Intervalley coherence and intrinsic spin-orbit coupling in rhombohedral trilayer graphene

Rhombohedral graphene multilayers provide a clean and highly reproducible platform to explore the emergence of superconductivity and magnetism in a strongly interacting electron system. Here we reveal a subtle competition between valley-imbalanced orbital ferromagnets and intervalley-coherent states in which electron wavefunctions in the two momentum-space valleys develop a macroscopically coherent relative phase. We focus on a rhombohedral trilayer in the quarter-metal regime-where there is a single Fermi surface that spontaneously breaks spin and valley-isospin symmetry-and employ local magnetometry and global charge sensing techniques. Comparing the in-plane spin susceptibility of the intervalley-coherent and valley-imbalanced phases reveals the influence of graphene's intrinsic spin-orbit coupling, which drives the emergence of a distinct correlated phase that is their hybrid. Spin-orbit coupling also suppresses the in-plane magnetic susceptibility of the valley-imbalanced phase, allowing us to extract the spin-orbit-coupling strength of approximately 50 mu eV for our hexagonal-boron-nitride-encapsulated graphene system. We discuss the implications of a finite spin-orbit coupling on the spin-triplet superconductors observed in both rhombohedral and twisted graphene multilayers. The role of electron-electron interactions in generating superconductivity in few-layer graphene remains controversial. Now, the observation of interaction-driven intervalley coherence may help to explain the Cooper pairing mechanism.