Intervalley coherence and flavor polarization in three-valley moiré systems

Abstract: We investigate interaction-induced symmetry breaking in moiré superlattices created by twisting two identical materials where the electronic low-energy degrees of freedom reside in the vicinity of the $M$ points. Based on general symmetry arguments, we identify and classify the possible candidate instabilities that, besides flavor polarized states, also involve a variety of intervalley-coherent (IVC) orders. This complexity is related primarily to the presence of three valleys, instead of the well-studied scenario of two, e.g., in graphene: IVC states can couple all three valleys identically, with a non-trivial sign structure, or even with different magnitudes. We study the energetics using an analytical strong-coupling framework and unrestricted Hartree-Fock applied to the full continuum model, with very good agreement between the two approaches. Interestingly, depending on stacking, IVC instabilities not only appear due to superexchange at moderate bandwidths, but also deep in the strong-coupling regime as a result of deviations from the flat-metric condition. Our work demonstrates that twisted $M$-point materials provide a rich playground for complex correlated physics and highlights differences and similarities to twisted multilayer graphene.