Symmetry-determined generalized ferromagnetism in multi-valley electron fluids

Abstract

Quantum electronic fluids with spin and valley degrees of freedom have a correlation driven tendency to flavor polarization (generalized ferromagnetism). To first order in the long-range Coulomb interactions -- i.e. in the Hartree-Fock approximation -- spin and valley polarization exhibit a spurious degeneracy. We show that to second order -- or more generally in the random-phase approximation -- this degeneracy is lifted in a way that depends only on the underlying symmetry relating the two valleys. In two spatial dimensions, if the valleys are related by an $n-$fold rotation ($n>2$) or by mirror reflection and each valley is invariant under $C_2$ or time reversal (as is the case in AlAs quantum wells) then valley polarization is preferred. If the valleys are related by time reversal or by $C_2$ rotation symmetry (as in multilayer graphene systems) then spin order is selected.