Ordering in the SU(4)-symmetric model of AA bilayer graphene

We examine possible ordered states of AA-stacked bilayer graphene arising due to electron-electron coupling. We show that under certain assumptions the Hamiltonian of the system possesses an SU(4) symmetry. The multicomponent order parameter is described by a $4\ifmmode\times\else\texttimes\fi{}4$ matrix $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$, for which a mean-field self-consistency equation is derived. This equation allows Hermitian and non-Hermitian solutions. Hermitian solutions can be grouped into three topologically distinct classes. First class corresponds to the charge density wave. Second class includes spin density wave, valley density wave, and spin-valley density wave. An ordered state in the third class is a combination of all the aforementioned density-wave types. For anti-Hermitian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ the ordered states are characterized by spontaneous interlayer loop currents flowing in the bilayer. Depending on the topological class of the solution these currents can carry charge, spin, valley, and spin-valley quanta. We also discuss the special case when matrix $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ is not Hermitian and not anti-Hermitian. Utility and weak points of the proposed SU(4)-based classification scheme of the ordered states are analyzed.