Landau level mixing effect on valley anisotropic interactions in the zeroth Landau level of graphene

Recent experiments (X. Liu et al.,Science,375(6578); H. Zhou et al.,PRX,12,021060) have revealed intriguing and complicated nature of the charge neutral monolayer graphene in the strong magnetic field. There is also evidence that the ground state ordering could depend on the dielectric environment of monolayer graphene (A. Coissard, Nature,605(7908)). Motivated by these observations, we study, using the second-order perturbation theory, how the long-range valley-independent Coulomb interaction renormalizes the lattice-scale valley-anisotropic interactions between two electrons in the zeroth Landau level. We show that the Landau level mixing can change the relative strength of different valley-anisotropic interaction channels and tip the balance between competing orders. We also study how this effect can be tuned by the dielectric screening.