Spin and valley ordering of fractional quantum Hall states in monolayer graphene

Abstract: We study spin and valley ordering in the quantum Hall fractions in monolayer graphene at Landau level filling factors $\nu_G=-2+n/3$ $(n=2,4,5)$. We use exact diagonalizations on the spherical as well as toroidal geometry by taking into account the effect of realistic anisotropies that break the spin/valley symmetry of the pure Coulomb interaction. We also use a variational method based on eigenstates of the fully $SU(4)$ symmetric limit. For all the fractions we study there are two-component states for which the competing phases are generalizations of those occurring at neutrality $\nu_G=0$. They are ferromagnetic, antiferromagnetic, charge-density wave and K\'ekul\'e phases, depending on the values of Ising or XY anisotropies in valley space. The varying spin-valley content of the states leads to ground state quantum numbers that are different from the $\nu_G=0$ case. For filling factor $\nu_G=-2+5/3$ there is a parent state in the $SU(4)$ limit which has a flavor content $(1,1/3,1/3,0)$ where the two components that are one-third filled form a two-component singlet. The addition of anisotropies leads to the formation of new states that have no counterpart at $\nu_G=0$. While some of them are predicted by the variational approach, we find notably that negative Ising-like valley anisotropy leads to the formation of a state which is a singlet in both spin and valley space and lies beyond the reach of the variational method. Also fully spin polarized two-component states at $\nu=-2+4/3$ and $\nu=-2+5/3$ display an emergent $SU(2)$ valley symmetry because they do not feel point-contact anisotropies. We discuss implications for current experiments concerning possible spin transitions.