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 νG = −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 νG = 0. They are ferromagnetic, antiferromagnetic, charge-density wave and Kékulé 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 νG = 0 case. For filling factor νG = −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 νG = 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 twocomponent states at ν = −2 + 4/3 and ν = −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.

Theory of Competing Charge Density Wave, Kekule and Antiferromagnetic ordered Fractional Quantum Hall states in Graphene aligned with Boron Nitride

“…We would like to also mention in passing, that while the above framework captures a large variety of multicomponent states, other multi-component states that deviate from this picture, and that to this date have not been understood theoretically, have been found in numerical studies 63,64 .…”