Graphene exhibits quantum Hall ferromagnetism in which an approximate $SU\left(4\right)$ symmetry involving spin and valley degrees of freedom is spontaneously broken. We construct a set of integer and fractional quantum Hall states that break the $SU\left(4\right)$ spin (valley) symmetry, and study their neutral and charged excitations. Several properties of these ferromagnets can be evaluated analytically in the $SU\left(4\right)$ symmetric limit, including the full collective-mode spectrum at integer fillings. By constructing explicit wave functions we show that the lowest-energy skyrmion states carry charge $\pm 1$ for any integer filling, and that skyrmions are the lowest-energy-charged excitations for graphene Landau-level index $\mid n\mid ⩽3$. We also show that the skyrmion lattice states which occur near integer-filling factors support four gapless collective-mode branches in the presence of full $SU\left(4\right)$ symmetry. Comparisons are made with the more familiar $SU\left(2\right)$ quantum Hall ferromagnets studied previously.

• Received 30 May 2006

DOI:https://doi.org/10.1103/PhysRevB.74.075423