Skyrme and Wigner crystals in graphene

At low energy, the band structure of graphene can be approximated by two degenerate valleys (K,K') about which the electronic spectra of the valence and conduction bands have linear dispersion relations. An electronic state in this band spectrum is a linear superposition of states from the A and B sublattices of the honeycomb lattice of graphene. In a quantizing magnetic field, the band spectrum is split into Landau levels with level N=0 having zero weight on the B(A) sublattice for the K(K') valley. Treating the valley index as a pseudospin and assuming the real spins to be fully polarized, we compute the energy of Wigner and Skyrme crystals in the Hartree-Fock approximation. We show that Skyrme crystals have lower energy than Wigner crystals (WCs), i.e., crystals with no pseudospin texture in some range of filling factor v around integer fillings. The collective mode spectrum of the valley-skyrmion crystal has three linearly dispersing Goldstone modes in addition to the usual phonon mode, while a WC has only one extra Goldstone mode with a quadratic dispersion. We comment on how these modes should be affected by disorder and how, in principle, a microwave absorption experiment could distinguish between Wigner and Skyrme crystals.