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## Skyrmion lattices in quantum Hall ferromagnets

In the presence of a strong magnetic field, and for an integer filling of the Landau levels, Coulomb interactions favor a ferromagnetic ground-state. It has been shown already more than twenty years ago, both theoretically and experimentally, that when extra charges are added or removed to such systems, the ferromagnetic state becomes unstable and is replaced by a Skyrmion crystal. Interest in these systems has been strongly renewed by the discovery of graphene. In this system, Coulomb interactions manifest an approximate SU(4) symmetry for spin and valley degrees of freedom.

We have focussed on the limit where we neglect coupling anisotropies in the N-dimensional spin and valley internal space, so that a perfect SU(N) symmetry is assumed to hold. In this case, minimal energy Skyrmion lattices may be described in terms of holomorphic maps from a torus (unit cell) to the Grassmannian manifold Gr(M,N), such that the associated topological charge density is as uniform as possible. The case of an undoped graphene layer corresponds to N = 4 and M = 2. The main outcome of this analysis is the existence of two regimes depending on whether the topological charge on the unit cell is smaller (unfrustrated case) or larger (frustrated case) than the number of internal states N accessible to electrons. We have shown that we can, to a large extent, identify minimal energy Skyrmion lattices by combining the solution of the M = 1 case with Atiyah’s explicit description of rank M vector bundles on a torus.

Our current work aims at understanding how spin waves are scattered by such Skyrmion lattices. This is motivated by recent experiments on magnon propagation by various groups, and we are beginning a collaboration with the experimental group led by P. Roulleau at CEA Saclay on this subject.