Two-dimensional electron gases under a strong magnetic field have tremendously expanded our understanding of many-body physics, with the discovery of integer and fractional quantum Hall effects, together with chiral edge states, fractional excitations, anyons. Another striking effect is the strong coupling between charge and spin/valley degrees of freedom, which takes place near integer filling M of the magnetic Landau levels. More precisely, because of the large energy gap associated to cyclotron motion, any slow spatial variation of the spin background induces a variation of the electronic density proportional to the topological density of the spin background. Minimizing Coulomb energy leads to an exotic class of two-dimensional crystals, which exhibit a periodic non-collinear spin texture called a Skyrmion lattice.
I will focus on the limit where a perfect SU(N) symmetry is assumed to hold in the space of internal electronic states. In this case, minimal energy Skyrmion lattices may be described in terms of holomorphic maps from a torus to the Grassmannian manifold Gr(M,N), such that the associated topological charge density is as uniform as possible. Such maps can be constructed by choosing N linearly independent sections of a rank M holomorphic vector bundle on a torus. The main outcome of such analysis is the existence of two regimes depending on whether the topological charge on the torus is smaller (unfrustrated case) or larger (frustrated case) than the number of internal states N accessible to electrons. In the former case, minimal energy Skyrmion lattices for M = 1 are obtained by choosing an orthonormal basis of sections of the associated line bundle with respect to the usual geometric quantization hermitian product. In the general M > 1 case, I will show 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.