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 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 spin textures called Skyrmions. We have generalized this notion to an arbitrary
number d of internal states for the electrons, which may correspond to the combination
of spin, valley, or layer indices.
The first step is to associate a many electron wave-function, projected on the
lowest Landau level, to any classical spin texture described by a smooth map from the plane
to the projective space CP(d-1). In the large magnetic field limit, we assume that the spin texture
is slowly varying on the scale of the magnetic length, which allows us
to evaluate the expectation value of the interaction Hamiltonian on these many electron quantum states.
The first non trivial term in this semi-classical expansion is the usual CP(d-1) non-linear sigma model,
which is known to exhibit a remarkable degeneracy of the many electron states obtained from holomorphic
textures. Surprisingly, this degeneracy is not lifted by reintroducing quantum fluctuations.
It is eventually lifted by the sub-leading term in the effective Hamiltonian, which
selects a hexagonal Skyrmion lattice and therefore breaks both translational and internal SU(d) symmetries.
I will show that these optimal classical textures can be interpreted in
an appealing way using geometric quantization.