Topological Electrostatics

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the Grassmannian Gr(M,N) manifold. These textures describe skyrmion lattices of N-component fermions in a quantising magnetic field, relevant to the physics of graphene, bilayer and other multicomponent quantum Hall systems near integer filling factors ν>1. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma model Gr(M,N) on a sphere and a torus, together with counting arguments which show that for any filling factor and number of components there is a critical value of topological charge dc above which there are no optimal textures. Below dc a solution of the optimality condition on a torus is unique, while in the case of a sphere one has, in general, a continuum of solutions corresponding to new {\it non-Goldstone} zero modes, whose degeneracy is not lifted (via a order from disorder mechanism) by any fermion interactions depending only on the distance on a sphere. We supplement our general theoretical considerations with the exact analytical results for the case of Gr(2,4), appropriate for recent experiments in graphene.