Topological electrostatics

Abstract

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the Grassmannian 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 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 above which there are no optimal textures. Below 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 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 , appropriate for recent experiments in graphene.