Topological Electrostatics

TL;DR

This paper develops a theory of optimal topological textures in Grassmannian sigma-models, describing skyrmion lattices in multicomponent quantum Hall systems, and provides analytical solutions relevant to graphene experiments.

Contribution

It introduces a new optimality condition for topological textures in Grassmannian sigma-models and analyzes their solutions on different geometries, including exact results for $ ext{Gr}(2,4)$.

Findings

Existence of a critical topological charge $d_c$ beyond which no optimal textures exist.

Unique solutions on a torus versus a continuum of solutions on a sphere.

Analytical results for $ ext{Gr}(2,4)$ relevant to graphene experiments.

Abstract

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the Grassmannian $\mathrm{Gr}(M,N)$ 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 $\nu>1$. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma model $\mathrm{Gr}(M,N)$ 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 $d_c$ above which there are no optimal textures. Below $d_c$ 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 $\mathrm{Gr}(2,4)$, appropriate for recent experiments in graphene.