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Topological electrostatics

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Moessner,  Roderich
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Doucot, B., Moessner, R., & Kovrizhin, D. L. (2023). Topological electrostatics. Journal of Physics: Condensed Matter, 35(7): 074001. doi:10.1088/1361-648X/ac9443.


Cite as: https://hdl.handle.net/21.11116/0000-000D-02CC-B
Abstract
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 nu > 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 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 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.