Vagson L. Carvalho-Santos, Rossen Dandoloff
We study the nonlinear $\sigma$-model in an external magnetic field applied on curved surfaces with rotational symmetry. The Euler-Lagrange equations derived from the Hamiltonian yield the double sine-Gordon equation (DSG) provided the magnetic field is tuned with the curvature of the surface. A $2\pi$ skyrmion appears like solution and surface deformations are predicted at the sector where the spins point in the opposite direction to the magnetic field. We also study some specific examples of such rotationally symmetric surfaces, e.g., the cylinder, the catenoid and the hyperboloid. The coupling between a magnetic field and the curvature of the substract is an interesting result and we believe that this issue may be important to be applied in condensed matter systems, e.g., superconductors, nematic liquid crystals, graphene and topological insulators.
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http://arxiv.org/abs/1206.6228
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