Coupling between magnetic field and curvature in Heisenberg spins on surfaces with rotational symmetry

We study the nonlinear σ-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 2π skyrmion appears like a solution for this model 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 by applying the model on three rotationally symmetric surfaces: the cylinder, the catenoid and the hyperboloid.

► Coupling between magnetic field and curvature in Heisenberg spins on curved surfaces.
► Obtaining of the homogeneous double sine-Gordon system if field is curvature tuned.
► Obtaining of a 2π skyrmion on an arbitrarily curved surface.
► Surface deformations are predicted from this tuning between field and curvature.
► Calculations of the energy and characteristic length of skyrmions on curved surfaces.