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

TL;DR

This paper investigates how magnetic fields interact with surface curvature in Heisenberg spin models, revealing a tunable double sine-Gordon equation and skyrmion solutions on curved surfaces with potential applications in condensed matter physics.

Contribution

It demonstrates the coupling between magnetic fields and surface curvature leading to a tunable DSG equation and predicts surface deformations and skyrmions on curved geometries.

Findings

Double sine-Gordon equation arises when magnetic field matches surface curvature.

Skyrmion solutions are found on curved surfaces.

Surface deformations occur where spins oppose the magnetic field.

Abstract

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 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. The coupling between a magnetic field and the curvature of the substract is an interesting result and we believe that this issue may be relevant to be applied in condensed matter systems, e.g., superconductors, nematic liquid crystals, graphene and topological insulators.