We study an almost planar membrane without overhangs which are described by a single-valued function h(x, y). The related Hamiltonian is given by the Helfrich model characterized by the bending modulus κ, the surface tension σ and a parameter μ. Furthermore the membrane surface is occupied by magnetic particles the spin orientation of which is indicated by the locally changing unit vector n. However due to presence of the membrane the spin orientation depends additionally on the local bending of the membrane leading to n(h). Assuming that the interaction of the localized spins is carried out by the mesoscopic Heisenberg model we demonstrate that the magnetic interaction is given by the non-linear sigma model-well established in magnetism-but the exchange coupling is strongly modified by the correlation of the height function h. As result the magnetic coupling depends on the bending strength κ and other parameters as σ and μ. In case of a strong bending of the membrane the effective exchange coupling is strongly reduced whereas in the opposite case of low bending force the effective coupling can be slightly enlarged. The effective model is characterized by a length scale ξ= (κ/μ)^1/4 . Additionally we discuss the influence of a magnetic field and the occurrence of inhomogeneous magnetic configurations on the membrane surface.

We study an almost planar membrane without overhangs which are described by a single-valued function h(x, y). The related Hamiltonian is given by the Helfrich model characterized by the bending modulus κ, the surface tension σ and a parameter μ. Furthermore the membrane surface is occupied by magnetic particles the spin orientation of which is indicated by the locally changing unit vector n. However due to presence of the membrane the spin orientation depends additionally on the local bending of the membrane leading to n(h). Assuming that the interaction of the localized spins is carried out by the mesoscopic Heisenberg model we demonstrate that the magnetic interaction is given by the non-linear sigma model-well established in magnetism-but the exchange coupling is strongly modified by the correlation of the height function h. As result the magnetic coupling depends on the bending strength κ and other parameters as σ and μ. In case of a strong bending of the membrane the effective exchange coupling is strongly reduced whereas in the opposite case of low bending force the effective coupling can be slightly enlarged. The effective model is characterized by a length scale ξ= (κ/μ)^1/4 . Additionally we discuss the influence of a magnetic field and the occurrence of inhomogeneous magnetic configurations on the membrane surface.