Coupling of Chiralities in Spin and Physical Spaces: The Möbius Ring as a Case Study

We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in the coupling of a topologically nontrivial magnetization pattern and topology of the object. The mechanism of this coupling is explored and illustrated by an example of a ferromagnetic Möbius ring, where a topologically induced domain wall appears as a ground state in the case of strong easy-normal anisotropy. For the Möbius geometry, the curvilinear form of the exchange interaction produces an additional effective Dzyaloshinskii-like term which leads to the coupling of the magnetochirality of the domain wall and chirality of the Möbius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the Möbius ring, respectively. In both cases, the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the Möbius ring on its geometrical parameters and on the value of the easy-normal anisotropy is explored numerically.

Figure 1

Geometrical notations of the problem: the 2D Möbius ring with radius $R$ and width $w$ is defined parametrically by (1) and the corresponding 3D Möbius ring of finite thickness $h$ is determined by (2). Möbius rings with opposite chiralities $\mathcal{C}$ are shown.

Figure 2

Diagram of ground states of magnetic Möbius rings with fixed radius $R=100\mathrm{nm}$ and width $w=80$, see (a). The magnetization distributions of possible ground states are shown in the middle part in a large scale: (b) vortex state (marked by disks on the diagram), (c) state with single transversal Bloch domain wall (open squares), (d) state with three transversal Bloch walls (filled triangles), (e) states with longitudinal domain wall (filled squares). The detailed structures of the transverse and longitudinal domain walls are shown in (${\mathrm{c}}^{\prime }$) and (${\mathrm{e}}^{\prime }$), respectively.