Magnetochiral symmetry breaking in a M\"obius ring

We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in 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 ferromagnetic M\"obius ring, where a topologically induced domain wall appears as a ground state in case of strong easy-normal anisotropy. For the M\"obius 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 M\"obius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the M\"obius ring, respectively. In both cases the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the M\"obius ring on its geometrical parameters and on the value of the easy-normal anisotropy is explored numerically.