Abstract

We study one-dimensional Majorana modes in topological Josephson junctions, which are controlled by the magnetization direction of a ferromagnet deposited in the junction. Keeping in mind that the magnetization is not perfectly rigid but exhibits dynamics governed by the Landau-Lifshitz-Gilbert equation, we derive a low-energy effective theory describing its coupling to the Majorana quasiparticles, incorporating the magnetic degrees of freedom on equal footing with the electronic ones. Within mean-field theory, an instability of the magnetic easy axis is identified, leading to a BCS-like self-consistency problem. We present an approach to finding its solitonic solutions, which carry Majorana zero modes, and estimate the corresponding energies and correlation lengths. We argue qualitatively that these solitons are responsible for the destruction of the long-range order similar to domain walls in the Ising model.