Magnetization Dynamics and Peierls Instability in Topological Josephson Structures

We study a long topological Josephson junction with a ferromagnetic stripe between the superconductors. The low-energy theory exhibits a non-local in time and space interaction between chiral Majorana fermions, mediated by the magnonic excitations in the ferromagnet. The spontaneous breaking of a Z2-symmetry at the mean-field level leads to a tilting of the magnetization and the opening of a fermionic gap (Majorana mass). This is equivalent to the Peierls instability in the commensurate Fröhlich model. Within a Gaussian fluctuation analysis, we identify critical values for the temporal and spatial non-locality of the interaction, beyond which the symmetry breaking is stable at zero temperature – despite the effective one-dimensionality of the model. We conclude that non-locality, i.e., the stiffness of the magnetization in space and time, stabilizes the symmetry breaking. In the stabilized regime, we expect the current-phase relation to exhibit an experimen- tally accessible discontinuous jump.
At nonzero temperatures, as usual in the 1D Ising model, the long-range order is destroyed by solitonic excitations, which in our case carry each a Majorana zero mode.