Magnetization dynamics and Peierls instability in topological Josephson structures

We study a long topological Josephson junction with a ferromagnetic strip between two superconductors. The low-energy theory exhibits a nonlocal in time and space interaction between chiral Majorana fermions, mediated by the magnonic excitations in the ferromagnet. We show that sufficiently strong and long-ranged interactions may induce a Z₂-symmetry breaking, leading to a tilting of the magnetization and the opening of a fermionic gap, analogous to the Peierls instability in the commensurate Fröhlich model. Within a Gaussian fluctuation analysis, we estimate critical values for the temporal and spatial nonlocality of the interaction, beyond which the symmetry breaking is stable at zero temperature. We compare and confirm these results with a renormalization group analysis of closely related models and conclude that nonlocality, 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 experimentally accessible discontinuous jump. At nonzero temperatures, the long-range order is destroyed by solitonic excitations, which in our case carry each a Majorana zero mode.