Binding zero modes with fluxons in Josephson junctions of time-reversal invariant topological superconductors

We study the joint dynamics of the phase bias and the propagating Majorana fermions of the edge modes in Josephson junctions containing two-dimensional time-reversal invariant topological superconductors (TRITOPS). We consider TRITOPS-TRITOPS junctions, as well as junctions between topological and nontopological superconductors (TRITOPS-S). Both types of junctions are described by effective Dirac Hamiltonians with a -dependent mass. We analyze the effect of the phase fluctuations in the junction, as well as solitonic solutions of generated by fluxons trapped in the junction. We show that these solitons generate a spatial-dependent mass with a sign change akin to the Jackiw-Rebbi model. This enables the formation of zero-energy fermionic states localized at the fluxon. For the TRITOPS-TRITOPS junction these consist of a Kramers pair of Majorana modes, while for the TRITOPS-S one, there is a single Majorana fermion. The localized bound states hybridize in soliton-antisoliton configurations. Depending on the occupation state, these modes generate an effective attraction or repulsion in the dynamics of the soliton-antisoliton collision.