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

We study the joint dynamics of the phase bias $\ensuremath{\phi}$ 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 $\ensuremath{\phi}$-dependent mass. We analyze the effect of the phase fluctuations in the junction, as well as solitonic solutions of $\ensuremath{\phi}$ 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.