Emergence of multiple zero modes bound to vortices in extended topological Josephson junctions

We study planar Josephson junctions formed on the surface of a three-dimensional topological insulator (Fu-Kane proposal). We examine the experimentally relevant parameter regimes in which the frequently used effective description in terms of two counter-propagating one-dimensional Majorana modes with hybridization dependent on the Josephson phase difference reaches its validity limit. This happens when the effective velocity of the emergent one-dimensional Majorana modes approaches zero. As parameters like the chemical potential or the width of the junction are tuned, instances of vanishing effective velocity mark the emergence of additional `Dirac cones' at zero energy and finite momentum. If the junction is subjected to an external magnetic field, Josephson vortices may then bind a number of zero modes in addition to the topological Majorana mode. The additional zero modes are `symmetry-protected' and can be lifted by a broken mirror symmetry (which is to be expected in realistic scenarios) as well as by an in-plane magnetization (or Zeeman field). We note that the ensuing presence of additional low-energy Andreev states can significantly contribute to measured quantities like the Josephson current or microwave absorption spectra.