The engineering of Majorana zero modes in topological superconductors, a new paradigm for the realization of topological quantum computing and topology-based devices, has been hampered by the absence of materials with sufficiently large superconducting gaps. Recent experiments, however, have provided enthralling evidence for the existence of topological surface superconductivity in the iron-based superconductor FeSe$_{0.45}$Te$_{0.55}$ possessing a full $s_pm$-wave gap of a few meV. Here, we propose a mechanism for the emergence of topological superconductivity on the surface of FeSe$_{0.45}$Te$_{0.55}$ by demonstrating that the interplay between the $s_pm$-wave symmetry of the superconducting gap, recently observed surface magnetism, and a Rashba spin-orbit interaction gives rise to several topological superconducting phases. Moreover, the proposed mechanism explains a series of experimentally observed hallmarks of topological superconductivity, such as the emergence of Majorana zero modes in the center of vortex cores and at the end of line defects, as well as of chiral Majorana edge modes along certain types of domain walls. We also propose that the spatial distribution of supercurrents near a domain wall is a characteristic signature measurable via a scanning superconducting quantum interference device that can distinguish between chiral Majorana edge modes and trivial in-gap states.
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