Competing topological superconducting phases in FeSe$_{0.45}$Te$_{0.55}$

We demonstrate that recent angle-resolved photo-emission spectroscopy experiments provide strong evidence for the existence of two competing topological superconducting phases in FeSe$_{0.45}$Te$_{0.55}$. The coupling of their underlying microscopic mechanisms -- one based on a three-dimensional topological insulator, one based on two-dimensional superconductivity -- stabilizes topological superconductivity over a wide range of parameters, and gives rise to two disjoint topological regions in the phase diagram of FeSe$_{0.45}$Te$_{0.55}$. We show that the topological origin of these regions can be identified by considering the form of Majorana edge modes at domain walls.