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Controlling Majorana modes by $p$-wave pairing in two-dimensional $p+id$ topological superconductors

We show that corner Majorana zero modes in a two-dimensional $p+id$ topological superconductor can be controlled by the manipulation of the parent $p-$wave superconducting order. Assuming that the $p$-wave superconducting order is in either a chiral or helical phase, we find that when a $d_{x^2-y^2}$ wave superconducting order is induced, the system exhibits quite different behavior depending on the nature of the parent $p$-wave phase. In particular, we find that while in the helical phase, a localized Majorana mode appears at each of the four corners, in the chiral phase, it is localized only along two of the four edges. We furthermore demonstrate that the Majoranas can be directly controlled by the form of the edges, as we explicitly show in case of the circular edges. We argue that the application of strain may provide additional means of fine-tuning the Majorana zero modes in the system, in particular, it can partially gap them out. Our findings may be relevant for probing the topology in two-dimensional mixed-pairing superconductors.