Abstract
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.
Highlights
Majorana zero modes (MZMs) represent a hallmark feature of topological superconductivity with several interesting properties [1–8]
For the parent helical p-wave order, we found that the effect of the d-wave order parameter is to gap out the edge states, leaving only zero-energy Majorana corner modes
In the case of the parent chiral p-wave pairing, we showed that the edge states can be partially gapped out above a critical value of the d-wave order parameter
Summary
Majorana zero modes (MZMs) represent a hallmark feature of topological superconductivity with several interesting properties [1–8]. Superconducting bulk gap, the corner zero modes can be obtained by gapping out the first-order edge states with a mass term that features a domain wall in momentum space, realizing a special case of the hierarchy of higher-order topological states [31,69]. In this respect, several concrete ways to realize corner MZMs in two dimensions have been proposed so far, for instance, by inducing a superconducting gap for the edge states of a topological insulator, either intrinsically [55] or via the proximity effect [35,38,40,56,58–60]. We assume that the p-wave superconducting order, in the absence of any other pairing, hosts a first-order topological state and exists in either a chiral or helical phase, referring to whether it breaks or preserves time-reversal symmetry, respectively. We note that both of the gapped regions d < p and d > p are topologically nontrivial and feature gapless edge states, which will be evident in the following
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