Tailoring phase transition from topological superconductor to trivial superconductor induced by magnetic textures of a spin-chain on a $p$-wave superconductor

Abstract: We theoretically investigate the phase transition from a non-trivial topological $p$-wave superconductor to a trivial $s$-wave like superconducting phase through a gapless phase, driven by different magnetic textures as an one-dimensional spin-chain impurity, e.g. Bloch-type, in-plane and out-of-plane N\'eel-type spin-chains etc. In our proposal, the chain of magnetic impurities is placed on a spin-triplet $p$-wave superconductor where we obtain numerically as well as analytically an effective $s$-wave like pairing due to spin rotation, resulting in gradual destruction of the Majorana zero modes present in the topological superconducting phase. In particular, when the impurity spins are antiferromagnetically aligned i.e. spiral wave vector $G_{s}=\pi$, the system becomes an effective $s$-wave superconductor without Majorana zero modes in the local density of states. The Shiba bands, on the other hand, formed due to the overlapping of Yu-Shiba-Rusinov states play a crucial role in this topological to trivial superconductor phase transition, confirmed by the sign change in the minigap within the Shiba bands. We also characterize this topological phase transition via gap closing and winding number analysis. Moreover, interference of the Shiba bands exhibiting oscillatory behavior within the superconducting gap, $-\Delta_{p}$ to $\Delta_{p}$, as a function of $G_{s}$, also reflects an important evidence for the formation of an effective $s$-wave pairing. Such oscillation is absent in the $p$-wave regime.