Tailoring the 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 nontrivial topological $p$-wave superconductor to a trivial $s$-wave-like superconducting phase through a gapless phase, driven by different magnetic textures as a 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 ${G}_{\mathrm{s}}=\ensuremath{\pi}$, 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 $\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{p}$ to ${\mathrm{\ensuremath{\Delta}}}_{p}$, 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. We also analyze the case of two-dimensional $p$-wave superconductor hosting Majorana edge modes (in absence of any magnetic chain) and show that initially Majorana zero modes (in presence of one-dimensional magnetic chain) can hybridize with such Majorana edge modes. Interestingly, in the topological regime with a fixed ${G}_{s}$ value, the Mazorana zero modes survive at the ends of the magnetic chain even when the Majorana edge states disappear at some critical value of the chemical potential and exchange coupling strength.