Engineering second order topological superconductor hosting tunable Majorana corner modes in magnet/$d$-wave superconductor hybrid platform

TL;DR

This paper proposes a hybrid platform combining magnet and $d$-wave superconductor to realize and tune second-order topological superconductors with Majorana corner modes, advancing control over topological quantum states.

Contribution

It introduces a method to tune Majorana corner modes in a hybrid system using noncollinear magnetic textures, expanding the design space for topological superconductors.

Findings

Majorana corner modes can be tuned by exchange strength and spin texture pitch.

Quadrupolar winding number characterizes the number of Majorana modes.

Effective Hamiltonian reveals in-plane Zeeman field and spin-orbit coupling effects.

Abstract

We theoretically study the noncollinear magnetic texture effect on second-order topological superconductor (SOTSC) phase generated in unconventional $d$-wave superconductors and two-dimensional (2D) quantum spin Hall insulators (QSHI). While the interplay of the $d$-wave superconductor and QSHI has been studied as a platform to realize Majorana corner modes (MCMs), we show that the addition of the spin texture enables the tunability of these MCMs. Each corner of this hybrid system can host one or two Majorana modes depending on the system parameters, in particular, exchange strength and pitch vector of the spin texture. To characterize the higher order bulk topology, we compute the quadrupolar winding number, which directly corresponds to the number of MCMs acquiring a value of one for four corner modes and two for eight corner modes. We investigate and show the close resemblance in the topological phase diagrams obtained from the low energy effective Hamiltonian that reveals an emergent in-plane Zeeman field and spin-orbit coupling induced by the spin texture, and the real space tight binding lattice model. The microscopic pairing mechanism responsible for the appearance of SOTSC phase is investigated via an effective bulk pairing analysis, while a low-energy edge theory captures the mechanism behind tunability of MCMs. Our result paves the way for realizing SOTC with multiple MCMs which can be tuned via system parameters.