Based on the Bogoliubov-de Gennes theory, we investigate the topological-defect states in a mesoscopic superconducting square with coexisted $d$-wave and extended $s$-wave condensates when the Dresselhaus spin-orbit (SO) interaction and competing antiferromagnetic order are introduced. In the topologically trivial time-reversal symmetry breaking $s+id$ superconducting regime, the single-flux quantum vortex with composite point-like cores undergoes splitting and favors to exhibit the coreless skyrmionic pattern with varying the SO coupling and next-nearest-neighbor (nnn) hopping strengths as well as the temperature. The one-quanta vortex splitting behavior can also be realized by suitable tuning the strength and direction of Zeeman fields. Moreover, when the $s+id$ system evolves into the second-order topological superconducting phase, the zero-energy Majorana corner states remain robust against vortex bound states near half filling. A continuous change in the chemical potential can lead to the modulation of the number and location of Majorana corner modes because of the nucleation of extra fractional-flux one-component vortices inside the mesoscopic sample. Interestingly, an appropriate nnn-hopping strength can lead to the formation of zero-energy fractional vortices, which are expected to host Majorana zero modes. The evolution of low-energy vortex states is also temperature dependent, and the application of an out-of-plane Zeeman field can give rise to a coexisting Majorana-skrymion mode.

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• Geometric & topological phases
• Superconducting phase transition
• Vortices in superconductors
• Unconventional superconductors
• Bogoliubov-de Gennes equations