Majorana bound states and phase transitions in first- and second-order topological superconductors

Based on the spin-generalized Bogoliubov-de Gennes theory, we investigate the existence of first- and second-order topological phases in Rashba superconducting ribbon and loop systems. The topological phase transitions driven by the strength of spin-orbit coupling and the spin correlation as well as the temperature are demonstrated. The formation of zero-energy Majorana edge or corner states is highly sensitive to these effects. Due to the modification of the relative strength of competing s- and d-wave pairing symmetries, the number and location of Majorana zero modes in the square loop may be tuned by the introduced spin correlation. Meanwhile, the second-order topological phase and emergent Majorana corner modes can remain robust for finite temperatures. We expect that our theoretical predictions may provide useful information for future experiments.