First- and higher-order topological superconductors on a square lattice

Topological systems with hybrid topology offer unique opportunities for exploring multiplexing topological phenomena and valuable applications. However, building a hybrid topological superconductor and achieving a controllable topological phase transition between different orders of topological superconductors remain a challenge. We propose a solution to unify both first- and higher-order topological superconductors on a square lattice, incorporating the Rashba spin&#x2212;orbit coupling, Zeeman field and <i>s</i>-wave superconducting pairing. By utilizing one-dimensional normal edge states, we construct a boundary-obstructed topological superconductor associated with closing the boundary energy gap. This leads to the emergence of Majorana corner modes, whose topological properties are characterized by the Berry phase. By tuning the amplitude of different intracell hoppings, we can control the localization of Majorana corner modes. We also generalize the Majorana polarization as a topological invariant to verify the existence of Majorana corner modes. Remarkably, the obtained phase diagram is well consistent with that described by the boundary energy gap, a quantized Berry phase and Chern number. With the further increase of Zeeman field, we observe a transition from a second- to first-order topological superconducting phase by closing the bulk gap. Its topology is protected by the bulk states and characterized by nonzero Chern number. Additionally, no Majorana corner modes are present and the topological boundary states are determined by nonzero Chern number in the region where both the quantized Berry phase and Chern number are nonzero. Furthermore, we achieve the hinge Majorana zero modes in a three-dimensional structure by stacking the two-dimensional square lattices. Our work unveils the physical mechanism to get a topological superconductor with different orders, and opens an avenue to characterize and detect different order topological superconductors on two-dimensional lattices.