Recent studies have discovered local potentials can induce nontrivial eigenmodes responding to the bulk topology for the system. While previous studies focused on conventional first-order topological states emerging from vacancy superlattice, here we study higher-order topological properties of the vacancy superlattice on a two-dimensional Chern insulator with particle-hole symmetry. The vacancy superlattice with alternate lattice spacings exhibits an emergent second-order topological phase characterized by the nontrivial edge polarization. This topological phase is robust against particle-hole symmetry-preserved perturbations as long as the energy gap remains open for the mid-gap sates. Our work generalizes the nontrivial higher-order topological properties to a Chern insulator with local defect vacancies and provides a controllable platform for engineering higher-order topological corner states.

Higher-order topological superconductors and superfluids (SFs) host lower-dimensional Majorana corner and hinge states since novel topology exhibitions on boundaries. While such topological nontrivial phases have been explored extensively, more possible schemes are necessary for engineering Majorana states. In this paper we propose Majorana corner states could be realized in a two-dimensional attractive quantum spin-Hall insulator with opposite in-plane Zeeman energy at two sublattice sites. The appropriate Zeeman field leads to the opposite Dirac mass for adjacent edges of a square sample, and naturally induce Majorana corner states. This topological phase can be characterized by Majorana edge polarizations, and it is robust against perturbations on random potentials and random phase fluctuations as long as the edge gap remains open. Our work provides a new possibility to realize a second-order topological SF in two dimensions and engineer Majorana corner states.