Geometry-Selective Higher-order topological insulators induced by $d_{xy}$ altermagnet

We show that proximity to a $d_{xy}$-wave altermagnet provides an effective route to engineering boundary-geometry-dependent higher-order topological phases. The boundary geometry plays a crucial role in determining the topological properties: axis-aligned edge states remain gapless, whereas diagonal edges acquire effective masses with alternating signs, giving rise to four corner modes localized at the mass domain walls. By projecting onto the edge-state subspace, we derive the effective edge Hamiltonians for different boundary geometries. Furthermore, $π$-flux defects introduced under periodic boundary conditions yield zero-energy bound states localized at the flux cores, providing complementary evidence for nontrivial topology without physical boundaries. The interplay between the Zeeman field and the altermagnetic order drives a topological transition through the closing and reopening of the edge gap, while the bulk gap remains open. These findings highlight the crucial role of boundary geometry in higher-order topology and establish proximity-induced $d_{xy}$-wave altermagnetism as a route to engineering higher-order topological phases.