Ultrastable ferrimagnetic second-order topological insulator in a two-dimensional metal-organic framework

Two-dimensional (2D) second-order topological insulators (SOTIs), characterized by zero-dimensional (0D) corner states, have garnered significant interest. However, such topological states have been primarily realized in nonmagnetic (NM), ferromagnetic (FM), and antiferromagnetic (AFM) systems. 2D ferrimagnetic (FiM) SOTIs, particularly those that simultaneously exhibit ultrastable corner states, are still lacking. Here, based on first-principles calculations and theoretical analysis, we reveal such SOTI state in a 2D metal-organic framework (MOF) material, $\mathrm{Cr}{(\mathrm{pyz})}_{2}$ (pyz = pyrazine). This material exhibits FiM ground state with an easy axis aligned along the [001] direction. It hosts a nontrivial real Chern number in the spin-up channel, enabled by $PT$ symmetry, with 0D corner states observable in the disk. In contrast, the spin-down channel exhibits a trivial gapped bulk state. Notably, the topological corner states in monolayer $\mathrm{Cr}{(\mathrm{pyz})}_{2}$ show high robustness, even if the symmetries are broken by introducing defects, the corner states persist. We also considered other external perturbations, including uniaxial/biaxial strain, ligand rotation, and electric fields; the corner states still remain stable. Even more, the energy positions of the corner states are also nearly unchanged. When spin-orbit coupling (SOC) is considered, the material maintains its corner states. This work identifies an ultrastable FiM SOTI state in the MOF system and provides an ideal platform for future experimental investigations and applications in spintronic devices.