Topological magnetic corner states in two-dimensional ferrimagnetic metal-organic frameworks

Topological corner states have been extensively studied in two-dimensional (2D) topological insulators (TIs), yet they remain rarely reported in 2D magnetic metal-organic frameworks (MOFs). In this work, we uncover the nontrivial corner states protected by ${C}_{2z}$ and ${C}_{4z}$ symmetry in three 2D ferrimagnetic MOFs. In the ferrimagnetic ${\mathrm{Cr}(\mathrm{ODZ})}_{2}$ (DZ = 1.2.5-diazole) with a magnetic space group (MSG) 14.2.87, the occupied electronic bands contain the essential band representations (BRs) of ${A}_{u}@2b$ and ${A}_{g}@2c$ with unconventional features, facilitating the formation of quantized fractional charge $(\frac{e}{2})$ in both spin-up and spin-down channels under ${C}_{2z}$ symmetry protection. Even in the presence of spin-orbit coupling (SOC), these nontrivial corner states are well preserved. Additionally, the ${C}_{4z}$ symmetry protected corner states are also identified in two other types of ferrimagnetic MOFs, specifically, ${\mathrm{Cr}(\mathrm{SDZ})}_{2}$ and ${\mathrm{Cr}(\mathrm{SeDZ})}_{2}$ monolayers in MSG 100.5.840 in the presence of SOC, contributed by the following basic BRs: ${A}_{u}@2b+{A}_{g}@2d$. In the former, the magnetic corner states only exist in the spin-up channel, whereas in the latter, they are present in both spin-up and spin-down channels. Our research findings offer an ideal material platform for exploring the application of magnetic corner states in a special topological device.