Ferroelectric switchable altermagnetic-like compensated ferrimagnets with charge ordering

Unconventional collinear magnets with almost zero magnetization but prominent nonrelativistic spin-splitting, such as altermagnets, can inherit the advantages of both ferromagnets and antiferromagnets. By incorporating more degrees of freedom such as ferroelectricity and charge ordering, these unconventional magnets can be even more interesting and functionalized. With this design principle, the Fe$_3$O$_5$ monolayer is predicted to exhibit a hybrid spin-splitting mechanism, with the superposition of the altermagnetic-like $k$-path alternating splitting and ferrimagnet-like Zeeman splitting. Benefiting from the hidden magnetoelectricity based on the spin-charge coupling, such spin-splitting can be fully switched by an electric field. Its conductivity is highly spin-polarized, with a polarization ratio above $99\%$, comparable to half-metals but with zero magnetization.