Multi-orbital physics in inverse Lieb lattice altermagnets

The inverse Lieb lattice has recently emerged as a promising platform for altermagnetism, with several materials with this structure proposed as $d$-wave altermagnetic candidates. Here, we develop a symmetry-based microscopic Hamiltonian for these materials that includes both sublattice and orbital degrees of freedom, going beyond the sublattice-only minimal models that have been extensively used to study such altermagnets. We apply these models to examine multi-orbital electron correlation physics in the vanadium oxychalcogenide family altermagnets, which contain dominant $xy$ and $xz/yz$ orbitals character at the Fermi level in the altermagnetic state. We demonstrate that $xy$ orbitals are crucial to stabilize the altermagnetic state observed within a single V$_2$O layer, and altermagnetic order in the $xz/yz$ orbitals is induced through Hund's coupling. Additionally, we show that these multi-orbital models reveal topological regimes in which topological edge states are naturally orbital selective.