Inverse Lieb Materials: Altermagnetism and More

The Lieb lattice, originally proposed for cuprate superconductors, has gained new attention in the emerging field of altermagnetism as a minimal analytical model for the latter. While initially the so-called inverse Lieb lattice (ILL) was deemed only a theoretical model, recently several real materials with this crystallographic motif have been found. The unique geometry of ILL can accommodate complex magnetic orderings arising from competing exchange interactions and geometric frustration, offering great tunability for magnetic properties. In this work, we provide comprehensive insights into magnetic phases in ILL materials and establish guidelines for efficient identification of altermagnetic materials within this family. We begin by constructing phase diagrams using a simple Heisenberg model to elucidate the fundamental mechanisms underlying altermagnetism and other complex magnetic phases observed experimentally. To bridge theory with experiment, we systematically investigate a series of existing ILL compounds using density functional theory (DFT) calculations to determine their magnetic ground states. Our computational results are in good agreement with experimental observations. Importantly, we identify a trend linking magnetic ordering to the $d$-shell filling of transition metal ions, with $d^{2-3}$ and $d^{5}$ configurations showing propensity for altermagnetic behavior. Additionally, we identify a promising metallic compound Sr$_{2}$CrO$_{2}$Cr$_{2}$OAs$_{2}$ as an altermagnet that is highly anisotropic in its $J_2$ exchange couplings with large N\'eel temperature ($\sim 600$ K). Using exchange coupling parameters extracted from DFT calculations, we compute the magnon spectra for altermagnetic systems. As expected, chiral splittings in the magnon dispersion are directly correlated with anisotropy between crystallographically inequivalent $J_{2}$ exchange interactions.