Orbital homology of p and t2g orbitals in models and materials

Abstract

The nominal divide between $p$- and $d$-electron systems often obscures a deep underlying unity in condensed matter physics. This review elucidates the orbital homology between the $p$ and $t_{2g}$ orbital manifolds, establishing the correspondence that extends from minimal model Hamiltonians to the complex behaviors of real quantum materials. We demonstrate that despite their distinct atomic origins, these orbitals host nearly identical hopping physics and spin-orbit coupling, formalized through an effective ${l=1}$ angular momentum algebra for the $t_{2g}$ case. This equivalence allows one to transpose physical intuition and theoretical models developed for $p$-orbital systems directly onto the more complex $t_{2g}$ materials, and vice versa. We showcase how this paradigm provides a unified understanding of emergent phenomena, including non-trivial band topology, itinerant ferromagnetism, and unconventional superconductivity, across a wide range of platforms, from transition metal compounds, two-dimensional oxide heterostructures, and iron-based superconductors, to $p$-orbital ultracold gases. Ultimately, this $p$-$t_{2g}$ homology serves not only as a tool for interpretation but also as a robust design principle for engineering novel quantum states.