Orbital homology of p and t2g orbitals in models and materials

TL;DR

This work argues that a deep orbital homology between $p$ and $t_{2g}$ orbitals unifies hopping and spin–orbit coupling across diverse quantum materials via an effective $L_{ ext{eff}}=1$ description. It shows how explicit mappings, e.g. $d_{yz} \leftrightarrow -|p_x\rangle$, allow transposing intuition and minimal models such as the Luttinger semimetal and BHZ/Kane–Mele frameworks between orbital classes. The correspondence is illustrated in real materials (e.g., half-Heuslers, pyrochlore iridates, iron-based superconductors), oxide interfaces, and ultracold-atom simulators, where quadratic band touchings, topological phases, and nematic superconductivity emerge. As a design principle, the p–t2g homology enables orbital engineering and unified exploration of topology and strong correlations across platforms.

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.

Figures (8)

• Figure 1: Schematic energy level diagram of $d$-electron splitting under an octahedral crystal field, followed by the effects of spin-orbit coupling. The crystal field lifts the degeneracy of the $d$-orbitals, splitting them into higher-energy $e_g$ and lower-energy $t_{2g}$ levels. With the inclusion of spin-orbit coupling, the $t_{2g}$ manifold further splits into a higher-energy $j_{\text{eff}} = 1/2$ doublet and a lower-energy $j_{\text{eff}} = 3/2$ quartet, corresponding to effective orbital angular momentum $L_{\text{eff}} = 1$.
• Figure 2: Orbital-specific hoppings in a square lattice. $d_{xz}$ and $d_{yz}$ orbitals form $\pi$-bonds along $x$ and $y$ directions, respectively, while $d_{xy}$ exhibits the $\pi$ bonding along both $x$ and $y$ directions. The symmetry-demanded $0$ hopping and the weak $\delta$ hopping are marked as well.
• Figure 3: Schematic picture of the band structure of GaAs near $k = 0$. CB means the conduction band, HH the heavy- hole band, LH the light-hole band, and SO the split-off band, respectively. When the small inversion symmetry breaking is neglected, all of them are doubly degenerate. The splitting $\Delta$ at $k = 0$ between the LH,HH and SO bands are due to the spin-orbit coupling. Figure is adapted from Ref. PhysRevB.69.235206.
• Figure 4: Comparison of the bulk band structures of CdTe and HgTe at the $\Gamma$ point. In CdTe, the $\Gamma_6$$s$-like conduction band lies above the $\Gamma_8$$p$-like valence band. In HgTe, strong spin-orbit coupling inverts this order, placing the $\Gamma_8$ (heavy-hole) band above the $\Gamma_6$ band. The Luttinger model Hamiltonian, parameterized by $\gamma_1$, $\gamma_2$, and $\gamma_3$, captures the degenerate $\Gamma_8$ valence band structure in both semiconductors. Figure is adapted from Ref. Bernevig_2006.
• Figure 5: The electronic band structure of Pr$_2$Ir$_2$O$_7$ from density functional theory (DFT). The Ir $t_{2g}$ electrons, under strong spin-orbit coupling, form $j_{\text{eff}}$ states. On the pyrochlore lattice, the four sublattices provide an internal degree of freedom that enables the formation of a quadratic band touching at the $\Gamma$ point, primarily derived from the $j_{\text{eff}}=1/2$ bands. This results in a Luttinger semimetal phase, where the low-energy physics is described by an effective Luttinger-type model but originates from the $j_{\text{eff}}=1/2$ manifold coupled to the lattice symmetry. Energy unit is in eV. Figure is adapted from Ref. Kondo_2015.