The ground-state phase diagrams of the three-orbital $t_{2\mathrm{g}}$ Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for $t_{2\mathrm{g}}$ models defined in terms of the effective total angular momentum $j_{\mathrm{eff}}$ is theoretically derived. These order parameters can classify off-diagonal orders between $j_{\mathrm{eff}}=1/2$ and $j_{\mathrm{eff}}=3/2$ manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the on-site Coulomb repulsion $U$, (2) the spin-orbit coupling $\lambda$, and (3) the number of electrons $n$ are mapped out. A variety of nontrivial quantum phases with $j_{\mathrm{eff}}$-diagonal and $j_{\mathrm{eff}}$–off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory, are discussed.

• Received 3 August 2021
• Revised 22 October 2021
• Accepted 1 November 2021

DOI:https://doi.org/10.1103/PhysRevB.104.235125