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Motivated by the recent activities on the Ni-based diamond latticeantiferromagnet NiRh$_2$O$_4$, we theoretically explore on a general ground theunique spin and orbital physics for the Ni$^{2+}$ ions with a $3d^8$ electronconfiguration in the tetrahedral crystal field environment and on a diamondlattice Mott insulator. The superexchange interaction between the local momentsusually favors magnetic orders. Due to the particular electron configuration ofthe Ni$^{2+}$ ion with a partially filled upper $t_{2g}$ level and a fullyfilled lower $e_g$ level, the atomic spin-orbit coupling becomes active at thelinear order and would favor a spin-orbital-entangled singlet with quenchedlocal moments in the single-ion limit. Thus, the spin-orbital entanglementcompetes with the superexchange and could drive the system to a quantumcritical point that separates the spin-orbital singlet and the magnetic order.We further explore the effects of magnetic field and uniaxial pressure. Thenon-trivial response to the magnetic field is intimately tied to the underlyingspin-orbital structure of the local moments. We discuss the future experimentssuch as doping and pressure, and point out the correspondence between differentelectron configurations.

Emergent quantum criticality from spin-orbital entanglement in $d^8$ Mott insulators: the case of a diamond lattice antiferromagnet