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Mott-insulating transition metal oxides containing $t_{2g}^4$ ions withstrong spin-orbit coupling were recently demonstrated to display unusualmagnetism due to a dynamical mixing of the low-energy multiplet states viaexchange processes. Here we derive exchange interactions in the situation wherea tetragonal or trigonal crystal field selects a single relevant excited stateon top of the singlet ionic ground state, producing an effectivespin-$\frac12$. We show that these moments universally obey antiferromagnetictransverse-field Ising model (TFIM) with an intrinsic transverse fieldgenerated by the splitting of the two ionic singlets. Using Ru$^{4+}$ as aexample ion, we provide quantitative estimates of the exchange and illustratethe emergent TFIM physics based on phase diagrams and excitation spectraobtained for several 2D lattices - square, honeycomb, as well as for thefrustrated triangular lattice.

Emergent transverse-field Ising model in $\boldsymbol{d}^\mathbf{4}$ spin-orbit Mott insulators