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

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