Emergent transverse-field Ising model in $d^4$ 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-$\frac{1}{2}$. 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 ${\mathrm{Ru}}^{4+}$ 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.