Octupolar order and Ising quantum criticality tuned by strain and dimensionality: Application to $d$-orbital Mott insulators*

Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the Osmate ($5d^2$) double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling driven by the interplay of two distinct intra-orbital hopping terms. Our work highlights the importance of higher order contributions to the exchange couplings coming from the small gap between the non-Kramers doublet and excited triplets in these systems. Using classical Monte Carlo simulations on the face-centered cubic lattice, we obtain a ferrooctupolar transition temperature ($T_c$) which is in good agreement with experiments on the osmate double perovskites. We also explore the impact of uniaxial strain and dimensional tuning via ultrathin films, which are shown to induce a transverse field on the Ising octupolar order. This suppresses $T_c$ and potentially allows one to access octupolar Ising quantum critical points. We discuss possible implications of our results for a broader class of materials which may host such non-Kramers doublet ions.