We investigate the magnetic behavior of the $6H$-perovskite dimer lattice ${\mathrm{Ba}}_3{\mathrm{Zn}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ru}}_2{\mathrm{O}}_9$ using analytical theory, density functional theory, inelastic neutron scattering, and modeling of historical magnetization and neutron-scattering data. A dimer mean-field theory built upon classical Luttinger–Tisza analysis generates a phase diagram revealing a transition from a nonmagnetic singlet to a finite-moment ground state as interdimer couplings increase. A (generalized) linear spin-wave theory captures multiplet mixing, excitation gap closing, and fluctuation-induced moment suppression. Density-functional-theory calculations on selected compounds, together with neutron spectroscopy of dilute ${\mathrm{Ba}}_3\mathrm{Zn}{\left({\mathrm{Ru}}_{1-x}{\mathrm{Sb}}_x\right)}_2{\mathrm{O}}_9$, confirm the exchange hierarchy, enabling quantification of previously published experiments within this framework. Our results identify three mechanisms for magnetic moment suppression—quantum fluctuations, ligand hybridization, and nonmagnetic-singlet/magnetic-multiplet mixing.

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• Exchange interaction
• Magnons
• Quantum phase transitions
• Antiferromagnets
• Perovskites
• Ruthenates
• Density functional theory
• Inelastic neutron scattering
• Linear spin wave theory
• Mean field theory