Lattice-Tuned Magnetism of Ru$^{4+}$(4d$^{4})$ Ions in Single-Crystals of the Layered Honeycomb Ruthenates: Li$_{2}$RuO$_{3}$ and Na$_{2}$RuO$_{3}$*

We synthesize and study single crystals of the layered honeycomb lattice Mott insulators Na$_{2}$RuO$_{3}$ and Li$_{2}$RuO$_{3}$ with magnetic Ru$^{4+}$(4d$^{4})$ ions. The newly found Na$_{2}$RuO$_{3}$ features a nearly ideal honeycomb lattice and orders antiferromagnetically at 30 K. Single-crystals of Li$_{2}$RuO$_{3}$ adopt a honeycomb lattice with either C2/m or more distorted P2$_{1}$/m below 300 K, depending on detailed synthesis conditions. We find that Li$_{2}$RuO$_{3}$ in both structures hosts a well-defined magnetic state, in contrast to the singlet ground state found in polycrystalline Li$_{2}$RuO$_{3}$. A phase diagram generated based on our results uncovers a new, direct correlation between the magnetic ground state and basal-plane distortions in the honeycomb ruthenates.