Li$_2$RuO$_3$, a valence bond liquid on the honeycomb lattice

Li$_2$RuO$_3$ has been known to form Ru-Ru dimers at low temperature, but was believed to be homogenous above the transition. We provide new experimental evidence for the melting of the low temperature dimer ordering by comparing x-ray diffraction probing the average crystal structure and the pair distribution function, which provides information about local order. We show that strong dimerization survives well above the ordering temperature $T_s\approx 450$ K and that the high temperature structure is a bond liquid of dynamically disordered dimers with the same 1:2 ratio of short and long bonds. Theoretically, we search for low energy structures of Li$_2$RuO$_3$ and find different long range orders of Ru dimer patterns. We can explain not only the low $T$ structure but also the ordering temperature: Due to strong covalency, dimerization leads to a large energy gain but the additional gain through dimer ordering is much smaller and compatible with the experimentally found $T_s$. Moreover, of the two holes present in Li$_2$RuO$_3$ one participates in a strongly covalent bond, which survives at all temperatures, and the other in a weak bond that breaks in the bond-liquid state. This explains why below $T_s$ the effective spin of Ru is 0, and above $1/2$, but never 1.