Quasi-one-dimensional spin-orbit-coupled correlated insulator in a multinuclear coordinated organometallic crystal

We show how quasi-one-dimensional correlated insulating states arise at two-thirds filling in organometallic multinuclear coordination complexes described by layered decorated honeycomb lattices. The interplay of spin-orbit coupling and electronic correlations leads to pseudospin-one moments arranged in weakly coupled chains with highly anisotropic exchange and a large trigonal splitting. We show that the in-plane exchange coupling is very different from the interlayer coupling; in particular the latter is much larger, despite the underlying hopping integrals being close to isotropic. Surprisingly, the effective dimensionality of the pseudospin model is strongly dependent on the strength of the electronic correlations: With increasing Hubbard $U$ the pseudospin-one model becomes increasingly one dimensional, even though the crystal is almost isotropic. We predict that the trigonal splitting leads to a quantum phase transition from a Haldane phase to a topologically trivial phase as the relative strength of the spin-orbit coupling increases.