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.

Figure 1

Crystal structure of ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$. In the $a-b$ plane the triangular molecules form a honeycomb lattice. As each complex contains three Wannier orbitals this results in a decorated lattice (a), which extrapolates between the kagome and graphene lattices [19]. Complexes stack in chains along the $c$ axis resulting in a triangular tube arrangement (b) of the Wannier orbitals. The hopping amplitudes: $t_c, t$, and $t_z$ entering model (1) are explicitly shown for clarity and tabulated in Table 1.

Figure 2

Effective pseudospin-one moments arising in isolated trimers in the presence of SOC. The lowest-lying eigenstates of the Hubbard-Heisenberg model describing trinuclear complexes with four electrons are shown for the pure Hubbard model ($U=10t_c,J_F=0$) (a) and for nonzero intracluster exchange (b) ($U=10t_c,J_F=-0.2t_c$). Only the $z$ component of the total angular momentum $j$ is conserved. The degeneracies of the eigenstates are labeled by the numbers on the right-hand side of the figure. As SOC, $\lambda ={\lambda }_{xy}={\lambda }_z$, increases the triplet state is split into a $j=0$ state and a doubly degenerate $j=\pm 1$ state. This is described by a pseudospin-one in the presence of a trigonal field generated by SOC.

Figure 3

Comparison of the exact eigenspectrum (lines) of two coupled triangular clusters with the eigenspectrum obtained from the second order perturbation theory effective model (crosses), $H_{\text{eff}}^{\left(2\right)}$. In (a) we show the case of two trimers coupled in the dumbbell arrangement through the hopping $t=0.785t_c$, while (b) corresponds to two trimers coupled in the tube arrangement through $t_z=0.683t_c$. In both cases we take $U=10t_c$. The black dot-dashed curves represent the lowest energy eigenstate of the exact solution that is not included in the effective spin model.

Figure 4

Anisotropic pseudospin exchange coupling $J_{\alpha \beta }^{\nu }$ and trigonal splitting $D$ induced by spin-orbit coupling in trinuclear complexes. Here $\nu =ab$ labels the exchange interaction in the basal plane (dumbbell model), while $\nu =c$ indicates the exchange interactions along the chains (tube model). We consider the effective interaction between the pseudospins formed in two neighboring clusters in the $a-b$ plane [panels (a)–(c)] and in the $c$ direction of the crystal [panel (d)]. In cases (a) and (b) the two molecules are related by inversion symmetry whereas in (c) they are related by ${\mathrm{C}}_2^{\left(z\right)}$ symmetry. In (d) the two molecules are related by translational symmetry. We take $U=10t_c$ and the intracluster Heisenberg exchange $J_F=0$ (Hubbard model) except in case (b) for which $J_F=-0.2t_c$. Anisotropic pseudospin exchange interactions arise when inversion symmetry is broken [cases (c) and (d)] and in the presence of inversion symmetry when $J_F\ne 0$ [case (b)]. We take $t=t_z=0.785t_c$ in all plots.

Figure 5

Effective one-dimensionality arising from electron correlations in trinuclear complexes. As the on-site Coulomb repulsion ($U$) increases spatial anisotropy of the exchange coupling, $r_{\alpha \alpha }=J_{\alpha \alpha }^c/J_{\alpha \alpha }^{ab}$, is enhanced driving the crystal to a quasi-one-dimensional system. We have taken $t=t_z=0.785t_c$.