Spin-orbital Entangled Molecular $j_{\rm eff}$ States in Lacunar Spinel Compounds

The entanglement of the spin and orbital degrees of freedom through the spin-orbit coupling has been actively studied in condensed matter physics. In several iridium-oxide systems, the spin-orbital entangled state, identified by the effective angular momentum $j_{\rm eff}$, can host novel quantum phases with the help of electron correlations. Here, we show that a series of lacunar spinel compounds, Ga$M_4X_8$ ($M$ = Nb, Mo, Ta, and W and $X$ = S, Se, and Te), gives rise to a $\textit{molecular}$ $j_{\rm eff}$ state as a new spin-orbital composite on which the low energy effective Hamiltonian is based. A wide range of electron correlations is accessible by tuning the bandwidth under external and/or chemical pressure, enabling us to investigate the interesting cooperation between spin-orbit coupling and electron correlations. As illustrative examples, a two-dimensional topological insulating phase and an anisotropic spin Hamiltonian are investigated in the weak and strong coupling regimes, respectively. Our finding can provide an ideal platform for exploring $j_{\rm eff}$ physics and the resulting emergent phenomena.