Orbital-Selective Spin-Orbit Mott Insulator in Fractional Valence Iridate La$_3$Ir$_3$O$_{11}$

The combination of strong spin-orbit coupling and Coulomb interactions makes the $5d$ iridates a unique platform for realizing novel correlated electronic states. Here, utilizing infrared spectroscopy, we demonstrate that a robust Mott insulating state persists in the $1/3$-hole self-doped system La$_3$Ir$_3$O$_{11}$, evidenced by the collapse of the Drude response and the emergence of sharp excitations across the Mott gap. Our theoretical calculations reveal that the insulating behavior arises from the cooperative interplay of structural distortions, spin-orbit coupling, and Coulomb interactions. Specifically, octahedral distortion and Ir-Ir dimerization split the $t_{2g}$ orbitals, driving the $J_{\mathrm{eff}} = 1/2$ bands toward half-filling while keeping the $J_{\mathrm{eff}} = 3/2$ bands away from it. Consequently, electron correlations induce an orbital-selective Mott transition in the $J_{\mathrm{eff}} = 1/2$ bands, whereas a band-insulating gap develops in the $J_{\mathrm{eff}} = 3/2$ bands, thereby stabilizing the unconventional insulating state in La$_3$Ir$_3$O$_{11}$. These findings provide new insights into the design and understanding of the insulating ground state of spin-orbit-coupled iridates.