Effects of Na deficiency on spin dynamics in the Mott insulating ${\mathrm{Na}}_{4-x}{\mathrm{Ir}}_3{O}_8$

We report on Raman scattering and ${}_{23}\mathrm{Na}$ nuclear magnetic resonance (NMR) investigations of $j_{\mathrm{eff}}=1/2$ hyperkagome antiferromagnet ${\mathrm{Na}}_{4-x}{\mathrm{Ir}}_3{O}_8$, which lies in the Mott insulating state. Our Raman scattering experiments unveil remarkable parallels between the magnetic excitations of the pristine ${\mathrm{Na}}_4{\mathrm{Ir}}_3{O}_8$ and the Kitaev honeycomb material ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, which are characterized by dominant fractional spinon excitations. In the case of moderate Na-deficient ${\mathrm{Na}}_{4-x}{\mathrm{Ir}}_3{O}_8$, however, a substantial suppression of the magnetic excitations is observed, alluding to the notable influence of charge fluctuations on spin dynamics. In addition, our site-specific ${}_{23}\mathrm{Na}$ NMR measurements offer further insights into the spin dynamics when a minor concentration of holes is introduced into a spin-orbit coupled Mott insulator. Specifically, the spin-lattice relaxation rate $1/T_1$ reveals the emergence of pseudogaplike correlations at the Na(2) site, alongside a critical slowing down behavior at the Na(1) and Na(3) sites. These findings showcase the intricate interplay between itinerant holes and magnetic correlations in a spin-liquid-like background.