Semimetal behavior and magnetic correlations in the doped hyper-kagome Na$_3$Ir$_3$O$_8$

We present a microscopic study of a doped quantum spin liquid candidate, the hyperkagome Na$_3$Ir$_3$O$_8$ compound by using $^{23}$Na NMR. We determine the intrinsic behavior of the uniform q = 0 susceptibility via shift measurements and the dynamical response by probing the spin-lattice relaxation rate. Throughout the studied temperature range, the susceptibility is consistent with a semimetal behavior. Remarkably, the antiferromagnetic fluctuations present in the insulating parent compound Na$_4$Ir$_3$O$_8$ survive in the studied compound. The dynamics are consistent with 120$^o$ excitations modes that are correlated over short ranges.