Spin liquid close to a quantum critical point in Na$_4$Ir$_3$O$_8$

Na$_4$Ir$_3$O$_8$ is a candidate material for a 3-dimensional quantum spin-liquid on the hyperkagome lattice. We present thermodynamic measurements of heat capacity $C$ and thermal conductivity $\kappa$ on high quality polycrystalline samples of Na$_4$Ir$_3$O$_8$ down to $T = 500$ mK and $75$ mK, respectively. Absence of long-range magnetic order down to $T = 75$ mK strongly supports claims of a spin-liquid ground state. The constant magnetic susceptibility $\chi$ below $T \approx 25$ K and the presence of a small but finite linear-$T$ term in $C(T)$ suggest the presence of gapless spin excitations. Additionally, the magnetic Gr$\ddot{\rm{u}}$neisen ratio shows a divergence as $T \rightarrow 0$ K and a scaling behavior which clearly demonstrates that Na$_4$Ir$_3$O$_8$ is situated close to a zero-field QCP.