Classification of $Z_2$ spin liquids in a hyperkagome lattice by projective symmetry groups

Being a rare candidate material supporting 3D spin liquid states, Na$_4$Ir$_3$O$_8$ has attracted much theoretical and experimental interest in the past decade. Propelled by such developments, we give a complete classification of $Z_2$ spin liquid states in the hyperkagome lattice formed by Ir$^{4+}$ ions in the projective symmetry group approach. A list of mean field states with different fractional quasi-particle excitations are correspondingly given, and their excitation gaps are analyzed. The effects of spin-orbit coupling due to the $5d$ electrons in Ir are also discussed. This work paves the way for further variational Monte-Carlo study of the spin liquid physics in hyperkagome lattices.