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Recent experiments suggest a quantum spin liquid ground state in the materialPbCuTe$_2$O$_6$, where $S=1/2$ moments are coupled by antiferromagneticHeisenberg interactions into a three dimensional structure of corner sharingtriangles dubbed the hyper-hyperkagome lattice. It exhibits a richerconnectivity, and thus likely a stronger geometric frustration, than therelatively well studied hyperkagome lattice. Here, we investigate the possiblequantum spin liquids in the $S=1/2$ hyper-hyperkagome magnet using the complexfermion mean field theory. Extending the results of a previous projectivesymmetry group analysis, we identify only two $\mathbb{Z}_2$ spin liquids and a$U(1)$ spin liquid that are compatible with the hyper-hyperkagome structure.The $U(1)$ spin liquid has a spinon Fermi surface. For the $\mathbb{Z}_2$ spinliquids, one has a small excitation gap, while the other is gapless andproximate to the $U(1)$ spin liquid. We show that the gapped and gapless spinliquids can in principle be distinguished by heat capacity measurements.Moreover, we calculate the dynamical spin structure factors of all three spinliquids and find that they highly resemble the inelastic neutron scatteringspectra of PbCuTe$_2$O$_6$. Implications of our work to the experiments, aswell as its relations to the existing theoretical studies, are discussed.

Theoretical study of quantum spin liquids in $S=1/2$ hyper-hyperkagome magnets: classification, heat capacity, and dynamical spin structure factor