  In the quest for quantum spin liquids in three spatial dimensions (3D), we
study the half-filled Hubbard model on the simple cubic lattice with hopping
processes up to third neighbors. Employing the variational cluster approach
(VCA), we determine the zero-temperature phase diagram: In addition to a
paramagnetic metal at small interaction strength $U$ and various
antiferromagnetic insulators at large $U$, we find an intermediate-$U$
antiferromagnetic metal. Most interestingly, we also identify a non-magnetic
insulating region, extending from intermediate to strong $U$. Using VCA results
in the large-$U$ limit, we establish the phase diagram of the corresponding
$J_1$-$J_2$-$J_3$ Heisenberg model. This is qualitatively confirmed - including
the non-magnetic region - using spin-wave theory. Further analysis reveals a
striking similarity to the behavior of the $J_1$-$J_2$ square-lattice
Heisenberg model, suggesting that the non-magnetic region hosts a 3D
spin-liquid phase.
