Projective symmetry group classifications of quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices

We perform extensive classifications of $\mathbb{Z}_2$ quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices using a spin-rotation invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group $O_h$, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third neighbor spin interactions and discuss the spinon dispersions, ground state energies and dynamical spin structure factors. Our results indicate that phases with non-uniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry protected line-like zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.