  The pyrochlore materials have long been predicted to harbor a quantum spin
liquid, an intrinsic long-range-entangled state supporting fractionalized
excitations. Existing pyrochlore experiments, on the other hand, have
discovered several weakly ordered states and a tendency of close competition
amongst them. Motivated by these facts, we give a complete classification of
spin-orbit-coupled $\mathbb{Z}_2$ spin-liquid states on the pyrochlore lattice
by using the projective symmetry group (PSG) approach for bosonic spinons. For
each spin liquid, we construct a mean-field Hamiltonian that can be used to
describe phase transitions out of the spin liquid via spinon condensation.
Studying these phase transitions, we establish phase diagrams for our
mean-field Hamiltonians that link magnetic orders to specific spin liquids. In
general, we find that seemingly unrelated magnetic orders are intertwined with
each other and that the conventional spin orders seen in the experiments are
accompanied by more exotic hidden orders. Our critical theories are categorized
into $z=1$ and $z=2$ types, based on their spinon dispersion and Hamiltonian
diagonalizability, and are shown to give distinct signatures in the heat
capacity and the spin structure factor. This study provides a clear map of
pyrochlore phases for future experiments and variational Monte Carlo studies in
pyrochlore materials.
