  Time Reversal Symmetry (TRS) broken topological phases provide gapless
surface states protected by topology, regardless of additional internal
symmetries, spin or valley degrees of freedom. Despite the numerous
demonstrations of 2D topological phases, few examples of 3D topological systems
with TRS breaking exist. In this article, we devise a general strategy to
design 3D Chern insulating (3D CI) cubic photonic crystals in a weakly TRS
broken environment with orientable and arbitrarly large Chern vectors. The
designs display topologically protected chiral and unidirectional surface
states with disjoint equifrequency loops. The resulting crystals present the
following novel characteristics: First, by increasing the Chern number,
multiple surface states channels can be supported. Second, the Chern vector can
be oriented along any direction simply changing the magnetization axis, opening
up larger 3D CI/3D CI interfacing possibilities as compared to 2D. Third, by
lowering the TRS breaking requirements, the system is ideal for realistic
photonic applications where the magnetic response is weak.
