  We study the robustness of 3D intrinsic topogical order under external
perturbations by investigating the paradigmatic microscopic model, the 3D toric
code in an external magnetic field. Exact dualities as well as variational
calculations reveal a ground-state phase diagram with first and second-order
quantum phase transitions. The variational approach can be applied without
further approximations only for certain field directions. In the general field
case, an approximative scheme based on an expansion of the variational energy
in orders of the variational parameters is developed. For the breakdown of the
3D intrinsic topological order, it is found that the (im-)mobility of the
quasiparticle excitations is crucial in contrast to their fractional
statistics.
