  Recently, level crossings in the energy bands of crystals have been
identified as a key signature for topological phase transitions. Using
realistic models we show that the parameter space controlling the occurrence of
level coincidences in energy bands has a much richer structure than anticipated
previously. In particular, we identify robust level coincidences that cannot be
removed by a small perturbation of the Hamiltonian compatible with the crystal
symmetry. Different topological phases that are insulating in the bulk are then
separated by a gapless (metallic) phase. We consider HgTe/CdTe quantum wells as
a specific example.
