From atomic semimetal to topological nontrivial insulator

Topological band insulators and (semi-) metals can arise out of atomic when the hopping strength between electrons increases. Such topological phases are separated from insulator by a bulk gap closing. In this work, we show that in many (magnetic) space groups, crystals with certain Wyckoff positions orbitals being occupied must be semimetal or limit, e.g. is infinite weak but not vanishing, which then termed (semi-)metals (ASMs). We derive sufficient condition for realizing ASMs spinless spinful systems. Remarkably, find increasing may transform an ASM into both symmetries electron fillings crystal preserved. The induced inevitably topologically non-trivial at least obstructed (OAIs) labeled as trivial quantum chemistry website. Particularly, using silicon example, criterion discover OAIs missed recently proposed filling enforced OAI. Our work only establishes efficient way to identify design also predicts group-IV elemental semiconductors ideal candidate materials