 Translation-symmetry breaking changes the band-filling constraint

A band filling constraint counts electrons per primitive unit cell of the actual translation symmetry. If an ordered state doubles the cell, a filling of one electron per original cell becomes two per new cell. The reduced Brillouin zone then permits reconstruction into different energy bands.

For spin-degenerate independent electrons, a filled band holds two electrons per primitive cell. Doubling can therefore remove the odd-filling obstruction to a band insulator, provided the reconstruction opens a global band gap. It does not guarantee a gap: folded bands can still overlap or retain symmetry-protected crossings. The original counting argument remains correct for its original symmetry assumptions; applying it after translation symmetry changes requires recounting using the new cell.