Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase

Phase transitions between the quantum spin Hall ( QSH) and the insulator phases in three dimensions ( 3D) are studied. We find that in inversion-asymmetric systems there appears a gapless phase between the QSH and insulator phases in 3D which is in contrast with the 2D case. Existence of this gapless phase stems from a topological nature of gapless points ( diabolical points) in 3D, but not in 2D.