2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139315Phase transitions between the quantum spin Hall and the insulator phases in three dimensions are studied. We find that in inversion-asymmetric systems there appears a gapless phase between the quantum spin Hall and insulator phases in three dimensions, which is in contrast with the two-dimensional case. Existence of this gapless phase stems from a topological nature of gapless points (diabolical points) in three dimensions, but not in two dimensions.16 pages, 5 figuresMesoscale and Nanoscale Physicstext