Phase 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.
@article{arxiv.0710.0930,
title = {Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase},
author = {Shuichi Murakami},
journal= {arXiv preprint arXiv:0710.0930},
year = {2007}
}