  ZrTe$_5$ and HfTe$_5$ have attracted increasingly attention recently since
the theoretical prediction of being topological insulators (TIs). However,
subsequent works show many contradictions about their topological nature.Three
possible phases, i.e. strong TI, weak TI, and Dirac semi-metal, have been
observed in different experiments until now. Essentially whether ZrTe$_5$ or
HfTe$_5$ has a band gap or not is still a question. Here, we present detailed
first-principles calculations on the electronic and topological properties of
ZrTe$_5$ and HfTe$_5$ on variant volumes and clearly demonstrate the
topological phase transition from a strong TI, going through an intermediate
Dirac semi-metal phase, then to a weak TI when the crystal expands.Our work
might give a unified explain about the divergent experimental results and
propose the crucial clue to further experiments to elucidate the topological
nature of these materials.
