Transition between strong and weak topological insulator in ZrTe$_5$ and HfTe$_5$

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.