Reentrant Quantum Spin Hall States in Charge Density Wave Phase of Doped Single-Layer Transition Metal Dichalcogenides

Using first-principles calculation methods, we reveal a series of phase transitions as a function of electron doping in single-layer 1T$'$-MoTe$_2$ and 1T$'$-WTe$_2$ exhibiting quantum spin Hall (QSH) edge states without doping. As increasing doping, we show that a phonon mediated superconducting phase first realizes and is followed by a charge density wave (CDW) phase with a nonsymmorphic lattice symmetry. The newly found CDW phase exhibits Dirac or Weyl energy bands with a spin-orbit coupling in case of a fractional band filling and re-enters into topological insulating phase with fully filled bands. The robust resurgence of QSH state coexisting with the CDW phase is shown to originate from band inversions induced by the nonsymmorphic lattice distortion through the strong electron-phonon interaction, thus suggesting a realization of various interfacial states between superconducting, density wave and topological states on a two-dimensional crystal only by doping.