  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.
