Lifshitz transition enhanced triplet $p_z$-wave superconductivity in hydrogen doped KCr$_3$As$_3$

The recently synthesized air-insensitive hydrogen doped KCr$_3$As$_3$ superconductor has aroused great research interests. This material has, for the first time in the research area of the quasi-one-dimensional Cr-based superconductivity (SC), realized a tunability through charge doping, which will potentially significantly push the development of this area. Here based on the band structure from first-principle calculations, we construct a six-band tight-binding (TB) model equipped with multi-orbital Hubbard interactions, and adopt the random-phase-approximation approach to study the hydrogen-doping dependence of the pairing symmetry and superconducting $T_c$. Under the rigid-band approximation, our pairing phase diagram is occupied by the triplet $p_z$-wave pairing through out the hydrogen-doping regime $x\in (0.4,1)$ in which SC has been experimentally detected. Remarkably, the $x$-dependence of $T_c$ shows a peak at the 3D-quasi-1D Lifshitz transition point, although the total density of state exhibit a dip there. A thorough investigation of the band structure reveals type-II van-Hove singularities (VHSs) in the $\gamma$ band, which favor the formation of the triplet SC. It turns out that the $\gamma$- Fermi surface (FS) comprises two flat quasi-1D FS sheets almost parallel to the $k_z=0$ plane and six almost perpendicular tube-like FS sheets, and the type-II VHS just lies in the boundary between these two FS parts. Furthermore, the $\left|k_z\right|$ of the VH planes reaches the maximum near the Lifshitz-transition point, which pushes the $T_c$ of the $p_z$-wave SC to the maximum. Our results appeal more experimental access into this intriguing superconductor.