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We study the pairing symmetry and the topological properties of the hydrogen-doped ACr$_3$As$_3$ superconductors. Based on our first-principle band structure with spin-orbit-coupling (SOC), we construct tight binding model including the on-site SOC terms, equipped with the multi-orbital Hubbard interactions. Then using the random-phase-approximation (RPA) approach, we calculate the pairing phase diagram of this model. Our RPA results yield the triplet pz-wave pairing in the ($\left(\uparrow\downarrow+\downarrow\uparrow\right)$) spin channel to be the leading pairing symmetry all over the experiment relevant hydrogen-doping regimes. This pairing state belongs to the spin-U(1)-symmetry protected time-reversal-invariant topological nodal-line superconductivity. Determined by the momentum-dependent topological invariant Z (kx; ky), the whole (001) surface Brillouin zone is covered with topological flat bands with different regimes covered with different numbers of flat surface bands, which can be detected by the scanning tunneling microscope experiments., Comment: 7 pages, 6 figures

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., Comment: 11 pages, 6 figures, accepted version by Phys. Rev. B

Bulk superconductivity (SC) has recently been observed in the Al-Zn-Mg quasicrystal (QC). To settle several fundamental issues of the SC on the QC, we use an attractive Hubbard model to perform a systematic study on the Penrose lattice. The first issue is the Cooper instability of the QC, i.e., no Fermi surface under an infinitesimal attractive interaction. Starting from the two-electron problem outside the filled Fermi-sea, we analytically prove that an infinitesimal Hubbard attraction can lead to the Cooper instability as long as the density of state is nonzero at the Fermi level, which provides the basis of the SC on the QC. Our numerical results yield that the Cooper pairing always takes place between the two time-reversal states, satisfying the Anderson's theorem. On this theorem, we perform a mean-field (MF) study at both zero and finite temperatures. The MF study also shows that an arbitrarily weak attraction can lead to the pairing order, with the resulted pairing state being well described by the BCS theory and the thermal dynamic behaviors being well consistent with experimental results. The second issue is about the superfluid density on the QC without translational symmetry. It's clarified that although the normal state of the system locates at the critical point of the metal-insulator transition, the pairing state exhibits real SC, carrying finite superfluid density that can be verified by the Meissner effect, consistent with experiment also. These revealed properties of the SC on the Penrose lattice are universal for all QCs., Comment: 11 pages, 6 figures

The recent discovery of quasi-one-dimensional Chromium-based superconductivity has generated much excitement. We study in this work the superconducting instabilities of a representative compound, the newly synthesized KCr$_3$As$_3$ superconductor. Based on inputs from density functional theory calculations, we first construct an effective multi-orbital tight-binding Hamiltonian to model its low-energy band structure. We then employ standard random-phase approximation calculations to investigate the superconducting instabilities of the resultant multi-orbital Hubbard model. We find various pairing symmetries in the phase diagram in different interaction parameter regimes, including the triplet $f$-wave, $p_z$-wave and singlet $s^\pm$-wave pairings. We argue that the singlet $s^\pm$-wave pairing, which emerges at intermediate interaction strength, is realized in this material. This singlet pairing is driven by spin-density wave fluctuations enhanced by Fermi-surface nesting. We point out that phase-sensitive measurement can distinguish the $s$-wave pairing in KCr$_3$As$_3$ from the $p_z$-wave previously proposed for a related compound K$_2$Cr$_3$As$_3$. The $s^{\pm}$-wave pairing in KCr$_3$As$_3$ shall also exhibit a subgap spin resonance mode near the nesting vector, which can be tested by inelastic neutron scattering measurements. Another intriguing property of the $s^\pm$-pairing is that it can induce time-reversal invariant topological superconductivity in a semiconductor wire with large Rashba spin-orbit coupling via proximity effect. Our study shall be of general relevance to all superconductors in the family of ACr$_3$As$_3$ (A=K, Rb, Cs).