Singlet $s^\pm$-wave pairing in quasi-one-dimensional ACr$_3$As$_3$ (A=K, Rb, Cs) superconductors.

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).