Magnetism and experimental consequences of $p_z$-wave spin triplet state in quasi-one-dimensional A$_2$Cr$_3$As$_3$ superconductors

The recently discovered quasi-one dimensional superconductors A$_2$Cr$_3$As$_3$(A=K,Rb,Cs), are found to possess strong frustrated magnetic fluctuations and are nearby a novel in-out co-planar magnetic ground state. Then, we find that the triplet $p_z$-wave pairing is strongly favored. Finally, with $p_z$ wave pairing state, we obtain the specific heat, superfluid density, Knight shift and spin relaxation rate and find that all these properties at low temperature ($T\ll T_c$) show powerlaw behaviors and are consistent available experiments. Particularly, the superfluid density determined by the $p_z$-wave pairing state in this quasi-one dimensional system is anisotropic: the in-plane superfluid density varies as $\Delta \rho_{\parallel} \sim T$ but the out-plane one varies as $\Delta \rho_{\perp} \sim T^3$ at low temperature. The anisotropic upper critical field reported in experiment is consistent with the $S_z=0$ (i.e,($\uparrow\downarrow+\downarrow\uparrow$)) $p_z$-wave pairing state. We also suggest the phase-sensitive dc-SQUID measurements to pin down the triplet $p_z$-wave pairing state. References: [1] X. Wu et al., Chin. Phys. Lett. 32, 057401 (2015). [2] X. Wu et al., Phys. Rev. B 92,104511 (2015). [3] X. Wu et al., arXiv: 1507.07451 (2015)