Superconductivity in the regime of attractive interactions in the Tomonaga-Luttinger liquid

While the vast majority of known physical realizations of the Tomonaga-Luttinger liquid (TLL) have repulsive interactions defined with the dimensionless interaction parameter $K_{\rm c}<1$, we here report that Rb$_2$Mo$_3$As$_3$ is in the opposite TLL regime of attractive interactions. This is concluded from a TLL-characteristic power-law temperature dependence of the $^{87}$Rb spin-lattice relaxation rates over broad temperature range yielding the TLL interaction parameter for charge collective modes $K_{\rm c}=1.4$. The TLL of the one-dimensional band can be traced almost down to $T_{\rm c} = 10.4 $~K, where the bulk superconducting state is stabilized by the presence of a three-dimensional band and characterized by the $^{87}$Rb temperature independent Knight shift and the absence of Hebel-Slichter coherence peak in the relaxation rates. The small superconducting gap measured in high magnetic fields reflects either the importance of the vortex core relaxation or the uniqueness of the superconducting state stemming from the attractive interactions defining the precursor TLL.