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Using first-principle calculations, we investigate the electronic,topological and superconducting properties of Nb$_3$X (X = Ge, Sn, Sb) andTa$_3$Y (Y = As, Sb, Bi) A15 compounds. We demonstrate that these compoundshost Dirac surface states which are related to a nontrivial Z$_2$ topologicalvalue. The spin-orbit coupling (SOC) splits the eightfold degenerate R pointclose to the Fermi level enhancing the amplitude of the spin Hall conductance.Indeed, despite the moderate spin-orbit of the Nb-compounds, a large spin Halleffect is also obtained in Nb$_3$Ge and Nb$_3$Sn compounds. We show that theCoulomb interaction opens the gap at the R point thus making more evident theoccurrence of Dirac surface states. We then investigate the superconductingproperties by determining the strength of the electron-phonon BCS coupling. Theevolution of the critical temperature is tracked down to the 2D limitindicating a reduction of the transition temperature which mainly arises fromthe suppression of the density of states at the Fermi level. Finally, wepropose a minimal tight-binding model based on three coupledSu-Schrieffer-Heeger chains with t$_{2g}$ Ta- and Nb-orbitals reproducing thespin-orbit splittings at the R point among the $\pi$-bond bands in this classof compounds. We separate the kinetic parameters in $\pi$ and $\delta$-bonds,in intradimer and interdimer hoppings and discuss their relevance for thetopological electronic structure. We point out that Nb$_3$Ge might represent aZ$_2$ topological metal with the highest superconducting temperature everrecorded.

Dirac surface states, multiorbital dimerization and superconductivity in Nb- and Ta-based A15 compounds