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We study the pairing symmetry and the topological properties of thehydrogen-doped ACr$_3$As$_3$ superconductors. Based on our first-principle bandstructure with spin-orbit-coupling (SOC), we construct tight binding modelincluding the on-site SOC terms, equipped with the multi-orbital Hubbardinteractions. Then using the random-phase-approximation (RPA) approach, wecalculate the pairing phase diagram of this model. Our RPA results yield thetriplet pz-wave pairing in the($\left(\uparrow\downarrow+\downarrow\uparrow\right)$) spin channel to be theleading pairing symmetry all over the experiment relevant hydrogen-dopingregimes. This pairing state belongs to the spin-U(1)-symmetry protectedtime-reversal-invariant topological nodal-line superconductivity. Determined bythe momentum-dependent topological invariant Z (kx; ky), the whole (001)surface Brillouin zone is covered with topological flat bands with differentregimes covered with different numbers of flat surface bands, which can bedetected by the scanning tunneling microscope experiments.

Topological $p_z$-wave nodal-line superconductivity with flat surface bands in the AH$_{x}$Cr${_3}$As${_3}$ (A=Na, K, Rb, Cs) superconductors