Hot-lines topology and the fate of the spin resonance mode in three-dimensional unconventional superconductors

In the quasi-two-dimensional (quasi-2D) copper- and iron-based superconductors, onset of superconductivity is accompanied by a prominent peak in magnetic spectrum at momenta close to wave-vector nearby antiferromagnetic state. Such well described terms spin resonance mode, i.e., spin-1 exciton theoretically predicted for quasi-2D superconductors with sign-changing gap. The same theories, however, indicate that such mode should be absent three-dimensional (3D) system spherical Fermi surface. This raises question fate layered unconventional are not strongly anisotropic, as certain heavy-fermion compounds potentially newly discovered nickelate superconductor NdNiO$_2$. Here, we use random-phase-approximation calculate dynamical susceptibility 3D $d_{x^2-y^2}$-wave gap symmetry corrugated cylindrical-like surfaces. By varying out-of-plane hopping anisotropy $t_z/t$, demonstrate appearance determined topology hot lines -- i.e. on surface connected wave-vector. For an in-plane wave-vector, undergo topological transition from open closed loops critical $t_z/t$ value. cross nodal superconducting lines, making overdamped incoherent. contrast, remain remains sharp. We discuss experimental implications our results dispersion and, more generally, inelastic neutron scattering experiments superconductors.