Neutron Spin Resonance Near a Lifshitz Transition in Overdoped Ba$_{0.4}$K$_{0.6}$Fe$_2$As$_2$

Elucidating the relationship between spin excitations and fermiology is essential for clarifying the pairing mechanism in iron-based superconductors (FeSCs). Here, we report inelastic neutron scattering results on the hole overdoped Ba$_{0.4}$K$_{0.6}$Fe$_2$As$_2$ near a Lifshitz transition, where the electron pocket at $M$ point is nearly replace by four hole pockets. In the normal state, the spin excitations are observed at incommensurate wave vectors with chimney-like dispersions. By cooling down to the superconducting state, a neutron spin resonance mode emerges with a peak energy of $E_r=$ 14-15 meV weakly modulated along $L$-direction. The incommensurability notably increases at low energies, giving rise to downward dispersions of the resonance mode. This behavior contrasts sharply with the upward dispersions of resonance observed in optimally doped Ba$_{0.67}$K$_{0.33}$Fe$_2$As$_2$ contributed by the hole to electron scattering, but resembles with the cases in KFe$_2$As$_2$ and KCa$_2$Fe$_4$As$_4$F$_2$ where the fermiology are dominated by hole pockets. These results highlight the critical role of electronic structure modifications near the Fermi level, especially in governing interband scattering under imperfect nesting conditions, which fundamentally shape the spin dynamics of FeSCs.