Mode-selective phonon effects on magnetism and superconductivity in trilayer nickelates

Abstract

The discovery of high-$T_c$ superconductivity in multilayer nickelates has stimulated intense interest, yet the microscopic origins of their correlated states remain debated. Recently the observed isotope effect of the magnetic state in trilayer nickelates highlights the important role of lattice vibrations in shaping these systems. Here, we investigate the impact of electron-phonon coupling (EPC) on magnetism and superconductivity in trilayer nickelates using ab initio and functional renormalization group calculations. We identify out-of-plane oxygen breathing modes with pronounced EPC and construct symmetry-resolved, layer-dependent couplings. Their relative phases across the three layers generate distinct retarded interactions, producing mode-selective enhancement or suppression of spin-density-wave order and $s_\pm$-wave superconductivity. This selectivity arises from screening-mediated spin-phonon interplay across a hierarchy of energy scales: EPC modifies charge screening at higher energies, reshapes magnetic fluctuations at lower energies, and ultimately influences pairing. Selected modes yield negative isotope coefficients for the magnetic state, qualitatively consistent with experiments. Our results establish how interlayer phonons modify electronically driven order and provide a microscopic framework for understanding isotope effects in trilayer nickelates.