Theory of non-linear electron-phonon coupling and its first-principles implementation

Despite a huge effort has been devoted to develop first principles methods to calculate efficiently the impact of lattice quantum effects and anharmonicity on the structural, vibrational, and superconducting properties of materials, the electron-phonon interaction is always calculated in the linear approximation, i.e., truncating the expansion of the electronic potential at first order. This approach is questionable, at least, whenever anharmonic effects on the vibrational properties are large, such as in superconducting hydrides and systems undergoing charge-density wave or ferroelectric transitions. Here we present a novel non-perturbative theory for the electron-phonon coupling that can be implemented from first principles. We apply the new theory to superconducting palladium hydrides and show that, remarkably, high-order non-linear effects are comparable in magnitude to the standard linear term. Non-linearity reveals crucial to explain the superconductivity as well as the inverse isotope effect in this system. Our new theory may have a large impact on the ab initio calculation of all properties related to the electron-phonon interaction, e.g. superconductivity and electrical conductivity, in highly anharmonic systems.