Nonlinear electron-phonon interactions in Migdal-Eliashberg theory

Superconducting systems based on attractive electron-phonon interactions are the ones which are best understood at a fundamental level. They are well described using Eliashberg theory, which, unlike BCS theory, explicitly takes into account phonon dynamics. It is most often assumed that only linear electron-phonon interactions are relevant. However, for some superconductors like MgB$_2$ or hydride based superconductors, nonlinear electron-phonon interactions are known to contribute significantly, which is not taken into account in conventional Eliashberg theory. We provide a modification to Eliashberg theory by introducing nonlinear electron-phonon interactions. We show that the Eliashberg equations remain unchanged apart from a nonlinear extension of the Eliashberg spectral function. This extended spectral function can be used as a baseline for future ab initio calculations. We use it to construct an analytical toy model and show that the nonlinear electron-phonon coupling affects the superconducting gap function on the imaginary and real axis and causes an increase in the superconducting critical temperature.