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 ${\mathrm{MgB}}_2$ 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.

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• Anharmonic lattice dynamics
• Electron-phonon coupling
• Superconducting gap
• Superconducting order parameter
• Superconducting phase transition
• Transition temperature
• Diagrammatic methods
• Eliashberg theory
• Green's function methods
• Perturbation theory
• Second quantization