Quantify the magnitude of nonlinear electron–phonon coupling (λ2) in real materials

Determine the attainable magnitude of the nonlinear electron–phonon coupling strength λ2 in superconductors that exhibit significant anharmonicity by computing the full 1‑electron–2‑phonon matrix element γ(p,k,q) for realistic materials and quantifying its contribution to the extended Eliashberg spectral function.

Background

The paper extends Migdal–Eliashberg theory by explicitly including a 1‑electron–2‑phonon (nonlinear) interaction, showing that the standard form of the Eliashberg equations is preserved while the Eliashberg spectral function acquires additional nonlinear contributions. In a dispersionless-phonon toy model, these contributions are parameterized by λ2, which augments the electron–phonon coupling and increases the superconducting critical temperature.

Despite the theoretical framework, the authors highlight a key gap: there is currently no computationally viable method to evaluate the full nonlinear matrix element γ(p,k,q) in realistic materials. Consequently, the size of λ2 in actual superconductors, particularly those with strong anharmonicity, remains undetermined. Establishing ab initio procedures to compute γ(p,k,q) would enable quantitative predictions of λ2 and its impact on Tc.