We solve the anisotropic, full-bandwidth, and nonadiabatic Eliashberg equations for phonon-mediated superconductivity by fully including the first vertex correction in the electronic self-energy. The nonadiabatic equations are solved numerically here without further approximations, for a one-band model system. We compare the results to those that we obtain by adiabatic full-bandwidth, as well as Fermi-surface-restricted Eliashberg-theory calculations. We find that nonadiabatic contributions to the superconducting gap can be positive, negative, or negligible, depending on the dimensionality of the considered system, the degree of nonadiabaticity, and the coupling strength. We further examine nonadiabatic effects on the transition temperature and the electron-phonon coupling constant. Our treatment emphasizes the importance of overcoming previously employed approximations in estimating the impact of vertex corrections on superconductivity and opens a pathway to systematically study vertex correction effects in systems such as high-$T_c$, flat band, and low-carrier density superconductors.

中文翻译：

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超导的全带宽Eliashberg理论超越Migdal近似

通过在电子自能中完全包含第一个顶点校正，我们解决了声子介导的超导性的各向异性，全带宽和非绝热Eliashberg方程。对于单频带模型系统，非绝热方程式在此处无需进一步近似即可在数值上求解。我们将结果与绝热全带宽以及费米面受限的Eliashberg理论计算得到的结果进行比较。我们发现，对超导间隙的非绝热贡献可以是正的，负的或微不足道的，这取决于所考虑系统的尺寸，非绝热的程度和耦合强度。我们进一步研究了非绝热对转变温度和电子-声子耦合常数的影响。${Ť}_C$，扁平带和低载流子密度超导体。