Electron-phonon interaction plays an important role in metals and can lead to superconductivity and other instabilities. Previous theoretical studies on superconductivity are largely based on the Migdal-Eliashberg theory, which neglects all the vertex corrections to electron-phonon coupling and breaks down in many unconventional superconductors. Here, we go beyond the Migdal-Eliashberg approximation and develop a nonperturbative Dyson-Schwinger equation approach to deal with the superconducting transition. Remarkably, we take into account all the vertex corrections by solving two coupled Ward-Takahashi identities derived from two global U(1) symmetries and rigorously prove that the fully renormalized electron propagator satisfies a self-closed integral equation that is directly amenable to numerical computations. Our approach works equally well in the weak and strong coupling regimes and provides an efficient method to determine superconducting $T_c$ and other quantities. As an application, our approach is used to investigate the high-$T_c$ superconductivity in one-unit-cell FeSe/${\mathrm{SrTiO}}_3$.

中文翻译：

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寻求电子-声子相互作用引起的超导Tc的精确解

电子-声子相互作用在金属中起重要作用，并可能导致超导性和其他不稳定性。先前有关超导性的理论研究主要基于Migdal-Eliashberg理论，该理论忽略了所有对电子-声子耦合的顶点校正，并在许多非常规超导体中分解了。在这里，我们超越了Migdal-Eliashberg逼近，并开发了一种非扰动的Dyson-Schwinger方程方法来处理超导转变。值得注意的是，我们通过求解源自两个全局U（1）对称性的两个Ward-Takahashi耦合恒等式来考虑所有顶点校正，并严格证明完全重正则化的电子传播子满足直接可用于数值计算的自闭合积分方程。 。${Ť}_C$和其他数量。作为一种应用，我们的方法用于调查高${Ť}_C$ 单元电池FeSe /中的超导性${\mathrm{钛酸锶}}_3$。