We analyze the validity of Eliashberg theory of phonon-mediated superconductivity in 2D systems in light of recent extensive Monte-Carlo studies of the Holstein model. Conventional wisdom says that Eliashberg theory is applicable as long as vertex corrections remain small. For small ratio of the phonon energy $\Omega_0$ and the Fermi energy $E_F$, this condition is supposed to hold even when the dimensionless electron-phonon coupling $\lambda$ is larger than one, i.e., in the strong coupling regime. A comparison between various quantities computed in the Migdal approximation and those computed by Quantum Monte Carlo prove that this belief is wrong, and we identify analytically some of the ways in which this breakdown occurs for various "normal state" properties at $\lambda = \lambda_{cr}$, where $\lambda_{cr} = O(1)$. The breakdown occurs at temperatures high enough that neither superconducting nor charge-density wave correlations extend over any significant range of distances, so it cannot be associated with the onset of an instability toward any of the relevant ordered ground-states - rather it is associated with the local physics of classical bipolaron formation. Still, we show that certain properties, including the superconducting $T_c$ and the superconducting gap structure below $T_c$, can be accurately inferred from the strong-coupling limit of Eliashberg theory at $\lambda \leq \lambda_{cr}$.

中文翻译：

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Eliashberg 声子介导超导理论——何时有效以及如何失效

根据最近对 Holstein 模型的广泛蒙特卡罗研究，我们分析了 Eliashberg 2D 系统中声子介导超导理论的有效性。传统观点认为，只要顶点修正量很小，Eliashberg 理论就适用。对于声子能量 $\Omega_0$ 和费米能量 $E_F$ 的小比率，即使无量纲电子 - 声子耦合 $\lambda$ 大于 1，即在强耦合状态下，该条件也应该成立。在 Migdal 近似中计算的各种量与由 Quantum Monte Carlo 计算的量之间的比较证明这种信念是错误的，我们通过分析确定了在 $\lambda = \ 的各种“正常状态”属性发生这种崩溃的一些方式lambda_{cr}$，其中 $\lambda_{cr} = O(1)$。击穿发生在足够高的温度下，超导和电荷密度波的相关性都不会延伸到任何显着的距离范围内，因此它不能与任何相关有序基态的不稳定性的开始有关 - 相反，它与经典双极子形成的局部物理学。尽管如此，我们仍然表明某些性质，包括超导 $T_c$ 和低于 $T_c$ 的超导间隙结构，可以从 Eliashberg 理论在 $\lambda\leq\lambda_{cr}$ 的强耦合极限中准确推断出来。所以它不能与对任何相关有序基态的不稳定性的开始有关——而是与经典双极子形成的局部物理学有关。尽管如此，我们仍然表明某些性质，包括超导 $T_c$ 和低于 $T_c$ 的超导间隙结构，可以从 Eliashberg 理论在 $\lambda\leq\lambda_{cr}$ 的强耦合极限中准确推断出来。所以它不能与对任何相关有序基态的不稳定性的开始有关——而是与经典双极子形成的局部物理学有关。尽管如此，我们仍然表明某些性质，包括超导 $T_c$ 和低于 $T_c$ 的超导间隙结构，可以从 Eliashberg 理论在 $\lambda\leq\lambda_{cr}$ 的强耦合极限中准确推断出来。