标题： 非线性电子-声子相互作用在米格达尔-埃利亚斯伯格理论中 显示英文标题

标题： Nonlinear electron-phonon interactions in Migdal-Eliashberg theory

摘要： 基于吸引性电子-声子相互作用的超导系统是在基本层面上最被理解的。 它们可以用埃利亚什伯格理论很好地描述，该理论不同于BCS理论，明确考虑了声子动力学。 通常假设只有线性电子-声子相互作用是相关的。 然而，对于一些超导体如MgB$_2$或基于氢化物的超导体，已知非线性电子-声子相互作用会显著贡献，而传统埃利亚什伯格理论并未考虑这一点。 我们通过引入非线性电子-声子相互作用对埃利亚什伯格理论进行了修改。 我们表明，除了埃利亚什伯格谱函数的非线性扩展外，埃利亚什伯格方程保持不变。 这个扩展的谱函数可以作为未来从头计算的基准。 我们用它构建了一个解析的玩具模型，并表明非线性电子-声子耦合会影响虚轴和实轴上的超导能隙函数，并导致超导临界温度的增加。 显示英文摘要

摘要： 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 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.