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Eliashberg theory in the weak-coupling limit: Results on the real frequency axis

机译：弱耦合极限中的Eliashberg理论：实频轴上的结果

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摘要

We formulate and solve the Eliashberg equations on the imaginary frequency axis at temperatures below T_c in the weak-coupling limit. We find an excellent scaling at all temperatures, for a fixed coupling strength, and the normalized order parameter exhibits a BCS-like temperature dependence. The hybrid real-imaginary axis equations are also solved to obtain numerically exact analytic continuations from the imaginary frequency axis to the real frequency axis. This provides a determination of the gap edge, which, in the weak-coupling limit, is identical to the order parameter from the imaginary axis. The analytical result for the zero-temperature gap edge deviates from the BCS result by a factor of 1/e~(1/2), which was also obtained for the transition temperature T_c. We show that the normalized gap function on both the real and imaginary frequency axes, for an electron-phonon Einstein spectrum (δ function) of a fixed strength, is a universal function of frequency, independent of temperature. The 1/e~(1/2) correction is a result of this nontrivial frequency dependence in the gap function. This modification, in the gap edge and in T_c serves to preserve various dimensionless ratios to their BCS values.
机译：我们在弱耦合极限内低于T_c的温度下，在虚频率轴上制定并求解Eliashberg方程。对于固定的耦合强度，我们发现在所有温度下均具有出色的缩放比例，并且归一化阶次参数显示出类似BCS的温度依赖性。还求解了混合实虚轴方程，以获得从虚频轴到实频轴的数值精确解析连续性。这提供了间隙边缘的确定，该间隙边缘在弱耦合极限中与虚轴上的阶次参数相同。零温度间隙边缘的分析结果与BCS结果的偏差为1 / e〜（1/2），这也是在转变温度T_c下获得的。我们表明，对于固定强度的电子声子爱因斯坦谱（δ函数），实，虚轴上的归一化间隙函数是频率的通用函数，与温度无关。 1 / e〜（1/2）校正是间隙函数中这种非平凡的频率依赖性的结果。在间隙边缘和在T_c中的这种修改用于保持与其BCS值的各种无量纲的比率。

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