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

TL;DR

This paper formulates and solves Eliashberg equations in the weak-coupling limit, revealing universal frequency dependence and precise analytic continuations, with results closely aligning with BCS theory but including specific deviations.

Contribution

It provides a detailed numerical and analytical analysis of Eliashberg equations on real and imaginary axes in the weak-coupling limit, including exact analytic continuations and universal gap functions.

Findings

Normalized order parameter exhibits BCS-like temperature dependence.

Analytic continuation yields exact gap edge on real axis.

Zero-temperature gap deviates from BCS by a factor of 1/√e.

Abstract

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 given 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 $1/\sqrt{e}$, 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 ($\delta$-function) of a given strength, is a universal function of frequency, independent of temperature. The $1/\sqrt{e}$ correction is a result of this non-trivial 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.