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

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 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 $1/\sqrt{e}$, 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 $(\ensuremath{\delta}$ function) of a fixed strength, is a universal function of frequency, independent of temperature. The $1/\sqrt{e}$ 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.

Locations

• Physical review. B./Physical review. B - View
• arXiv (Cornell University) - View - PDF
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