Electrical conductivity and NMR relaxation rate of Eliashberg superconductors in the weak-coupling limit

Electrical conductivity is an important transport response in superconductors, enabling clear signatures of dynamical interactions to be observed. Of primary interest is to study signatures of the electron-phonon interaction in weak-coupling Eliashberg theory (Eth), and to note the distinctions from BCS theory. Recent analysis of weak-coupling Eth has shown that while there are modifications from the BCS results, certain dimensionless ratios are in agreement. Here we show that the conductivities in BCS theory and Eth fundamentally differ, with the latter having an imaginary gap component that damps a divergence. We focus on the dirty limit, and for both BCS theory and Eth we derive expressions for the small-frequency limit of the real conductivity. For Eth specifically, there are two limits to consider, depending on the relative size of the frequency and the imaginary part of the gap. In the case of identically zero frequency, we derive an analytical expression for the NMR relaxation rate. Our analysis of the conductivity complements the previous study of the Meissner response and provides a thorough understanding of weak-coupling Eth.