Drude Conductivity of a Granular System

TL;DR

This paper derives the Drude conductivity for granular systems using diagrammatic methods, addressing convergence issues and providing a formula consistent with Einstein's relation and Fermi's golden rule.

Contribution

It offers a complete derivation of granular Drude conductivity, highlighting convergence challenges and resolving them through analytic continuation and integration techniques.

Findings

Derived the formula for granular Drude conductivity

Identified and addressed convergence issues in the derivation

Confirmed the validity of naive momentum summation with proper interpretation

Abstract

We present a complete derivation of the granular analogue to Drude conductivity using diagrammatic methods. The convergence issues arising when changing the order of momentum and frequency summation are more severe than in the homogeneous case. This is because there are now two momentum sums rather than one, due to the intragrain momentum scrambling in tunnelling events. By careful analytic continuation of the frequency sum, and use of integration by parts, we prove that the system is in the normal (non-superconducting) state, and derive the formula for the granular Drude conductivity expected from Einstein's relation and Fermi's golden rule. We also show that naively performing the momentum sums first gives the correct result, provided that we interpret a divergent frequency sum by analytic continuation using the Hurwitz zeta function.