On the δ-function broadening in Kubo-Greenwood equation

We propose a solution to a long-standing issue in the calculation of Kubo-Greenwood DC conductivity in finite-sized quantum systems. The Kubo-Greenwood equation contains a sum of delta-functions, which are usually broadened. The estimate of the DC conductivity depends significantly on the type of broadening applied, making the estimate ambiguous. We eliminate the ambiguity by mapping the broadening onto a specific way of introduction of the electric field and making a correction based on Drude equation. We also consider the influence of delta-function broadening on conductivity averaged over disorder. We demonstrate the influence can be reduced to a convolution of the average conductivity with the broadening function over the frequency. Even though in finite systems the average conductivity is an oscillatory function of frequency, if convolved with a slowly varying function, it behaves similarly to a Lorentzian. We propose a procedure of extracting the parameters of the Lorentzian by extrapolation similarly to a single system.