Modeling and Computation of Kubo Conductivity for 2D Incommensurate Bilayers

This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. Firstly, we derive an expression for the large-body limit of Kubo-Greenwood conductivity in terms of an integral of the conductivity function with respect to a current-current correlation measure. We then observe that the incommensurate structure can be exploited to decompose the current-current correlation measure into local contribution and deduce an approximation scheme which is exponentially convergent in terms of domain size. Secondly, we analyze the cost of computing local conductivities via Chebyshev approximation. Our main finding is that if the inverse temperature $\beta$ is sufficiently small compared to the inverse relaxation time $\eta$, namely $\beta \lesssim \eta^{-1/2}$, then the dominant computational cost is $\mathcal{O}\bigl(\eta^{-3/2}\bigr)$ inner products for a suitably truncated Chebyshev series, which significantly improves on the $\mathcal{O}\bigl(\eta^{-2}\bigr)$ inner products required by a naive Chebyshev approximation. Thirdly, we propose a rational approximation scheme for the low temperature regime $\eta^{-1/2} \lesssim \beta$, where the cost of the polynomial method increases up to $\mathcal{O}\bigl(\beta^2\bigr),$ but the rational scheme scales much more mildly with respect to $\beta$.