Multiscale Modeling &Simulation, Volume 18, Issue 4, Page 1525-1564, January 2020.
This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. First, 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 contributions and deduce an approximation scheme which is exponentially convergent in terms of domain size. Second, 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 $\mathcal{O}(\eta^{-3/2})$ inner products for a suitably truncated Chebyshev series, which significantly improves on the $\mathcal{O}(\eta^{-2})$ inner products required by a naive Chebyshev approximation. Third, 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 $\mathcal{O}(\beta^2),$ but the rational scheme scales much more mildly with respect to $\beta$.

中文翻译：

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二维非对称双层Kubo电导率的建模与计算

多尺度建模与仿真，第18卷，第4期，第1525-1564页，2020年1月。
本文提出了一种统一的方法，用于有限温度下非对称双层异质结构的久保电导率的建模和计算。首先，我们根据电导率函数相对于电流-电流相关性度量的积分，得出久保-格林伍德电导率的大体极限表达式。然后，我们观察到可以利用不等价的结构将电流-电流相关性度量分解为局部贡献，并得出在域大小方面呈指数收敛的近似方案。其次，我们通过Chebyshev近似分析计算局部电导率的成本。我们的主要发现是，如果逆温度$ \ beta $与逆弛豫时间$ \ eta $相比足够小，即$ \ beta \ lesssim \ eta ^ {-1/2} $，那么对于适当截断的Chebyshev系列，主要的计算成本是$ \ mathcal {O}（\ eta ^ {-3/2}）$内部产品，这在$ \ mathcal {O}（\ eta ^ {- 2}）$天真切比雪夫近似所需的内部乘积。第三，我们针对低温状态$ \ eta ^ {-1/2} \ lesssim \ beta $提出了一种有理逼近方案，其中多项式方法的成本增加到$ \ mathcal {O}（\ beta ^ 2 ），$，但是相对于$ \ beta $，理性方案的伸缩幅度要小得多。