分支割上格点格林函数的一般计算方法
A general method for calculating lattice Green functions on the branch cut
摘要：
提出了一种计算晶格上任意两个格点$i$和$j$之间任意实频率$ω$下复格林函数$G_{ij} (ω)$的方法。从正方形，立方，蜂窝状，三角形，体心立方，面心立方和菱形晶格上的行走数开始，我们推导出$G_{ij} (ω)$的切比雪夫展开系数。Chebyshev级数的收敛可以通过构造函数$f(ω)$来加速，该函数模仿$G_{ij} (ω)$中的van Hove奇点，并从原始系数中减去它们的Chebyshev系数。我们证明了这一点明确的正方形晶格和体心立方晶格。我们的算法使用1000个系列项达到6- 9个有效数字的典型精度。
英文摘要：
We present a method for calculating the complex Green function $G_{ij} (ω)$ at any real frequency $ω$ between any two sites $i$ and $j$ on a lattice. Starting from numbers of walks on square, cubic, honeycomb, triangular, bcc, fcc, and diamond lattices, we derive Chebyshev expansion coefficients for $G_{ij} (ω)$. The convergence of the Chebyshev series can be accelerated by constructing functions $f(ω)$ that mimic the van Hove singularities in $G_{ij} (ω)$ and subtracting their Chebyshev coefficients from the original coefficients. We demonstrate this explicitly for the square lattice and bcc lattice. Our algorithm achieves typical accuracies of 6--9 significant figures using 1000 series terms.
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