存在费米海时实空间格点传播子的高效计算
Efficient calculation of real-space lattice propagators in the presence of a Fermi sea
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中文总结 AI 辅助
本研究提出一种与维度无关的高效方法，用于计算存在费米海时任意紧束缚哈密顿量的实空间格点传播子，经超立方晶格验证，可体现费米面形状对传播子空间结构的影响。
中文摘要 AI 辅助
我们提出一种高效方法，用于计算存在载流子浓度为$x$的费米海时，任意紧束缚哈密顿量的实空间传播子（格点格林函数）。该方法适用于任意晶格、色散关系和维度，只要已知对应$x=0$时的传播子即可。我们证明，有限$x$下粒子添加传播子的适当组合，其实部或虚部与$x=0$时的对应量存在简单关系，剩余部分则遵循克拉默斯-克勒尼希关系，可通过快速傅里叶变换一次性对所有能量进行计算。因此，该方法的计算成本与维度无关，这与直接布里渊区积分的计算成本不同。粒子移除传播子可通过通用恒等式得到。我们针对一维、二维和三维超立方晶格，将该方法与直接积分结果对比进行验证，并利用二维正方晶格说明费米面的形状如何印刻在传播子的空间结构上。特别地，在远离开带的能量下，传播子映射会收敛到夫琅禾费衍射图样，其孔径为布里渊区的未占据部分。
英文摘要
We present an efficient method for computing the real-space propagators (lattice Green's functions) of any tight-binding Hamiltonian in the presence of a Fermi sea with carrier concentration $x$. The method is valid for any lattice, dispersion, and dimension, provided the corresponding $x=0$ propagators are known. We show that suitable combinations of the finite-$x$ particle-addition propagators have a real or imaginary part trivially related to their $x=0$ counterpart, while the remaining part follows from a Kramers-Kronig relation that can be evaluated for all energies at once using the fast Fourier transforms. The computational cost is therefore independent of the dimensionality, unlike that of direct Brillouin-zone integration. The particle-removal propagators follow from general identities. We validate the method against direct integration for hypercubic lattices in one, two, and three dimensions, and use the 2D square lattice to illustrate how the shape of the Fermi surface is imprinted on the spatial structure of the propagators. In particular, at energies far outside the band, the propagator maps converge to the Fraunhofer diffraction pattern whose aperture is the unoccupied part of the Brillouin zone.
