论无平移不变性的格点和的计算
On the computation of lattice sums without translational invariance
AI总结：
针对带边界几何中因平移不变性丧失导致格点和计算困难的问题，提出基于推广Riemann zeta函数的指数收敛数值算法，其运行时间与粒子数无关，可高效计算宏观凝聚态系统的相关求和。
摘要：
本文提出了一种高效计算带边界几何结构中振荡多维格点和的新方法。这类求和在纯数学与应用数学中普遍存在，在凝聚态物理和拓扑量子物理中有直接应用。这类求和的计算难点在于奇异长程相互作用，以及边界导致的平移不变性丧失，使得标准工具失效。本研究表明，这些格点和可通过将Riemann zeta函数推广到多维非周期格点和来生成。我们提出了该zeta函数的一种新表示形式，以及一种能在广泛几何结构中实现指数收敛的数值算法。值得注意的是，我们方法的运行时间仅受所考虑几何结构的复杂度影响，与粒子数量无关，为宏观凝聚态系统的高效模拟奠定了基础。我们通过计算含$3\times 10^{23}$个粒子的三维晶体结构中的相互作用能，展示了该方法的实用性。大量数值实验验证了我们方法的准确性。本文随文在线提供了参考实现。
英文摘要：
This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with $3\times 10^{23}$ particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.
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