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.

翻译：本文提出了一种用于高效计算边界几何中振荡多维晶格求和的新方法。此类求和在纯数学与应用数学中普遍存在，并可直接应用于凝聚态物理与拓扑量子物理领域。其评估挑战源于奇异长程相互作用与边界导致的平移不变性缺失的双重叠加，使得标准工具失效。我们的研究表明，这些晶格求和可通过将黎曼ζ函数推广至多维非周期晶格求和来生成。我们提出了该ζ函数的新表示形式，并配以数值算法，确保在广泛的几何结构中实现指数级收敛。值得注意的是，该方法的运行时间仅受所考虑几何结构的复杂度影响，而与粒子数量无关，从而为宏观凝聚态系统的高效模拟奠定基础。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价值。广泛的数值实验验证了其精度。本文附带了参考实现的在线代码。