发表机构
Saarland University; ETH Zürich; German Aerospace Center (DLR)（萨尔兰州立大学; 苏黎世联邦理工学院; 德国航空航天中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义并分析各向异性Epstein zeta函数，推导其稳定可计算表示，用于高效精确计算各向异性幂律核晶格求和，并开发数值算法，在多种晶格、指数和阶数下达到机器精度。
AI 中文摘要
涉及幂律核的大规模晶格求和的精确高效评估，是模拟具有长程相互作用的经典和量子系统中的一个基本计算问题。虽然针对空间各向同性核的方法（其中一些基于Epstein zeta函数）近年来取得了显著进展，但各向异性情形却相对滞后，尽管其与基础及有效相互作用（如磁性材料中的偶极相互作用）具有广泛相关性。在本工作中，我们通过定义和分析各向异性Epstein zeta函数来解决这一问题，并推导出可由各向同性相互作用核的晶格求和的波矢导数获得的稳定可计算表示。这些函数可直接应用于各向异性相互作用晶格系统的解析和数值研究。进一步地，它们在经典Euler-Maclaurin求和公式推广到涉及幂律核的晶格及被加项的近期泛化中，提供了离散晶格与其连续类比之间精确等价的修正项。它们与zeta函数高阶导数的联系可用于提高数值算法的收敛速度，例如在微磁学中，或提供适合广义zeta函数预计算的快速收敛展开。我们推导了各向异性Epstein zeta函数的稳定可计算表示，包括解析去除Rayleigh-Wood奇异性的可能性，并开发了一种数值算法，用于对任意晶格、幂律衰减指数和各向异性阶数进行稳定评估。我们将该算法与闭式恒等式、直接求和及多精度结果进行基准测试，在各种晶格、幂律指数和各向异性阶数下均获得机器精度。
英文摘要
The precise and efficient evaluation of large-scale lattice sums involving power-law kernels is a fundamental computational problem in the simulation of classical and quantum systems with long-range interactions. While methods for spatially isotropic kernels, some based on Epstein zeta functions, have advanced considerably in recent years, the anisotropic case has lagged behind, despite its broad relevance to both fundamental and effective interactions such as the dipole interaction in magnetic materials. In this work, we solve this issue by defining and analyzing anisotropic Epstein zeta functions for which we derive stably computable representations obtained from wave vector derivatives of lattice sums over isotropic interaction kernels. These functions find direct application in the analytical and numerical study of anisotropically interacting lattice systems. Going further, they provide the correction term in an exact equivalence between discrete lattices and their continuous analogs in a recent generalization of the classical Euler-Maclaurin summation formula to lattices and summands involving power-law kernels. Their connection to high-order derivatives of zeta functions can be used to improve convergence rates of numerical algorithms, for instance in micromagnetics, or to provide rapidly convergent expansions suitable for precomputations of generalized zeta functions. We derive a stably computable representation of anisotropic Epstein zeta functions, including the possibility for analytically removing Rayleigh--Wood singularities, and we develop a numerical algorithm for their stable evaluation for any lattice, power-law decay exponent and anisotropy order. We benchmark the algorithm against closed-form identities, direct summation and multi-precision results, obtaining machine precision across various lattices, power-law exponents, and anisotropy orders.
