标题： Epstein zeta 方法用于多体晶格求和 显示英文标题

标题： Epstein zeta method for many-body lattice sums

摘要： 多体相互作用在经典和量子多体系统的微扰处理中自然出现，并在描述凝聚态系统中起着关键作用。 在三体相互作用的情况下，Axilrod-Teller-Muto（ATM）势对于材料性质的定量预测非常重要。 在d维晶格系统中计算由此产生的能量具有挑战性，因为需要以高精度评估高维晶格求和。 本工作解决了这一长期存在的问题。 我们提出了一种高效可计算的多体晶格求和表示形式，该形式基于Epstein zeta函数乘积上的奇异积分。 对于三维中的三体相互作用，这种方法将计算ATM晶格求和的时间从几周减少到几分钟。 我们的方法进一步扩展到一大类n体晶格求和。 我们证明了我们方法的计算成本随着n线性增长，避免了直接求和的复杂性指数增长。 在二维晶格上评估51体相互作用，对应于100维求和，可以在笔记本电脑上在几秒内完成。 我们讨论了计算所出现的奇异积分的技术，并将我们的结果准确性与可计算的基准进行比较，对于大于系统维度的指数实现了完全精度。 最后，我们将我们的方法应用于研究包含有限压力下ATM三体项的Lennard-Jones二体相互作用的三维晶格系统的稳定性，发现随着ATM耦合强度的增加，从面心立方结构向体心立方结构发生了转变。 本工作建立了对物质稳定性受多体相互作用影响的持续研究的数学基础。 显示英文摘要

摘要： Many-body interactions arise naturally in the perturbative treatment of classical and quantum many-body systems and play a crucial role in the description of condensed matter systems. In the case of three-body interactions, the Axilrod-Teller-Muto (ATM) potential is highly relevant for the quantitative prediction of material properties. The computation of the resulting energies in d-dimensional lattice systems is challenging, as a high-dimensional lattice sum needs to be evaluated to high precision. This work solves this long-standing issue. We present an efficiently computable representation of many-body lattice sums in terms of singular integrals over products of Epstein zeta functions. For three-body interactions in 3D, this approach reduces the runtime for computing the ATM lattice sum from weeks to minutes. Our approach further extends to a broad class of n-body lattice sums. We demonstrate that the computational cost of our method only increases linearly with n, evading the exponential increase in complexity of direct summation. The evaluation of 51-body interactions on a two-dimensional lattice, corresponding to a 100-dimensional sum, can be performed within seconds on a laptop. We discuss techniques for computing the arising singular integrals and compare the accuracy of our results against computable benchmarks, achieving full precision for exponents greater than the system dimension. Finally, we apply our method to study the stability of a three-dimensional lattice system with Lennard-Jones two-body interactions under the inclusion of an ATM three-body term at finite pressure, finding a transition from the face-centered-cubic to the body-centered-cubic lattice structure with increasing ATM coupling strength. This work establishes the mathematical foundation for an ongoing investigation into the influence of many-body interactions on the stability of matter.