Epstein zeta method for many-body lattice sums

TL;DR

The paper tackles the long-standing challenge of efficiently computing high-dimensional, slowly converging many-body lattice sums, including the Axilrod–Teller–Muto three-body potential. It develops an Epstein zeta representation that rewrites $n$-body lattice sums as singular integrals over products of Epstein zeta functions and provides a robust numerical scheme with meromorphic continuation, a Duffy-based quadrature, and Taylor-expansion for small-argument behavior. The method achieves full precision for 1- and 2-body cases, reliable results for 3-body ATM sums, and, crucially, linear-in-$n$ scaling up to at least $n=51$, enabling $100$-dimensional sums on a laptop. Demonstrations include accurate ATM energies and a phase-transition analysis in 3D Lennard–Jones systems, underscoring the approach’s practical impact for ab initio perturbative studies in chemistry and condensed matter physics.

Abstract

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.

Figures (9)

• Figure 1: The many-body zeta function $\zeta^{(n)}_\Lambda$ described as a circular graph for (a) $n=2$, (b) $n=3$, and $(c)$ general $n$.
• Figure 2: Minimum of absolute and relative error $E$ of the many-body zeta function $\zeta^{(n)}_\Lambda(\bm \nu)$ for $n\in\{1,2\}$ for different one- and two dimensional lattices $\Lambda\in\{\mathds Z,\Lambda_{\rm sq},\Lambda_{\rm hex}\}$.
• Figure 3: (a)--(c) Function values of the meromorphic continuation of the three-body zeta function for $\Lambda=\mathds Z$ where the dashed black lines indicate the singularities. (d)--(f) Error values of the meromorphic continuation shown in (a)--(c). (d) The error is magnified for small values of $\nu_1+\nu_2+\nu_3$.
• Figure 4: Minimum of absolute and relative error $E$ of the three-body zeta function $\zeta^{(3)}_{\Lambda}(\bm \nu)$ on the integer lattice $\Lambda=\mathds Z$ by comparision with exact summation for sufficiently large $\bm \nu$.
• Figure 5: Minimum of absolute and relative error $E$ of the three-body zeta function $\zeta^{(3)}_{\Lambda}(\bm \nu)$ by comparison with exact summation for sufficiently large $\nu_i$ for $\Lambda=\Lambda_{\rm sq}$ in (a)--(c) and for the hexagonal lattice $\Lambda=\Lambda_{\rm hex}$ in (d)--(f).

Theorems & Definitions (14)

• definition 1.1: Axilrod-Teller-Muto potential and cohesive energy
• definition 2.1: Lattices
• definition 2.2: Epstein zeta function
• definition 2.3: Many-body zeta function
• proposition 2.4
• proof
• lemma 2.5
• proof
• theorem 2.6: Epstein representation
• proof