Epstein zeta method for many-body lattice sums

This paper introduces an efficient Epstein zeta function-based method that transforms the computation of many-body lattice sums from exponentially complex direct summation to linear-cost singular integrals, enabling high-precision studies of three-body interactions like the Axilrod-Teller-Muto potential and revealing pressure-induced structural transitions in condensed matter systems.

The Gist

Imagine you are trying to calculate the total "happiness" (or energy) of a massive crowd of people standing in a perfect grid, like soldiers in a parade. In the real world, these people aren't just standing still; they are constantly interacting with their neighbors.

Usually, we only worry about how much Person A likes Person B (a two-body interaction). But in the complex world of materials science, things get tricky: Person A's mood might also depend on who Person C is standing next to, even if A and C aren't touching. This is called a three-body interaction.

The problem is that when you try to add up all these complicated interactions for a crystal lattice (a repeating 3D grid of atoms), the math becomes a nightmare. It's like trying to count every grain of sand on a beach, but the sand keeps multiplying the more you look at it. Traditionally, doing this calculation for a 3D crystal took a supercomputer weeks to finish, and even then, the answer wasn't perfectly precise.

The "Magic Lens" Solution

The authors of this paper, Andreas Buchheit and Jonathan Busse, have invented a new mathematical "lens" to solve this. Instead of trying to count every single interaction one by one (which is slow and prone to errors), they found a way to rewrite the entire problem using a special mathematical tool called the Epstein Zeta function.

Think of the old method as trying to walk through a dense forest, counting every single tree individually. It takes forever, and you might trip over roots (mathematical singularities) along the way.

The new method is like taking a helicopter ride. Instead of walking, you look at the forest from above. You realize that the trees follow a specific pattern. By using this pattern (the Epstein Zeta function), you can calculate the total number of trees in seconds rather than weeks.

How They Did It (The Analogy)

1. The Problem: The math involves "singularities," which are like mathematical black holes where the numbers blow up to infinity. Standard calculators choke on these.
2. The Trick: The authors realized that if you look at the problem from a different angle (using something called a "Fourier transform" and integrating over a "Brillouin zone," which is just a fancy way of looking at the grid's frequency), those scary black holes turn into manageable bumps.
3. The Result: They broke the massive, impossible sum down into a series of smaller, smooth integrals. They then used a clever mathematical "stretching" technique (called the Duffy transformation) to flatten out the bumps so a computer could easily measure them.

The Big Wins

• Speed: What used to take weeks on a single computer processor now takes minutes on a standard laptop.
• Precision: They can now get answers with "full precision" (meaning the computer knows the answer down to the very last decimal point), whereas the old methods often had to guess or stop early.
• Scalability: Usually, if you add more people to the interaction (going from 3-body to 4-body to 5-body), the math gets exponentially harder (like trying to solve a puzzle where the pieces double every time you add a new one). Their method is different: the difficulty only increases linearly. It's like adding a new row to a spreadsheet; it takes a little more time, but it doesn't break the computer. They successfully calculated a "100-dimensional" sum (a concept that sounds impossible) in just a few seconds.

What They Discovered

Using this new super-fast calculator, they looked at a specific type of crystal (like solid Argon or graphene). They found that when you include these tricky three-body interactions, the crystal doesn't always stay in its favorite shape.

• The Finding: Under certain conditions (specifically, when the three-body "coupling strength" gets high), the crystal prefers to change its shape. It switches from a Face-Centered Cubic (FCC) structure (a very common, tight packing) to a Body-Centered Cubic (BCC) structure.
• Why it matters: This explains why some materials might change their structure under pressure or different conditions, a detail that was previously too hard to calculate accurately.

Summary

In short, the authors built a mathematical "super-tool" that turns a slow, impossible calculation into a fast, precise one. They used it to prove that three-body interactions can force crystals to change their shape, solving a problem that has been stuck in the "too hard" pile for a long time. This tool is now open for other scientists to use to understand how matter holds itself together.

Technical Summary

Technical Summary: Epstein Zeta Method for Many-Body Lattice Sums

Problem Statement
The accurate prediction of material properties in condensed matter physics and quantum chemistry often requires the evaluation of many-body interactions. While two-body interactions are additive, higher-order corrections (three-body and beyond) are crucial for describing systems such as ultracold noble gas crystals and bilayer graphene. A primary example is the Axilrod-Teller-Muto (ATM) potential, which arises from quantum dipole fluctuations in third-order perturbation theory.

The central challenge addressed in this work is the numerical computation of the resulting cohesive energies in $d$-dimensional lattice systems. These energies are defined by high-dimensional singular lattice sums that converge extremely slowly. Direct summation of these terms is computationally prohibitive; for instance, computing the ATM cohesive energy for a three-dimensional lattice to a few digits can take weeks on a single CPU. Furthermore, the complexity of direct summation increases exponentially with the number of interacting particles $n$, rendering the evaluation of $n$-body sums for $n > 3$ effectively impossible using standard techniques.

Methodology
The authors propose a method to represent many-body lattice sums as integrals over products of Epstein zeta functions, thereby transforming the problem from a high-dimensional discrete sum into a lower-dimensional integral problem.

1. Many-Body Zeta Functions: The authors define $n$-body zeta functions, $\zeta^{(n)}_\Lambda(\nu)$, as lattice sums over $n-1$ independent lattice vectors with power-law interactions.
2. Epstein Representation: The core theoretical result (Theorem 2.2) establishes that an $n$-body zeta function can be expressed as an integral over the Brillouin zone (BZ) of the product of $n$ simple Epstein zeta functions:
   $$\zeta^{(n)}_\Lambda(\nu) = V_\Lambda \int_{BZ} \prod_{i=1}^n Z_{\Lambda, \nu_i}(k) \, dk$$
   where $V_\Lambda$ is the volume of the elementary cell and $Z_{\Lambda, \nu}(k)$ is the Epstein zeta function. This representation allows the $n$-body sum to be decomposed into a product of 1D functions integrated over a $d$-dimensional domain, rather than a sum over a $(n-1)d$-dimensional lattice.
3. Handling Singularities: The Epstein zeta function exhibits power-law singularities at $k=0$. To handle this numerically, the authors employ a Duffy transformation (Corollary 3.1). This transformation maps the integration domain into a sum of pyramids, converting the $d$-dimensional singularity into a set of one-dimensional singular integrals (in variable $u$) and $(d-1)$-dimensional smooth integrals (in variable $v$).
4. Numerical Integration:
   • The smooth part (integral over $v$) is evaluated using tensorized Gauss-Legendre quadrature. The authors prove that the integrand admits a holomorphic extension to a complex neighborhood, ensuring exponential convergence of the quadrature error with respect to the number of points.
   • The singular part (integral over $u$) is handled via adaptive integration. For cases requiring meromorphic continuation (where exponents $\nu_i \le d$), the regularized Epstein zeta function is expanded in a Taylor series near the origin, and the resulting singular integrals are evaluated analytically.

Key Contributions

• Efficient Representation: The paper provides an analytically rigorous and numerically efficient representation of general $n$-body lattice sums in terms of products of Epstein zeta functions.
• Linear Scaling: A critical contribution is the demonstration that the computational cost of this method scales linearly with the number of bodies $n$. This stands in stark contrast to the exponential scaling of direct summation.
• Meromorphic Continuation: The method robustly handles cases where interaction exponents lead to divergences in the original sum by utilizing the meromorphic continuation of the Epstein zeta function, allowing for the computation of physically relevant parameters (e.g., $\nu = -1$ in the ATM potential) that are inaccessible to direct summation.
• ATM Decomposition: The authors explicitly derive the decomposition of the Axilrod-Teller-Muto cohesive energy into a linear combination of three-body zeta functions with specific exponents, enabling its precise calculation for any lattice.

Results

• Precision and Speed: For three-body interactions in three dimensions, the method reduces the runtime from weeks to minutes while achieving full machine precision.
• Benchmarking: The method was benchmarked against analytically computable cases (1- and 2-body sums) and direct summation for low dimensions and large exponents. In all tested cases, the method achieved errors below $10^{-14}$.
• High-Dimensional Sums: The authors successfully computed $n$-body zeta functions for $n$ up to 51 (corresponding to a 100-dimensional sum for a 2D lattice) in seconds on a standard laptop. The results showed that the normalized many-body zeta function decays algebraically with $n$.
• Application to Stability: The method was applied to study the stability of 3D crystal lattices (fcc vs. bcc) under Lennard-Jones two-body interactions and an ATM three-body term. The results identified a phase transition where increasing the ATM coupling strength destabilizes the face-centered-cubic (fcc) lattice in favor of the body-centered-cubic (bcc) structure.

Significance
The paper establishes both the numerical and analytical foundation for investigating the influence of many-body interactions on the stability of matter. By solving the long-standing issue of computing high-dimensional, slowly converging lattice sums, the method enables the precise calculation of minute energy differences between lattice structures that were previously inaccessible. The authors note that this work serves as the basis for further investigations into the stability of cuboidal crystal lattices (Ref. [27]) and offers a general tool for perturbative treatments of quantum many-body systems, including quantum spin systems with long-range interactions. The ability to compute these sums efficiently opens the door to rigorous theoretical studies of material properties where many-body effects are dominant.