Zeta Expansion for Long-Range Interacting Classical and Quantum Lattices

Large singular sums play a central role in many areas of pure and applied mathematics and arise prominently in the simulation of classical and quantum systems with long-range interactions. In this talk, I present a collection of numerical methods, based on generalized zeta functions and their derivatives, that enable the efficient and precise simulation of long-range interacting lattice models in previously inaccessible parameter regimes. The Epstein zeta function, a higher-dimensional analogue of the Riemann zeta function that describes oscillatory lattice sums, has recently become an indispensable tool in the numerical treatment of long-range interacting many-body systems.

I describe an efficient and stable method for its evaluation [1], implemented for general real parameters in the open-source package EpsteinLib [4]. Next, I discuss the computation of high-dimensional sums appearing in systems with many-body interactions, with direct impact in theoretical chemistry and in perturbative expansions of quantum spin systems. Here, direct summation approaches quickly become ineffective due to exponential increase of numerical work with the number of interaction partners. I show how a large class of these lattice sums can be represented in terms of efficiently computable integrals involving products of Epstein zeta functions [3], reducing the scaling from exponential to linear in the number of interaction partners. This method forms the basis for a rigorous investigation of the stability of matter under the inclusion of many-body interactions with first results in [6]. In certain parameter regimes, derivatives of Epstein zeta functions are required for obtaining full precision. I then introduce an algorithm for the stable evaluation of high-order derivatives of generalized zeta functions in machine-precision arithmetic [2]. Benchmarks demonstrate that full precision is retained for derivative orders up to order 8.

In the final part of the talk, I address the efficient computation of power-law based interaction potentials of homogeneous d-dimensional bodies with an infinite n-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in the simulation of magnetic textures with periodic boundary conditions and related fields, where it is common practice to truncate the arising sums. We show that the error due to truncation can be completely removed by a correction terms given in terms of efficiently computable derivatives of generalized zeta functions, which yields the exact value at the cost of a small direct sum [5].

References:

1. JB, Andreas A. Buchheit et al. (2024): Computation and properties of the Epstein zeta function with high-performance implementation in EpsteinLib. arXiv:2412.16317.
2. JB, Andreas A. Buchheit and Jonathan K. Busse (soon on arXiv): Computation of Large-Scale Singular Sums Using Stable Derivatives of Generalized Zeta Functions.
3. JB, Andreas A. Buchheit (2025): Epstein Zeta Method for Many-Body Lattice Sums. arXiv:2504.11989.
4. JB, Andreas A. Buchheit et al. (2024): EpsteinLib: High-performance computation of the Epstein zeta function, including a Python, Julia and Mathematica bindings. github.com/epsteinlib/epsteinlib.
5. JB, Andreas A. Buchheit et al. (2025): Zeta Expansion for Long-Range Interactions under Periodic Boundary Conditions with Applications to Micromagnetics. arXiv:2509.26274.
6. Andres Robles-Navarro, JB, et al. Exact Lattice Summations for Lennard-Jones Potentials Coupled to a Three-Body Axilrod-Teller-Muto Term Applied to Cuboidal Phase Transitions (2025). J. Chem. Phys. doi.org/10.1063/5.027667.Joint work with A. A. Buchheit; supervised by Prof. Sergej Rjasanow.

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