On the computation of lattice sums without translational invariance

TL;DR

This paper presents a novel, efficient numerical method for computing oscillatory lattice sums in bounded geometries, overcoming challenges posed by long-range interactions and lack of translational symmetry, with applications in condensed matter physics.

Contribution

It introduces a generalized multidimensional zeta function and a convergent numerical algorithm for boundary-influenced lattice sums, enabling large-scale simulations.

Findings

Method achieves exponential convergence across various geometries.

Runtime depends only on geometry complexity, not particle number.

Successfully computed energies for a crystal with 3×10^{23} particles.

Abstract

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with $3\times 10^{23}$ particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.