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On the computation of lattice sums without translational invariance

Andreas A. Buchheit, Torsten Keßler, Kirill Serkh

TL;DR

This work tackles the challenge of computing oscillatory lattice sums in finite geometries where translational invariance breaks down. It develops a unified framework based on a set zeta function $Z_{L,\nu}$ and a generalized Crandall representation, enabling exponential convergence for non-translationally invariant sums over corners and parallelepipeds via Hadamard integral representations. The authors derive non-oscillatory integral forms and a stable numerical algorithm that scales with geometric complexity rather than particle count, demonstrated by simulating a macroscopic 3D spin structure with approximate particle number $3\times 10^{23}$ on a conventional laptop and providing open-source code. This approach yields accurate long-range interaction energies in bounded lattices and offers a pathway to efficient boundary-aware simulations in condensed matter and topological quantum physics, including potential applications to Majorana-based devices and quasi-crystalline systems.

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.

On the computation of lattice sums without translational invariance

TL;DR

This work tackles the challenge of computing oscillatory lattice sums in finite geometries where translational invariance breaks down. It develops a unified framework based on a set zeta function and a generalized Crandall representation, enabling exponential convergence for non-translationally invariant sums over corners and parallelepipeds via Hadamard integral representations. The authors derive non-oscillatory integral forms and a stable numerical algorithm that scales with geometric complexity rather than particle count, demonstrated by simulating a macroscopic 3D spin structure with approximate particle number on a conventional laptop and providing open-source code. This approach yields accurate long-range interaction energies in bounded lattices and offers a pathway to efficient boundary-aware simulations in condensed matter and topological quantum physics, including potential applications to Majorana-based devices and quasi-crystalline systems.

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 particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.
Paper Structure (20 sections, 25 theorems, 182 equations, 6 figures, 4 tables)

This paper contains 20 sections, 25 theorems, 182 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Epstein zeta and reformulation of Crandall's formula
  3. Main result: Computing non-translationally invariant lattice sums
  4. Numerical application: Boundary effects in a macroscopic 3D spin structure
  5. Outlook and Conclusions
  6. Derivation
  7. Preliminaries
  8. Derivation of the generalized Crandall formula
  9. Derivation of the Hadamard integral representation
  10. Integral representation for corner geometry
  11. Numerical algorithm
  12. Choice of the Riemann splitting parameter $\lambda$
  13. Evaluation of the function $\psi_{xL-x}$
  14. Evaluation of the singular Hadamard finite-part integral
  15. Numerical experiments
  16. ...and 5 more sections

Key Result

Theorem 2.3

Let $\Lambda=A \mathds Z^d$ with $A\in \mathds R^{d\times d}$ regular, $\bm x,\bm y\in \mathds R^d$, and $\nu\in \mathds C\setminus\{d\}$. Then for any $\lambda>0$, with the reciprocal lattice $\Lambda^\ast = A^{-T} \mathds Z^d$. The function $G_\nu$ is defined as and $G_\nu(\bm 0) = -2/\nu$. Here $\Gamma(\nu,z)$ denotes the upper incomplete Gamma function.

Figures (6)

  • Figure 1: (a) Schematic depiction of a macroscopic three-dimensional spin circuit with inner edge length $1\,\mathrm{cm}$, outer edge length $2\,\mathrm{cm}$, and thickness $1\,\mathrm{mm}$. (b) Microscopic structure with spins (blue arrows) polarized in the $x_3$ direction. The material exhibits a cubic lattice structure and lattice constant $a=10^{-10}\,m$, amounting to $N=3\times 10^{23}$ particles in total. (b) Potential energy due to long-range interactions with exponent $\nu = 3.001$ of a spin defect obtained from inverting the spin orientation as a function of position $\bm x=(x_1,x_2,0)^T$ with $x_3 =0$ corresponding to the symmetry plane. Position is written in units of the lattice constant and potential energy in units of the interaction energy of neighboring parallel spins. Additional contours close to the energy minimum are highlighted at values $-206$ (yellow), $-207.0$ (orange), $-207.1$ (blue), and $-207.2$ (white). The figure excludes the immediate boundary layer, where the potential energy increases sharply.
  • Figure 2: Runtime for evaluating the potential energy for introducing a defect in the spin circuit as a function of particle number $N$. Particle number is modified by rescaling the geometry while keeping the lattice constant fixed. While the numerical effort for exact summation increases linearly (black), our approach (red) yields an effectively constant runtime even up to macroscopic particle numbers.
  • Figure 3: The error $E=\min(E_\text{abs},E_\text{rel})$ of our algorithm for certain lattice sums.
  • Figure 4: The lattice parameters indicated on the unit cell $E_\Lambda$.
  • Figure 5: The error plotted for a range of $\nu$, for $d=1,2,3$.
  • ...and 1 more figures

Theorems & Definitions (54)

  • Definition 2.1: Lattices
  • Definition 2.2: Epstein zeta function
  • Theorem 2.3: Crandall's formula
  • Definition 3.1: Uniformly discrete sets
  • Definition 3.2: Set zeta function
  • Remark 3.3
  • Definition 3.4: Fourier transformation
  • Definition 3.5: Generalized Dirac comb and form factor
  • Theorem 3.6: Set zeta Crandall
  • Theorem 3.7: Hadamard integral representation
  • ...and 44 more