Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics

TL;DR

This work tackles the long-standing problem of computing infinite lattice sums for power-law interactions under periodic boundary conditions with high precision. It introduces a zeta expansion that splits the lattice sum into a small near-field part and a far-field correction expressed through derivatives of generalized zeta functions, with the far-field term controlled by an explicit differential operator $\mathcal{D}_\Omega$ and a Crandall representation that yields superexponential convergence. The authors provide a complete computational pipeline, including analytic derivative formulas and an efficient algorithm for evaluating Crandall functions and incomplete Bessel functions, achieving machine-precision results at a cost comparable to naive truncation. They also derive new asymptotic corrections to the demagnetization field and supply high-precision benchmarks, establishing a robust reference for micromagnetic solvers. The framework generalizes beyond micromagnetics, with broad applicability to molecular dynamics, ferroelectrics, and related lattice-sum problems requiring accurate long-range interactions under periodicity.

Abstract

We 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 micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. We show that the resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes. In order to compute the generalized zeta functions efficiently, we provide a superexponentially convergent algorithm for their evaluation, as well as for all required special functions, such as incomplete Bessel functions. Magnetic fields can thus be evaluated to machine precision in arbitrary cuboidal domains periodically extended along one or two dimensions. We benchmark our method against known formulas for magnetic interactions and against direct summation for Riesz potentials with large exponents, consistently achieving full precision. In addition, we identify new corrections to the asymptotic limit of the demagnetization field and tabulate high-precision benchmark values that can be used as a reliable reference for micromagnetic solvers. The techniques developed are broadly applicable, with direct impact in other areas such as molecular dynamics.

Figures (6)

• Figure 1: Sketch illustrating a 3D system of of $N_x\times N_y\times N_z=5\times 3\times 4$ interacting cuboids repeated infinitely in $x$ direction (a), and both in $x$ and $y$ direction (b). The green cuboid is influenced by the red cuboid indicated by the arrow, as well as by the total influence of all opaque red cuboids
• Figure 2: Relative error between the three-dimensional potential $U^{(0,0,2)}(\bm r)$ centered at $\bm r=(1,1,1)/2$ for the lattice at $\mathds Z^2\times\{0\}$ as obtained by the zeta representation in \ref{['thm:sem']} in comparison with direct summation over the truncated grid $\{-N_{\rm{cut}},\ldots,N_{\rm{cut}}\}^2\times\{0\}$ as dots a function of the the truncation length integers $1\le N_{\rm{cut}}\le 10$. The error scaling $C_\nu N_{\rm{cut}}^{-\nu}$ for some fitted parameter $C_\nu$ is shown as lines of the in the same color as the corresponding the error dots.
• Figure 3: Cuboid $\Omega=[1/(2N),1/(2N)]^3$ and its infinite repetitions $\bm z+\Omega$ along the two-dimensional lattice embedded in three dimensions $L=\mathds Z^2\times \{0\}$ for $N=2$ (a), $N=5$ (b) and $N=10$ (c).
• Figure 4: Difference between demagnetization factor $D_z$ for 3D cuboids $\Omega=[-1/(2N),1/(2N)]^3$ interacting with a lattice $\Lambda = \mathds Z^2$, computed from our method, and the known asymptotic formula. The asymptotic behavior is correctly reproduced, as well as the corner cases ($D_z = 1$ for $N=1$ and $D_z \to 1/3$ for $N\to\infty$). The next order correction from the known asymptotics as $N\to \infty$ scales as $N^{-7}$ (obtained by a fit).
• Figure 5: Regions of \ref{['alg']}. The white line indicates the split, where above the dashed white line we employ the reciprocal values. The purple region where the upper bound of the incomplete Bessel function is smaller than $10^{-16}$ is shown for $\nu = 0$.

Theorems & Definitions (20)

• definition 1.1: Generalized potential with periodic boundary conditions
• definition 2.1: Fourier transformation
• definition 2.2: Set zeta function
• theorem 2.3: Zeta expansion for micromagnetics
• proof
• corollary 2.4: Error scaling
• proof
• theorem 2.5: Crandall representation of set zeta functions
• proof
• corollary 2.6