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Titel: On the computation of lattice sums without translational invariance
VerfasserIn: Buchheit, Andreas A.
Keßler, Torsten
Serkh, Kirill
Sprache: Englisch
Titel: Mathematics of Computation
Verlag/Plattform: American Mathematical Society
Erscheinungsjahr: 2024
DDC-Sachgruppe: 510 Mathematik
Dokumenttyp: Journalartikel / Zeitschriftenartikel
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 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 like Ewald summation 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 and precise 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.
DOI der Erstveröffentlichung: 10.1090/mcom/4024
URL der Erstveröffentlichung: https://doi.org/10.1090/mcom/4024
Link zu diesem Datensatz: urn:nbn:de:bsz:291--ds-445279
hdl:20.500.11880/39738
http://dx.doi.org/10.22028/D291-44527
ISSN: 1088-6842
0025-5718
Datum des Eintrags: 27-Feb-2025
Fakultät: MI - Fakultät für Mathematik und Informatik
Fachrichtung: MI - Mathematik
Professur: MI - Prof. Dr. Sergej Rjasanow
Sammlung:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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