- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Efficient Predecision Scheme for Metropolis Monte Carlo Simulation of Long-Range Interacting Lattice Systems

Fabio Müller* and Wolfhard Janke†

  • *Contact author: fabio.mueller@itp.uni-leipzig.de
  • †Contact author: wolfhard.janke@itp.uni-leipzig.de
PDF

Phys. Rev. Lett. 136, 227101 – Published 2 June, 2026

DOI: https://doi.org/10.1103/29yw-9grr

Abstract

We propose a fast and general predecision scheme for Metropolis Monte Carlo simulation of d-dimensional long-range interacting lattice models with N constituents. For potentials of the form V(r)=r−d−σ, this reduces the computational complexity from O(N2) to O(N2−σ/d) for σ<d and to O(N) for σ>d, respectively. The algorithm is implemented and tested for several O(n) spin models ranging from the Ising over the XY to the Edwards-Anderson spin-glass model. With the same random-number sequence it produces exactly the same Markov chain as a simulation with explicit summation of all terms in the Hamiltonian. Because of its generality, its simplicity, and its reduced computational complexity it has the potential to find broad application and thus lead to a deeper understanding of the role of long-range interactions in the physics of lattice models, especially in nonequilibrium settings.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (36)

  1. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21, 1087 (1953).
  2. B. K. Chakrabarti and M. Acharyya, Dynamic transitions and hysteresis, Rev. Mod. Phys. 71, 847 (1999).
  3. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, England, 2010).
  4. A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51, 481 (2002).
  5. R. Livi and P. Politi, Nonequilibrium Statistical Physics: A Modern Perspective (Cambridge University Press, Cambridge, England, 2017).
  6. W. H. Zurek, Cosmological experiments in superfluid helium?, Nature (London) 317, 505 (1985).
  7. G. Teza, J. Bechhoefer, A. Lasanta, O. Raz, and M. Vucelja, Speedups in nonequilibrium thermal relaxation: Mpemba and related effects, Phys. Rep. 1164, 1 (2026).
  8. H. Christiansen, S. Majumder, and W. Janke, Phase ordering kinetics of the long-range Ising model, Phys. Rev. E 99, 011301(R) (2019).
  9. H. Christiansen, S. Majumder, M. Henkel, and W. Janke, Aging in the long-range Ising model, Phys. Rev. Lett. 125, 180601 (2020).
  10. R. Agrawal, F. Corberi, E. Lippiello, P. Politi, and S. Puri, Kinetics of the two-dimensional long-range Ising model at low temperatures, Phys. Rev. E 103, 012108 (2021).
  11. H. Christiansen, S. Majumder, and W. Janke, Zero-temperature coarsening in the two-dimensional long-range Ising model, Phys. Rev. E 103, 052122 (2021).
  12. R. Agrawal, F. Corberi, F. Insalata, and S. Puri, Asymptotic states of Ising ferromagnets with long-range interactions, Phys. Rev. E 105, 034131 (2022).
  13. F. Müller, H. Christiansen, and W. Janke, Phase-separation kinetics in the two-dimensional long-range Ising model, Phys. Rev. Lett. 129, 240601 (2022).
  14. F. Müller, H. Christiansen, and W. Janke, Nonuniversality of aging during phase separation of the two-dimensional long-range Ising model, Phys. Rev. Lett. 133, 237102 (2024).
  15. F. Müller, H. Christiansen, S. Schnabel, and W. Janke, Fast, hierarchical, and adaptive algorithm for metropolis Monte Carlo simulations of long-range interacting systems, Phys. Rev. X 13, 031006 (2023).
  16. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. (Berlin) 369, 253 (1921).
  17. S. Schnabel and W. Janke, Accelerating polymer simulation by means of tree data-structures and a parsimonious Metropolis algorithm, Comput. Phys. Commun. 256, 107414 (2020).

  18. An imperfect ordering in |Ji,j|, where subsequences are additionally ordered to minimize cache misses, can lead to further improved wall-clock times.

  19. We have carefully checked that, compared to a simple Metropolis implementation, the increase of the algorithmic prefactor stemming from the slightly more complicated algorithmic procedure is of the order of unity and thus negligible.

  20. See Supplemental Material at http://link.aps.org/supplemental/10.1103/29yw-9grr where the computational complexity of the new method is derived under minimal assumptions and a full implementation of the algorithm in the julia programming language is provided. The code can also be downloaded from https://gitlab.com/flabons/predecisionmetropolis.
  21. E. Luijten and H. W. J. Blöte, Boundary between long-range and short-range critical behavior in systems with algebraic interactions, Phys. Rev. Lett. 89, 025703 (2002).
  22. M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Relations between short-range and long-range Ising models, Phys. Rev. E 89, 062120 (2014).
  23. M. Picco, Critical behavior of the Ising model with long range interactions, arXiv:1207.1018.
  24. C. Behan, L. Rastelli, S. Rychkov, and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys. A 50, 354002 (2017).
  25. G. Giachetti, N. Defenu, S. Ruffo, and A. Trombettoni, Berezinskii-Kosterlitz-Thouless phase transitions with long-range couplings, Phys. Rev. Lett. 127, 156801 (2021).
  26. C. Behan, E. Lauria, M. Nocchi, and P. van Vliet, Analytic and numerical bootstrap for the long-range Ising model, J. High Energy Phys. 03 (2024) 136.
  27. T. Xiao, D. Yao, C. Zhang, Z. Fan, and Y. Deng, Two-dimensional XY ferromagnet induced by long-range interaction, Chin. Phys. Lett. 42, 070002 (2025).
  28. T. Xiao, Z. Liu, Z. Fan, and Y. Deng, On Sak’s criterion for statistical models with long-range interaction, arXiv:2512.04805.
  29. B. Hetényi, K. Bernacki, and B. Berne, Multiple “time step” Monte Carlo, J. Chem. Phys. 117, 8203 (2002).
  30. S. C. Kapfer and W. Krauth, Cell-veto Monte Carlo algorithm for long-range systems, Phys. Rev. E 94, 031302(R) (2016).
  31. X. Gao, S. Jiang, J. Liang, and Q. Zhou, An O(logN) Monte Carlo method for periodic Coulomb systems, arXiv:2601.09288.
  32. K. De’Bell, A. B. MacIsaac, and J. P. Whitehead, Dipolar effects in magnetic thin films and quasi-two-dimensional systems, Rev. Mod. Phys. 72, 225 (2000).
  33. A. J. Bray and A. D. Rutenberg, Growth laws for phase ordering, Phys. Rev. E 49, R27 (1994).
  34. K. Du, X. Fang, C. Won, C. De, F.-T. Huang, W. Xu, H. You, F. J. Gómez-Ruiz, A. del Campo, and S.-W. Cheong, Kibble-Zurek mechanism of Ising domains, Nat. Phys. 19, 1495 (2023).
  35. G. Schaller, F. Queisser, S. P. Katoorani, C. Brand, C. Kohlfürst, M. R. Freeman, A. Hucht, P. Kratzer, B. Sothmann, M. H.-v. Hoegen, and R. Schützhold, Kibble-Zurek dynamics in the anisotropic Ising model of the Si(001) surface, Phys. Rev. Lett. 134, 246202 (2025).
  36. G. Ódor, Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004).

Outline

Information

Phys. Rev. Lett. 136, 227101– Published 2 June, 2026
  • Received 2 October 2025
  • Revised 17 February 2026
  • Accepted 10 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation