- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Universal exotic dynamics in critical mesoscopic systems: Simulating the square root of Avogadro’s number of spins

Mauro Bisson1,*, Alexandros Vasilopoulos2,*, Massimo Bernaschi3, Massimiliano Fatica1, Nikolaos G. Fytas2,†, Isidoro González-Adalid Pemartín3, and Víctor Martín-Mayor4

  • *These authors contributed equally to this work.
  • †Contact author: nikolaos.fytas@essex.ac.uk
PDF

Phys. Rev. Research 7, 033218 – Published 4 September, 2025

DOI: https://doi.org/10.1103/ngkf-7816

Abstract

We explicitly demonstrate the universality of critical dynamics through unprecedented large-scale Graphics Processing Units (GPU)-based simulations of two out-of-equilibrium processes, comparing the behavior of spin-1/2 Ising and spin-1 Blume-Capel models on a square lattice. In the first protocol, a completely disordered system is instantaneously brought into contact with a thermal bath at the critical temperature, allowing it to evolve until the coherence length exceeds 103 lattice spacings. Finite-size effects are negligible due to the mesoscopic scale of the lattice sizes studied, with linear dimensions up to L=222 and 219 for the Ising and Blume-Capel models, respectively. Our numerical data, and the subsequent analysis, demonstrate a strong dynamic universality between the two models and provide the most precise estimate to date of the dynamic critical exponent for this universality class, z=2.1676(1). In the second protocol, we corroborate the role of the universal ratio of dynamic and static length scales in achieving an exponential acceleration in the approach to equilibrium just above the critical temperature, through a time-dependent variation of the thermal bath temperature. The results presented in this work leverage our Compute Unified Device Architecture (CUDA)-based numerical code, breaking the world record for the simulation speed of the Ising model.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (58)

  1. K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud, and P. Nordblad, Memory and chaos effects in spin glasses, Phys. Rev. Lett. 81, 3243 (1998).
  2. M. Baity-Jesi et al., Memory and rejuvenation effects in spin glasses are governed by more than one length scale, Nat. Phys. 19, 978 (2023).
  3. I. Paga et al. (Janus Collaboration), Quantifying memory in spin glasses, Phys. Rev. Lett. 133, 256704 (2024).
  4. Aristotle, Metaphysics (English translation by W.D. Ross) (Clarendon, Oxford, 1981).
  5. M. Jeng, The Mpemba effect: When can hot water freeze faster than cold? Am. J. Phys. 74, 514 (2006).
  6. E. B. Mpemba and D. G. Osborne, Cool? Phys. Educ. 4, 172 (1969).
  7. J. Bechhoefer, A. Kumar, and R. Chétrite, A fresh understanding of the Mpemba effect, Nat. Rev. Phys. 3, 534 (2021).
  8. A. Gal and O. Raz, Precooling strategy allows exponentially faster heating, Phys. Rev. Lett. 124, 060602 (2020).
  9. A. Lapolla and A. Godec, Faster uphill relaxation in thermodynamically equidistant temperature quenches, Phys. Rev. Lett. 125, 110602 (2020).
  10. G. Teza, J. Bechhoefer, A. Lasanta, O. Raz, and M. Vucelja, Speedups in nonequilibrium thermal relaxation: Mpemba and related effects, arXiv:2502.01758.
  11. Z. Lu and O. Raz, Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse, Proc. Natl. Acad. Sci. USA 114, 5083 (2017).
  12. G. Teza, R. Yaacoby, and O. Raz, Eigenvalue crossing as a phase transition in relaxation dynamics, Phys. Rev. Lett. 130, 207103 (2023).
  13. I. G.-A. Pemartín, E. Mompó, A. Lasanta, V. Martín-Mayor, and J. Salas, Shortcuts of freely relaxing systems using equilibrium physical observables, Phys. Rev. Lett. 132, 117102 (2024).
  14. I. González-Adalid Pemartín, E. Mompó, A. Lasanta, V. Martín-Mayor, and J. Salas, Slow growth of magnetic domains helps fast evolution routes for out-of-equilibrium dynamics, Phys. Rev. E 104, 044114 (2021).
  15. M. Baity-Jesi et al., The Mpemba effect in spin glasses is a persistent memory effect, Proc. Natl. Acad. Sci. USA 116, 15350 (2019).
  16. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Clarendon Press, Oxford, 2005).
  17. P. Hohenberg and B. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  18. R. Folk and G. Moser, Critical dynamics: a field-theoretical approach, J. Phys. A: Math. Gen. 39, R207 (2006).
  19. B. M. McCoy and T. T. Wu, The Two Dimensional Ising Model (Harvard University Press, Cambridge, 1973).
  20. M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, 1999).
  21. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 2nd ed. (Cambridge University Press, Cambridge, 2005).
  22. C. Bonati, A. Pelissetto, and E. Vicari, Critical relaxational dynamics at the continuous transitions of three-dimensional spin models with z2 gauge symmetry, Phys. Rev. B 111, 115129 (2025).
  23. M. Hasenbusch, A. Pelissetto, and E. Vicari, Relaxational dynamics in 3D randomly diluted Ising models, J. Stat. Mech. (2007) P11009.
  24. L. F. da Silva, U. L. Fulco, and F. D. Nobre, The two-dimensional site-diluted Ising model: a short-time-dynamics approach, J. Phys.: Condens. Matter 21, 346005 (2009).
  25. W. Zhong, G. T. Barkema, and D. Panja, Super slowing down in the bond-diluted Ising model, Phys. Rev. E 102, 022132 (2020).
  26. M. Blume, Theory of the first-order magnetic phase change in UO2, Phys. Rev. 141, 517 (1966).
  27. H. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting, Physica 32, 966 (1966).
  28. J. Zierenberg, N. G. Fytas, M. Weigel, W. Janke, and A. Malakis, Scaling and universality in the phase diagram of the 2D Blume-Capel model, Eur. Phys. J. Spec. Top. 226, 789 (2017).
  29. M. Nightingale and H. Blöte, Dynamic exponent of the two-dimensional Ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix, Phys. Rev. Lett. 76, 4548 (1996).
  30. M. Nightingale and H. Blöte, Monte Carlo computation of correlation times of independent relaxation modes at criticality, Phys. Rev. B 62, 1089 (2000).
  31. M. Hasenbusch, Dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model, Phys. Rev. E 101, 022126 (2020).
  32. Z. Liu, E. Vatansever, G. T. Barkema, and N. G. Fytas, Critical dynamical behavior of the Ising model, Phys. Rev. E 108, 034118 (2023).
  33. F.-G. Wang and C.-K. Hu, Universality in dynamic critical phenomena, Phys. Rev. E 56, 2310 (1997).
  34. C. Duclut and B. Delamotte, Frequency regulators for the nonperturbative renormalization group: A general study and the model a as a benchmark, Phys. Rev. E 95, 012107 (2017).
  35. L. Adzhemyan, D. Evdokimov, M. Hnatič, E. Ivanova, M. Kompaniets, A. Kudlis, and D. Zakharov, The dynamic critical exponent z for 2d and 3d Ising models from five-loop ε expansion, Phys. Lett. A 425, 127870 (2022).
  36. D. A. Levin and Y. Peres, Markov Chains and Mixing Times (American Mathematical Society, Providence, USA, 2017), Vol. 107.
  37. A. D. Sokal, Monte Carlo methods in statistical mechanics: Foundations and new algorithms, in Functional Integration: Basics and Applications (1996 Cargèse School), edited by C. DeWitt-Morette, P. Cartier, and A. Folacci (Plenum, New York, 1997), pp. 131–192.
  38. L. A. Fernández, E. Marinari, V. Martín-Mayor, G. Parisi, and J. Ruiz-Lorenzo, Out-of-equilibrium 2D Ising spin glass: Almost, but not quite, a free-field theory, J. Stat. Mech. (2018) 103301.
  39. L. A. Fernández, E. Marinari, V. Martín-Mayor, G. Parisi, and J. Ruiz-Lorenzo, An experiment-oriented analysis of 2D spin-glass dynamics: a twelve time-decades scaling study, J. Phys. A: Math. Theor. 52, 224002 (2019).
  40. A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43, 357 (1994).
  41. M. Bisson, M. Bernaschi, M. Fatica, N. G. Fytas, I. González-Adalid Pemartín, V. Martín-Mayor, and A. Vasilopoulos, Massive-scale simulations of 2D Ising and Blume-Capel models on rack-scale multi-GPU systems, Comput. Phys. Commun. 315, 109690 (2025).
  42. J. Romero, M. Bisson, M. Fatica, and M. Bernaschi, High performance implementations of the 2D Ising model on GPUs, Comput. Phys. Commun. 256, 107473 (2020).
  43. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ngkf-7816 for (1) simulation details, (2) computation of the critical temperature of the Δ=0 square-lattice Blume-Capel model, (3) analysis of the correlation function, (4) outline of the fitting procedure, and (5) thermodynamic limit illustrations.
  44. P. Di Francesco, H. Saleur, and J. Zuber, Critical Ising correlation functions in the plane and on the torus, Nucl. Phys. B 290, 527 (1987).
  45. P. Di Francesco, H. Saleur, and J. Zuber, Correlation functions of the critical Ising model on a torus, Europhys. Lett. 5, 95 (1988).
  46. J. Salas and A. D. Sokal, Universal amplitude ratios in the critical two-dimensional Ising model on a torus, J. Stat. Phys. 98, 551 (2000).
  47. M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, Irrelevant operators in the two-dimensional Ising model, J. Phys. A: Math. Gen. 35, 4861 (2002).
  48. R. H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86 (1987).
  49. F. Belletti et al. (Janus Collaboration), Nonequilibrium spin-glass dynamics from picoseconds to one tenth of a second, Phys. Rev. Lett. 101, 157201 (2008).
  50. L. A. Fernandez, E. Marinari, V. Martin-Mayor, I. Paga, and J. J. Ruiz-Lorenzo, Dimensional crossover in the aging dynamics of spin glasses in a film geometry, Phys. Rev. B 100, 184412 (2019).
  51. D. J. Amit and V. Martín-Mayor, Field Theory, the Renormalization Group and Critical Phenomena, 3rd ed. (World Scientific, Singapore, 2005).
  52. L. A. Fernández and V. Martín-Mayor, Testing statics-dynamics equivalence at the spin-glass transition in three dimensions, Phys. Rev. B 91, 174202 (2015).
  53. Y. Ozeki and N. Ito, Nonequilibrium relaxation method, J. Phys. A: Math. Theor. 40, R149 (2007).
  54. H. Shao, W. Guo, and A. W. Sandvik, Quantum criticality with two length scales, Science 352, 213 (2016).
  55. B. Nienhuis, Analytical calculation of two leading exponents of the dilute Potts model, J. Phys. A: Math. Gen. 15, 199 (1982).
  56. A. E. Ferdinand and M. E. Fisher, Bounded and inhomogeneous Ising models. I. specific-heat anomaly of a finite lattice, Phys. Rev. 185, 832 (1969).
  57. M. Bernaschi, I. González-Adalid Pemartín, V. Martín-Mayor, and G. Parisi, The QISG suite: High-performance codes for studying quantum Ising spin glasses, Comput. Phys. Commun. 298, 109101 (2024).
  58. L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ramaswamy, and J. Toner, The inconvenient truth about flocks, arXiv:2503.17064.

Outline

Information

Phys. Rev. Research 7, 033218– Published 4 September, 2025
  • Received 17 March 2025
  • Accepted 29 July 2025
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