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Nonconserved critical dynamics of the two-dimensional Ising model as a surface kinetic roughening process

Héctor Vaquero del Pino1,* and Rodolfo Cuerno2,†

  • 1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
  • 2Universidad Carlos III de Madrid, Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos (GISC), Avenida de la Universidad 30, 28911 Leganés (Madrid), Spain

  • *Contact author: hector001@e.ntu.edu.sg
  • †Contact author: cuerno@math.uc3m.es
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Phys. Rev. Research 7, 043192 – Published 20 November, 2025

DOI: https://doi.org/10.1103/y9b7-z6zq

Abstract

We have revisited the nonconserved (or model A) critical dynamics of the two-dimensional Ising model through numerical simulations of its lattice and continuum formulations—Glauber dynamics and the time-dependent Ginzburg-Landau (TDGL) equation, respectively—to analyze them with current tools from surface kinetic roughening. Our study examines two critical quenches, one from an ordered and other one from a disordered initial state, for both of which we assess the dynamic scaling ansatz, the critical exponent values, and the fluctuation field statistics that occur. Notably, the dynamic scaling ansatz followed by the system strongly depends on the initial condition: A critical quench from the ordered phase follows Family-Vicsek (FV) scaling, while a critical quench from the disordered phase shows an initial overgrowth regime with intrinsic anomalous roughening, followed by relaxation to equilibrium. This behavior is explained in terms of the dynamical instability of the stochastic Ginzburg-Landau equation at the critical temperature, whereby the linearly unstable term is eventually stabilized by nonlinear interactions. For both quenches, we have determined the occurrence of probability distribution functions for the field fluctuations, which are time independent along the nonequilibrium dynamics when suitably normalized by the time-dependent fluctuation amplitude. Additionally, we have developed a related interface model for a field that scales as the space integral of the TDGL field (integral GL model). Numerical simulations of this model reveal either FV or faceted anomalous roughening, depending on the critical quenched performed, as well as an emergent symmetry in the fluctuation statistics for a critical quench from the ordered phase.

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References (76)

  1. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford University Press, Oxford, 1992).
  2. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (CRC Press, Boca Raton, 2018).
  3. M. Kardar, Statistical Physics of Fields (Cambridge University Press, Cambridge, 2007).
  4. T. Ising, R. Folk, R. Kenna, B. Berche, and Y. Holovatch, The fate of Ernst Ising and the fate of his model, J. Phys. Stud. 21, 3002 (2017).
  5. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  6. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  7. J. García-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems (Springer, New York, 1999).
  8. S.-K. Ma, Modern Theory of Citical Phenomena (Routledge, New York, 2018).
  9. G. F. Mazenko, Nonequilibrium Statistical Mechanics (Wiley-VCH, Weinheim, 2006).
  10. U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-equilibrium Scaling Behavior (Cambridge University Press, Cambridge, 2014).
  11. G. Ódor, Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004).
  12. A.-L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995).
  13. J. Krug, Origins of scale invariance in growth processes, Adv. Phys. 46, 139 (1997).
  14. E. E. Mozo Luis, T. A. de Assis, and F. A. Oliveira, Unveiling the connection between the global roughness exponent and interface fractal dimension in EW and KPZ lattice models, J. Stat. Mech. (2022) 083202.
  15. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56, 889 (1986).
  16. T. Kriecherbauer and J. Krug, A pedestrian's view on interacting particle systems, KPZ universality and random matrices, J. Phys. A: Math. Theor. 43, 403001 (2010).
  17. T. Halpin-Healy and K. A. Takeuchi, A KPZ cocktail-shaken, not stirred, J. Stat. Phys. 160, 794 (2015).
  18. K. A. Takeuchi, An appetizer to modern developments on the Kardar–Parisi–Zhang universality class, Physica A 504, 77 (2018).
  19. Q. Fontaine, D. Squizzato, F. Baboux, I. Amelio, A. Lemaître, M. Morassi, I. Sagnes, L. Le Gratiet, A. Harouri, M. Wouters, I. Carusotto, A. Amo, M. Richard, A. Minguzzi, L. Canet, S. Ravets, and J. Bloch, Kardar-Parisi-Zhang universality in a one-dimensional polariton condensate, Nature (London) 608, 687 (2022).
  20. D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  21. S. Mu, J. Gong, and G. Lemarié, Kardar-Parisi-Zhang physics in the density fluctuations of localized two-dimensional wave packets, Phys. Rev. Lett. 132, 046301 (2024).
  22. R. Gutiérrez and R. Cuerno, Kardar-Parisi-Zhang universality class in the synchronization of oscillator lattices with time-dependent noise, Phys. Rev. E 110, L052201 (2024).
  23. E. Bertin and M. Clusel, Generalized extreme value statistics and sum of correlated variables, J. Phys. A: Math. Gen. 39, 7607 (2006).
  24. G. Makey, S. Galioglu, R. Ghaffari, E. D. Engin, G. Yıldırım, Ö. Yavuz, O. Bektaş, Ü. S. Nizam, Ö. Akbulut, Ö. Şahin, K. Güngör, D. Dede, H. V. Demir, F. Ö. Ilday, and S. Ilday, Universality of dissipative self-assembly from quantum dots to human cells, Nat. Phys. 16, 795 (2020).
  25. R. A. L. Almeida, S. O. Ferreira, T. J. Oliveira, and F. D. A. Aarão-Reis, Universal fluctuations in the growth of semiconductor thin films, Phys. Rev. B 89, 045309 (2014).
  26. T. Halpin-Healy and G. Palasantzas, Universal correlators and distributions as experimental signatures of (2 + 1)-dimensional Kardar-Parisi-Zhang growth, Europhys. Lett. 105, 50001 (2014).
  27. P. A. Orrillo, S. N. Santalla, R. Cuerno, L. Vázquez, S. B. Ribotta, L. M. Gassa, F. J. Mompean, R. C. Salvarezza, and M. E. Vela, Morphological stabilization and KPZ scaling by electrochemically induced co-deposition of nanostructured NiW alloy films, Sci. Rep. 7, 17997 (2017).
  28. M. Schroeder, M. Siegert, D. E. Wolf, J. D. Shore, and M. Plischke, Scaling of growing surfaces with large local slopes, Europhys. Lett. 24, 563 (1993).
  29. S. Das Sarma, S. V. Ghaisas, and J. M. Kim, Kinetic super-roughening and anomalous dynamic scaling in nonequilibrium growth models, Phys. Rev. E 49, 122 (1994).
  30. J. M. López, M. A. Rodríguez, and R. Cuerno, Superroughening versus intrinsic anomalous scaling of surfaces, Phys. Rev. E 56, 3993 (1997).
  31. J. J. Ramasco, J. M. López, and M. A. Rodríguez, Generic dynamic scaling in kinetic roughening, Phys. Rev. Lett. 84, 2199 (2000).
  32. R. Cuerno and L. Vázquez, Universality issues in surface kinetic roughening of thin solid films, in Advances in Condensed Matter and Statistical Physics, edited by E. Korutcheva and R. Cuerno (Nova Science Publishers, New York, 2004).
  33. B. G. Barreales, J. J. Meléndez, R. Cuerno, and J. J. Ruiz-Lorenzo, Large-scale kinetic roughening behavior of coffee-ring fronts, Phys. Rev. E 106, 044801 (2022).
  34. E. Rodríguez-Fernández, S. N. Santalla, M. Castro, and R. Cuerno, Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022).
  35. R. Gutiérrez and R. Cuerno, Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices, Phys. Rev. Res. 5, 023047 (2023).
  36. R. Gutiérrez and R. Cuerno, Kardar-Parisi-Zhang fluctuations in the synchronization dynamics of limit-cycle oscillators, Phys. Rev. Res. 6, 033324 (2024).
  37. E. Rodríguez-Fernández and R. Cuerno, Transition between chaotic and stochastic universality classes of kinetic roughening, Phys. Rev. Res. 3, L012020 (2021).
  38. J. M. Marcos, P. Rodríguez-López, J. J. Meléndez, R. Cuerno, and J. J. Ruiz-Lorenzo, Spreading fronts of wetting liquid droplets: Microscopic simulations and universal fluctuations, Phys. Rev. E 105, 054801 (2022).
  39. J. A. Plascak and P. H. Martins, Probability distribution function of the order parameter: Mixing fields and universality, Comput. Phys. Commun. 184, 259 (2013).
  40. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 4th ed. (Cambridge University Press, Cambridge, 2014).
  41. M. Newman and G. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999).
  42. R. Toral and P. Colet, Stochastic Numerical Methods: An Introduction for Students and Scientists (Wiley-VCH, Weinheim, 2014).
  43. M. Hairer, M. D. Ryser, and H. Weber, Triviality of the 2D stochastic Allen-Cahn equation, Electron. J. Probability 17, 1 (2012).
  44. M. D. Ryser, N. Nigam, and P. F. Tupper, On the well-posedness of the stochastic Allen–Cahn equation in two dimensions, J. Comput. Phys. 231, 2537 (2012).
  45. D. L. Monroy and G. G. Naumis, Description of mesoscale pattern formation in shallow convective cloud fields by using time-dependent Ginzburg-Landau and Swift-Hohenberg stochastic equations, Phys. Rev. E 103, 032312 (2021).
  46. Note, the letters α and β are frequently employed for the critical exponents characterizing the critical behavior of the Ising model at its Curie phase transition, and/or to denote inverse temperature. From now on, they are to be understood as kinetic roughening exponents.
  47. E. Vivo, M. Nicoli, and R. Cuerno, Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models, Phys. Rev. E 86, 051611 (2012).
  48. I. S. S. Carrasco and T. J. Oliveira, Universality and geometry dependence in the class of the nonlinear molecular beam epitaxy equation, Phys. Rev. E 94, 050801(R) (2016).
  49. Y. Saito, M. Dufay, and O. Pierre-Louis, Nonequilibrium cluster diffusion during growth and evaporation in two dimensions, Phys. Rev. Lett. 108, 245504 (2012).
  50. M. Prähofer and H. Spohn, Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett. 84, 4882 (2000).
  51. S. Prolhac and H. Spohn, Height distribution of the Kardar-Parisi-Zhang equation with sharp-wedge initial condition: Numerical evaluations, Phys. Rev. E 84, 011119 (2011).
  52. E. Rodríguez-Fernández and R. Cuerno, Non-KPZ fluctuations in the derivative of the Kardar-Parisi-Zhang equation or noisy Burgers equation, Phys. Rev. E 101, 052126 (2020).
  53. E. Rodríguez-Fernández and R. Cuerno, Gaussian statistics as an emergent symmetry far from equilibrium, Phys. Rev. E 99, 042108 (2019).
  54. K. B. Lauritsen and H. C. Fogedby, Critical exponents from power spectra, J. Stat. Phys. 72, 189 (1993).
  55. G. Besold, W. Schleier, and K. Heinz, Determination of the dynamic critical exponent by quench kinetics simulations, Phys. Rev. E 48, 4102 (1993).
  56. M. Campellone and J.-P. Bouchaud, Self-consistent screening approximation for critical dynamics, J. Phys. A: Math. Gen. 30, 3333 (1997).
  57. E. Medina, T. Hwa, M. Kardar, and Y. C. Zhang, Burgers equation with correlated noise: Renormalization-group analysis and applications to directed polymers and interface growth, Phys. Rev. A 39, 3053 (1989).
  58. K. G. Wilson, Feynman-graph expansion for critical exponents, Phys. Rev. Lett. 28, 548 (1972).
  59. E. Katzav and M. Schwartz, Exponent inequalities in dynamical systems, Phys. Rev. Lett. 107, 125701 (2011).
  60. E. Katzav and M. Schwartz, Dynamical inequality in growth models, Europhys. Lett. 95, 66003 (2011).
  61. C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, 2001).
  62. M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge University Press, Cambridge, 2009).
  63. K. Binder, Time-dependent Ginzburg-Landau theory of nonequilibrium relaxation, Phys. Rev. B 8, 3423 (1973).
  64. D. A. Huse, Remanent magnetization decay at the spin-glass critical point: A new dynamic critical exponent for nonequilibrium autocorrelations, Phys. Rev. B 40, 304 (1989).
  65. J. López, M. Castro, and R. Gallego, Scaling of local slopes, conservation laws, and anomalous roughening in surface growth, Phys. Rev. Lett. 94, 166103 (2005).
  66. C. Cartes, E. Tirapegui, R. Pandit, and M. Brachet, The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. Roy. Soc. A 380, 2021009 (2022).
  67. C. Fontaine, F. Vercesi, M. Brachet, and L. Canet, Unpredicted scaling of the one-dimensional Kardar-Parisi-Zhang equation, Phys. Rev. Lett. 131, 247101 (2023).
  68. J. López, Scaling approach to calculate critical exponents in anomalous surface roughening, Phys. Rev. Lett. 83, 4594 (1999).
  69. I. Balog, A. Rançon, and B. Delamotte, Critical probability distributions of the order parameter from the functional renormalization group, Phys. Rev. Lett. 129, 210602 (2022).
  70. T. Schäfer and V. Skokov, Dynamics of non-Gaussian fluctuations in model A, Phys. Rev. D 106, 014006 (2022).
  71. A. Pimpinelli and J. Villain, Physics of Crystal Growth (Cambridge University Press, Cambridge, 1998).
  72. H. Janssen, On critical exponents and the renormalization of the coupling constant in growth models with surface diffusion, Phys. Rev. Lett. 78, 1082 (1997).
  73. E. Katzav, Self-consistent expansion for the molecular beam epitaxy equation, Phys. Rev. E 65, 032103 (2002).
  74. C. Steinbock and E. Katzav, Dynamics of fluctuating thin sheets under random forcing, Phys. Rev. E 107, 025002 (2023).
  75. C. Steinbock and E. Katzav, Thermally driven elastic membranes are quasi-linear across all scales, J. Phys. A: Math. Theor. 56, 215002 (2023).
  76. B. I. Halperin, P. C. Hohenberg, and S.-K. Ma, Calculation of dynamic critical properties using Wilson's expansion methods, Phys. Rev. Lett. 29, 1548 (1972).

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Phys. Rev. Research 7, 043192– Published 20 November, 2025
  • Received 1 July 2025
  • Accepted 3 October 2025
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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.

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