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  • Open Access

Langer’s Nucleation Rate Reproduced on the Lattice

Joonas Hirvonen* and Oliver Gould†

  • *Contact author: joonas.hirvonen@nottingham.ac.uk
  • †Contact author: oliver.gould@nottingham.ac.uk
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Phys. Rev. Lett. 136, 081601 – Published 24 February, 2026

DOI: https://doi.org/10.1103/c75g-xbw4

Abstract

We show that Langer’s rate of bubble nucleation is quantitatively correct up to small higher-loop corrections, in comparison to lattice simulations. These results are a significant advancement on decades of lattice studies showing only qualitative trends, and the first showing agreement for any conservative field theory. We confirm that the failure to fully thermalize the metastable phase explains discrepancies with recent lattice studies that found disagreement with Langer’s rate. The key theoretical development is the translation of Langer’s perturbative definition of a thermal metastable phase into a nonperturbative statement that can be implemented on the lattice. Our statistical and systematic errors are small enough to allow us to measure on the lattice the coefficient of the two-loop contribution, missing from the perturbative prediction. Our conclusions also exclude a possible systematic uncertainty in He3 experiments.

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References (68)

  1. J. W. Gibbs, On the equilibrium of heterogeneous substances, Trans. Connecticut Acad. Arts Sci. 3, 343 (1874).
  2. R. Becker and W. Döring, Kinetische behandlung der keimbildung in übersättigten dämpfen, Ann. Phys. (Berlin) 416, 719 (1935).
  3. E. Wigner, The transition state method, Trans. Faraday Soc. 34, 29 (1938).
  4. Y. B. Zeldovich, On the theory of new phase formation. Cavitation, in Selected Works of Yakov Borisovich Zeldovich, Volume I: Chemical Physics and Hydrodynamics, Vol. 1, edited by J. P. Ostriker, G. I. Barenblatt, R. A. Sunyaev, A. Granik, and E. Jackson (Princeton University Press, Princeton, NJ, 1992), pp. 120–137.
  5. H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical re actions, Physica (Utrecht) 7, 284 (1940).
  6. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys. 31, 688 (1959).
  7. J. S. Langer, Theory of the condensation point, Ann. Phys. (N.Y.) 41, 108 (1967); 281, 941(E) (2000).
  8. J. S. Langer, Statistical theory of the decay of metastable states, Ann. Phys. (N.Y.) 54, 258 (1969).
  9. A. Berera, J. Mabillard, B. W. Mintz, and R. O. Ramos, Formulating the Kramers problem in field theory, Phys. Rev. D 100, 076005 (2019).
  10. A. Ekstedt, Bubble nucleation to all orders, J. High Energy Phys. 08 (2022) 115.
  11. O. Gould and J. Hirvonen, Effective field theory approach to thermal bubble nucleation, Phys. Rev. D 104, 096015 (2021).
  12. J. Hirvonen, Real-time nucleation and off-equilibrium effects in high-temperature quantum field theories, Phys. Rev. D 111, 116020 (2025).
  13. See Supplemental Material at http://link.aps.org/supplemental/10.1103/c75g-xbw4 for discussion of the connection to high-temperature quantum field theory, the continuum limit, lattice details, the thermalization algorithm, the numerical tests we performed, and the priors for model fits, which includes Refs. [12,14–27].
  14. G. Aarts and J. Smit, Finiteness of hot classical scalar field theory and the plasmon damping rate, Phys. Lett. B 393, 395 (1997).
  15. G. Aarts and J. Smit, Classical approximation for time dependent quantum field theory: Diagrammatic analysis for hot scalar fields, Nucl. Phys. B511, 451 (1998).
  16. D. Bodeker, Classical real time correlation functions and quantum corrections at finite temperature, Nucl. Phys. B486, 500 (1997).
  17. C. Greiner and B. Muller, Classical fields near thermal equilibrium, Phys. Rev. D 55, 1026 (1997).
  18. J.-P. Blaizot and E. Iancu, The Quark gluon plasma: Collective dynamics and hard thermal loops, Phys. Rep. 359, 355 (2002).
  19. S. Jeon, The Boltzmann equation in classical and quantum field theory, Phys. Rev. C 72, 014907 (2005).
  20. M. Laine, Exact relation of lattice and continuum parameters in three-dimensional SU(2) + Higgs theories, Nucl. Phys. B451, 484 (1995).
  21. D. Pîrvu, A. Shkerin, and S. Sibiryakov, Thermal false vacuum decay in (1+1)-dimensions: Evidence for non-equilibrium dynamics, Int. J. Mod. Phys. A 39, 2445007 (2024).
  22. E. Forest and R. D. Ruth, Fourth order symplectic integration, Physica (Amsterdam) 43D, 105 (1990).
  23. H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150, 262 (1990).
  24. S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195, 216 (1987).
  25. M. Tuckerman, Statistical Mechanics: Theory And Molecular Simulation (Oxford University Press, New York, 2001), pp. 121–124.
  26. 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).
  27. W. K. Hastings, Monte carlo sampling methods using markov chains and their applications, Biometrika 57, 97 (1970).
  28. D. W. Oxtoby, Homogeneous nucleation: Theory and experiment, J. Phys. Condens. Matter 4, 7627 (1992).
  29. J. S. Langer and A. J. Schwartz, Kinetics of nucleation in near-critical fluids, Phys. Rev. A 21, 948 (1980).
  30. R. B. Nellas, S. J. Keasler, J. I. Siepmann, and B. Chen, Exploring the discrepancies between experiment, theory, and simulation for the homogeneous gas-to-liquid nucleation of 1-pentanol, J. Chem. Phys. 132 (2010).
  31. M. Hindmarsh et al. (QUEST-DMC Collaboration), A-B transition in superfluid He3 and cosmological phase transitions, J. Low Temp. Phys. 215, 495 (2024).
  32. O. Fialko, B. Opanchuk, A. I. Sidorov, P. D. Drummond, and J. Brand, Fate of the false vacuum: Towards realization with ultra-cold atoms, Europhys. Lett. 110, 56001 (2015).
  33. O. Fialko, B. Opanchuk, A. I. Sidorov, P. D. Drummond, and J. Brand, The universe on a table top: Engineering quantum decay of a relativistic scalar field from a metastable vacuum, J. Phys. B 50, 024003 (2017).
  34. T. P. Billam, K. Brown, and I. G. Moss, False-vacuum decay in an ultracold spin-1 Bose gas, Phys. Rev. A 105, L041301 (2022).
  35. Y. Tian et al., Supercooling of the A phase of He3, Nat. Commun. 14, 148 (2023).
  36. A. C. Jenkins, J. Braden, H. V. Peiris, A. Pontzen, M. C. Johnson, and S. Weinfurtner, Analog vacuum decay from vacuum initial conditions, Phys. Rev. D 109, 023506 (2024).
  37. A. C. Jenkins, I. G. Moss, T. P. Billam, Z. Hadzibabic, H. V. Peiris, and A. Pontzen, Generalized cold-atom simulators for vacuum decay, Phys. Rev. A 110, L031301 (2024).
  38. S. Darbha, M. Kornjača, F. Liu, J. Balewski, M. R. Hirsbrunner, P. L. S. Lopes, S.-T. Wang, R. Van Beeumen, D. Camps, and K. Klymko, False vacuum decay and nucleation dynamics in neutral atom systems, Phys. Rev. B 110, 155103 (2024).
  39. B. Song, S. Dutta, S. Bhave, J.-C. Yu, E. Carter, N. Cooper, and U. Schneider, Realizing discontinuous quantum phase transitions in a strongly correlated driven optical lattice, Nat. Phys. 18, 259 (2022).
  40. A. Zenesini, A. Berti, R. Cominotti, C. Rogora, I. G. Moss, T. P. Billam, I. Carusotto, G. Lamporesi, A. Recati, and G. Ferrari, Observation of false vacuum decay via bubble formation in ferromagnetic superfluids, Nat. Phys. 20, 558 (2023).
  41. D. Y. Grigoriev and V. A. Rubakov, Soliton pair creation at finite temperatures. Numerical study in (1+1)-dimensions, Nucl. Phys. B299, 67 (1988).
  42. A. I. Bochkarev and P. de Forcrand, The study of Kink—Antikink pair production at finite temperature by means of the Langevin equation, Phys. Rev. Lett. 63, 2337 (1989).
  43. M. G. Alford, H. Feldman, and M. Gleiser, Thermal nucleation of kink—anti-kink pairs, Phys. Rev. Lett. 68, 1645 (1992).
  44. O. T. Valls and G. F. Mazenko, Nucleation in a time-dependent Ginzburg-Landau model: A numerical study, Phys. Rev. B 42, 6614 (1990).
  45. S. Borsanyi, A. Patkos, J. Polonyi, and Z. Szep, Fate of the classical false vacuum, Phys. Rev. D 62, 085013 (2000).
  46. M. G. Alford, H. Feldman, and M. Gleiser, Thermal activation of metastable decay: Testing nucleation theory, Phys. Rev. D 47, R2168 (1993).
  47. M. G. Alford and M. Gleiser, Metastability in two-dimensions and the effective potential, Phys. Rev. D 48, 2838 (1993).
  48. G. D. Moore and K. Rummukainen, Electroweak bubble nucleation, nonperturbatively, Phys. Rev. D 63, 045002 (2001).
  49. G. D. Moore, K. Rummukainen, and A. Tranberg, Nonperturbative computation of the bubble nucleation rate in the cubic anisotropy model, J. High Energy Phys. 04 (2001) 017.
  50. O. Gould, S. Güyer, and K. Rummukainen, First-order electroweak phase transitions: A nonperturbative update, Phys. Rev. D 106, 114507 (2022).
  51. A. Ekstedt, Higher-order corrections to the bubble-nucleation rate at finite temperature, Eur. Phys. J. C 82, 173 (2022).
  52. A. Ekstedt, O. Gould, and J. Hirvonen, BubbleDet: A Python package to compute functional determinants for bubble nucleation, J. High Energy Phys. 12 (2023) 056.
  53. D. Pîrvu, A. Shkerin, and S. Sibiryakov, Thermal false vacuum decay is not what it seems, arXiv:2407.06263.
  54. O. Gould, A. Kormu, and D. J. Weir, Nonperturbative test of nucleation calculations for strong phase transitions, Phys. Rev. D 111, L051901 (2025).
  55. A. Shkerin and S. Sibiryakov, Simple third order operator-splitting schemes for stochastic mechanics and field theory, arXiv:2504.03844.
  56. L. Batini, A. Chatrchyan, and J. Berges, Real-time dynamics of false vacuum decay, Phys. Rev. D 109, 023502 (2024).
  57. S. Y. Khlebnikov and M. E. Shaposhnikov, The statistical theory of anomalous fermion number nonconservation, Nucl. Phys. B308, 885 (1988).
  58. G. D. Moore, Measuring the broken phase sphaleron rate nonperturbatively, Phys. Rev. D 59, 014503 (1999).
  59. D. W. Scott, On optimal and data-based histograms, Biometrika 66, 605 (1979).
  60. M. Carosi and B. Garbrecht, False vacuum decay beyond the quadratic approximation: Summation of nonlocal self-energies, Phys. Rev. D 111, 085002 (2025).
  61. J. Braden, M. C. Johnson, H. V. Peiris, A. Pontzen, and S. Weinfurtner, New semiclassical picture of vacuum decay, Phys. Rev. Lett. 123, 031601 (2019) 129, 059901(E) (2022).
  62. T. P. Billam, R. Gregory, F. Michel, and I. G. Moss, Simulating seeded vacuum decay in a cold atom system, Phys. Rev. D 100, 065016 (2019).
  63. M. P. Hertzberg, F. Rompineve, and N. Shah, Quantitative analysis of the stochastic approach to quantum tunneling, Phys. Rev. D 102, 076003 (2020).
  64. P. Hanggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after kramers, Rev. Mod. Phys. 62, 251 (1990).
  65. K. Enqvist, J. Ignatius, K. Kajantie, and K. Rummukainen, Nucleation and bubble growth in a first order cosmological electroweak phase transition, Phys. Rev. D 45, 3415 (1992).
  66. A. Kurkela and G. D. Moore, Thermalization in weakly coupled nonabelian plasmas, J. High Energy Phys. 12 (2011) 044.
  67. A. Ekstedt, O. Gould, and J. Löfgren, Radiative first-order phase transitions to next-to-next-to-leading order, Phys. Rev. D 106, 036012 (2022); 110, 019901(E) (2024).
  68. J. Hirvonen and O. Gould, Langer’s nucleation rate reproduced on the lattice, dataset, 2025, 10.5281/zenodo.15424939.

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Phys. Rev. Lett. 136, 081601– Published 24 February, 2026
  • Received 19 June 2025
  • Revised 18 December 2025
  • Accepted 3 February 2026
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