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Apparent bistability from weak long-range interactions

Achilleas Lazarides1 and Andrea Pizzi2

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Phys. Rev. B 113, 224116 – Published 26 June, 2026

DOI: https://doi.org/10.1103/6c4l-kv48

Abstract

Bistability, or the coexistence of two stable phases, can be broken by a bias field h destabilizing one of the phases via the nucleation and growth of defects. Strong long-range interactions, 1/rα with α less than the system's dimensionality d, can suppress the proliferation of defects and restore bistability. The case of weak long-range interactions d<α<d+1 remains instead poorly understood. Here, we show that it supports apparent bistability: While the system has in principle a unique stable phase, it appears bistable for all practical purposes for α<αc, with αc>d behaving as a genuine critical point. At the core of this is an exponential scaling of the critical droplet size Rc∼h−1/(α−d) with α, which makes nucleating destabilizing droplets extremely unlikely for α<αc, and such that αc is mostly independent of system size. In support of these conclusions we provide field-theoretical arguments and numerics on a probabilistic cellular automaton. Overall, our results offer a way to rethink phase stability in systems with long-range interactions as well as another route to achieve practical bistability.

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References (69)

  1. K. Binder, Theory of first-order phase transitions, Rep. Prog. Phys. 50, 783 (1987).
  2. D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Wetting and spreading, Rev. Mod. Phys. 81, 739 (2009).
  3. D. N. Sibley, B. D. Goddard, F. F. Ouali, D. J. Fairhurst, and A. J. Archer, Coexisting multiphase and interfacial behavior of ouzo, Phys. Fluids 37, 042118 (2025).
  4. J. S. Langer, Statistical theory of the decay of metastable states, Ann. Phys. 54, 258 (1969).
  5. M. E. Fisher, The theory of condensation and the critical point, Phys. Phys. Fiz. 3, 255 (1967).
  6. J. S. Langer, Theory of the condensation point, Ann. Phys. 41, 108 (1967).
  7. N. Gunther, D. Wallace, and D. Nicole, Goldstone modes in vacuum decay and first-order phase transitions, J. Phys. A: Math. Gen. 13, 1755 (1980).
  8. H. Tomita and S. Miyashita, Statistical properties of the relaxation processes of metastable states in the kinetic Ising model, Phys. Rev. B 46, 8886 (1992).
  9. P. A. Rikvold, H. Tomita, S. Miyashita, and S. W. Sides, Metastable lifetimes in a kinetic Ising model: Dependence on field and system size, Phys. Rev. E 49, 5080 (1994).
  10. A. Bray, Domain-growth scaling in systems with long-range interactions, Phys. Rev. E 47, 3191 (1993).
  11. A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51, 481 (2010).
  12. R. B. Griffiths, C.-Y. Weng, and J. S. Langer, Relaxation times for metastable states in the mean-field model of a ferromagnet, Phys. Rev. 149, 301 (1966).
  13. O. Penrose and J. L. Lebowitz, Rigorous treatment of metastable states in the van der Waals-Maxwell theory, J. Stat. Phys. 3, 211 (1971).
  14. P. Chandra, Nucleation in the presence of long-range interactions, Phys. Rev. A 39, 3672 (1989).
  15. M. Antoni and S. Ruffo, Clustering and relaxation in Hamiltonian long-range dynamics, Phys. Rev. E 52, 2361 (1995).
  16. Y. Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, and S. Ruffo, Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model, Physica A 337, 36 (2004).
  17. D. Mukamel, S. Ruffo, and N. Schreiber, Breaking of ergodicity and long relaxation times in systems with long-range interactions, Phys. Rev. Lett. 95, 240604 (2005).
  18. A. Campa, T. Dauxois, and S. Ruffo, Statistical mechanics and dynamics of solvable models with long-range interactions, Phys. Rep. 480, 57 (2009).
  19. A. Gabrielli, M. Joyce, and B. Marcos, Quasistationary states and the range of pair interactions, Phys. Rev. Lett. 105, 210602 (2010).
  20. M. Nishino, C. Enachescu, S. Miyashita, P. A. Rikvold, K. Boukheddaden, and F. Varret, Macroscopic nucleation phenomena in continuum media with long-range interactions, Sci. Rep. 1, 162 (2011).
  21. S. Gupta and S. Ruffo, The world of long-range interactions: A bird's eye view, Int. J. Mod. Phys. A 32, 1741018 (2017).
  22. M. Kastner, Diverging equilibration times in long-range quantum spin models, Phys. Rev. Lett. 106, 130601 (2011).
  23. N. Defenu, Metastability and discrete spectrum of long-range systems, Proc. Natl. Acad. Sci. USA 118, e2101785118 (2021).
  24. N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys. 95, 035002 (2023).
  25. N. Defenu, A. Lerose, and S. Pappalardi, Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions, Phys. Rep. 1074, 1 (2024).
  26. R. Mattes, I. Lesanovsky, and F. Carollo, Long-range interacting systems are locally noninteracting, Phys. Rev. Lett. 134, 070402 (2025).
  27. For instance, for the two-dimensional Ising model, the critical droplet radius Rc and metastable lifetime τ diverge for h→0. The existence of more stable phases in this case is unsurprising, indeed unavoidable given the Z2 symmetry for h=0. By contrast, we want here to focus on bistability in spite of a bias field h, that is, not directly relying on an underlying symmetry.
  28. C. H. Bennett and G. Grinstein, Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems, Phys. Rev. Lett. 55, 657 (1985).
  29. A. L. Toom, Stable and attractive trajectories in multicomponent systems, in Multicomponent Random Systems, edited by R. L. Dobrushin, Advances in Probability Vol. 6 (Dekker, New York, 1980), pp. 549–575.
  30. G. Grinstein, Can complex structures be generically stable in a noisy world, IBM J. Res. Dev. 48, 5 (2004).
  31. A. Lazarides, S. Roy, F. Piazza, and R. Moessner, On time crystallinity in dissipative Floquet systems, Phys. Rev. Res. 2, 022002(R) (2020).
  32. F. M. Gambetta, F. Carollo, A. Lazarides, I. Lesanovsky, and J. P. Garrahan, Classical stochastic discrete time crystals, Phys. Rev. E 100, 060105(R) (2019).
  33. F. M. Gambetta, F. Carollo, M. Marcuzzi, J. P. Garrahan, and I. Lesanovsky, Discrete time crystals in the absence of manifest symmetries or disorder in open quantum systems, Phys. Rev. Lett. 122, 015701 (2019).
  34. A. Pizzi, A. Nunnenkamp, and J. Knolle, Bistability and time crystals in long-ranged directed percolation, Nat. Commun. 12, 1061 (2021).
  35. M. McGinley, S. Roy, and S. A. Parameswaran, Absolutely stable spatiotemporal order in noisy quantum systems, Phys. Rev. Lett. 129, 090404 (2022).
  36. F. Machado, Q. Zhuang, N. Y. Yao, and M. P. Zaletel, Absolutely stable time crystals at finite temperature, Phys. Rev. Lett. 131, 180402 (2023).
  37. R. J. McCraw, Metastability in a long range one-dimensional Ising model, Phys. Lett. A 75, 379 (1980).
  38. A. C. van Enter, B. Kimura, W. Ruszel, and C. Spitoni, Nucleation for one-dimensional long-range Ising models, J. Stat. Phys. 174, 1327 (2019).
  39. P. Horikx, Metastability in the two dimensional long-range Ising model, Master's thesis, Utrecht University, 2020, https://studenttheses.uu.nl/handle/20.500.12932/38364.
  40. V. Jacquier, W. Ruszel, and C. Spitoni, Isoperimetric inequality for nonlocal bi-axial discrete perimeter, arXiv:2412.13005.
  41. V. Jacquier, Exploring metastability in Ising models: Critical droplets, energy barriers and exit time, arXiv:2501.06163.
  42. Z.-H. Zhu, Y. Liu, G. Lagnese, F. M. Surace, W.-Y. Zhang, M.-G. He, J. C. Halimeh, M. Dalmonte, S. C. Morampudi, F. Wilczek, et al., Probing false vacuum decay on a cold-atom gauge-theory quantum simulator, arXiv:2411.12565.
  43. D. Gonzalez-Cuadra, M. Hamdan, T. V. Zache, B. Braverman, M. Kornjača, A. Lukin, S. H. Cantú, F. Liu, S.-T. Wang, A. Keesling, et al., Observation of string breaking on a (2+ 1)D Rydberg quantum simulator, Nature (London) 642, 321 (2025).
  44. D.-S. Xiang, P. Zhou, C. Liu, H.-X. Liu, Y.-W. Zhang, D. Yuan, K. Zhang, B. Xu, M. Dalmonte, D.-L. Deng, et al., Real-time scattering and freeze-out dynamics in Rydberg-atom lattice gauge theory, arXiv:2508.06639.
  45. J. Vodeb, J.-Y. Desaules, A. Hallam, A. Rava, G. Humar, D. Willsch, F. Jin, M. Willsch, K. Michielsen, and Z. Papić, Stirring the false vacuum via interacting quantized bubbles on a 5,564-qubit quantum annealer, Nat. Phys. 21, 386 (2025).
  46. R. C. Farrell, M. Illa, A. N. Ciavarella, and M. J. Savage, Scalable circuits for preparing ground states on digital quantum computers: The Schwinger model vacuum on 100 qubits, PRX Quantum 5, 020315 (2024).
  47. N. A. Zemlevskiy, Scalable quantum simulations of scattering in scalar field theory on 120 qubits, Phys. Rev. D 112, 034502 (2025).
  48. In a volume Ld there fit Nℓ∼(L/ℓ)d objects of linear extent ℓ, with total surface area Nℓℓd−1=Ld/ℓ, so the wall area per unit volume is ∼ℓ−1.
  49. M. Schick, Introduction to wetting phenomena, in Liquids at Interfaces, edited by J. Charvolin, J. F. Joanny, and J. Zinn-Justin, Les Houches Summer School Proceedings Vol. 48 (Elsevier, Amsterdam, 1990), pp. 415–497.
  50. S. Dietrich, Wetting phenomena, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1988), Vol. 12, pp. 1–218.
  51. G. Grinstein, C. Jayaprakash, and Y. He, Statistical mechanics of probabilistic cellular automata, Phys. Rev. Lett. 55, 2527 (1985).
  52. J. L. Lebowitz, C. Maes, and E. R. Speer, Statistical mechanics of probabilistic cellular automata, J. Stat. Phys. 59, 117 (1990).
  53. Note that Fig. 4 assumes a single droplet placed in a square box of size L. With periodic boundary conditions, this is equivalent to a periodic array of droplets, namely, not a single droplet but rather a mixed set of both phases, not dissimilar from Fig. 3.
  54. C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, et al., Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021).
  55. M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  56. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  57. T. Manovitz, S. H. Li, S. Ebadi, R. Samajdar, A. A. Geim, S. J. Evered, D. Bluvstein, H. Zhou, N. U. Koyluoglu, J. Feldmeier, et al., Quantum coarsening and collective dynamics on a programmable simulator, Nature (London) 638, 86 (2025).
  58. D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature (London) 393, 440 (1998).
  59. G. Passarelli, P. Lucignano, R. Fazio, and A. Russomanno, Dissipative time crystals with long-range Lindbladians, Phys. Rev. B 106, 224308 (2022).
  60. F. Machado, D. V. Else, G. D. Kahanamoku-Meyer, C. Nayak, and N. Y. Yao, Long-range prethermal phases of nonequilibrium matter, Phys. Rev. X 10, 011043 (2020).
  61. A. Pizzi, A. Nunnenkamp, and J. Knolle, Classical approaches to prethermal discrete time crystals in one, two, and three dimensions, Phys. Rev. B 104, 094308 (2021).
  62. B. Ye, F. Machado, and N. Y. Yao, Floquet phases of matter via classical prethermalization, Phys. Rev. Lett. 127, 140603 (2021).
  63. A. Pizzi, J. Knolle, and A. Nunnenkamp, Higher-order and fractional discrete time crystals in clean long-range interacting systems, Nat. Commun. 12, 2341 (2021).
  64. H.-K. Jin, A. Pizzi, and J. Knolle, Prethermal nematic order and staircase heating in a driven frustrated Ising magnet with dipolar interactions, Phys. Rev. B 106, 144312 (2022).
  65. S. Yi-Thomas and J. D. Sau, Theory for dissipative time crystals in coupled parametric oscillators, Phys. Rev. Lett. 133, 266601 (2024).
  66. D. Luo, F. M. Surace, A. De, A. Lerose, E. R. Bennewitz, B. Ware, A. Schuckert, Z. Davoudi, A. V. Gorshkov, O. Katz, et al., Quantum simulation of bubble nucleation across a quantum phase transition, arXiv:2505.09607.
  67. C. Yin, F. M. Surace, and A. Lucas, Theory of metastable states in many-body quantum systems, Phys. Rev. X 15, 011064 (2025).
  68. G. Lagnese, F. M. Surace, S. Morampudi, and F. Wilczek, Detecting a long-lived false vacuum with quantum quenches, Phys. Rev. Lett. 133, 240402 (2024).
  69. K. Macieszczak, M. Guţă, I. Lesanovsky, and J. P. Garrahan, Towards a theory of metastability in open quantum dynamics, Phys. Rev. Lett. 116, 240404 (2016).

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Phys. Rev. B 113, 224116– Published 26 June, 2026
  • Received 16 June 2025
  • Revised 3 March 2026
  • Accepted 25 March 2026
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