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  • Open Access

Quantum Kibble-Zurek mechanism: The role of boundary conditions, endpoints, and kink types

Jose Soto-Garcia and Natalia Chepiga

  • The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom and Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

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Phys. Rev. B 113, 085430 – Published 20 February, 2026

DOI: https://doi.org/10.1103/s1pr-8vlp

Abstract

Quantum phase transitions are characterized by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble-Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism was used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and quantum three-state Potts, both in one dimension—we study the effect of boundary conditions, the location of the endpoints, and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or antisymmetric. We also show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions. Finally, we test our kink definition for the Ising transition on the period-2 phase of the Rydberg model and show that it is more robust against the endpoint than the standard definition.

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References (61)

  1. S. Sachdev, Quantum Phase Transitions 2nd ed. (Cambridge University Press, 2011).
  2. J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996), Vol. 5.
  3. W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett. 95, 105701 (2005).
  4. A. Polkovnikov, Universal adiabatic dynamics in the vicinity of a quantum critical point, Phys. Rev. B 72, 161201(R) (2005).
  5. M. Uhlmann, R. Schützhold, and U. R. Fischer, Vortex quantum creation and winding number scaling in a quenched spinor Bose gas, Phys. Rev. Lett. 99, 120407 (2007).
  6. J. Dziarmaga, Dynamics of a quantum phase transition and relaxation to a steady state, Adv. Phys. 59, 1063 (2010).
  7. A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, et al., Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator, Nature (London) 568, 207 (2019).
  8. M. Uhlmann, R. Schützhold, and U. R. Fischer, O (N) symmetry-breaking quantum quench: Topological defects versus quasiparticles, Phys. Rev. D 81, 025017 (2010).
  9. M. Uhlmann, R. Schützhold, and U. R. Fischer, System size scaling of topological defect creation in a second-order dynamical quantum phase transition, New J. Phys. 12, 095020 (2010).
  10. A. del Campo and W. H. Zurek, Universality of phase transition dynamics: Topological defects from symmetry breaking, Int. J. Mod. Phys. A 29, 1430018 (2014).
  11. T. W. Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen. 9, 1387 (1976).
  12. T. W. Kibble, Some implications of a cosmological phase transition, Phys. Rep. 67, 183 (1980).
  13. W. H. Zurek, Cosmological experiments in superfluid helium? Nature (London) 317, 505 (1985).
  14. J. Dziarmaga, Dynamics of a quantum phase transition: Exact solution of the quantum Ising model, Phys. Rev. Lett. 95, 245701 (2005).
  15. M. Kolodrubetz, B. K. Clark, and D. A. Huse, Nonequilibrium dynamic critical scaling of the quantum Ising chain, Phys. Rev. Lett. 109, 015701 (2012).
  16. A. Chandran, A. Erez, S. S. Gubser, and S. L. Sondhi, Kibble-Zurek problem: Universality and the scaling limit, Phys. Rev. B 86, 064304 (2012).
  17. A. Francuz, J. Dziarmaga, B. Gardas, and W. H. Zurek, Space and time renormalization in phase transition dynamics, Phys. Rev. B 93, 075134 (2016).
  18. D. Sadhukhan, A. Sinha, A. Francuz, J. Stefaniak, M. M. Rams, J. Dziarmaga, and W. H. Zurek, Sonic horizons and causality in phase transition dynamics, Phys. Rev. B 101, 144429 (2020).
  19. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  20. P. Laguna and W. H. Zurek, Density of kinks after a quench: When symmetry breaks, how big are the pieces? Phys. Rev. Lett. 78, 2519 (1997).
  21. H. Saito, Y. Kawaguchi, and M. Ueda, Kibble-Zurek mechanism in a quenched ferromagnetic Bose-Einstein condensate, Phys. Rev. A 76, 043613 (2007).
  22. G. De Chiara, A. Del Campo, G. Morigi, M. B. Plenio, and A. Retzker, Spontaneous nucleation of structural defects in inhomogeneous ion chains, New J. Phys. 12, 115003 (2010).
  23. A. del Campo, G. De Chiara, G. Morigi, M. B. Plenio, and A. Retzker, Structural Defects in Ion Chains by Quenching the External Potential: The Inhomogeneous Kibble-Zurek Mechanism, Phys. Rev. Lett. 105, 075701 (2010).
  24. D. Jaschke, K. Maeda, J. D. Whalen, M. L. Wall, and L. D. Carr, Critical phenomena and Kibble–Zurek scaling in the long-range quantum Ising chain, New J. Phys. 19, 033032 (2017).
  25. J. Soto Garcia and N. Chepiga, Resolving chiral transitions in one-dimensional Rydberg arrays with quantum Kibble-Zurek mechanism and finite-time scaling, Phys. Rev. B 110, 125113 (2024).
  26. C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis, and B. P. Anderson, Spontaneous vortices in the formation of Bose–Einstein condensates, Nature (London) 455, 948 (2008).
  27. S. M. Griffin, M. Lilienblum, K. T. Delaney, Y. Kumagai, M. Fiebig, and N. A. Spaldin, Scaling behavior and beyond equilibrium in the hexagonal manganites, Phys. Rev. X 2, 041022 (2012).
  28. G. Lamporesi, S. Donadello, S. Serafini, F. Dalfovo, and G. Ferrari, Spontaneous creation of Kibble–Zurek solitons in a Bose–Einstein condensate, Nat. Phys. 9, 656 (2013).
  29. M. Mielenz, J. Brox, S. Kahra, G. Leschhorn, M. Albert, T. Schaetz, H. Landa, and B. Reznik, Trapping of topological-structural defects in Coulomb crystals, Phys. Rev. Lett. 110, 133004 (2013).
  30. K. Pyka, J. Keller, H. Partner, R. Nigmatullin, T. Burgermeister, D. Meier, K. Kuhlmann, A. Retzker, M. B. Plenio, W. Zurek, et al., Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals, Nat. Commun. 4, 2291 (2013).
  31. S. Ulm, J. Roßnagel, G. Jacob, C. Degünther, S. Dawkins, U. Poschinger, R. Nigmatullin, A. Retzker, M. Plenio, F. Schmidt-Kaler, et al., Observation of the Kibble–Zurek scaling law for defect formation in ion crystals, Nat. Commun. 4, 2290 (2013).
  32. J. Beugnon and N. Navon, Exploring the Kibble–Zurek mechanism with homogeneous Bose gases, J. Phys. B: At. Mol. Opt. Phys. 50, 022002 (2017).
  33. B. Ko, J. W. Park, and Y.-i. Shin, Kibble–Zurek universality in a strongly interacting Fermi superfluid, Nat. Phys. 15, 1227 (2019).
  34. J.-M. Cui, F. J. Gómez-Ruiz, Y.-F. Huang, C.-F. Li, G.-C. Guo, and A. del Campo, Experimentally testing quantum critical dynamics beyond the Kibble–Zurek mechanism, Commun. Phys. 3, 44 (2020).
  35. Y. Bando, Y. Susa, H. Oshiyama, N. Shibata, M. Ohzeki, F. J. Gómez-Ruiz, D. A. Lidar, S. Suzuki, A. del Campo, and H. Nishimori, Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond, Phys. Rev. Res. 2, 033369 (2020).
  36. K. Lee, S. Kim, T. Kim, and Y. Shin, Universal Kibble–Zurek scaling in an atomic Fermi superfluid, Nat. Phys. 20, 1570 (2024).
  37. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  38. A. del Campo, F. J. Gómez-Ruiz, and H.-Q. Zhang, Locality of spontaneous symmetry breaking and universal spacing distribution of topological defects formed across a phase transition, Phys. Rev. B 106, L140101 (2022).
  39. R. Samajdar and D. A. Huse, Quantum and classical coarsening and their interplay with the Kibble-Zurek mechanism, arXiv:2401.15144.
  40. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer, New York, 1997).
  41. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  42. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  43. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  44. F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  45. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. 411, 167998 (2019).
  46. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  47. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  48. B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New J. Phys. 12, 025012 (2010).
  49. C.-Y. Xia and H.-B. Zeng, Kibble Zurek mechanism in rapidly quenched phase transition dynamics, arXiv:2110.07969.
  50. H.-B. Zeng, C.-Y. Xia, and A. del Campo, Universal breakdown of Kibble-Zurek scaling in fast quenches across a phase transition, Phys. Rev. Lett. 130, 060402 (2023).
  51. At the boundaries between the KZ regime and the other two regimes, we observe a transient behavior in which the kinks density increases more rapidly with the sweep rate than the KZ scaling would predict. The transient regime between the fast-quench and the KZ regime is the presaturated regime, which occurs when the initial point is in the adiabatic regime but the ending point is inside the nonadiabatic regime [61]. The size of both transient regimes is model dependent.
  52. J. Dziarmaga and M. M. Rams, Kink correlations, domain-size distribution, and emptiness formation probability after a Kibble-Zurek quench in the quantum Ising chain, Phys. Rev. B 106, 014309 (2022).
  53. D. Sen and S. Vishveshwara, Quenching across quantum critical points: Role of topological patterns, Europhys. Lett. 91, 66009 (2010).
  54. S. Sharma and A. Dutta, One-and two-dimensional quantum models: Quenches and the scaling of irreversible entropy, Phys. Rev. E 92, 022108 (2015).
  55. S. Hegde, V. Shivamoggi, S. Vishveshwara, and D. Sen, Quench dynamics and parity blocking in Majorana wires, New J. Phys. 17, 053036 (2015).
  56. G. G. Cabrera and R. Jullien, Role of boundary conditions in the finite-size Ising model, Phys. Rev. B 35, 7062 (1987).
  57. M. Campostrini, A. Pelissetto, and E. Vicari, Quantum Ising chains with boundary fields, J. Stat. Mech.: Theory Exp. (2015) P11015.
  58. A. Franchi, D. Rossini, and E. Vicari, Critical crossover phenomena driven by symmetry-breaking defects at quantum transitions, Phys. Rev. E 105, 034139 (2022).
  59. I. Affleck, N. Laflorencie, and E. S. Sørensen, Entanglement entropy in quantum impurity systems and systems with boundaries, J. Phys. A: Math. Theor. 42, 504009 (2009).
  60. F. J. Gómez-Ruiz, D. Subires, and A. del Campo, Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition, Phys. Rev. B 106, 134302 (2022).
  61. H.-C. Kou and P. Li, Varying quench dynamics in the transverse Ising chain: The Kibble-Zurek, saturated, and presaturated regimes, Phys. Rev. B 108, 214307 (2023).

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Phys. Rev. B 113, 085430– Published 20 February, 2026
  • Received 2 September 2025
  • Revised 8 December 2025
  • Accepted 26 January 2026
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