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ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

Statistics of Topological Defects in Close-Ended 1D Structures based on the Kibble Zurek Mechanism

Journal of Vibration Testing and System Dynamics 9(4) (2025) 391--402 | DOI:10.5890/JVTSD.2025.12.006

Vasileios Vachtsevanos$^{1,2,3}$, Hariton M. Polatoglou$^1$

$^1$ Physics Department, Aristotle University of Thessaloniki, Greece

$^2$ Physics Department, University of Thessaly

$^3$ Department of Computer Science, DEI college, University of Sunderland

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Abstract

The expected behaviour and number of topological defects that appear within a one dimensional object that undergoes a second order phase transition is well known and well defined by the Kibble- Zurek Mechanism. However one issue that is not usually tackled is whether or not a ring (close ended) structure has a differing behaviour from a linear (open ended) structure. The divergent behaviour could mean that classical statistical approaches to the problem might not be enough to fully describe the nature of this process. This ansatz is due to the fact that open ended structures do not have the effect of the lattice coupling at their boundary lattice sites. We theorise that the divergence due to this characteristic is meaningful for structures with a very small number of lattice sites. In this paper we present a comparison for the boundary condition of extremely small structures, approximately up to 22 lattice sites, where the shape of the lattice and the boundaries have a visible divergence in their behaviour. We also study the antiphase coupling case, where at least one defect should arise due the symmetry of the coupling being broken with a repulsive behaviour.

References

  1. [1]  Kibble, T. (2007), Phase-transition dynamics in the lab and the universe, Physics Today, 60(9), 47.
  2. [2]  Kibble, T.W.B. (1980), Some implications of a cosmological phase transition, Physics Reports, 67(1), 183-199.
  3. [3]  Zurek, W.H. and Dorner, U. (2008), Phase transition in space: how far does a symmetry bend before it breaks?, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366(1877), 2953-2972.
  4. [4]  Kibble, T.W.B. and Volovik, G.E. (1997), On phase ordering behind the propagating front of a second-order transition, Journal of Experimental and Theoretical Physics Letters, 65(1), 102-107.
  5. [5]  Zurek, W.H. (1996), Cosmological experiments in condensed matter systems, Physics Reports, 276(4), 177-221.
  6. [6]  Zurek, W.H., Dorner, U., and Zoller, P. (2005), Dynamics of a quantum phase transition, Physical Review Letters, 95(10), 105701.
  7. [7]  Zurek, W.H. (1985), Cosmological experiments in superfluid helium?, Nature, 317(6037), 505-508.
  8. [8]  Zurek, W.H. (1993), Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions, Acta Physica Polonica B, 24(7), 1301-1311.
  9. [9]  Kibble, T.W.B. (1976), Topology of cosmic domains and strings, Journal of Physics A: Mathematical and General, 9(8), 1387.
  10. [10]  Zurek, W. (2009), Causality in Condensates: Grey Solitons as Remnants of BEC Formation, Physical Review Letters, 102, 105702.
  11. [11]  Reichhardt, C.J., del Campo, A., and Reichhardt, C. (2022), Kibble-Zurek mechanism for nonequilibrium phase transitions in driven systems with quenched disorder, Communications Physics, 5(1). DOI:10.1038/s42005-022-00952-w.
  12. [12]  Cui, J.M., Gomez-Ruiz, F.J., Huang, Y.F., Li, C.F., Guo, G.C., and del Campo, A. (2020), Experimentally testing quantum critical dynamics beyond the Kibble-Zurek mechanism, Communications Physics, 3(1). DOI:10.1038/s42005-020-0306-6.
  13. [13]  Mayo, J.J., Fan, Z., Chern, G.W., and Del Campo, A. (2021), Distribution of kinks in an Ising ferromagnet after annealing and the generalized Kibble-Zurek Mechanism, Physical Review Research, 3(3). DOI:10.1103/physrevresearch.3.033150.
  14. [14]  Revathy, B.S., Mukherjee, V., Divakaran, U., and del Campo, A. (2020), Universal finite-time thermodynamics of many-body quantum machines from Kibble-Zurek scaling, Physical Review Research, 2(4), 043247.
  15. [15]  Bando, Y., Susa, Y., Oshiyama, H., Shibata, N., Ohzeki, M., Gomez-Ruiz, F.J., Lidar, D.A., Suzuki, S., Del Campo, A., and Nishimori, H. (2020), Probing the universality of topological defect formation in a Quantum annealer: Kibble-Zurek Mechanism and beyond, Physical Review Research, 2(3), 033369. DOI:10.1103/physrevresearch.2.033369.
  16. [16]  del Campo, A., Gómez-Ruiz, F.J., and Zhang, H.-Q. (2022), Locality of spontaneous symmetry breaking and universal spacing distribution of topological defects formed across a phase transition, Physical Review B, 106(14), L140101. DOI:10.1103/physrevb.106.l140101.
  17. [17]  del Campo, A., Gómez-Ruiz, F.J., Li, Z.H., Xia, C.Y., Zeng, H.B., and Zhang, H.Q. (2021), Universal statistics of vortices in a newborn holographic superconductor: Beyond the Kibble-Zurek mechanism, Journal of High Energy Physics, 2021(6), 1-21. DOI:10.1007/jhep06(2021)061.
  18. [18]  Maegochi, S. (2024), Kibble-Zurek mechanism for the dynamical ordering transition, Springer Theses, 83–94.
  19. [19]  Vachtsevanos, V. and Polatoglou, H.M. (2024), Statistics of Topological Defects in Finite One-Dimensional Structures based on the Kibble-Zurek Mechanism, Journal of Vibration Testing and System Dynamics, 8(02), 173-181.
  20. [20]  Kittel, C. (1996), Introduction to Solid State Physics, New York: Wiley.
  21. [21]  Chicone, C. (1999), Ordinary Differential Equations with Applications, Springer-Verlag, New York.
  22. [22]  Skyrme, T.H.R. and Schonland, B.F.J. (1961), A non-linear field theory, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 260(1300), 127–138.
  23. [23]  Tsallis, C. (1994), What are the numbers that experiments provide?, Química Nova, 17, 468.
  24. [24]  Duda, J. (2024), Phase space maximal entropy random walk: Langevin-like ensembles of physical trajectories, https://doi.org/10.48550/arXiv.2401.01239.
  25. [25]  Dziarmaga, J. (2005), Dynamics of a quantum phase transition: Exact solution of the quantum Ising model, Physical Review Letters, 95(24), 245701.
  26. [26]  Damski, B. (2005), The simplest quantum model supporting the Kibble-Zurek mechanism of topological defect production: Landau-Zener transitions from a new perspective, Physical Review Letters, 95(3), 035701.
  27. [27]  Imry, Y. and Wortis, M. (1979), Influence of quenched impurities on first-order phase transitions, Physical Review B, 19(7), 3580.
  28. [28]  Ojovan, M.I. (2013), Ordering and structural changes at the glass–liquid transition, Journal of Non-Crystalline Solids, 382, 79-86.
  29. [29]  Polkovnikov, A. (2005), Universal adiabatic dynamics in the vicinity of a quantum critical point, Physical Review B, 72(16), 161201.
  30. [30]  Hindmarsh, M. and Rajantie, A. (2000), Defect formation and local gauge invariance, Physical Review Letters, 85(22), 4660.
  31. [31]  Dziarmaga, J. (2010), Dynamics of a quantum phase transition and relaxation to a steady state, Advances in Physics, 59(6), 1063-1189.
  32. [32]  Deutschländer, S., Dillmann, P., Maret, G., and Keim, P. (2015), Kibble–Zurek mechanism in colloidal monolayers, Proceedings of the National Academy of Sciences, 112(22), 6925-6930.
  33. [33]  del Campo, A. (2018), Universal statistics of topological defects formed in a quantum phase transition, Physical Review Letters, 121(20), 200601.
  34. [34]  Gomez-Ruiz, F.J., Mayo, J.J., and del Campo, A. (2020), Full counting statistics of topological defects after crossing a phase transition, Physical Review Letters, 124(24), 240602.
  35. [35]  Del Campo, A., De Chiara, G., Morigi, G., Plenio, M.B., and Retzker, A. (2010), Structural defects in ion chains by quenching the external potential: The inhomogeneous Kibble-Zurek mechanism, Physical Review Letters, 105(7), 075701.
  36. [36]  del Campo, A. and Zurek, W.H. (2014), Universality of phase transition dynamics: Topological defects from symmetry breaking, Symmetry and Fundamental Physics: Tom Kibble at 80, 31-87.
  37. [37]  del Campo, A., Kibble, T.W.B., and Zurek, W.H. (2013), Causality and non-equilibrium second-order phase transitions in inhomogeneous systems, Journal of Physics: Condensed Matter, 25(40), 404210.
  38. [38]  Kirzhinits, D.A. and Shpatakovskaya, G.V. (1972), Atomic structure oscillation effects, Soviet Physics JETP, 35(6).
  39. [39]  Kirzhinits, D.A. and Linde, A.D. (1972), Macroscopic consequences of the Weinberg model, Physics Letters B, 42(4), 471-474.
  40. [40]  Rajantie, A. (2002), Formation of topological defects in gauge field theories, International Journal of Modern Physics A, 17(1), 1-43.
  41. [41]  Antunes, N.D., Gandra, P., and Rivers, R.J. (2006), Is domain formation decided before or after the transition?, Physical Review D, 73(12), 125003.
  42. [42]  Arnold, M. and Nigmatullin, R. (2021), Dynamics of vortex defect formation in two dimensional Coulomb crystals, [Preprint], arXiv:2105.11680.
  43. [43]  Jaeger, G. (1998), The Ehrenfest classification of phase transitions: introduction and evolution, Archive for History of Exact Sciences, 54(1), 51-81.
  44. [44]  Landau, L. (1936), The theory of phase transitions, Nature, 138(3498), 840-841.
  45. [45]  Kloeden, P.E. and Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, Springer, Berlin.

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