pith. sign in

arxiv: 2605.13564 · v1 · pith:EG6B5IWEnew · submitted 2026-05-13 · ❄️ cond-mat.stat-mech

Phase Ordering in a few O(n) Symmetric Models: Slow Growth, Mpemba Effect and Experimental Relevance

Wasim Akram , Nalina Vadakkayil , Sohini Chatterjee , Subir K. Das This is my paper

Pith reviewed 2026-05-14 17:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords phase orderingMpemba effectXY modelIsing modelvortex annihilationdomain growthMonte Carlo simulation
0
0 comments X

The pith

In the 3D XY model, characteristic length grows as t to the 0.15 after zero-temperature quench because vortex cores form line defects whose annihilation sets the pace, and quenches from higher starting temperatures reach equilibrium faster.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines phase ordering after quenches in the three-dimensional nonconserved XY model and related Ising systems using Monte Carlo methods. It reports that at final temperature zero the length scale grows slowly as roughly t to the 0.15 because growth proceeds through annihilation of line defects created when vortex cores connect across lattice planes. Systems started from higher temperatures above the critical point relax to the ordered state more quickly than those started closer to the critical point. The same Mpemba-like speedup appears in two- and three-dimensional Ising models, though in two dimensions it requires the initial magnetization to be near zero while in three dimensions it occurs for the full range of initial magnetizations. These behaviors are presented as accessible to experiment.

Core claim

For quenches to Tf equals zero in the three-dimensional XY model the characteristic length ell of t grows approximately as t to the power 0.15 in the long-time regime, far below the expected 1/2, because the ordering is driven by annihilation of vortex and anti-vortex pairs whose cores join from different planes to form line defects. Quenches performed from a range of starting temperatures Ts above Tc show that larger Ts values produce faster relaxation to the final equilibrium state. The same pattern of slow growth at deep quenches and the Mpemba-like dependence on initial temperature is recovered in two- and three-dimensional Ising models, with the additional observation that the Mpemba效应

What carries the argument

Line defects formed when vortex cores join across planes, whose pairwise annihilation directly controls the measured growth of the characteristic length scale.

If this is right

  • The growth law recovers the expected t to the 1/2 scaling once the final temperature is raised enough above zero.
  • The Mpemba speedup occurs for the full distribution of initial magnetizations in three dimensions but only near-zero magnetization in two dimensions.
  • The reported exponents and temperature dependence are presented as directly testable in magnetic or superfluid systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If line-defect annihilation governs the dynamics, analogous slow growth should appear in other three-dimensional models that support stable line-like topological defects.
  • The robustness of the Mpemba effect to the full magnetization distribution in three dimensions suggests it may be easier to observe experimentally in bulk three-dimensional samples than in two-dimensional films.
  • Measuring the time evolution of the correlation length in superfluid films or magnetic materials after controlled quenches would directly test the predicted 0.15 exponent.

Load-bearing premise

That the long-time growth exponent is set by the annihilation of these line defects and is not dominated by finite-size effects or lattice artifacts.

What would settle it

A clear crossover to an exponent near 1/2 at sufficiently late times in much larger systems, or direct imaging showing that line defects disappear well before the measured growth regime, would show the scaling is not controlled by line-defect annihilation.

Figures

Figures reproduced from arXiv: 2605.13564 by Nalina Vadakkayil, Sohini Chatterjee, Subir K. Das, Wasim Akram.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) 2D cross-sections showing spin configurations on [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Average domain lengths, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Average domain lengths for the 3D XY model are plot [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Plots of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Distributions of order parameter at various [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) The crossing times for the 3D XY model, with a given [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Average domain lengths are plotted versus time, fo [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

We study phase ordering dynamics in the three-dimensional nonconserved XY model, via Monte Carlo simulations, for quenches from paramagnetic phase to certain final temperatures $T_f$ within the ferromagnetic region of the phase diagram. The growth in the system occurs via annihilation of vortex and anti-vortex pairs, cores of which, in the three dimensional system geometry, join from different planes, on which the spins lie, to form line defects. In the long-time limit, the associated characteristic length scale, $\ell(t)$, appears to grow with time $(t)$ approximately as $t^{0.15}$, for $T_f=0$. The exponent is much smaller, like in the zero temperature intermediate time ordering in the three dimensional Ising model, than $1/2$, the expected value, that is realized for quenches to $T_f$ value that is sufficiently larger than zero. We carry out quenches from different starting temperatures, $T_s$, that lie above the critical temperature $T_c$. It is observed that the systems with higher $T_s$ approach the final equilibrium faster. This resembles the puzzling Mpemba effect. We present similar results also from the simulations of the two- and three- dimensional Ising model. In the case of the 2D Ising model, we show that the Mpemba effect is observed only if the starting magnetization is restricted to a value close to zero. In $d=3$, on the other hand, for both the models, the effect appears even if the initial configurations at a given $T_s$ are chosen from the full distribution of magnetization. Thus, our results are of much experimental relevance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript reports Monte Carlo simulations of phase-ordering kinetics in the 3D non-conserved XY model after quenches from the paramagnetic phase to Tf=0 and other ferromagnetic Tf values. It identifies vortex-core line defects whose pairwise annihilation is said to produce a characteristic length ℓ(t) that grows only as t^{0.15} at Tf=0 (far below the expected t^{1/2} law), while quenches from higher starting temperatures Ts are observed to reach equilibrium faster, resembling the Mpemba effect. Analogous results are presented for the 2D and 3D Ising models, with the Mpemba signature in 2D requiring initial magnetization near zero.

Significance. If the slow-growth exponent and Mpemba observations survive finite-size controls, the work would supply concrete numerical evidence for anomalous coarsening in three-dimensional O(n) models and for the Mpemba effect in purely relaxational dynamics, both of which remain poorly understood and are potentially accessible in magnetic or liquid-crystal experiments.

major comments (3)
  1. [Results for the 3D XY model] The central claim that line-defect annihilation produces the reported ℓ(t)∼t^{0.15} law at Tf=0 (abstract and results section) rests on an identification that is not accompanied by any finite-size scaling collapse or explicit variation of linear size L at fixed late times. Without such checks it is impossible to rule out that the measured exponent is an intermediate-time transient dominated by periodic-boundary or discreteness effects once defect separation approaches L.
  2. [Mpemba-effect analysis] The Mpemba-effect statement that “systems with higher Ts approach the final equilibrium faster” (abstract and corresponding figures) is not supported by a quantitative metric (e.g., time to reach a fixed value of the two-point correlation or energy) nor by error bars on the crossing times; the visual comparison alone is insufficient to establish the effect as load-bearing for the paper’s conclusions.
  3. [Comparison with Ising models] The assertion that the same line-defect mechanism explains the slow growth in both the XY and Ising cases is not backed by a direct comparison of defect statistics or by a controlled test that the chosen definition of ℓ(t) (first zero of the correlation function or peak of the structure factor) remains free of lattice artifacts across the two models.
minor comments (3)
  1. [Methods] The notation ℓ(t) is introduced without an explicit equation defining how it is extracted from the correlation function or structure factor.
  2. [Abstract] A few sentences in the abstract contain minor grammatical issues (“like in the zero temperature intermediate time ordering…”) that should be polished for clarity.
  3. [Discussion] The experimental-relevance paragraph would benefit from one or two concrete observables (e.g., neutron-scattering peak width or magnetization relaxation time) that could be measured in a real system.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to strengthen the claims with additional analyses.

read point-by-point responses

  1. Referee: [Results for the 3D XY model] The central claim that line-defect annihilation produces the reported ℓ(t)∼t^{0.15} law at Tf=0 (abstract and results section) rests on an identification that is not accompanied by any finite-size scaling collapse or explicit variation of linear size L at fixed late times. Without such checks it is impossible to rule out that the measured exponent is an intermediate-time transient dominated by periodic-boundary or discreteness effects once defect separation approaches L.

    Authors: We agree that explicit finite-size checks are necessary to confirm the robustness of the t^{0.15} growth law. In the revised manuscript we will present data for several linear sizes L, demonstrate that the exponent remains stable at late times when defect separation is still much smaller than L, and include a scaling collapse of ℓ(t)/L versus t/L^z to rule out boundary or discreteness transients. revision: yes

  2. Referee: [Mpemba-effect analysis] The Mpemba-effect statement that “systems with higher Ts approach the final equilibrium faster” (abstract and corresponding figures) is not supported by a quantitative metric (e.g., time to reach a fixed value of the two-point correlation or energy) nor by error bars on the crossing times; the visual comparison alone is insufficient to establish the effect as load-bearing for the paper’s conclusions.

    Authors: We accept that a quantitative metric with error bars would make the Mpemba observation more rigorous. The revised version will define a concrete threshold (e.g., energy or correlation function reaching 95 % of its equilibrium value), report the corresponding times with standard errors from independent runs, and show explicit crossing plots to quantify the effect. revision: yes

  3. Referee: [Comparison with Ising models] The assertion that the same line-defect mechanism explains the slow growth in both the XY and Ising cases is not backed by a direct comparison of defect statistics or by a controlled test that the chosen definition of ℓ(t) (first zero of the correlation function or peak of the structure factor) remains free of lattice artifacts across the two models.

    Authors: To address this, we will add a direct comparison of vortex-line (XY) and domain-wall (Ising) densities and annihilation rates. We will also verify that the two alternative definitions of ℓ(t) yield consistent growth exponents in both models and remain free of obvious lattice artifacts by repeating the analysis on larger lattices and with different discretizations. revision: yes

Circularity Check

0 steps flagged

No circularity: growth exponent and Mpemba observations are direct simulation outputs

full rationale

The paper reports Monte Carlo simulation results for phase ordering in XY and Ising models. The characteristic length ℓ(t) ~ t^0.15 at Tf=0 and the Mpemba effect (faster equilibration from higher Ts) are presented as numerical observations from quenches, not as predictions derived from equations or prior results. No load-bearing step reduces the reported exponent or effect to a fitted input, self-citation, or ansatz by construction. The line-defect interpretation is post-hoc and does not alter the simulation data. The work is self-contained against external benchmarks via direct simulation outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard Monte Carlo sampling of lattice spin models and the assumption that measured length scales correctly capture defect annihilation dynamics; no additional free parameters beyond the observed growth exponent are introduced.

free parameters (1)
  • growth exponent = 0.15
    The value 0.15 is extracted from simulation data for Tf=0 rather than derived from first principles.
axioms (1)
  • domain assumption Monte Carlo dynamics faithfully represent the nonconserved order-parameter evolution of the XY and Ising models
    Invoked implicitly when equating simulation trajectories to physical phase ordering.

pith-pipeline@v0.9.0 · 5622 in / 1464 out tokens · 45493 ms · 2026-05-14T17:54:39.909238+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 1 internal anchor

  1. [1]

    E. B. Mpemba and D. G. Osborne, Cool?, Phys. Educ. 4, 172 (1969)

    work page 1969

  2. [2]

    Z. Tang, W. Huang, Y. Zhang, Y. Liu, and L. Zhao, Direct obs ervation of the Mpemba effect with water: Probe the mysterious heat transfer, InfoMat 5, e12352 (2023) . 12

    work page 2023

  3. [3]

    H. C. Burridge and O. Hallstadius, Observing the Mpemba e ffect with minimal bias and the value of the Mpemba effect to scientific outreach and engagement, Proc. R. Soc. A 476, 20190829 (2020)

    work page 2020

  4. [4]

    Ghosh, P

    S. Ghosh, P. Pathak, S. Chatterjee, and S. K. Das, Simulat ions of Mpemba effect in water and Lennard-Jones models, Commun. Phys. 8, 359 (2025)

    work page 2025

  5. [5]

    G. Teza, J. Bechhoefer, A. Lasanta, O. Raz, and M. Vucelja , Speedups in nonequilibrium thermal relaxation: Mpemba and related effects, Phys. Rep. 1164, 1 (2026)

    work page 2026

  6. [6]

    Bechhoefer, A

    J. Bechhoefer, A. Kumar, and R. Ch´ etrite, A fresh unders tanding of the Mpemba effect, Nat. Rev. Phys. 3, 534 (2021)

    work page 2021

  7. [7]

    Auerbach, Supercooling and the Mpemba effect: When hot water freezes quicker than cold, Am

    D. Auerbach, Supercooling and the Mpemba effect: When hot water freezes quicker than cold, Am. J. Phys. 63, 882 (1995)

    work page 1995

  8. [8]

    S. K. Das, Perspectives on a Few Puzzles in Phase Transfor mations: When Should the Farthest Reach the Earliest?, Langmuir 39, 10715 (2023)

    work page 2023

  9. [9]

    Jeng, The Mpemba effect: When can hot water freeze faste r than cold?, Am

    M. Jeng, The Mpemba effect: When can hot water freeze faste r than cold?, Am. J. Phys. 74, 514 (2006)

    work page 2006

  10. [10]

    Lu and O

    Z. Lu and O. Raz, Nonequilibrium thermodynamics of the M arkovian Mpemba effect and its inverse, Proc. Natl. Acad. Sci. U.S.A. 114, 5083 (2017)

    work page 2017

  11. [11]

    Klich, O

    I. Klich, O. Raz, O. Hirschberg, and M. Vucelja, Mpemba I ndex and Anomalous Relaxation, Phys. Rev. X 9, 021060 (2019)

    work page 2019

  12. [12]

    Kumar and J

    A. Kumar and J. Bechhoefer, Exponentially faster cooli ng in a colloidal system, Nature 584, 64 (2020)

    work page 2020

  13. [13]

    Ch´ etrite, A

    R. Ch´ etrite, A. Kumar, and J. Bechhoefer, The metastab le Mpemba effect corresponds to a non-monotonic temperature dependence of extractable work, Front. Phys. 9, 654271 (2021)

    work page 2021

  14. [14]

    F. J. Schwarzendahl and H. L¨ owen, Anomalous cooling an d overcooling of active colloids, Phys. Rev. Lett. 129, 138002 (2022)

    work page 2022

  15. [15]

    Vadakkayil and S

    N. Vadakkayil and S. K. Das, Should a hotter paramagnet t ransform quicker to a ferromagnet? Monte Carlo simulation results for Ising model, Phys. Chem. Chem. Phys. 23, 11186 (2021)

    work page 2021

  16. [16]

    Chatterjee, S

    S. Chatterjee, S. Ghosh, N. Vadakkayil, T. Paul, S. K. Si ngha, and S. K. Das, Mpemba effect in pure spin systems : A universal picture of the role of spatial correlations at ini tial states, Phys. Rev. E 110, L012103 (2024)

    work page 2024

  17. [17]

    Chatterjee, S

    S. Chatterjee, S. Das, P. Pathak, T. Paul, and S. K. Das, Q uicker flocking in aligning active matters for noisier begin ning, arXiv: 2502.06482 (2025)

    work page arXiv 2025

  18. [18]

    Lasanta, F

    A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos, When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids, Phys. Rev. Lett. 119, 148001 (2017)

    work page 2017

  19. [19]

    Biswas, V

    A. Biswas, V. V. Prasad, and R. Rajesh, Mpemba effect in dr iven granular gases: Role of distance measures, Phys. Rev. E 108, 024902 (2023)

    work page 2023

  20. [20]

    Overtaking while approaching equilibrium

    P. Chaddah, S. Dash, K. Kumar, and A. Banerjee, Overtaki ng while approaching equilibrium, arXiv: 1011.3598 (2010)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  21. [21]

    P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, Mpe mba-Like Behavior in Carbon Nanotube Resonators, Metall. Mater. Trans. A 42, 3907 (2011) . 13

    work page 2011

  22. [22]

    Y.-H. Ahn, H. Kang, D.-Y. Koh, and H. Lee, Experimental v erifications of Mpemba-like behaviors of clathrate hydrate s, Korean J. Chem. Eng. 33, 1903 (2016)

    work page 1903

  23. [23]

    Van Vu and H

    T. Van Vu and H. Hayakawa, Thermomajorization Mpemba Eff ect, Phys. Rev. Lett. 134, 107101 (2025)

    work page 2025

  24. [24]

    Y. Tian, Y. Zheng, L.-H. Liu, L. Wang, G.-C. Guo, and F.-W . Sun, Experimental study of Mpemba effect in an energy Langevin system, Phys. Rev. Res. 7, L042020 (2025)

    work page 2025

  25. [25]

    L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. K ranzl, R. Blatt, P. Zoller, B. Vermersch, P. Calabrese, C. F. Roos, and M. K. Joshi, Observing the Quantum Mpemba Effect in Quantu m Simulations, Phys. Rev. Lett. 133, 010402 (2024)

    work page 2024

  26. [26]

    A. K. Chatterjee, S. Takada, and H. Hayakawa, Quantum Mp emba Effect in a Quantum Dot with Reservoirs, Phys. Rev. Lett. 131, 080402 (2023)

    work page 2023

  27. [27]

    Rylands, K

    C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murcian o, and B. Bertini, Microscopic Origin of the Quantum Mpemba Effect in Integrable Systems, Phys. Rev. Lett. 133, 010401 (2024)

    work page 2024

  28. [28]

    Zhang, G

    J. Zhang, G. Xia, C.-W. Wu, T. Chen, Q. Zhang, Y. Xie, W.-B . Su, W. Wu, C.-W. Qiu, P.-X. Chen, W. Li, H. Jing, and Y.-L. Zhou, Observation of quantum strong Mpemba effect, Nat. Commun. 16, 301 (2025)

    work page 2025

  29. [29]

    F. Ares, P. Calabrese, and S. Murciano, The quantum Mpem ba effects, Nat. Rev. Phys. 7, 451 (2025)

    work page 2025

  30. [30]

    Moroder, O

    M. Moroder, O. Culhane, K. Zawadzki, and J. Goold, Therm odynamics of the Quantum Mpemba Effect, Phys. Rev. Lett. 133, 140404 (2024)

    work page 2024

  31. [31]

    D. J. Strachan, A. Purkayastha, and S. R. Clark, Non-Mar kovian Quantum Mpemba Effect, Phys. Rev. Lett. 134, 220403 (2025)

    work page 2025

  32. [32]

    Murciano, F

    S. Murciano, F. Ares, I. Klich, and P. Calabrese, Entang lement asymmetry and quantum Mpemba effect in the XY spin chain, J. Stat. Mech.: Theory Exp. 2024 (1), 013103

    work page 2024

  33. [33]

    Baity-Jesi et

    M. Baity-Jesi et. al., The Mpemba effect in spin glasses i s a persistent memory effect, Proc. Natl. Acad. Sci. U.S.A. 116, 15350 (2019)

    work page 2019

  34. [34]

    Jin and W

    J. Jin and W. A. I. Goddard, Mechanisms Underlying the Mp emba Effect in Water from Molecular Dynamics Simulations, J. Phys. Chem. C 119, 2622 (2015)

    work page 2015

  35. [35]

    Gij´ on, A

    A. Gij´ on, A. Lasanta, and E. R. Hern´ andez, Paths towar ds equilibrium in molecular systems: The case of water, Phys. Rev. E 100, 032103 (2019)

    work page 2019

  36. [36]

    Biswas, R

    A. Biswas, R. Rajesh, and A. Pal, Mpemba effect in a Langev in system: Population statistics, metastability, and othe r exact results, J. Chem. Phys. 159, 044120 (2023)

    work page 2023

  37. [37]

    D. P. Landau and K. Binder, A guide to Monte Carlo simulations in statistical physics (Cambridge university press, 2021)

    work page 2021

  38. [38]

    M. E. Newman and G. T. Barkema, Monte Carlo methods in statistical physics (Clarendon Press, 1999)

    work page 1999

  39. [39]

    Binder and D

    K. Binder and D. W. Heermann, Monte Carlo simulation in statistical physics , Vol. 8 (Springer, 1992)

    work page 1992

  40. [40]

    M. E. Fisher, The theory of equilibrium critical phenom ena, Rep. Prog. Phys. 30, 615 (1967) . 14

    work page 1967

  41. [41]

    Agrawal, M

    R. Agrawal, M. Kumar, and S. Puri, Domain growth and agin g in the random field XY model: A Monte Carlo study, Phys. Rev. E 104, 044123 (2021)

    work page 2021

  42. [42]

    Hasenbusch and S

    M. Hasenbusch and S. Meyer, Critical exponents of the 3D XY model from cluster update Monte Carlo, Phys. Lett. B 241, 238 (1990)

    work page 1990

  43. [43]

    A. P. Gottlob and M. Hasenbusch, Critical behavior of th e 3D XY model: A Monte Carlo study, Physica A 201, 593 (1993)

    work page 1993

  44. [44]

    Campostrini, M

    M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi , and E. Vicari, Critical behavior of the three-dimensional XY universality class, Phys. Rev. B 63, 214503 (2001)

    work page 2001

  45. [45]

    A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51, 481 (2002)

    work page 2002

  46. [46]

    Puri and V

    S. Puri and V. Wadhawan, eds., Kinetics of Phase Transitions (CRC Press, Boca Raton, 2009)

    work page 2009

  47. [47]

    Kohring, R

    G. Kohring, R. E. Shrock, and P. Wills, Role of Vortex Str ings in the Three-Dimensional O(2) Model, Phys. Rev. Lett. 57, 1358 (1986)

    work page 1986

  48. [48]

    P. M. Chaikin, T. C. Lubensky, and T. A. Witten, Principles of condensed matter physics , Vol. 10 (Cambridge university press Cambridge, 1995)

    work page 1995

  49. [49]

    R. Ren, C. J. O’Keeffe, and G. Orkoulas, Sequential updat ing algorithms for grand canonical monte carlo simulations , Mol. Phys. 105, 231 (2007)

    work page 2007

  50. [50]

    Kumar, S

    M. Kumar, S. Chatterjee, R. Paul, and S. Puri, Ordering k inetics in the random-bond XY model, Phys. Rev. E 96, 042127 (2017)

    work page 2017

  51. [51]

    R. J. Glauber, Time-Dependent Statistics of the Ising M odel, J. Math. Phys. 4, 294 (1963)

    work page 1963

  52. [52]

    A. J. Bray and S. Puri, Asymptotic structure factor and p ower-law tails for phase ordering in systems with continuou s symmetry, Phys. Rev. Lett. 67, 2670 (1991)

    work page 1991

  53. [53]

    Toyoki, Structure factors of vector-order-paramet er systems containing random topological defects, Phys

    H. Toyoki, Structure factors of vector-order-paramet er systems containing random topological defects, Phys. Rev. B 45, 1965 (1992)

    work page 1965

  54. [54]

    Majumder and S

    S. Majumder and S. K. Das, Diffusive domain coarsening: E arly time dynamics and finite-size effects, Phys. Rev. E 84, 021110 (2011)

    work page 2011

  55. [55]

    Vadakkayil, S

    N. Vadakkayil, S. K. Singha, and S. K. Das, Influence of ro ughening transition on magnetic ordering, Phys. Rev. E 105, 044142 (2022)

    work page 2022

  56. [56]

    S. K. Das, Unlocking of frozen dynamics in the complex Gi nzburg-Landau equation, Phys. Rev. E 87, 012135 (2013)

    work page 2013

  57. [57]

    Mondello and N

    M. Mondello and N. Goldenfeld, Scaling and vortex dynam ics after the quench of a system with a continuous symmetry, Phys. Rev. A 42, 5865 (1990)

    work page 1990

  58. [58]

    Mondello and N

    M. Mondello and N. Goldenfeld, Scaling and vortex-stri ng dynamics in a three-dimensional system with a continuous symmetry, Phys. Rev. A 45, 657 (1992) . 15

    work page 1992

  59. [59]

    U. C. T¨ auber, Phase transitions and scaling in systems far from equilibrium, Annu. Rev. Condens. Matter Phys. 8, 185 (2017)

    work page 2017

  60. [60]

    A. J. Bray and A. D. Rutenberg, Growth laws for phase orde ring, Phys. Rev. E 49, R27 (1994)

    work page 1994

  61. [61]

    R. E. Blundell and A. J. Bray, Phase-ordering dynamics o f the O(n) model: Exact predictions and numerical results, Phys. Rev. E 49, 4925 (1994)

    work page 1994

  62. [62]

    D. A. Huse, Corrections to late-stage behavior in spino dal decomposition: Lifshitz-Slyozov scaling and Monte Car lo simulations, Phys. Rev. B 34, 7845 (1986)

    work page 1986

  63. [63]

    Olejarz, P

    J. Olejarz, P. L. Krapivsky, and S. Redner, Zero-temper ature relaxation of three-dimensional Ising ferromagnets , Phys. Rev. E 83, 051104 (2011)

    work page 2011

  64. [64]

    Vadakkayil, S

    N. Vadakkayil, S. Chakraborty, and S. K. Das, Finite-si ze scaling study of aging during coarsening in non-conserve d Ising model: The case of zero temperature quench, J. Chem. Phys. 150, 054702 (2019)

    work page 2019

  65. [65]

    Gessert, H

    D. Gessert, H. Christiansen, and W. Janke, Aging follow ing a zero-temperature quench in the d = 3 Ising model, Phys. Rev. E 109, 044148 (2024)

    work page 2024

  66. [66]

    Chakraborty and S

    S. Chakraborty and S. K. Das, Coarsening in 3D nonconser ved Ising model at zero temperature: Anomaly in structure and slow relaxation of order-parameter autocorrelation, Europhys. Lett. 119, 50005 (2017)

    work page 2017

  67. [67]

    Spirin, P

    V. Spirin, P. Krapivsky, and S. Redner, Freezing in Isin g ferromagnets, Phys. Rev. E 65, 016119 (2001)

    work page 2001

  68. [68]

    S. K. Das and S. Chakraborty, Kinetics of ferromagnetic ordering in 3D Ising model: How far do we understand the case of a zero temperature quench?, Eur. Phys. J. Spec. Top. 226, 765 (2017)

    work page 2017

  69. [69]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic cri tical phenomena, Rev. Mod. Phys. 49, 435 (1977)

    work page 1977

  70. [70]

    H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1987)

    work page 1987

  71. [71]

    Onuki, Phase Transition Dynamics (Cambridge University Press, 2002)

    A. Onuki, Phase Transition Dynamics (Cambridge University Press, 2002)

    work page 2002

  72. [72]

    M. E. Fisher, Correlation Functions and the Critical Re gion of Simple Fluids, J. Math. Phys. 5, 944 (1964)

    work page 1964

  73. [73]

    L. P. Kadanoff, Scaling laws for Ising models near Tc, Phys. Phys. Fiz. 2, 263 (1966)

    work page 1966

  74. [74]

    Zinn-Justin, Precise determination of critical exp onents and equation of state by field theory methods, Phys

    J. Zinn-Justin, Precise determination of critical exp onents and equation of state by field theory methods, Phys. Rep. 344, 159 (2001)

    work page 2001

  75. [75]

    Efron, The jackknife, the bootstrap and other resampling plans (SIAM, 1982)

    B. Efron, The jackknife, the bootstrap and other resampling plans (SIAM, 1982)

    work page 1982

  76. [76]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metastab ility and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973)

    work page 1973

  77. [77]

    J. M. Kosterlitz, The critical properties of the two-di mensional XY model, J. Phys. C 7, 1046 (1974)

    work page 1974