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REVIEW 4 major objections 6 minor 61 references

Kibble-Zurek Dynamics & Statistics of Topological Defects in Chiral Superfluid $^3$He Films

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Pith's one-line read In a two-dimensional chiral superfluid 3He film, quench-generated vortices obey Kibble-Zurek scaling, with freeze-out exponent 0.436 ± 0.003.

desk verdict First KZ quench simulations for a chiral p-wave film with domain walls; solid work, but the reported exponent rests on a single freeze-out criterion and the asymmetry needs error bars. read the letter →

arxiv 2412.03544 v2 pith:JVF5LNMG submitted 2024-12-04 cond-mat.supr-con

Noble Gluscevich , J. A. Sauls This is my paper · ORCID
classification cond-mat.supr-con
keywords Kibble-Zurekmechanismchiralsuperfluid3He-Atime-dependentGinzburg-Landautheorytopologicaldefectsvortexcountingstatisticsdomainwallstemperaturequenchcorestructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the Kibble-Zurek mechanism, the prediction that defect density after a quench is set by the crossing rate of a phase transition, survives when the ordered phase is a chiral p-wave superfluid with an additional discrete degeneracy between two time-reversed states. Using a time-dependent Ginzburg-Landau theory for strong-coupling 3He, with Langevin noise and damping, the authors simulate temperature quenches in a thin film and find that the freeze-out time and the mean number of phase vortices follow the Kibble-Zurek power laws, with freeze-out exponent $\delta=0.436\pm0.003$. The variance and skewness of the vortex-number distribution also scale with quench rate, but their exponents differ measurably from one another, which is only qualified support for independent-defect models of full counting statistics. The simulations also show that after freeze-out the two inequivalent vortex core structures, triangular and crescent, are produced in unequal numbers, an asymmetry attributed to vortex interactions with domain walls. If the results hold, quenched $^3$He films become a test bed for Kibble-Zurek statistics in a system whose defects carry topologically nontrivial internal structure.

What carries the argument

The argument is carried by the time-dependent Ginzburg-Landau (TDGL) field theory for the spin-triplet p-wave order parameter of 3He, reduced to a two-component complex vector in the film plane, with strong-coupling coefficients, Langevin noise, and dissipative damping (Eqs. (20)--(22)). Quenches from $1.5T_c$ to $0.5T_c$ are integrated on a periodic grid; the freeze-out time is defined operationally as the time of maximum growth rate of the spatially averaged order parameter, and vortex numbers are counted from $2\pi$ phase windings around plaquettes. The defect taxonomy uses the pair of winding numbers $(p,m)$ of the dominant and time-reversed chiral amplitudes, constrained by $p+l=m-l$ with $l$ the Chern number of the chiral ground state, which yields the triangular and crescent core structures. This machinery lets the authors separate Kibble-Zurek scaling of vortex number from the effects of the extra $Z_2$ domain-wall sector and test whether the statistics of defect formation remain those of independent events.

What would settle it

A concrete calculation that would settle the model-dependence of the central claim is to repeat the same quench protocol with the chiral $Z_2$ degeneracy removed, so the order parameter is a single complex U(1) field in two dimensions. If the three cumulant exponents then coincide, the measured differences are caused by the extra chiral and domain-wall sector; if they remain different, the deviations originate elsewhere. An experimental counterpart would be to measure the vortex-count distribution of a quenched $^3$He-A film as a function of cooling rate and compare the fitted exponents with $0.436\pm0.003$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is that the Kibble-Zurek mechanism remains intact when the symmetry-broken ground state is a doubly degenerate chiral condensate rather than a simple U(1) scalar field. Over about two decades of quench rate, the freeze-out time obeys $\hat{t}\sim \tau_Q^{\delta}$ with $\delta=0.436\pm0.003$, below the mean-field value $\delta=1/2$. The mean, variance, and skewness of the number of $2\pi$ phase vortices each scale as powers of $1/\tau_Q$; the fitted exponents are close to one another but not equal, which the paper reports as small but measurable deviations from the prediction that all cumulants share a single Kibble-Zurek exponent. In the late-stage dynamics, two types of singly quantized vortices form in a given chiral domain, triangular cores with winding numbers $(p,m)=(+1,+3)$ and crescent cores with $(-1,+1)$, and their post-freeze-out populations are asymmetric. The paper argues this asymmetry emerges from the interactions of vortices with domain walls between time-reversed chiral domains and is absent in scalar U(1) theories.

Load-bearing premise

The load-bearing premise is that the time-dependent Ginzburg-Landau equations, with coefficients computed near the transition temperature, continue to describe the order parameter faithfully all the way down to half that temperature, where vortex numbers and core structures are counted.

Editorial extensions

If this is right

  • If correct, thin films of superfluid 3He-A provide a quantitative test bed for the Kibble-Zurek mechanism in a chiral condensate, where the predicted freeze-out exponent is 0.436 and higher cumulants scale with quench rate.
  • The measured power-law scaling of variance and skewness means that quench experiments should see the full distribution of vortex counts narrow and skew in a predictable way as the cooling rate changes.
  • Because vortices and anti-vortices in a given chiral domain are structurally inequivalent, any Kibble-Zurek prediction for a chiral superfluid must specify which core type is being counted; the triangular-versus-crescent asymmetry is a concrete prediction.
  • The differences among the cumulant exponents imply that independent-defect binomial models are only approximations for this system, so vortex-domain-wall interactions belong in the theory of full counting statistics for chiral condensates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not run a control simulation with the chiral $Z_2$ degeneracy switched off; such a U(1)-only quench in two dimensions would show whether the small differences among $\beta_1,\beta_2,\beta_3$ are caused by the domain-wall sector or by something else.
  • The triangular-versus-crescent population asymmetry could serve as a dynamical probe of vortex-domain-wall interactions in future experiments, since equilibrium probes such as chiral edge currents do not encode quench history.
  • If the freeze-out exponent near 0.436 is robust across settings, it hints that the Kibble-Zurek exponent is insensitive to the internal degeneracy structure of the order parameter, while the distributional statistics are not; this extrapolation goes beyond what the paper itself claims.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript reports numerical simulations of temperature quenches in thin films of superfluid 3He-A described by a time-dependent Ginzburg-Landau (TDGL) theory with strong-coupling coefficients and Langevin noise. The authors identify the freeze-out time from the maximum slope of the spatially averaged order parameter and extract Kibble-Zurek exponents for the freeze-out time and for the first three cumulants of the vortex number distribution. They also analyze late-time vortex core structures and report a triangular/crescent vortex asymmetry. The central claimed results are (i) KZ scaling with delta_KZM = 0.436 +/- 0.003, (ii) cumulant exponents beta_1, beta_2, beta_3 that deviate from the mean-field value 1/2 and from each other by small but measurable amounts, and (iii) a pronounced excess of triangular vortices over crescent vortices.

Significance. If confirmed, these results would provide the first strong-coupling TDGL calculation of KZ dynamics in a chiral superfluid film, including the role of domain walls, and would extend full counting statistics tests to a two-dimensional multicomponent order parameter. The large number of noise realizations (20,000 for most quench rates) and the use of microscopically computed strong-coupling coefficients are clear strengths. However, the headline exponents depend on an untested freeze-out criterion and on fitting choices that are not documented in sufficient detail; the central quantitative claims therefore require additional robustness analysis before the paper can serve as a definitive test of KZ scaling in this system.

major comments (4)
  1. [III, Eq. (24), Fig. 1] The freeze-out time is defined exclusively as the time at which the spatially averaged order parameter has maximum slope, yet the paper itself cites Ref. 29 as finding KZ exponents that varied between 0.43 and 0.57 depending on the numerical criterion for identifying freeze-out. No alternative criterion is tested, and the vortex counts in Fig. 2 are all evaluated at this same freeze-out time, so the exponents beta_1, beta_2, beta_3 are not independent of this definition. The quoted uncertainty delta_KZM = 0.436 +/- 0.003 reflects only the fit to this one definition. Please report the dependence of delta_KZM and the beta_i on at least two alternative freeze-out criteria (e.g., first appearance of net phase winding, or onset of phase coherence across the domain, as in Ref. 29), and document which data points were excluded as 'saturated' and the exact fit range.
  2. [III, Fig. 2] The claim of 'small, albeit measurable' differences among beta_1, beta_2, and beta_3 is not supported by any displayed uncertainty. The text mentions that delta beta_3/beta_3 ~ 10% for N = 20,000, but the figure has no error bars or confidence intervals, and no statistical test is reported. Without these, the reader cannot judge whether the deviations are significant or within the fit noise. Add error bars (e.g., bootstrap or covariance) and a test of the null hypothesis beta_1 = beta_2 = beta_3, and also specify the fit range for each cumulant, since the scaling range in Fig. 2 appears shorter than that in Fig. 1.
  3. [III.3, Fig. 6] The triangular/crescent asymmetry is based on only 100 quench realizations and is presented as spline fits without error bars or a significance test. Since one of the suggested explanations is explicitly labeled a conjecture needing further study, the claim of a 'significant asymmetry' must be quantified. Provide standard errors or confidence bands for the vortex populations and, if possible, a test of sensitivity to the vortex-detection thresholds used in the DoG and phase-winding procedure.
  4. [II.3 and App. A] The dynamical equations use a constant damping gamma appropriate to the gapless regime near Tc, and strong-coupling beta_p coefficients whose temperature scaling was fitted near the transition. However, the freeze-out and counting occur at or after the quench has reached T = 0.5 Tc (App. A: freeze-out near t = 60 while the final temperature is reached at t = 50). At T = 0.5 Tc the quasiparticle damping and strong-coupling corrections may differ substantially from the near-Tc values used in Eqs. (20)-(21). This is a load-bearing approximation for the final vortex populations and asymmetry. Please either justify the extrapolation to T = 0.5 Tc or test one case with modified damping and strong-coupling coefficients to show that the exponents and asymmetry are insensitive to this approximation.
minor comments (6)
  1. [II.1, Eq. (22)] The combination K23 is used in Eq. (22) but defined only in the following sentence; define K23 = K2 + K3 at first use.
  2. [II.2, Eq. (26)] The index structure of the mass current expression is unclear; please clarify the summation convention and verify the expression is gauge invariant.
  3. [App. A] The video animation is an important part of the presentation, but a clickable link in a manuscript may not be persistent; please state that the animation is available as supplementary material.
  4. [Abstract and Introduction] There are several typographical errors, including 'paramter' and 'transiton'; the manuscript needs a careful proofreading pass.
  5. [Fig. 2 caption] The phrase 'the quench rate (time) is in units of 1/tau_GL (tau_GL)' is confusing; specify clearly the units of the horizontal and vertical axes.
  6. [Table I] The first two columns would be clearer if the headers explicitly said 'A+ amplitude' and 'A- amplitude', with row entries indicating whether that component is dominant or subdominant, to avoid ambiguity with the vortex labels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the KZ exponents and vortex counting statistics are emergent outputs of the TDGL simulations, and the self-citations provide model infrastructure rather than the predicted scaling results.

full rationale

The paper's central claims are the measured KZ scaling of the freeze-out time and vortex cumulants (Figs. 1-2) and the triangular/crescent vortex asymmetry (Fig. 6). These are outputs of explicit numerical solution of the TDGL equations (Eqs. (20)-(22)) with Langevin noise, not inputs. The freeze-out time is defined operationally as the maximum of ∂_t ⟨Δ⟩, and the exponent δ_KZM = 0.436 ± 0.003 is obtained by fitting the simulated t̂ versus τQ; the scaling law is not assumed in the dynamics. Likewise, the vortex-number distribution is built from histograms of 20,000 noise realizations, and no binomial or Poisson-binomial distribution is imposed on the simulation; the comparison with independent-defect statistics in Sec. I.3 is a post-hoc null model, not a fitted input. The cited prior work by the authors (strong-coupling GL coefficients from Refs. 8 and 48, the TDGL framework in Ref. 7, and the vortex winding constraint p+1=m−1 from Ref. 57) supplies the physical model and vortex classification, but does not contain or force the KZ exponents or the measured asymmetry; those emerge from the dynamics. The sensitivity of the freeze-out criterion to its operational definition is a robustness concern that the paper itself flags by citing Antunes et al. (Ref. 29) and by removing saturating fast-quench points, but it is not a circularity because the criterion is not defined in terms of the defect densities it is used to predict. No equation in the manuscript reduces a predicted quantity to a fitted parameter or to a self-citation by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data in this paper. The simulation uses material coefficients (alpha, beta_p, mu, K_i, gamma) inherited from microscopic strong-coupling calculations, and the quench protocol, grid spacing, and noise realizations are stated numerical choices rather than fitted inputs. No new particles, forces, dimensions, or conserved quantities are introduced; the vortex core structures are known from prior work, and the domain-wall vortices are composites of existing defects.

assumptions (4)
  • domain assumption Film confinement with specular surfaces and thickness D <= 9 xi0 stabilizes the chiral A phase and suppresses out-of-plane orbital components, reducing the order parameter to the two-dimensional complex vector A(r,t) = (A_x, A_y) with dipole locking d || z.
    Sec. II.2 and Eqs. (18)-(19); justified by Refs. 33, 53, 54. If surfaces are not specular or dipolar locking is broken, the defect dynamics and even the ground state manifold change.
  • domain assumption The same Langevin TDGL dynamics (Eqs. 20-22) with constant gamma and mean-field GL coefficients remains quantitatively valid through the quench down to the final temperature T = 0.5Tc.
    The GL expansion and gapless-regime damping gamma are derived near Tc; the quench protocol in App. A and all quantitative results extend them to 0.5Tc without renormalization or suppression of additional modes.
  • domain assumption The freeze-out time is operationalized as the maximum slope of the spatially averaged order parameter amplitude, and this definition yields the KZ time scale.
    Sec. III and Fig. 1; Ref. 29 showed that the extracted KZ exponent can depend on the numerical criterion for freeze-out, and the paper does not test the sensitivity of delta_KZM = 0.436 to this choice.
  • domain assumption Strong-coupling beta_p coefficients from 4th-order polynomial fits to the values reported in Ref. 48 correctly describe the free-energy landscape of the film for all pressures.
    Eq. (12) and Sec. II.1; these are external inputs from prior calculations and are not independently derived here.

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Pith. "Pith review of Kibble-Zurek Dynamics & Statistics of Topological Defects in Chiral Superfluid $^3$He Films." pith.science (2026). https://pith.science/paper/JVF5LNMG

@misc{pith2026241203544,
  author       = {Pith},
  title        = {Pith review of: Kibble-Zurek Dynamics & Statistics of Topological Defects in Chiral Superfluid $^3$He Films},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JVF5LNMG}},
  note         = {Machine review of arXiv:2412.03544}
}
abstract

In equilibrium, confined films of superfluid $^3$He-A have the chiral axis, $\hat{\ell}$, locked normal to the surface of the film. There are two degenerate ground states $\hat{\ell}\;||\pm\hat{z}$. However, for a temperature quench, i.e. cool down through the phase transition at a finite rate, causally disconnected regions of order parameter fluctuations develop and evolve into an inhomogeneous ordered phase that hosts both domain walls between time-reversed chiral phases as well as vortices with winding numbers $p\in\mathbb{Z}$. We present simulations based on a time-dependent generalization of Ginzburg-Landau theory for strong-coupling $^3$He that reveal both types of topological defects to be present following the temperature quench. Results for the dynamics of vortices interacting with anti-vortices as well as domain walls are presented. The vortex number density as a function of quench rate agrees well with the scaling predicted by Kibble and Zurek. We also present results for the number distribution and compare with other theoretical models for full counting statistics of the topological defect density. Finally, we present results for an asymmetry in the post-freeze-out populations of inequivalent vortex core structures that are characteristic of a chiral superfluid.

Figures

Figures reproduced from arXiv: 2412.03544 by the authors.

Figure 1
Figure 1. FIG. 1. Left panel: Spatially averaged order parameter (Eq. (24), in units [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Counting statistics and moments of the probability dis [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Late-stage development of order order parameter for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Structure of singly-quantized vortices with global winding numbers [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Mass currents associated with vortices and domain walls [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Asymmetry in the number of singly-quantized vortex [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. An early time frame of an animation of order parameter dynamics for a thin film of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Structure of topological defects trapped by a domain wall. Left panel: the four plots are the amplitude and phase of order [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Reference graph

Works this paper leans on

61 extracted references · 31 canonical work pages

  1. [1]

    Mizushima \ and\ author K

    author author T. Mizushima \ and\ author K. Machida ,\ Multifaceted properties of Andreev bound states: interplay of symmetry and topology title Multifaceted properties of Andreev bound states: interplay of symmetry and topology ,\ 10.1098/rsta.2015.0355 journal journal Phil. Trans. Roy. Soc. A \ volume 376 ,\ pages 20150355 ( year 2018 ) NoStop

    arXiv

  2. [2]

    author author G. E. \ Volovik ,\ Topological Superfluids title Topological Superfluids ,\ 10.48550/arXiv.1602.02595 journal journal Zh. Eskp. Teor. Fiz. \ volume 156 ,\ pages 700 ( year 2019 ) NoStop

    work page Pith review arXiv doi:10.48550/arxiv.1602.02595 2019

  3. [3]

    author author Y. Nambu ,\ Fermion-Boson relations in BCS-type theories title Fermion-Boson relations in BCS-type theories ,\ 10.1016/0167-2789(85)90157-5 journal journal Physica D: Nonlinear Phenomena \ volume 15 ,\ pages 147 ( year 1985 ) NoStop

    work page doi:10.1016/0167-2789(85)90157-5

  4. [4]

    author author G. E. \ Volovik \ and\ author M. A. \ Zubkov ,\ Scalar excitation with Leggett frequency in ^3 H e- B and the 125 G e V Higgs particle in top quark condensation models as pseudo- G oldstone bosons title Scalar excitation with Leggett frequency in ^3 H e- B and the 125 G e V Higgs particle in top quark condensation models as pseudo- G oldston...

    work page doi:10.1103/physrevd.92.055004

  5. [5]

    author author J. A. \ Sauls \ and\ author T. Mizushima ,\ On the Nambu Fermion-Boson Relations for Superfluid ^3 He title On the Nambu Fermion-Boson Relations for Superfluid ^3 He ,\ 10.1103/PhysRevB.95.094515 journal journal Phys. Rev. B \ volume 95 ,\ pages 094515 ( year 2017 ) NoStop

    doi:10.1103/physrevb.95.094515

  6. [6]

    author author M. B. \ Hindmarsh \ and\ author T. W. B. \ Kibble ,\ Cosmic Strings title Cosmic Strings ,\ 10.1088/0034-4885/58/5/001 journal journal Rep. Prog. Phys. \ volume 58 ,\ pages 477 ( year 1995 ) NoStop

    doi:10.1088/0034-4885/58/5/001 1995

  7. [7]

    Hindmarsh , author J

    author author M. Hindmarsh , author J. A. \ Sauls , author K. Zhang , \ and\ author et al. ,\ AB Transition in Superfluid and Cosmological Phase Transitions title AB Transition in Superfluid and Cosmological Phase Transitions ,\ 10.1007/s10909-024-03151-9 journal journal J. Low Temp. Phys. \ volume 215 ,\ pages 495 ( year 2024 ) NoStop

    doi:10.1007/s10909-024-03151-9

  8. [8]

    author author J. J. \ Wiman \ and\ author J. A. \ Sauls ,\ Strong-Coupling and the Stripe Phase of ^3 He title Strong-Coupling and the Stripe Phase of ^3 He ,\ 10.1007/s10909-016-1632-7 journal journal J. Low Temp. Phys. \ volume 184 ,\ pages 1054 ( year 2016 ) NoStop

    doi:10.1007/s10909-016-1632-7

Show all 61 references
  1. [9]

    Mizushima \ and\ author J

    author author T. Mizushima \ and\ author J. A. \ Sauls ,\ Bosonic Surface States and Acoustic Spectroscopy of Confined Superfluid ^3 He-B title Bosonic Surface States and Acoustic Spectroscopy of Confined Superfluid ^3 He-B ,\ https://arxiv.org/abs/1801.02277 journal journal a...

    2018 arXiv

  2. [10]

    author author R. C. \ Regan , author J. J. \ Wiman , \ and\ author J. A. \ Sauls ,\ The Vortex Phase Diagram of Rotating Superfluid ^3 He-B title The Vortex Phase Diagram of Rotating Superfluid ^3 He-B ,\ 10.1103/PhysRevB.101.024517 journal journal Phys. Rev. B \ volume 101 ,\...

    doi

  3. [11]

    author author T. W. B. \ Kibble ,\ Topology of Cosmic Domains and Strings title Topology of Cosmic Domains and Strings ,\ 10.1088/0305-4470/9/8/029 journal journal Journal of Physics A: Mathematical and General \ volume 9 ,\ pages 1387 ( year 1976 ) NoStop

    doi

  4. [12]

    author author T. W. B. \ Kibble ,\ Some Implications of a Cosmological Phase Transition title Some Implications of a Cosmological Phase Transition ,\ 10.1016/0370-1573(80)90091-5 journal journal Physics Reports \ volume 67 ,\ pages 183 ( year 1980 ) NoStop

    doi

  5. [13]

    author author W. H. \ Zurek ,\ Cosmological Experiments in Superfluid Helium? title Cosmological Experiments in Superfluid Helium? \ 10.1038/382296a0 journal journal Nature \ volume 317 ,\ pages 505 ( year 1985 ) NoStop

    1985 doi

  6. [14]

    author author W. H. \ Zurek ,\ Cosmic Strings in Laboratory Superfluids and the Topological Remnants of other Phase Transitions title Cosmic Strings in Laboratory Superfluids and the Topological Remnants of other Phase Transitions ,\ https://www.actaphys.uj.edu.pl/R/24/7/1301/...

    1993

  7. [15]

    Bäuerle , author Y

    author author C. Bäuerle , author Y. M. \ Bunkov , author S. N. \ Fisher , author H. Godfrin , \ and\ author G. R. \ Pickett ,\ Laboratory Simulation of Cosmic String Formation in the Early Universe Using Superfluid ^3 He title Laboratory Simulation of Cosmic String Formation ...

    doi

  8. [16]

    author author V. M. H. \ Ruutu , author V. B. \ Eltsov , author A. J. \ Gill , author T. W. B. \ Kibble , author M. Krusius , author Y. G. \ Makhlin , author B. Plaçais , author G. E. \ Volovik , \ and\ author W. Xu ,\ Vortex Formation in Neutron-Irradiated Superfluid ^3 He as...

    doi

  9. [17]

    author author V. M. \ Ruutu , author V. B. \ Eltsov , author M. Krusius , author Y. G. \ Makhlin , author B. Plaçais , \ and\ author G. E. \ Volovik ,\ Defect Formation in Quench-Cooled Superfluid Phase Transition title Defect Formation in Quench-Cooled Superfluid Phase Transi...

    doi

  10. [18]

    author author V. B. \ Eltsov \ and\ author M. Krusius ,\ title Broken Symmetry and Non-Equilibrium Superfluid ^3 He ,\ in\ @noop booktitle Topological Defects and Non-Equilibrium Symmetry Breaking Phase Transitions - Lecture Notes for the 1999 Les Houches Winter School ,\ edit...

    1999

  11. [19]

    Maniv , author E

    author author A. Maniv , author E. Polturak , \ and\ author G. Koren ,\ Observation of Magnetic Flux Generated Spontaneously During a Rapid Quench of Superconducting Films title Observation of Magnetic Flux Generated Spontaneously During a Rapid Quench of Superconducting Films...

    doi

  12. [20]

    Chuang , author R

    author author I. Chuang , author R. Durrer , author N. Turok , \ and\ author B. Yurke ,\ Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals title Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals ,\ 10.1126/science.251.4999.1336 journal journal Scien...

  13. [21]

    author author M. J. \ Bowick , author L. Chandar , author E. A. \ Schiff , \ and\ author A. M. \ Srivastava ,\ The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals title The Cosmological Kibble Mechanism in the Laboratory: String Formation i...

    doi

  14. [22]

    author author P. C. \ Hendry , author N. S. \ Lawson , author R. A. M. \ Lee , author P. V. E. \ McClintock , \ and\ author C. D. H. \ Williams ,\ Generation of Defects in Superfluid ^4 He as an Analogue of the Formation of Cosmic Strings title Generation of Defects in Superfl...

    doi

  15. [23]

    Autti , author V

    author author S. Autti , author V. V. \ Dmitriev , author J. T. \ M\"akinen , author A. A. \ Soldatov , author G. E. \ Volovik , author A. N. \ Yudin , author V. V. \ Zavjalov , \ and\ author V. B. \ Eltsov ,\ Observation of Half-Quantum Vortices in Topological Superfluid ^3 H...

    doi

  16. [24]

    Rysti , author J

    author author J. Rysti , author J. T. \ M\"akinen , author S. Autti , author T. Kamppinen , author G. E. \ Volovik , \ and\ author V. B. \ Eltsov ,\ Suppressing the Kibble-Zurek Mechanism by a Symmetry-Violating Bias title Suppressing the Kibble-Zurek Mechanism by a Symmetry-V...

    doi

  17. [25]

    author author C. N. \ Weiler , author T. W. \ Neely , author D. R. \ Scherer , author A. S. \ Bradley , author M. J. \ Davis , \ and\ author B. P. \ Anderson ,\ Spontaneous Vortices in the Formation of Bose–Einstein Condensates title Spontaneous Vortices in the Formation of Bo...

    doi

  18. [26]

    Lamporesi , author S

    author author G. Lamporesi , author S. Donadello , author S. Serafini , author F. Dalfovo , \ and\ author G. Ferrari ,\ Spontaneous Creation of Kibble–Zurek Solitons in a Bose–Einstein Condensate title Spontaneous Creation of Kibble–Zurek Solitons in a Bose–Einstein Condensate...

    doi

  19. [28]

    author author W. H. \ Zurek ,\ Cosmological Experiments in Condensed Matter Systems title Cosmological Experiments in Condensed Matter Systems ,\ 10.1016/S0370-1573(96)00009-9 journal journal Phys. Rep. \ volume 276 ,\ pages 177 ( year 1996 ) NoStop

    doi

  20. [29]

    author author N. D. \ Antunes , author L. M. A. \ Bettencourt , \ and\ author W. H. \ Zurek ,\ Vortex String Formation in a 3D U(1) Temperature Quench title Vortex String Formation in a 3D U(1) Temperature Quench ,\ 10.1103/PhysRevLett.82.2824 journal journal Phys. Rev. Lett. ...

    doi

  21. [30]

    author author A. del Campo ,\ Universal Statistics of Topological Defects Formed in a Quantum Phase Transition title Universal Statistics of Topological Defects Formed in a Quantum Phase Transition ,\ 10.1103/PhysRevLett.121.200601 journal journal Phys. Rev. Lett. \ volume 121...

    doi

  22. [31]

    author author F. J. \ G\'omez-Ruiz , author J. J. \ Mayo , \ and\ author A. del Campo ,\ Full Counting Statistics of Topological Defects after Crossing a Phase Transition title Full Counting Statistics of Topological Defects after Crossing a Phase Transition ,\ 10.1103/PhysRev...

    doi

  23. [32]

    For a discussion of the higher order cumulants see Ref

    note The first three cumulants are the mean, _1 = n , variance, _2 = (n - n )^2 , and skewness, _3 = (n - n )^3 . For a discussion of the higher order cumulants see Ref. @citealpnum gom20 . Stop

  24. [33]

    author author P. J. \ Heikkinen , author L. V. \ Levitin , author X. Rojas , author A. Singh , author N. Eng , author A. Casey , author J. Saunders , author A. Vorontsov , author N. Zhelev , author A. T. \ Sebastian , \ and\ author J. M. \ Parpia ,\ Chiral Superfluid Helium-3 ...

    doi

  25. [34]

    note Rotations about the chiral axis reduce to an overall phase factor which can be undone with a gauge transformation. Stop

  26. [35]

    author author J. A. \ Sauls ,\ Surface states, Edge Currents, and the Angular Momentum of Chiral p -wave Superfluids title Surface states, Edge Currents, and the Angular Momentum of Chiral p -wave Superfluids ,\ 10.1103/PhysRevB.84.214509 journal journal Phys. Rev. B \ volume ...

    doi

  27. [36]

    author author T. A. \ Tokuyasu , author D. W. \ Hess , \ and\ author J. A. \ Sauls ,\ Vortex states in an unconventional superconductor and the mixed phases of UPt _3 title Vortex states in an unconventional superconductor and the mixed phases of UPt _3 ,\ 10.1103/PhysRevB.41....

    doi

  28. [37]

    author author J. A. \ Sauls \ and\ author M. Eschrig ,\ Vortices in Chiral, Spin-Triplet Superconductors and Superfluids title Vortices in Chiral, Spin-Triplet Superconductors and Superfluids ,\ 10.1088/1367-2630/11/7/075008 journal journal New J. Phys. \ volume 11 ,\ pages 07...

    doi

  29. [38]

    Abrahams \ and\ author T

    author author E. Abrahams \ and\ author T. Tsuneto ,\ Time Variation of the Ginzburg - Landau Order Parameter title Time Variation of the Ginzburg - Landau Order Parameter ,\ 10.1103/PhysRev.152.416 journal journal Phys. Rev. \ volume 152 ,\ pages 416 ( year 1966 ) NoStop

    doi

  30. [39]

    author author N. B. \ Kopnin ,\ Vortex dynamics and mutual friction in superconductors and Fermi superfluids title Vortex dynamics and mutual friction in superconductors and Fermi superfluids ,\ 10.1088/0034-4885/65/11/202 journal journal Reports on Progress in Physics \ volum...

    doi

  31. [40]

    author author I. S. \ Aranson , author N. B. \ Kopnin , \ and\ author V. M. \ Vinokur ,\ Nucleation of Vortices by Rapid Thermal Quench title Nucleation of Vortices by Rapid Thermal Quench ,\ 10.1103/PhysRevLett.83.2600 journal journal Phys. Rev. Lett. \ volume 83 ,\ pages 260...

    doi

  32. [41]

    author author N. B. \ Kopnin \ and\ author E. V. \ Thuneberg ,\ Time-Dependent Ginzburg--Landau Analysis of Inhomogeneous Normal-Superfluid Transitions title Time-Dependent Ginzburg--Landau Analysis of Inhomogeneous Normal-Superfluid Transitions ,\ 10.1103/PhysRevLett.83.116 j...

    doi

  33. [42]

    Kleinert ,\ Collective Q uantum F ields title Collective Q uantum F ields ,\ 10.1002/prop.19780261102 journal journal Forts

    author author H. Kleinert ,\ Collective Q uantum F ields title Collective Q uantum F ields ,\ 10.1002/prop.19780261102 journal journal Forts. Phys. \ volume 26 ,\ pages 565 ( year 1978 ) NoStop

    1978 doi

  34. [43]

    note The spin basis, \ e _ | =x',y',z'\ , is an orthogonal triad of unit vectors in spin space, while the orbital basis functions, p _i= e _i p , are the direction cosines of the orbital momentum, p , defined on the Fermi surface, and referenced to the unit vectors \ e _i | i=...

  35. [44]

    K_ = i [ Tr \ A A^ \ - Tr \ A A ^ \ ]

    note Violation of particle-hole symmetry by the parent Fermi-liquid allows for an additional invariant in the Lagrangian that is first-order in _t A , i.e. K_ = i [ Tr \ A A^ \ - Tr \ A A ^ \ ] . This C-violating term is small so we omit it in the dynamical simulations that fo...

  36. [45]

    Rainer \ and\ author J

    author author D. Rainer \ and\ author J. W. \ Serene ,\ Free Energy of Superfluid ^3He title Free Energy of Superfluid ^3He ,\ 10.1103/PhysRevB.13.4745 journal journal Phys. Rev. B \ volume 13 ,\ pages 4745 ( year 1976 ) NoStop

    1976 doi

  37. [46]

    author author J. A. \ Sauls \ and\ author J. W. \ Serene ,\ Potential Scattering Models for the Quasiparticle Interactions in Liquid ^3 He title Potential Scattering Models for the Quasiparticle Interactions in Liquid ^3 He ,\ 10.1103/PhysRevB.24.183 journal journal Phys. Rev....

    doi

  38. [47]

    author author J. A. \ Sauls \ and\ author J. W. \ Serene ,\ Higher-Order Strong Coupling Effects in Superfluid ^3 He title Higher-Order Strong Coupling Effects in Superfluid ^3 He ,\ 10.1016/0378-4363(81)90869-X journal journal Physica B+C \ volume 108 ,\ pages 1137 ( year 198...

    doi

  39. [48]

    Wiman ,\ title Quantitative Superfluid He-Three from Confinement to Bulk ,\ https://search.proquest.com/docview/2164903630?pq-origsite=gscholar Ph.D

    author author J. Wiman ,\ title Quantitative Superfluid He-Three from Confinement to Bulk ,\ https://search.proquest.com/docview/2164903630?pq-origsite=gscholar Ph.D. thesis ,\ school Northwestern University , address Evanston, Illinois ( year 2019 ) NoStop

    2019

  40. [49]

    eq-tau_vs_T because the latter defines the time scale for pair fluctuations in the critical region in which the order parameter has not yet developed

    note Note that temperature-dependent scaling of _ 5 6 GL (T) differs from that in Eq. eq-tau_vs_T because the latter defines the time scale for pair fluctuations in the critical region in which the order parameter has not yet developed. Stop

  41. [50]

    note In a magnetic field with B || and B B_c 2.5 \, mT the Zeeman energy dominates, is minimized for d B , which ``unlocks'' d from \! . Stop

  42. [51]

    @citealpnum sau15

    note This Lagrangian and its extension to p-wave superconductors with tetragonal symmetry was used to investigate the collective modes of the chiral p-wave model for Sr _ 2 RuO _ 4 \ in Ref. @citealpnum sau15 . Stop

  43. [52]

    Rantanen \ and\ author V

    author author R. Rantanen \ and\ author V. Eltsov ,\ Competition of vortex core structures in superfluid title Competition of vortex core structures in superfluid ,\ 10.1103/PhysRevResearch.6.043112 \ volume 6 ,\ pages 043112 ( year 2024 ) NoStop

    doi

  44. [53]

    author author L. V. \ Levitin , author R. G. \ Bennett , author A. Casey , author B. Cowan , author J. Saunders , author D. Drung , author T. Schurig , \ and\ author J. M. \ Parpia ,\ Phase Diagram of the Topological Superfluid ^3 He Confined in a Nano-scale Slab Geometry titl...

    doi

  45. [54]

    Zhelev , author T

    author author N. Zhelev , author T. S. \ Abhilash , author E. N. \ Smith , author R. G. \ Bennett , author X. Rojas , author L. Levitin , author J. Saunders , \ and\ author J. M. \ Parpia ,\ The A-B transition in superfluid Helium-3 under confinement in a thin slab geometry ti...

    doi

  46. [55]

    author author J. C. \ Strikwerda ,\ 10.1137/1.9780898717938 title Finite Difference Schemes and Partial Differential Equations ,\ edition second edition \ ed.\ ( publisher Society for Industrial and Applied Mathematics ,\ year 2004 ) NoStop

    doi

  47. [56]

    author author G. E. P. \ Box \ and\ author M. E. \ Muller ,\ A Note on the Generation of Random Normal Deviates title A Note on the Generation of Random Normal Deviates ,\ 10.1214/aoms/1177706645 journal journal The Annals of Mathematical Statistics \ volume 29 ,\ pages 610–61...

  48. [57]

    author author J. A. \ Sauls ,\ The Order Parameter for the Superconducting Phases of UPt _3 title The order parameter for the superconducting phases of UPt _3 ,\ https://arxiv.org/abs/1812.09984 journal journal Adv. Phys. \ volume 43 ,\ pages 113 ( year 1994 ) NoStop

    1994 arXiv

  49. [58]

    Ichioka , author K

    author author M. Ichioka , author K. Machida , \ and\ author J. A. \ Sauls ,\ Vortex States of Chiral P-Wave Superconductors title Vortex States of Chiral P-Wave Superconductors ,\ 10.1088/1742-6596/400/2/022031 journal journal J. Phys. (IOP) \ volume 400 ,\ pages 022031 ( yea...

    doi

  50. [59]

    Shevtsov \ and\ author J

    author author O. Shevtsov \ and\ author J. A. \ Sauls ,\ Electrons and Weyl Fermions in Superfluid ^3 He-A title Electrons and Weyl Fermions in Superfluid ^3 He-A ,\ 10.1103/PhysRevB.94.064511 journal journal Phys. Rev. B \ volume 94 ,\ pages 064511 ( year 2016 ) NoStop

    doi

  51. [60]

    Wu \ and\ author J

    author author H. Wu \ and\ author J. A. \ Sauls ,\ Weyl Fermions and Broken Symmetry Phases in Laterally Confined Films title Weyl Fermions and Broken Symmetry Phases in Laterally Confined Films ,\ https://doi.org/10.1088/1361-648X/acf42b journal journal J. Phys. Cond. Mat. \ ...

    doi

  52. [61]

    author author D. G. \ Lowe ,\ Distinctive Image Features from Scale-Invariant Keypoints title Distinctive image features from scale-invariant keypoints ,\ 10.1023/b:visi.0000029664.99615.94 journal journal International Journal of Computer Vision \ volume 60 ,\ pages 91–110 ( ...

  53. [62]

    author author J. A. \ Sauls , author H. Wu , \ and\ author S. B. \ Chung ,\ Anisotropy and Strong-Coupling Effects on the Collective Mode Spectrum of Chiral Superconductors: Application to Sr _2 RuO _4 title Anisotropy and Strong-Coupling Effects on the Collective Mode Spectru...

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