Pith. sign in

REVIEW 3 major objections 4 minor 48 references

Competing Bosonic Reactions: Insight from Exactly Solvable Time-Dependent Models

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two competing decay channels of a bosonic condensate produce near-total final asymmetry even when the energy split between the products is arbitrarily small, according to the exact solution of the driven Tavis-Cummings model.

desk verdict Solid exact two-channel asymmetry result, but the second phase transition and phi^4 universality claim rest on an unproven effective Hamiltonian that drops remaining-molecule occupancy factors. read the letter →

arxiv 2504.13027 v2 pith:YOQAHAIP submitted 2025-04-17 quant-ph cond-mat.stat-mechgr-qchep-th

Fumika Suzuki , Rajesh K. Malla , Nikolai A. Sinitsyn This is my paper · ORCID
classification quant-phcond-mat.stat-mechgr-qchep-th
keywords Landau-ZenertransitionsTavis-Cummingsmodelbosonicreactionsparticleproductionasymmetrydynamicphasetransitionuniversalityclasseffectivetemperaturecoherentmoleculardissociation
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

Two competing decay channels of a bosonic condensate can produce a nearly perfect final asymmetry even when the energy splitting between the channels is arbitrarily small. Using the exact solution of the driven two-channel Tavis-Cummings model, the paper shows that in the quasi-adiabatic regime—sweeps slow enough to dissociate most molecules but not slow enough to be strictly adiabatic—essentially all dissociated pairs end up in the lower-energy channel, with the population of the upper channel growing linearly with the sweep rate. The asymmetry emerges because post-resonance dynamics thermalizes the product pairs within each sector of fixed total pair number at an effective temperature set by the sweep rate rather than by the energy splitting. The paper also argues that the decay of an unstable vacuum is generically accomplished by two phase transitions, the second belonging to the universality class of a complex $\phi^4$ field theory. A sympathetic reader would care because this gives a parameter-free mechanism by which a tiny mass- or CP-type asymmetry can dominate the outcome of coherent particle production.

What carries the argument

The load-bearing object is the two-channel driven bosonic Tavis-Cummings Hamiltonian (59), whose solubility comes from its membership in an integrable multistate Landau-Zener family: it is one member of a $t/\tau$-pair of Hamiltonians (33)-(34), and path deformation in the two-time plane turns the many-body transition into a product of independent two-level Landau-Zener crossings. After the fast dissociation step, the dynamics in each sector with a fixed total number of dissociated pairs $n$ is captured by an effective Hamiltonian (72) with a coupling that decays as $1/t$, and the exact solution of that effective model yields the Gibbs distribution (75), $P_{\delta n} = e^{-\delta n/k_B T}/Z$ with $k_B T = \beta/(\pi g^2)$. The identity that carries the asymmetry is the detailed-balance ratio (74), $P(n_1,n_2|n)/P(n_1+1,n_2-1|n) = x = e^{-2\pi g^2/\beta}$, which fixes the effective temperature and hence the linear scaling of $\langle n_2\rangle$.

What would settle it

Measure the number of pairs ending in the higher-energy channel, $\langle n_2\rangle$, versus sweep rate $\beta$ in a coherent molecular dissociation experiment swept through a Feshbach resonance: the paper predicts $\langle n_2\rangle = \beta/(2\pi g^2)$ for quasi-adiabatic sweeps, linear with no logarithmic correction and independent of the energy splitting $\varepsilon$. Observing a logarithmic correction, a plateau at slow sweeps, or a strong $\varepsilon$ dependence would falsify the effective-Hamiltonian and $\phi^4$ claims; a numerical evolution of the full Hamiltonian (59) that includes higher-order processes and shows deviation from Eq. (69) would also do.

Watch

Extended reading notes

Core claim

The paper's central claim is that the final asymmetry parameter for the two-channel reaction, $\eta = (\langle n_1\rangle - \langle n_2\rangle)/\langle n\rangle$, approaches $1 - 2\beta/(2\pi g^2 N)$ in the quasi-adiabatic regime, so for fixed molecule number $N$ and sufficiently slow sweeps nearly all dissociated pairs pass through the lower-energy channel (Eq. 70). This result follows exactly from the joint distribution $P_{n_1,n_2} = P_{n_1} x^{N-n_1-n_2} (x^{N-n_1-n_2+1},x)_{n_2}$ (Eq. 61), which the paper derives as a special case of the solvable multichannel Tavis-Cummings solution. The average number of pairs in the upper channel, $\langle n_2\rangle = \beta/(2\pi g^2)$, is independent of the energy splitting $\varepsilon$; the splitting only sets the time scale $\sim 1/\varepsilon$ at which the asymmetry freezes in. The paper further claims that the evolution passes through a second phase transition, described by the effective Hamiltonian (72), and that this transition falls in the universality class of complex $\phi^4$ theory, with the undissociated molecules acting as massive 'Higgs' excitations ($\sim \beta \ln\beta$) and the upper-channel atoms as near-massless 'Goldstone' excitations ($\sim \beta$).

Load-bearing premise

The central assumption is that after the first resonance the effective Hamiltonian (72)—derived from second-order perturbation theory, with the total number of dissociated pairs fixed and different pair-number sectors evolving independently—faithfully describes the rest of the evolution.

Editorial extensions

If this is right

  • In quasi-adiabatic sweeps, essentially all dissociated pairs occupy the lower-energy channel; the upper-channel population is $\beta/(2\pi g^2)$ and vanishes as the sweep slows, independently of the energy splitting $\varepsilon$.
  • The exact solution predicts a dynamic phase transition in the sweep rate near $\beta_c \sim 2\pi g^2 N / \ln N$, below which the distribution's peak moves from zero dissociation to finite dissociation, with $m_{\max} \sim N(1 - \beta/\beta_c)$.
  • Within each sector of fixed total pair number, the post-dissociation redistribution is a Gibbs distribution with effective temperature $k_B T = \beta/(\pi g^2)$, giving a detailed-balance ratio $e^{-2\pi g^2/\beta}$ between adjacent occupation numbers.
  • The decay of an unstable vacuum in this model proceeds through two phase transitions: a fast one at the molecular resonance and a slower one at time scale $\sim 1/\varepsilon$; the second belongs to the universality class of complex $\phi^4$ theory, with molecular excitations scaling as $\beta\ln\beta$ and atomic excitations as $\beta$.
  • The semiclassical analysis indicates that the asymmetry and its scaling are robust against integrability-breaking perturbations to the Tavis-Cummings Hamiltonian.

Reading between the lines

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

  • The mechanism suggests a generic route to matter-antimatter-like asymmetries in coherent decays: a tiny mass- or CP-type asymmetry can be amplified to order-one final asymmetry whenever the decay products pass through a near-degenerate multi-channel resonance, because the effective temperature is set by the sweep rate, not by the energy splitting.
  • The two-stage picture—fast dissociation followed by slow pseudo-thermalization—implies a measurable delay between molecule depletion and the buildup of mode asymmetry; checking whether the upper-channel population appears on a time scale $\sim 1/\varepsilon$ after the first resonance would test the second phase transition directly.
  • Since the effective Hamiltonian (72) is related to time-dependent Gaudin magnets, the same Gibbs-distribution structure may appear in other driven integrable systems with decaying $1/t$ couplings, suggesting a broader class of coherent sweep dynamics that end in pseudo-thermal states.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies competing bosonic reaction channels in driven multistate Landau-Zener models, focusing on the two-channel driven bosonic Tavis-Cummings model of Eq. (59). After reviewing the integrability machinery of MLZ models and the known single-channel solution, it presents exact asymptotic probability distributions for the two-channel problem, Eqs. (61)-(62), and analyzes the resulting asymmetry parameter η. The central quantitative claim is that in the quasi-adiabatic regime the final population is almost completely asymmetric, η≈1, even when the energy splitting ε between the two product channels is arbitrarily small; this is summarized by Eqs. (69)-(70). The paper also proposes that the dynamics passes through a second phase transition associated with pseudo-thermalization inside fixed total-pair sectors, described by the effective Hamiltonian (72), and assigns this transition to the universality class of complex φ^4 theory. The manuscript is partly a review of previously solved models and partly a new application of those exact solutions to reaction competition.

Significance. If the asymmetry result holds, the paper gives an exact solvable demonstration that an arbitrarily small product-channel energy difference can be coherently amplified into near-total final asymmetry, which is a striking and physically relevant statement. The exact two-channel distributions (61)-(62), the quasi-adiabatic averages (46), (68), (69), and the asymmetry formula (70) are concrete, checkable results, and the numerical evaluations in Figs. 5 and 6 support them. The proposal that unstable-vacuum decay is generically accompanied by a second phase transition in the complex-φ^4 universality class is interesting, but it is less firmly established because it depends on an effective Hamiltonian that is asserted rather than derived. The paper is likely to be useful to researchers working on solvable Landau-Zener models and on coherent particle-production dynamics, provided the effective-model analysis is either completed or appropriately qualified.

major comments (3)
  1. [Section VI.A, Eqs. (72)-(73)] The effective Hamiltonian (72) is introduced as the result of second-order perturbation theory for large t in the full model (59), but no derivation is shown. In a sector with n0=N-n un-dissociated molecules, a direct Schrieffer-Wolff reduction of (59) produces virtual amplitudes proportional to (n0+1). For example, the |n1,n2> to |n1+1,n2-1> amplitude is g^2 (n0+1) n1(n2+1)/(β t), with a corresponding diagonal shift proportional to (n0+1)(n1^2+n2^2) g^2/(β t). Equation (73) contains these expressions with n0+1 replaced by 1, so (72) is justified at most in the fully dissociated sector n=N. In the quasi-adiabatic regime, Eq. (68) gives n0=(β/2πg^2)[ln(β/2πg^2)-ψ1(2)], which can be tens to hundreds for the allowed parameter range. Consequently the pseudo-thermalization result (75), the linear scaling (76), and the mean-field transition in Section VI.B are not established for the original two-channel model (59); they are properties of an unverified surrogate model. This does not affect the exact asymmetry result (70).
  2. [Section VI.B and Section VII.B] The claim that the two-channel reaction belongs to the universality class of complex φ^4 field theory is not derived from the exact solution. It rests on the mean-field decoupling of the effective Hamiltonian (72) and on the semiclassical Hamiltonian (89). Even if this mean-field analysis correctly describes the ground state of (72), the same issue as above applies: (72) has not been shown to describe the post-resonance evolution of the full Hamiltonian, particularly when n0 is not small. The agreement of Eq. (76) with the exact result (69) is not independent evidence: both expressions scale as β/(2πg^2), so the match does not discriminate between the true dynamics and the effective model. Please either derive (72) including the molecular-occupation factors and revisit the transition within that corrected derivation, or explicitly restrict the two-phase-transition and universality-class statements to the n=N sector. I note that this revision would not affect the paper's main exact asymmetry claim.
  3. [Section VI.A, time-scale separation] The assumption that the total number of dissociated pairs n is fixed after τ_LZ and that each n-sector evolves independently is asserted but not justified from the exact solution. The exact distribution (61) is a statement about the final state at t→+∞; it does not directly imply conservation of n at intermediate times. Moreover, Figs. 7 and 8 compare the predictions of Eq. (75) and Eq. (76) only with numerical solutions of the effective Hamiltonian (72), not with numerical solutions of the original model (59). A numerical comparison of the effective-model dynamics with the full two-channel model in the quasi-adiabatic regime, or an analytical estimate of the error in the second-order reduction, would be needed to support the claimed phase transition.
minor comments (4)
  1. [Section IV.D, Eq. (58)] The scattering phases in Eq. (58) are presented without derivation, as the text acknowledges. Since these phases are not load-bearing for the asymmetry result, a short derivation in an appendix or a precise reference to the source of the formula would strengthen the paper.
  2. [Figs. 7 and 8] The captions and surrounding text should state clearly that these figures validate the effective Hamiltonian (72) alone, not the reduction from the full Hamiltonian (59). Without such clarification, a reader may infer that the numerical checks validate the time-scale separation in Section VI.A.
  3. [Section V, notation] The symbol n is used both for the number of un-dissociated molecules, n=N-n1-n2, and for the total number of dissociated pairs, n=n1+n2. This dual use is confusing; for example, Eq. (68) uses n in the former sense while Eq. (76) uses n in the latter. Please introduce distinct notation, e.g., n0 for the molecular remainder.
  4. [Eq. (76)] The effective-model result (76) contains a -1/2 correction relative to the exact average (69). If this constant is a genuine subleading correction, the text should say so; if it is an artifact of the kBT≪n approximation, the approximation error should be stated. The current wording makes the relation between (69) and (76) ambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exact two-channel asymmetry is derived from a cited integrable solution and numerically checked, and the effective-model thermal distribution is explicitly imported from and verified against that same exact distribution rather than being a fitted prediction.

full rationale

No circular step is found. The exact two-channel distributions, Eqs. (61)-(62), are taken from the recursive formula Eq. (36) of the authors' published integrable solution [19]; this is an external input, not fitted in this paper, and the derived asymmetry Eq. (70) and average Eq. (69) follow by algebra and are checked numerically in Figs. 5-6. In Sec. VI, the effective Hamiltonian Eq. (72) is introduced by a perturbation-theory claim, and the thermal distribution Eq. (75) is explicitly obtained by conditioning the same exact distribution Eq. (61) via the detailed-balance ratio Eq. (74), not by fitting; Fig. 7 independently verifies Eq. (75) against a numerical solution of Eq. (72). The agreement of Eq. (76) with Eq. (69) is a consistency check, not a fitted prediction. The phi^4 universality assignment is imported from the authors' prior work [27], but the scalings it labels are independently derived here (Eqs. (68)-(69)), so the citation is corroborating rather than load-bearing in a circular sense. Omitted derivations, such as the statement that Eq. (58) is presented without derivation and the unshown second-order reduction leading to Eq. (72), are support and correctness concerns, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results depend on previously established exact solutions of the driven Tavis-Cummings model, primarily from the authors' own PRL 129, 033201 (2022), plus a second-order perturbative effective Hamiltonian and semiclassical or mean-field analysis. No new particles, fields, or conserved quantities are introduced, and no parameters are fitted to data.

assumptions (5)
  • domain assumption The exact transition probability formula in Eq. (36) for the multi-channel driven Tavis-Cummings model holds as stated.
    Sections III.C and V use Eq. (36) without re-derivation, citing the authors' PRL 129, 033201 (2022); all new two-channel results inherit this result.
  • standard math Standard properties of q-Pochhammer symbols and q-digamma functions used in Eqs. (40) through (68) are correct.
    The derivations of the Euler distribution, average excitation numbers, and critical sweep rate rely on these special-function identities without proof.
  • domain assumption The energy levels and couplings in the driven bosonic Tavis-Cummings Hamiltonian in Eq. (33) faithfully represent coherent dissociation of bosonic molecules into pairs.
    The physical mapping from the cubic Lagrangian in Eq. (4) to Eq. (33) is asserted in Section I; the cosmological and ultracold-atom applications depend on this mapping.
  • domain assumption Equation (72), derived from second-order perturbation theory in g/(beta t) for large times, captures the subsequent evolution within each fixed-n sector.
    Section VI.A uses this effective Hamiltonian to derive the Gibbs distribution in Eq. (75) and the second phase transition; higher-order terms and continuing dissociation are neglected.
  • domain assumption Semiclassical and mean-field analyses in Sections VI.B and VII reproduce the infinite-N phase transition and universality class.
    The phi^4 universality classification is inferred from the classical Hamiltonian in Eq. (89) and the mean-field decoupling in Eq. (77), not from a rigorous limit of the quantum model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Competing Bosonic Reactions: Insight from Exactly Solvable Time-Dependent Models." pith.science (2026). https://pith.science/paper/YOQAHAIP

@misc{pith2026250413027,
  author       = {Pith},
  title        = {Pith review of: Competing Bosonic Reactions: Insight from Exactly Solvable Time-Dependent Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOQAHAIP}},
  note         = {Machine review of arXiv:2504.13027}
}
read the original abstract

We discuss the progress on exactly solvable multistate Landau-Zener models from a perspective of their application to competing reactions of particle creation from a false vacuum. Such models generally predict that, even with identical initial conditions, and for nearly the same other particle parameters, a quantum coherent evolution results in a final particle distribution with significant asymmetry. We use an exact solution of the driven bosonic Tavis-Cummings model for two reaction pathways in order to quantify this effect, reveal a corresponding phase transition, and identify its universality class.

Figures

Figures reproduced from arXiv: 2504.13027 by the authors.

Figure 1
Figure 1. FIG. 1. Diabatic energy levels of the 3-state Demkov-Osherov model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. An integration path [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The distribution of the number of the dissociated molecules, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical test for Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The number of particle pairs in the higher energy mode, [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 42 canonical work pages

  1. [1]

    Brundobler and V

    S. Brundobler and V . Elser, Journal of Physics A: Mathematical and General 26, 1211 (1993)

    work page 1993

  2. [2]

    reaction channels

    By equating P0→0 obtained atε = 0 and forε≫g, we find that P (1) 0→0(2γ) = P (1) 0→0(γ) 2 , which has a solution P (1) 0→0 = ecγ, where the constant c = −2π can be calculated perturbatively and trivially in the limit of large β. Knowing the solution of the Demkov-Osherov model, the transition probabilities (18) for the model with de- caying coupling can b...

    work page

  3. [3]

    This is the Hamiltonian of a classical particle with two de- grees of freedom, which has already been investigated in [27]

    (89) where V (X) =−βt|X|2 + γ 4|X|4 +ε(X2 2−X2 1). This is the Hamiltonian of a classical particle with two de- grees of freedom, which has already been investigated in [27]. The potential energy for this motion is time-dependent. For ε = 0 , the system clearly experiences a phase transition be- cause, for t <0, the potential energy has a single minimum a...

    work page

  4. [4]

    D. A. Harmin and P. N. Price, Phys. Rev. A 49, 1933 (1994)

    work page 1994

  5. [5]

    D. A. Harmin, Phys. Rev. A 56, 232 (1997)

    work page 1997

  6. [6]

    Shevchenko, S

    S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1 (2010)

    2010

  7. [7]

    X. Mi, S. Kohler, and J. R. Petta, Phys. Rev. B 98, 161404 (2018)

    work page 2018

  8. [8]

    Ginzel, A

    F. Ginzel, A. R. Mills, J. R. Petta, and G. Burkard, Phys. Rev. B 102, 195418 (2020)

    work page 2020

Show all 48 references
  1. [9]

    Werther, F

    M. Werther, F. Grossmann, Z. Huang, and Y . Zhao, The Journal of Chemical Physics 150, 234109 (2019), https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/1.5096158/14120774/234109 1 online.pdf

    2019 doi

  2. [10]

    Bello, N

    F. Bello, N. Kongsuwan, J. F. Donegan, and O. Hess, Nano Let- ters 20, 5830 (2020)

    2020

  3. [11]

    Kervinen, J

    M. Kervinen, J. E. Ram ´ırez-Mu˜noz, A. V ¨alimaa, and M. A. Sillanp¨a¨a, Phys. Rev. Lett. 123, 240401 (2019)

    2019

  4. [12]

    Z. Sun, J. Ma, X. Wang, and F. Nori, Phys. Rev. A 86, 012107 (2012)

    2012

  5. [13]

    P. Y . Wen, O. V . Ivakhnenko, M. A. Nakonechnyi, B. Suri, J.-J. Lin, W.-J. Lin, J. C. Chen, S. N. Shevchenko, F. Nori, and I.-C. Hoi, Phys. Rev. B 102, 075448 (2020)

    2020

  6. [14]

    R. K. Malla and M. E. Raikh, Phys. Rev. B 97, 035428 (2018)

    2018

  7. [15]

    N. A. Sinitsyn, E. A. Yuzbashyan, V . Y . Chernyak, A. Patra, and C. Sun, Phys. Rev. Lett. 120, 190402 (2018)

    2018

  8. [16]

    E. A. Yuzbashyan, Annals of Physics 392, 323 (2018)

    2018

  9. [17]

    N. A. Sinitsyn and F. Li, Phys. Rev. A 93, 063859 (2016)

    2016

  10. [18]

    V . Y . Chernyak, F. Li, C. Sun, and N. A. Sinitsyn, Journal of Physics A: Mathematical and Theoretical 53, 295201 (2020)

    2020

  11. [19]

    R. K. Malla, V . Y . Chernyak, and N. A. Sinitsyn, Phys. Rev. B 103, 144301 (2021)

    2021

  12. [20]

    Barik, L

    S. Barik, L. Bakker, V . Gritsev, and E. Yuzbashyan, arXiv preprint arXiv:2409.17053 (2024)

    2024 arXiv

  13. [21]

    R. K. Malla, V . Y . Chernyak, C. Sun, and N. A. Sinitsyn, Phys. Rev. Lett. 129, 033201 (2022)

    2022

  14. [22]

    Sun and N

    C. Sun and N. A. Sinitsyn, Phys. Rev. A 94, 033808 (2016)

    2016

  15. [23]

    C. Sun, V . Y . Chernyak, A. Piryatinski, and N. A. Sinitsyn, Phys. Rev. Lett. 123, 123605 (2019)

    2019

  16. [24]

    Zhang, L

    Z. Zhang, L. Chen, K.-X. Yao, and C. Chin, Nature 592, 708 (2021)

    2021

  17. [25]

    K. D. Lozanov, in Proceedings of the Les Houches Summer School: Post-Planck Cosmology , V ol. 100, edited by C. Def- fayet, E. Masso, and G. Veneziano (Oxford University Press,

  18. [26]

    Damski, Phys

    B. Damski, Phys. Rev. Lett. 95, 035701 (2005)

    2005

  19. [27]

    T. W. B. Kibble, Journal of Physics A: Mathematical and Gen- eral 9, 1387 (1976)

    1976

  20. [28]

    W. H. Zurek, Physics Reports 276, 177 (1996)

    1996

  21. [29]

    V . M. G. Silveira, C. A. Z. Vasconcellos, E. G. S. Luna, and D. Hadjimichef, Journal of High Energy Physics 2021, 285 (2021)

    2021

  22. [30]

    Tyagi, F

    B. Tyagi, F. Suzuki, V . A. Chernyak, and N. A. Sinitsyn, Phys. Rev. A 111, 032205 (2025)

    2025

  23. [31]

    Carenza and M

    P. Carenza and M. D. Marsh, Journal of Cosmology and As- troparticle Physics 2023 (04), 021

    2023

  24. [32]

    N. A. Sinitsyn, J. Lin, and V . Y . Chernyak, Phys. Rev. A 95, 012140 (2017)

    2017

  25. [33]

    B. Yan, V . Y . Chernyak, W. H. Zurek, and N. A. Sinitsyn, Phys. Rev. Lett. 126, 070602 (2021)

    2021

  26. [34]

    Y . N. Demkov and V . I. Osherov, Soviet Journal of Experimen- tal and Theoretical Physics 26, 916 (1968)

    1968

  27. [35]

    N. A. Sinitsyn, Phys. Rev. A 90, 062509 (2014). 18

    2014

  28. [36]

    V . A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and Y . B. Band, Journal of Physics B: Atomic, Molecular and Optical Physics 32, 1845 (1999)

    1999

  29. [37]

    A. A. Rangelov, J. Piilo, and N. V . Vitanov, Phys. Rev. A 72, 053404 (2005)

    2005

  30. [38]

    Sun, arXiv preprint arXiv:2504.02576 (2025)

    C. Sun, arXiv preprint arXiv:2504.02576 (2025)

    2025

  31. [39]

    Faddeev and L

    L. Faddeev and L. Takhtajan, Hamiltonian methods in the the- ory of solitons (Springer Science & Business Media, 2007)

    2007

  32. [40]

    A. S. Fokas, A. R. Its, A. A. Kapaev, and V . Y . Novok- shenov, Painlev´e Transcendents: The Riemann-Hilbert Ap- proach, Mathematical Surveys and Monographs (American Mathematical Society, 2006)

    2006

  33. [41]

    F. Li, V . Y . Chernyak, and N. A. Sinitsyn, Phys. Rev. Lett.121, 190601 (2018)

    2018

  34. [42]

    Itin and P

    A. Itin and P. T ¨orm¨a, arXiv preprint arXiv:0901.4778 (2009)

    2009 arXiv

  35. [43]

    V . G. Sadhasivam, F. Suzuki, B. Yan, and N. A. Sinitsyn, Nature Communications 15, 10246 (2024)

    2024

  36. [44]

    Altland, V

    A. Altland, V . Gurarie, T. Kriecherbauer, and A. Polkovnikov, Phys. Rev. A 79, 042703 (2009)

    2009

  37. [45]

    Zabalo, A.-K

    A. Zabalo, A.-K. Wu, J. H. Pixley, and E. A. Yuzbashyan, Phys. Rev. B 106, 104513 (2022)

    2022

  38. [46]

    A. P. Itin and P. T¨orm¨a, Phys. Rev. A 79, 055602 (2009)

    2009

  39. [48]

    Suzuki and W

    F. Suzuki and W. H. Zurek, Phys. Rev. Lett. 132, 241601 (2024)

    2024

  40. [2020]

    419–476, arXiv:1907.04402 [astro-ph.CO]

    pp. 419–476, arXiv:1907.04402 [astro-ph.CO]

    1907 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.