Pith. sign in

REVIEW 4 major objections 4 minor 82 references

The full counting statistics of the sine-Gordon model can be computed from thermodynamic Bethe Ansatz data, and the topological-charge cumulants vary fractally with the coupling strength while the energy and momentum cumulants do not.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 11:45 UTC pith:YYSAP6SL

load-bearing objection A careful numerical extension of BFT to sine-Gordon with a genuinely new fractal-fluctuation result, but the load-bearing analyticity assumption is explicitly unproven. the 4 major comments →

arxiv 2601.05079 v2 pith:YYSAP6SL submitted 2026-01-08 cond-mat.stat-mech

Full counting statistics in the sine-Gordon model

Botond C. Nagy , M\'arton Kormos , G\'abor Tak\'acs This is my paper
classification cond-mat.stat-mech PACS 05.60.-k05.30.-d
keywords full counting statisticssine-Gordon modelthermodynamic Bethe ansatzballistic fluctuation theorytopological chargefractal transportcumulantsultracold atoms
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that the full counting statistics of the integrable sine-Gordon field theory — the entire distribution of space-time integrated conserved charges and currents — can be obtained from thermodynamic Bethe Ansatz data through the Ballistic Fluctuation Theory, once the eta sign factors of the nested-Bethe-Ansatz magnons are put into the cumulant formulas. The authors compute the first four cumulants for energy, momentum, and topological charge across temperatures, couplings, and spacetime ray angles, and compare them against analytic results in low- and high-temperature limits. Their central finding is that the topological charge cumulants depend on the coupling in an irregular, fractal-like way, while energy and momentum cumulants stay smooth; they trace this to whether charge-carrying magnon excitations can cross the chosen spacetime ray. If right, the fractal signal and the low-temperature Skellam distribution are quantitative predictions testable in tunnel-coupled ultracold-atom experiments.

Core claim

On its own terms, the paper reports that the TBA-based cumulant expressions, derived from the Ballistic Fluctuation Theory and amended with the eta signs of the magnon nodes, correctly describe the full counting statistics of the sine-Gordon model in all studied regimes. The main discovery is the fractal-like dependence of the topological charge cumulants on the renormalised coupling, which disappears only when the ray angle is large enough that no charge-carrying excitation can cross the ray; no such irregularity appears for energy or momentum. In analytically tractable limits the method yields a Skellam distribution at low temperatures and reflectionless couplings, explicit low-temperature

What carries the argument

The central object is the dynamical free energy, or FCS, F_alpha(lambda), the scaled cumulant-generating function defined by integrating charge and current averages along a path through generalised Gibbs ensembles; it is generated by the flow equation on the pseudo-energies with a sign function that selects which quasiparticles cross the spacetime ray. The cumulants are evaluated from the TBA integral expressions, whose content is that every cumulant is a rapidity integral of filling fractions, dressed charges, and effective velocities, with the sign function and the species signs eta tracked through the dressing operations. The coupling enters through the continued fraction that fixes how m

Load-bearing premise

The whole numerical program rests on the unproven analyticity conditions, stated before Eq. (16), that guarantee the characteristic function decays exponentially in length and that the cumulants are derivatives of the FCS, together with the carried-over rule of discarding all terms containing derivatives of the sign function when differentiating the flow equations (Appendix A).

What would settle it

Measure the distribution of the phase difference in a tunnel-coupled quasi-condensate at a reflectionless coupling and low temperature: the low-temperature prediction is a Skellam distribution with all cumulants equal to 2Ω. If the measured higher cumulants violate this equality, the exponential-scaling and cumulant formulas are wrong; alternatively, an independent numerical computation of the topological-charge variance as a function of coupling at fixed temperature and ray angle should reproduce the irregular oscillations of Fig. 5(a).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The topological-charge FCS at low temperature and reflectionless couplings is the Skellam distribution, so all cumulants equal 2Ω and a full distribution is available, not just moments.
  • At high temperature, topological charge fluctuations become Gaussian with variance set by the free-boson CFT result, and the energy cumulants follow simple power laws in temperature and ray angle.
  • The fractal dependence of charge cumulants on the coupling is a quantitative prediction: the structure appears only for ray angles within the effective light cone set by the last charge-carrying particle's dressed velocity.
  • Numerical differentiation of the FCS integral is unreliable beyond the first cumulant; direct dressing of the cumulant integrands is the dependable route to higher cumulants.

Where Pith is reading between the lines

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

  • If the fractal signal survives, then in a cold-atom realisation a slight shot-to-shot variation in the effective coupling could scramble the measured topological-charge variance while energy and momentum cumulants remain stable; coupling calibration should be treated as a critical uncertainty for charge statistics.
  • The mechanism behind the fractal structure — magnonic effective velocities crossing the ray angle — suggests a general diagnostic: in any integrable model, plot a ballistic-transport cumulant against a control parameter and look for kinks where the dressed velocity of the slowest charge carrier crosses tan(alpha).
  • The eta-sign generalisation of the cumulant formulas is likely to carry over to other nested Bethe Ansatz theories with sign-carrying auxiliary particles, where similar fractal transport fluctuations could appear.
  • A direct testable extension is to measure the vertex-operator two-point function at imaginary counting field for non-reflectionless couplings: the paper's claim implies its exponential decay rate is the FCS, and its fourth cumulant should show the same fractal structure as the second.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper presents a TBA/BFT-based numerical study of the full counting statistics of energy, momentum, and topological charge in the sine-Gordon model. The authors derive explicit TBA expressions for the first four scaled cumulants (Eqs. 23a-d), compute them numerically over a wide range of temperatures, couplings, and spacetime ray angles, and compare with analytic results in three limits: low-temperature reflectionless points, level-one repulsive couplings, and the high-temperature CFT limit. The headline claim is that the topological-charge cumulants exhibit an irregular, fractal-like dependence on the coupling β^2/8π, while energy and momentum cumulants are smooth, and that this fractal structure disappears once charge-carrying excitations cannot reach the integration ray. The paper explicitly relies on the BFT assumption of exponential scaling in Eq. (15), states that the required analyticity conditions are assumed to hold, and calls for independent numerical/experimental verification.

Significance. If the BFT/TBA framework is valid for sine-Gordon with non-diagonal soliton-magnon scattering, this is a substantial advance: it provides quantitative, parameter-free predictions for full counting statistics in an interacting integrable QFT, including the striking prediction of fractal coupling dependence of topological-charge cumulants. The analytic checks in Sections IV.A.1–IV.A.3 are genuine and nontrivial, especially the low-temperature Skellam distribution, the level-one repulsive formulas (28)-(32), and the high-temperature CFT limits (35)-(38). The paper is also transparent about its numerical-differentiation limitations and about the fact that external verification is still needed. However, the central claim is conditional on an unproven analyticity/extensivity assumption, and the numerical evidence is not accompanied by convergence criteria, error bars, or code.

major comments (4)
  1. [Section III, before Eq. (16)] The entire FCS program rests on the exponential scaling (15) and on analyticity conditions 'which we assume to hold' (Section III, before Eq. (16)). The cited references [23,26,48] establish such scaling for other settings, not for sine-Gordon with non-diagonal soliton-magnon scattering. If the true characteristic function does not scale extensively, the cumulant formulas (23a-d) and the fractal structure in Figs. 5, 6, and 14 lose their physical meaning. The authors' own discussion (Section V) notes that semiclassical approaches [50,51] predict anomalous, non-ballistic charge transport at reflective couplings, which would contradict this scaling. This is a load-bearing gap: please either provide a derivation or independent numerical evidence of exponential scaling for the sine-Gordon TBA, or clearly delineate the parameter regime in which it is expected to hold.
  2. [Appendix A, after Eq. (A8)] The derivation of the cumulant expressions (23a-d) discards all terms involving derivatives of the sign function s, following [27]. For the sine-Gordon nested Bethe Ansatz, the flow equation (21) contains non-trivial sign factors η_a, and the dressed velocities of magnons can have non-monotonic rapidity dependence. The validity of discarding s' is not demonstrated for this multi-species magnon case. Since c3 and c4 depend on the details of this step, a proof or at least a careful discussion of when the sign-function derivative terms vanish is needed.
  3. [Section IV.B and Figs. 5-7] The paper reports a fractal-like coupling dependence of topological-charge cumulants without providing any resolution study, quantitative measure (e.g., self-similarity or dimension), or error estimates. The differences plotted in Fig. 6 are at the 1e-6 level, and without convergence criteria or numerical error bars it is difficult to distinguish genuine irregular non-analyticity from discretization noise. Please provide the numerical grid parameters, iteration tolerances, and a convergence test with decreasing step size, or make the code available. This is essential to support the central numerical claim.
  4. [Table I and Eq. (28)] Table I shows that the approximate formula (28) has relative error 0.42 at T=1 for ξ=3. The text describes this as valid at 'low and medium temperatures,' but no quantitative criterion for 'medium' is given. Since the comparison is used to validate the numerical framework, please specify the range of validity and, if possible, provide an estimate of the numerical errors in the TBA evaluation at high temperature.
minor comments (4)
  1. [Appendix B] The symbol p is used both for momentum in Eq. (23a) and as an integer index in the low-temperature formulas (e.g., Eq. (B14)-(B15)). Please rename one of them to avoid confusion.
  2. [Section IV.B.1] The term 'fractal-like' is used as a qualitative description. In the conclusion it is strengthened to 'fractal dependence.' Since no quantitative criterion is provided, consider using a more measured phrasing such as 'irregular, apparently fractal' or adding a scaling analysis.
  3. [Appendix C, Fig. 10] For α=0, c3^e is stated to be zero, but the plot shows nonzero values of order 1e-19 at T=0.2. Please indicate whether these are numerical noise and how such zeros are identified.
  4. [General] The paper would benefit from a short paragraph in the numerical section describing the discretization of rapidity, the treatment of the light-cone singularities in the flow equations, and the stopping criterion for the iterative solution of Eq. (5).

Circularity Check

0 steps flagged

No significant circularity: the cumulant expressions are parameter-free derivatives of an externally supplied BFT flow, and the analytic limits are independent checks.

full rationale

Walking the derivation: Eqs. (23a-d) are obtained by differentiating the BFT dynamical free energy (20) with the flow equation (21), which is taken from refs. [27,28]; the eta factors are fixed by the TBA sign structure [42], not by matching cumulants. The analytic limits in Sec. IV A are computed from the same TBA equations but in regimes where the equations simplify (low T free-fermionic/Maxwell-Boltzmann, high T CFT), and are compared with independent semiclassical [50,51] and CFT [52,53] results. The fractal coupling dependence is a numerical output; no parameter is fitted to it, and the same code reproduces smooth energy/momentum cumulants and the analytic limits, ruling out an output-only artifact. Self-citations [41,42,55] supply the TBA machinery and earlier fractal-Drude-weight observations; however, the FCS/numerical program rests on the externally developed BFT and on explicit analyticity assumptions (stated before Eq. (16)) that are not equivalent to the obtained cumulants. No equation in the paper reduces by construction to a fitted parameter or to a self-citation. Thus no circular step is present; at most there are minor self-citations that are not load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central calculations rest on the prior TBA description of sine-Gordon (from the authors' earlier papers) and on the Ballistic Fluctuation Theory framework. No ad hoc fitted parameters or new entities are introduced. The main unproven input is the BFT analyticity assumption, plus standard TBA/CFT machinery.

axioms (5)
  • domain assumption BFT analyticity conditions ensuring exponential scaling of the generating function (Eq. 15) hold for sine-Gordon.
    Stated explicitly in Section III before Eq. (16): 'Under certain analyticity conditions [23, 26, 48] which we assume to hold'. All cumulant formulas depend on this.
  • domain assumption The nested Bethe Ansatz/magnon TBA solution of [41,42] correctly describes the thermodynamics and dressed quantities of sine-Gordon for generic couplings.
    Used as the numerical foundation for Eqs. (4)-(12); not re-derived here, and the authors are also the authors of [41,42].
  • domain assumption Terms involving derivatives of the sign function s = sign(cosα v_eff - sinα) vanish when computing cumulants.
    Appendix A: 'special care must be taken... effectively requires discarding all terms that explicitly involve derivatives of s.' This is carried over from [27] without a proof for the magnon case.
  • domain assumption Baker-Campbell-Hausdorff corrections for time-like separations in the vertex-operator representation (Eq. 19) produce only power-law corrections to the leading exponential decay.
    Section III: 'one can argue that even for time-like separations, the corrections ... yield power-law corrections and do not affect the leading exponential decay [49].'
  • standard math The high-temperature reduction of the TBA free energy using Rogers dilogarithm identities is valid for the sine-Gordon TBA with chemical potential.
    Section IV A 3 uses this to obtain Eq. (33); the paper extends a standard derivation to finite chemical potential.

pith-pipeline@v1.3.0-alltime-deepseek · 42366 in / 12332 out tokens · 122243 ms · 2026-08-03T11:45:57.665756+00:00 · methodology

0 comments
read the original abstract

Full counting statistics (FCS) is a dynamical generalisation of the free energy, encapsulating detailed information about the distribution and large-scale correlation functions of conserved charges and their associated currents. In this work, we present a comprehensive numerical study of the FCS and the cumulants of the three lowest charges across the full parameter space of the sine-Gordon field theory. To this end, we extend the thermodynamic Bethe Ansatz (TBA) formulation of the FCS to the sine-Gordon model, emphasise the methodological subtleties for a reliable numerical implementation, and compare numerical results with analytical predictions in certain limits.

Figures

Figures reproduced from arXiv: 2601.05079 by Botond C. Nagy, G\'abor Tak\'acs, M\'arton Kormos.

Figure 1
Figure 1. Figure 1: FIG. 1: Variance [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Variance [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Charge cumulants as a function of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: High temperature limit of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Variance [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: displays c q 2 , c q 4 and the maximum of the effective velocity of the last charge-carrying particle (the soliton at reflectionless points and the last magnon at reflective points) as functions of the coupling strength, for two different values of α. At reflectionless couplings, charge degrees of freedom always propagate at a maximum velocity 1, whereas for reflective couplings, the speed of charge propag… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Variances [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: High temperature limit of the energy cumulants as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Skewness of the energy-related distributions. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Third cumulant of energy related distributions, [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Third cumulant of momentum related distributions, [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Fourth cumulant of energy related distributions, [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Fourth cumulant of momentum related distributions, [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Fourth cumulant of topological charge related distributions, [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗

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

82 extracted references · 3 canonical work pages

  1. [1]

    At very low temperatures, the source terms dominate, and the convolution terms can be neglected, resulting in free-fermionic filling fractions and root densities

    Reflectionless points in the low temperature limit For integer values of 1/ξ, the scattering of solitons is diagonal and the TBA equations (5) can be written in terms of the massive soliton, antisoliton and breather particles, and there is no need to introduce magnons. At very low temperatures, the source terms dominate, and the convolution terms can be n...

  2. [2]

    Level-one repulsive couplings at low temperatures Forα= 0, at low temperatures and integerξcouplings when there is a single magnonic level, we can compute the variance of the current (see Appendix B 2) cq 2(α= 0) = r 8 mS T π 3/2 e−mS /T tan π 2ξ ,(28) which is compared with the numerical results in Table I and in Fig. 2. 8 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆...

  3. [3]

    In this limit, the root densities become bimodal, concentrated aroundθ≈log(2T /m S) corresponding to left and right movers

    High temperature limit In the high temperature limit, we should recover the behaviour of the UV limit of the theory, which is the free massless boson conformal field theory. In this limit, the root densities become bimodal, concentrated aroundθ≈log(2T /m S) corresponding to left and right movers. Extending the standard derivation employing Roger’s dilogar...

  4. [4]

    Topological charge: fractal structure Due to the topological charge distribution being an even function atµ= 0, the odd cumulants of the topological charge are identically zero. The even cumulants show a fractal-like behaviour as a function of coupling up to someα, 5 In the cited works, the authors studied non-equilibrium steady states after joining two s...

  5. [5]

    These quantities are continuous functions of the coupling in all parameter regimes where we gathered data

    Energy and momentum In this section, we present our numerical results for the second cumulants of energy and momentum. These quantities are continuous functions of the coupling in all parameter regimes where we gathered data. This is in contrast to the cumulants of topological charge, which exhibit fractal behaviour in certain regimes. This also demonstra...

    2024

  6. [6]

    The large deviation approach to statistical mechanics,

    H. Touchette, “The large deviation approach to statistical mechanics,”Physics Reports478(2009) 1–69, arXiv:0804.0327 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2009

  7. [7]

    Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,

    M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,”Rev. Mod. Phys.81(2009) 1665–1702,arXiv:0811.3717 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2009

  8. [8]

    Observations of density fluctuations in an elongated bose gas: Ideal gas and quasicondensate regimes,

    J. Esteve, J.-B. Trebbia, T. Schumm, A. Aspect, C. I. Westbrook, and I. Bouchoule, “Observations of density fluctuations in an elongated bose gas: Ideal gas and quasicondensate regimes,”Phys. Rev. Lett.96(Apr, 2006) 130403. https://link.aps.org/doi/10.1103/PhysRevLett.96.130403

    work page doi:10.1103/physrevlett.96.130403 2006

  9. [9]

    Probing three-body correlations in a quantum gas using the measurement of the third moment of density fluctuations,

    J. Armijo, T. Jacqmin, K. V. Kheruntsyan, and I. Bouchoule, “Probing three-body correlations in a quantum gas using the measurement of the third moment of density fluctuations,”Phys. Rev. Lett.105(Nov, 2010) 230402. https://link.aps.org/doi/10.1103/PhysRevLett.105.230402

    work page doi:10.1103/physrevlett.105.230402 2010

  10. [10]

    Spectroscopy of collective excitations in interacting low-dimensional many-body systems using quench dynamics,

    V. Gritsev, E. Demler, M. Lukin, and A. Polkovnikov, “Spectroscopy of collective excitations in interacting low-dimensional many-body systems using quench dynamics,”Phys. Rev. Lett.99(2007) 200404, arXiv:cond-mat/0702343 [cond-mat.other]

    Pith/arXiv arXiv 2007

  11. [11]

    Experimental characterization of a quantum many-body system via higher-order correlations,

    T. Schweigler, V. Kasper, S. Erne, I. Mazets, B. Rauer, F. Cataldini, T. Langen, T. Gasenzer, J. Berges, and J. Schmiedmayer, “Experimental characterization of a quantum many-body system via higher-order correlations,”Nature 545(2017) 323–326,arXiv:1505.03126 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2017

  12. [12]

    Quantum gas microscopy of kardar-parisi-zhang superdiffusion,

    D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher, “Quantum gas microscopy of kardar-parisi-zhang superdiffusion,”Science376(2022) 716–720, arXiv:2107.00038 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2022

  13. [13]

    Full counting statistics of particle distribution on a digital quantum computer,

    Y.-Z. Fan and D.-B. Zhang, “Full counting statistics of particle distribution on a digital quantum computer,”Phys. Rev. A109(2024) 012412,arXiv:2308.01255 [quant-ph]

    Pith/arXiv arXiv 2024

  14. [14]

    Dynamics of magnetization at infinite temperature in a Heisenberg spin chain,

    E. Rosenberg et al., “Dynamics of magnetization at infinite temperature in a Heisenberg spin chain,”Science384(2024) 48–53,arXiv:2306.09333 [quant-ph]

    Pith/arXiv arXiv 2024

  15. [15]

    Quantum turnstiles for robust measurement of full counting statistics,

    R. Samajdar, E. McCulloch, V. Khemani, R. Vasseur, and S. Gopalakrishnan, “Quantum turnstiles for robust measurement of full counting statistics,”Phys. Rev. Lett.133(2024) 240403,arXiv:2305.15464 [quant-ph]

    Pith/arXiv arXiv 2024

  16. [16]

    Charge distribution in quantum shot noise,

    L. S. Levitov and G. B. Lesovik, “Charge distribution in quantum shot noise,”JETP Lett.58(1993) 230

    1993

  17. [17]

    Non-equilibrium steady states in the Klein–Gordon theory,

    B. Doyon, A. Lucas, K. Schalm, and M. J. Bhaseen, “Non-equilibrium steady states in the Klein–Gordon theory,”J. Phys. A Math. Theor.48(2015) 095002,arXiv:1409.6660 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2015

  18. [18]

    Full counting statistics in the free Dirac theory,

    T. Yoshimura, “Full counting statistics in the free Dirac theory,”J. Phys. A Math. Theor.51(2018) 475002, arXiv:1802.01154 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2018

  19. [19]

    Conformal field theory out of equilibrium: a review,

    D. Bernard and B. Doyon, “Conformal field theory out of equilibrium: a review,”J. Stat. Mech. Theor. Exp.2016(2016) 064005,arXiv:1603.07765 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2016

  20. [20]

    Order parameter statistics in the critical quantum Ising chain,

    A. Lamacraft and P. Fendley, “Order parameter statistics in the critical quantum Ising chain,”Phys. Rev. Lett.100 (2008) ,arXiv:0802.1246 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2008

  21. [21]

    Full counting statistics in the Haldane-Shastry chain,

    J.-M. St´ ephan and F. Pollmann, “Full counting statistics in the Haldane-Shastry chain,”Phys. Rev. B95(2017) 035119, arXiv:1608.06856 [cond-mat.str-el]

    Pith/arXiv arXiv 2017

  22. [22]

    Full counting statistics in the spin-1/2 Heisenberg XXZ chain,

    M. Collura, F. H. L. Essler, and S. Groha, “Full counting statistics in the spin-1/2 Heisenberg XXZ chain,”J. Phys. A Math. Theor.50(2017) 414002,arXiv:1706.07939 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2017

  23. [23]

    Full counting statistics in the transverse field ising chain,

    S. Groha, F. Essler, and P. Calabrese, “Full counting statistics in the transverse field ising chain,”Scipost Phys.4(2018) 043,arXiv:1803.09755 [cond-mat.stat-mech]. 15

    Pith/arXiv arXiv 2018

  24. [24]

    Exact Local Correlations and Full Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas,

    A. Bastianello, L. Piroli, and P. Calabrese, “Exact Local Correlations and Full Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas,”Phys. Rev. Lett.120(2018) 190601

    2018

  25. [25]

    Full counting statistics and large deviations in a thermal 1d bose gas,

    M. Arzamasovs and D. M. Gangardt, “Full counting statistics and large deviations in a thermal 1d bose gas,”Phys. Rev. Lett.122(Mar, 2019) 120401.https://link.aps.org/doi/10.1103/PhysRevLett.122.120401

    work page doi:10.1103/physrevlett.122.120401 2019

  26. [26]

    Quench action and large deviations: Work statistics in the one-dimensional Bose gas,

    G. Perfetto, L. Piroli, and A. Gambassi, “Quench action and large deviations: Work statistics in the one-dimensional Bose gas,”Phys. Rev. E100(2019) 032114,arXiv:1904.06259 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2019

  27. [27]

    Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain,

    F. Ares, M. A. Rajabpour, and J. Viti, “Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain,”Phys. Rev. E103(2021) 042107,arXiv:2012.14012 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2021

  28. [28]

    Universal anomalous fluctuations in charged single-file systems,

    ˇZ. Krajnik, J. Schmidt, V. Pasquier, T. Prosen, and E. Ilievski, “Universal anomalous fluctuations in charged single-file systems,”Phys. Rev. Research6(2024) 013260,arXiv:2208.01463 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2024

  29. [29]

    Efficient computation of cumulant evolution and full counting statistics: Application to infinite temperature quantum spin chains,

    A. Valli, C. P. Moca, M. A. Werner, M. Kormos, ˇZ. Krajnik, T. Prosen, and G. Zar´ and, “Efficient computation of cumulant evolution and full counting statistics: Application to infinite temperature quantum spin chains,”Phys. Rev. Lett.135(2025) 100401,arXiv:2409.14442 [cond-mat.stat-mech]

    arXiv 2025

  30. [30]

    Full counting statistics of charge in chaotic many-body quantum systems,

    E. McCulloch, J. De Nardis, S. Gopalakrishnan, and R. Vasseur, “Full counting statistics of charge in chaotic many-body quantum systems,”Phys. Rev. Lett.131(2023) 210402,arXiv:2302.01355 [quant-ph]

    Pith/arXiv arXiv 2023

  31. [31]

    Distinct universality classes of diffusive transport from full counting statistics,

    S. Gopalakrishnan, A. Morningstar, R. Vasseur, and V. Khemani, “Distinct universality classes of diffusive transport from full counting statistics,”Phys. Rev. B109(2024) 024417,arXiv:2203.09526 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2024

  32. [32]

    Transport fluctuations in integrable models out of equilibrium,

    J. Myers, J. Bhaseen, R. J. Harris, and B. Doyon, “Transport fluctuations in integrable models out of equilibrium,” SciPost Phys.8(2020) 007,arXiv:1812.02082 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  33. [33]

    Fluctuations in Ballistic Transport from Euler Hydrodynamics,

    B. Doyon and J. Myers, “Fluctuations in Ballistic Transport from Euler Hydrodynamics,”Ann. Henri Poincar´ e21 (2020) 255–302,arXiv:1902.00320

    Pith/arXiv arXiv 2020

  34. [34]

    Giamarchi,Quantum physics in one dimension

    T. Giamarchi,Quantum physics in one dimension. International series of monographs on physics. Clarendon Press, Oxford, 2004

    2004

  35. [35]

    Application of Massive Integrable Quantum Field Theories to Problems in Condensed Matter Physics,

    F. H. L. Essler and R. M. Konik, “Application of Massive Integrable Quantum Field Theories to Problems in Condensed Matter Physics,” inFrom Fields to Strings: Circumnavigating Theoretical Physics: Ian Kogan Memorial Collection (in 3 Vols), M. Shifman et al., ed., pp. 684–830. World Scientific, 2005.arXiv:cond-mat/0412421 [cond-mat.str-el]

    Pith/arXiv arXiv 2005

  36. [36]

    Excitation hierarchy of the quantum sine-gordon spin chain in a strong magnetic field,

    S. A. Zvyagin, A. K. Kolezhuk, J. Krzystek, and R. Feyerherm, “Excitation hierarchy of the quantum sine-gordon spin chain in a strong magnetic field,”Phys. Rev. Lett.93(2004) 027201,arXiv:cond-mat/0403364 [cond-mat.str-el]

    Pith/arXiv arXiv 2004

  37. [37]

    Dynamical Properties of One Dimensional Mott Insulators,

    D. Controzzi, F. H. L. Essler, and A. M. Tsvelik, “Dynamical Properties of One Dimensional Mott Insulators,” inNew Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik, ed., p. 25–46. Springer Netherlands, Dordrecht, 2001.arXiv:cond-mat/0011439 [cond-mat.str-el]

    Pith/arXiv arXiv 2001

  38. [38]

    Linear response theory for a pair of coupled one-dimensional condensates of interacting atoms,

    V. Gritsev, A. Polkovnikov, and E. Demler, “Linear response theory for a pair of coupled one-dimensional condensates of interacting atoms,”Phys. Rev. B75(2007) 174511,arXiv:cond-mat/0701421 [cond-mat.other]

    Pith/arXiv arXiv 2007

  39. [39]

    Cold Atom Simulation of Interacting Relativistic Quantum Field Theories,

    J. I. Cirac, P. Maraner, and J. K. Pachos, “Cold Atom Simulation of Interacting Relativistic Quantum Field Theories,” Phys. Rev. Lett.105(2010) 190403,arXiv:1006.2975 [cond-mat.str-el]

    Pith/arXiv arXiv 2010

  40. [40]

    Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons,

    E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichs¨ ollner, M. Gustavsson, M. Dalmonte, G. Pupillo, and H.-C. N¨ agerl, “Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons,”Nature466(2010) 597–600, arXiv:1004.3168 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2010

  41. [41]

    Preparing and analyzing solitons in the sine-gordon model with quantum gas microscopes,

    E. Wybo, A. Bastianello, M. Aidelsburger, I. Bloch, and M. Knap, “Preparing and analyzing solitons in the sine-gordon model with quantum gas microscopes,”PRX Quantum4(2023) 030308,arXiv:2303.16221 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2023

  42. [42]

    Quantum sine-Gordon dynamics in coupled spin chains,

    E. Wybo, M. Knap, and A. Bastianello, “Quantum sine-Gordon dynamics in coupled spin chains,”Phys. Rev. B106 (2022) 075102,arXiv:2203.09530

    Pith/arXiv arXiv 2022

  43. [43]

    The quantum sine-Gordon model with quantum circuits,

    A. Roy, D. Schuricht, J. Hauschild, F. Pollmann, and H. Saleur, “The quantum sine-Gordon model with quantum circuits,”Nucl. Phys. B968(2021) 115445,arXiv:2007.06874 [quant-ph]

    Pith/arXiv arXiv 2021

  44. [44]

    Extracting the field theory description of a quantum many-body system from experimental data,

    T. V. Zache, T. Schweigler, S. Erne, J. Schmiedmayer, and J. Berges, “Extracting the field theory description of a quantum many-body system from experimental data,”Phys. Rev. X10(2020) 011020,arXiv:1909.12815 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2020

  45. [45]

    Sine-Gordon model from coupled condensates: A generalized hydrodynamics viewpoint,

    A. Bastianello, “Sine-Gordon model from coupled condensates: A generalized hydrodynamics viewpoint,”Phys. Rev. B 109(2024) 035118,arXiv:2310.04493 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2024

  46. [46]

    Thermodynamics and fractal Drude weights in the sine-Gordon model,

    B. C. Nagy, M. Kormos, and G. Tak´ acs, “Thermodynamics and fractal Drude weights in the sine-Gordon model,”Phys. Rev. B108(2023) L241105,arXiv:2305.15474 [cond-mat.str-el]

    Pith/arXiv arXiv 2023

  47. [47]

    Thermodynamic Bethe Ansatz and generalised hydrodynamics in the sine-Gordon model,

    B. C. Nagy, G. Tak´ acs, and M. Kormos, “Thermodynamic Bethe Ansatz and generalised hydrodynamics in the sine-Gordon model,”SciPost Phys.16(2024) 145,arXiv:2312.03909 [cond-mat.str-el]

    Pith/arXiv arXiv 2024

  48. [48]

    Exact Large-Scale Fluctuations of the Phase Field in the Sine-Gordon Model,

    G. Del Vecchio Del Vecchio, M. Kormos, B. Doyon, and A. Bastianello, “Exact Large-Scale Fluctuations of the Phase Field in the Sine-Gordon Model,”Phys. Rev. Lett.131(2023) 263401,arXiv:2305.10495 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2023

  49. [49]

    Exact Thermodynamics and Transport in the Classical Sine-Gordon Model,

    R. Koch and A. Bastianello, “Exact Thermodynamics and Transport in the Classical Sine-Gordon Model,”SciPost Phys. 15(2023) 140,arXiv:2303.16932 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2023

  50. [50]

    Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium,

    O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, “Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium,”Phys. Rev. X6(2016) 041065,arXiv:1605.07331 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2016

  51. [51]

    Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents,

    B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, “Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents,”Phys. Rev. Lett.117(2016) 207201,arXiv:1605.09790 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2016

  52. [52]

    Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence,

    M. Borsi, B. Pozsgay, and L. Pristy´ ak, “Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence,”Phys. Rev. X10(2020) 011054,arXiv:1908.07320 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  53. [53]

    Exact anomalous current fluctuations in a deterministic 16 interacting model,

    ˇZ. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, and T. Prosen, “Exact anomalous current fluctuations in a deterministic 16 interacting model,”Phys. Rev. Lett.128(2022) 160601

    2022

  54. [54]

    The hydrodynamic theory of dynamical correlation functions in the XX chain,

    G. Del Vecchio Del Vecchio and B. Doyon, “The hydrodynamic theory of dynamical correlation functions in the XX chain,”J. Stat. Mech. Theor. Exp.2022(2022) 053102,arXiv:2111.08420 [math-ph]

    Pith/arXiv arXiv 2022

  55. [55]

    Universal Relaxational Dynamics of Gapped One-Dimensional Models in the Quantum Sine-Gordon Universality Class,

    K. Damle and S. Sachdev, “Universal Relaxational Dynamics of Gapped One-Dimensional Models in the Quantum Sine-Gordon Universality Class,”Phys. Rev. Lett.95(2005) 187201,arXiv:cond-mat/0507380 [cond-mat.str-el]

    Pith/arXiv arXiv 2005

  56. [56]

    Finite-temperature dynamics in gapped one-dimensional models in the sine-Gordon family,

    M. Kormos, D. V¨ or¨ os, and G. Zar´ and, “Finite-temperature dynamics in gapped one-dimensional models in the sine-Gordon family,”Phys. Rev. B106(2022) 205151,arXiv:2208.08406 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2022

  57. [57]

    Energy flow in non-equilibrium conformal field theory,

    D. Bernard and B. Doyon, “Energy flow in non-equilibrium conformal field theory,”J. Phys. A Math. Gen.45(2012) 362001,arXiv:1202.0239 [cond-mat.str-el]

    Pith/arXiv arXiv 2012

  58. [58]

    Non-Equilibrium Steady States in Conformal Field Theory,

    D. Bernard and B. Doyon, “Non-Equilibrium Steady States in Conformal Field Theory,”Ann. Henri Poincar´ e(2014) 45, arXiv:1302.3125 [math-ph]

    Pith/arXiv arXiv 2014

  59. [59]

    Dynamical separation of charge and energy transport in one-dimensional Mott insulators,

    F. Møller, B. C. Nagy, M. Kormos, and G. Tak´ acs, “Dynamical separation of charge and energy transport in one-dimensional Mott insulators,”Phys. Rev. B109(2024) L161112,arXiv:2311.16234 [cond-mat.str-el]

    Pith/arXiv arXiv 2024

  60. [60]

    Anomalous charge transport in the sine-Gordon model,

    F. Møller, B. C. Nagy, M. Kormos, and G. Tak´ acs, “Anomalous charge transport in the sine-Gordon model,”Phys. Rev. B111(2025) 115121,arXiv:2411.11473 [cond-mat.stat-mech]

    arXiv 2025

  61. [61]

    Matter-wave interferometry in a double well on an atom chip,

    T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Kr¨ uger, “Matter-wave interferometry in a double well on an atom chip,”Nature Physics1(2005) 57–62, arXiv:quant-ph/0507047 [quant-ph]

    Pith/arXiv arXiv 2005

  62. [62]

    Non-equilibrium coherence dynamics in one-dimensional Bose gases,

    S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,”Nature449(2007) 324–327,arXiv:0706.2259 [cond-mat.other]

    Pith/arXiv arXiv 2007

  63. [63]

    Projective phase measurements in one-dimensional Bose gases,

    Y. D. van Nieuwkerk, J. Schmiedmayer, and F. Essler, “Projective phase measurements in one-dimensional Bose gases,” SciPost Phys.5(2018) 046,arXiv:1806.02626 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2018

  64. [64]

    Transport in the sine-Gordon field theory: From generalized hydrodynamics to semiclassics,

    B. Bertini, L. Piroli, and M. Kormos, “Transport in the sine-Gordon field theory: From generalized hydrodynamics to semiclassics,”Phys. Rev. B100(2019) 035108,arXiv:1904.02696 [cond-mat.stat-mech]. 17 Appendix A: TBA expressions of cumulants In general, the cumulant generating function of a random variableXis defined as F(λ) = log⟨e λX ⟩= X n cn λn n! .(A...

    Pith/arXiv arXiv 2019

  65. [65]

    III, for space-like separations, the FCS is given by the difference of generalised free energies

    Space-like case (x > t, α > π/4) As discussed at the end of Sec. III, for space-like separations, the FCS is given by the difference of generalised free energies. At low temperatures, this yields in the case of the topological charge, F q(λ) = ˜f| ¯µ→¯µ−λ− ˜f| ¯µ= 2(coshλ−1) Z dθ 2π Mcoshθe −Mcoshθ/T = 2 sinh2(λ/2)n ¯µ=0,(B12) so eλQx ≍e −2 sinh2(λ/2)nx ....

  66. [66]

    V ariance of the topological current Let us turn now to the caseα= 0 corresponding to the traditional full counting statistics. The second cumulant is cq 2 = pX j=0 Z dθρj(θ)|veff j (θ)|qdr j (θ)2[1−ϑ j(θ)].(B14) For zero chemical potential, only the last two magnons have non-zero dressed topological charges equal to±ξ. Their filling fractions are constan...

  67. [67]

    For simplicity, we assumeζ >0

    Arbitrary ray (α̸= 0) For any non-zeroαand low enough temperatures, for the magnons|v eff j (θ, λ)|< ζ=x/twill hold for allθandλ. For simplicity, we assumeζ >0. Then their flow equations become ∂λεk =h dr k (θ;λ).(B16) Usingh dr k =−η k∂¯µεk,this implies εk(θ;λ) =ε k(θ; 0)|h→h−ηkλ ,(B17) that is, the chemical potential gets shifted. For simplicity, we foc...

  68. [68]

    •c e 3, see Fig

    Note that we omitc q 3 here, because it is zero due to the symmetry of the topological charge distribution. •c e 3, see Fig. 10. –c e 3 is negative, and the graphs show its absolute value. –Although the plots seem to show some structure forα= 0, the magnitudes show thatc e 3 is zero atα= 0, corresponding to the probability density function of the time-int...

  69. [69]

    × 10-43 α c3 p (c)T= 0.01,ξ= 1 3+1/2 0 π 12 π 6 π 4 π 3 5 π 12 π 2 0.00 0.05 0.10 0.15 α c3 p (d)T= 0.5,ξ= 1/3 0 π 12 π 6 π 4 π 3 5 π 12 π 2 0 10 20 30 40 50 α c3 p (e)T= 2.0,ξ= 1/3 ◆◆◆ ◆◆◆◆◆◆◆◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■       0.0 0.2 0.4 0.6 0.8 1.0

  70. [70]

    × 10-4 0.001 0.002 β2/8π c3 p (f)T= 0.2,α= 0 ◆◆◆ ◆◆◆◆◆◆◆◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■       0.0 0.2 0.4 0.6 0.8 1.0

  71. [71]

    × 10-4 0.001 0.002 β2/8π c3 p (g)T= 0.2,α= π 6 ◆◆◆ ◆◆◆◆◆◆◆◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■       0.0 0.2 0.4 0.6 0.8 1.0

  72. [72]

    × 10-4 0.001 0.002 β2/8π c3 p (h)T= 0.2,α= π 4 ◆◆◆ ◆◆◆◆◆◆◆◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■       0.0 0.2 0.4 0.6 0.8 1.0

  73. [73]

    11: Third cumulant of momentum related distributions,c p 3

    × 10-4 0.001 0.002 β2/8π c3 p (i)T= 0.2,α= π 3 ◆◆◆ ◆◆◆ ◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ ◆◆ ◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ■ ■ ■ ■■■■ ■ ■■ ■■■■■■■■■       0.0 0.2 0.4 0.6 0.8 1.0 46 47 48 49 50 β2/8π c3 p (j)T= 2.0,α= 0 ◆◆◆ ◆◆◆ ◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ ◆◆ ◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ■ ■ ■ ■■■■ ■ ■■ ■■■■■■■■■      ...

  74. [74]

    12: Fourth cumulant of energy related distributions,c e 4

    × 10-40 α c4 e (c)T= 0.01,ξ= 1 3+1/2 0 π 12 π 6 π 4 π 3 5 π 12 π 2 0.30 0.35 0.40 α c4 e (d)T= 0.5,ξ= 1/3 0 π 12 π 6 π 4 π 3 5 π 12 π 2 280 300 320 340 360 380 400 α c4 e (e)T= 2.0,ξ= 1/3 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■ ■ ■■ ■■■ ■■■ ■■■      0.0 0.2 0.4 0.6 0.8 1.0 0.002 0.003 0.0...

  75. [75]

    × 10-43 1.5 × 10-43 α c4 p (c)T= 0.01,ξ= 1 3+1/2 0 π 12 π 6 π 4 π 3 5 π 12 π 2 0.24 0.26 0.28 0.30 0.32 0.34 0.36 α c4 p (d)T= 0.5,ξ= 1/3 0 π 12 π 6 π 4 π 3 5 π 12 π 2 280 300 320 340 360 380 400 α c4 p (e)T= 2.0,ξ= 1/3 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■      0.0 0.2...

  76. [76]

    × 10-4 0.001 0.002 β2/8π c4 p (f)T= 0.2,α= 0 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■      0.0 0.2 0.4 0.6 0.8 1.0

  77. [77]

    × 10-4 0.001 0.002 β2/8π c4 p (g)T= 0.2,α= π 6 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■      0.0 0.2 0.4 0.6 0.8 1.0

  78. [78]

    × 10-4 0.0012 0.0018 0.0024 β2/8π c4 p (h)T= 0.2,α= π 4 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆■ ■ ■ ■■ ■ ■■ ■■ ■■■ ■■■ ■■■      0.0 0.2 0.4 0.6 0.8 1.0 0.001 0.002 0.003 β2/8π c4 p (i)T= 0.2,α= π 3 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ ◆ ◆◆ ◆◆◆ ◆ ◆◆ ◆ ◆ ◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ■ ■ ■ ■■■■ ■ ■■ ■■■■■■■■...

  79. [79]

    × 10-45 α c4 q (c)T= 0.01,ξ= 1 3+1/2 0 π 12 π 6 π 4 π 3 5 π 12 π 2 0.013 0.014 0.015 0.016 0.017 0.018 0.019 α c4 q (d)T= 0.5,ξ= 1/3 0 π 12 π 6 π 4 π 3 5 π 12 π 2 0.000 0.005 0.010 0.015 0.020 0.025 α c4 q (e)T= 2.0,ξ= 1/3 ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ■ ■ ■ ■■■■ ■ ■ ■ ■ ■ ■ ■■■ ■■■ ...

  80. [80]

    × 10-4 0.001 β2/8π c4 q (f)T= 0.2,α= 0 ◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆◆◆ ◆ ◆ ◆◆◆◆■ ■ ■ ■■■ ■ ■ ■■ ■■■ ■ ■■■■■  0.0 0.2 0.4 0.6 0.8 1.0

Showing first 80 references.