Role of Noise on Defect Formation and Correlations in a Long-Range Ising Model Under Adiabatic Driving

Santanu Dhara; Suhas Gangadharaiah

REVIEW 4 major objections 4 minor 58 references

Role of Noise on Defect Formation and Correlations in a Long-Range Ising Model Under Adiabatic Driving

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

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Pith's one-line read Long-range Ising chain under adiabatic driving: noise reverses the effect of interaction range on defect density and shifts the controlling momentum mode from k=π to k=0.

desk verdict Useful subfield contribution to KZ/AKZ physics in a long-range Ising model, but a wrong k=π expansion and a factor-of-two in the Landau–Zener exponent make the quantitative predictions unreliable as written. read the letter →

arxiv 2505.02661 v1 pith:CRX7UW6Q submitted 2025-05-05 cond-mat.other

Santanu Dhara , Suhas Gangadharaiah This is my paper · ORCID

classification cond-mat.other

keywords long-rangeIsingmodelKibble-Zurekmechanismanti-Kibble-ZurekscalingLandau-Zenertransitionwhitenoisespincorrelationsfullcountingstatisticstransverse-fieldchain

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 how the range of a power-law interaction changes the defects and correlations produced when a transverse-field Ising chain is swept through its critical region, with and without noise. It establishes that in the noiseless long-range regime (interaction exponent $1<\alpha<2$) the defect density scales as $n\propto \tau_Q^{-1/2}$, so the Kibble–Zurek exponent is independent of $\alpha$, while the prefactor grows as $\alpha$ decreases. In the presence of fast white noise, the trend reverses: longer-range couplings suppress defect formation, and the defect density follows the anti-Kibble–Zurek form $n\simeq \eta_0^2 \tau_Q R(\alpha)$ for small $\eta_0^2\tau_Q$. The same analysis yields the spatial profiles of fermionic and spin correlators and the full counting statistics of kinks from an exactly solvable fermionic mapping. A sympathetic reader would care because it gives a complete, parameter-free picture of which momentum modes control defect formation in long-range systems and how noise shifts that control from $k=\pi$ to $k=0$.

What carries the argument

The machinery is the Jordan-Wigner mapping of the long-range cluster Ising model with couplings $J_r = 1/(\zeta(\alpha)r^\alpha)$ to a quadratic Kitaev-like fermionic chain whose momentum-space Hamiltonian breaks into independent two-level Landau-Zener problems. The load-bearing object is the antisymmetric pairing function $f_\infty^\alpha(k)=\frac{1}{2i\zeta(\alpha)}[\mathrm{Li}_\alpha(e^{ik})-\mathrm{Li}_\alpha(e^{-ik})]$, whose asymmetric form for $1<\alpha<2$ is responsible for the dominance of the $k=\pi$ and later $k=0$ modes. The final state is the decohered density matrix $\rho^s_k = p_k|0_k\rangle\langle 0_k| +(1-p_k)|k,-k\rangle\langle k,-k|$ built from the Landau-Zener probability $p_k$. For the noisy case, the argument uses the fast-noise approximation in which the noise-averaged equation's oscillatory phase is replaced by $\cos[\bar{\omega}(t)(t-t_1)]$ and the time integral is extended to infinity, turning the noise into a delta-correlated damping term proportional to $\eta_0^2 (f_\infty^\alpha)^2$; this produces the closed-form noisy transition probability $p^{\eta_0}_k = \frac{1}{2}[1+e^{-4\pi\tau_Q\eta_0^2(f_\infty^\alpha)^2}(2p^0_k-1)]$.

What would settle it

Compute the full noise-averaged Landau-Zener dynamics from Eq. (B5) at finite noise correlation time $\gamma$ without the fast-noise replacement $\cos[\bar{\omega}(t)(t-t_1)]$ and without extending the integral limits to infinity. If the noisy transition probability no longer peaks near $k=0$, or the defect density does not decrease as $\alpha$ goes from $1.8$ to $1.25$ at fixed $\eta_0^2\tau_Q$, the central noisy claim is falsified.

Watch

Extended reading notes

Core claim

The central discovery is that the long-range interaction exponent $\alpha$ does not change the universality class of defect production but does determine both the magnitude of defect density and the momentum modes responsible for it. Under a noiseless adiabatic ramp crossing two quantum critical points, defect density obeys $n\propto \tau_Q^{-1/2}$ for every $\alpha>1$, with modes near $k=\pi$ dominating; decreasing $\alpha$ increases $n$. Under delta-correlated white noise, the dominant modes move to $k=0$, the defect density shows anti-Kibble–Zurek behavior (it grows with drive speed and noise strength) and is increasingly suppressed as $\alpha$ decreases, which is the opposite of the noiseless trend. The paper also finds that two-point fermionic correlators in the long-range regime first decay as Gaussians and then are quadratically suppressed with separation, while longitudinal spin correlators decay exponentially; and the kink-number distribution remains approximately Gaussian, with higher cumulants proportional to the mean in the noiseless case and a characteristic crossover in the noisy case.

Load-bearing premise

The load-bearing premise is that the noise is effectively instantaneous (delta-correlated white noise), so the memory time of the noise can be ignored; if the noise has a finite correlation time, the predicted shift of the dominant modes to $k=0$ and the suppression of defect density with decreasing $\alpha$ could change.

Editorial extensions

If this is right

For noiseless adiabatic ramps the Kibble–Zurek exponent stays $1/2$ for every $\alpha>1$, while the defect-density prefactor grows as the interaction range increases.
Under white noise the defect density increases with $\tau_Q$ and $\eta_0^2$ (anti-Kibble–Zurek behavior), but longer-range interactions suppress that increase.
The optimal quench time retains the universal $\tau_Q^{\rm O}\propto \eta_0^{-4/3}$ scaling, with an $\alpha$-dependent shift of where the minimum occurs.
In the long-range regime the two-point fermionic correlator shows Gaussian decay followed by quadratic suppression rather than a power law, for both noisy and noiseless protocols.
The kink-number distribution remains approximately Gaussian; in the slow-drive noisy regime the variance-to-mean ratio is independent of quench time.

Reading between the lines

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

Editorial inference: because the noiseless enhancement is tied to the $k=\pi$ critical point closing more gradually as $\alpha$ decreases, a protocol that crosses only that single critical point should reproduce the enhancement; this would separate the two-critical-point geometry from the long-range effect itself.
Editorial inference: the fast-noise approximation becomes questionable exactly where $\bar{\omega}(t)\to 0$ near the critical point; a finite-$\gamma$ calculation should reveal a crossover from anti-Kibble-Zurek to Kibble-Zurek behavior as the noise slows, and the location of that crossover is testable.
Editorial inference: because the noise couples through $f_\infty^\alpha(k)$, which shrinks with decreasing $\alpha$, the suppression of defects is a direct consequence of the interaction-range dependence of the pairing function; measuring the two-point correlator's Gaussian-to-quadratic crossover in an ion-trap or Rydberg simulator would provide a direct check.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper deserves a serious referee, but not because the numbers are ready to use. The combination of a two-critical-point long-range cluster Ising model, white noise, correlation functions, and kink statistics is genuinely new, and the qualitative pictures—α-independent n∝τ_Q^{−1/2}, anti-Kibble–Zurek behavior under noise, the shift of dominant modes to k=0, and Gaussian kink distributions—look robust. The paper is clearly written and the analytic LZ framework is applied honestly, with the fast-noise approximation stated openly. The shift of the noise-induced dominant mode from k=π to k=0 as α decreases is the most interesting result.

The soft spots are real. First, the stress-test is correct: near k=π the expansion of f∞α(k) has coefficient η(α−1)/ζ(α)=(1−2^{2−α})ζ(α−1)/ζ(α), not the ζ(1−α)/ζ(α) stated in Eq. (A2). Eq. (11) then uses ζ(α−1)/ζ(α), which is still not the right coefficient. This corrupts the prefactor B(α), the correlation length ξ0, and the noisy suppression term in Eq. (17). The τ_Q^{−1/2} scaling survives because the linear-in-δ form is correct, but the α-dependence of prefactors and correlation lengths is off. Second, the Landau–Zener exponent in Eq. (10) is inconsistent with Eq. (5). A standard LZ calculation with the Hamiltonian in Eq. (5) gives exp(−4πτ_Q f^2), not exp(−2πτ_Q f^2). This rescales all rates. Third, the abstract says the two-point fermionic correlator shows “quadratic suppression instead of power-law decay,” but the body explicitly gives an algebraic tail I1∝1/r^{2α−1} at large r. The abstract should be aligned. There is also a minor notation slip in Eq. (9), where p_k is written as the ground-state weight but later used as the excitation probability; the intended meaning is clear from the rest of the text.

These are mechanical errors, not deep conceptual ones. The qualitative claims and the overall structure should survive correction. The paper is for researchers working on quench dynamics in long-range or noisy systems, and it adds useful predictions for correlations and defect statistics beyond the standard short-range results. I would send it to peer review with the expectation that the authors fix the expansion coefficients and the LZ prefactor before publication. For my own work, I would not rely on the quoted numbers until those corrections appear.

Referee Report

4 major / 4 minor

Summary. The manuscript studies a one-dimensional long-range transverse-field Ising chain driven linearly across two quantum critical points, with and without time-dependent noise. Using a Jordan-Wigner mapping and a Landau-Zener description, the authors derive approximate analytic expressions for the defect density, two-point and spin correlation functions, and the full counting statistics of defects as functions of the quench time tau_Q, the interaction decay exponent alpha, and the noise strength. The central noiseless claim is the Kibble-Zurek scaling n proportional to tau_Q^{-1/2} with an alpha-dependent prefactor that increases as alpha decreases; in the noisy case the paper reports anti-Kibble-Zurek behavior with suppression as alpha decreases, Gaussian-to-quadratic crossover in fermionic correlators, exponential decay of spin correlators, and near-Gaussian kink statistics.

Significance. The topic is timely: extending Kibble-Zurek and anti-Kibble-Zurek phenomenology to long-range interacting systems is of active interest, and the paper provides an exactly solvable free-fermion setting with closed-form Landau-Zener and full-counting-statistics formulas. The derivations are analytic and contain no fitted free parameters, which is a genuine strength. However, the quantitative alpha-dependent statements rest on a polylogarithm expansion in Appendix A that is incorrect, and the noisy results inherit that error through the expansion of f_alpha(k). The n proportional to tau_Q^{-1/2} exponent and the Gaussian/exponential forms of the correlations are likely robust, but the alpha-dependent prefactors, correlation lengths, and the reported suppression magnitudes need to be recomputed before the main conclusions can be trusted.

major comments (4)

The small-delta expansion of f_inf_alpha(k) near k=pi is incorrect. Since f_inf_alpha(k) = zeta(alpha)^{-1} sum_r r^{-alpha} sin(rk), writing k=pi-delta gives f_inf_alpha(pi-delta) = [eta(alpha-1)/zeta(alpha)] delta + O(delta^3), with eta(s) = (1 - 2^{1-s}) zeta(s). For alpha=1.5 the correct coefficient is +0.232, whereas Eq. (A2) gives zeta(1-alpha)/zeta(alpha) = -0.080. Moreover, Eq. (11) defines phi(alpha)=pi[zeta(alpha-1)/zeta(alpha)]^2, which is neither the coefficient from Eq. (A2) nor the correct coefficient; the correct phi is pi[eta(alpha-1)/zeta(alpha)]^2 = (1-2^{2-alpha})^2 times the stated value. This error propagates into the prefactor B(alpha) in Eq. (12), the correlation length xi_0 = sqrt(4 phi tau_Q), the noisy suppression term in Eq. (17), and the correlators I_2 and G_eta0 in Eqs. (19) and (22). The tau_Q^{-1/2} scaling is unaffected, but the alpha-dependent magnitudes and correlation lengths reported in the figures and discussion are quantitatively unreliable and must be recomputed.
The expansion of f_inf_alpha(k) around k=0 also has an incorrect linear term. The second term in Eq. (A1) should be zeta(alpha-1)/zeta(alpha) k, not zeta(1-alpha)/zeta(alpha) k. This is not a minor notational difference: zeta(alpha-1) diverges as alpha approaches 2, whereas zeta(1-alpha) does not, and the two coefficients have opposite signs in part of the range 1<alpha<2. Because F(k,alpha) in Eq. (14) and the prefactor R(alpha) in Eq. (15) are constructed from the square of this expansion, the noisy defect density and the anti-Kibble-Zurek magnitude inherit the error. The derivation should be redone with the correct second coefficient, and the comparison shown in Fig. 5 should be regenerated.
The manuscript does not state clearly whether the defect-density figures are computed from the exact Landau-Zener expressions p0_k and p_eta0_k or from the approximate formulas in Eqs. (11)-(17). This distinction matters because the approximate formulas contain the incorrect expansion coefficients. If the figures use exact numerics, the analytic equations should be corrected and compared with the exact curves; if the figures use the approximations, the plotted alpha-dependence may be an artifact. Please specify the numerical source and provide exact-versus-approximation comparisons of n(tau_Q) for at least two values of alpha in the long-range regime.
The noisy Landau-Zener result rests on the white-noise and fast-noise approximations: the phase is replaced by cos[bar(omega)(t)(t-t1)] and the integral limits are extended to infinity. The paper should state the quantitative conditions under which these replacements are valid for the parameter ranges used in Figs. 4-6, and should comment on how finite gamma or non-Markovian corrections could affect the two central noisy claims: the shift of dominant modes toward k=0 and the suppression of defects with decreasing alpha. Without such a discussion, the robustness of the anti-Kibble-Zurek conclusions to finite noise correlation time is not established.

minor comments (4)

The abstract describes the drive as going from a paramagnetic phase with all spins down to one with all spins up; because the protocol crosses two quantum critical points and the final transverse field is large, the spin-state convention and the initial and final field values should be stated explicitly.
The approximation for I1(r) retains only the first two terms of a moment expansion; the text should state the condition under which this truncation is controlled, for example r^gamma/(chi tau_Q) much less than 1, and should note where the expansion begins to fail.
The longitudinal correlation function Cxx(r) is expressed as a Toeplitz determinant with g(r) defined by Eq. (28), but the sign convention is not fixed; the log-scale plots appear to show |Cxx|. Please state explicitly whether the plotted quantity is Cxx or its absolute value.
There are minor typographical issues: 'noise effects the coherence' should read 'affects', and in Eq. (B5) the noise correlation function should be written with explicit angle brackets after averaging over eta(t).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are self-contained analytic calculations, with self-citations used only for context.

full rationale

The derivation chain is self-contained and analytic. The noiseless defect density is obtained by inserting the exact Landau-Zener probability p0_k = exp(-2πτ_Q(f∞_α(k))^2) into n = (1/π)∫dk p0_k; the α-dependence enters only through the momentum-space pairing function f∞_α(k) fixed by the Hamiltonian, and no parameter is fitted to the defect-density result. The noisy LZ probability is derived in Appendix B from the von Neumann equation with the stated white-noise (γ→∞) limit, and the anti-Kibble-Zurek suppression follows by expanding that derived expression; it is not imposed by ansatz or by the cited Ref. [41], although that reference is cited for context. Similarly, the FCS section re-derives the characteristic function rather than importing the cumulant scaling. The self-citations to Refs. [41] and [32] are therefore not load-bearing. The apparent α-dependent coefficient error in Appendix A (Eq. A2) is a quantitative accuracy concern that affects prefactors and correlation lengths, but it does not make any step circular: the τ_Q^{-1/2} exponent and the Gaussian-versus-exponential correlation statements follow from the functional forms, not from the specific coefficient. No self-definitional, fitted-input-as-prediction, or author-imported-uniqueness step was found.

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

No numerical fitting is performed: the prefactors A(alpha), B(alpha), chi(alpha), phi(alpha), and R(alpha) are analytic functions of alpha. The non-trivial assumptions are the free-fermion mapping, the white/fast-noise limit, and the dephased steady-state ansatz.

assumptions (5)

standard math Jordan-Wigner transformation maps the long-range cluster Ising model to a free-fermion Kitaev chain.
Used in Sec. II, Eqs. (2)-(3); exact for alpha > 1 in the thermodynamic limit.
domain assumption The white-noise limit gamma -> infinity and the fast-noise approximation in the noise-averaged von Neumann equation.
Introduced in Sec. II and Appendix B, around Eqs. (B6)-(B10); this controls the noisy LZ probability.
standard math Landau-Zener formula for independent linear two-level crossings.
Used in Sec. III, Eq. (10); the model is reduced to independent momentum modes.
domain assumption Neglect of off-diagonal density-matrix elements in the long-time nonequilibrium steady state.
Sec. III, after Eq. (8); required to write rho_k^s as a dephased mixture of ground and excited states.
standard math Wick's theorem and Toeplitz determinant representation for spin correlators.
Used in Sec. IV, Eqs. (26)-(28), and Appendix C; standard for free-fermionic post-quench states.

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Cite this review

Pith. "Pith review of Role of Noise on Defect Formation and Correlations in a Long-Range Ising Model Under Adiabatic Driving." pith.science (2026). https://pith.science/paper/CRX7UW6Q

@misc{pith2026250502661,
  author       = {Pith},
  title        = {Pith review of: Role of Noise on Defect Formation and Correlations in a Long-Range Ising Model Under Adiabatic Driving},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRX7UW6Q}},
  note         = {Machine review of arXiv:2505.02661}
}

abstract

We study an exactly solvable long-range (LR) transverse-field Ising model (TFIM) with a power-law decaying interaction characterized by a decay exponent {\alpha}. In the thermodynamic limit, the system is adiabatically driven in the presence of noise, from a paramagnetic phase with all spins down to one with all spins up. Our study examines the role of long-range interactions on the defect density, its distribution, and spin correlations, comparing noisy and noiseless scenarios. In the noiseless case, within the long-range regime, the steady-state properties are primarily influenced by modes near the k = {\pi} region. However, in the presence of noise, the dominant contributions shift to the modes near k = 0. This differs from the SR model, where previous studies have shown that modes around k = {\pi}/2 play a significant role under noisy conditions. In the absence of noise, defect density scales as $n\propto \tau_Q^{-1/2}$, implying scaling exponent independent of decay exponent. However, we find that decreasing the value of {\alpha} (i.e., increasing the range) enhances the defect density, whereas in the presence of noise, it is suppressed. In the LR regime, two-point fermionic correlators initially exhibit Gaussian decay, followed by quadratic suppression instead of power-law decay for both noisy and noiseless scenarios. Meanwhile, spin correlators, expressed as a string of fermionic operators, undergo purely exponential decay with no crossover behavior. Furthermore, our analysis of defect formation reveals the influence of LR interaction on the kink-number distribution and its cumulants.

Figures

Figures reproduced from arXiv: 2505.02661 by the authors.

**Figure 2.** FIG. 2. The Landau-Zener transition probability has been [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. The plot shows the defect density as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. The LZ transition probability in the presence of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. The comparison between the exact, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7. The two-point correlation [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 6.** Figure 6: FIG. 6. The interplay of noise and LR exponents in the defect [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: FIG. 8. In the LR regime for [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9. The rescaled longitudinal spin correlation ( [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. The matrix elements of [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11. The influence of LR interactions on cumulants is [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 12.** Figure 12: FIG. 12. The plot presents the kink distribution for different [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

**Figure 13.** Figure 13: FIG. 13. The behavior of the cumulants as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]

**Figure 15.** Figure 15: FIG. 15. The kink distribution has been plotted for the noisy [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]

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The plot exhibits the universal scaling of the optimal quench time

for different α. The plot exhibits the universal scaling of the optimal quench time. c† x are the fermionic creation and annihilation operators. In the final decohered state, only the cross correlators G(n−n′) = ⟨AnBn′⟩ remain nonzero while the corre- lators of the type, ⟨AnAn′⟩, and⟨BnBn′⟩ vanish. Since the Wick’s theorem holds, the spin correlators can ...

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