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REVIEW 4 major objections 5 minor 85 references

Two-Parameter Ansatz for the Violation of Eigenstate Thermalization

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-parameter ansatz for the violation of eigenstate thermalization places disordered and Stark J1-J2 spin chains in the trapped-ergodicity class, where the apparent finite-size breakdown of thermalization disappears at large L.

desk verdict Useful two-parameter framework for ETH violation, but the 'trapped ergodicity' classification in the J1-J2 chains is not yet established because the fitted theta is not zero and no error bars are reported. read the letter →

arxiv 2608.02744 v1 pith:WJQ3YGCK submitted 2026-08-03 cond-mat.stat-mech cond-mat.dis-nnquant-ph

Kohei Ohgane , Lev Vidmar This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nnquant-ph
keywords eigenstatethermalizationhypothesistrappedergodicityfadingergodizationlengthdisorderedJ1-J2spinchainStarkmany-bodylocalizationfinite-sizescalingquantumchaos
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

The paper introduces a two-parameter ansatz for how the eigenstate thermalization hypothesis (ETH) fails as a Hamiltonian parameter moves away from the ergodic regime, and argues that the failure comes in two distinct flavors. In fading ergodicity the ergodization length $\xi(h,L)$ grows with system size and a true ergodicity-breaking phase transition exists; in trapped ergodicity $\xi(h,L)=\lambda(h)$ is independent of $L$, so the variance of eigenstate-to-eigenstate fluctuations decays like the inverse Hilbert-space dimension once $L$ exceeds $\lambda(h)$, and the ETH violation is a finite-size effect that disappears in the thermodynamic limit. The paper presents numerical evidence that the disordered $J_1$-$J_2$ spin-$1/2$ chain and the Stark $J_1$-$J_2$ chain both realize trapped, not fading, ergodicity. If correct, the widely studied disorder-induced ETH breakdown in these chains is a crossover rather than the precursor of a phase transition.

What carries the argument

The load-bearing object is the variance of diagonal matrix elements, $\sigma^2=\rho^{-1}|f_0|^2$, where $\rho\sim e^{L/\eta_0}$ is the many-body density of states and $|f_0|^2$ is the squared ETH envelope at zero frequency. The paper factors $|f_0|^2=|f_0(h,L)|^2 e^{\xi/\eta_0}$ and parametrizes the ergodization length as $\xi(h,L)=\theta(h)L+\lambda(h)$. The ergodization length is the system size above which the conventional ETH scaling $\sigma^2\propto e^{-L/\eta_0}$ becomes visible; $\theta(h)$ controls whether the decay rate changes with $h$ (fading ergodicity) and $\lambda(h)$ controls a horizontal shift of the decay curve (trapped ergodicity). The numerical analysis extracts $\theta$ and $\lambda$ from linear fits of $\ln\sigma^2$ versus $L$ and tests the ansatz by scaling collapse of $\ln\sigma^2$ against $L-\lambda_\infty(h)$.

What would settle it

Compute $\sigma^2(L)$ for a disordered $J_1$-$J_2$ chain at a fixed $h$ inside the trapped regime for $L$ beyond 22. Trapped ergodicity predicts that $\ln\sigma^2$ becomes linear in $L$ with slope $-(1-\theta_0)/\eta_0$ and that the fitted $\lambda(h)$ is stable when the fitting window changes; if instead $\sigma^2$ saturates to an $h$-dependent constant, or if the fitted $\lambda(h)$ shifts systematically with $L$ by more than the fit error, the two-parameter ansatz is falsified. A second decisive check is to measure $|f_0(h,L)|^2$ directly: trapped ergodicity requires it to grow no faster than polynomially, while an exponential growth would invalidate the factorization.

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Extended reading notes

Core claim

The central claim is that the breakdown of the conventional ETH near the boundary of ergodicity can be captured by writing the variance of diagonal matrix elements as $\sigma^2 \propto \exp\{-(L-\xi(h,L))/\eta_0\}$ with $\xi(h,L)=\theta(h)L+\lambda(h)$, where $h$ is the disorder or field strength and $\eta_0=1/\ln 2$. The paper's new scenario, trapped ergodicity, corresponds to $\theta(h)=0$ and $\lambda(h)=\xi(h)$ growing algebraically with $h$ while remaining independent of $L$. In that scenario the conventional ETH scaling $\sigma^2 \propto D^{-1}$ reappears at $L\gg\lambda(h)$, and the apparent complete breakdown at accessible sizes is a finite-size crossover that moves to larger $h$ as $L$ grows; no ergodicity-breaking transition occurs in the thermodynamic limit. The authors show that in both the disordered and Stark $J_1$-$J_2$ chains, fits of $\ln \sigma^2$ versus $L$ give $\theta\approx 0.11$ and $\theta\approx 0.15$, respectively, with $\lambda(h)\approx a_0 h^\mu+a_1$, and the scaled variance collapses when plotted against $L-\lambda_\infty(h)$.

Load-bearing premise

The classification rests on the assumption that all exponential dependence of the fluctuation variance is captured by the single length scale $\xi$, that this length scale can be written as $\theta L+\lambda$ with $\theta$ and $\lambda$ independent of $L$, and that the remaining prefactor grows at most polynomially in $L$. If that prefactor carries its own exponential size dependence, or if $\theta$ and $\lambda$ drift with $L$, the trapped-ergodicity interpretation is imposed by the fitting form rather than established by the data.

Editorial extensions

If this is right

  • In the disordered $J_1$-$J_2$ model, $\sigma^2(h,L)$ at fixed $h$ decays as $\exp\{-(L-\lambda_\infty(h))/\eta_0\}$ once $L\gtrsim\lambda_\infty(h)$, so data for different $h$ and $L$ collapse onto one curve when plotted against $L-\lambda_\infty(h)$.
  • The extracted $\theta\approx0.11$ (disordered) and $\theta\approx0.15$ (Stark) are close to zero and do not increase with $h$, so these models do not follow the fading-ergodicity route to a thermodynamic-limit breakdown.
  • In trapped ergodicity the conventional ETH scaling $\sigma^2\propto D^{-1}$ is restored for sufficiently large $L$; any finite-size crossing to $\sigma^2=O(1)$ moves to larger $h$ as $L$ grows, meaning no ergodicity-breaking phase transition occurs at finite $h$.
  • The ergodization length grows algebraically, $\xi(h)\propto h^\mu$ with $\mu\approx1.55$ and $\mu\approx1.83$ in the two models, so the crossover field $h^*_L$ diverges with $L$.
  • Short-range spectral statistics remain GOE-like in exactly the $L$-independent trapped regime, so the breakdown of conventional ETH precedes the breakdown of GOE statistics, supporting a generic sequence of breakdowns near the boundary of ergodicity.

Reading between the lines

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

  • One consequence the authors leave implicit: for any fixed finite $L$, the apparent breakdown disorder $h^*_L$ keeps drifting upward with $L$, so experiments or numerics that infer a phase transition from a crossing of finite-size curves can misclassify trapped ergodicity as a genuine transition.
  • The ansatz could be tested independently by computing $|f_0(h,L)|^2$ at $\omega=0$ over a wider range of $L$; the classification stands only if this prefactor is sub-exponential in $L$, and measuring it separately would turn a fitting assumption into a check.
  • The same two-parameter decomposition may apply to other ETH-violation candidates such as fragmented, disorder-free, or Stark-localized models; applying the scaling collapse of $\ln\sigma^2$ against $L-\lambda_\infty(h)$ would show whether their finite-size breakdown shares the $L$-independent ergodization length.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a two-parameter ansatz for the violation of eigenstate thermalization, writing the ergodization length as xi(h,L)=theta(h)L+lambda(h) and the variance of diagonal matrix elements as sigma^2 proportional to exp{-(L-xi)/eta0}. Within this framework, the authors distinguish fading ergodicity (theta increases with the model parameter) from trapped ergodicity (theta=0 and lambda(h) grows with the parameter, so conventional ETH scaling sigma^2 proportional to D^{-1} is restored for sufficiently large L). They apply the ansatz to the disordered and Stark J1-J2 spin-1/2 chains using exact diagonalization for L=12-22, observe horizontal shifts of ln sigma^2 versus L with approximately preserved slopes, and conclude that both models exhibit trapped ergodicity. The central claim is that the finite-size ETH violation in these models is controlled by an L-independent ergodization length and does not signal an ergodicity-breaking phase transition.

Significance. If established, the framework would provide a useful unified language for classifying finite-size ETH violations and would change the interpretation of disordered and Stark spin chains as systems with ergodicity-breaking transitions. The ansatz is simple and yields falsifiable predictions, and the numerical data do show the qualitative hallmark of horizontal shifts with roughly preserved slopes. However, the central classification as trapped ergodicity hinges on the condition theta=0, and the reported fits give theta0=0.11 (disordered) and 0.15 (Stark) with no uncertainties. The significance of the paper therefore rests on a quantitative claim that is currently not supported by the evidence presented.

major comments (4)
  1. [Trapped ergodicity; Eq. (8); Figs. 3(b), 6(b)] The defining condition of trapped ergodicity is theta(h)=0, because Eq. (8) gives sigma^2 proportional to exp{-(1-theta)L/eta0}, and restoration of conventional ETH scaling sigma^2 proportional to D^{-1} requires theta=0. The fits in Figs. 3(b) and 6(b) instead report a constant theta0=0.11 for the disordered model and theta0=0.15 for the Stark model, and the analysis then subtracts theta0 L in defining xi_tilde and ln sigma_tilde^2 in Figs. 3(d,e) and 6(d,e). This subtraction removes from the data exactly the quantity whose absence defines trapped ergodicity. Without error bars on theta(h) or a demonstration that theta tends to zero as L grows, the data are equally consistent with sigma^2 proportional to D^{-0.89} or D^{-0.85}, which are not conventional ETH scalings. The central prediction that conventional ETH reappears in the thermodynamic limit is therefore not established by the present analysis.
  2. [Eqs. (4)-(8) and Appendix A] The ergodization length xi is extracted from the very same sigma^2 data that are later used to demonstrate the scaling collapse: Eq. (6) defines xi = eta0 ln sigma^2 + L, and lambda_infinity(h) is fitted to xi_tilde(h,L) in Figs. 3(d) and 6(d). Consequently, the collapse of ln sigma_tilde^2 versus L-lambda_infinity(h) in Figs. 3(e) and 6(e) is a consistency check of the assumed functional form rather than an independent test of the two-parameter ansatz. For any data in which ln sigma^2 is approximately linear in L at fixed h, choosing lambda as the intercept will produce a collapse by construction. An independent test would require, for example, predicting lambda(h) from a separate observable or from an independently determined polynomial prefactor in Eq. (4), or demonstrating that the fitted parameters are stable when different L ranges are used.
  3. [Eq. (4)] The factorization |f0|^2 = |f0(h,L)|^2 exp{xi/eta0} assumes that the prefactor |f0(h,L)|^2 depends on L at most polynomially. No microscopic argument is provided that this assumption holds for the disordered or Stark J1-J2 chains. If the prefactor itself carries exponential L-dependence, then the fitted theta and lambda are effective parameters that do not correspond to an ergodization length, and the trapped-versus-fading classification would be an artifact of the assumed scaling form. The authors should at least check the plausibility of the polynomial-prefactor assumption, for instance by examining the residuals of the fits or by estimating |f0|^2 independently from the off-diagonal matrix elements.
  4. [Appendix A and Figs. 3-6] The fitting procedure selects the L range post hoc by dropping the smallest system sizes until the fitting error chi^2_n falls below the threshold chi^2_thr (4e-4 for the disordered model, 2e-2 for the Stark model). With only six system sizes L=12...22, this procedure can bias the extracted slope toward smaller theta when the data have curvature, and it can make a vanishing theta look plausible. No confidence intervals are reported for the fitted a(h) and b(h), and no sensitivity study with respect to chi^2_thr or to the moving-average width M is given. This is particularly concerning for the Stark model, where Fig. 6(e) shows a noticeably less clean collapse than Fig. 3(e).
minor comments (5)
  1. [Trapped ergodicity section, p. 3] There is a typo: 'exact diagonalizaiton' should be 'exact diagonalization'.
  2. [Introduction, p. 1] In the sentence 'We refer to the later as the complete breakdown of the ETH', 'later' should be 'latter'.
  3. [End Matter, Appendix C] In the text describing the Stark model, the notation lambda_infinity(h) and xi_tilde(h,L) is used even though the control parameter is F; using lambda_infinity(F) and xi_tilde(F,L) throughout would avoid confusion.
  4. [End Matter, Fig. 6(e) caption] The caption defines ln sigma_tilde^2 = ln sigma^2 + eta0 L, but the main text for the analogous quantity in Fig. 3(e) defines ln sigma_tilde^2 = ln sigma^2 + theta0 L/eta0. These definitions are inconsistent and the figure 6(e) definition appears dimensionally inconsistent; please clarify the intended subtraction.
  5. [Figs. 3(b) and 6(b)] The y-axis labels in panels (b) are difficult to read (the tick labels appear to run from 0 to 1 while the fitted values are around 0.1-0.15); showing the axis range explicitly and zooming into the relevant interval would make the closeness of theta to zero more transparent.

Circularity Check

3 steps flagged · score 5.0 of 10

Central 'trapped ergodicity' classification is partially circular: the ergodization length is defined from the measured variance, the fitted nonzero slope is subtracted before declaring θ=0, and the scaling collapse re-plots the same linear fit.

  1. fitted input called prediction [Numerical examples, Fig. 3(d); Appendix A, Eq. (10)]
    "A complementary approach to the fitting procedure that yields θ(h) and λ(h), is to directly extract the ergodization length ξ(h, L), assuming equality instead of proportionality in Eq. (6), i.e., to calculate ξ(h, L) = η0 lnσ2 + L."

    Equation (6) is just the parametrization σ² ∝ exp{-(L-ξ)/η0}; inverting it defines ξ as a one-to-one function of the measured lnσ². Hence the later finding that ξ̃(h,L)=ξ(h,L)-θ0L is L-independent is not an independent measurement but a restatement of the linear slope of lnσ² versus L. The 'ergodization length' is the logarithm of the variance relabeled, so any conclusion extracted from its L-dependence is inherited from the same data used to fit the ansatz.

  2. fitted input called prediction [Numerical examples, Fig. 3(e); End Matter Fig. 6(e)]
    "Our analysis of trapped ergodicity identifies the relevant variable for the scaling of variance to be L−λ∞(h). This can be tested numerically by plotting lnσ2 vs L−λ∞(h). As in Fig. 3(d), we minimize finite-size effects by studying the scaled variance ln σ̃² = ln σ² + θ0L/η0."

    The curve λ∞(h)=a0h^μ+a1 is fitted to ξ̃(h,L)=η0 lnσ²+L-θ0L, i.e., to the same σ² data. The plotted quantity lnσ̃²=lnσ²+θ0L/η0 equals -(L-ξ)/η0, so the claimed collapse onto -(L-λ∞)/η0 is precisely the linear fit from which θ0 and λ were extracted. The collapse is therefore a display of the fit residuals, not a test of an independent prediction of Eq. (8).

1 more flagged steps
  1. self definitional [Trapped ergodicity section; Numerical examples, Fig. 3(b)]
    "We note that small finite-size effects can be removed if one also takes into account the small nonzero value of θ(h) = θ0, and studies the scaled ergodization length ˜ξ(h, L) = ξ(h, L) − θ0L."

    Trapped ergodicity is defined by θ(h)=0, but the fits give θ0=0.11 (disordered) and θ0=0.15 (Stark), which are nonzero. The analysis then subtracts this fitted θ0L from the data before judging whether the ergodization length is L-independent. If the true exponent were 1-θ0, the variance would decay as D^{-(1-θ0)}, not D^{-1}; the D^{-1} restoration that defines trapped ergodicity is thus restored by removing the measured slope, so the classification is not independently established.

full rationale

The paper does not rely on a load-bearing self-citation chain: the ansatz ξ=θL+λ is a legitimate two-parameter parametrization, and the framework has some falsifiable content, e.g., the separation between L-dependent and L-independent ergodization lengths and the coincidence with the GOE gap-ratio regime. However, the central numerical evidence for trapped ergodicity is partially circular. The ergodization length is defined by inverting the measured variance, so its L-independence is a restatement of the linear fit rather than a new prediction. The scaling collapse in Figs. 3(e) and 6(e) uses λ∞(h) fitted from the same σ² data, making the collapse a consistency check of the fit. Most importantly, the scenario is defined by θ=0, yet the extracted θ0 is 0.11 and 0.15; subtracting this fitted nonzero value before assessing the θ=0 prediction is a self-definitional move. With no reported error bars on θ, the data are also compatible with a small genuine positive θ, which would correspond to a different scaling. These issues do not make the entire paper vacuous, but the headline claim that conventional ETH is restored at large L in these models is not yet independently supported.

Assumptions & free parameters 7 free parameters · 4 assumptions · 1 invented entities

The framework rests on the Srednicki ansatz, an exponential density-of-states parametrization, and the postulated two-parameter form for xi. The numerical extraction then adds several hand-chosen thresholds. The ledger below lists the fitted numbers and the structural assumptions that the central claim depends on.

free parameters (7)
  • theta(h) (slope parameter) = theta0 ~ 0.11 (disordered), theta0 ~ 0.15 (Stark)
    Extracted from linear fits to ln sigma^2 via Eq (10); classification as trapped ergodicity depends on theta being zero, but fitted values are nonzero.
  • lambda(h) (intercept parameter) = increases with h; fitted lambda_infinity(h)=a0 h^mu + a1
    Extracted per h from linear fits; the growth with h is the trapped-ergodicity signature.
  • lambda_infinity(h) fit coefficients = disordered: a0=1.72, a1=2.24, mu=1.55; Stark: a0=10.58, a1=1.90, mu=1.83
    Fitted to the L-independent branch of the rescaled ergodization length; used to build the scaling variable L-lambda_infinity.
  • eta0 for Stark model = approximately 1.509
    Obtained from a fit to the Hilbert-space dimension D in the Stark sector, whereas the disordered case uses eta0=1/ln2 from the binomial asymptotics.
  • chi^2_thr (fit-quality threshold) = 4e-4 (disordered), 2e-2 (Stark)
    Hand-chosen cutoff that determines which L values enter the linear fits; the values are calibrated to the conventional ETH regime, making the L-range selection post hoc.
  • delta r_tilde_thr (gap-ratio threshold) = e^{-3} ~ 0.05
    Hand-chosen threshold above numerical noise, used to define the GOE regime and hence the filled symbols in Fig 3(d).
  • Moving average width M = 30
    Hand-chosen width in the variance estimator; affects the numerical value of sigma^2 but not the exponential scaling.
assumptions (4)
  • domain assumption Srednicki ansatz for matrix elements, Eq (1)
    The entire framework starts from the Srednicki form of ETH for matrix elements; standard in the field but not derived in this paper.
  • domain assumption Density of states rho = rho_bar(L) exp{L/eta0} with polynomial subleading corrections neglected
    The density of states is assumed to grow exponentially with L with subleading polynomial corrections neglected; used to convert sigma^2 = rho^{-1}|f0|^2 into Eq (6).
  • ad hoc to paper Factorization |f0|^2 = |f0(h,L)|^2 exp{xi/eta0} with polynomial prefactor
    This is a definitional reparametrization in Eq (4); the nontrivial assumption is that all exponential L-dependence of |f0|^2 is contained in xi and that the prefactor is at most polynomial.
  • ad hoc to paper Linear ansatz xi = theta L + lambda with L-independent theta and lambda
    Eq (7) postulates the two-parameter form; the conclusions about trapped versus fading ergodicity follow from this assumed form, not from a microscopic derivation.
invented entities (1)
  • Trapped ergodicity independent evidence
    purpose: A proposed scenario for ETH violation in which sigma^2 decays as D^{-1} for L larger than an L-independent ergodization length xi=lambda(h), so apparent ETH breakdown in finite systems is restored in the thermodynamic limit.
    It yields testable signatures: theta(h)=0, lambda(h) independent of L, and scaling collapse in L-lambda_infinity(h). These can be checked in other models, although the evidence here is from the same data used to fit the parameters.

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Cite this review

Pith. "Pith review of Two-Parameter Ansatz for the Violation of Eigenstate Thermalization." pith.science (2026). https://pith.science/paper/WJQ3YGCK

@misc{pith2026260802744,
  author       = {Pith},
  title        = {Pith review of: Two-Parameter Ansatz for the Violation of Eigenstate Thermalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJQ3YGCK}},
  note         = {Machine review of arXiv:2608.02744}
}
abstract

The eigenstate thermalization hypothesis (ETH) provides the prevailing framework for understanding quantum thermalization and ergodicity in isolated many-body systems. Yet, no general theory describes the continuous onset of ETH violation between the conventional ETH and its complete breakdown. Here, we introduce a two-parameter ansatz for the ETH violation that unifies and distinguishes two mechanisms: fading ergodicity and trapped ergodicity. While fading ergodicity captures the established route to ergodicity breaking, trapped ergodicity describes a distinct scenario in which ETH is violated in finite systems but restored in the thermodynamic limit. We test this framework in the spin-1/2 $J_1$-$J_2$ chains with on-site disorder and linear potential. In both cases, we find that the observed ETH violation is consistent with trapped ergodicity.

Figures

Figures reproduced from arXiv: 2608.02744 by the authors.

Figure 1
Figure 1. FIG. 1. The ETH breakdown in finite systems upon tuning [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Two scenarios for the ETH violation: (a,c) fading er [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Evidence of trapped ergodicity in the disordered [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Fitting error [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Evidence of trapped ergodicity in the Stark [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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