{"id":"1d56963b-5427-4e76-839a-bd7ea6af5ded","arxiv_id":"2608.02744","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Trapped ergodicity, a finite-size scenario for eigenstate thermalization violation, is introduced and argued to describe disordered and Stark J1-J2 spin chains.","lead":"This paper proposes a two-parameter formula that describes how thermalization (ETH) starts to fail in quantum systems as a disorder or field strength is increased. It distinguishes a new scenario, trapped ergodicity, in which the apparent thermalization failure is a finite-size effect that disappears for large systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed trapped ergodicity requires θ=0, but fits yield θ0≈0.11 (disordered) and 0.15 (Stark); without error bars demonstrating θ is consistent with zero, the D^{-1} restoration claim is unsupported.","rationale":"The reader's weakest assumption correctly identifies that the factorization in Eq. (4) and the L-independence of parameters in Eq. (7) are load-bearing. My concern is more specific: even granting the ansatz, the numerical classification as trapped ergodicity requires θ=0, but the fits yield θ0=0.11 and 0.15. The paper treats these as negligible, but the difference between θ=0 and θ=0.11 changes the predicted large-L scaling from D^{-1} to D^{-0.89}, which is the key physical distinction between trapped ergodicity and other scenarios. The scaling collapse in Fig. 3(e) is partly constructed by subtracting the fitted θ0L; this does not independently confirm θ=0. The proposed test—bootstrap confidence intervals for θ and a check against L-range dependence—would settle whether θ is consistent with zero. I agree with the reader's CONDITIONAL verdict: the framework is plausible and the horizontal-shift signature is real, but the central classification is not yet supported. Since this concern reinforces the reader's conditions rather than changing the verdict, I recommend UNCHANGED.","tokens_in":14603,"tokens_out":3801,"duration_ms":35671,"concrete_test":"Perform a weighted least-squares fit of the raw variance data to ln σ² = −(1−θ)L/η0 − λ/η0 using the full L range {12,...,22}, and obtain a confidence interval for θ at fixed h (or F) by bootstrap resampling over disorder realizations. Do this for representative parameters in the claimed trapped regime, e.g., h=1.5 and 2.0 for the disordered chain and F=0.5 and 0.6 for the Stark chain. If the 95% confidence interval for θ excludes 0 at any of these parameters, trapped ergodicity is falsified for that parameter. Additionally, repeat the fit using only L≥18; if the estimated θ does not decrease toward 0 when small-L points are removed, the nonzero θ is a true thermodynamic exponent rather than a finite-size artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the disordered and Stark J1-J2 chains exhibit trapped ergodicity, defined in the section 'Trapped ergodicity' as θ(h)=0, with σ² ∝ D^{-1} at large L. However, the extracted θ(h) in Fig. 3(b) is fitted by a constant θ0=0.11 over h∈[0,2], and θ0=0.15 in the Stark model over F∈[0,0.7]. These values are not zero. The text calls them 'very close to zero' and subtracts θ0L in defining the scaled quantities ξ̃(h,L) and ln σ̃² (Figs. 3(d,e) and 6(d,e)), effectively removing a nonzero slope from the data. If the true scaling exponent is 1−θ0≈0.89, the variance decays as D^{-0.89}, not D^{-1}, so the defining prediction of trapped ergodicity—restoration of conventional ETH at sufficiently large L—does not follow. The distinction between θ=0 and θ=0.11 is central, yet no uncertainties are reported for the fitted parameters; Figs. 3(b) and 6(b) show no error bars, and the linear fits in Appendix A use a post hoc L-range selection with threshold χ²_thr. With only L∈[12,22] and an average over many disorder realizations, the data may be equally consistent with a small but genuine positive θ (which would be fading or an intermediate scenario) or with θ→0 at larger L. Thus the classification as trapped ergodicity is not yet established; the central claim is conditional on θ being exactly zero or vanishing asymptotically.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14985,"tokens_out":5325,"duration_ms":51584,"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":[{"comment":"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.","section":"Trapped ergodicity; Eq. (8); Figs. 3(b), 6(b)"},{"comment":"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.","section":"Eqs. (4)-(8) and Appendix A"},{"comment":"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.","section":"Eq. (4)"},{"comment":"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).","section":"Appendix A and Figs. 3-6"}],"minor_comments":[{"comment":"There is a typo: 'exact diagonalizaiton' should be 'exact diagonalization'.","section":"Trapped ergodicity section, p. 3"},{"comment":"In the sentence 'We refer to the later as the complete breakdown of the ETH', 'later' should be 'latter'.","section":"Introduction, p. 1"},{"comment":"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.","section":"End Matter, Appendix C"},{"comment":"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.","section":"End Matter, Fig. 6(e) caption"},{"comment":"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.","section":"Figs. 3(b) and 6(b)"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written and the two-parameter framework is a potentially useful organizing principle. The main obstacle is quantitative: the fitted theta0 values are not zero and no uncertainties are given. I would be willing to reconsider after the authors provide a statistically honest analysis of theta(h), including error bars and a demonstration that the asymptotic slope is consistent with zero (or a reformulation of trapped ergodicity that does not require theta=0). The current manuscript is a solid draft but the central claim is not yet supported by the data as presented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it packages the variance of diagonal matrix elements into a two-parameter ansatz, xi = theta L + lambda, and uses it to separate two routes away from conventional ETH. The distinction between fading ergodicity (theta growing to 1) and trapped ergodicity (theta = 0, lambda growing) is a clean phenomenology, and the numerical observation of horizontal shifts in ln sigma^2 with roughly preserved slopes is a real and interesting pattern. The idea that apparent MBL-like behavior in disordered and Stark J1-J2 chains might be a finite-size effect is worth taking seriously, and the authors are transparent about the fitting procedure.\n\nThe soft spot is the one the stress-test flags, and it is load-bearing. Trapped ergodicity is defined by theta(h) = 0, but the fits give theta0 = 0.11 (disordered) and 0.15 (Stark) — not zero. The paper calls these \"very close to zero\" and subtracts theta0 L before doing the scaling collapse. That is legitimate as a data-cleaning step, but it means the central claim that conventional D^{-1} scaling is restored at large L is not actually shown. If theta is a genuine small positive constant, the variance still decays to zero but as D^{-(1-theta)}, which is not conventional ETH. With no error bars on the fitted parameters, I cannot tell whether the data are consistent with theta = 0 or with theta = 0.1. That distinction is the whole point of the paper.\n\nThe second issue is circularity, as the reader notes: xi is extracted from the same sigma^2 that is then fitted and collapsed. The scaling collapse is therefore a consistency check, not an independent confirmation. This is acceptable if framed as such, and the paper mostly frames it that way, but the summary overstates the conclusion.\n\nThere is also a minor typo in End Matter: the definition of ln~sigma^2 for the Stark model says \"+ eta0 L\" where the main text uses \"+ theta0 L / eta0.\"\n\nOn balance: the framework is worth publishing, and the numerical signature of horizontal shifts is a valuable observation. But the specific claim of trapped ergodicity in these chains needs a direct test of theta = 0 versus small positive theta, with uncertainties, and a robustness check of the L-range selection. This is addressable, not fatal. I would send it to peer review and ask for those checks.\n\nThe paper deserves a serious referee; it is coherent, honest about the fitting, and the ansatz will likely be cited as a useful organizing framework regardless of what the final theta values turn out to be.","headline":"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.","tokens_in":15555,"tokens_out":2956,"would_cite":true,"duration_ms":31080,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["eigenstate thermalization hypothesis","trapped ergodicity","fading ergodicity","ergodization length","disordered J1-J2 spin chain","Stark many-body localization","finite-size scaling","quantum chaos"],"falsifier":"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.","tokens_in":14311,"feed_emoji":"⚛️","tokens_out":10228,"duration_ms":100974,"temperature":0.7,"pith_summary":"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.","feed_headline":"Thermalization returns at larger sizes in disordered spin chains","feed_subtitle":"The apparent eigenstate-thermalization breakdown is a finite-size crossover, not an ergodicity-breaking transition.","key_machinery":"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 $L-\\lambda_\\infty(h)$.","core_discovery":"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 $L-\\lambda_\\infty(h)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Gives the conventional ETH matrix-element ansatz whose zero-frequency envelope defines the $|f_0|^2$ that the paper parametrizes.","marker":"[41]"},{"why":"Introduces the fading-ergodicity scenario that the two-parameter ansatz generalizes and contrasts with trapped ergodicity.","marker":"[65]"},{"why":"Establishes the variance of diagonal matrix elements as the finite-size diagnostic used to quantify ETH violation.","marker":"[11, 25]"},{"why":"Defines the disordered $J_1$-$J_2$ spin-$1/2$ chain, the main numerical model in which trapped ergodicity is exhibited.","marker":"[77, 78]"},{"why":"Provides the polynomially filtered exact-diagonalization method used to obtain eigenstates up to $L=22$.","marker":"[79, 80]"}],"fun_headline_variants":["ETH violation is a finite-size mirage, not a transition","Trapped ergodicity explains apparent ETH breakdown in spin chains","No ergodicity breaking: thermalization returns at larger sizes","Stark and disorder chains show ETH recovery in thermodynamic limit","Finite-size crossover, not true ETH violation in J1-J2 chains"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["ETH violation is a finite-size mirage, not a transition","Trapped ergodicity explains apparent ETH breakdown in spin chains","No ergodicity breaking: thermalization returns at larger sizes","Stark and disorder chains show ETH recovery in thermodynamic limit","Finite-size crossover, not true ETH violation in J1-J2 chains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000284,"raw_usage":{"total_tokens":1684,"prompt_tokens":965,"completion_tokens":719,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":631}},"tokens_in":581,"tokens_out":719,"duration_ms":7853,"temperature":1.0,"reasoning_tokens":631,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:09:55.496015+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J","cited_arxiv_id":null,"evidence_quote":"Gives the conventional ETH matrix-element ansatz whose zero-frequency envelope defines the $|f_0|^2$ that the paper parametrizes."}],"review_version":1}