{"id":"f2e8c674-e52b-471e-b00b-31cefa4f2aa1","arxiv_id":"2509.06358","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Driven localization transitions in the Aubry-André model from an extended initial state obey finite-time scaling, with new power laws for the inverse participation ratio and excitation energy.","lead":"Starting a quantum system from a spread-out, gapless state, this paper slowly pushes it through a localization transition and finds that the standard finite-time scaling laws still apply. The work extends the Kibble-Zurek mechanism to a previously excluded initial condition and gives testable predictions for cold-atom experiments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed KZM/FTS applicability rests on an untested sufficiency criterion (z'<r); varying the initial-state preparation would determine whether initial-phase excitations contaminate the critical scaling.","rationale":"The paper's main quantitative results—the scaling forms for I and D in Eqs. (7) and (12)—are internally consistent: the small-R and large-R branches match at the crossover R L^r ~ 1, and the exponents (s/(ν), (s-ν)/(rν), z/r, 2r-z) follow from the independently known critical exponents. The numerical collapses in Figs. 3, 5, 6, and 7 are visually good and use zero free parameters, which is genuine evidence. However, the theoretical justification for applying KZM/FTS to a gapless initial state rests on the imported inequality z'<r from Ref. [97]. The reader's weakest assumption identifies exactly this point: the paper does not test the adiabatic-impulse separation or the irrelevance of initial-phase excitations; it only demonstrates consistency of the final data collapse. I agree with this assessment. The most direct way to settle the concern is to vary the initial preparation—specifically the initial distance from the critical point δ and the quasiperiodic phase φ—and check whether the scaling functions remain universal. If they do, the criterion is validated; if they do not, the central claim would need to be weakened or rejected. A secondary concern is that the large-R branch for D, D ∝ R^{z/r}, cannot hold for arbitrarily large R because the single-particle spectrum is bounded; the data should show saturation in the instantaneous-quench limit, and the paper does not discuss this cutoff. But the primary, load-bearing issue is the sufficiency of the z'<r criterion. The reader's CONDITIONAL verdict is appropriate, and my read does not change it.","tokens_in":14999,"tokens_out":14256,"duration_ms":132488,"concrete_test":"For fixed R and L, repeat the driven evolution starting from initial ground states at λ0 = λ_c - δ for δ = 0.5, 1, 2, and 4, with several quasiperiodic phases φ. Rescale the ε=0 values of I and D according to Eqs. (7) and (12), and also test the off-critical ε collapse at fixed R L^r. If the scaling functions f_i, g_i and the extracted exponents are independent of δ (up to a constant shift in the scaling variable), the assumption that initial-phase excitations are irrelevant is supported. If the collapse degrades or exponents shift with δ, the z'<r criterion is insufficient and the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, stated in Sec. III A, is that driven dynamics starting from an extended gapless state satisfies the criterion for KZM/FTS because the extended-phase dynamical exponent z'=2 is less than r=3.37 (Eq. 3). This sufficiency criterion is imported from Ref. [97], which derived it for Dirac-like gapless initial states. The paper does not independently verify that the criterion guarantees the irrelevance of excitations produced in the initial extended phase; it only checks the inequality and then observes data collapse for I and D. The scaling ansatze in Eqs. (7) and (12) are two-branch forms with exponents fixed by the critical point, and the collapse at ε=0 and along ε at fixed R L^r is consistent, but it is a necessary rather than sufficient test. If initial-phase excitations contribute with the same power-law form over the simulated R and L ranges, the collapse could still occur while the universal claim is false. Thus the load-bearing assumption is the unsupported sufficiency of z'<r for the AA extended phase, not the quality of the numerical collapse.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the driven dynamics of the localization-delocalization transition in the one-dimensional Aubry-André (AA) model, with the initial state chosen as a gapless extended state rather than the localized states used in earlier work. The authors linearly ramp the quasiperiodic potential strength across the critical point and compute the inverse participation ratio (I) and the deviation from the instantaneous ground-state energy (D). They claim that the driven dynamics starting from the extended state satisfies the criterion z' < r, where z' is the dynamical exponent of the extended phase and r = z + 1/ν is the combination of critical exponents that controls finite-time scaling (FTS). They propose two-branch scaling forms for I and D, valid in the small- and large-driving-rate regimes, and validate these forms by data collapse in Figs. 3, 5, 6 and 7. The exponents appearing in the scaling functions are not fitted to the data but are combinations of known critical exponents (s = 0.333, ν = 1, z = 2.37, r = 3.37). The paper reports, for the first time, a quadratic dependence D ∝ R² in the small-R regime.","tokens_in":15226,"tokens_out":5739,"duration_ms":53444,"significance":"If the central claim is correct, the paper extends the applicability of Kibble-Zurek theory and finite-time scaling to localization transitions starting from gapless extended states, a regime previously thought to violate the adiabatic-impulse hypothesis. The scaling forms for I and D provide concrete, testable predictions that could be checked in ultracold-atom realizations of the AA model. A notable strength is that the scaling exponents are not free parameters fitted to the collapses; they are derived from independent static critical exponents, which makes the validation nontrivial. Another strength is the systematic numerical verification across system sizes L = 8 to 987 and a wide range of driving rates. The work thus goes beyond a purely empirical scaling analysis and offers an experimentally accessible framework.","major_comments":[{"comment":"The central claim that the driven extended-state dynamics satisfies the criterion for KZM/FTS applicability rests entirely on the imported inequality z' < r from Ref. [97], but the paper does not establish that this criterion is sufficient for the AA model. The criterion was derived for Dirac-like gapless systems (linear dispersion), whereas the AA extended phase has z' = 2, indicating a different low-energy structure. The authors check the inequality and then observe data collapse, but the collapse is a necessary rather than a sufficient test: initial-phase excitations might contribute with the same power-law forms over the simulated ranges and still produce a collapse while the universal critical-region interpretation is wrong. A direct test is needed, for example by varying the starting point λ₀ within the extended phase (changing the initial-state gap) or by comparing with a case where z' > r to see whether the scaling forms break down. Without such a test, the paper has not demonstrated the sufficiency of the criterion for the AA model; it has only shown consistency with an assumed sufficiency.","section":"Sec. III A, Eq. (3)"},{"comment":"The two-branch scaling ansatz for I is introduced as a reconciliation of the small-R and large-R asymptotic behaviors, but it is not derived from the FTS framework. In particular, it is unclear whether the two branches are asymptotic limits of a single homogeneous scaling function or represent genuinely different scaling forms. The paper also does not provide a quantitative crossover condition between the small-R and large-R regimes; the boundary between them is identified by eye in Fig. 3(a). A derivation or at least a more explicit statement of how the crossover scale behaves (e.g., R_c ∝ L^{-r}) would make the scaling claim stronger and more testable.","section":"Sec. III B, Eq. (7)"},{"comment":"The small-R scaling D ∝ R² L^{2r-z} is justified by a parity argument: 'D must remain positive regardless of whether R is positive or negative, so the expansion should contain only even powers of R.' However, the driving protocol always has R > 0 by construction (λ increases linearly with time), so positivity does not force the odd terms to vanish. If the authors intend to invoke adiabatic perturbation theory for a finite system, they should state the explicit condition under which the quadratic term dominates, especially because the initial state is in a gapless phase and the adiabatic theorem does not directly apply. As written, the derivation of this scaling law is not convincing, although the numerical data are consistent with it.","section":"Sec. III C, Eq. (12)"}],"minor_comments":[{"comment":"The caption of Fig. 7 refers to rescaling according to Eq. (7) and Eq. (8), but the data shown are for the dynamic deviation D and should be rescaled according to Eq. (12) and Eq. (13), respectively. This typo should be corrected.","section":"Fig. 7 caption"},{"comment":"The phrase 'the system satisfies the following precondition z' < r' is stated without explanation of the physical mechanism behind the criterion. A reader unfamiliar with Ref. [97] is left with no intuition for why an inequality between two dynamical exponents guarantees that initial-phase excitations are irrelevant. At least a qualitative description of the mechanism would improve the readability.","section":"Sec. III A"},{"comment":"The numerical data are averaged over only 10 samples of the phase φ, and no error bars or standard deviations are shown. Given that the scaling collapses are the central evidence, a quantitative measure of the collapse quality (e.g., residuals or confidence intervals for the power-law fits) would strengthen the claims. At present, the collapses are assessed by eye.","section":"Figs. 3, 5, 6, 7"},{"comment":"The text states that 'for sufficiently large L, Eq. (12) can be simplified to D = R^{z/r} g₃(...)', but Eq. (12) has two branches. It is not explained why the small-R branch becomes negligible in the large-L limit; the crossover between the two branches should be quantified, otherwise the simplification appears to select one branch arbitrarily.","section":"Sec. III C, after Eq. (12)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it does a clean numerical job of extending finite-time scaling (FTS) to the Aubry-André localization transition starting from an extended, gapless initial state — a case the previous KZM studies skipped because they all started from localized states. Second, the headline finding is a new scaling law, D ~ R^2 in the small-R regime, plus two-branch scaling functions for both I and D that differ from the localized-initial-state forms. The central claim is that the dynamics still obeys KZM/FTS because the extended-state dynamical exponent z'=2 is less than r=3.37.\n\nWhat the paper does well: the scaling exponents are not fitted to the dynamics but taken from independently known critical exponents (s=0.333, ν=1, z=2.37), and the data collapses in Figs. 3, 5, 6, and 7 are visually convincing. The authors also test the scaling at ε=0 and along the ε direction, which is more than many FTS papers do. The comparison with localized initial states is useful and the experimental predictions are concrete. The citation pattern is appropriate; the heavy overlap with the group's own prior work on the localized case is natural given the direct extension.\n\nSoft spots, in order of importance. The sufficiency criterion z'<r is imported from Ref. [97], which derived it for Dirac-like gapless systems. The paper just checks the inequality and then observes collapse. That is necessary but not sufficient: if initial-phase excitations happen to contribute with the same power-law form over the simulated L and R ranges, the collapse would not reveal it. The stress-test note is right to flag this as the load-bearing assumption. I would not call it fatal — the single-particle AA model is integrable, and the clean collapse is encouraging — but the authors should test it by varying the initial state's distance from the critical point or its phase, and show that the scaling functions are insensitive. Second, there are no error bars anywhere, and the dynamics are averaged over only 10 phase samples; the static data use 1000 but the driven data do not. Third, the crossover between small-R and large-R regimes is identified by eye, and the derivation of the L^{2r-z} factor in the small-R D scaling is hand-wavy (the parity argument gives R^2, but the L dependence is asserted rather than derived). Minor: Fig. 7's caption says 'Eq. (7)' where it should be 'Eq. (12)', and the text around Eq. (8) mislabels the scaling variable.\n\nWho is this for? People working on KZM/FTS in localization transitions, and experimentalists with cold atoms in quasiperiodic lattices. It is a useful extension, not a breakthrough. I would send it to peer review — the core result is likely correct and the caveats are addressable. I would ask for error bars, a direct check of the initial-state dependence, and a cleaner derivation for the D scaling before accepting.","headline":"A solid, workmanlike FTS extension to extended initial states in the AA model, with a new D~R^2 law; the main caveat is an imported sufficiency criterion that the numerics support but don't prove.","tokens_in":15734,"tokens_out":1789,"would_cite":true,"duration_ms":18015,"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":"Linearly driving the Aubry-André model across its localization transition from a gapless extended state still obeys Kibble-Zurek and finite-time scaling, with newly derived two-regime scaling functions for the inverse participation ratio…","keywords":["Aubry-André model","localization transition","Kibble-Zurek mechanism","finite-time scaling","gapless initial state","inverse participation ratio","driven quantum dynamics","quantum criticality"],"falsifier":"Take the same linear ramp but engineer the extended phase so its dynamical exponent satisfies $z' \\geq r$ (for instance by adding long-range hopping that changes the low-energy dispersion); the criterion predicts the data collapse of $\\mathcal{I}$ and $\\mathcal{D}$ should fail, so a clean collapse in that regime would refute the explanation. Within the Aubry-André model itself, measuring $\\mathcal{D}(\\varepsilon=0)$ at fixed small $R$ and fitting its $L$ and $R$ exponents would also settle the small-$R$ ansatz: any systematic departure from $\\mathcal{D} \\propto R^2 L^{2r-z}$ at the smallest rates would falsify Eq. (12).","tokens_in":14793,"feed_emoji":"⚛️","tokens_out":11153,"duration_ms":94161,"temperature":0.7,"pith_summary":"The paper asks whether the Kibble-Zurek mechanism, whose adiabatic-impulse picture normally assumes a gapped ground state, can still describe a localization phase transition when the ramp begins in a gapless extended state. It argues yes for the Aubry-André model because the extended phase's dynamical exponent $z' = 2$ is smaller than $r = z + 1/\\nu = 3.37$, the criterion imported from the finite-time-scaling literature. On that basis it derives and numerically validates scaling functions for the inverse participation ratio $\\mathcal{I}$ and the deviation from instantaneous ground-state energy $\\mathcal{D}$: each takes one form for slow ramps and another for fast ramps, and both differ from the known results for localized initial states. The payoff is a set of explicit power laws in system size $L$ and driving rate $R$ that cold-atom experiments can test directly.","feed_headline":"Extended-state ramps keep Kibble-Zurek scaling in the AA model","feed_subtitle":"Ramps from extended states give universal collapse and a new R^2 energy-deviation law.","key_machinery":"The load-bearing machinery is the finite-time scaling (FTS) ansatz combined with the applicability criterion $z' < r$, where $r = z + 1/\\nu$. Here $z'$ is the dynamical exponent of the gapless extended phase ($z' = 2$), and $z = 2.37$, $\\nu = 1$ are the critical exponents at the localization transition, so $r = 3.37$; the criterion says that excitations generated while the state is still extended do not dominate the universal critical dynamics. The scaling ansatz treats the driving rate $R$ as a scaling field, asserting that observables at the critical point are homogeneous functions of $R L^r$ and $\\varepsilon R^{-1/(r\\nu)}$, with prefactors that change between the small- and large-$R$ limits. The numerical verification is a sequence of data collapses: rescaled $\\mathcal{I}$ and $\\mathcal{D}$ curves for different $L$ and $R$ fall onto single master curves, which is the evidence that the ansatz, not just the individual power laws, is correct.","core_discovery":"The central claim is that the driven dynamics from an extended initial state across the Aubry-André localization transition is governed by the same universal scaling as KZM, even though the initial state is gapless, provided $z' < r$. For the inverse participation ratio the scaling functions are $\\mathcal{I} = L^{-s/\\nu} f_1(R L^r, \\varepsilon R^{-1/(r\\nu)})$ for small $R$ and $\\mathcal{I} = L^{-1} R^{(s-\\nu)/(r\\nu)} f_2(R L^r, \\varepsilon R^{-1/(r\\nu)})$ for large $R$; for the energy deviation they are $\\mathcal{D} = R^2 L^{2r-z} g_1(R L^r, \\varepsilon R^{-1/(r\\nu)})$ and $\\mathcal{D} = R^{z/r} g_2(R L^r, \\varepsilon R^{-1/(r\\nu)})$. The large-$R$ inverse participation ratio behaves like an extended state ($\\mathcal{I} \\propto L^{-1}$), while the small-$R$ energy deviation grows as $R^2$, a behavior the paper reports for the first time. The paper validates these forms through data collapse over system sizes from $L = 8$ to $987$ and driving rates spanning many decades.","pith_inferences":["A stress test the paper does not perform is to tune the extended phase until $z' \\geq r$; the criterion predicts the data collapse should fail, so a model with long-range hopping would directly probe whether the collapse is caused by the criterion or by the broader applicability of FTS.","The small-$R$ $\\mathcal{D} \\propto R^2$ law looks like the leading even term in a power-series expansion of an observable that must be positive for both signs of $R$; if so, it should appear in other drives of finite-size gapped single-particle states, and its presence alone may not be unique to the gapless-start criterion.","Because the paper works at the single-particle level, an immediate extension is to test the same scaling functions in interacting or non-Hermitian Aubry-André models; deviations from the predicted collapse would show where many-body effects or non-Hermitian terms enter the universal scaling.","Experimentally, the $R^2$ law for $\\mathcal{D}$ could be checked by measuring the overlap with the instantaneous ground state during slow ramps in an ultracold-atom realization, giving a time-resolved test of the scaling ansatz rather than only a final-state measurement."],"forward_implications":["At the critical point $\\varepsilon = 0$, the inverse participation ratio crosses over from the static finite-size form $\\mathcal{I} \\propto L^{-s/\\nu}$ at slow driving to a rate-dominated extended-like form $\\mathcal{I} \\propto L^{-1} R^{(s-\\nu)/(r\\nu)}$ at fast driving, so either regime can be used to extract $s$, $\\nu$, and $r$.","The energy deviation $\\mathcal{D}$ at $\\varepsilon = 0$ obeys $\\mathcal{D} \\propto R^2 L^{2r-z}$ for slow ramps and $\\mathcal{D} \\propto R^{z/r}$ for fast ramps; the $R^2$ law is a new diagnostic for extended-initial-state ramps.","Because the scaling functions hold across $L = 8$ to $987$ and a wide range of $R$, experiments at moderate system sizes can use the collapse to extrapolate to the thermodynamic limit.","The difference between these forms and the localized-initial-state formulas (for example $\\mathcal{I} \\propto R^{s/(r\\nu)}$) means the initial-state character is visible in the final scaling, not washed out by the critical dynamics.","In any model satisfying the criterion $z' < r$, the same FTS-based scaling description should apply, so the present Aubry-André results are a template for other quasiperiodic or disordered localization transitions."],"supporting_citations":[{"why":"Supplies the sufficiency criterion $z' < r$ that licenses KZM and FTS for gapless initial states; the paper's central argument checks this condition for the Aubry-André model.","marker":"[97]"},{"why":"Introduces the finite-time scaling framework with the driving rate as a scaling field, from which the paper builds its scaling ansätze.","marker":"[94]"},{"why":"Provides the single-particle KZM study of the Aubry-André localization transition from localized initial states, the baseline whose scaling forms the paper contrasts with and extends.","marker":"[106]"},{"why":"Gives the previous driven-dynamics study whose localized-initial-state scaling for $\\mathcal{I}$ and $\\mathcal{D}$ is compared against the extended-initial-state results.","marker":"[107]"},{"why":"Fixes the critical exponent $s = 0.333$ for the inverse participation ratio used in the $\\mathcal{I}$ scaling functions.","marker":"[25]"},{"why":"Provides the dynamical critical exponent $z = 2.37$ used to set $r = z + 1/\\nu = 3.37$.","marker":"[112]"}],"fun_headline_variants":["Extended-state ramps hold KZM scaling; new R^2 law appears","For extended starts, ramping across AA transition keeps KZM scaling","Ramping extended states: universal collapse and new R^2 law","Extended initial states obey KZM scaling under AA ramps","New energy-deviation law for ramps from extended AA states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a gapless starting phase with fast-enough low-energy dynamics (dynamical exponent below $r = z + 1/\\nu$) produces excitations that never matter for the universal scaling at the transition; the paper checks only that the final curves collapse consistently with this, without directly measuring whether the adiabatic and impulse stages really separate.","fun_headline_variants_meta":{"raw":{"variants":["Extended-state ramps hold KZM scaling; new R^2 law appears","For extended starts, ramping across AA transition keeps KZM scaling","Ramping extended states: universal collapse and new R^2 law","Extended initial states obey KZM scaling under AA ramps","New energy-deviation law for ramps from extended AA states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001378,"raw_usage":{"total_tokens":5648,"prompt_tokens":1075,"completion_tokens":4573,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":4483}},"tokens_in":691,"tokens_out":4573,"duration_ms":30910,"temperature":1.0,"reasoning_tokens":4483,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:16:42.083641+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the same linear ramp but engineer the extended phase so its dynamical exponent satisfies $z' \\geq r$ (for instance by adding long-range hopping that changes the low-energy dispersion); the criterion predicts the data collapse of $\\mathcal{I}$ and $\\mathcal{D}$ should fail, so a clean collapse in that regime would refute the explanation. Within the Aubry-André model itself, measuring $\\mathcal{D}(\\varepsilon=0)$ at fixed small $R$ and fitting its $L$ and $R$ exponents would also settle the small-$R$ ansatz: any systematic departure from $\\mathcal{D} \\propto R^2 L^{2r-z}$ at the smallest rates would falsify Eq. (12).","supporting_citations":[{"cited_title":"Zeng, Y.-K","cited_arxiv_id":null,"evidence_quote":"Supplies the sufficiency criterion $z' < r$ that licenses KZM and FTS for gapless initial states; the paper's central argument checks this condition for the Aubry-André model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the finite-time scaling framework with the driving rate as a scaling field, from which the paper builds its scaling ansätze."},{"cited_title":"Wei, Fidelity susceptibility in one-dimensional disordered lattice models, Phys","cited_arxiv_id":null,"evidence_quote":"Provides the dynamical critical exponent $z = 2.37$ used to set $r = z + 1/\\nu = 3.37$."}],"review_version":2}