{"id":"76e12c67-47cb-4804-981d-1546939ac3f7","arxiv_id":"2504.14661","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"After a quench to a sine-square deformed Hamiltonian, symmetry-resolved entanglement entropy grows as log t, with a subleading charge-dependent correction that breaks equipartition.","lead":"This paper computes how a special kind of entanglement entropy, split into charge sectors, grows after switching a one-dimensional system from a uniform to a spatially deformed (sine-square) Hamiltonian. It finds logarithmic growth at late times and a tiny charge-dependent correction that breaks the usual 'equipartition' of entanglement across sectors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equipartition-breaking term in Eq. (4.33) is O(1/log^2), not O(1/log) as claimed, because b_1(t)+b(t) is a time-independent constant.","rationale":"The paper's main object—the exact time-dependent SREE after an SSD quench—is a plausible and useful result, and the leading log t growth plus leading-order equipartition are not in question. The stress-test concern is narrow: the subleading q-dependent term in Eq. (4.33), which is the advertised signature of equipartition breaking, has a coefficient whose L,t dependence is fixed by (4.24)-(4.25) to be 1/b_1^2, while the text repeatedly says 1/log L. This is a factual inconsistency in the central result, not a matter of taste or consensus. It does not by itself invalidate the computation of (4.33); it changes the claimed order and therefore the quantitative prediction. I do not follow the reader's additional claim that dropped O(alpha^4) terms contribute at the same order: a cumulant treatment shows they shift the variance at relative order c/b^2 and, after the n-derivative, the q^2 entropy coefficient at O(1/log^3), which is subdominant to the retained C/b_1^2 term; the error is internal to the retained term. The numerical section is also thin (only q=1, no stated theta in (5.6)), but the decisive correctable flaw is the scaling statement. Verdict remains CONDITIONAL: accept the construction pending correction of the scaling claim and a q=0,2 benchmark.","tokens_in":15008,"tokens_out":11858,"duration_ms":108333,"concrete_test":"Use the exact lattice free-fermion computation behind Fig. 1 (Eqs. (5.1)-(5.12)) at q=0,1,2 for L=300,600,1200 and several t. Because (4.33) predicts S_A(q,t)-S_A(0,t) = -[(b_1+b)/(2 b_1^2)] q^2 with b_1+b constant, compute the left-hand side and b_1(t) from (4.24). Then fit the q-dependence to A/b_1 + B/b_1^2 and test whether B/(b_1+b) stays constant as t and L vary. If A is required, the paper's 1/log claim is correct; if only B survives, the correction is O(1/log^2) and the text must be corrected. The same data also tests whether q=2 agrees with the formula, which Fig. 1 does not show.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula (4.33) is internally inconsistent with the stated O(1/log L) scaling of the equipartition breaking. Let M(t)=8L^2/pi^2(r(t)^2+r(t)s(t)). Equations (4.24)-(4.25) give b_1(t)=1/(4pi^2) log M(t) - gamma(1) and b(t)=-1/(4pi^2) log M(t) - gamma'(1), so b_1(t)+b(t)=-gamma(1)-gamma'(1), a constant independent of t and L. The q^2 term in (4.33) is therefore C/(2 b_1(t)^2) q^2 with C=-gamma(1)-gamma'(1), i.e. O(1/log^2 L) at fixed t and O(1/log^2 t) at late times, not O(1/log L) as claimed in the abstract, Sec. 4.4, and the conclusion. The advertised strength of the equipartition breaking is off by one power of the logarithm. The O(alpha^4) terms dropped in (4.20) are not the origin of the problem: for q=O(1) they shift the q^2 coefficient at O(1/log^3) after the n->1 derivative, subdominant to C/(2 b_1^2); the error is in the retained term itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the time evolution of symmetry-resolved entanglement entropy (SREE) after a quench from the uniform Hamiltonian to a Möbius/SSD deformed Hamiltonian in a critical free-fermion chain with open boundaries. The authors use the fluxed twist-field method, bosonization, and a sequence of conformal maps to obtain the time-dependent charged moments (Eq. (4.19)), approximate their α-dependence by a Gaussian (Eqs. (4.22)–(4.23)), and derive a closed-form SREE, Eq. (4.33). They claim logarithmic growth S_A(q,t) ∼ log t at late times, leading-order equipartition, and a subleading q² correction that breaks equipartition at O(1/log L). The predictions are benchmarked against exact lattice numerics in Fig. 1.","tokens_in":15265,"tokens_out":15618,"duration_ms":134134,"significance":"If the derivation holds, the paper would provide one of the first exact CFT treatments of symmetry-resolved entanglement in an inhomogeneous quench, with an analytic formula containing no fitted parameters and an independent exact numerical benchmark. The broad physical picture — leading-order equipartition with subleading symmetry dependence — is plausible and consistent with the free-fermion structure. However, the specific order of the equipartition breaking is misidentified in the text, and once that order is corrected the Gaussian truncation leaves uncontrolled terms of the same order in 1/log L. These issues affect the central quantitative claim and require revision before the paper can be accepted.","major_comments":[{"comment":"The asserted O(1/log L) equipartition-breaking scaling is inconsistent with the displayed formula. From Eqs. (4.24) and (4.25), b_1(t)+b(t) = −γ(1)−γ′(1) is independent of t and L. The q² term in Eq. (4.33) is therefore C/[2 b_1(t)²] with C = −γ(1)−γ′(1), which is O(1/log² L) at fixed t and O(1/log² t) at late times, not O(1/log L) as stated in Sec. 4.4, in the sentence following Eq. (4.33), and in the Conclusion. The numerical comparison in Fig. 1 plots S_A(q,t) versus t and is unlikely to distinguish 1/log t from 1/log² t; the authors should correct the scaling statements and present data or a scaling analysis that actually tests the corrected order.","section":"Sec. 4.4 and discussion after Eq. (4.33)"},{"comment":"Once the q² term is recognized as O(1/log² L), the Gaussian truncation is not sufficient to determine the SREE at that order. The full Υ_n(α) in Eq. (4.18) contains O(α⁴) terms; writing log Z_n(α) = log Z_n(0) − b_n α²/2 + c_n α⁴ + ⋯ with c_n = O(1), the Fourier transform in Eq. (4.28) generates a q⁴ contribution with coefficient c_n/b_n² = O(1/log² L). After the n → 1 derivative this contributes to S_A(q,t) at the same order in 1/log L as the retained q² term of Eq. (4.33), for fixed q = O(1). Thus Eq. (4.33), truncated at q², is not the complete SREE to the order at which equipartition is broken unless the α⁴ and higher terms are shown to be subleading in the appropriate regime, which the manuscript does not do.","section":"Secs. 4.2–4.3, Eqs. (4.20)–(4.23) and (4.33)"}],"minor_comments":[{"comment":"The phrase \"we derive an exact expression for the SREE\" overstates the result: Eq. (4.22) is obtained by expanding Υ_n(α) through order α², and the subsequent SREE formula inherits this approximation.","section":"Abstract"},{"comment":"The error-function arguments appear to be mistyped: the Fourier integral over [−π, π] gives arguments (b_n π ± i q)/√(2 b_n), not b_n π ± i q√(2 b_n). The large-b_n asymptotics are unaffected, but the finite-b_n expression should be corrected.","section":"Eq. (4.29)"},{"comment":"There are several typographical errors, including \"Specificly\" (Sec. 4), \"aprears\" (after Eq. (4.33)), \"Guassian\" (after Eq. (4.23)), \"coulumn\" (Sec. 5), and \"finial\" (after Eq. (4.13)).","section":"Throughout"},{"comment":"The statement that q ≥ 2 requires larger L would benefit from a quantitative convergence test or an estimate of the required system size, since the q-dependent corrections are the central object of the paper.","section":"Footnote 6"}],"recommendation":"major_revision","confidential_remarks":"The core conformal-mapping computation appears sound and the work is worth pursuing, but the quantitative claim about the order of equipartition breaking is internally inconsistent with Eq. (4.33), and the Gaussian truncation issue needs to be addressed once the scaling is corrected. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is the first symmetry-resolved entanglement entropy (SREE) computation for the uniform-to-SSD (Möbius) quench in a free fermion CFT. The authors use fluxed twist fields plus the known Möbius conformal map, get a closed-form charged moment, and derive an explicit SREE formula with log t growth and a q^2 correction to equipartition. That is a genuine new result, and the t=0 limit correctly reduces to the known equilibrium charged moments, which is a good sanity check.\n\nThe soft spots are real but contained. The stress-test note is right: the paper claims the equipartition-breaking term is O(1/log L), but from their own Eqs. (4.24)-(4.25), b_1(t)+b(t) is a time- and L-independent constant. So the q^2 term in (4.33) is suppressed as 1/log^2 L (or 1/log^2 t at late times), not 1/log. This appears in the abstract, Sec. 4.4, and the conclusion. The O(alpha^4) terms dropped in (4.20) are not the culprit; for fixed q they contribute at 1/log^3 after the n->1 derivative. So it is an internal inconsistency in the retained expression, not a neglected correction. The formula itself may still be correct; the claimed order of the symmetry-breaking effect is off by one log.\n\nTwo other issues. First, the finite-theta Möbius result is asserted but never shown; the paper jumps straight to the SSD limit after saying the general expression is too complicated to report. That is a gap in derivability, though not fatal. Second, the numerical benchmark only shows q=1. The q^2 term is the whole point, and footnote 6 admits q>=2 needs larger L than they could handle. They also never state which theta their lattice Hamiltonian uses; if they did not actually take the SSD limit in the numerics, the comparison is weaker than claimed.\n\nWho is this for? People working on symmetry-resolved entanglement dynamics and inhomogeneous quenches. The method is a straightforward but useful combination of existing machinery, and the log t growth plus suppressed revivals is consistent with earlier SSD quench results. The central formula (4.33) is probably correct, and the flaws are addressable in revision.\n\nRecommendation: send to peer review. A referee should ask the authors to correct the scaling statement, include the finite-theta expression or a clear derivation, and provide at least some q=0 and q=2 data with a specified lattice Hamiltonian. With those changes it would be a solid paper; as it stands, the advertised main effect is overstated.","headline":"First SREE result for the SSD quench, but the advertised O(1/log) equipartition breaking is actually O(1/log^2), and the numerics never test the q-dependence.","tokens_in":15845,"tokens_out":1762,"would_cite":false,"duration_ms":16558,"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":"The paper derives exact symmetry-resolved entanglement entropy after a uniform-to-SSD quench in a free-fermion CFT: log t growth with a subleading q^2 term that breaks equipartition.","keywords":["symmetry-resolved entanglement entropy","sine-square deformation","inhomogeneous quantum quench","free-fermion CFT","charged moments","equipartition of entanglement","Möbius deformation","conformal field theory"],"falsifier":"Compute the coefficient of $q^2$ in $S_A(q,t)-S_A(t)$ numerically at fixed large $t$ and increasing $L$ (for example $L$ from $10^3$ to $10^5$ with $q=1,2$). Equations (4.24)-(4.25) give $b_1(t)+b(t)=$ constant, so within the paper's Gaussian approximation the coefficient should scale as $1/\\log^2 L$, not $1/\\log L$; measuring the exponent directly separates the two. Independently, evaluating the $\\alpha^4$ term in $\\Upsilon_n(\\alpha)$ and asking whether it shifts the $q^2$ coefficient at that same order would settle whether Eq. (4.33) is the exact asymptotic form or only the Gaussian leading approximation.","tokens_in":14717,"feed_emoji":"⚛️","tokens_out":12046,"duration_ms":100673,"temperature":0.7,"pith_summary":"This paper studies what happens to symmetry-resolved entanglement entropy when a one-dimensional critical free-fermion system, initially in the ground state of the uniform Hamiltonian, is suddenly evolved with the Möbius (or sine-square-deformed, SSD) Hamiltonian. The authors derive a closed-form expression for the charged moments of the reduced density matrix and, from them, the entropy $S_A(q,t)$ in each charge sector. They find that at late times the sector-resolved entropy grows as $\\log t$, matching the total entropy, so equipartition holds at leading order; the first sector-dependent correction is proportional to $q^2$ and breaks equipartition at subleading order. Exact free-fermion numerics agree with the formula.","feed_headline":"Entropy after SSD quench grows as log t; equipartition breaks","feed_subtitle":"Each charge sector's entropy grows without saturating; a q² term breaks equipartition.","key_machinery":"The load-bearing object is the fluxed twist field $T_{n,\\alpha}$, which inserts both the replica permutation and the U(1) phase $e^{i\\alpha Q_A}$; diagonalizing the replica twist matrix and bosonizing turns its correlation function into products of vertex-operator two-point functions. The second ingredient is the Möbius conformal map that rewrites evolution under $H_1(\\theta)$ as a dilatation in the $\\zeta$-plane with scale $\\lambda = e^{2\\pi\\tau/(L\\cosh 2\\theta)}$. In the SSD limit $\\theta\\to\\infty$ these steps produce the Gaussian charged moments of Eq. (4.23), and the $n$-derivative of $b_n(t)$ at $n=1$, denoted $b(t)$, fixes both the $q$-independent subleading terms and the $q^2$ coefficient in Eq. (4.33).","core_discovery":"The central claim is that after the uniform-to-SSD quench, the charged moments $Z_n(\\alpha,t)$ are Gaussian in the flux $\\alpha$, $$Z_n(\\$\\alpha$,t) \\simeq Z_n(0,t) $e^{{-\\alpha^2 b_n(t)/2}}$,$$ with $b_n(t)$ given by a logarithm of the time-dependent geometric factors $r(t),s(t)$ minus the non-universal constant $\\gamma(n)$. Fourier transforming this Gaussian and taking the replica limit $n\\to 1$ yields $$S_A(q,t) = S_A(t) - \\frac{1}{2}\\log(2\\pi b_1(t)) + \\frac{b(t)}{2 b_1(t)} - \\frac{b(t)+b_1(t)}{2 b_1(t)^2} $q^{2}$.$$ The leading term grows as $\\log t$ and is independent of $q$; the $q^2$ term is the first sector-dependent correction, which the paper reports at order $1/\\log L$. The same calculation also explains why revivals are absent: the SSD boundary suppresses quasiparticle reflection, so the entropy never saturates.","pith_inferences":["Because $b_1(t)+b(t)$ is a constant in the paper's formulas, the $q^2$ term in Eq. (4.33) decays as $1/\\log^2 L$ rather than $1/\\log L$; one can check numerically which scaling the exact free-fermion result follows.","The Gaussian form of the charged moments implies the full counting statistics of the charge $Q_A$ in the subsystem is approximately Gaussian with variance $b_1(t)\\sim (4\\pi^2)^{-1}\\log L$, linking symmetry-resolved entropy directly to charge fluctuations.","The same Gaussian machinery should give the symmetry-resolved Rényi entropies for $n\\neq 1$ and charged entanglement negativity after this quench; their $q$-dependence will be controlled by the same $b_n(t)$ and can be derived without new ingredients.","A natural extension is a quench between two different Möbius parameters or a local joining quench governed by the SSD Hamiltonian; the conformal map changes but the calculation structure should survive, with a different non-universal constant."],"forward_implications":["At late times and for large $L$, every charge sector's entanglement entropy grows as $\\log t$ with the same leading coefficient as the total entropy, so there is no saturation and no revival peak.","Equipartition holds at leading order in $1/\\log L$; sector dependence first enters through the $q^2$ term, so observables that mix sectors will see the breaking only at subleading order.","The coefficient of the $q^2$ correction is explicitly computable from $r(t)$, $s(t)$, and the non-universal constants $\\gamma(1)$ and $\\gamma'(1)$, giving a parameter-free prediction for any subsystem length $l$ and time $t$.","The same fluxed twist-field computation works for the whole Möbius family of quenches (finite $\\theta$), not only the SSD limit.","The free-fermion numerical method, evolving the correlation matrix as $C(t)=e^{iht}C(0)e^{-iht}$, applies to arbitrary inhomogeneous post-quench Hamiltonians and can benchmark other protocols."],"supporting_citations":[{"why":"Supplies the uniform-to-nonuniform SSD quench setup and the conformal transformation used to evaluate the time evolution.","marker":"[21]"},{"why":"Provides the Möbius quantization and Möbius coordinate formalism that turns the deformed Hamiltonian into a dilatation generator.","marker":"[16]"},{"why":"Gives the equilibrium charged moments with boundaries used to fix the UV cutoff by matching the t=0 limit.","marker":"[50]"},{"why":"Supports truncating the flux-dependent constant at order alpha squared, i.e. the Gaussian approximation of the charged moments.","marker":"[51]"},{"why":"Provides the non-universal constant through the Fisher-Hartwig conjecture, which enters the total entropy and the n-derivative terms.","marker":"[52]"},{"why":"Serves as the exact free-fermion quench benchmark for symmetry-resolved entanglement and motivates the numerical comparison.","marker":"[37]"},{"why":"Defines equipartition of entanglement, the property whose leading-order validity and subleading breakdown are the main results.","marker":"[53]"},{"why":"Gives the correlation-matrix evolution formula used in the free-fermion numerics for arbitrary post-quench Hamiltonians.","marker":"[54]"}],"fun_headline_variants":["Symmetry-resolved entropy grows as log t after SSD quench","Log t growth for entropy sectors, equipartition broken by q² term","Inhomogeneous quench: entropy per charge grows logarithmically, no revivals","Subleading q² correction breaks entanglement equipartition after quench"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the flux dependence of the non-universal constant $\\Upsilon_n(\\alpha)$ is fully captured by its $\\alpha^2$ term, so the charged moments are exactly Gaussian; if the neglected $\\alpha^4$ corrections contribute to the $q^2$ coefficient at the same order in $1/\\log L$, the predicted equipartition breaking is not a controlled asymptotic result.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry-resolved entropy grows as log t after SSD quench","Log t growth for entropy sectors, equipartition broken by q² term","Inhomogeneous quench: entropy per charge grows logarithmically, no revivals","Subleading q² correction breaks entanglement equipartition after quench"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1493,"prompt_tokens":865,"completion_tokens":628,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":548}},"tokens_in":481,"tokens_out":628,"duration_ms":5682,"temperature":1.0,"reasoning_tokens":548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:46:00.650609+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the coefficient of $q^2$ in $S_A(q,t)-S_A(t)$ numerically at fixed large $t$ and increasing $L$ (for example $L$ from $10^3$ to $10^5$ with $q=1,2$). Equations (4.24)-(4.25) give $b_1(t)+b(t)=$ constant, so within the paper's Gaussian approximation the coefficient should scale as $1/\\log^2 L$, not $1/\\log L$; measuring the exponent directly separates the two. Independently, evaluating the $\\alpha^4$ term in $\\Upsilon_n(\\alpha)$ and asking whether it shifts the $q^2$ coefficient at that same order would settle whether Eq. (4.33) is the exact asymptotic form or only the Gaussian leading approximation.","supporting_citations":[{"cited_title":"Jin and V.E","cited_arxiv_id":null,"evidence_quote":"Provides the non-universal constant through the Fisher-Hartwig conjecture, which enters the total entropy and the n-derivative terms."}],"review_version":1}