{"id":"e29d83d9-99c7-4773-af7d-aaed64894228","arxiv_id":"2507.02301","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A review arguing that the quantum Mpemba effect in closed systems is driven by unequal thermalization rates of different symmetry sectors, with entanglement asymmetry and charge variance playing complementary roles.","lead":"This review explains the quantum Mpemba effect, where a system starting further from equilibrium can still reach equilibrium faster, through symmetry restoration in closed many-body quantum systems. It argues that entanglement asymmetry and charge variance reveal different versions of the effect, one tied to symmetric dynamics and one to symmetry-breaking Hamiltonians.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (9)-(10) expansion is unverified and uncontrolled at the t≈2 crossing; the CV-QME mechanism is the main weak point.","rationale":"The reader's weakest assumption already identified the small-time truncation as the central risk, and I agree that this is the most load-bearing unsupported step. My stress-test adds that the coefficient itself is asserted without derivation in Eq. (9), so the concern is not only about the radius of convergence but also about the correctness and completeness of the leading term. If Eq. (9) has an algebraic or boundary-condition error, the sign of the curvature—and hence the concave-versus-convex mechanism—changes. The numerical crossing in Fig. 11(a) may still exist, but the paper's explanation of why the CV-QME emerges only in non-symmetric Hamiltonian dynamics would be unsupported. The rest of the review is a survey with broadly consistent literature citations, and the EA-QME claims are supported by multiple referenced studies. Since the reader's CONDITIONAL verdict already requires the authors to derive or cite Eq. (9) and to verify the truncation, my read does not change the verdict. The concrete test above would settle whether the concern lands: a direct re-derivation plus exact-diagonalization comparison at and beyond the crossing time.","tokens_in":22363,"tokens_out":18969,"duration_ms":208140,"concrete_test":"Independently re-derive Eq. (9) from the Heisenberg equations of motion for H1 in Eq. (6) with γ≠1, stating whether OBC or PBC are used. Then compute exact σ_Q^2(t)/L by exact diagonalization for L=12 and compare the truncation Eq. (10) at t=1,2,3 for θ=0.2π and θ=0.5π. If the exact t=0 curvature disagrees with Eq. (9), or if the crossing time predicted from Eq. (10) differs from the full-dynamics crossing in Fig. 11(a) by more than 10%, the O(t^2)-based mechanism is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central dichotomy in Section V turns on the claim that the charge-variance QME arises only for non-symmetric Hamiltonian dynamics, with the mechanism attributed to the sign of the t^2 coefficient in Eq. (10). That coefficient comes from Eq. (9), which is introduced with 'Without presenting the full derivation' and is not supported by a cited source or a numerical check. Since the crossing in Fig. 11(a) occurs near t≈2 for L=12, the O(t^2) truncation of σ_Q^2(t) must remain valid up to that time; the paper gives no radius-of-convergence estimate and no comparison of Eq. (10) to full exact dynamics. For the displayed parameters (γ=0.7, Δ=0.4), the t^2 term changes σ_Q^2/L by roughly +0.3 for θ=0.2π and −0.18 for θ=0.5π at t=2, so the correction is comparable to the initial values whose ordering is being reversed. If higher-order terms contribute at this scale, the concave-versus-convex explanation does not follow even if the numerical crossing is real. In addition, Eq. (9) is stated for H1 with L=12 without explicitly giving the boundary conditions used in the calculation, so an independent re-derivation is needed to rule out missing boundary or commutator terms. This is the load-bearing weak point: the only new analytic content supporting the CV branch of the central claim is an unverified small-time expansion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reviews the quantum Mpemba effect (QME) in closed quantum many-body systems from the perspective of symmetry breaking and restoration. It introduces entanglement asymmetry (EA) and charge variance (CV) as diagnostics, summarizes their early- and late-time dynamics under random unitary circuits and Hamiltonian evolution, and covers integrable, chaotic, and many-body-localized regimes. The paper's central synthetic claim, stated in Section V, is that the EA version of the QME arises under symmetric evolution and remains robust against weak symmetry-breaking perturbations, whereas the CV version emerges only in non-symmetric Hamiltonian dynamics. To support the CV half of this claim, the paper presents a small-time expansion of the charge variance, Eq. (10), whose t^2 coefficient is argued to control the early-time crossing in Fig. 11(a).","tokens_in":22660,"tokens_out":7742,"duration_ms":82587,"significance":"If the classification in Section V is correct, the paper offers a useful organizing principle: the QME is not a single anomaly but a family of observable-dependent effects tied to the sector structure and the symmetry properties of the dynamics. The review is comprehensive, and the tabular summaries in Tables I and II are helpful for navigating the literature. The paper consolidates several peer-reviewed results and clearly separates the EA and CV diagnostics. However, the only new analytic content, Eqs. (9)-(10), is presented without derivation or numerical validation, and it is load-bearing for the CV half of the central claim. The paper also lacks error estimates on power-law fits and leaves part of Table II unreadable. These issues are fixable and do not undermine the review portions that summarize prior work.","major_comments":[{"comment":"Equation (9) is introduced with the phrase \"Without presenting the full derivation,\" and no citation or numerical verification is provided. This equation is the only analytic support for the claim that the CV-QME is controlled by the sign of the t^2 coefficient in Eq. (10), so it is load-bearing. Please give the full derivation, explicitly stating the boundary conditions used for H1 at L=12 (Eq. (6) distinguishes open and periodic boundary conditions), and validate Eq. (10) against exact dynamics. The crossing in Fig. 11(a) occurs near t=2, and for the displayed parameters the t^2 term changes sigma_Q^2/L by approximately +0.3 for theta=0.2*pi and -0.18 for theta=0.5*pi at t=2, so the O(t^2) truncation must be justified up to the crossing time, e.g., by a radius-of-convergence estimate or by comparing with higher-order terms and the full evolution.","section":"Section IV.B, Eqs. (9)-(10)"},{"comment":"The central dichotomy states that the charge-variance version of the QME \"emerges only in non-symmetric Hamiltonian dynamics.\" However, CV dynamics are not reported for U(1)-symmetric Hamiltonian evolution: Section III.C studies EA only, and the circuit results of Section III.B are not Hamiltonian dynamics. Because Eq. (10) contains a factor (1-gamma), its t^2 term vanishes at gamma=1, but that does not exclude a crossing generated by higher-order terms or by other initial states. Please add explicit exact CV dynamics for a symmetric Hamiltonian (e.g., H1 with gamma=1), or qualify the claim to the specific models and parameter ranges studied.","section":"Section V, first paragraph; Section III.C"}],"minor_comments":[{"comment":"The rows \"EA(late time)\" and \"CV(late time)\" in Table II appear empty in the rendered text, and the arrows mentioned in the caption are missing; please complete the table so the late-time summary is readable.","section":"Table II"},{"comment":"The power-law fits y = a x^b are reported without error bars, fit ranges, or goodness-of-fit measures, and the text immediately notes that the scaling holds only for P_Haar < 0.1; please specify the fitting range and report uncertainties on a and b.","section":"Section III.B, Fig. 4(b)"},{"comment":"Panels (a) and (b) use different system sizes (N=16 and N=8), so the comparison of QME presence and absence couples the symmetry-sector structure with the system size; please state whether the conclusions are robust at matched sizes.","section":"Section III.A, Fig. 3"},{"comment":"The sentence explaining the ordering of the two layers in Eq. (3) is confusing (\"the second (first) bracket represents the operations applied in the first (second) layer\"); please rephrase so the time-ordering of the brick-wall layers is unambiguous.","section":"Section II, Eq. (3)"},{"comment":"The parameters t1 and t2 used for long-time averaging (e.g., t1=2000 and t2=40000 in Fig. 9) are not defined; please specify their meaning and whether they use the same time units as the plotted axes.","section":"Section IV.B"},{"comment":"There are several typographical issues, including \"F erromagnetic\" in Table II and inconsistent rendering of accented characters such as \"R'enyi\"; a careful proofread is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is closer to a review of the authors' own and closely related work (Refs. [53,70,76] supply a substantial fraction of the presented numerical results) than to a standalone original research paper. The only new analytic claim, Eq. (9), is unverified, and the editor may wish to have that derivation checked against the companion paper Ref. [76] before committing to publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a review, not a primary research result, and it knows it. What it does well: it organizes the quantum Mpemba literature around symmetry sectors, lays out the entanglement-asymmetry and charge-variance diagnostics clearly, and gives a clean taxonomy (Tables I and II) of when each version of QME does or does not appear. The only genuinely new analysis is the small-time expansion of charge variance, Eq. (10), and the crossing table; most figures are reprinted from published work, which is normal for a review.\n\nThe surveyed material is presented accurately. The quasi-particle mechanism for integrable systems and the sector-dimension mechanism for chaotic systems are described properly, and the MBL section is a fair summary. The authors are also transparent: they mark which figures are reprinted and they admit Eq. (9) is presented without a full derivation.\n\nThat admission points at the paper's real soft spot. Eq. (9) is load-bearing for the CV-QME story: it gives the t^2 coefficient that is supposed to produce the concave-versus-convex crossover. It is stated without derivation, not cited to a prior source, and the boundary conditions for H1 (open or periodic) are not explicit in the text. More important, Eq. (10) is then used to explain a crossing near t=2 in Fig. 11(a), but the O(t^2) truncation is never checked against the full dynamics at that time. The stress-test numbers suggest the t^2 correction is comparable to the initial values at the crossing, so higher-order terms could easily matter. Without a radius-of-convergence estimate or a direct numerical comparison, the CV-QME mechanism is a plausible guess, not an established explanation.\n\nMinor complaints: the power-law fits in Fig. 4(b) have no error bars, the code and data are not deposited, and the self-citation load is heavy—though that's expected for a review consolidating one group's line of work. The central dichotomy (\"CV-QME only in non-symmetric Hamiltonian dynamics\") is stated as a general finding, but the supporting evidence is a handful of small-system numerics; a little more modesty would be appropriate.\n\nBottom line: as a review, this deserves serious referee time. A referee should ask the authors to either derive or cite Eq. (9), verify the truncation numerically at the crossing, and make supporting data available. Conditional acceptance is the right outcome.","headline":"A solid review of the QME organized around symmetry sectors, but the paper's only new analytic claim—the small-time charge-variance expansion—is unproven and needs to be nailed down.","tokens_in":23188,"tokens_out":3849,"would_cite":true,"duration_ms":39942,"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 quantum Mpemba effect in closed systems is governed by unequal thermalization rates of symmetry sectors, with the entanglement-asymmetry version appearing under symmetric evolution and the charge-variance version only under…","keywords":["quantum Mpemba effect","entanglement asymmetry","charge variance","symmetry restoration","random quantum circuits","many-body localization","nonequilibrium thermalization","U(1) symmetry"],"falsifier":"Compute the exact charge-variance dynamics of a tilted ferromagnetic state under the anisotropic Hamiltonian at gamma=0.7 without truncating at second order, and check whether the curves for theta=0.2 pi and theta=0.5 pi cross near t=2 exactly as the expansion predicts; if the crossing disappears, shifts in time, or higher-order terms dominate before the crossing, the concave-versus-convex mechanism is refuted. A second check is to run the same initial states under the symmetric Hamiltonian gamma=1, where the review predicts no charge-variance QME, so observing a crossing there would falsify the claimed separation between the two versions.","tokens_in":22123,"feed_emoji":"⚛️","tokens_out":5794,"duration_ms":67746,"temperature":0.7,"pith_summary":"This review argues that the quantum Mpemba effect, in which a more symmetry-broken initial state restores symmetry faster than a less broken one, is not a single anomaly but a family of effects controlled by the symmetry-sector structure of the dynamics. Its central claim is that the entanglement-asymmetry version appears under symmetric evolution and survives weak symmetry-breaking perturbations, while the charge-variance version appears only when the Hamiltonian itself breaks the symmetry. The authors attribute both versions to unequal thermalization rates across charge sectors, whose sizes differ and therefore relax at different speeds. The review uses this sector-rate mechanism to organize observations from quenches, random circuits, integrable and chaotic Hamiltonians, and many-body-localized systems into one diagnostic framework.","feed_headline":"Two probes give two quantum Mpemba effects, split by symmetry","feed_subtitle":"Entanglement asymmetry crosses under symmetric dynamics; charge variance only when the Hamiltonian breaks symmetry.","key_machinery":"The central objects are the entanglement asymmetry, defined as the difference between the Rényi entropy of the symmetry-projected reduced density matrix and that of the original reduced density matrix, and the charge variance, defined as the spread of charge over symmetry sectors. The argument is carried by the unequal thermalization rates of different charge sectors: a subsystem sector with larger Hilbert-space dimension equilibrates faster, so the initial overlap with slow sectors controls how quickly symmetry is restored. For the charge-variance version, the load-bearing identity is the second-order small-time expansion with coefficient obtained from the equation of motion, whose sign changes with tilt angle and produces the concave-versus-convex behavior behind the early-time crossing.","core_discovery":"For closed quantum systems quenched from U(1)-asymmetric initial states, the question that decides whether the quantum Mpemba effect occurs is which symmetry sectors the initial state populates and how fast each sector thermalizes. Under U(1)-symmetric random circuits or Hamiltonians, the entanglement asymmetry decays faster for more asymmetric states because those states have larger overlap with high-dimensional sectors that thermalize quickly, while mildly tilted states remain pinned by slow sectors such as the Q_A=0 sector. Under Hamiltonians with anisotropy that breaks U(1), symmetry breaking persists in the steady state, and the charge variance instead develops early-time crossings: weakly tilted ferromagnetic states show concave growth while strongly tilted states show convex decay, an effect traced through the second-order time expansion of the variance. The two probes, entanglement asymmetry and charge variance, thus capture different manifestations of the same underlying symmetry-sector dynamics.","pith_inferences":["This reader's inference: because which probe shows a crossing depends on whether the dynamics preserves the symmetry, reports of the quantum Mpemba effect should always specify the observable and the symmetry-sector structure; apparent contradictions between experiments or numerics may simply reflect different probes.","This reader's inference: the sector-thermalization-rate mechanism suggests a practical tuning knob, namely preparing initial states with controlled overlap onto slow sectors to engineer or suppress crossings, which could be tested directly in the small 12-site Hamiltonian used here.","This reader's inference: the second-order coefficient criterion for charge variance should be testable for other symmetry groups, such as Z2 or SU(2), and other drive protocols such as Floquet evolution, by checking whether the sign of the t^2 coefficient consistently predicts the presence of an early-time crossing."],"forward_implications":["Under U(1)-symmetric random circuits, early-time entanglement-asymmetry dynamics produce crossings for tilted ferromagnetic and tilted domain-wall states but not for tilted antiferromagnetic states.","The entanglement-asymmetry version of the quantum Mpemba effect persists under weak symmetry-breaking perturbations, with a threshold that depends on the initial state: about gamma>=0.8 for ferromagnetic states and gamma>=0.4 for antiferromagnetic states in the studied Hamiltonian.","The charge-variance version appears only under non-symmetric Hamiltonian dynamics, and for tilted ferromagnetic states the early-time crossing persists to late times, implying an odd number of crossings over the full time evolution.","Late-time entanglement asymmetry vanishes under symmetric evolution in chaotic systems and in random circuits, but remains nonzero for integrable systems described by a generalized Gibbs ensemble and for symmetry-breaking Hamiltonians.","In many-body-localized systems the quantum Mpemba effect occurs universally for tilted product states, with the QME timescale growing exponentially with subsystem size and full subsystem symmetry restoration occurring without thermalization."],"supporting_citations":[{"why":"Introduces the entanglement asymmetry metric and reports Hamiltonian-dynamics QME crossings that the review organizes and extends.","marker":"[52]"},{"why":"Shows that the QME in symmetric random circuits arises from different thermalization rates across charge sectors and that Z2-symmetric circuits lack it because their sector dimensions match.","marker":"[53]"},{"why":"Demonstrates lack of symmetry restoration in integrable quenches, supporting the review's claim of nonzero late-time entanglement asymmetry in the generalized Gibbs ensemble.","marker":"[65]"},{"why":"Analyzes dynamical symmetry restoration in the Heisenberg chain and supports the early-time QME mechanism tied to the small ZZ term and gapless regime.","marker":"[67]"},{"why":"Extends the QME to many-body-localized systems and supplies the universal tilted-product-state result with an exponentially growing QME timescale.","marker":"[70]"},{"why":"Introduces the symmetry-breaking protocols for Hamiltonian and random-circuit dynamics on which the review's robustness and non-symmetric-dynamics claims rest.","marker":"[76]"},{"why":"Provides the quasiparticle microscopic origin of the QME in integrable systems, which the review uses to explain state-dependent relaxation speeds.","marker":"[80]"},{"why":"Provides the experimental demonstration that intrinsic quantum fluctuations, rather than environmental decoherence, drive the symmetry-restoration crossings.","marker":"[81]"}],"fun_headline_variants":["Symmetry decides which probe shows the quantum Mpemba effect","Two quantum Mpemba effects, split by symmetry","Quantum Mpemba: symmetry dictates the probe","Break symmetry to see Mpemba in charge variance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explanation of the charge-variance QME assumes that the second-order Taylor expansion of the charge variance remains valid up to the crossing time around t=2, and the review does not establish the expansion's radius of convergence or verify it against the full dynamics at the crossing.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry decides which probe shows the quantum Mpemba effect","Two quantum Mpemba effects, split by symmetry","Quantum Mpemba: symmetry dictates the probe","Break symmetry to see Mpemba in charge variance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001595,"raw_usage":{"total_tokens":6309,"prompt_tokens":847,"completion_tokens":5462,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":5410}},"tokens_in":463,"tokens_out":5462,"duration_ms":37001,"temperature":1.0,"reasoning_tokens":5410,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:33:03.514740+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact charge-variance dynamics of a tilted ferromagnetic state under the anisotropic Hamiltonian at gamma=0.7 without truncating at second order, and check whether the curves for theta=0.2 pi and theta=0.5 pi cross near t=2 exactly as the expansion predicts; if the crossing disappears, shifts in time, or higher-order terms dominate before the crossing, the concave-versus-convex mechanism is refuted. A second check is to run the same initial states under the symmetric Hamiltonian gamma=1, where the review predicts no charge-variance QME, so observing a crossing there would falsify the claimed separation between the two versions.","supporting_citations":[{"cited_title":"Rylands, K","cited_arxiv_id":null,"evidence_quote":"Provides the quasiparticle microscopic origin of the QME in integrable systems, which the review uses to explain state-dependent relaxation speeds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the experimental demonstration that intrinsic quantum fluctuations, rather than environmental decoherence, drive the symmetry-restoration crossings."}],"review_version":1}