{"id":"fd2b098a-1786-48bb-8719-812d7b59d93d","arxiv_id":"2502.09711","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Fidelity susceptibility peaks track the ergodicity-breaking critical point and mobility edge in the quantum sun model, with peak height scaling as the square of the density of states.","lead":"Using exact diagonalization of the quantum sun model, the authors show that the fidelity susceptibility of excited eigenstates peaks at the ergodicity-breaking critical point and at the many-body mobility edge, and that the peak height grows like the square of the density of states. The result offers a sensitive numerical probe for locating ergodicity-breaking transitions in interacting quantum systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The peak-coincidence and η(α) predictions are anchored to the imported critical point αc = 0.734 from the same-group preprint ref. [80]; if that value is biased, the central agreement is not independently established.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing point: the imported αc = 0.734 is the anchor for all quantitative comparisons, and the paper provides no independent verification. I considered two other candidate concerns. (1) The µ-regularization dependence of χav (Appendix A): while it shifts the χav peak position, the unregularized χtyp also peaks near αc and shows ζtyp → 2, so the central claim does not rest solely on the regularized quantity. (2) The linear-in-1/L extrapolations lack error bars and use only L = 9–13: this is a data-quality limitation, but it only matters if αc is uncertain. The imported αc is the root cause because it enters the extrapolated differences and the theoretical η(α) in Eq. (18), so even perfect extrapolations cannot separate a systematic bias in αc from a genuine coincidence. The paper's own admission in Sec. IIA that αc may differ from ᾱc and that this is beyond its scope makes the vulnerability explicit. The proposed test—an independent determination of αc from a different observable—directly settles whether the peak positions and the η(α) collapse are internally consistent or merely tied to a possibly biased reference value. Since this is a condition that can be satisfied by releasing data or performing a standard finite-size scaling of ⟨r⟩, it does not invalidate the paper; it strengthens the need for the conditional verdict already assigned.","tokens_in":21845,"tokens_out":8119,"duration_ms":91488,"concrete_test":"Independently determine αc from the same Hamiltonian realizations using a different probe—e.g., finite-size scaling of the mean gap ratio ⟨r⟩ or the participation entropy of eigenstates—and recompute the differences αpeak(ε) − αc and the collapse using Eq. (18) with this new αc. Alternatively, treat αc as a free parameter in a joint fit of the fidelity susceptibility peaks and the η(α) curve, and check whether the best-fit αc is consistent with 0.734 within error bars. If an independent estimate shifts αc by more than the extrapolation uncertainty (≈0.01), the central claim of peak-critical-point coincidence and the fidelity of Eq. (18) are not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central quantitative claim—that the fidelity susceptibility peak marks the ergodicity-breaking critical point and that χmax ∝ ρ²—is tested against αc = 0.734, which is imported from a data collapse of single-site entanglement entropy in ref. [80] (same group, arXiv:2412.15331). This value appears in every comparison: the vertical lines in Figs. 2(a), 4, and 11, the extrapolated differences in the inset of Fig. 2(a) and Fig. 5, the analytical η(α) in Eq. (18), and the mobility edge αc(ε) in Eq. (B1). The authors themselves state in Sec. IIA that they have no argument that αc equals the analytically predicted ᾱc = 1/√2 and leave the distinction open. Consequently, if the true thermodynamic αc differs from 0.734 by an amount larger than the (unreported) uncertainty of the data collapse, then (i) the differences αpeak − αc would not extrapolate to zero, (ii) the extracted exponent ζ would be compared at the wrong point, and (iii) Eq. (18) would systematically mispredict the fluctuation exponent. The paper's finite-size extrapolations are linear in 1/L for L = 9–13 with no error bars, so they cannot rule out a ~0.01–0.02 shift in αc. The secondary concern about µ-regularization in χav (Appendix A) is real but less threatening because the unregularized χtyp supports the same peak and ρ² scaling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the quantum sun model and reports that the fidelity susceptibility of excited eigenstates exhibits a peak at the interaction strength corresponding to the ergodicity-breaking critical point, and, at fixed energy away from mid-spectrum, at the many-body mobility edge. The authors interpret this within the 'fading ergodicity' framework, where the fluctuation exponent η(α) interpolates between the ETH value η=2 and η→∞ at the critical point. Their central quantitative claims are: (i) the peak positions α_c^typ and α_c^av extrapolate linearly in 1/L to the imported critical point α_c=0.734 (Fig. 2 and inset); (ii) the peak height scales as χ ∝ ρ^2, termed 'maximal chaos' (Figs. 3 and 6); and (iii) the same picture holds for energy-dependent mobility edges, with χ_typ^max(ϵ) ∝ ρ(ϵ)^2 and η(α,ϵ) diverging along the mobility edge (Figs. 4–7). The theoretical anchor is Eq. (14), χ_fading ∝ ρ^{2−2/η}, combined with η(α) from Eq. (18), and its energy-dependent generalization Eq. (22). The numerical tests use exact diagonalization up to L_tot=16, with multiple observables and with a non-ETH-sensitive operator as a control.","tokens_in":22051,"tokens_out":6886,"duration_ms":72868,"significance":"If the claims hold, the paper establishes fidelity susceptibility and adiabatic gauge potential norms as sharp, quantitative probes of ergodicity-breaking transitions and many-body mobility edges, and connects the fading ergodicity scenario to a specific 'maximal chaos' scaling. The work is numerically careful in several respects: it tests three different perturbation operators (Fig. 3), it provides a closed-form prediction Eq. (14) that collapes data in Fig. 2(b), and it extends the analysis to nonzero energy densities with a convincing ρ(ϵ)^2 scaling in Fig. 6. The mobility-edge heat map of η in Fig. 7(a) is visually compelling. However, the significance is conditional on the imported critical point α_c=0.734 and on the parameterization of η(α) and η(α,ϵ): the theoretical curves are not fully parameter-free, and the average-fidelity-susceptibility peak position is regulator dependent. These issues do not by themselves invalidate the main observation, but they need to be addressed for the quantitative claims to be fully convincing.","major_comments":[{"comment":"The central coincidence of the fidelity-susceptibility peak with the critical point is tested against α_c=0.734, which is imported from a same-group preprint (Ref. [80]) and not derived or independently verified in this manuscript. The extrapolations of α_c^typ−α_c and α_c^av−α_c in the inset of Fig. 2(a) and of α_max(ϵ)−α_c(ϵ) in Fig. 5 are linear in 1/L over L=9–13 with no reported uncertainty, so they cannot distinguish a zero difference from a small nonzero shift. Since Eq. (18) and Eq. (B1) also use this α_c, every quantitative comparison is anchored to the same value. I request that the authors either (a) provide an independent determination of α_c within this paper (e.g., from level statistics or entanglement entropy with error bars), or (b) perform a robustness analysis that treats α_c as a free parameter (e.g., over the range 0.72–0.75) and shows that the peak-coincidence and scaling conclusions are unchanged. Without this, the statement that the peak 'coincides' with the critical point is only as strong as the imported value.","section":"Sec. IIA and Sec. IIIA (Figs. 2 and 5)"},{"comment":"The position of the peak of the average fidelity susceptibility χ_av depends on the regularization cutoff μ. Figure 9 shows that for μ>ω_H the peak approaches α_c from the right and for μ<ω_H from the left, and only the choice μ∝ω_H gives a peak near α_c with the weakest L-dependence. Consequently, the claim that 'the' fidelity susceptibility peaks at the critical point is not regulator independent for the χ_av branch; only χ_typ provides such a statement. The abstract and introduction should be phrased accordingly, and the analysis should either focus on χ_typ as the primary quantity or provide a principled criterion for μ∝ω_H beyond the observed weakest L-dependence.","section":"Appendix A (Fig. 9)"},{"comment":"The 'prediction' curves for χ_fading and ζ_fading are not parameter-free: Eq. (18) is obtained by replacing the analytically predicted ᾱ_c=1/√2 with the numerically imported α_c=0.734, and Eq. (22) is further modified by a heuristic rescaling η(α,ϵ)→η(α,ϵ)^ν with fitted prefactor A∈(0.95,1.25) and ν≈1.3. As a result, the good data collapse in Fig. 2(b) and the agreement in Fig. 3 partly reflect that the same α_c is used both to locate the peak and to define the theoretical curve. I ask the authors to state explicitly which parameters are fitted and to show the comparison with α_c treated as an unknown (e.g., by plotting χ/χ_fading for several α_c values) or by deriving ν from a microscopic argument. The current presentation overstates the predictive content of the fading ergodicity ansatz.","section":"Eqs. (18) and (22), Figs. 2(b), 3, and 7(b)"},{"comment":"The energy-dependent generalization assumes that energy dependence enters only through ω_H(ϵ) and not through Γ(ϵ), as stated in Sec. IIIB. Moreover, the extraction of Γ in Appendix C relies on Lorentzian fits that deviate significantly from the coarse-grained matrix elements for ϵ≪0.5 (Fig. 12(a)–(c)), and the integrated-spectral-function method Γ_2 is used instead. Given these ambiguities, the claim that fading ergodicity 'accurately describes' the ETH breakdown at the mobility edge is supported mainly by the rescaled fit η(α,ϵ)^ν rather than by the a priori expression Eq. (22). The authors should quantify the fit quality (e.g., residuals or confidence intervals for ν and A) and show that the qualitative conclusions do not depend on the heuristic rescaling.","section":"Sec. IIIB and Appendix C (Eqs. (21)–(22), Figs. 7 and 12)"}],"minor_comments":[{"comment":"The phrase 'gives rise to to the maximally divergent fidelity susceptibility' contains a duplicated 'to' that appears both in the abstract and in the introduction.","section":"Abstract and Sec. I"},{"comment":"The inset caption reads 'Inset of Fig. 4(b)' but there is only one panel in Fig. 4; this should be 'Inset of Fig. 4'.","section":"Fig. 4 caption and inset"},{"comment":"The text says μ is set proportional to ω_H, while Sec. III.A reports the specific choice μ=√L_tot/D. The connection between this expression and ω_H for the 50% spectral window is only explained in Appendix A; it should be stated at the point of use.","section":"Sec. II.B and Sec. III.A"},{"comment":"The legend entry 'r · 500' is unexplained; it presumably denotes a scaled quantity or a typo, and should be clarified.","section":"Fig. 2(b) legend"},{"comment":"The term 'maximal chaos' is used to describe χ ∝ ρ^2, but the formal upper bound that this saturates is not defined precisely. I suggest stating explicitly that χ_fading ≤ C ρ^2 with a constant C, so that 'maximal' is meaningful.","section":"Sec. II.B (definition of maximal chaos)"},{"comment":"The determination of peak positions via fifth-order polynomial fits is mentioned repeatedly, but no details are given about the α-grid spacing, the number of fit points, or the statistical uncertainty of the fitted peak positions; adding this information would strengthen the extrapolations.","section":"Appendix A (numerical details)"},{"comment":"Reference [45] is an arXiv preprint (Lim et al.); if a published version now exists, it should be cited instead.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main quantitative anchor of the paper is the critical point α_c=0.734, which is imported from a same-group preprint (Ref. [80]) and used in every comparison, including the theoretical curves Eqs. (18) and (B1). If this preprint is not yet accepted or if its uncertainty is not reported, the central claim is not independently established within the present manuscript. I would advise asking the authors to either include the relevant data collapse and its error analysis in the supplementary material or to demonstrate that all conclusions are robust to the uncertainty in α_c. The μ-dependence of the χ_av peak is also worth flagging to the editor as a point that may require a clearer separation between regulator-dependent and regulator-independent claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead the new Świętek-Łydżba-Vidmar on fidelity susceptibility in the quantum sun model. The paper does a clean numerical job: it shows that the typical fidelity susceptibility peaks at the interaction-driven ergodicity-breaking critical point and that the peak tracks the many-body mobility edge away from mid-spectrum. The new content is the high-precision finite-size extrapolation of the peak to αc = 0.734, the energy-resolved peak following the mobility edge, and the clean χmax ∝ ρ² scaling, which they call maximal chaos. The fading ergodicity ansatz gives a simple prediction for χ that collapses the data reasonably on the ergodic side. This is a useful and honest piece of work; the authors flag several limitations themselves.\n\nThe soft spot is the imported critical point. The entire comparison rests on αc = 0.734 taken from the same group's preprint (ref. [80]), and that value also feeds the analytical η(α) in Eq. (18) and the mobility edge formula in Eq. (B1). If the true thermodynamic αc is off by roughly 0.01–0.02, the linear 1/L extrapolations without error bars cannot rule it out, and the peak-coincidence claim would be weaker. This is not a fatal flaw, but it does mean the central quantitative claim is conditional on an external input the paper does not independently verify. The µ regularization in χav is a lesser issue: the peak position does depend on µ, and the chosen µ is partly justified by bringing the peak to αc, but the unregularized χtyp supports the same peak and ρ² scaling, so the central claim does not rest on µ alone. No code or data is provided, which makes it harder to check the extrapolations.\n\nNet: this is a solid numerical demonstration, not a breakthrough. The fading ergodicity description is plausible and fits the data, but the evidence is not fully self-contained. It deserves a serious referee: the referee should ask for data/code, uncertainty estimates on αc and peak positions, and ideally a test against an independently determined critical point. If you work on ETH breakdown or fidelity probes, this is worth citing.\n\nBest,\n[Your name]","headline":"Solid numerical evidence that fidelity susceptibility peaks at the ergodicity-breaking critical point and mobility edge in the quantum sun model, but the central comparison is anchored to an imported αc from the same group's preprint.","tokens_in":22726,"tokens_out":2572,"would_cite":true,"duration_ms":26204,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In the quantum sun model, the fidelity susceptibility peaks exactly at the ergodicity-breaking critical point and grows as the square of the density of states, the signature of maximal chaos.","keywords":["fading ergodicity","fidelity susceptibility","eigenstate thermalization hypothesis","many-body mobility edge","quantum sun model","maximal chaos","ergodicity breaking transition","adiabatic gauge potential"],"falsifier":"At system sizes L = 14, 15, 16 in the quantum sun model, compute χ_typ(ε) versus α at a fixed energy density ε = 0.3 with the same µ = ω_H regularization, and fit a fifth-order polynomial to locate the peak. Maximal chaos predicts the peak heights for all L fall on the line 0.05 ρ(ε)² with a spread comparable to the L ≤ 13 data, and the extrapolated peak position α_max(ε) approaches α_c(ε) = 0.734 exp[(a²(ε−1/2)²)/(4b²)] with a = 1.05, b = 0.45 to within the 1/L trend. Any systematic bending of χ_max versus ρ², or a peak position that misses α_c(ε) beyond the finite-size drift, would falsify the maximal-chaos claim.","tokens_in":21505,"feed_emoji":"⚛️","tokens_out":7204,"duration_ms":69195,"temperature":0.7,"pith_summary":"The paper aims to show that the breakdown of the eigenstate thermalization hypothesis (ETH) in the quantum sun model is accompanied by a fidelity susceptibility that diverges maximally at the critical point: its peak position extrapolates to the ergodicity-breaking critical point αc = 0.734, and its peak value scales as χ ∝ ρ², the square of the density of states. It extends the same logic to finite energy densities, where the peak tracks the many-body mobility edge αc(ε). The authors argue that fading ergodicity—a framework in which the fluctuation exponent η interpolates between the ETH value η = 2 and η → ∞ at the transition—accounts for the full scaling on the ergodic side. If correct, fidelity susceptibility becomes a sharp, quantitative detector of ergodicity breaking and mobility edges in interacting quantum systems.","feed_headline":"Quantum thermalization collapses at a maximal-chaos peak","feed_subtitle":"The fidelity susceptibility's peak marks the ergodicity-breaking point and grows as the square of the density of states.","key_machinery":"The central object is the fidelity susceptibility χ_n = Σ_{m≠n} |V_nm|² / (E_m − E_n)², a measure of eigenstate sensitivity to perturbations; when averaged with the AGP regularization µ ∝ ω_H it becomes the norm of the adiabatic gauge potential. The argument runs through the fading ergodicity ansatz |V_nm|² ∝ $ρ^{{−2/η}}$ with fluctuation exponent η, which combined with µ = ω_H ∝ $ρ^{{−1}}$ yields χ_fading ∝ $ρ^{{2−2/η}}$. At the ETH limit η = 2 this gives χ ∝ ρ; as η → ∞ at the critical point it reaches χ ∝ ρ², the maximal-chaos scaling. The analytical form η(α) = 2(1 − ln α / ln αc)^{−1} and its energy-dependent generalization η(α, ε) are then tested against numerics.","core_discovery":"The central claim is that the breakdown of the conventional ETH at the interaction-driven ergodicity-breaking critical point of the quantum sun model produces a fidelity susceptibility whose peak coincides with the critical point and whose maximal value scales as χ ∝ ρ², saturating the upper bound (dubbed 'maximal chaos'). For the operator of the most distant spin, the typical susceptibility χ_typ and the regularized AGP norm χ_av both peak at values that extrapolate linearly in 1/L to αc = 0.734, the critical point obtained from entanglement-entropy data collapse. Away from mid-spectrum, the peak position α_max(ε) follows the many-body mobility edge αc(ε) from the analytical formula, and the peak height collapses as χ_typ_max(ε) ∝ ρ(ε)². The paper also extracts the fluctuation exponent η(α, ε) and finds its divergence tracks the mobility edge, showing fading ergodicity holds away from the middle of the spectrum.","pith_inferences":["A natural, testable extension is to apply the same analysis to disordered spin chains with a well-characterized many-body localization transition: if the χ ∝ ρ² scaling is universal, fidelity susceptibility could replace level statistics as the primary finite-size probe of the transition.","The fidelity susceptibility of a distant boundary spin can be read as the system's response to locally coupling the dot, suggesting a quantitative link between maximal chaos and the avalanche picture of ergodicity breaking.","The asymmetry between χ_av (scale-invariant on both sides) and χ_typ (only on the ergodic side) hints at a distinction between mean and typical eigenstate sensitivity; explaining why typical sensitivity fails to collapse on the nonergodic side may require modeling rare resonant spots.","If the heuristic exponent ν ≈ 1.3 needed to match the energy-resolved η(α, ε) is more than a fitting artifact, it would constitute a new universal exponent for the many-body mobility edge; checking its constancy across quantum sun variants would settle this."],"forward_implications":["If correct, the peak of the typical or regularized-average fidelity susceptibility—not just spectral statistics—pinpoints the ergodicity-breaking critical point in the quantum sun model, with the agreement improving linearly in 1/L.","At the critical point the susceptibility reaches its maximal possible scaling, χ ∝ ρ²; the same maximal scaling χ_typ_max ∝ ρ(ε)² holds at every energy density, so the mobility edge is itself a locus of maximal chaos.","The fading ergodicity formula χ_fading ∝ ρ^{2−2/η} collapses data across the whole ergodic phase for transition-sensitive operators, and the divergence of η(α, ε) tracks the analytical mobility edge throughout the spectrum.","Operators insensitive to the transition keep the ETH exponent ζ = 1 across the phase diagram, showing the effect requires a perturbation that couples to the ergodicity-breaking degrees of freedom."],"supporting_citations":[{"why":"Supplies the numerically extracted critical point αc = 0.734 from entanglement-entropy data collapse, used as the reference for all peak-position comparisons.","marker":"[80]"},{"why":"Introduces fading ergodicity and the fluctuation exponent η, along with the ansatz |V_nm|² = ω_H/Γ underlying Eq. (18).","marker":"[46]"},{"why":"Provides the many-body mobility edge formula αc(ε) used for the energy-driven analysis.","marker":"[51]"},{"why":"Introduces the quantum sun model and the Thouless-energy scaling Γ ∝ exp(−L ln(1/α²)) used in deriving η(α).","marker":"[47]"},{"why":"Defines the regularized AGP norm / fidelity susceptibility as a probe of chaos and sets the µ-regularization convention.","marker":"[38]"},{"why":"Establishes the expectation that the AGP-norm peak signals the ergodicity-breaking transition, which this paper verifies at high precision.","marker":"[39]"}],"fun_headline_variants":["Maximal chaos peaks at ergodicity breakdown in quantum sun model","Fidelity susceptibility diverges at maximal chaos transition","ETH collapse leads to quadratic fidelity susceptibility peak","Quantum sun model reveals maximal chaos at ergodicity edge","Fading ergodicity by maximal chaos: a quantum sun model signature"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The key premise is that the thermodynamic critical point of the model is αc = 0.734, a value imported from a data collapse of single-site entanglement entropy in a companion preprint; the peak-position agreement is only as strong as that imported number and the linear-in-1/L extrapolations.","fun_headline_variants_meta":{"raw":{"variants":["Maximal chaos peaks at ergodicity breakdown in quantum sun model","Fidelity susceptibility diverges at maximal chaos transition","ETH collapse leads to quadratic fidelity susceptibility peak","Quantum sun model reveals maximal chaos at ergodicity edge","Fading ergodicity by maximal chaos: a quantum sun model signature"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000636,"raw_usage":{"total_tokens":2899,"prompt_tokens":880,"completion_tokens":2019,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":1938}},"tokens_in":496,"tokens_out":2019,"duration_ms":14423,"temperature":1.0,"reasoning_tokens":1938,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:45:06.344407+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At system sizes L = 14, 15, 16 in the quantum sun model, compute χ_typ(ε) versus α at a fixed energy density ε = 0.3 with the same µ = ω_H regularization, and fit a fifth-order polynomial to locate the peak. Maximal chaos predicts the peak heights for all L fall on the line 0.05 ρ(ε)² with a spread comparable to the L ≤ 13 data, and the extrapolated peak position α_max(ε) approaches α_c(ε) = 0.734 exp[(a²(ε−1/2)²)/(4b²)] with a = 1.05, b = 0.45 to within the 1/L trend. Any systematic bending of χ_max versus ρ², or a peak position that misses α_c(ε) beyond the finite-size drift, would falsify the maximal-chaos claim.","supporting_citations":[{"cited_title":"Pawlik, P","cited_arxiv_id":null,"evidence_quote":"Provides the many-body mobility edge formula αc(ε) used for the energy-driven analysis."},{"cited_title":"Šuntajs and L","cited_arxiv_id":null,"evidence_quote":"Introduces the quantum sun model and the Thouless-energy scaling Γ ∝ exp(−L ln(1/α²)) used in deriving η(α)."},{"cited_title":"Pandey, P","cited_arxiv_id":null,"evidence_quote":"Defines the regularized AGP norm / fidelity susceptibility as a probe of chaos and sets the µ-regularization convention."},{"cited_title":"Sels and A","cited_arxiv_id":null,"evidence_quote":"Establishes the expectation that the AGP-norm peak signals the ergodicity-breaking transition, which this paper verifies at high precision."}],"review_version":1}