{"id":"417cfbb0-672a-42e7-ad65-5f9af3007944","arxiv_id":"2607.15798","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Thermal averaging strongly suppresses the mid-phase entropic-complexity peak of eigenstates in both random-matrix models, leaving at most a ~10% shadow, while a distinct edge feature appears only in thermal-state complexity.","lead":"This paper asks whether the complexity of individual quantum eigenstates survives once states are mixed into a thermal (Gibbs) state, across three disordered models. The answer is mostly no: a sharp mid-phase eigenstate complexity peak leaves at most a faint ~10% shadow in thermal observables, while a separate thermal-only \"edge\" feature appears near the ergodic boundary.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central taxonomy hinges on un-recomputed self-cited Heisenberg C_eig result; finite-size drift documented in PLBRM makes the risk concrete.","rationale":"The paper's strongest claim is the taxonomy: a mid-phase C_eig peak in two random-matrix models, absent in the Heisenberg chain. The RP and PLBRM computations are internally consistent, with bootstrap estimates, a random-basis control, and explicit tests of the D2=1/2 hypothesis. The Heisenberg leg, however, is not demonstrated here. Sec. III E explicitly imports the result from Ref. [18] and labels the absence 'structural rather than numerical.' Since the conclusion 'confined to random-matrix models' depends on a null finding for the interacting chain, and since self-citation limits independent audit, this is the least secure load-bearing assumption. The paper's own PLBRM data show that peak locations can drift with N (b_edge ∝ N^0.6), so finite-size location shifts are a real mechanism. A direct recomputation of C_eig(W) on the same chain, with the same definition and pointer basis, would settle the issue. The PLBRM thermal-shadow concern is secondary: even if the 10–13% bump were an artifact, the mid-phase eigenstate feature in two RM models would remain, and the conclusion would soften only from 'suppresses but does not eliminate' to 'suppresses' — less central than losing the Heisenberg contrast. Therefore the reader's CONDITIONAL verdict is appropriate and should be explicitly conditioned on this missing recomputation.","tokens_in":20256,"tokens_out":6376,"duration_ms":43887,"concrete_test":"Independently recompute C_eig(W) for the random-field Heisenberg chain with L=10,12,14,16 (and, if feasible, L=18 via shift-invert exact diagonalization) using the same definition C=S1−S2, the same S^z-product pointer basis, the same bulk spectral window, and at least several hundred disorder realizations per W. Locate the maximum by parabolic interpolation with bootstrap errors, as done for b* in Sec. III D. If the peak remains at W≈4–5 across L and no second maximum appears near W≈2–3, the taxonomy is supported; if the peak drifts toward mid-phase or develops a shoulder/plateau, the 'confined to random-matrix models' claim must be revised. Comparing directly against the data/code from Ref. [18] on the same disorder ensemble would settle the question.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central structural claim — 'This mid-phase eigenstate feature is confined to the two random-matrix models' (Sec. III G, echoed in Abstract and Conclusion) — rests on the Sec. III E assertion that the Heisenberg-chain C_eig(W) peaks at the ETH–MBL crossover (W≈4–5) rather than at an interior point. That assertion is not computed in this paper; it is imported from Ref. [18], a self-cited paper by the first author. The text marks this as 'structural rather than numerical,' but that distinction is exactly the load-bearing part: if the Heisenberg maximum actually sits at W≈2–3 for larger L, or if C_eig develops a mid-phase shoulder, the 'confined to random-matrix models' classification fails, and the strong version of the paper's conclusion — clean separation of thermal-state and eigenstate physics — is substantially weakened.\n\nThe risk is not hypothetical. The same paper documents an N-dependent peak location in PLBRM (b_edge ∝ N^{0.60±0.07}, Sec. III D), and the MBL crossover location itself is known to drift with L. The Heisenberg result is quoted only for L≤16, the same size range used throughout, and no direct comparison with [18]'s data or code is provided. Since [18] is self-cited, the reader cannot audit the finite-size scaling. The central claim therefore has a missing leg: the absence of a Heisenberg mid-phase feature is a null result that this manuscript does not itself establish.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the entropic complexity C = S_1 − S_2 (Shannon minus second-order Rényi entropy of a normalized distribution) across the ergodic-to-localized crossover of three disordered models: the Rosenzweig–Porter (RP) ensemble, the power-law banded random matrix (PLBRM) ensemble, and the random-field Heisenberg chain. It compares the single-eigenstate quantity C_eig with thermal-state quantities — the basis-independent trace complexity C_tr, the diagonal complexity C_diag of the dephased Gibbs state in the site (pointer) basis, the coherence gap C_gap = C_diag − C_tr, and the relative entropy of coherence C_rel. Principal findings: (i) C_eig develops a pronounced mid-phase maximum in RP at γ* ≈ 1.5–1.6 (1.561±0.008 at N=1600), located at D_2 ≈ 0.44, invisible to D_2(γ) and ⟨r⟩(γ); (ii) the same feature appears in PLBRM at b* = 0.31±0.01, N-independent over N=200–1600; (iii) thermal averaging strongly suppresses this feature — no resolvable shadow in RP at matched statistics, but a ~10–13% bump at b* in high-statistics PLBRM scans, attributed to a slower-than-1/k microcanonical washout (measured α≈0.85 vs. the α=1 independence prediction of Eq. (12)); (iv) a distinct thermal edge feature (γ≈0.8–0.9 in RP; W*/J≈0.5–0.7 in the Heisenberg chain) has no eigenstate-level counterpart and recedes with N in PLBRM (b*_edge ∝ N^{0.60±0.07}); (v) T*_tr and T*_diag merge upon localization, defining the scale-invariant indicator L = ln(T*_tr/T*_diag). The paper concludes that the mi","tokens_in":20470,"tokens_out":24370,"duration_ms":176608,"significance":"If the central taxonomy holds, the paper offers a useful organizing distinction: entropic complexity computed on individual eigenstates and on the thermal state tracks different physics, with a mid-phase wavefunction-level maximum reproduced in two structurally independent random-matrix ensembles at a comparable fractal dimension (D_2 ≈ 0.44) and cleanly separated from a thermal edge feature. The numerical discipline is a genuine strength: the Haar-random-basis rotation control flattens C_eig to the analytic GOE value 0.366±0.001; the D_2 = 1/2 hypothesis for the peak location is tested and ruled out at accessible sizes; the paper's analytic predictions are falsified rather than fitted — Eq. (10) predicts a flat plateau and Eq. (12) predicts α = 1, both contradicted by data, with the measured α ≈ 0.85 supplying the mechanism for the surviving PLBRM shadow; peak locations carry bootstrap errors; and the coherence-resource distinction (C_rel monotone under dephasing, C_gap not) is verified dynamically. The main risk to the significance of the classification is the Heisenberg-chain leg: the absence of a mid-phase eigenstate feature is imported from the self-cited Ref. [18] and not rec","major_comments":[{"comment":"The claim that the mid-phase eigenstate feature is 'confined to the two random-matrix models' (Abstract; Sec. III G; Conclusion) rests entirely on the assertion that Heisenberg-chain C_eig(W) peaks at the ETH–MBL transition (W≈4–5), not at an interior point. That assertion is not computed here: it is imported from Ref. [18], a prior paper by the first author, and the text marks it 'structural rather than numerical.' The gap is concrete: the paper already exact-diagonalizes the same chain at L=10–16 for the thermal diagnostics and ⟨r⟩ correlations of Sec. III E, so C_eig(W) on that ensemble is modest additional work. The same paper documents finite-size drift of the PLBRM edge feature (b*_edge ∝ N^{0.60±0.07}, Sec. III D), so an L≤16 null result cannot be assumed stable. Recommend either computing C_eig(W) directly, or re-scoping the abstract/conclusion to make the confinement claim condi","section":"Sec. III E / III G; Abstract; Conclusion"}],"minor_comments":[{"comment":"Eq. (12) is labeled twice (Sec. III B and again in Appendix A). Renumber the second occurrence.","section":"Sec. III B / Appendix A"},{"comment":"At N=200 the RP peak height (C*_eig ≈ 0.70) is only ~1% above the flat mean-field value ln 2 ≈ 0.693; the phrase 'rises well above ln 2' is accurate only at larger N. State the N-dependence of the excess explicitly.","section":"Sec. III A"},{"comment":"The 'essentially independent of N' statement for γ* rests on coarse production grids for N<1600; the high-resolution scan (Δγ≈0.03) is at N=1600 only. A high-resolution scan at one additional size, or a table of peak locations with bootstrap errors per size, would make the fixed-point claim quantitative.","section":"Sec. III A / Fig. 3"},{"comment":"The ~10–13% PLBRM thermal bump is described as 'right at b*' but no uncertainty or width is given for its location; a bootstrap estimate in the same style as the C_eig peak would strengthen the shadow identification. Relatedly, the matched-statistics RP null of Sec. III B is at N=200 only; state this explicitly wherever 'no thermal trace' is asserted.","section":"Sec. III D / Sec. III B"},{"comment":"The per-sample Pearson correlations (|ρ|≲0.1) are quoted without standard errors; with 2000 samples the standard error of ρ is ≈0.022, so ρ≈0.06 is ~3σ from zero. 'Negligible' is defensible in variance-explained terms (~1%), but should be stated that way.","section":"Sec. III E"},{"comment":"The Heisenberg edge location W*/J ≈ 0.5–0.7 is 'set by adjacent points of the W-grid rather than a bootstrap-resolved location,' in contrast to the peak-location methodology used for RP and PLBRM. A parabolic interpolation on the log grid would improve cross-model comparability.","section":"Sec. III E"}],"recommendation":"major_revision","confidential_remarks":"This is a careful numerical paper with unusually good internal controls; the RP/PLBRM observations appear solid and publishable. The single load-bearing issue for the central taxonomy is the imported, self-cited Heisenberg-chain C_eig statement. I have asked for a direct computation or an explicit re-scoping; given that the authors already diagonalize the same chain at L=10–16 for their thermal analysis, the recomputation is a modest addition and should be feasible in revision. I would not reject on these grounds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result is worth knowing: the entropic complexity C = S1 − S2 develops a pronounced mid-phase peak in both the Rosenzweig–Porter and PLBRM models at a matching fractal dimension D2 ≈ 0.44, and thermal averaging suppresses it strongly but not completely — a ~10% shadow survives in high-statistics PLBRM, none in RP. The paper also documents a separate, purely thermal edge feature that recedes with system size in PLBRM. That separation of eigenstate-level from thermal-state-level complexity is a genuinely new observation, and the cross-model reproducibility is the strongest point.\n\nThe numerical controls are unusually solid. A Haar-random-basis rotation flattens the RP peak to the analytic GOE value, so the feature is not a basis artifact. The D2 = 1/2 hypothesis is explicitly tested and rejected at accessible sizes. Peak locations carry bootstrap errors. And the authors are honest where their own analytic guesses fail: Eq. (10) predicts a flat profile and Eq. (12) predicts a 1/k washout, both contradicted by the numerics (measured α ≈ 0.85). That is how numerical papers should report.\n\nThe main soft spot is the Heisenberg chain. The claim that the mid-phase eigenstate feature is confined to the two random-matrix models rests on the assertion that C_eig(W) in the Heisenberg chain peaks at the ETH–MBL crossover, not at an interior point. That result is not computed here; it is imported from Ref. [18], a self-cited paper by the first author, and the finite-size behavior is not independently established. Given that the same manuscript shows the PLBRM edge feature drifts with N, a skeptic could worry the Heisenberg peak might also move if pushed to larger L. This does not sink the paper, but it is load-bearing for the abstract's \"confined to the two random-matrix models\" and needs either recomputation or a much clearer justification.\n\nMinor issues: the PLBRM thermal shadow is quoted as ~10–13% without propagated error bars, and no code or data is shipped, so the decisive high-statistics scans cannot be audited. The thermodynamic-limit question — whether γ* converges to the D2 = 1/2 point — is left open, which the authors acknowledge.\n\nOverall: the RP/PLBRM finding is solid and the paper is more honest than most. It deserves a serious referee. I would send it out, with a request to address the Heisenberg import head-on and to quantify the shadow errors.","headline":"A careful, honest numerical study showing an eigenstate-level entropic complexity peak in two random-matrix models that largely washes out under thermal averaging — worth engaging, but the Heisenberg leg rests on a self-cited result you should check.","tokens_in":21177,"tokens_out":2443,"would_cite":true,"duration_ms":63007,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B44","15B52","81Q50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that the entropic complexity C = S1 − S2 separates two kinds of physics in disordered quantum systems: a mid-phase eigenstate feature, reproducible across two random-matrix models and pinned to a fixed parameter value,","keywords":["entropic complexity","Rosenzweig-Porter ensemble","power-law banded random matrices","many-body localization","thermal state","quantum coherence","decoherence","multifractality"],"falsifier":"Exact-diagonalize the random-field Heisenberg chain at L = 18–20 and scan C_eig(W) finely; if the maximum moves away from the ergodic-to-MBL crossover (W/J ≈ 4–5) into the ergodic phase as L grows, the paper's claim that the mid-phase feature is confined to random-matrix models fails. Alternatively, repeat the RP thermal scan near γ* with PLBRM-level statistics; a resolvable bump at γ* would show the RP 'no shadow' is a statistical limit rather than a structural difference.","tokens_in":19934,"feed_emoji":"🎲","tokens_out":6830,"duration_ms":51971,"temperature":0.7,"pith_summary":"The paper asks whether thermal averaging destroys the fingerprints of individual eigenstate complexity in disordered quantum systems. It studies the entropic complexity C = S1 − S2 (Shannon minus second-order Rényi entropy) across the ergodic-to-localized crossover in three models: two random-matrix ensembles with an extended multifractal phase, and the random-field Heisenberg chain. Its central finding is that a sharp mid-phase maximum in the single-eigenstate complexity — located at γ* ≈ 1.5–1.6 in the Rosenzweig–Porter model and b* ≈ 0.31 in the power-law banded model, both at fractal dimension D2 ≈ 0.44 — is strongly but not completely suppressed by thermal mixing: a ~10–13% thermal shadow survives in one model, while a distinct low-temperature 'edge' feature has no eigenstate counterpart. The paper argues that entropic complexity thus cleanly separates thermal-state physics from eigenstate physics, and that care is needed in stating which quantity a measurement sees.","feed_headline":"Thermal averaging nearly erases a sharp eigenstate signature","feed_subtitle":"A mid-phase complexity peak in two random-matrix models leaves only a faint ~10% trace in thermal observables.","key_machinery":"The central object is C = S1 − S2, evaluated in three ways: on individual eigenstates (C_eig), on the thermal state's eigenvalue populations (C_tr), and on its dephased site-basis populations (C_diag). The coherence gap C_gap = C_diag − C_tr decomposes as C_rel − ΔS2, where C_rel is the relative entropy of coherence (a genuine monotone) and ΔS2 obeys ΔS2/ln N → D2 as T → 0, tying the thermal probe to the eigenstate fractal dimension. The washout mechanism is quantified by expanding C to quadratic order in site-intensity fluctuations, giving C ≈ v(γ)/(2k) for k-state mixtures, with a measured exponent α ≈ 0.85 rather than 1 due to eigenstate correlations.","core_discovery":"Entropic complexity C = S1 − S2, the difference between Shannon and second-order Rényi entropy of a distribution in a fixed site basis, is claimed to separate two structurally distinct features across the ergodic-to-localized crossover. In the Rosenzweig–Porter and power-law banded random-matrix ensembles, the single-eigenstate quantity C_eig peaks deep inside the fractal phase — γ* ≈ 1.561 ± 0.008 in RP and b* = 0.311 ± 0.008 in PLBRM, both size-independent — at fractal dimension D2 ≈ 0.44, invisible to D2 and to level statistics. Thermal averaging strongly suppresses the peak: no shadow is resolvable in RP, while a weak (~10–13%) bump survives in PLBRM’s temperature-maximized diagonal comp","pith_inferences":["If the mid-phase peak is driven by multifractal fluctuations rather than by the specific RP or PLBRM construction, a similar C_eig maximum should appear in any model with a broad fractal regime — e.g., quasiperiodic systems, Bethe-lattice Anderson models, or interacting models at criticality — a testable extension.","The RP/PLBRM difference in thermal shadow suggests a quantitative link: models with stronger energy-eigenstate correlations should show larger surviving shadows. Measuring the washout exponent α for other ensembles would predict where shadows appear.","The PLBRM edge feature receding with N hints that the RP edge at γ ≈ 0.8–0.9, reported as size-independent in the accessible range, may itself drift at larger N; checking this would test whether the two edges share a finite-size origin.","Because C_rel orders by density-of-states class rather than collapsing universally, a common scaling variable (e.g., an effective dimension) remains to be found; if found, it could collapse both edge and mid-phase features across models."],"forward_implications":["Because C_eig peaks where D2 and level statistics are featureless, it offers a new, independent marker of deep-fractal-phase structure that could identify extended fractal phases in other disordered systems.","The strong but incomplete thermal suppression means energy-resolved, few-state observables, not full Gibbs states, are the right place to look for eigenstate-complexity signatures.","The scale-invariant log-ratio L = ln(T*_tr/T*_diag) vanishes on localization in all three models, giving a simple crossover indicator that is independent of the temperature rescaling.","In systems without an extended fractal phase, such as the Heisenberg chain, the eigenstate complexity peaks at the ergodic-to-MBL transition, so the mid-phase feature can serve to distinguish models with a genuine multifractal regime from those without.","The edge feature's N-dependent drift in PLBRM warns that thermal-edge locations extracted at finite size may not persist in the thermodynamic limit."],"fun_headline_variants":["Thermal state nearly erases eigenstate complexity peak in random matrices","Complexity measure C = S1−S2: thermal shadow is only ~10% of eigenstate peak","Eigenstate complexity peak survives thermal averaging as faint ~10% trace","Random-matrix complexity peak leaves only weak thermal imprint"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The classification of the Heisenberg chain as lacking a mid-phase eigenstate peak depends on a previously published, cited result that C_eig(W) is maximal at the ergodic-to-MBL crossover for L ≤ 16; that result is not recomputed in this paper, and if it drifts with system size the 'random-matrix-only' conclusion would need revision.","fun_headline_variants_meta":{"raw":{"variants":["Thermal state nearly erases eigenstate complexity peak in random matrices","Complexity measure C = S1−S2: thermal shadow is only ~10% of eigenstate peak","Eigenstate complexity peak survives thermal averaging as faint ~10% trace","Random-matrix complexity peak leaves only weak thermal imprint"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1431,"prompt_tokens":900,"completion_tokens":531,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":449}},"tokens_in":644,"tokens_out":531,"duration_ms":5324,"temperature":1.0,"reasoning_tokens":449,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T22:18:20.683389+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exact-diagonalize the random-field Heisenberg chain at L = 18–20 and scan C_eig(W) finely; if the maximum moves away from the ergodic-to-MBL crossover (W/J ≈ 4–5) into the ergodic phase as L grows, the paper's claim that the mid-phase feature is confined to random-matrix models fails. Alternatively, repeat the RP thermal scan near γ* with PLBRM-level statistics; a resolvable bump at γ* would show the RP 'no shadow' is a statistical limit rather than a structural difference.","supporting_citations":[],"review_version":1}