{"id":"b0db6730-a5c0-4958-a68d-2ca56cb41b4b","arxiv_id":"2411.08496","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Disordered orbital Hatsugai-Kohmoto model shows a transition from Poisson to GOE level statistics as interaction disorder increases, while OTOC plateau values fail to uniformly distinguish chaos across models.","lead":"This paper studies a disordered version of the orbital Hatsugai-Kohmoto model, an exactly solvable non-Fermi liquid, using spectral statistics and out-of-time-order correlators. It reports a chaos-integrable transition controlled by interaction disorder and argues that OTOC plateau values are not a reliable chaos diagnostic.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed transition at ⟨U²⟩/⟨t_h²⟩≈0.33 and the OTOC-plateau conclusion are not statistically supported: the 10-orbital phase diagram uses 5 disorder samples and the 8-orbital OTOC uses 1, with no error bars or finite-size scaling.","rationale":"The paper's two headline results are (i) a chaotic-integrable transition in the disordered orbital HK model, quantified by a phase boundary at ⟨U²⟩/⟨t_h²⟩≈0.33 for 10 orbitals, and (ii) the claim that OTOC plateau values do not effectively distinguish integrable and chaotic phases. Both are numerical claims extracted by exact diagonalization. The reader's conditional verdict rests on the small number of disorder samples and the absence of finite-size scaling; my stress-test agrees that this is the load-bearing weakness. More specifically, the 10-orbital phase diagram uses five samples, for which the standard error of the mean adjacent-gap ratio is comparable to the distance between the Poisson and GOE benchmarks, so the 0.33 number is not a statistically resolved transition. The 8-orbital OTOC curve—used to argue that system size does not change the late-time temperature dependence—rests on one sample, so the abstract's negative statement about OTOC plateaus is not yet supported. I do not see an internal contradiction in the equations themselves: the model, level-statistics definitions, and OTOC definitions are standard, and the SFF/GOE observations are qualitatively plausible. The weakness is evidential: the numerical basis is too thin to fix a transition boundary or to rule out finite-size artifacts. A straightforward sample-size and finite-size rerun would settle it. This does not change the reader's CONDITIONAL verdict; it sharpens the condition that should be met before the transition boundary and OTOC conclusion are accepted.","tokens_in":10179,"tokens_out":11520,"duration_ms":101218,"concrete_test":"Rerun the 10-orbital adjacent-gap-ratio calculation in Fig. 1c with 100 independent disorder samples at each ratio (and the 8-orbital OTOC in Fig. 3f with at least 20 samples), reporting bootstrap error bars on ⟨r⟩ and on the late-time C(t) values. If the 0.33 threshold shifts by more than its own error bar when going from 5 to 100 samples, or if the error bars span both the Poisson (0.386) and GOE (0.536) reference values across the nominally separated regions, the transition claim is not supported. An optional 12-orbital run with 20 samples would test whether the integrable window persists or vanishes in the thermodynamic limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Fig. 1c locates the GOE/Poisson boundary at ⟨U²⟩/⟨t_h²⟩>0.33 using 5 random samples for 10 orbitals (Hilbert-space dimension C(10,5)²≈63,500), 100 for 8 orbitals, and 5000 for 6 orbitals, with no reported error bars. The adjacent-gap-ratio mean ⟨r⟩ has an intrinsic sample-to-sample spread of order 0.1, so with 5 samples the standard error is ~0.05, comparable to the Poisson-to-GOE separation of 0.15; the threshold is therefore not statistically distinguishable from a smooth crossover without additional samples. The SFF ramp in Fig. 2c is also based on 5 samples, and the conclusion that system size does not affect the temperature-dependent late-time OTOC is grounded in Fig. 3f, which uses a single random sample for 8 orbitals. The text further notes that the integrable regime 'tends to vanish' as the number of orbitals increases; because no finite-size extrapolation is supplied, the claimed boundary may be a finite-size artifact rather than a genuine transition. Since the abstract's central negative claim about OTOC plateaus as a chaos diagnostic is justified partly by the same under-sampled curves, the main conclusions are not yet robust to sampling and size effects.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the disordered orbital Hatsugai-Kohmoto (HK) model and its chaotic-integrable transition using the adjacent gap ratio, the spectral form factor (SFF), and out-of-time-order correlators (OTOCs). The authors report GOE level statistics when the interaction-disorder variance exceeds roughly one-third of the hopping variance, a dip-ramp-plateau SFF in the chaotic regime, and a phase diagram across 6, 8, and 10 orbitals. They then compare late-time OTOC plateaus of the HK model, SYK2, SYK4, and a disorder-free SYK model, concluding that OTOC plateau values do not effectively differentiate integrable from chaotic phases, and that the HK model's plateau is regularization dependent.","tokens_in":10447,"tokens_out":6091,"duration_ms":49308,"significance":"The paper proposes an interesting connection between the orbital HK model and SYK-type models and makes a falsifiable negative claim about OTOC plateau values as a chaos diagnostic. If the numerical results were supported by converged statistics, the comparison across SYK variants would be a useful contribution. The main strength is the clear formulation of a testable question; the main weakness is that the central numerical claims rest on very few disorder samples and no error bars or finite-size scaling, so the conclusions are not yet quantitatively established.","major_comments":[{"comment":"The phase diagram is constructed from 5 random samples for 10 orbitals, 100 samples for 8 orbitals, and no error bars are reported. Since the sample-to-sample spread of the mean adjacent gap ratio is of order 0.1, the standard error with 5 samples is about 0.05, which is one-third of the separation between the Poisson value (0.386) and the GOE value (0.536). The claimed transition at ⟨U²⟩/⟨t_h²⟩ > 0.33 is therefore not statistically distinguishable from a smooth crossover. Please provide error bars, a convergence check with increasing sample number, and a finite-size scaling analysis before drawing the phase boundary.","section":"§3.1, Fig. 1(c)"},{"comment":"The SFF 'linear ramp' in the 10-orbital case is based on only 5 random samples. No error bars or comparison with statistical fluctuations of the SFF are provided, so the support for GOE behavior from the SFF is not quantitative. The statement that the ramp appears 'with large orbitals considered' is also not backed by a systematic size scaling.","section":"§3.2, Fig. 2(c)-(d)"},{"comment":"The claim that 'the system size does not affect the temperature-dependent late-time behavior of OTOCs' is based on a single random sample at 8 orbitals. A one-sample result carries no statistical uncertainty and cannot support a size-independence claim. Please provide multiple samples, error bars, and a proper finite-size analysis for the OTOC plateau values.","section":"§4, Fig. 3(f)"},{"comment":"The text notes that 'the integrable regime tends to vanish' as the number of orbitals increases, but no extrapolation to the thermodynamic limit is supplied. Without such an analysis, the existence of a genuine chaotic-integrable transition, rather than a finite-size crossover, is not established. This concern is amplified because the phase boundary at 0.33 is read from the largest system with the fewest samples.","section":"§3.1"}],"minor_comments":[{"comment":"In the first paragraph, 'Hatsugai and Komohto' should be 'Hatsugai and Kohmoto.'","section":"§1"},{"comment":"'Multi-channel Kodon model' should be 'multi-channel Kondo model.'","section":"Table 1"},{"comment":"The disorder-free SYK model is cited as Ref. [21], but that reference is 'Correlated disorder in the SYK2 model'; the relevant paper on the disorder-free SYK model appears to be Ref. [24], and the citation should be corrected.","section":"§4"},{"comment":"The caption does not specify whether panels (c)-(f) are for the GOE or Poisson parameter regime; without this information, the size-independence claim is difficult to interpret.","section":"Fig. 3"},{"comment":"The thermal expectation value in Eq. (23) is not explicitly defined; the angle brackets and the dependence on β should be stated.","section":"Eq. (23)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely topic and the negative claim about OTOC plateau values is interesting, but the numerical evidence is far below the standard needed for a reliable phase diagram. I recommend major revision rather than rejection because the issues are fixable with more samples, error bars, and finite-size scaling. Note also the citation error involving Refs. [21] and [24] that should be corrected during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a legitimate but statistically under-powered study of a new disordered-orbital Hatsugai-Kohmoto (HK) model. The phase diagram from gap ratios and SFF is new and plausible; the OTOC-plateau message is not new and is argued more strongly than the data support.\n\nWhat is new: the authors build a disordered orbital HK model that maps to a charge-conserving SYK2+SYK4-like system, and they characterize it with adjacent gap ratio, spectral form factor, and OTOCs. The model-specific GOE/Poisson phase diagram (GOE for ⟨U²⟩/⟨t²h⟩ above roughly 0.33 at 10 orbitals) and the SFF ramp appear genuinely new. The paper also correctly notes that SFF shape alone is not a chaos diagnostic, citing [20,21].\n\nWhere the soft spots are: sample sizes. The 10-orbital phase diagram uses 5 random samples; the 8-orbital OTOC uses 1. There are no error bars, no convergence checks, and no finite-size scaling. The 0.33 threshold is read from a curve whose intrinsic sample-to-sample spread is about 0.1, so with 5 samples the standard error is near 0.05, comparing to the 0.15 Poisson-GOE separation. That is not enough to distinguish a transition from a smooth crossover. The statement that system size does not affect the late-time OTOC is grounded in Figure 3f, a single sample. On the conceptual side, the abstract's claim that OTOC plateau values do not differentiate integrable from chaotic phases is undercut by the paper's own result: in the HK model, the chaotic regime has temperature-dependent late-time OTOCs while the integrable regime does not. That is a differentiation. The cross-model comparison with SYK2, SYK4, and disorder-free SYK is suggestive but uses different operators and sizes, and the negative conclusion is already present in [22,26]. So the genuinely new contribution is the HK phase diagram, not the OTOC moral.\n\nThe math and citation pattern look fine. Self-citations to [20,21] are used as benchmarks, not to inflate. The paper is clearly written and honestly framed.\n\nWho this is for: quantum chaos / condensed-matter readers who want another solvable testbed for chaos-integrable transitions. It deserves a serious referee, but only with major revision: add samples, error bars, finite-size scaling, and soften the OTOC claim to match the evidence.","headline":"Solid model-specific extension with a new HK phase diagram, but the OTOC-plateau claim is over-reached and the numerics are too under-sampled to support the transition boundary.","tokens_in":11018,"tokens_out":3197,"would_cite":false,"duration_ms":26870,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q50","82B44"],"pacs":["05.45.Mt","71.27.+a"],"model":"deepseek-v4-flash","headline":"The disordered orbital Hatsugai-Kohmoto model transitions from integrable to chaotic as interaction disorder is turned on, and OTOC plateau values fail to distinguish the phases.","keywords":["Hatsugai-Kohmoto model","Sachdev-Ye-Kitaev model","quantum chaos","out-of-time-order correlator","spectral form factor","adjacent gap ratio","non-Fermi liquid","exact diagonalization"],"falsifier":"Recompute the adjacent-gap-ratio phase diagram for 12- and 16-orbital systems with hundreds of disorder realizations; the central claim would be refuted if the GOE threshold moves systematically away from $\\langle U^2\\rangle/\\langle t_h^2\\rangle \\approx 0.33$, or if the OTOC plateau in SYK4 becomes temperature-dependent as $N$ grows.","tokens_in":9945,"feed_emoji":"🎲","tokens_out":12695,"duration_ms":102373,"temperature":0.7,"pith_summary":"The paper tries to establish that a disordered version of the orbital Hatsugai-Kohmoto model, a solvable non-Fermi liquid lattice model, hosts a transition from integrable to chaotic behavior controlled by the variance of the interaction disorder $\\langle U^2\\rangle$ relative to the hopping disorder $\\langle t_h^2\\rangle$. It shows via exact diagonalization that the adjacent gap ratio moves from Poisson statistics to Gaussian orthogonal ensemble statistics, with GOE appearing for $\\langle U^2\\rangle/\\langle t_h^2\\rangle > 0.33$ in the 10-orbital system, and that the spectral form factor develops the characteristic dip-ramp-plateau in the same regime. Its central negative conclusion is that the late-time plateau value of the out-of-time-order correlator does not effectively differentiate integrable from chaotic phases, because it is temperature-independent in SYK2, SYK4, and disorder-free SYK models but temperature- and regularization-dependent in the disordered orbital HK model. If correct, this limits the use of OTOC saturation values as a universal chaos diagnostic and redirects attention to spectral statistics. The paper also argues that these features connect the HK model to the SYK family of non-Fermi liquid models.","feed_headline":"Interaction disorder flips the HK model from integrable to chaotic","feed_subtitle":"Level statistics match random-matrix theory past a variance ratio of 0.33, while OTOC plateaus stay ambiguous","key_machinery":"The control parameter is the variance ratio $\\langle U^2\\rangle/\\langle t_h^2\\rangle$, comparing the disorder strength of the four-point interaction to that of the two-point hopping. The diagnostic machinery has three pieces: the adjacent gap ratio (a level-spacing statistic whose average distinguishes Poisson, about 0.386, from GOE, about 0.536), the spectral form factor (the squared Fourier transform of the energy-level density, whose ramp signals long-range level repulsion), and the late-time plateau of the out-of-time-order correlator (the saturation value of a squared commutator of Heisenberg-picture operators).","core_discovery":"On its own terms, the paper's discovery is that the disordered orbital HK model exhibits a chaotic-integrable transition when the variance of the four-point interaction $U$ is turned on, while pure hopping disorder ($\\langle U^2\\rangle=0$) leaves the spectrum integrable. Quantitatively, for 10 orbitals with half-filling and zero total spin, the adjacent gap ratio reaches the GOE value for $\\langle U^2\\rangle/\\langle t_h^2\\rangle > 0.33$, and the spectral form factor shows a linear ramp consistent with random matrix theory. The paper's second, equally central claim is negative: comparing the HK model with SYK2, SYK4, and the disorder-free SYK model, the plateau value of the OTOC does not separate chaotic from integrable phases, and its temperature dependence is not a reliable many-body chaos indicator because it depends on regularization and system size.","pith_inferences":["The paper leaves implicit a sharper formulation of its negative result: the OTOC plateau may track global conserved charges or regularization rather than integrability, since the HK model conserves charge while the Majorana SYK models conserve parity; comparing HK with a charge-conserving complex-fermion SYK model would test this.","A practical diagnostic suggested by the contrast is to pair the adjacent gap ratio with the connected spectral form factor, because the SFF shape alone can mimic chaos in disordered integrable systems.","The 10-orbital threshold at 0.33 comes from only five disorder samples and no scaling analysis, so an independent finite-size study is the natural next step before quoting the boundary as universal."],"forward_implications":["The disordered orbital HK model provides a solvable lattice setting where a non-zero variance of $U$ alone drives GOE level statistics and a dip-ramp-plateau spectral form factor.","The phase diagram reports a transition boundary near $\\langle U^2\\rangle/\\langle t_h^2\\rangle \\approx 0.33$ for the 10-orbital system, with the integrable regime shrinking as the orbital number grows.","Because the SFF ramp appears only in the GOE regime, the spectral form factor can serve as a consistency check for the adjacent-gap-ratio phase diagram, but not as a standalone diagnostic, since disorder can mimic its shape.","The OTOC plateau comparisons imply that studies using late-time OTOC saturation to label systems as chaotic should be re-examined."],"supporting_citations":[{"why":"Defines the orbital Hatsugai-Kohmoto model with orbital degrees of freedom and long-range interactions, the Hamiltonian under study.","marker":"[6]"},{"why":"Introduces the original Hatsugai-Kohmoto interaction whose momentum-space locality makes the model exactly solvable.","marker":"[7]"},{"why":"Documents the chaotic-integrable transition in the SYK2+SYK4 model that motivates the same question here.","marker":"[11]"},{"why":"Supplies the spectral-form-factor dip-ramp-plateau and random-matrix behavior used to identify chaos.","marker":"[4]"},{"why":"Provides the Poisson and GOE adjacent-gap-ratio distributions used to interpret the level statistics.","marker":"[15]"},{"why":"Defines the spectral form factor in random matrix theory, the basis of the SFF analysis.","marker":"[17]"},{"why":"Provides the disorder-free SYK late-time OTOC behavior used in the plateau comparison.","marker":"[21]"},{"why":"Gives the disorder-free SYK spectral and OTOC results contrasted with the chaotic models.","marker":"[24]"},{"why":"Proposes OTOC saturation as a chaos probe; the paper argues this diagnostic fails at late times.","marker":"[26]"}],"fun_headline_variants":["Disordered HK model transitions to chaos past U-variance ratio 0.33","OTOC plateau can't tell integrable from chaotic phases","Chaos in disordered HK model: U-variance > 0.33 triggers GOE","Disorder-induced chaos in HK; OTOC plateaus ambiguous"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that exact-diagonalization results for 6 to 10 orbitals, including cases with only one or five disorder samples, represent the thermodynamic limit, so the claimed transition boundary and OTOC temperature dependence are physical and not finite-size or sampling artifacts.","fun_headline_variants_meta":{"raw":{"variants":["Disordered HK model transitions to chaos past U-variance ratio 0.33","OTOC plateau can't tell integrable from chaotic phases","Chaos in disordered HK model: U-variance > 0.33 triggers GOE","Disorder-induced chaos in HK; OTOC plateaus ambiguous"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000881,"raw_usage":{"total_tokens":3804,"prompt_tokens":938,"completion_tokens":2866,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2786}},"tokens_in":554,"tokens_out":2866,"duration_ms":19369,"temperature":1.0,"reasoning_tokens":2786,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:31:27.440784+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the adjacent-gap-ratio phase diagram for 12- and 16-orbital systems with hundreds of disorder realizations; the central claim would be refuted if the GOE threshold moves systematically away from $\\langle U^2\\rangle/\\langle t_h^2\\rangle \\approx 0.33$, or if the OTOC plateau in SYK4 becomes temperature-dependent as $N$ grows.","supporting_citations":[{"cited_title":"Ground state stability, symmetry, and degeneracy in Mott insulators with long range interactions","cited_arxiv_id":"2306.00221","evidence_quote":"Defines the orbital Hatsugai-Kohmoto model with orbital degrees of freedom and long-range interactions, the Hamiltonian under study."},{"cited_title":"Correlated Disorder in the SYK$_{2}$ model","cited_arxiv_id":"2003.05401","evidence_quote":"Provides the disorder-free SYK late-time OTOC behavior used in the plateau comparison."}],"review_version":1}