{"id":"e31039b2-48b7-485f-8ecd-e3e17b32724d","arxiv_id":"2603.25724","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Krylov spread complexity scales as N_H (ergodic) versus N_H^α with α<1 (MBL), with stretched-exponential Krylov profiles dominated by a vanishing fraction of resonant eigenstates.","lead":"Infinite-time Krylov spread complexity of a disordered spin chain scales linearly with Hilbert-space size in the ergodic phase but sublinearly in the many-body localized phase. The Krylov-chain profile and eigenstate statistics give a basis-optimized diagnostic of localization and rare resonances.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged phenomenological assumption.","rationale":"The reader's weakest_assumption correctly isolates the only soft spot: the convenient exponential distributions P_\\xi|E and P_\\xi d used to derive the stretch exponents γ=1/2 or 1/3. Because that construction is phenomenological and not required for the scaling or large-deviation results that constitute the strongest claim, it does not threaten the central numerical distinction between ergodic and MBL Krylov anatomy. The ED data, sum rules, and saddle-point analysis of Σ(x) are internally consistent and re-implementable. Consequently the CONDITIONAL verdict with high confidence stands; no adjustment is warranted.","tokens_in":24061,"tokens_out":496,"duration_ms":5413,"concrete_test":"Recompute the disorder-averaged profile \\langle\\Lambda_n\\rangle and the first-moment scaling of S_{K,\\infty} for the same model at L=16 (N_H=65536) with at least 50 independent mid-spectrum initial states; if the extracted \\alpha remains <1 for W=8,10 and the log-log plot of -ln f(x) continues to show slope ≲ 1/2, the numerical core of the claim is robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on exact-diagonalization scaling of S_{K,\\infty} ~ N_H^\\alpha (\\alpha=1 ergodic; \\alpha<1 MBL) and the associated \\Lambda_n profile (Figs. 4–7), together with the large-deviation entropy density \\Sigma(x) of eigenstate complexities (Sec. V, Fig. 9, Table I). These are direct numerical observables obtained from the Lanczos-generated Krylov basis of the standard tilted-field Ising chain; they do not depend on the exponential length-scale ansatz of Sec. IV C. That ansatz is offered only as a post-hoc rationalization of the stretch exponent and is already identified by the reader as the weakest assumption. No internal inconsistency, hidden normalization error, or untested regime appears that would overturn the ED distinction between phases. Finite-size caveats on MBL numerics remain, but they are generic and already acknowledged.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the Krylov-space anatomy of states and the infinite-time Krylov spread complexity S_{K,∞} for the disordered tilted-field Ising chain, contrasting the ergodic and MBL regimes. Using the Lanczos-generated Krylov basis from a mid-spectrum product state, the authors show that ⟨S_{K,∞}⟩ scales linearly with Fock-space dimension N_H in the ergodic phase (occupying a finite fraction of the Krylov chain) and sublinearly as N_H^α with α<1 in the MBL phase (occupying a vanishing fraction). The disorder-averaged profile ⟨Λ_n⟩ collapses onto a scaling form with the same α and, in the MBL regime, exhibits a stretched-exponential decay. A large-deviation analysis of the eigenstate contributions S_{K,|E⟩} further shows that the ergodic sum is carried by a finite fraction of eigenstates while the MBL sum is dominated by a vanishing (but still exponentially large) fraction of anomalously complex eigenstates, with multifractal IPRs of those contributions. A phenomenological theory based on exponential length-scale distributions is offered to rationalize the stretch exponent.","tokens_in":24330,"tokens_out":1393,"duration_ms":11027,"significance":"If the reported scalings hold, the work supplies a clean, basis-optimized diagnostic that sharply separates ergodic and MBL phases on a one-dimensional Krylov chain whose length is exactly N_H. The combination of direct ED scaling of S_{K,∞} and Λ_n (Figs. 4–7), the large-deviation entropy density Σ(x) (Fig. 9, Table I), and the multifractal IPR of eigenstate complexities (Fig. 10) is internally consistent and goes beyond earlier operator-Krylov studies that are less sensitive to MBL. The explicit mapping between Krylov orbitals and Fock-space Hamming shells (Sec. III B, Fig. 2) also clarifies why the Krylov chain remains a faithful probe even when most orbitals have support over the entire Fock graph. Finite-size caveats on MBL numerics are generic and already acknowledged; the central numerical distinction itself is robust within accessible sizes.","major_comments":[{"comment":"Sec. IV C, Eqs. (33)–(47): the phenomenological theory assumes pure exponential decay of contributing eigenstate amplitudes with lengths ξ_|E⟩ drawn from an exponential distribution of mean ξ_d, and that the ξ_d themselves are exponentially distributed over disorder. This functional form is chosen for analytic convenience and is only a posteriori consistent with the observed stretch exponents (γ ≃ 1/2 at accessible sizes, asymptotically 1/3). Because the central claims of the paper rest on the ED scalings of S_{K,∞} and Λ_n (Figs. 4–7) and on the large-deviation analysis (Sec. V), not on this ansatz, the theory should be clearly labeled as a post-hoc rationalization rather than a derivation. A short numerical check of the actual distribution of effective decay lengths extracted from individual eigenstates would strengthen or falsify the assumption.","section":null},{"comment":"Sec. IV A and Fig. 4: the reported exponents α(W) are extracted from system sizes L ≲ 14–16. While the ergodic α = 1 result is solid, the MBL values α < 1 (and the associated claim that the long-time state occupies a vanishing fraction of the Krylov chain) remain subject to the usual finite-size caveats of MBL numerics. The manuscript should state more explicitly the largest L used for each W, the number of disorder realizations, and whether any drift of α with L is visible; a brief comparison with an independent localization diagnostic (e.g., half-chain entanglement or Fock-space IPR) on the same samples would help calibrate how deep into the putative MBL regime the data sit.","section":null}],"minor_comments":[{"comment":"Fig. 7 and surrounding text: the stretch exponent is quoted as γ ≃ 1/2 for the bulk of the data and γ = 1/3 for the largest n/N_H^α. A single sentence clarifying that the asymptotic analytic result is γ = 1/3 while finite-size data remain closer to 1/2 would remove residual ambiguity.","section":null},{"comment":"Appendix A, Fig. 11: the scaling of ⟨b_n⟩ and the effective disorder W_n are shown but not used later. Either a brief remark on why these bare Krylov-matrix statistics do not distinguish the phases, or a pointer to future work, would improve cohesion.","section":null},{"comment":"Eq. (9) and the definition of Λ_n: the sum rule ∑_n Λ_n = 1 is stated, but it would help the reader to note explicitly that the infinite-time average eliminates the off-diagonal E \neq E' terms, so that Λ_n is strictly a sum of |c_n,E|^2 |c_0,E|^2.","section":null},{"comment":"References: a few recent works on state Krylov complexity near the MBL transition (e.g., those already cited as [54–59]) could be more explicitly contrasted in the introduction to highlight what is new in the infinite-time anatomy and large-deviation analysis.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, carefully executed ED study that fits well within the journal's scope for disordered quantum many-body systems. The phenomenological section is the weakest part but is not load-bearing; a light revision that frames it as rationalization rather than derivation, plus the usual finite-size caveats, should be sufficient. No novelty or citation concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: infinite-time Krylov spread complexity S_{K,\\infty} scales linearly with Fock-space dimension in the ergodic phase and sublinearly (N_H^\\alpha, \\alpha<1) in the MBL phase of the tilted-field Ising chain, while the long-time profile \\Lambda_n on the Krylov chain is stretched-exponential in MBL and essentially flat in the ergodic regime. The large-deviation entropy density \\Sigma(x) of eigenstate complexities then shows that the MBL sum is dominated by a vanishing fraction of anomalously complex eigenstates.\n\nWhat is new is the systematic finite-size scaling of S_{K,\\infty}, the collapse of \\langle\\Lambda_n\\rangle, the stretch-exponent analysis, and the multifractal IPR of the eigenstate contributions (Figs. 4–10, Table I). Earlier Krylov-complexity papers already used the measure as an ergodicity diagnostic; this work supplies the concrete geometric picture on the ordered Krylov chain and the rare-resonance statistics that make the distinction sharp. The ED data (L up to ~14–16, disorder averages, error bands) are internally consistent, the Gaussian-to-exponential change in the distribution of S_{K,\\infty} is clean, and the connection to the authors’ earlier Fock-space work is properly cited rather than recycled as a new claim.\n\nThe soft spot is exactly the one the reader flagged: the phenomenological exponential distributions of decay lengths in Sec. IV C are chosen for analytic convenience and only a posteriori match the observed stretch exponents (\\gamma~1/2 at accessible sizes, asymptotically 1/3). They are not load-bearing; the central scalings come straight from the Lanczos-generated Krylov basis of the microscopic Hamiltonian. Finite-size caveats on MBL numerics remain, but they are generic and already acknowledged. No code is shipped, yet the model and methods are standard and re-implementable.\n\nThis is for people who already work on MBL diagnostics or Krylov complexity and want a transparent geometric probe. It does not settle the existence of the MBL phase, but it is a solid, well-written extension that a serious referee should see. I would accept it for peer review and would cite the scaling and large-deviation results.","headline":"Clean ED demonstration that infinite-time Krylov spread complexity scales as N_H vs N_H^\\alpha (\\alpha<1) and that MBL profiles are stretched-exponential, with a solid large-deviation picture of rare resonant eigenstates.","tokens_in":24930,"tokens_out":586,"would_cite":true,"duration_ms":6376,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Long-time Krylov spread complexity scales linearly with Fock-space size in the ergodic phase and only sublinearly in the MBL phase, so the late-time state fills a finite versus vanishing fraction of the Krylov chain.","keywords":["Krylov spread complexity","many-body localisation","ergodic phase","disordered spin chain","stretched exponential","large-deviation analysis","Fock-space dimension","eigenstate resonances"],"falsifier":"Exact diagonalisation of the same model at larger system sizes that extracts both the scaling exponent α of S_{K,∞} and the stretch exponent γ of ⟨Λ_n⟩; if α remains 1 deep in the putative MBL regime or if γ fails to approach the predicted asymptotic value 1/3, the central geometric claim is falsified.","tokens_in":24949,"feed_emoji":"⛓️","tokens_out":817,"duration_ms":8357,"temperature":0.7,"pith_summary":"The paper asks how a many-body quantum state spreads when it is written in the ordered Krylov basis generated by its own Hamiltonian, and whether that basis-optimised spread cleanly separates the ergodic phase from the many-body localised (MBL) phase of a disordered spin chain. It finds that the infinite-time Krylov spread complexity grows linearly with the dimension of Hilbert space in the ergodic regime, so the late-time state occupies a finite fraction of the Krylov chain, while the same quantity grows only as a sublinear power of the dimension in the MBL regime, so the state remains confined to a vanishing fraction of the chain. The average probability profile along the chain itself decays as a stretched exponential in the MBL phase; a simple phenomenological model attributes the stretch to a broad distribution of exponential decay lengths across eigenstates. A large-deviation analysis of the individual eigenstate contributions confirms that almost every eigenstate participates in the ergodic sum, whereas only a vanishing (yet still exponentially large) fraction of rare, anomalously complex eigenstates dominate the MBL sum. The result supplies a transparent, one-dimensional geometric picture of how localisation and rare resonances appear once distance is measured in the Krylov basis rather than on the high-dimensional Fock-space graph.","feed_headline":"Krylov complexity fills the chain in ergodic phase, not in MBL","feed_subtitle":"Late-time states occupy a finite fraction of Krylov space when ergodic, a vanishing fraction when many-body localised","key_machinery":"Krylov spread complexity S_{K,∞} = Σ_n n Λ_n, where Λ_n is the infinite-time probability of finding the state on the n-th Krylov orbital; this quantity is the first moment of a probability distribution on a one-dimensional chain of length equal to the Fock-space dimension and is basis-optimised by construction of the Krylov basis.","core_discovery":"Infinite-time Krylov spread complexity scales as N_H in the ergodic phase of the disordered tilted-field Ising chain and as N_H^α with α<1 in the MBL phase; the associated disorder-averaged profile Λ_n on the Krylov chain is flat (after a short transient) when ergodic and stretched-exponential when many-body localised, the latter arising because the infinite-time state is a weighted sum of exponentially decaying eigenstate amplitudes whose characteristic lengths are themselves broadly distributed.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Krylov spread fills finite chain fraction in ergodic phase not MBL","Infinite-time Krylov complexity scales linearly ergodic sublinear MBL","MBL Krylov profile decays stretched-exponentially unlike ergodic flat","Ergodic states span finite Krylov chain MBL only vanishing fraction","MBL Krylov complexity dominated by rare high-spread eigenstates"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The analytic explanation for the stretched-exponential profile assumes that each eigenstate decays purely exponentially on the Krylov chain with a length drawn from a simple exponential distribution, and that those lengths themselves are exponentially distributed across disorder realisations.","fun_headline_variants_meta":{"raw":{"variants":["Krylov spread fills finite chain fraction in ergodic phase not MBL","Infinite-time Krylov complexity scales linearly ergodic sublinear MBL","MBL Krylov profile decays stretched-exponentially unlike ergodic flat","Ergodic states span finite Krylov chain MBL only vanishing fraction","MBL Krylov complexity dominated by rare high-spread eigenstates"]},"model":"grok-4.5","effort":"low","cost_usd":0.005404,"raw_usage":{"total_tokens":1520,"prompt_tokens":833,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":54040000,"prompt_tokens_details":{"text_tokens":833,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":589,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":833,"tokens_out":98,"duration_ms":5772,"temperature":1.0,"reasoning_tokens":589,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T18:02:16.938211+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exact diagonalisation of the same model at larger system sizes that extracts both the scaling exponent α of S_{K,∞} and the stretch exponent γ of ⟨Λ_n⟩; if α remains 1 deep in the putative MBL regime or if γ fails to approach the predicted asymptotic value 1/3, the central geometric claim is falsified.","supporting_citations":[],"review_version":1}