{"id":"9a8c0557-9c49-411e-9166-09b58864e585","arxiv_id":"2605.24115","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Stroboscopic Floquet mapping of Anderson and Aubry-André dynamics yields operator Krylov signatures that distinguish localized, delocalized, and critical regimes via distributions, wavefronts, and power spectra.","lead":"The paper applies operator Krylov space methods to Anderson localization by mapping continuous-time Hamiltonian dynamics to stroboscopic Floquet times. This may enable more efficient spectral computations and new phase diagnostics in operator space.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Stroboscopic Floquet mapping may not faithfully reproduce continuous-time localization transition","rationale":"The reader's weakest assumption directly pinpoints the unverified fidelity of the stroboscopic-to-Floquet reduction. No other internal inconsistency (e.g., in the recursive parameter generation or the preference for disorder-averaged autocorrelation) appears load-bearing for the phase correspondence. Because the full text was not supplied in the query, the concern remains open and the verdict stays UNVERDICTED.","tokens_in":1799,"tokens_out":390,"duration_ms":25545,"concrete_test":"For the Aubry-André model at fixed quasiperiodic strength λ near 2, compute the disorder-averaged inverse participation ratio both from continuous-time evolution (via exact diagonalization or Trotterization) and from the stroboscopic Floquet operator at the same T; if the extracted critical λ or scaling of IPR with system size differs by more than the reported statistical error, the mapping does not faithfully capture the localization physics.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central claim equates delocalized/localized phases of the Aubry-André Hamiltonian to Porter-Thomas distributions, ballistic/localized wavefronts, and smooth/discrete Bernstein-Szegő spectra in operator Krylov space. This equivalence rests on the stroboscopic mapping (dynamics at t = nT) to an effective Floquet operator whose Krylov parameters are generated recursively from an inhomogeneous transverse-field Ising chain. The mapping is asserted to preserve localization physics, yet the abstract and construction provide no direct check that the critical point (e.g., λ=2), localization length, or multifractal exponents match those of the original continuous-time Schrödinger evolution. If the discrete sampling introduces commensurate resonances or alters the effective disorder distribution, the observed Krylov signatures could be artifacts of the Floquet reduction rather than intrinsic to Anderson localization.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies Anderson localization and the Aubry-André localization-delocalization transition via operator Krylov space. Dynamics are sampled at stroboscopic times and mapped to an effective Floquet operator whose Krylov parameters are generated recursively from an inhomogeneous transverse-field Ising chain. The delocalized phase is reported to exhibit a Porter-Thomas distribution, a ballistically propagating wavefront, and a smooth Bernstein-Szegő power spectrum in Krylov space, while the localized phase shows the absence of these features; the transition and multifractal scaling at criticality are also claimed to appear in the Krylov description. A moment method extracts parameters from the discrete-time autocorrelation, and disorder-averaged autocorrelation (rather than averaged parameters) is argued to yield a more physical spectral function.","tokens_in":1961,"tokens_out":563,"duration_ms":21730,"significance":"If the central correspondences hold, the work supplies a computationally lighter Krylov-space route to localization diagnostics and links the problem to an effective Floquet Ising chain with recursively generated parameters. The explicit demonstration of wavefront propagation, Porter-Thomas statistics, and Bernstein-Szegő spectra as phase indicators, together with the multifractal signature at criticality, would constitute a concrete new perspective on Anderson localization.","major_comments":[{"comment":"The central claim equates Krylov-space signatures to the localization transition of the original continuous-time Aubry-André Hamiltonian, yet the manuscript provides no quantitative comparison of the critical point (λ=2), localization length, or multifractal exponents between the stroboscopic Floquet construction and the continuous-time Schrödinger evolution. Without such a check, it remains unclear whether the observed Porter-Thomas distribution, ballistic wavefront, and spectral features are intrinsic to Anderson localization or artifacts of the discrete-time reduction.","section":"Floquet mapping and Krylov construction (abstract and § on effective Ising model)"},{"comment":"The assertion that the disorder-averaged autocorrelation function produces a “more physical” spectral function than the disorder-averaged Krylov parameters is load-bearing for the reported phase distinctions, but the manuscript does not demonstrate that this choice recovers the known continuous-time spectral properties or localization length scaling.","section":"Spectral function extraction via moment method"}],"minor_comments":[{"comment":"The spelling “Berstein-Szegő” appears in the abstract; the standard term is Bernstein-Szegő.","section":"Abstract"},{"comment":"Notation for the effective Floquet Ising parameters and the recursion step index should be introduced with explicit equations rather than described only in prose.","section":"Krylov parameter recursion"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the constructive major comments. The points raised highlight the need for explicit validation of the stroboscopic mapping against continuous-time benchmarks. We address each comment below and have revised the manuscript to include the requested quantitative comparisons.","responses":[{"response":"We agree that a direct side-by-side comparison strengthens the central claim. The stroboscopic Floquet operator is constructed exactly from the time-evolution operator at integer periods, so the localization-delocalization transition remains at the same critical value λ=2 as in the continuous-time Aubry-André model. In the revised manuscript we add a new subsection that extracts the localization length from the Krylov wavefront velocity and from the inverse participation ratio of the Krylov basis states, and we compare these scalings quantitatively with the known continuous-time results (both analytic and numerical) for the Aubry-André model. The multifractal exponents at criticality are likewise recomputed from the Krylov-space IPR and shown to match the literature values within numerical precision. These additions demonstrate that the reported signatures are not discretization artifacts.","revision_made":"yes","referee_comment":"[Floquet mapping and Krylov construction (abstract and § on effective Ising model)] The central claim equates Krylov-space signatures to the localization transition of the original continuous-time Aubry-André Hamiltonian, yet the manuscript provides no quantitative comparison of the critical point (λ=2), localization length, or multifractal exponents between the stroboscopic Floquet construction and the continuous-time Schrödinger evolution. Without such a check, it remains unclear whether the observed Porter-Thomas distribution, ballistic wavefront, and spectral features are intrinsic to Anderson localization or artifacts of the discrete-time reduction."},{"response":"We acknowledge that the manuscript did not previously contain an explicit validation of the spectral function obtained from the disorder-averaged autocorrelation. In the revised version we include a direct comparison: the spectral function reconstructed via the moment method from the averaged autocorrelation is shown to reproduce (i) the expected power-law scaling of the localization length near λ=2 and (ii) the known continuous-time density of states features of the Aubry-André model. This comparison is presented both for the delocalized and localized regimes and at criticality, confirming that the choice yields physically consistent results.","revision_made":"yes","referee_comment":"[Spectral function extraction via moment method] The assertion that the disorder-averaged autocorrelation function produces a “more physical” spectral function than the disorder-averaged Krylov parameters is load-bearing for the reported phase distinctions, but the manuscript does not demonstrate that this choice recovers the known continuous-time spectral properties or localization length scaling."}],"tokens_in":1554,"tokens_out":569,"duration_ms":24121,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core move is to sample the Aubry-André dynamics only at discrete times, recast it as a Floquet problem, and build the operator Krylov space from the resulting effective operator. This produces a recursive construction that maps the problem onto the edge operator of an inhomogeneous Floquet transverse-field Ising chain whose parameters are generated step by step.\n\nWhat the work actually delivers is a moment method that pulls Krylov parameters straight from the discrete autocorrelation function, plus the observation that averaging the autocorrelation first and then extracting the spectral function is cleaner than averaging the parameters themselves. The delocalized phase is tied to a Porter-Thomas distribution, ballistic spread in Krylov space, and a smooth Bernstein-Szegő spectrum; the localized phase lacks these features. At the critical point they recover multifractal scaling in the long-time dynamics and inverse participation ratio.\n\nThe computational claim is that the stroboscopic route reduces resources needed for the spectral function. The narrowing of the disorder-averaged Krylov-parameter distribution with recursion depth is also shown.\n\nThe soft spot is the untested assumption that the Floquet stroboscopic map preserves the localization physics of the original continuous-time problem. The abstract does not report a direct comparison of the critical point, localization length, or multifractal exponents against the known continuous-time values, so it remains possible that the observed Krylov signatures are partly shaped by the discrete sampling. If the full text contains such checks, the concern shrinks; otherwise it stays material.\n\nThis is for people already working on Krylov methods or Floquet approaches to localization. It is technically grounded and engages the relevant literature without obvious internal contradictions. It deserves a serious referee.","headline":"The paper maps Anderson localization in the Aubry-André model to operator Krylov space via stroboscopic Floquet dynamics and reports phase-specific signatures in distributions, wavefronts, and spectra.","tokens_in":2417,"tokens_out":425,"would_cite":false,"duration_ms":25063,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Stroboscopic Floquet mapping lets operator Krylov space diagnose Anderson localization.","keywords":["Anderson localization","Aubry-André model","operator Krylov space","Floquet dynamics","Porter-Thomas distribution","localization transition","spectral function","wavefront propagation"],"falsifier":"A direct numerical check, in a small system whose continuous-time localization properties are known independently, whether the stroboscopic Krylov wavefront remains ballistic in a regime that should be localized.","tokens_in":2709,"feed_emoji":"","tokens_out":769,"duration_ms":42664,"temperature":0.7,"pith_summary":"The paper shows how to study Anderson localization and the Aubry-André transition by mapping Hamiltonian dynamics at discrete times to an effective Floquet problem. This mapping produces an operator Krylov space description in which the spectral function and Krylov parameters can be obtained with reduced computational cost via a moment method applied to the discrete autocorrelation. In the resulting picture the delocalized phase produces a Porter-Thomas distribution, a wavefront that moves ballistically through Krylov space, and a smooth Bernstein-Szegő power spectrum, while the localized phase produces none of these features. The same diagnostics locate the localization-delocalization transition and reveal multifractal scaling at the critical point.","feed_headline":"Krylov space detects Anderson localization via wavefront and spectrum","feed_subtitle":"Stroboscopic Floquet mapping produces ballistic spread and Porter-Thomas statistics only in the delocalized phase.","key_machinery":"The recursively generated Krylov parameters of the edge operator under the effective Floquet transverse-field Ising map, extracted from the discrete-time autocorrelation via the moment method.","core_discovery":"By recasting stroboscopic evolution as the dynamics of an edge operator in an inhomogeneous Floquet transverse-field Ising chain whose parameters are generated recursively, the delocalized phase is marked by the appearance of a Porter-Thomas distribution, a ballistically propagating wavefront in operator Krylov space, and a smooth power spectrum; the localized phase shows the absence of these signatures together with a stationary wavefront and a discrete spectrum. Disorder averaging performed on the autocorrelation function rather than on the Krylov parameters yields the more physical spectral function. The transition is visible directly in Krylov space, and the critical point itself display","pith_inferences":["Tracking the speed of the Krylov wavefront could supply a dynamical estimate of the localization length without requiring full eigenstate analysis.","The recursive construction of the effective Ising parameters may be viewed as an explicit renormalization flow whose fixed-point structure encodes the localization transition.","The same Krylov diagnostics could be applied to interacting Floquet systems to test whether many-body localization produces an analogous absence of ballistic wavefronts."],"forward_implications":["The spectral function computed from the disorder-averaged autocorrelation is physically more relevant than the one obtained from disorder-averaged Krylov parameters.","The localization-delocalization transition appears in Krylov space as the change from a discrete to a smooth power spectrum and from a stationary to a propagating wavefront.","At the critical point a Porter-Thomas distribution coexists with multifractal scaling of the inverse participation ratio and long-time dynamics.","The narrowing of the distribution of Krylov parameters with recursion depth occurs in both phases but does not erase the phase distinction."],"fun_headline_variants":["Floquet operator Krylov space probes Anderson localization","Krylov space maps stroboscopic dynamics to Floquet Ising edge","Ballistic Krylov wavefront indicates delocalized Anderson phase","Porter-Thomas stats emerge in delocalized operator Krylov space","Critical point shows multifractal Krylov scaling with size"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That sampling the continuous-time Hamiltonian evolution only at stroboscopic instants produces Krylov-space quantities that still correctly distinguish localized from delocalized behavior.","fun_headline_variants_meta":{"raw":{"variants":["Floquet operator Krylov space probes Anderson localization","Krylov space maps stroboscopic dynamics to Floquet Ising edge","Ballistic Krylov wavefront indicates delocalized Anderson phase","Porter-Thomas stats emerge in delocalized operator Krylov space","Critical point shows multifractal Krylov scaling with size"]},"model":"grok-4.3","cost_usd":0.006372,"raw_usage":{"total_tokens":3052,"prompt_tokens":791,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":63724500,"prompt_tokens_details":{"text_tokens":791,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2178,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":791,"tokens_out":83,"duration_ms":25815,"temperature":1.0,"reasoning_tokens":2178,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T14:28:33.283730+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct numerical check, in a small system whose continuous-time localization properties are known independently, whether the stroboscopic Krylov wavefront remains ballistic in a regime that should be localized.","supporting_citations":[],"review_version":1}