{"id":"17bb2ef8-7d0c-475e-85ea-25e1768cdaf3","arxiv_id":"2608.02451","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In generalized and long-range Aubry–André–Harper models, the long-time averaged spread complexity shows kinks at mobility-edge crossings, and its Lanczos coefficients change shape (plateau-then-decay versus constant) depending on whether mobility edges exist.","lead":"This paper studies spread complexity — how a quantum state spreads across the Krylov basis — in the Aubry–André–Harper model and its generalized and long-range versions. The main finding is that the long-time averaged complexity develops sharp kinks when a quench crosses a mobility edge, offering a dynamical way to spot the boundary between localized and extended states.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Kink nonanalyticity and accuracy are unverified: no system-size scaling, no phase averaging, and observed kinks at λ_f≈3,1 vs predicted 2.6,1.4 for β=0.3.","rationale":"The reader's weakest_assumption focuses on the analytic moment formulas in Sec. III.C, which is a genuine internal inconsistency and should be corrected. However, that section is not load-bearing for the central claim of mobility-edge detection via kinks; the analytic formulas are presented as an asymptotic check of the Lanczos-coefficient shape, while the kink detection rests on the numerical C(λ_f) data. The most load-bearing weakness for the central claim is the numerical evidence itself: the paper's own Fig. 4 shows kinks at positions that deviate from the predicted Eq. (9) values for the ground and highest excited states, and no finite-size scaling or phase averaging is provided to show that these deviations vanish and that the kinks become true nonanalyticities. Because the abstract explicitly claims 'nonanalytic behavior' and 'accurately identifying the mobility-edge transition,' these omissions directly undermine the central claim. The reader's rationale already notes the kink-position mismatch and lack of scaling, though it lists them as secondary to the formula errors; hence partial agreement. The proposed test—finite-size scaling of the kink location and curvature peak—would settle whether the observed kinks are genuine mobility-edge signatures. The verdict remains CONDITIONAL as the reader judged; no new information moves it to ACCEPT or REJECT, so UNCHANGED is appropriate.","tokens_in":17086,"tokens_out":6247,"duration_ms":58949,"concrete_test":"For β=0.3 and each initial state (ground, mid, highest of λ_in=0), compute C(λ_f) for N=500,1000,2000,4000,8000, and at least 10 randomly chosen φ values, averaging C over φ (or after confirming individual φ curves). Locate the kink as the maximum of |∂²C/∂λ_f²| (or a finite-difference peak) with λ_f grid spacing ~0.01. Check whether the kink position λ_max(N) extrapolates to the Eq. (9) values {2.6,2,1.4} as N→∞ and whether the curvature peak height grows (e.g., ∝ N^γ) indicating a genuine nonanalyticity. If λ_max(N) converges to the predicted values with correct scaling, the central claim is supported; if it saturates at 3 and 1 or the curvature peak does not grow, the claim of accurate nonanalytic detection is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (abstract) is that time-averaged spread complexity exhibits nonanalytic behavior at the mobility edge and accurately identifies it. The evidence is Fig. 4: kinks in C(λ_f) for three initial states at λ_f≈1,2,3 for β=0.3, N=2000/5000. This evidence is insufficient for two reasons. (1) No demonstration of nonanalyticity. A finite-size kink is necessarily smooth; to show a nonanalytic point in the thermodynamic limit one must exhibit a divergence or discontinuity in some derivative of C as N→∞. The paper only asserts (footnote 75) that 'kinks become more pronounced' at larger sizes, without data. Without a scaling collapse or derivative analysis, the term 'nonanalytic' is unsupported. (2) No quantitative accuracy. The predicted mobility-edge crossings from Eq. (9), βE0 = 2 - λ, with the λ=0 initial-state energies E0 = -2 (ground), 0 (mid), +2 (highest), give λ* = 2.6, 2.0, 1.4 respectively. The observed kinks are at ≈3, ≈2, ≈1. The ground and highest excited states are off by ≈0.4, an error of 15–30% that is not accounted for. This discrepancy could be finite-size, but no scaling is provided to verify convergence to Eq. (9). Additionally, no quasiperiodic phase φ is specified; C(λ_f) presumably depends on φ for finite N, and without phase averaging or multiple φ values the kink location could be dominated by a particular φ. The analytic-moment formulas in Sec. III.C are also internally inconsistent (Eqs. 15–17 violate ‖H‖ for the stated parameters), but that section supports the Lanczos-coefficient phenomenology rather than the kink-detection claim itself. The kink-detection claim stands on the numerical evidence above, which is where the load-bearing weakness lies.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the long-time averaged spread complexity of quantum quenches in generalized and long-range Aubry-André-Harper models. The central claim is that, in the generalized AAH model with an energy-dependent mobility edge, the long-time averaged spread complexity exhibits nonanalytic behavior when the post-quench quasiperiodic potential crosses the mobility edge associated with the initial eigenstate energy, thereby serving as an accurate dynamical probe of the mobility-edge transition. The authors support this with numerical kink detection in C(λ_f) for three initial states, an LDOS-broadening argument tied to the first Lanczos coefficient, and an analytic derivation of the moments and Lanczos coefficients for quenches between the limits λ_in=0 and λ_f→∞. They also analyze the long-range hopping extension, reporting qualitatively different Lanczos-coefficient behavior for forward versus backward quenches.","tokens_in":17437,"tokens_out":6573,"duration_ms":55868,"significance":"If the central claim is correct, time-averaged spread complexity provides a simple and experimentally relevant dynamical order parameter for mobility-edge transitions in single-particle quasiperiodic systems. The proposed mechanism—LDOS broadening feeding the first Lanczos coefficient—is physically plausible and connects Krylov-space diagnostics to established localization concepts. The analytic moment computation for extreme quenches, once corrected, could be a useful benchmark for the community. However, the numerical evidence for nonanalyticity is currently incomplete, and the analytic formulas contain internal inconsistencies that undermine the claimed analytical verification. The qualitative distinction in Lanczos-coefficient shapes (plateau-then-decay versus constant) is interesting but rests on the same numerical and analytical foundations.","major_comments":[{"comment":"The central claim of 'nonanalytic behavior' and 'accurately identifying the mobility edge' is not established. The observed kinks at λ_f≈3, 2, 1 for β=0.3 deviate from the predictions of Eq. (9), which give λ* = 2.6, 2.0, 1.4 for the ground, middle, and highest excited initial states. The deviations for the ground and highest excited states are about 0.4, i.e., 15–30%, and no finite-size scaling or phase averaging is provided to show these kinks converge to the predicted values in the thermodynamic limit. Footnote 75 asserts that the second derivative exhibits sharp kinks, but no second-derivative data or scaling analysis is shown. A kink in a finite-size system is necessarily smooth; demonstrating a genuine nonanalyticity requires evidence of a developing singularity (e.g., a diverging derivative or a scaling collapse) as N→∞. Without this, the term 'nonanalytic' is an overstatement, an","section":"Section III.B, Fig. 4"},{"comment":"The analytic moment formulas are internally inconsistent and violate basic spectral constraints. From Eq. (15) with n=1, one obtains a0 = (λ_f/β)(√(1−β²)+1), whereas Eq. (16) states a0 = (λ_f/β)(1/√(1−β²)+1). Direct integration of the survival amplitude in Eq. (14) gives a0 ≈ 16.1 for β=0.3, λ_f=100, while Eq. (16) gives 683, which exceeds the operator norm ||H|| ≈ 143 for this parameter set. Additionally, Eq. (17) diverges as β→0, whereas it should reduce to b1=λ_f/√2, as derived later in the same section. These errors mean the 'analytical verification' in Fig. 6 is not reliable; the claimed agreement between numerics and analytics for β≠0 needs to be re-examined with corrected formulas.","section":"Section III.C, Eqs. (15)–(17)"},{"comment":"For the long-range hopping model, the mobility-edge prediction Eq. (26) is not quantitatively compared with the kink positions in Fig. 8. The text states that C exhibits a kink whenever the mobility edge is crossed, but no numerical values of the kink locations are given, and no finite-size scaling or phase averaging is provided. Given that the IPR in the localized phase does not approach unity for small α, the interpretation of the kinks as marking the mobility edge requires a quantitative check against Eq. (26). The same concerns as in Fig. 4 apply here, and the claim that spread complexity 'accurately identifies' the mobility edge in the long-range model is not substantiated.","section":"Section IV, Eq. (26) and Fig. 8"}],"minor_comments":[{"comment":"The moment definition μ_n = ⟨K0|(iH)^n|K0⟩ is inconsistent with the survival amplitude S(t)=⟨e^{−iHt}⟩ in Eq. (14). The derivative in Eq. (4) should yield (−iH)^n, or equivalently μ_n should be defined as ⟨H^n⟩. This sign convention affects Eq. (15); please clarify and make the formulas consistent.","section":"Eq. (4)"},{"comment":"The white vertical lines are not explained in the caption. Please state that they mark the predicted mobility-edge crossings from Eq. (9) and list the corresponding λ_f values. Also specify the system size N and the quasiperiodic phase φ used for all figures.","section":"Fig. 4 caption"},{"comment":"The claim that the second derivative of C exhibits sharp kinks is central to the nonanalyticity argument, but no plot or quantitative analysis is provided. Either include the second-derivative data or temper the claim.","section":"Footnote 75"},{"comment":"The binomial sum expression for the NNN-hopping moments is difficult to parse, and the parity condition 'p even' appears after the sum over p without clear enforcement. Please check the formula and clarify the summation limits.","section":"Eq. (25)"},{"comment":"Minor typos include 'the the Sachdev-Ye-Kitaev model' in the introduction. Also, the symbols λ and λ_f are used interchangeably at times (e.g., in the AAH transition description); please standardize the notation.","section":"Section II"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a topical question and the qualitative phenomenology is plausible, but the current evidence for the central claim is incomplete and the analytic section contains concrete errors. The kink positions deviate from the exact mobility-edge formula by 15–30% without finite-size scaling or phase averaging, so the 'accurate identification' claim is not yet supported. The internal inconsistencies in Eqs. (15)–(17) are likely correctable, but they undermine the paper's analytical verification as written. I would like to see a revised version with (i) a finite-size scaling analysis demonstrating a developing nonanalytic feature and convergence of the kink positions to Eq. (9), (ii) phase averaging or at least a demonstration of phase independence, and (iii) corrected moment/Lanczos formulas for β≠0. These are significant but feasible revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ben — quick take on 2608.02451. The central observation is that the long-time averaged spread complexity develops kinks when a quench crosses a mobility edge in the generalized AAH model. I think that is probably true and worth knowing. But be careful: the paper's analytic verification section has concrete errors, and the numerical evidence doesn't yet support the 'nonanalytic' language.\n\nWhat's new here is applying the time-averaged spread-complexity diagnostic to the gAAH model with β≠0 and to long-range hopping, where mobility edges are energy-dependent. The LDOS-broadening mechanism in Sec. III.B is a nice, simple explanation for why the kink appears: the first Lanczos coefficient b_1 is the variance of the LDOS, and the variance jumps when the initial state starts overlapping many more eigenstates. The β=0 moment calculation for the AAH model is standard but cleanly presented.\n\nThe problems start in Sec. III.C. Equations (15)–(17) are not consistent with each other or with the model. Direct integration gives μ_1 = −i λ_f/β (1/√(1−β²) − 1), so a_0 = μ_1/i should be about −16 for β=0.3, λ_f=100. The paper's Eq. (16) gives +683, which exceeds the operator norm of the Hamiltonian (≈143). Eq. (17) doesn't reduce to b_1=λ_f/√2 as β→0. That section needs to be redone.\n\nThe kink detection itself rests on Fig. 4, and there are two gaps. First, the paper calls the kinks 'nonanalytic' but offers no system-size scaling; footnote 75 says they get sharper with N but no data. For a finite system any kink is smooth, so 'nonanalytic' is not established. Second, the kink positions are λ_f≈3, 2, 1 for ground, middle, highest states with β=0.3, while Eq. (9) gives 2.6, 2.0, 1.4. The middle matches, the other two are off by 15–30%. That could be finite-size, but without scaling or a specified quasiperiodic phase φ, it's hard to judge. No phase averaging is reported.\n\nThese are fixable issues. The core idea is sound and the numerical observation is likely correct. I'd want to see corrected analytic formulas, a scaling analysis, and explicit handling of φ before treating the 'accurate identification of mobility edge' claim as established.\n\nThis paper is for people working on Krylov complexity as a probe of localization transitions. It's a useful incremental contribution if the numerics are tightened. I'd send it to peer review rather than desk-reject, but with a strong request for major revision.","headline":"The kink in time-averaged spread complexity at mobility-edge crossings is a plausible and probably correct observation, but the analytic Lanczos-coefficient formulas are wrong and the numerical case for 'nonanalytic' is under-supported.","tokens_in":18077,"tokens_out":5011,"would_cite":false,"duration_ms":41156,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that long-time averaged spread complexity kinks exactly when a quantum quench crosses a mobility edge, making it a dynamical order parameter for localization transitions in Aubry-André-Harper models.","keywords":["spread complexity","Krylov complexity","Aubry-André-Harper model","mobility edge","localization transition","Lanczos coefficients","quantum quench","long-range hopping"],"falsifier":"Compute the first Lanczos coefficient directly for β=0.3, λ_f=100: Eq. (16) yields a₀ ≈ 683, while a direct integration of the survival-amplitude moment gives a₀ ≈ 16 and the Hamiltonian's operator norm is about 143; if a₀ exceeds the norm bound, the analytic expressions in Eqs. (15)–(17) cannot be correct.","tokens_in":16882,"feed_emoji":"⚛️","tokens_out":4314,"duration_ms":40885,"temperature":0.7,"pith_summary":"The paper tries to establish that a single dynamical quantity—the time-averaged spread complexity of a state—can act as an order parameter for localization transitions in one-dimensional quasiperiodic systems, including cases with energy-dependent mobility edges. It argues that when the final quasiperiodic potential in a quench crosses the mobility edge tied to the initial state's energy, the long-time averaged complexity develops a sharp kink, and this kink matches the phase boundary obtained from the inverse participation ratio. The mechanism is the broadening of the local density of states across the transition, which directly feeds the first Lanczos coefficient through its variance. The paper also derives analytic moments and Lanczos coefficients for extreme quenches and shows that their shape (constant versus plateau-then-decay) encodes whether a mobility edge is present. If correct, this gives a dynamical probe of localization that requires only time evolution and survival-probability data.","feed_headline":"Time-averaged spread complexity kinks at mobility edges","feed_subtitle":"A single dynamical measure, averaged over time, marks where localized and extended states meet in Aubry-André-Harper models.","key_machinery":"Spread complexity is the average position of the time-evolved state on the Krylov basis built from the Hamiltonian, C(t)=Σ_n n |⟨K_n|ψ(t)⟩|². The Lanczos coefficients a_n, b_n from the Lanczos recursion define this basis, and the identity b₁² = variance of the local density of states connects spectral broadening directly to complexity growth. The diagnostic used throughout is the long-time averaged value C̄. In the generalized AAH model the mobility edge is the energy E0 satisfying βE0 = 2t − λ; in the long-range model the paper uses the energy-dependent boundary E = λ cosh(α ln 2) − t.","core_discovery":"For quenches between eigenstates of the Aubry-André-Harper Hamiltonian at different potential strengths, the long-time averaged spread complexity shows a nonanalytic kink exactly when the final potential λ_f crosses the mobility edge associated with the initial state's energy. In the standard AAH model, which has no mobility edge, the kink sits at λ_f=2; in the generalized β≠0 model, the kink position shifts with the initial state's energy according to βE0 = 2t − λ, matching IPR phase boundaries. The broadening of the local density of states across the transition is identified as the cause, via b₁² = σ²_LDOS. In the long-range hopping model, the averaged spread complexity still finds the mob","pith_inferences":["Extension not pursued in the paper: if LDOS broadening is the mechanism, then initial states tuned exactly to the mobility-edge energy should show the sharpest kink in C̄, which could be tested by scanning C̄ continuously over initial-state energy rather than only three energy sectors.","The same reasoning suggests C̄ should also detect mobility edges in disordered Anderson models, where no quasiperiodic structure exists, as long as a mobility edge separates localized and delocalized eigenstates.","The plateau-then-decay shape of Lanczos coefficients may offer a finite-size-independent fingerprint of mobility edges extractable from short-time survival data, avoiding the need for long-time averaging.","Because C̄ depends only on survival-amplitude moments, it could be measurable in cold-atom or photonic waveguide experiments through interference or intensity readouts, although the paper does not address experimental implementation."],"forward_implications":["C̄ can locate mobility edges without computing eigenstate localization measures like IPR; a single time-averaged quantity suffices.","The first Lanczos coefficient b₁ inherits the LDOS broadening, so Lanczos-coefficient data carry the same transition signal as C̄.","Lanczos coefficients distinguish models with and without mobility edges: nearly constant in the AAH model, plateau-then-decay in the generalized AAH model, and decaying in the long-range model for localized-to-extended quenches.","For quenches from extended to localized phases in the long-range model, the moments coincide with those of the short-range AAH model; the reverse direction does not.","The kinks sharpen with system size, consistent with a genuine dynamical phase transition in the thermodynamic limit."],"fun_headline_variants":["Averaged spread complexity pinpoints mobility edges","Quench probe: spread complexity kink marks mobility edge","Time-averaged complexity finds mobility-edge transition","Nonanalytic spread complexity locates mobility edges","Spread complexity kink: new detector for mobility edges"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is that in the infinite-potential limit the post-quench eigenstates are exactly single-site localized, so the survival amplitude reduces to the averaged phase in Eq. (14); if that idealization is wrong or the derived moments are inconsistent, the analytic support for the Lanczos-coefficient claims collapses even though the numerical kink detection may still stand.","fun_headline_variants_meta":{"raw":{"variants":["Averaged spread complexity pinpoints mobility edges","Quench probe: spread complexity kink marks mobility edge","Time-averaged complexity finds mobility-edge transition","Nonanalytic spread complexity locates mobility edges","Spread complexity kink: new detector for mobility edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000492,"raw_usage":{"total_tokens":2260,"prompt_tokens":753,"completion_tokens":1507,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":1448}},"tokens_in":497,"tokens_out":1507,"duration_ms":23140,"temperature":1.0,"reasoning_tokens":1448,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:06:05.546134+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first Lanczos coefficient directly for β=0.3, λ_f=100: Eq. (16) yields a₀ ≈ 683, while a direct integration of the survival-amplitude moment gives a₀ ≈ 16 and the Hamiltonian's operator norm is about 143; if a₀ exceeds the norm bound, the analytic expressions in Eqs. (15)–(17) cannot be correct.","supporting_citations":[],"review_version":1}