{"id":"3c1803d2-7c78-4bce-862c-76e1c8fcd278","arxiv_id":"2411.16302","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"For bosonic and fermionic Schrödinger fields with chemical potential μ≤0, the Lanczos coefficients grow linearly and the Krylov complexity grows exponentially with an extracted rate near 2.75/β, below the 4/β slope prediction.","lead":"This paper computes Krylov complexity, a measure of how quantum operators spread, for non-relativistic Schrödinger fields with a chemical potential. It reports exponential spreading with a slower rate than expected, which tests how universal chaos bounds are outside relativistic quantum field theory.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim rests on a 0.05β fit window never shown to be asymptotic; the linear a_n term sits at the critical Stark/SL(2,R) ratio γ/α=2, where the paper's own analogue gives non-exponential growth, so λ_K≈2.746/β is likely a transient.","rationale":"The reader's rejection is based on the finite-time fit not establishing the asymptotic regime; I agree this is the most load-bearing weakness. However, the reader's rationale that a_n 'drops out of |φ_n|² dynamics' is not quite right: in the unitary Krylov chain, a linear a_n acts as a Stark field, affecting the relative phases of neighboring sites and therefore the probability flow. The paper's own SL(2,R) example shows that at the critical ratio γ/α=2 the growth is not exponential, and the fitted Lanczos coefficients sit precisely at that ratio. This strengthens the concern: the claimed λ_K≈2.746/β may be fitting a transient before Stark localization or other non-asymptotic behavior sets in. The analytic derivations of the Wightman spectrum and the linear Lanczos coefficients appear consistent and may be salvageable, but the central asymptotic claim is not supported by the presented numerics. The verdict should remain REJECT, as the reader concluded.","tokens_in":24789,"tokens_out":34070,"duration_ms":303027,"concrete_test":"Extend the Runge-Kutta evolution of Eq. (2.24) for the fitted coefficients to t/β≥5, with nmax large enough that the wavefront (n_front~e^{4t/β}) stays far below the cutoff, and compute d log K/dt in sliding windows. Also run the same calculation with a_n set to zero; for a_n=0 the exact solution is K=cosh²(2t/β), and fitting the same t/β∈[0.45,0.5] window gives a slope near 3/β, not 4/β. If the instantaneous slope does not plateau near 2.746/β at longer times, the reported λ_K is a transient artifact and the claim should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline result is that K(t) grows exponentially with asymptotic rate λ_K≈2.746/β<4/β, attributed to the diagonal Lanczos coefficients a_n. The numerical evidence is a linear fit of log K(t) over t/β∈[0.45,0.5] (Figures 5 and 11, Table 2). No longer-time data or timescale estimate is provided. With βa_n≈−4n−4 and βb_n≈2n+1, the site-to-site potential drop is F≈4/β, giving a Bloch period 2π/F≈1.57β; the simulation ends at t/β=0.5, before one period. This matters because at the critical slope ratio γ/α=2 the authors' own SL(2,R) example (Eqs. 4.9–4.10) degenerates: in the limit γ→2α the sinh² factor becomes quadratic growth, not exponential growth. The numerically extracted coefficients satisfy exactly this critical ratio. Moreover, even with a_n=0, the same short window gives a fitted slope around 3/β, so observing a slope below 4/β in this window is not evidence of a suppressed asymptotic rate. The reader's statement that a_n 'drops out' of the |φ_n|² dynamics is too strong—a linear a_n contributes relative phases that can Stark-localize the chain—but the conclusion that the asymptotic claim is unverified is correct. The central claim therefore lacks support.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript computes the Wightman power spectrum for the free non-relativistic Schrödinger field in d=5 at finite temperature with chemical potential μ≤0, extracts the Lanczos coefficients {a_n,b_n} by the moment method, and solves the resulting discrete Schrödinger equation numerically. It reports that βa_n≈−4(n+1)−μ and βb_n≈2n+1 for both bosons and fermions, and that the Krylov complexity grows exponentially with an asymptotic rate λ_K≈2.746/β, substantially below twice the slope of b_n. The authors attribute this suppression to the non-Hermitian nature of the field operator and to the diagonal Lanczos coefficients a_n.","tokens_in":25147,"tokens_out":9230,"duration_ms":85196,"significance":"If the asymptotic claim were established, the paper would provide a QFT example in which the Krylov exponent is not set by twice the slope of b_n, and it would also generalize the no-staggering conditions of Camargo et al. The derivations of the spectral function, moments, and Lanczos coefficient recurrences are careful, and the paper includes a useful check of the normalization of φ_n(t) in Fig. 2. However, the central quantitative claim rests on a short finite-time fit and is not supported by the analysis as presented; the paper's own SL(2,R) analogy degenerates exactly at the fitted slope ratio, which is a serious internal tension.","major_comments":[{"comment":"The fitted coefficients (4.4)–(4.5), (4.16), (4.20)–(4.21), and (4.37)–(4.38) give βa_n≈−4n+O(1) and βb_n≈2n+O(1), so −a_n/(2b_n)→1. In the SL(2,R) example used to explain the suppression, the growth rate is 2α√(1−γ²/(4α²)); at γ=2α, Eq. (4.10) degenerates to a quadratic function of time, not an exponential. Since the extracted coefficients sit exactly at this critical ratio, the proposed explanation, taken at face value, predicts the opposite of the reported exponential asymptotic growth. A controlled argument that the discrete chain differs from this limit, or a revised interpretation, is required.","section":"§4.1.1, Eq. (4.10)"},{"comment":"The 'asymptotic' slope is extracted by fitting log K(t) on t/β∈[0.45,0.5] only. No timescale estimate for the onset of the asymptotic regime is given, no longer-time run is shown, and no comparison with the b_n-only prediction λ_K=4/β is made. With βa_n≈−4n−4, the local tilt is F≈4/β, so the Bloch period is 2π/F≈1.57β; the simulation ends at t/β=0.5, before a single period. A slope below 4/β over this window is therefore equally consistent with transient behavior, and the claim that λ_K≈2.746/β is the true asymptotic Krylov exponent is not established.","section":"§4.1.2, Fig. 5, Table 2; §4.2.3, Fig. 11"},{"comment":"The attribution of the reduced λ_K to a_n is qualitative. In Eq. (2.24), the diagonal term i a_n φ_n cancels in ∂_t|φ_n|², so a_n influences K(t)=1+Σ n|φ_n|² only indirectly through phases in the off-diagonal currents. No calculation shows that this phase effect converts the λ_K=4/β rate of the b_n-only chain into λ_K≈2.746/β, and the SL(2,R) analogy does not apply at the fitted critical ratio. A direct WKB or continuum analysis, or an explicit solution for the discrete chain, is needed to support the central claim.","section":"§2.1, Eq. (2.24); §5"}],"minor_comments":[{"comment":"The same symbol μ is used for the moments and for the chemical potential; the warning in footnote 3 appears only later, and a remark at Eq. (2.26) would reduce confusion.","section":"Eq. (2.26) and Section 3"},{"comment":"The caption and vertical axis do not make clear whether the plotted quantity is Σ_n|φ_n(t)|² or a single component; this should be stated explicitly.","section":"Figure 2"},{"comment":"The prefactor b(0)/b(x) in the continuum-limit solution suggests a dependence on the lattice parameter ε that is not spelled out; a sentence explaining the ε-scaling of b(x) would help.","section":"§4.2.2, Eq. (4.33)"},{"comment":"The conditions from [9] are cited in the main text as conditions for the absence of staggering, but Appendix C shows that they are not complete; the main text should flag this at the first mention.","section":"§2.1.1 and Appendix C"},{"comment":"The reported slopes are given without fit uncertainties or a statement of the number of fitted points; adding this information would help the reader judge the convergence of the bosonic and fermionic values to 2.746.","section":"Table 2"}],"recommendation":"reject","confidential_remarks":"I recommend rejection rather than major revision because the headline asymptotic result is not merely under-supported; it conflicts with the paper's own SL(2,R) critical limit. The Lanczos coefficient computations and the staggering discussion in Appendix C are useful and could form the basis of a future paper that does not claim an asymptotic exponential Krylov rate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look for the machinery, not for the headline. The paper derives the Wightman power spectrum for the Schrödinger field in five dimensions, uses the moment method to extract the two Lanczos sequences {a_n} and {b_n}, and finds linear coefficients with b_n independent of the chemical potential. That part is careful and reproducible, and the generalization of the staggering conditions in Appendix C is a genuine small contribution.\n\nThe problem is the central claim: that K(t) grows exponentially with asymptotic rate λ_K ≈ 2.746/β, well below the 4/β expected from twice the slope of {b_n}, and that the suppression comes from {a_n}. The numerical evidence is a fit of log K(t) over t/β ∈ [0.45, 0.5] in Figures 5 and 11, with no longer-time data or timescale estimate. With βb_n ≈ 2n+1 and βa_n ≈ -4(n+1)-μ, the ratio of slopes is exactly 2, which is the critical point in the authors’ own SL(2,R) example: Eq. (4.10) degenerates to quadratic growth in the limit γ → 2α. The Bloch period for the potential drop is about 1.57β, yet the simulation stops at 0.5β. So the extracted slope is at best a transient. The stress-test note is right that the reader’s claim that a_n “drops out” of |φ_n|² is too strong—a linear a_n Stark-localizes the chain—but the conclusion stands: the asymptotic claim is unverified.\n\nI would not want this published as is, but I would send it to a referee. The analytic derivations are solid, the Lanczos coefficients are likely correct, and the staggering criterion is useful. The fix is clear: run the time evolution longer (well past one Bloch period), estimate the asymptotic regime, and either confirm or retract the reduced rate. If the rate is genuinely 4/β, the paper becomes a straightforward application with a nice appendix; if a plateau emerges, that would be interesting. Either way, a referee can extract value from the core computation.","headline":"Useful analytic machinery for Krylov complexity in Schrödinger field theory, but the headline asymptotic exponent is not supported by the short-time numerics.","tokens_in":25667,"tokens_out":3133,"would_cite":true,"duration_ms":136400,"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":"The paper claims that Schrödinger-field Krylov complexity grows exponentially at roughly $2.75/\\beta$, below the $4/\\beta$ expected from twice the slope of $b_n$, because the non-Hermitian nature of the field makes $a_n$ nonzero and slows…","keywords":["Krylov complexity","operator growth","Lanczos coefficients","Schrödinger field theory","chemical potential","non-Hermitian operators","Wightman power spectrum","Lanczos algorithm"],"falsifier":"Integrate the same discrete Schrödinger equation to $t/\\beta>0.5$ (or refit the log-slope over $[0.5,1]$) and read the slope of $\\log K(t)$; if it moves from about $2.75/\\beta$ toward $4/\\beta$, the reported suppression is a transient of the finite fit window rather than the asymptotic Krylov exponent.","tokens_in":24561,"feed_emoji":"⚛️","tokens_out":7856,"duration_ms":68903,"temperature":0.7,"pith_summary":"This paper works out Krylov complexity—the average spread of a Heisenberg operator on its Krylov chain—for the non-relativistic Schrödinger field at finite temperature and chemical potential, in both bosonic and fermionic versions. It claims that in five spacetime dimensions the Lanczos coefficients are linear in $n$, with $\\beta b_n \\approx 2n+1$ independent of the chemical potential and $\\beta a_n \\approx -4(n+1)-\\mu$ shifting with $\\mu$. Numerically solving the resulting discrete Schrödinger equation, the authors find that late-time complexity is exponential, $K(t)\\sim e^{\\lambda_K t}$, with $\\lambda_K \\approx 2.75/\\beta$ for large $|\\mu|$. That value is noticeably smaller than twice the slope of $b_n$, namely $4/\\beta$, the rate the universal operator growth hypothesis would predict for linear $b_n$; the paper attributes the gap to non-Hermiticity, which makes $a_n$ nonzero and slows operator growth. If right, this is a concrete counterexample to the simple $\\lambda_K=2\\alpha$ rule outside Hermitian setups.","feed_headline":"Krylov complexity grows at 2.75/β, not expected 4/β","feed_subtitle":"In Schrödinger field theory, a nonzero Lanczos coefficient a_n slows operator growth below the usual 4/β rate.","key_machinery":"The central object is the pair of Lanczos coefficients $\\{a_n\\}$ and $\\{b_n\\}$, the on-site and hopping amplitudes in the one-dimensional Krylov chain where a Heisenberg operator spreads. They are obtained from the moments of the Wightman power spectrum $f^W(\\omega)$, which the paper computes from the spectral function $\\rho(\\omega,k)=2\\pi\\delta(\\xi_k-\\omega)$ and the KMS relation; in five spacetime dimensions this yields $f^W(\\omega)\\propto(\\mu-\\omega)^2\\,\\Theta(\\mu-\\omega)/(\\sinh \\beta\\omega/2)$ for bosons and a similar form with $\\cosh$ for fermions. The discrete Schrödinger equation $\\partial_t\\varphi_n = i a_n\\varphi_n + b_n\\varphi_{n-1} - b_{n+1}\\varphi_{n+1}$ then propagates the wavefunction, and the Krylov complexity $K(t)=1+\\sum_n n|\\varphi_n(t)|^2$ is the mean position on the chain. The nonzero $a_n$ is what carries the claimed suppression of the exponential rate.","core_discovery":"The paper's central discovery is that for the Schrödinger field, operator growth is controlled by two linear Lanczos sequences rather than one: $\\beta a_n \\approx -4(n+1)-\\mu$ and $\\beta b_n \\approx 2n+1$, for both bosons and fermions, so the chemical potential enters only through $a_n$. The corresponding autocorrelation functions have very similar squared moduli, which is why the bosonic and fermionic complexities nearly coincide. At late times the Krylov complexity grows as $e^{\\lambda_K t}$ with $\\lambda_K \\approx 2.75/\\beta$, and the asymptotic slope stays below $4/\\beta$, the value expected from $2\\times\\text{slope}(b_n)$. The authors interpret the reduction as a genuine effect of the non-Hermitian operator: a nonzero $a_n$ contributes to the discrete Schrödinger equation and, by analogy with the SL(2,R) spread-complexity example, lowers the exponential rate.","pith_inferences":["Beyond the paper: one could integrate the discrete Schrödinger equation past $t/\\beta=0.5$; if the log-slope of $K(t)$ then climbs toward $4/\\beta$, the reported $\\lambda_K\\approx2.75/\\beta$ would be a pre-asymptotic artifact rather than the true Krylov exponent.","Beyond the paper: the proposed mechanism suggests a general effective-rate formula for linear Lanczos data, $b_n=\\alpha n$ and $a_n=\\gamma n+\\delta$; other non-Hermitian models with such coefficients could be checked to see whether the rate is typically $2\\sqrt{\\alpha^2-\\gamma^2/4}$-like, as in the SL(2,R) analogy.","Beyond the paper: the staggering rule stated in Appendix C is presented as a conjecture rather than a proven theorem, and testing it on families of random Wightman power spectra would be a cheap numerical check.","Beyond the paper: because $b_n$ is independent of $\\mu$, the paper implicitly suggests that chaotic or scrambling bounds tied to operator growth in such non-relativistic theories may be insensitive to density, a claim that could be probed with out-of-time-order correlators at finite chemical potential."],"forward_implications":["If the central claim is right, the standard operator-growth relation $\\lambda_K=2\\alpha$ for linear $b_n\\approx \\alpha n$ does not hold for non-Hermitian operators; the slope of $a_n$ must be included in the growth rate.","The chemical potential shifts the on-site Lanczos term $a_n$ but leaves the hopping $b_n$ intact, so the fastest possible operator growth in this theory is independent of charge density.","Bosonic and fermionic Schrödinger fields share nearly the same late-time Krylov complexity because their $|\\varphi_0(t)|^2$ profiles and $b_n$ coefficients match, despite different Wightman power spectra.","For large $|\\mu|$, the asymptotic rate approaches the same universal value, about $2.746/\\beta$, in both statistics.","The paper's Appendix C conjecture predicts a simple rule for Lanczos staggering: symmetric power spectra with a zero at the symmetry axis give staggered $b_n$, while asymmetric spectra do not, a rule that generalizes earlier criteria."],"supporting_citations":[{"why":"Supplies the universal operator growth hypothesis that linear $b_n$ gives $\\lambda_K=2\\alpha$; this is the baseline the paper's smaller $\\lambda_K$ is contrasted with.","marker":"[1]"},{"why":"Provides the Krylov-complexity computation for scalar field theories with bounded power spectrum and the staggering conditions that Appendix C generalizes.","marker":"[9]"},{"why":"The authors' earlier scalar-field treatment whose power-spectrum expansion method is adapted here; its relativistic free-scalar result $\\lambda_K\\approx2\\alpha$ is the comparison case.","marker":"[10]"},{"why":"Shows exponential Krylov complexity in free and rational CFTs, supporting the view that exponential growth is not a chaos signature and setting the quantum-field-theory context.","marker":"[18]"},{"why":"Gives the SL(2,R) spread-complexity formula in which a nonzero $a_n$ lowers the exponential rate; this is the paper's analogy for explaining the suppressed $\\lambda_K$.","marker":"[31]"},{"why":"The recursion-method reference for converting moments of the Wightman power spectrum into Lanczos coefficients.","marker":"[63]"}],"fun_headline_variants":["In Schrödinger field theory, Krylov complexity grows at 2.75/β instead of 4/β","Bosonic and fermionic Krylov complexities nearly coincide in Schrödinger fields","Chemical potential alters only the a_n coefficients, not the b_n, in Krylov complexity","Exponential Krylov growth at rate 2.75/β in Schrödinger field theory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central number rests on treating the fit of $\\log K(t)$ over $t/\\beta\\in[0.45,0.5]$ as already asymptotic, and the paper provides no longer-time calculation or timescale estimate to justify that assumption.","fun_headline_variants_meta":{"raw":{"variants":["In Schrödinger field theory, Krylov complexity grows at 2.75/β instead of 4/β","Bosonic and fermionic Krylov complexities nearly coincide in Schrödinger fields","Chemical potential alters only the a_n coefficients, not the b_n, in Krylov complexity","Exponential Krylov growth at rate 2.75/β in Schrödinger field theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000681,"raw_usage":{"total_tokens":3127,"prompt_tokens":1016,"completion_tokens":2111,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":2024}},"tokens_in":632,"tokens_out":2111,"duration_ms":14765,"temperature":1.0,"reasoning_tokens":2024,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:18:19.968981+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Integrate the same discrete Schrödinger equation to $t/\\beta>0.5$ (or refit the log-slope over $[0.5,1]$) and read the slope of $\\log K(t)$; if it moves from about $2.75/\\beta$ toward $4/\\beta$, the reported suppression is a transient of the finite fit window rather than the asymptotic Krylov exponent.","supporting_citations":[],"review_version":1}