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Krylov Complexity in the Schr\"odinger Field Theory

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Pith's one-line read 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…

desk verdict Useful analytic machinery for Krylov complexity in Schrödinger field theory, but the headline asymptotic exponent is not supported by the short-time numerics. read the letter →

arxiv 2411.16302 v3 pith:5GU4CLVB submitted 2024-11-25 hep-th cond-mat.stat-mech

Peng-Zhang He , Hai-Qing Zhang This is my paper · ORCID
classification hep-thcond-mat.stat-mech
keywords KrylovcomplexityoperatorgrowthLanczoscoefficientsSchrödingerfieldtheorychemicalpotentialnon-HermitianoperatorsWightmanpowerspectrumalgorithm
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§4.1.1, Eq. (4.10)] 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.
  2. [§4.1.2, Fig. 5, Table 2; §4.2.3, Fig. 11] 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.
  3. [§2.1, Eq. (2.24); §5] 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.
minor comments (5)
  1. [Eq. (2.26) and Section 3] 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.
  2. [Figure 2] 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.
  3. [§4.2.2, Eq. (4.33)] 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.
  4. [§2.1.1 and Appendix C] 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.
  5. [Table 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported Krylov exponent is a numerical output, not an input, and the only self-citation is methodological.

full rationale

The derivation chain is self-contained. The Wightman power spectrum is obtained from the thermal propagator and spectral function, the moments are computed from that spectrum, the Lanczos coefficients are obtained by the recursion method, and the Krylov complexity is obtained by numerically solving the discrete Schrödinger equation with those coefficients. The reported asymptotic slope λ_K≈2.746/β is extracted from the resulting K(t), so it is an output of the computation rather than a fitted input used to force the result. The paper does not fit any parameter to reproduce λ_K, nor does it define a_n or b_n in terms of the final complexity. The self-citation to the authors' prior work [10] concerns the series-expansion method for the power spectrum, which is a computational technique and not a load-bearing citation of the paper's central claim; the cited method is independently implementable from the equations given here. The statements that the discrepancies are difficult to explain precisely and that the relation to chemical potential is only visible numerically are explicit limitations, not disguised inputs. The main caveat is that the 'asymptotic' fit is performed on a short window t/β∈[0.45,0.5] and the paper's attribution of the suppressed exponent to the a_n Lanczos coefficients is questionable because |φ_n|² evolves independently of a_n; however, these are issues of numerical evidence and physical interpretation, not circularity. The derivation does not reduce to its own inputs by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, mediators, forces, or dimensions. Its free parameters are numerical fits of computed output quantities, not ad hoc inputs. The main axioms are standard thermal-field-theory and Krylov-space assumptions, plus the tractability restrictions to d=5 and μ≤0.

free parameters (5)
  • linear fit intercept of βb_n = 0.980 to 0.999 (bosonic), 0.999 to 1.004 (fermionic)
    Outcome of fitting computed Lanczos coefficients up to n=80; not an input, but used to infer the empirical formula βb_n≈2n+1.
  • linear fit slope of βb_n = 2.0000 within 2e-5
    Empirical slope of computed b_n; would predict λ_K=4/β under the standard operator growth hypothesis.
  • linear fit intercept of βa_n = from -3.96 at μ=0 to -104 at μ=-100 (bosonic and fermionic)
    Depends on μ; fitted from computed a_n and central to the claimed μ-dependence of a_n.
  • asymptotic Krylov exponent λ_K = 2.669/β to 2.746/β (bosonic), 2.783/β to 2.746/β (fermionic)
    Numerically fitted slope of log K(t) over t/β∈[0.45,0.5]; the central claimed result.
  • chemical potential μ = 0, -0.1, -1, -10, -50, -100
    Input control parameter, not fitted; only μ≤0 is considered because the method of Ref. [10] is not applied for μ>0.
assumptions (5)
  • domain assumption The inner product (2.29) defines a positive definite Hilbert space for non-Hermitian fields and makes L Hermitian.
    Standard in Krylov complexity; needed for real Lanczos coefficients. Positivity for non-Hermitian O is assumed without proof.
  • standard math KMS relations and spectral representation connect the thermal propagator to the Wightman power spectrum.
    Used in Section 2.2 and Appendix B to derive Eq. (3.14).
  • domain assumption The spectral function of the free Schrödinger field is a delta function, Eq. (3.9).
    Derived in Appendix B from the thermal propagator (3.8); assumes free field, no internal degrees of freedom, and exact delta-function support.
  • standard math The normalization constant N is fixed by Eq. (2.33) so that (O|O)=1.
    Defines the normalized Wightman power spectrum and makes the initial operator unit normalized.
  • ad hoc to paper d=5 and μ≤0 are chosen for tractability.
    Even spacetime dimensions and μ>0 make the moment integrals hard; this restricts the scope of the central claim and is acknowledged in the text.

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Pith. "Pith review of Krylov Complexity in the Schr\"odinger Field Theory." pith.science (2026). https://pith.science/paper/5GU4CLVB

@misc{pith2026241116302,
  author       = {Pith},
  title        = {Pith review of: Krylov Complexity in the Schr\"odinger Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GU4CLVB}},
  note         = {Machine review of arXiv:2411.16302}
}
abstract

We investigate the Krylov complexity of Schr\"odinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble that includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator $[H,\cdot]$ on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients $\{b_{n}\}$ are independent of the chemical potential, while $\{a_{n}\}$ exhibits a dependence on it. Both $\{a_{n}\}$ and $\{b_{n}\}$ show linear relationships with respect to $n$. For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions $\abs{\varphi_{0}(t)}^{2}$. The Krylov complexity grows exponentially with time, but its asymptotic scaling factor $\lambda_{K}$ is significantly smaller than the twice of the slope of the $\{b_{n}\}$ coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of $\{b_{n}\}$.

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