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In the Dicke model, long-time-averaged Krylov complexity and entropy change sharply at a critical coupling, separating regular oscillatory dynamics from chaotic, spaghettified dynamics.

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2026-08-01 06:59 UTC pith:WE6WGOMC

load-bearing objection A new numerical observation about Krylov complexity in the Dicke model, but the 'nonanalytic transition' claim is stronger than the finite-N, finite-time evidence supports. the 3 major comments →

arxiv 2607.21583 v1 pith:WE6WGOMC submitted 2026-07-23 cond-mat.quant-gas quant-ph

Complexity transition in the Dicke model of light-matter interaction

Yicheng Zhang , Erhai Zhao This is my paper
classification cond-mat.quant-gas quant-ph
keywords Krylov complexityDicke modelquantum quench dynamicsdynamical phase transitionquantum chaosLanczos coefficientswave packet dynamicslight-matter interaction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that Krylov complexity, a measure of how an evolving quantum state spreads through Hilbert space, can detect a genuine dynamical transition in a realistic many-body model. In the Dicke model of two-level atoms coupled to a cavity mode, the long-time-averaged Krylov complexity and its Shannon entropy first decrease smoothly as the dimensionless coupling g̃ increases, then change slope abruptly near g̃_c ≈ 0.5 and rise steeply into a fluctuating regime. The authors interpret this as a transition between regular dynamics, where the Krylov wave packet is confined and oscillates, and chaotic dynamics, where the packet is stretched and destroyed by Rindler-like hopping. The transition is accompanied by distinct system-size scalings, C̄_K ∼ N^{3/2} versus ∼ N^{5/2}, and coincides broadly with the onset of chaos inferred from level statistics and Lyapunov exponents. The paper positions Krylov measures as an independent, state-based order parameter for dynamical phase transitions, with a clean signal where traditional observables become noisy.

Core claim

On the paper's own terms, the central discovery is that the time-averaged Krylov complexity C̄_K of the Dicke model after a quench from a spin-coherent initial state is a non-monotonic function of the dimensionless coupling g̃. Approaching from below, it decreases smoothly; near g̃_c ≈ 0.5 its slope changes suddenly, while the Krylov entropy S̄_K jumps, and above that point both quantities become large and fluctuate strongly with g̃. The same behavior is seen for several initial states and detunings, and the data collapse onto distinct power laws in the spin number N — C̄_K ∼ N^{3/2} on the regular side and ∼ N^{5/2} on the chaotic side. The paper identifies the mechanism as wave-packet comp

What carries the argument

Krylov complexity: expand the time-evolving state in the Lanczos basis generated from the initial state; the Hamiltonian becomes a one-dimensional tight-binding chain with site energies a_k and hopping amplitudes b_k. C_K is the center of mass of the occupation probabilities along this chain, and S_K is their Shannon entropy. The paper's explanatory engine is the Rindler-Krylov toy Hamiltonian, a_k = k, b_k = ηk, where η is the slope ratio: linear onsite energy confines the particle, while positional hopping mimics Rindler spacetime, and η = 1/2 is a delocalization critical point. Deviations from this linear form — deterministic disorder δa_k, δb_k produced by the Lanczos procedure — create

Load-bearing premise

The load-bearing premise is that the numerically computed finite-N, finite-time-averaged Krylov quantities (taken over gt ∈ [50,200]) are representative of the true long-time, thermodynamic limit, so that the sharpening with N is a genuine nonanalytic transition rather than a finite-size crossover that would saturate at larger N.

What would settle it

Compute C̄_K and S̄_K for N = 50, 80, 100 (and, if possible, trapped-ion data at N up to 100) while extending the time average to gt ∈ [200, 1000]. If the slope change at g̃_c remains smooth, the transition width stops shrinking with N, or the N^{3/2}/N^{5/2} scalings fail to persist, the central claim of a sharp complexity transition is refuted, leaving only a crossover.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Krylov complexity and Krylov entropy can act as an independent, state-based diagnostic of dynamical phase transitions, complementing time-averaged magnetizations, entanglement entropy, and level statistics that become noisy near chaos.
  • The complexity phase diagram in the (g̃, Ω/δ) plane gives experiment a map: trapped-ion or cavity-QED realizations of the Dicke model can look for the regular, chaotic, and spin-locked regimes identified here.
  • The distinct scalings C̄_K ∼ N^{3/2} and ∼ N^{5/2} provide a quantitative signature that can be checked at larger N and in other models.
  • The spaghettification mechanism is not specific to the Dicke model: any Hamiltonian whose Lanczos coefficients approach a_k ∝ k, b_k ∝ ηk near η = 1/2 should show a similar complexity transition, so the criteria can be exported to other many-body systems.
  • Because level statistics and Lyapunov exponents give fuzzy or parameter-dependent boundaries, the paper positions C̄_K and S̄_K as the cleaner, directly observable order parameters for the transition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the transition is truly nonanalytic, it should appear as a sharpening feature in a controlled finite-size scaling: computing C̄_K for N up to 100 or more and extending the time average beyond gt ∈ [50,200] would reveal whether the slope change converges to a step or saturates as a crossover.
  • The different N-scaling suggests C̄_K may quantify the classical simulation cost of a quench: below g̃_c the state is effectively confined to a small Krylov window, while above it the N^{5/2} growth signals a much larger, harder-to-simulate Hilbert-space sector.
  • One could test the Rindler picture directly by fitting Lanczos coefficients in other models and looking for a dynamical transition whenever the fitted η approaches 1/2, effectively turning η into a universal control parameter for complexity transitions.
  • The off-resonance spin-locked region of the phase diagram may be a distinct third regime worth probing experimentally, since it predicts extremely low complexity and essentially frozen spin dynamics for weak coupling away from resonance.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. This letter studies quench dynamics in the Dicke model through Krylov complexity. It claims that the long-time-averaged Krylov complexity C̅_K and Krylov entropy S̅_K undergo a sharp, nonanalytic transition as the dimensionless coupling g̃ is tuned, near g̃_c ≈ 0.5, separating a 'regular' confined oscillatory regime from a 'chaotic' spaghettified regime. Evidence is presented as finite-N data (N = 20, 30, 40), a complexity phase diagram in (g̃, Ω/δ), N-scaling C̅_K ∼ N^{3/2} from below and ∼ N^{5/2} above, and an interpretation in terms of competition between linear confinement, Rindler-like hopping, and deterministic disorder in Lanczos coefficients, supported by a solvable linear toy model.

Significance. If fully established, the result would be significant: Krylov complexity would provide a state-dependent, direct order parameter for dynamical transitions, with a transparent wave-packet picture and distinct finite-size scaling. The paper has notable strengths: it studies a realistic, experimentally relevant model; it carefully compares the original Lanczos coefficients with a smoothed version to isolate the role of deterministic disorder; and it provides an analytic eigenfunction solution of the toy model at η = 1/2. However, the central nonanalyticity claim is not yet supported by the evidence, and the current significance is conditional on closing that gap.

major comments (3)
  1. [Transition in complexity / Fig. 1(a,c)] The claim of a 'sharp (hence nonanalytic)' transition is not established. The data are for N=20,30,40 with a single time average over gt∈[50,200]; no error bars, no stated boson-number truncation, and no finite-size scaling of the transition width are given. For fixed N and fixed truncation, C̅_K(g̃) is analytic in g̃, so nonanalyticity must be demonstrated via the N→∞ and/or t→∞ limit. The statement that the transition 'gets sharper for larger N' is qualitative. A scaling collapse of dC̅_K/dg̃ (peak height and width vs N) and a convergence check in time are needed to rule out a crossover that saturates. The toy model Eq. (3) does not fill this gap: Fig. S7 shows η(g̃) stays below 1/2 and gives 'no hint' of the C̅_K transition.
  2. [Fig. 2(a), Fig. S3, definition of C̅_K] The long-time average is taken over gt∈[50,200], but the regular regime shows persistent oscillations of C_K(t). The averaged value can depend on the chosen window and may shift with oscillation phase/period as g̃ varies; the dip near g̃_c could be a window artifact. Please show for representative g̃ in both regimes that C̅_K(T) converges as T increases (e.g., varying the window and checking for a plateau), and provide estimates of the sampling fluctuations. This is essential because C̅_K is used as the order parameter for the transition.
  3. [Supplemental Material 'SCALING OF C_K', Fig. S4(b)] The claimed N^{3/2}/N^{5/2} scaling is not quantitative. Only three N values are used for a power-law fit, and the Supplemental Material explicitly states that neither N^{3/2} nor N^{5/2} 'fully captures the N dependence' and leaves the scaling coefficients for future work. If the scaling behavior is meant to corroborate the transition, the exponents and their uncertainties must be determined with a systematic finite-size analysis; otherwise the claim should be softened to 'consistent with' a crossover, which would change the paper's central message.
minor comments (3)
  1. [Fig. 1 caption] The caption lists panel (e) for the Krylov entropy S̅_K, but the figure contains panels (a)–(d). The entropy panel should be labeled (d), not (e).
  2. [Numerical methods / reproducibility] The cutoff in the bosonic Hilbert space used for the finite-N simulations is not stated. This is needed to reproduce the numerics and to confirm that the results are not affected by truncation.
  3. [Fig. 5(a)-(b)] In the Rindler-Krylov toy model, the time axis in Fig. 5(a)-(b) is not defined with units (e.g., in units of 1/a_1). Adding the time unit would improve clarity.

Circularity Check

0 steps flagged

No significant circularity: the complexity transition is an empirical finite-N observation, and the supporting toy model and disorder analysis are independent of the claimed transition point.

full rationale

The paper's central claim is that the long-time-averaged Krylov complexity and entropy change sharply as a function of the Dicke coupling. This is presented as a direct numerical observation (Fig. 1) rather than as a quantity derived from a prior definition of the transition; the same observable is used both to locate the feature and to characterize the two regimes, but that is the normal role of an order parameter, not a circular derivation. The Rindler-Krylov toy model (Eq. 3) is independent of the Dicke model and, as the paper explicitly states, gives 'no hint of the C_K transition,' so the toy model is not used to force the transition. The disorder analysis compares the full Lanczos coefficients to a smoothed model and shows that the smoothed model misses the transition; this is a post-hoc mechanistic explanation, not a fitted input used as a prediction. Self-citations (Refs. [8] and [47]) are used only for background and motivation, not as uniqueness theorems or load-bearing justifications. The absence of finite-size scaling or error bars is a scientific robustness concern, not a circularity. No step in the paper reduces by construction to its own input.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central claim is a numerical observation supported by a fitted toy model and a post-hoc disorder mechanism. No new physical entity is introduced. The free parameters are mostly fitting/analysis choices; none is elevated to a predictive theory. The paper does not ship code or data, and the infinite-dimensional truncation is left unstated.

free parameters (5)
  • η (Rindler slope ratio) = 0.42–0.50, increasing with g̃
    Slope ratio b_k/a_k fitted to Lanczos coefficients; used to characterize the regular regime, but the fitted values stay below 0.5 and do not locate g_c.
  • Scaling exponents α (N^{3/2}, N^{5/2}) = ≈1.5 and ≈2.5
    Fitted to the N-dependence of C̅_K in the two regimes; no error bars or formal collapse analysis are provided.
  • Time-averaging window gt ∈ [50,200] = None (chosen)
    The long-time average is taken over this finite window; no convergence study is shown, and the window may not be representative near g_c.
  • Sliding-average window size (k±10) = 10 sites
    Used to construct the smoothed model and define disorder δa_k, δb_k; the choice affects the disorder profiles in Fig. 2.
  • Microcanonical window size (6N states) and Lyapunov threshold (λ>0.01) = 6N, 0.01
    Used in the level-spacing and Lyapunov analyses; the paper acknowledges these choices affect the inferred chaos onset.
axioms (5)
  • domain assumption A truncated bosonic Hilbert space and finite Krylov chain capture the dynamics faithfully for the reported N and times.
    The Dicke model has an infinite bosonic Hilbert space, but the paper never states the truncation dimension or shows convergence tests. The central numerics depend on this.
  • domain assumption The long-time average over gt∈[50,200] is representative of the infinite-time average.
    Used to define C̅_K and S̅_K as 'order parameters'; no time-convergence analysis is provided.
  • ad hoc to paper The linear toy model a_k = k, b_k = ηk captures the essential wave-packet dynamics of the Dicke model.
    Introduced in Eq. (3) as a 'crude approximation' of the actual (a_k,b_k). It is used to motivate the Rindler/spaghettification picture, but the authors show it does not reproduce the C̅_K transition.
  • domain assumption Level-spacing ratio in a microcanonical window near the initial energy diagnoses integrability/chaos.
    Standard tool, but the finite window size (6N states) is a choice; results in Fig. 3(c) depend on it.
  • ad hoc to paper The disorder (δa_k, δb_k) is the cause of the confinement and the complexity transition.
    This is inferred from the smoothed-model comparison (Figs. 5c-d), a numerical ablation rather than a parameter-free derivation.

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read the original abstract

Tuning the coupling strength $g$ of an interacting quantum system may drive a sudden change in its ground-state or thermal properties. To identify and grasp non-analytical, or even discontinuous, transitions in far-from-equilibrium dynamics proves more challenging. Recently Krylov complexity $C_K$ has offered fresh insights about operator growth, thermalization, and chaos in quantum dynamics. Yet it remains unclear if, and how, changing $g$ can trigger a sharp transition in the complexity measures. Here we present evidence for such a transition by mapping out the complexity phase diagram of the paradigmatic Dicke model describing two-level atoms coupled to a cavity photon mode. Two qualitatively different regimes of dynamics are identified and characterized. At the transition, the slope of $C_K$ changes suddenly to coincide with a jump in the Krylov entropy. We elucidate the nature of the regime change from the wave packet dynamics in Krylov space, where a particle is confined by a roughly linear potential but hops as if it lives in a Rindler reference frame. The competition between confinement, which leads to bouncing, and deconfinement by Rindler hopping, which leads to the destruction of wave packet analogous to gravitational spaghettification, is sensitive to the disorder in Lanczos coefficients. The framework outlined here can be applied to other quantum many-body systems.

Figures

Figures reproduced from arXiv: 2607.21583 by Erhai Zhao, Yicheng Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. The Krylov dynamics of Dicke model from the initial [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Left column: (a) The probability [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The average of magnetization [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The robustness of the complexity transition. (a) The [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

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