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Positive quantum Lyapunov exponents in experimental systems with a regular classical limit

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Pith's one-line read Exponential growth of the fidelity OTOC occurs in the regular Dicke model and integrable LMG model because of unstable stationary points, not because of chaos.

desk verdict Core claim is right but the Dicke-model numerics need one more validation pass at the reported parameters. read the letter →

arxiv 1909.02578 v4 pith:RJJLJOXZ submitted 2019-09-05 cond-mat.stat-mech nlin.CDquant-ph

Saúl Pilatowsky-Cameo , Jorge Chávez-Carlos , Miguel A. Bastarrachea-Magnani , Pavel Stránský , Sergio Lerma-Hernández , Lea F. Santos , Jorge G. Hirsch This is my paper · ORCID
classification cond-mat.stat-mechnlin.CDquant-ph MSC 81Q5037D4581V80 PACS 05.45.Mt03.65.-w
keywords out-of-time-ordercorrelatorsquantumchaosDickemodelLipkin-Meshkov-GlickLyapunovexponentsunstablestationarypointsexcited-statephasetransitionfidelityOTOC
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 challenges the widespread identification of exponential out-of-time-order-correlator (OTOC) growth with quantum chaos. It shows that the fidelity OTOC can grow exponentially in the Dicke model when the classical limit is regular, and in the classically integrable Lipkin-Meshkov-Glick model. The growth rate matches the positive classical Lyapunov exponent of an unstable stationary point even for initial states centered away from that point. The paper concludes that in experimentally accessible systems OTOC growth can signal the presence of unstable stationary points, not chaos, and so exponential OTOC growth is not by itself a reliable signature of quantum chaos.

What carries the argument

The mechanism is the unstable stationary point of the classical Hamiltonian: the saddle point $(Q,P)=(0,0)$ in the LMG model and $(q,p,Q,P)=(0,0,0,0)$ in the Dicke model, whose Jacobian matrix has a positive real eigenvalue that defines the classical Lyapunov exponent $\lambda$. In the quantum dynamics, an initially localized coherent state broadens into this unstable direction; the fidelity OTOC is computed as the variance of canonical operators, $\sigma_Q^2+\sigma_P^2$ for LMG and $\sigma_Q^2+\sigma_P^2+\sigma_q^2+\sigma_p^2$ for Dicke, whose initial growth rate is $2\Lambda \approx 2\lambda$. Exact quantum evolution validates the truncated Wigner approximation for sizes where exact diagonalization is feasible, and the approximation is then used for larger $j$, where the correspondence improves as $j$ grows from 500 to 5000.

What would settle it

Perform an exact quantum simulation of the fidelity OTOC for the Dicke model at $\omega_0=3$, $\gamma=0.66$, $\omega=0.5$, $j=500$, for coherent states centered at $Q=0.1,0.2,0.3$ at energy $E=-j\omega_0$, and check whether the initial growth rate $2\Lambda$ still matches $2\lambda$ from Eq. (6); if no exponential window appears, the quantum-activation claim for the regular regime would fail.

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Extended reading notes

Core claim

The central claim is that positive quantum Lyapunov exponents, defined through the exponential growth rate of the fidelity OTOC, occur in quantum systems whose classical limits are regular or integrable. At an unstable stationary point the quantum exponent $\Lambda$ coincides with the classical Lyapunov exponent $\lambda$ in both models; more importantly, coherent states centered in the surrounding regular region exhibit the same initial exponential rate $2\Lambda \approx 2\lambda$, even though those classical orbits have zero Lyapunov exponents. The exponential regime lasts shorter and saturates lower as the center moves away from the unstable point, but it remains present for generic nearby states. For the Dicke model this happens at parameters used in trapped-ion experiments, so those experiments may observe the effect of an unstable point rather than of classical chaos.

Load-bearing premise

The Dicke-model results in the regular regime rely on the truncated Wigner approximation, and its validation against exact quantum evolution is performed at a different parameter value where the unstable point lies inside a chaotic region, so the accuracy at the regular-regime parameters is assumed rather than directly demonstrated.

Editorial extensions

If this is right

  • Exponential OTOC growth alone cannot certify quantum chaos, because it also occurs in regular and integrable systems around unstable stationary points.
  • Trapped-ion experiments on the Dicke model that measure the fidelity OTOC may observe exponential growth caused by an unstable stationary point rather than by classical chaos.
  • The quantum-classical correspondence between OTOC growth rate and classical Lyapunov exponent holds at the unstable point but breaks in its neighborhood, where quantum mechanics generates instability in a classically stable region.
  • The LMG model, being classically integrable, provides a simple platform to observe the same effect without the complications of a chaotic regime.

Reading between the lines

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

  • A testable extension would be to repeat the fidelity-OTOC measurement around hyperbolic fixed points of other collective models, such as the two-mode Bose-Hubbard dimer, where a similar exponential window should appear despite integrability.
  • The paper's mechanism implies that an OTOC growth-rate measurement alone cannot certify chaos; a practical diagnostic would combine the rate with level statistics or phase-space structure, a step the authors do not take.
  • Finite-size scaling is a natural next probe: one expects the duration of the exponential window to grow like $\log j / \lambda$, which could be checked numerically or in ion-trap experiments.
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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

2 major / 4 minor

Summary. This manuscript argues that exponential growth of out-of-time-ordered correlators is not by itself a reliable signature of classical chaos. For the LMG and Dicke models in parameter regimes where the classical limit is regular, the authors compute the Lyapunov exponent of an unstable stationary point from the Jacobian (Eqs. (3) and (6)) and compare it with the growth rate Λ extracted from the fidelity OTOC, defined as the variance of canonical coordinates. For the LMG model the comparison is made with exact quantum evolution; for the Dicke model it is made with the truncated Wigner approximation at large j. The main numerical finding (Fig. 3) is that coherent states centered not only at the unstable point but also at nearby points with zero classical Lyapunov exponents show an initial exponential FOTOC growth with rate 2Λ ≈ 2λ. The paper concludes that in these experimentally relevant systems the exponential OTOC growth is caused by unstable stationary points rather than by chaos.

Significance. If the Dicke part of the evidence is properly validated, this is a significant cautionary result for the current practice of identifying quantum chaos with OTOC scrambling. The analytical formulas are clear, the comparison with classical Lyapunov exponents is parameter-free and not obtained by fitting, and the LMG half of the claim is backed by exact quantum evolution. The paper also identifies a concrete experimental context, the trapped-ion Dicke simulator, where a positive quantum Lyapunov exponent can be observed in a regular classical regime. The main obstacle is the missing direct TWA benchmark at the regular-regime parameters used in the central Dicke figure; this is a fixable but load-bearing gap.

major comments (2)
  1. [SM III A, Fig. S1; main text 'Quantum-classical correspondence' and Fig. 3(d)] The validation of the truncated Wigner approximation against exact quantum evolution is performed only at ω0=0.649, where, as the main text states, the unstable point is immersed in a chaotic region. The central regular-regime Dicke result in Fig. 3(d) uses ω0=3, γ=0.66, ω=0.5, j=500. The benchmark in Fig. S1 also uses only σ_q², whereas the Dicke FOTOC in Figs. 2(b) and 3(d) is the four-term sum σ_Q²+σ_P²+σ_q²+σ_p². Because the Dicke model is the experimentally relevant part of the title and abstract claim, an exact-vs-TWA comparison is needed at ω0=3, for example at j=100, for the full sum and for the states O, A, B, and C, over the time window of exponential growth. Without this, the statement that the exact quantum evolution and the TWA match is not supported at the parameters where the regular-regime exponential behavior is demonstrated.
  2. [Discussion, first paragraph; Fig. 3 caption] The sentence 'given that the exact quantum evolution and the TWA match' is stronger than what the SM currently demonstrates, since the existing match is at ω0=0.649 and only for one quadrature. The statement should be qualified until the ω0=3 benchmark is supplied, and the Fig. 3 caption should state explicitly that panel (d) is computed with TWA while panel (b) is exact. This is a presentation issue that becomes load-bearing because it is used to justify the Dicke model's place in the paper's main conclusion.
minor comments (4)
  1. [Fig. 2 caption] The caption should state explicitly that the Dicke panel uses TWA while the LMG panel uses exact quantum evolution; the current wording 'the TWA is used' appears only in the body text.
  2. [SM Eq. (S5) and nearby text] Several equations in the Supplemental Material contain doubled norm bars such as '||e^{At}||'; this appears to be a LaTeX typo and should be corrected.
  3. [Main text, 'Quantum activation of the instability'] The phrase 'The choices of A, B, and C are done such that' should be 'The choices of A, B, and C are made such that'.
  4. [Fig. 3(b) and 3(d)] The exponential growth rates are described as 'exactly the same' on the basis of visual inspection; reporting the extracted slopes and the fitted time windows for each of the four states would make the claim quantitative and reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the FOTOC growth rate is extracted from independent dynamics and compared with analytically computed classical Lyapunov exponents; the Dicke TWA validation gap is a numerical caveat, not a circular reduction.

full rationale

The claimed derivation chain is self-contained rather than circular. The classical Lyapunov exponents for the stationary points are derived analytically from the Jacobian eigenvalues (main-text Eqs. (3) and (6); SM Eqs. (S4), (S14), (S18)), with no input from OTOC or FOTOC data. The quantum Lyapunov exponent is then read off from the growth of the variance sigma_G^2(t) computed by exact evolution for the LMG model and by truncated Wigner approximation for the Dicke model. The classical lambda and the extracted quantum Lambda are obtained by independent routes: lambda is not fitted to the FOTOC slope, and Lambda is not defined as lambda. The central new claim, exponential FOTOC growth at points O, A, B, and C in the regular regime, is a numerical result and not an identity forced by any definition. The one potentially load-bearing self-citation is Ref. [18] for the OTOC/classical-LE correspondence, but the paper also cites independent works for that correspondence, and the present conclusion about unstable stationary points does not depend on that theorem. The most serious caveat is numerical rather than circular: the exact-vs-TWA validation in SM Fig. S1 is performed at omega0=0.649 (where the unstable point sits in a chaotic region) and for sigma_q^2 only, while Fig. 3 uses TWA at omega0=3 and the full sum sigma_Q^2+sigma_P^2+sigma_q^2+sigma_p^2. That is a validation/extrapolation gap that should be weighed as a correctness risk, but it does not make the prediction equivalent to its inputs by construction. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameter is fitted to the target result. The model parameters (Omega, xi, omega0, gamma, j) are physical inputs or chosen to place the systems in regular regimes; the quantum LE is extracted from the FOTOC slope, not fitted to the classical formula. The load-bearing assumptions are the mean-field classical limit, the FOTOC-variance identity, the regularity of the chosen phase-space regions, and the validity of TWA at the main Dicke parameters.

assumptions (4)
  • domain assumption The classical Hamiltonians (2) and (5) are obtained by taking expectation values on coherent states and neglecting O(1/j) terms.
    Standard mean-field/semiclassical limit; it defines the regular classical limit and the stationary points used for the Lyapunov exponents.
  • domain assumption In the perturbative limit delta_phi << 1, the FOTOC equals the variance sigma_G^2(t).
    Established in Ref. [25] and SM Eq. (S24); the paper computes the FOTOC as a variance and reads the quantum LE from its growth.
  • ad hoc to paper The truncated Wigner approximation reproduces the exact quantum FOTOC evolution for the Dicke model at the parameters and times used in Figs. 2(b) and 3(d).
    Validation against exact evolution is shown only at omega0=0.649 in SM Fig. S1, while the main regular-regime Dicke results use omega0=3 and j=500; the transfer of validity is assumed.
  • domain assumption The phase-space regions surrounding the chosen unstable points are classically regular, with zero LEs except on the stable and unstable manifolds.
    Supported by the LE map in Fig. 1(c) and Refs. [67,81]; if hidden chaos were present, the regular-regime interpretation would weaken.

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Pith. "Pith review of Positive quantum Lyapunov exponents in experimental systems with a regular classical limit." pith.science (2026). https://pith.science/paper/RJJLJOXZ

@misc{pith2026190902578,
  author       = {Pith},
  title        = {Pith review of: Positive quantum Lyapunov exponents in experimental systems with a regular classical limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJJLJOXZ}},
  note         = {Machine review of arXiv:1909.02578}
}
read the original abstract

Quantum chaos refers to signatures of classical chaos found in the quantum domain. Recently, it has become common to equate the exponential behavior of out-of-time order correlators (OTOCs) with quantum chaos. The quantum-classical correspondence between the OTOC exponential growth and chaos in the classical limit has indeed been corroborated theoretically for some systems and there are several projects to do the same experimentally. The Dicke model, in particular, which has a regular and a chaotic regime, is currently under intense investigation by experiments with trapped ions. We show, however, that for experimentally accessible parameters, OTOCs can grow exponentially also when the Dicke model is in the regular regime. The same holds for the Lipkin-Meshkov-Glick model, which is integrable and also experimentally realizable. The exponential behavior in these cases are due to unstable stationary points, not to chaos.

Figures

Figures reproduced from arXiv: 1909.02578 by the authors.

Figure 1
Figure 1. Top: Energy surface for the classical LMG model for two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The classical LE λ (solid line) and the quantum LE Λ (symbols) for the LMG (a) and the Dicke (b) model at the unstable point. The results for Λ for the LMG model are obtained with the exact quantum evolution and for the Dicke model, the TWA is used. For the LMG model, the FOTOC corresponds to σ 2 Q(t) + σ 2 P (t), ξ = −1, and j = 500. For the Dicke model, the FOTOC is σ 2 Q(t) + σ 2 P (t)+σ 2 q (t)+σ 2 p(t), ω = 0.5… view at source ↗
Figure 3
Figure 3. Energy surface of the LMG model (a) and of the Dicke [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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