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Chaos exponents of SYK traversable wormholes

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Pith's one-line read In two coupled SYK models, the traversable-wormhole phase is weakly chaotic, with a small nonzero Lyapunov exponent that decays as $e^{-(q/2-1)\beta E_{\mathrm{gap}}/2}$, the same exponential law as the decay rate of the first spectral…

desk verdict Plausible exponential relation between λ_L and Egap, but the wormhole-phase nonzero λ_L needs error bars and the quasi-particle ratio doesn't match the numerics. read the letter →

arxiv 2009.10759 v1 pith:HKI574OP submitted 2020-09-22 hep-th cond-mat.stat-mechcond-mat.str-elnlin.CDquant-ph

Tomoki Nosaka , Tokiro Numasawa This is my paper · ORCID
classification hep-thcond-mat.stat-mechcond-mat.str-elnlin.CDquant-ph
keywords SYKmodeltraversablewormholechaosexponentout-of-time-orderedcorrelatorquantumfirst-orderphasetransitionenergygapLyapunov
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 studies the chaos exponent, the exponential growth rate of out-of-time-ordered correlators, in two coupled SYK models that undergo a first-order phase transition from a high-temperature black-hole-like phase to a low-temperature gapped traversable-wormhole phase. Its central claim is that at the transition the chaos exponent drops discontinuously from near the universal bound $2\pi/\beta$, and that deep in the wormhole phase it remains small but nonzero, obeying $\lambda_L \sim e^{-(q/2-1)\beta E_{\mathrm{gap}}/2}$, the same exponential temperature dependence as the decay rate of the first spectral peak. If correct, the wormhole phase is not integrable: it scrambles, but with a parametrically small Lyapunov exponent tied to the energy gap, consistent with the earlier observation that two-point functions decay in this phase. The paper also analyzes a single-sided mass-deformed SYK model and finds that its chaos exponent is always greater than that of the two-coupled model at the same parameters.

What carries the argument

The central machinery is the retarded real-time Schwinger-Dyson equation together with the ladder-kernel eigenvalue problem for the out-of-time-ordered correlator, solved numerically by power iteration to find where the largest eigenvalue crosses unity. In the low-temperature wormhole phase the ladder equation is reduced to a second-order differential equation by approximating the spectral function as a single peak plus its mirror with finite width $\Gamma$, and by assuming the slowly varying envelope $g(t)$ can be treated as constant over the gap period. This reduction produces a Bessel equation whose smoothness condition at $t=0$ determines the c-number ratio $\lambda_L/\Gamma$, giving $\lambda_L/\Gamma \approx 2.706$ for $q=4$.

What would settle it

Compute the out-of-time-ordered correlator in the wormhole phase of the two-coupled model for $q=6$ or $q=8$ with converged real-time numerics, which the paper reports were not reached, and check whether $\lambda_L$ follows $e^{-(q/2-1)\beta E_{\mathrm{gap}}/2}$ with the predicted ratio to the first-peak decay rate, or whether the exponentially small nonzero value disappears once infrared cutoff artifacts are removed.

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Extended reading notes

Core claim

The authors establish, by numerically solving the real-time Schwinger-Dyson equations and the ladder equation for the out-of-time-ordered four-point function, that the chaos exponent $\lambda_L$ of the two-coupled model stays close to the universal bound $2\pi/\beta$ at high temperature, then decreases above the transition temperature and jumps downward at the first-order transition. In the wormhole phase $\lambda_L$ is small but nonzero and follows the same exponential law as the decay width of the first spectral peak: $\lambda_L \sim e^{-(q/2-1)\beta E_{\mathrm{gap}}/2}$. In a quasi-particle approximation where the spectral function is dominated by a single narrow peak and its mirror, the ladder equation reduces to a differential equation whose solution is a Bessel function; matching at the origin fixes the ratio $\lambda_L/\Gamma \approx 2.706$ for $q=4$, independent of temperature and coupling. The same exponential law holds at sufficiently low temperature even when the coupling is large enough that no phase transition exists, and the single-sided mass-deformed SYK model, which has no phase transition, obeys the same formula with its own gap while always having a larger chaos exponent than the two-coupled model.

Load-bearing premise

The analytic exponential law and the constant ratio $\lambda_L/\Gamma \approx 2.706$ rest on the assumption that at low temperature the wormhole spectral function is essentially one narrow peak plus its mirror and that the slowly varying envelope $g(t)$ is constant over a gap period; if higher peaks contribute significantly, the predicted proportionality would not follow.

Editorial extensions

If this is right

  • The wormhole phase scrambles, but weakly: information spreads with a Lyapunov exponent exponentially small in $\beta E_{\mathrm{gap}}$, so the system thermalizes on time scales set by the inverse decay rate while still exhibiting quasi-particle oscillations.
  • The first-order transition between black-hole and wormhole phases is also a sharp drop in chaos: $\lambda_L/(2\pi/\beta)$ falls from order one to exponentially small, distinguishing the two phases as strongly versus weakly chaotic.
  • When the coupling is large enough that no phase transition exists, the same exponential law still holds at low temperature, so the relation $\lambda_L \sim e^{-(q/2-1)\beta E_{\mathrm{gap}}/2}$ is governed by the energy gap itself rather than by the transition.
  • The ratio $\lambda_L/\Gamma$ is predicted to be a fixed number independent of temperature and couplings, giving a quantitative link between scrambling and the decay rate of the quasi-particle peak.
  • In the single-sided mass-deformed SYK model, the absence of a phase transition and the smaller gap lead to a chaos exponent that is always larger than in the two-coupled model, showing that stronger correlation between the two sides suppresses chaos.

Reading between the lines

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

  • If the quasi-particle ratio $\lambda_L/\Gamma \approx 2.706$ is universal across gapped large-$N$ models, the same Bessel matching argument might apply to any system whose spectral function is a single decaying peak, turning two-point decay data into a probe of weak chaos.
  • The exponential suppression of $\lambda_L$ in $\beta E_{\mathrm{gap}}$ suggests that direct large-$q$ expansions, which are blind to effects of order $e^{-(q/2-1)\beta E_{\mathrm{gap}}/2}$, would see an apparently integrable wormhole phase; capturing chaos would require nonperturbative corrections.
  • A natural testable extension is the partially correlated two-coupling model mentioned in the discussion: if the chaos exponent decreases monotonically as the left-right random couplings become more correlated, the two coupled and single-sided models bracket the scrambling behavior of all intermediate models.
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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 / 4 minor

Summary. The paper computes the large-N out-of-time-ordered correlator chaos exponent for two models: the Maldacena-Qi two-coupled SYK model (2.1) and a single mass-deformed SYK model (2.3). Using the real-time GΣ formalism, the authors derive Schwinger-Dyson and ladder equations, solve them numerically for q=4, and extract λ_L by locating the crossing of the leading ladder eigenvalue through unity. Their central results are: (i) in the black-hole phase λ_L/(2π/β) is near the chaos bound and drops discontinuously at the first-order transition; (ii) in the wormhole phase λ_L is small but claimed to be nonzero and to obey the same exponential temperature dependence as the first-peak decay width, λ_L ∼ e^{-β E_gap/2} for q=4, with a quasi-particle derivation predicting λ_L/Γ ≈ 2.706; (iii) the single-sided model exhibits no phase transition and is claimed to have λ_L always larger than that of the two-coupled model. The paper also reports critical exponents for ∂_T λ_L near the transition endpoints.

Significance. If the wormhole-phase λ_L is genuinely nonzero, this is an important result: it would show that the traversable-wormhole phase of the coupled SYK model is not integrable but scrambles with a parametrically small Lyapunov exponent set by the gap, and it would provide a concrete quantitative link between a two-point decay rate and a four-point chaos exponent. The manuscript has clear strengths: the real-time Schwinger-Dyson and ladder equations are derived in detail; the Euclidean and real-time solutions are cross-checked through (3.19); λ_L, E_gap, and Γ are extracted by independent procedures; and the comparison to the external work [1] by Qi and Zhang is appropriate rather than circular. The main uncertainty is numerical: the "small but nonzero" claim has no error bars or convergence study, and the analytic quasi-particle support is quantitatively imperfect. These issues are fixable within the scope of the paper, but they currently prevent full confidence in the headline claim.

major comments (3)
  1. [§4.1.3, Fig. 11, Appendix A.1] The headline "small but non-zero" claim rests on a numerical eigenvalue crossing that is not accompanied by any convergence or error analysis. The method is described only as "power iteration" with no stopping tolerance, and the only robustness checks are the two (Λ_L,T_L) settings shown for μ=0.1. Because λ_L at β≈50 is of order 10^-3, comparable to the fitted peak width Γ at the same β, a discretization or finite-window artifact cannot be excluded a priori. The fragility of this numerical regime is underlined by Appendix A.1, where the q=6,8 wormhole real-time iterations did not converge. Please add a systematic study of the eigenvalue crossing as a function of Λ_L and T_L, state the power-iteration tolerance and convergence criterion, and provide a quantitative uncertainty estimate for λ_L. Without this, the nonzero conclusion is not fully established.
  2. [§4.1.4, Eqs. (4.10), (4.17), (4.20)] The quasi-particle derivation used to support the exponential law relies on two uncontrolled approximations: the truncation of the spectral function to a single δ-function peak and its mirror (4.10), and the replacement of cos^{q-2} and sin^{q-2} by their period averages (4.17). These approximations can be tested against the numerics, and the test is not favorable: the predicted λ_L/Γ ≈ 2.706 is not reproduced by the fitted prefactors, which give e^{-1.04}/e^{-0.788} ≈ 0.78, a discrepancy of roughly a factor 3.5. The sentence "it is not easy to reproduce this value precisely" understates this mismatch. Since the analytic argument is the main theoretical reason to expect a nonzero λ_L independent of the numerical extraction, please quantify the error introduced by (4.10) and (4.17), or restrict the claim to the exponential temperature dependence and present the constant ratio as a model-dependent prediction rather than a numerical match.
  3. [§4.2, Fig. 15, Abstract] The abstract and §4.2 state that the single-sided model's chaos exponent is "always greater" than that of the two-coupled model in the entire parameter space. In the printed Fig. 15, however, the single-sided data (▼) appear to lie slightly below the two-coupled data (●) at small β for both μ=0.12 and μ=0.3. If this visual impression is correct, the claim is false in the high-temperature regime; if it is not, the figure should be redrawn so that the relative ordering is unambiguous. Please reconcile the stated claim with the actual data, including the asymptotic high-temperature region, and adjust the abstract if needed.
minor comments (4)
  1. [Throughout] There are many typographical errors that should be corrected, including "sence" for "sense", "exhibhts" for "exhibits", "simpliy" for "simplify", "inntegration" for "integration", "funcntion" for "function", "copuled" for "coupled", and "equivanlent" for "equivalent".
  2. [Fig. 12 caption] The caption says "with (Λ_L,T_L) = (2×10^5,10^5) and ∆T = 10^3, 10^4, 10^5"; this should presumably read ∆T = ±10^-3, ±10^-4, ±10^-5. Please correct the sign and exponent notation.
  3. [Eq. (1.1)] The exponential expression in Eq. (1.1) is garbled in the rendering; the intended form appears to be λ_L ∼ e^{-(q/2-1)βE_gap/2}. Please ensure the equation is typeset unambiguously.
  4. [§4.1.4, Eq. (4.16)] The notation "q∈4N" is nonstandard; it would be clearer to write "q divisible by 4" or "q ∈ 4ℕ" with an explanation, since the simplification does not apply for q = 4n+2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: λ_L, Egap, and Γ are extracted independently, and the quasi-particle derivation does not reduce to the claimed result.

full rationale

The central claim λ_L ∼ e^{-(q/2-1)βEgap/2} is not circular. The three quantities entering it are obtained independently: λ_L is found as the test value where the largest eigenvalue of the real-time ladder kernel (3.36) crosses unity by power iteration (§4.1.3); Egap is fitted from the exponential decay of the Euclidean propagator (4.4) at T = 0.001 (§4.1.1); Γ_1st is fitted from the width of the first spectral peak of ρ_LL using a Lorentzian δΓ fit (Fig. 8, §4.1.2). The exponential relation is then a comparison of separately computed numerical data. The analytic derivation in §4.1.4 starts from the δ-peak spectral ansatz (4.10) and uses the quasi-particle expression for Γ from the external reference [1]; the genuinely new step is the reduction to the differential equation (4.19) and the constant ratio λ_L/Γ ≈ 2.706 from (4.20). The ansatz does encode the Boltzmann factor e^{-βEgap/2} in G(β/2+it), but that is an input about the spectral function, not about λ_L, and it does not force the ratio λ_L/Γ. The self-citations to [31] and [38] are used for background, reproduced phase-diagram data, and model motivation, not as the load-bearing derivation; [43] is a 'Work in Progress' citation in the Discussion for a unification claim, but the main result does not depend on it. Numerical concerns such as IR-cutoff artifacts and non-convergence of the q = 6, 8 wormhole runs are correctness risks, not circularity.

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

No free parameters are introduced into the central derivation; q, J=1, μ, and T are model inputs, and E_gap, Γ, λ_L are computed observables. The main assumptions are the standard large-N saddle point and melonic dominance, the KMS-based analytic continuation, and the symmetry ansatz (3.20); the quasi-particle ansatz for the low-temperature spectral functions is the only ad hoc-to-paper modeling choice.

assumptions (5)
  • domain assumption Large N limit and melonic dominance justify the GΣ saddle point and the ladder resummation.
    Used throughout §2 and §3; standard in SYK literature, but an assumption about the disorder average and diagrammatic selection.
  • domain assumption The KMS condition and the analytic continuation ansatz (2.22)-(2.23) close the real-time Schwinger-Dyson equations via (3.18).
    Standard thermal field theory; invoked in §3.1.1 to relate G> to GR.
  • domain assumption The Z4 symmetry ansatz (3.20) is consistent and selects the relevant saddle.
    Imposed in §3.1.2 to reduce the equations and split the ladder kernel into even/odd sectors.
  • domain assumption At late time, only the second term of the ladder equation (3.32) survives, and other contributions are suppressed.
    Standard late-time OTOC analysis; stated in §3.1.4 before (3.34).
  • ad hoc to paper The quasi-particle approximation: the low-temperature spectral function is dominated by a single peak and its mirror, approximated by δ-functions (4.10), and g(t) varies slowly enough to average cos/sin powers (4.17).
    Introduced in §4.1.4 to derive the differential equation for λ_L; this is the main modeling assumption for the analytic formula.

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Cite this review

Pith. "Pith review of Chaos exponents of SYK traversable wormholes." pith.science (2026). https://pith.science/paper/HKI574OP

@misc{pith2026200910759,
  author       = {Pith},
  title        = {Pith review of: Chaos exponents of SYK traversable wormholes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HKI574OP}},
  note         = {Machine review of arXiv:2009.10759}
}
abstract

In this paper we study the chaos exponent, the exponential growth rate of the out-of-time-ordered four point functions, in a two coupled SYK models which exhibits a first order phase transition between the high temperature black hole phase and the low temperature gapped phase interpreted as a traversable wormhole. We see that as the temperature decreases the chaos exponent exhibits a discontinuous fall-off from the value of order the universal bound $2\pi/\beta$ at the critical temperature of the phase transition, which is consistent with the expected relation between black holes and strong chaos. Interestingly, the chaos exponent is small but non-zero even in the wormhole phase. This is surprising but consistent with the observation on the decay rate of the two point function [arXiv:2003.03916], and we found the chaos exponent and the decay rate indeed obey the same temperature dependence in this regime. We also studied the chaos exponent of a closely related model with single SYK term, and found that the chaos exponent of this model is always greater than that of the two coupled model in the entire parameter space.

Figures

Figures reproduced from arXiv: 2009.10759 by the authors.

Figure 1
Figure 1. Left: Keldysh contour for the insertion of single operator. Right: Contours [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Top/Middle: The diagrammatic representation of the ladder equation/ retarded [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Euclidean propagators for the black hole phase (∆ [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Top left/top right/bottom left: free energy/energy/entropy for the black hole [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: The critical exponents νBH, νWH of the specific heat (4.3) of the two coupled model with q = 4, J = 1. We have determined νBH, νWH by fitting (∂E/∂T) −1 near the discontinuity of E(T) ( [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]
Figure 6
Figure 6. Figure 6: Left: Euclidean propagator for the wormhole phase around [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]
Figure 7
Figure 7. Figure 7: Top left/right: propagators/spectral functions [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: Top left/top right/bottom left: first/second/third peak of the spectral function [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]
Figure 9
Figure 9. Figure 9: Left: Fitting of decay width of the first peak of [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]
Figure 10
Figure 10. Figure 10: Chaos exponent of the two coupled model with [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]
Figure 11
Figure 11. Figure 11: Top left: chaos exponent for the black hole solution and the wormhole solution, [PITH_FULL_IMAGE:figures/full_fig_p035_11.png]
Figure 12
Figure 12. Figure 12: The chaos exponent of the two coupled model near [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]
Figure 13
Figure 13. Figure 13: The critical exponent ηBH and ηWH of the chaos exponent (4.9) of the two coupled model with q = 4, J = 1. We have determined ηBH, ηWH by fitting (∂T λL) −1 by the ansatz (∂T λL) −1 = c(T −Tc,BH) ηBH and (∂T λL) −1 = c(Tc,WH−T) ηWH with the fitting parameters (c, ηBH) …
Figure 14
Figure 14. Figure 14: Top left: Euclidean propagator Gab(τ ) of the single sided model with µ = 0.05, T = 0.001, 0.05. Top right: Free energy. Bottom left: Euclidean propagator at low temperature µ = 0.05, T = 0.001 where the exponential decay is significant. Bottom right: Egap of the sing…
Figure 15
Figure 15. Figure 15: Comparison of the chaos exponent between the two copuled model and the single [PITH_FULL_IMAGE:figures/full_fig_p041_15.png]
Figure 16
Figure 16. Figure 16: Top left/top right: propagator and spectral function of the single sided model [PITH_FULL_IMAGE:figures/full_fig_p042_16.png]
Figure 17
Figure 17. Figure 17: Top left: Chaos exponent [PITH_FULL_IMAGE:figures/full_fig_p043_17.png]
Figure 18
Figure 18. Figure 18: Schematic picture for the expected behavior of the chaos exponent in the unstable [PITH_FULL_IMAGE:figures/full_fig_p045_18.png]
Figure 21
Figure 21. Figure 21: As in the case of q = 4, there are no phase transition. At low temperature we found that the chaos exponent obeys the following formula λL ∼ e − q/2−1 2 βEgap . (A.1) 11 Precisely speaking, the GΣ effective action and its first variation are identical for the two mode…
Figure 19
Figure 19. Figure 19: Phase diagram (left) and the chaos exponent (right) of the two coupled model with [PITH_FULL_IMAGE:figures/full_fig_p047_19.png]
Figure 20
Figure 20. Figure 20: Phase diagram (left) and the chaos exponent (right) of the two coupled model with [PITH_FULL_IMAGE:figures/full_fig_p047_20.png]
Figure 21
Figure 21. Figure 21: Top left/right: The chaos exponent of the single sided model with [PITH_FULL_IMAGE:figures/full_fig_p048_21.png]

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Forward citations

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.