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Hot wormholes and chaos dynamics in a two-coupled SYK model

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The unstable 'hot wormhole' phase of the two-coupled SYK model has computable chaos exponents, obtained from two non-equilibrium protocols, and they interpolate smoothly between the black hole and wormhole phases.

desk verdict First Lyapunov exponents for the hot wormhole phase, computed by non-equilibrium protocols; the effective-temperature labeling is asserted more than shown, but the claim is plausible and the paper is honest about its limits. read the letter →

arxiv 2501.04660 v3 pith:C3EUJERD submitted 2025-01-08 hep-th cond-mat.str-elgr-qcquant-ph

Martí Berenguer , Javier Mas , Juan Santos-Suárez , Alfonso V. Ramallo This is my paper · ORCID
classification hep-thcond-mat.str-elgr-qcquant-ph
keywords Sachdev-Ye-KitaevmodeltraversablewormholehotphaseLyapunovexponentout-of-time-ordercorrelatorSchwinger-KeldyshformalismFloquetdrivingSchwarzianapproximation
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

The paper aims to show that the unstable 'hot wormhole' phase of the two-coupled Sachdev-Ye-Kitaev (SYK) model—a tractable quantum toy model of a traversable wormhole—can be probed dynamically even though equilibrium simulations cannot reach it, and that its chaotic properties can be extracted. Using the Schwinger-Keldysh formalism, the authors cool a high-temperature black-hole state into a wormhole by coupling it to a cold bath, and separately inject energy by periodically driving the coupling between the two SYK sides. Along each non-equilibrium trajectory they assign an effective temperature from the fluctuation-dissipation ratio and compute the Lyapunov exponent from the retarded out-of-time-order kernel. They find that the hot-wormhole Lyapunov exponent interpolates smoothly between the two stable phases, and that periodic driving reaches only the upper segment of the unstable branch, with additional non-thermal excited states. If correct, this gives a concrete numerical handle on a phase that is stable only in the microcanonical ensemble and has a gravitational dual.

What carries the argument

The central object is the retarded out-of-time-order kernel $K_{abcd}(\omega,\omega')$ of Eqs. (3.10)-(3.11), built from the spectral functions $\rho_{ab}(\omega)$ and the effective inverse temperature $\beta_{\rm eff}$; the Lyapunov exponent $\lambda_L$ is the value at which its largest eigenvalue crosses one. The non-equilibrium protocols are the second pillar: the Kadanoff-Baym equations (2.13)-(2.14) produce the real-time propagators, and the Wigner-transformed fluctuation-dissipation fit (2.18) converts each time slice into an effective equilibrium saddle point. The third element is the Schwarzian effective action (4.3) with the matter correction (4.7), whose potential develops a local maximum corresponding to the hot wormhole.

What would settle it

Run the same two-coupled SYK model in the microcanonical ensemble, where the hot wormhole is stable, locate the saddle point, and compute its Lyapunov exponent from the equilibrium kernel at the same $\beta_{\rm eff}$; if the value differs from the one obtained by the cooling and Floquet protocols, the effective-temperature assignment is not reproducing the unstable saddle point.

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Extended reading notes

Core claim

The paper claims that the unstable hot wormhole phase carries computable chaos exponents, even though it is inaccessible in equilibrium. Feeding spectral functions and effective temperatures from non-equilibrium simulations into the retarded out-of-time-order kernel yields Lyapunov exponents along the unstable branch, and those exponents match the smoothly interpolating curve conjectured in earlier work. In the driven protocol, thermal states form only for sufficiently large energy injections ($n\geq 14$ half-cycles); these states lie on the upper half of the hot wormhole branch and their Lyapunov exponents are those of the original model, with no bath. Smaller injections produce two classes of non-thermal states: one is interpreted as excited states of the cold wormhole, and the other shows a slowly varying effective temperature whose extrapolated endpoint would fall on the wrong side of the transition, a puzzle the paper leaves open. The Schwarzian approximation with a matter contribution reproduces the unstable maximum but does not capture all observed behaviors.

Load-bearing premise

The argument assumes that during the slow cooling or after the driving stops, each instant of the non-equilibrium evolution is close enough to a thermal state that a single effective temperature $\beta_{\rm eff}$ from fitting the fluctuation-dissipation ratio labels the same equilibrium (unstable) saddle point whose spectral functions feed the chaos kernel; if that mapping fails, the extracted exponents are not hot-wormhole exponents.

Editorial extensions

If this is right

  • The Lyapunov exponent of the hot wormhole phase is numerically accessible and interpolates smoothly between the black hole and wormhole values along the unstable branch, confirming the interpolation conjectured in [16].
  • Periodic driving of the inter-side coupling $\mu$ reaches only the upper segment of the hot wormhole branch; energy injections below a threshold ($n<14$ half-cycles) yield non-thermal excited states of the cold wormhole rather than hot wormhole states.
  • The cooling-with-bath protocol extracts chaos exponents for a modified system, namely the model coupled to the bath, so only the driven and combined protocols give exponents of the original model.
  • The Schwarzian potential with the matter contribution develops a local maximum corresponding to the hot wormhole, explaining qualitatively why energy injections above a threshold thermalize while smaller ones remain trapped.
  • The combined cooling-then-decoupling protocol reproduces the same picture: only the upper segment of the hot wormhole branch thermalizes, and larger energy extractions produce non-thermal states.

Reading between the lines

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

  • A direct check of the effective-temperature map would be to compute the kernel eigenvalue at $\beta_{\rm eff}$ from the true equilibrium unstable saddle point and compare it with the non-equilibrium value; agreement would make the smooth Lyapunov interpolation a statement about the phase diagram itself.
  • The red non-thermal states, whose extrapolated inverse temperatures fall above $\beta_c$, hint at a slow thermalization connected to the power-law revival envelope noted in [22]; extending the simulations beyond $t_{\rm max}=1500$ could decide whether the fluctuation-dissipation fits eventually become exact.
  • A microcanonical simulation, where the negative-heat-capacity phase is stable, is the natural place to populate the lower segment of the hot wormhole branch that the driving protocol cannot reach, and to benchmark the non-equilibrium exponents.
  • The effective-temperature labeling and kernel machinery could be carried over to other holographic quench or Floquet systems, and to the tighter chaos bounds cited in [38,39] that the paper leaves for future work.
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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 / 6 minor

Summary. The paper studies chaotic dynamics in the two-coupled SYK model, focusing on the thermodynamically unstable "hot wormhole" phase, which is not accessible in equilibrium canonical simulations. The authors use two non-equilibrium Schwinger-Keldysh protocols: coupling the system to a cold bath, and periodically driving the inter-copy coupling mu. From the real-time Green functions they extract a time-dependent effective inverse temperature beta_eff(T) via a fluctuation-dissipation fit, Eq. (2.18), together with the associated spectral functions. These are fed into the retarded ladder kernel, Eqs. (3.10)-(3.11), and the Lyapunov exponent lambda_L is obtained from the eigenvalue crossing condition. For the bath protocol lambda_L interpolates smoothly between the black-hole and wormhole values; for the Floquet protocol only sufficiently large energy injections (n >= 14) thermalize, and these land on the upper segment of the hot wormhole branch. Non-thermal states at smaller n are classified and interpreted using a Schwarzian effective potential, Eq. (4.5), supplemented by a matter contribution, Eq. (4.7). The paper concludes that the hot wormhole phase has computable chaos exponents and a richer structure than the equilibrium phase diagram alone suggests.

Significance. If the identification of the simulated states with the unstable hot-wormhole saddle is correct, the paper provides the first numerical Lyapunov exponents for this phase and a direct test of the conjecture in Ref. [16] that lambda_L interpolates smoothly across the stable/unstable branches. The frequency-space kernel in Eq. (3.11) is a standard and explicit tool, and the use of the NESSi library for long-time simulations is appropriate. The paper is also transparent about the bath protocol modifying the model and about which Floquet states satisfy the fluctuation-dissipation relation. A notable strength is that the Floquet protocol, when it works, returns to the original Hamiltonian and produces thermal final states without any bath, giving a physically clean route to the hot branch. However, the central attribution of the extracted exponents to the hot wormhole phase relies on an equivalence between non-equilibrium states and equilibrium unstable saddles that is asserted but not demonstrated in detail, and the numerical lambda_L values are reported without error estimates or convergence tests.

major comments (2)
  1. [Section 2.1, Eq. (2.18); Section 3, Eq. (3.11)] The load-bearing assumption is that each non-equilibrium state used in the calculation coincides with the (possibly bath-modified) equilibrium unstable saddle at the fitted beta_eff(T). For the cooling protocol this is asserted in Section 2.1 with the sentence 'We have checked numerically that this is the case,' but no comparison is shown: the non-equilibrium spectral functions rho(T,omega) are never directly compared with the spectral functions of the equilibrium hot-wormhole solution at the same beta. Since Eq. (3.11) uses equilibrium thermal Wightman functions with a single beta, the lambda_L values in Figs. 5 and 8 are only interpretable as hot-wormhole exponents if this equivalence holds. For the Floquet thermal states, the FD relation and the energy location in Fig. 7(c) are suggestive, but FD alone does not select among the three equilibrium saddle solutions in the coexistence region. A direct comparison of the final spectral functions (or of G^R and G^W) with those of the hot-wormhole saddle would close this gap and should be added.
  2. [Section 3, Figs. 5 and 8] No error bars, convergence estimates, or numerical parameters are reported for the Lyapunov exponents. Solving the eigenvalue problem in Eqs. (3.10)-(3.11) requires discretizing omega, truncating the spectral functions, and fitting beta_eff(T); the sensitivity of the eigenvalue crossing to these choices is not discussed. Since the main quantitative claim is the smooth interpolation and the specific values of lambda_L on the hot branch, the authors should provide convergence checks (e.g., grid spacing, time-window length, fitting range for beta_eff) or at least representative error bars.
minor comments (6)
  1. [Section 4, paragraph after Fig. 10] The phrase 'the thermalized purple solutions should be rather called cold black holes' is confusing because the purple solutions were previously defined as states for which the fluctuation-dissipation relation is not satisfied; the intended wording is likely 'non-thermal purple solutions.'
  2. [Fig. 6 caption] The caption 'initially in a black hole solution (T = 0)' is ambiguous: T here appears to denote the average time, not temperature, but the same symbol is used for temperature throughout the paper; please disambiguate (e.g., T_avg or t_avg).
  3. [Eq. (2.18) and Fig. 7] The paper does not state the frequency range over which the tanh(beta omega/2) fit is performed, nor the goodness of fit for the thermal states. Adding this information would make the beta_eff extraction reproducible and would strengthen the distinction between thermal and non-thermal states.
  4. [Section 3.1] The simulation parameters state J = J_B = 1, mu = 0.1 and V = 0.2, but the number of bath fermions M and any convergence checks with respect to M or the time step are not reported; a brief statement would help reproducibility.
  5. [Section 4, Eq. (4.1)] For the red non-thermal solutions, the exponential fit beta_eff(T) = A e^{-gamma T} + beta_infty is used to place points in Fig. 7(c), but no fit quality or fitted values of A and gamma are reported; without these, the claim beta_infty > beta_c is difficult to assess.
  6. [Section 4.1.1, Eq. (4.7)] The proposed attractor mechanism for explaining why a range of energy injections leads to long-time residence at the local maximum of the modified Schwarzian potential is presented as a speculation ('may give rise to an attractor mechanism'); this is acceptable, but the sentence 'it is ultimately dictated by the Schwinger-Dyson equations, which admit a single solution for a given energy' needs a brief justification or a reference, because the coexistence region by definition contains multiple stationary solutions at a given temperature.

Circularity Check

0 steps flagged · score 1.0 of 10

The reported Lyapunov exponents come from a standard kernel eigenvalue calculation, not from fitting the target result; the paper is essentially non-circular, with only a minor unverified mapping between non-equilibrium states and equilibrium saddle points.

full rationale

The central quantity, the hot-wormhole Lyapunov exponent, is obtained by solving the retarded-kernel eigenvalue condition (3.10)-(3.11) for the value of lambda_L at which the largest eigenvalue crosses 1. The inputs are the spectral functions rho_ab(omega,T) and the effective inverse temperature beta_eff(T) extracted from the non-equilibrium simulation via the fluctuation-dissipation fit (2.18), but lambda_L is not a fitted parameter and no equation defines beta_eff in terms of lambda_L or vice versa. The Floquet protocol is cited from the authors' prior work [19], yet that citation supplies a driving setup rather than the theoretical conclusion, and the protocol is explicitly described in Section 3.2, making the numerical output an independent result. The paper also honestly discards the n<14 Floquet states because the FD relation is not satisfied, so those non-thermal solutions are not mislabeled as hot-wormhole states for the purpose of the exponent calculation. The main weakness is that Section 2.1 asserts 'We have checked numerically that this is the case' when claiming that the non-equilibrium configurations correspond to the unstable equilibrium saddle-point solutions, but no direct comparison of non-equilibrium and equilibrium spectral functions is shown; this is an evidence gap affecting whether lambda_L is exactly the hot-wormhole exponent, not a circularity in the derivation. No uniqueness theorem is imported from the authors' prior work, and no known empirical pattern is merely renamed as the central claim. Overall, the Lyapunov computation is self-contained and non-circular.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, dimensions, or conserved quantities are introduced. The 'hot wormhole' and 'cold black hole' are relabelings of existing unstable solutions of the coupled SYK model. The main fitted inputs are the effective temperature used to label the trajectory and the asymptotic temperature extrapolated for the unresolved red solutions.

free parameters (2)
  • beta_eff(T) effective inverse temperature = time-dependent fits at each average time T
    Central to the extraction of Lyapunov exponents. Obtained by fitting iG^K/rho to tanh(beta*omega/2) in Eq. (2.18). This is a standard diagnostic definition, not a parameter tuned to the target Lyapunov exponent.
  • beta_infinity from exponential fits of red solutions = values above beta_c ~ 27.3, shown as red dots in Fig. 7(c)
    Eq. (4.1) fits beta_eff(T) = A exp(-gamma T) + beta_infinity for the red non-thermal solutions; the resulting beta_infinity > beta_c underpins the claim that these solutions are suspicious from the phase diagram. This is a diagnostic fit to numerical data, acknowledged by the authors as not a demonstrated thermalization.
assumptions (6)
  • domain assumption Large-N saddle point and factorization of correlators in the two-coupled SYK model.
    The effective action (2.5), Schwinger-Dyson equations (2.6)-(2.7), and the 1/N expansion (3.2) assume large-N dominance. Standard in the SYK literature but not proven in this paper.
  • domain assumption Kadanoff-Baym equations (2.13)-(2.14) integrated with the NESSi package give accurate real-time Green's functions.
    The numerical method is taken from the NESSi library and prior works [18,19]; no convergence or error analysis is provided.
  • domain assumption The retarded chaos kernel (3.5) remains valid in the presence of the bath.
    Justified by the bath action being linear in the system two-point functions (Eq. (2.10)), so quadratic fluctuations are unchanged. Relies on that linearity and on large-N factorization.
  • domain assumption The Schwarzian action (4.2) and effective potential (4.5) capture the low-energy dynamics.
    Used for qualitative explanation in Section 4.1; the authors state that it does not capture all observed behaviors.
  • domain assumption Matter contribution to the Schwarzian potential (4.7) from the lightest fermion with Delta = 1/4.
    Taken from [18]; the barrier it creates is used to explain the metastable hot wormhole solutions.
  • domain assumption Driving at the resonant frequency Omega = 1.0 of the wormhole phase injects energy exponentially.
    Protocol taken from [19]; the exponential energy growth is assumed for the Floquet injection mechanism.

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Pith. "Pith review of Hot wormholes and chaos dynamics in a two-coupled SYK model." pith.science (2026). https://pith.science/paper/C3EUJERD

@misc{pith2026250104660,
  author       = {Pith},
  title        = {Pith review of: Hot wormholes and chaos dynamics in a two-coupled SYK model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3EUJERD}},
  note         = {Machine review of arXiv:2501.04660}
}
read the original abstract

We study the dynamics of chaos across the phase transition in a 2-coupled Sachdev-Ye-Kitaev (SYK) model, with a focus on the unstable "hot wormhole" phase. Using the Schwinger-Keldysh formalism, we employ two non-equilibrium protocols that allow access to this phase, which is inaccessible through equilibrium simulations: one involves cooling the system via a coupling to a thermal bath, while in the other we periodically drive the coupling parameter between the two sides. We numerically compute the Lyapunov exponents of the hot wormhole for the two cases. Our results uncover a rich structure within this phase, including both thermal and non-thermal solutions. These behaviors are analyzed in detail, with partial insights provided by the Schwarzian approximation, which captures certain but not all aspects of the observed dynamics.

Figures

Figures reproduced from arXiv: 2501.04660 by the authors.

Figure 1
Figure 1. Left: Phase diagram of the coupled model for [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Contours used in the numerical integration. At [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Usual contour used for the evaluation of the OTOC in the chaos regime. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Wormhole formation in real time [18]. We show the cooling process that allows us to transition from the black hole to the wormhole phase by turning on a coupling to a cold bath. The system explores the hot wormhole solutions of the new phase diagram. On the left we sho…
Figure 5
Figure 5. Figure 5: Chaos exponents of the hot wormhole phase obtained at each average time [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Wormhole formation in real time. By coupling the system, initially in a black hole solution [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Figs. 7(a), 7(b): Effective inverse temperature for the non-thermal (n ∈ [1, 13] in [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Chaos exponents of the final equilibrium solutions that thermalize within our simulation [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: The figure displays the time evolution of the inverse effective temperature and energy, [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Transmission amplitudes of four different final states within the unstable phase. We [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Shape of the potential (4.5) for different values of ∆. The case ∆ = 1/2 can be solved analytically. looking at the potential we see that these will be bounded and oscillating solutions when E < 0, but they are unbounded when E > 0, with the energy given by E = 2αS J …
Figure 12
Figure 12. Figure 12: Typical bounded (E < 0) and unbounded (E ≥ 0) solutions of the potential (4.5). The constant blue line corresponds to the solution sitting at the minimum. Let’s focus on the constant solution at the minimum of the potential. We aim to demon￾strate that we can bring it…
Figure 13
Figure 13. Figure 13: Solid lines: numerical solutions of the equation of motion ( [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Original (orange, dashed) and modified (black) potential for two different temperatures. [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

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  1. Size Operator and Spectral Clustering in the Two Coupled SYK Model

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.