{"id":"dbdb3733-1938-4845-b791-ce4dd7fa53f2","arxiv_id":"2501.04660","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"First numerical Lyapunov exponents for the unstable hot wormhole phase of the two-coupled SYK model, obtained via cooling and periodic-driving protocols.","lead":"This paper simulates two coupled SYK quantum systems out of equilibrium and extracts the chaos (Lyapunov) exponents of an unstable 'hot wormhole' phase that equilibrium methods cannot reach. It finds both thermal and non-thermal states in that phase, with a Schwarzian model explaining part but not all of the behavior.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The effective-temperature labeling is not independently verified: non-equilibrium spectral functions are never compared to the unstable equilibrium hot-wormhole saddle, so the reported λ_L may not be hot-wormhole exponents.","rationale":"The reader's weakest_assumption correctly identifies the effective-temperature labeling as the load-bearing step. My stress-test sharpens this into a concrete, checkable requirement: the non-equilibrium spectral functions must equal those of the equilibrium unstable hot-wormhole saddle. The paper asserts this has been checked but provides no direct comparison, and the FD relation alone is insufficient because the equilibrium model has multiple saddles at the same β. The cooling protocol further complicates the mapping because the bath modifies the Hamiltonian, but the Floquet thermal solutions are the ones claimed to correspond to the original model's hot wormhole, so testing those is the decisive check. The proposed test is feasible with existing numerical methods and would settle the concern without ambiguity. No internal inconsistency is found; the concern is about verification, not soundness. Therefore the conditional verdict stands, and no verdict change is warranted.","tokens_in":17527,"tokens_out":3574,"duration_ms":36843,"concrete_test":"For a representative Floquet thermal solution (e.g., n=25), extract the asymptotic β_eff and spectral functions ρ_ab(ω) from the simulation. Independently solve the equilibrium Schwinger-Dyson equations (2.6)-(2.7) at the same β and μ=0.1 using a root-finding method (e.g., Newton iteration on the Matsubara axis) to obtain the unstable hot-wormhole saddle. Compare ρ_LL(ω) and ρ_LR(ω) between the simulation output and this saddle. If they agree within the numerical tolerance used for the FD check, the identification is validated. If they differ, recompute λ_L using the equilibrium saddle's spectral functions and check whether it matches Fig. 8; a mismatch would show the reported exponents are not those of the hot wormhole.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Lyapunov exponents are computed for the hot wormhole phase by labeling non-equilibrium trajectories with an effective temperature from Eq. (2.18) and using the corresponding spectral functions in the kernel equation (3.11). This requires that the instantaneous non-equilibrium state coincides, at each average time T, with the unstable equilibrium solution of the Schwinger-Dyson equations at β_eff(T). Section 2.1 asserts 'We have checked numerically that this is the case,' but no such comparison is shown in the paper. For the Floquet protocol, the thermal solutions (n≥14) satisfy the FD relation, so the two-point functions are thermal; however, the equilibrium model at the same β has three saddle solutions (black hole, hot wormhole, wormhole) in the coexistence region, and FD alone does not identify which saddle the simulated state corresponds to. The energy location in Fig. 7(c) is suggestive but relies on the same simulation data being plotted against the equilibrium curve, not on an independent verification of the spectral functions. If the simulated state is not exactly the hot-wormhole saddle, then λ_L extracted from Eq. (3.11) is not the hot-wormhole exponent, even though it may be a meaningful effective exponent for the actual non-equilibrium state. The paper also provides no error bars or convergence checks for the numerical eigenvalues, compounding the uncertainty.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17631,"tokens_out":8206,"duration_ms":84887,"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":[{"comment":"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.","section":"Section 2.1, Eq. (2.18); Section 3, Eq. (3.11)"},{"comment":"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.","section":"Section 3, Figs. 5 and 8"}],"minor_comments":[{"comment":"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.'","section":"Section 4, paragraph after Fig. 10"},{"comment":"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).","section":"Fig. 6 caption"},{"comment":"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.","section":"Eq. (2.18) and Fig. 7"},{"comment":"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.","section":"Section 3.1"},{"comment":"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.","section":"Section 4, Eq. (4.1)"},{"comment":"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.","section":"Section 4.1.1, Eq. (4.7)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and addresses a timely question in SYK holography. The main technical concern is not internal inconsistency but missing verification of the state identification that is load-bearing for the central claim. I do not see grounds for rejection; the required additions are concrete comparisons and convergence checks. No code or data availability statement is included, which would be useful given the numerical nature of the work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the computation: first Lyapunov exponents for the unstable hot wormhole phase of the two-coupled SYK model, via two non-equilibrium protocols (cooling through a bath, and Floquet driving) using standard large-N Schwinger-Dyson and OTOC kernel technology. The smooth interpolation of lambda_L along the hot branch, and the thermal/non-thermal split in the driven protocol, are new and useful for the SYK/holography subfield. The authors are also candid about what they cannot explain: the red non-thermal solutions, the bath-modified phase diagram, and the partial Schwarzian picture are all flagged in the text. That transparency earns real credit.\n\nThe main soft spot is exactly what the stress-test note says: the mapping from non-equilibrium trajectories to equilibrium hot-wormhole saddles rests on an effective-temperature fit (Eq. 2.18) and a numerical check that is asserted but not shown. For the Floquet protocol, FD-thermalized states are located on the equilibrium phase diagram in Fig. 7(c), which is suggestive but not a direct comparison of spectral functions to the unstable equilibrium saddle. So the reported lambda_L could be effective exponents for the actual non-equilibrium states rather than the hot-wormhole saddle exponents. That is a genuine caveat, but not a fatal one: the protocol is explicit, and the smooth interpolation in Fig. 8 is a reasonable consistency check. Missing error bars or convergence estimates for the numerical eigenvalues are a minor issue, but worth requesting in revision.\n\nI see no circularity problem: lambda_L comes from a kernel eigenvalue condition, not from fitting the claimed result. The bath protocol modifies the model by construction, and the authors acknowledge this and use the Floquet protocol to address it. The Schwarzian analysis is explicitly qualitative and partial, which is fine.\n\nWho is this for? SYK/holography researchers interested in non-equilibrium dynamics, scrambling, and the hot wormhole phase. It deserves a serious referee: novel claims, standard enough methods to be checkable, and honest about limitations. I would send it to peer review, asking for the missing spectral-function comparison and error estimates rather than rejecting it.","headline":"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.","tokens_in":18329,"tokens_out":2264,"would_cite":true,"duration_ms":22988,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Sachdev-Ye-Kitaev model","traversable wormhole","hot wormhole phase","Lyapunov exponent","out-of-time-order correlator","Schwinger-Keldysh formalism","Floquet driving","Schwarzian approximation"],"falsifier":"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.","tokens_in":17182,"feed_emoji":"🕳️","tokens_out":12571,"duration_ms":105330,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hot wormhole chaos exponents computed out of equilibrium","feed_subtitle":"Cooling and periodic driving reach the unstable phase; Lyapunov values run smoothly from black hole to wormhole.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It defines the two-coupled SYK model, introduces the hot wormhole phase, and conjectures its microcanonical stability.","marker":"[10]"},{"why":"It introduces the cooling-with-bath protocol and the effective-temperature fit through the fluctuation-dissipation relation.","marker":"[18]"},{"why":"It provides the retarded OTOC kernel equation and the Lyapunov exponents for the two stable phases that this paper extends.","marker":"[16]"},{"why":"It introduces the periodic driving of the inter-side coupling used to reach the hot wormhole without a bath.","marker":"[19]"},{"why":"It derives the quadratic fluctuations of the effective action and the Schwarzian limit underlying the chaos kernel.","marker":"[7]"},{"why":"It supplies the numerical solver used for the Kadanoff-Baym equations in the non-equilibrium simulations.","marker":"[34]"},{"why":"It notes that the retarded kernel remains valid in the presence of the linear bath coupling.","marker":"[35]"},{"why":"It provides the analytic Schwarzian solutions for the $\\Delta=1/2$ case used in the qualitative analysis.","marker":"[37]"}],"fun_headline_variants":["Hot wormhole chaos mapped via non-equilibrium protocols","Cooling and driving reveal hot wormhole Lyapunov exponents","Chaos in hot wormholes computed out of equilibrium","Hot wormhole instability yields computable chaos exponents","Non-equilibrium paths expose hot wormhole chaos dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hot wormhole chaos mapped via non-equilibrium protocols","Cooling and driving reveal hot wormhole Lyapunov exponents","Chaos in hot wormholes computed out of equilibrium","Hot wormhole instability yields computable chaos exponents","Non-equilibrium paths expose hot wormhole chaos dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000154,"raw_usage":{"total_tokens":1174,"prompt_tokens":872,"completion_tokens":302,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":225}},"tokens_in":488,"tokens_out":302,"duration_ms":3321,"temperature":1.0,"reasoning_tokens":225,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:27:16.540694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Chaos exponents of SYK traversable wormholes","cited_arxiv_id":"2009.10759","evidence_quote":"It provides the retarded OTOC kernel equation and the Lyapunov exponents for the two stable phases that this paper extends."},{"cited_title":"NESSi: The Non-Equilibrium Systems Simulation package","cited_arxiv_id":"1911.01211","evidence_quote":"It supplies the numerical solver used for the Kadanoff-Baym equations in the non-equilibrium simulations."},{"cited_title":"Gravitational collapse in SYK models and Choptuik-like phenomenon","cited_arxiv_id":"1812.03979","evidence_quote":"It provides the analytic Schwarzian solutions for the $\\Delta=1/2$ case used in the qualitative analysis."}],"review_version":1}