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Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit

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Pith's one-line read A mixed quench of the large-q Majorana SYK model reaches thermal equilibrium in the thermodynamic limit.

desk verdict Finite-rate thermalization in a mixed large-q SYK quench is a real result with good numerics, but the thermal identification rests on a fit rather than an independent KMS check. read the letter →

arxiv 2411.12421 v4 pith:4I4U2W3N submitted 2024-11-19 cond-mat.str-el cond-mat.stat-mechquant-ph

Santiago Salazar Jaramillo , Rishabh Jha , Stefan Kehrein This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mechquant-ph
keywords Sachdev-Ye-KitaevmodelthermalizationofclosedquantumsystemsquenchKadanoff-Baymequationslarge-qlimitthermodynamicnon-equilibriumGreen'sfunctionsrandomhopping
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

A large-$q$ Majorana Sachdev-Ye-Kitaev model prepared in thermal equilibrium is suddenly coupled to a random-hopping term, and the paper asks whether this closed system, evolving unitarily in the thermodynamic limit, settles into a new thermal state. The paper's answer is yes: after enough time the real-time Green's function (the two-time fermion correlation function) becomes stationary and obeys the same second-order ordinary differential equation that equilibrium Green's functions obey, with a final inverse temperature $\beta_f$ that depends on the initial temperature and the quench strength. The approach to that state is quantified by fitting the relaxation of the kinetic energy to an exponential, giving thermalization rates that span orders of magnitude and drop sharply for weak quenches. The significance is that this is genuine finite-time thermalization in a closed, non-integrable model solved directly in the thermodynamic limit, with no bath and no finite-size extrapolation.

What carries the argument

The carrier of the argument is the large-$q$ exponential ansatz $G^>(t_1,t_2) = -\frac{i}{2} e^{g(t_1,t_2)/q}$, which reduces the Schwinger-Dyson equations to a single nonlinear, non-Markovian Volterra integro-differential equation for $g(t_1,t_2)$, Eq. (15). That equation is solved on the two-time plane by causal predictor-corrector stepping, which propagates the known equilibrium initial state forward without retro-causal input. Stationarity is then checked in Wigner coordinates, and thermality is judged by whether a late-time slice satisfies the equilibrium differential equation $\frac{d^2g}{dt^2} = -2J_q^2 e^g - 2J_2^2$, with the initial slope supplied by the same numerical solution. The same machinery produces the interaction and kinetic energy densities from which the final inverse temperature and the thermalization rate are extracted.

What would settle it

Take the stored late-time Green's function at a reported $\beta_f$ and test the thermal periodicity $g(t)=g(-t-i\beta_f)$; alternatively solve Eq. (17) with $g(0)=0$ and the slope fixed by that same $\beta_f$ and compare with the Kadanoff-Baym solution. A mismatch beyond the stated 1% energy-error budget would show that the stationary state is not the thermal state at the fitted temperature.

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Extended reading notes

Core claim

The central claim is that the mixed quench—switching on the random-hopping term $J_2$ at $t=0$ in the Hamiltonian $H = \mathrm{SYK}_q + \mathrm{SYK}_2$—does not thermalize instantaneously, but its Kadanoff-Baym solution nevertheless flows to the thermal equilibrium state of the post-quench Hamiltonian. In Wigner coordinates $(T,t)$, the late-time Green's function becomes independent of the average time $T$, and the remaining function $g_{\beta_f}(t)$ satisfies the equilibrium ordinary differential equation $\frac{d^2 g}{dt^2} = -2J_q^2 e^{g} - 2J_2^2$. The paper fixes $\beta_f$ by matching the long-time interaction energy density to the equilibrium energy at that temperature, and extracts the thermalization rate $\gamma$ from an exponential fit of the kinetic energy density. It reports that stronger quenches yield higher final temperatures, approaching the infinite-temperature regime at $J_2 \approx 0.07$, and that thermalization rates fall by orders of magnitude as the final temperature decreases. The conclusion is that a closed SYK system in the thermodynamic limit exhibits real-time thermalization with respect to Green's functions and energy, with a finite, quench-dependent rate.

Load-bearing premise

The paper's identification of the late-time state as thermal rests on the premise that stationarity plus satisfaction of the equilibrium differential equation Eq. (17), with the initial slope taken from the same numerical solution, is enough to prove thermal equilibrium; the KMS thermal-periodicity condition is not checked independently of the energy fit.

Editorial extensions

If this is right

  • The final state of the mixed quench is a thermal state at a computable $\beta_f$ that interpolates between the initial inverse temperature and the infinite-temperature limit as $J_2$ grows.
  • Thermalization timescales are finite and quench-dependent: strong quenches relax quickly and approach $\beta_f \sim 10^{-4}$, while weak quenches give rates orders of magnitude smaller.
  • Because the model is solved directly in the thermodynamic limit, the result provides a concrete instance of a closed non-integrable quantum system thermalizing without a bath and without finite-size extrapolation.
  • The reported final temperatures are obtained from the interaction energy without assuming fluctuation-dissipation, so the thermal identification is inferred from the dynamics rather than imposed by the fitting procedure.
  • The predictor-corrector algorithm reproduces both analytically solved limits and conserves total energy to within 1%, so the observed relaxation is not an artifact of the integration scheme.

Reading between the lines

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

  • If the late-time solution is genuinely thermal, the sharp drop in $\gamma$ near $J_q\beta_f \approx 10^{-1}$ could signal a crossover in the relaxation mechanism; computing a time-dependent spectral function at low final temperatures would test whether this is a dynamical crossover or a numerical/energy-budget effect.
  • The same method could be re-run with a slow ramp instead of an instantaneous quench to ask whether the final temperature follows an adiabatic path; that would connect the result to general questions about quench thermodynamics.
  • An independent check the paper leaves implicit is the KMS periodicity $g(t)=g(-t-i\beta_f)$; verifying it directly on the stored numerical data would settle whether the fitted $\beta_f$ is the true temperature or only an energy-matching parameter.
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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 / 6 minor

Summary. The manuscript studies thermalization after a mixed quench in a closed large-q Majorana Sachdev-Ye-Kitaev (SYK) system in the thermodynamic limit. The authors derive Kadanoff-Baym equations from the large-N Schwinger-Dyson equations, reduce them via the large-q ansatz to a single nonlinear integro-differential equation for g(t1,t2), solve this equation numerically with a predictor-corrector scheme with causal stepping, and benchmark the solver against two exactly solvable limits (pure large-q SYK in equilibrium and a kinetic quench to SYK_2). They then analyze the late-time solution in Wigner coordinates and claim that it becomes stationary, satisfies the equilibrium ODE (17), and therefore reaches a thermal state with final inverse temperature β_f and thermalization rate γ. The manuscript also provides a detailed derivation of the energy in the Keldysh contour, an estimation protocol for β_f and γ, and an appendix with additional quench cases.

Significance. The paper is valuable as a concrete demonstration of finite-timescale thermalization in a closed, solvable model in the thermodynamic limit, complementing known instantaneous-thermalization results for pure SYK quenches. The first-principles derivation of the non-equilibrium energy, the analytic benchmarks, the transparent predictor-corrector implementation, and the public availability of data and code are strengths. The thermalization-rate curves (Fig. 9b) reveal a nontrivial dependence on final temperature that is worth reporting. However, the core claim that the final state is thermal rather than merely stationary is not yet independently established, as detailed in the major comments.

major comments (3)
  1. [Sec. VI A, Eq. (66), Fig. 8] The identification of the late-time state as thermal is not independently verified. The 'remarkable match' in Fig. 8 compares the numerical late-time Wigner slice to the solution of Eq. (17) obtained with the initial slope taken from the same numerical data via Eq. (66). By the Picard-Lindelöf theorem, any function satisfying Eq. (17) with that initial slope is exactly that ODE solution, so the comparison can only fail through numerical integration error. Eq. (17) is the stationarity condition, not the KMS/detailed-balance condition; non-thermal stationary or prethermal states also satisfy it. The only quantitative link between the late-time state and a temperature is the energy fit in Sec. IV C, which matches a single observable (interaction energy) and can always be satisfied by choosing β_f. The paper explicitly avoids a fluctuation-dissipation test (Sec. VII) because of a leading-order delta function in 1/q. The central claim therefore currently rests on stationarity plus an energy fit rather than on a genuine thermal-equilibrium check. I recommend testing the late-time solution against the real-time analytic continuation of the imaginary-time solution of Eq. (19) for the fitted β_f (without using Eq. (66)), or performing an alternative KMS check that is well-defined in the large-q limit.
  2. [Sec. IV A and Fig. 6/Fig. 13] The numerical acceptance criterion is not consistently met for the weak-quench data used in the main quantitative results. Sec. IV A states that the change in energy is required to be below 1% at every point in time, but Fig. 6(c) reports |(E_f - E_0)/E_0| = 0.026 for J2=0.003, and Fig. 13 shows errors exceeding 1% for several other weak quenches. These points nevertheless contribute to Fig. 9 and to the thermalization-rate analysis. Please either enforce the stated criterion by excluding such runs, provide a quantitative uncertainty propagation for these points, or explicitly document and justify the relaxed tolerance for weak quenches; as written, the weakest-quench results are not backed by the paper's own quality standard.
  3. [Sec. IV C, Fig. 9a] The final inverse temperatures are reported without uncertainties, even though they are determined by a one-parameter fit of V_EQ(β_f) to the late-time interaction energy V(t1). At least for quenches with larger energy residuals (e.g., J2=0.003), the fitted β_f can carry substantial error. Please report confidence intervals or bootstrap/error-propagation estimates for β_f, and state how the late-time fitting window (T_max) affects the result. Without this, the quantitative characterization of the final state is incomplete.
minor comments (6)
  1. [Sec. III E 1] There is a typo: 'intantly thermalize' should read 'instantly thermalize'.
  2. [Sec. VI A] There is a typo: 'knwo' should read 'know' (in the sentence before Eq. (66)).
  3. [Appendix G] The paragraph describing Figs. 14 and 15 is duplicated verbatim after Fig. 13; the repetition should be removed.
  4. [Data and Code Availability] The statement says the data are 'available at Zenodo upon reasonable request', which is internally inconsistent: the data are either openly available via the Zenodo DOI [62] or available on request, not both. Please rephrase.
  5. [Sec. IV C] Reference [59] is an unpublished bachelor thesis; if possible, provide a permanent identifier or archived version, or fully describe the solver in an appendix, to make the equilibrium Green's function calculation reproducible.
  6. [Fig. 3 caption] The caption contains a typo: 'max([Error)' should read 'max(|Error|)'.'

Circularity Check

3 steps flagged · score 6.0 of 10

Thermal identification is made tautological by fitting βf to the late-time energy and by constructing gβf(t) from the numerical slope (Eq. 66); Eq. (17) checks stationarity, not KMS.

  1. fitted input called prediction [Section IV C (Estimation of the final temperature), Eqs. (59)-(61)]
    "Here βf is a free parameter. Having recursively solved for the equilibrium Green's function ˜g(x), the correct βf for the quenched system can be determined by fitting the equilibrium energy to the long time limit of non-equilibrium energy in Eq. (25). ... Therefore we fit this VEQ to the non-equilibrium interaction energy density V(t1) in Eq. (26) that has been evolved for a long time for it to relax to a constant value. This fitting gives us the estimate for the final temperature βf of the quenched system if it thermalizes."

    βf is not independently predicted: it is defined as the value that makes the equilibrium interaction energy equal to the numerically evolved late-time interaction energy. Any run in which V(t1) saturates yields a βf by construction. The later claim that the final state is thermal equilibrium with this βf therefore rests on this fit, since no independent KMS or fluctuation-dissipation check is performed; Section VII explicitly discards the FDT route. The fit is an input chosen to match the data, not a test of thermalization.

  2. self definitional [Section VI A (Stationary limit), Eq. (66) and Fig. 8]
    "In order to avoid guess-work, the initial condition of the derivative of g(t) is calculated via dgβf(t)/dt|t=0 = dg(Tmax,t)/dt|t=0, (66), which is then used to solve Eq. (17) numerically via a Runge-Kutta-4 method. Fig. 8 shows that g(T, t) converges towards the thermal gβf(t) as T increases."

    Eq. (17) is a second-order ODE with gβf(0)=0 and with its initial slope taken from the same numerical solution at Tmax. By Picard-Lindelöf, gβf(t) is the unique solution with those initial data, i.e., the local continuation of the numerical solution itself. Comparing g(Tmax,t) with gβf(t), or checking that g(T,t) satisfies Eq. (17), is therefore a consistency check that cannot fail once the integrator is stable; it does not independently establish that the stationary solution is the thermal state at the fitted βf.

1 more flagged steps
  1. other [Section VI A, Eqs. (64)-(65)]
    "we can verify whether or not the stationary state gβf(t) in Eq. (64) corresponds to thermal equilibrium. The way this is done is that we wait for a long time T such that the relaxation dynamics have happened and then we substitute the Green's function g(T → large, t) into the equilibrium ordinary differential equation for gβf(t) given in Eq. (17)."

    Eq. (17) is derived purely from the stationarity assumption g(t,t′)=g(t−t′) and contains no β or KMS condition; every stationary solution of the large-q Kadanoff-Baym equations satisfies it. Thus satisfying Eq. (17) is equivalent to stationarity, not to thermality. The label 'thermal' is imported from the fitted βf of Section IV C; without an independent KMS test, the stationarity check cannot certify equilibrium, so the thermal conclusion reduces to the energy fit plus stationarity.

full rationale

The paper's first-principles derivation of the Kadanoff-Baym equations (Section III and Appendices A-F) and its numerical integration are self-contained: no load-bearing self-citation is involved, and the benchmark against the two analytic limiting cases is genuine evidence. The stationarity result (dg/dT→0, Fig. 7) and the saturated energy curves are independent numerical findings, and the thermalization rates are extracted by an exponential fit to K(t1), which is a fitted output but not itself circular. However, the central claim that the stationary state is the thermal state at the fitted inverse temperature reduces by construction at two points. First, βf is obtained by fitting the equilibrium interaction energy V_EQ(βf) to the long-time non-equilibrium interaction energy, so βf is an input chosen to match the data rather than a predicted quantity. Second, gβf(t) is defined by solving Eq. (17) with its initial slope taken from the same numerical solution at Tmax via Eq. (66); by Picard-Lindelöf this is the unique continuation of that solution, making the Fig. 8 match tautological. Moreover, Eq. (17) is only the stationarity condition and contains no β or KMS relation, so satisfying it cannot certify thermality; the paper explicitly declines the FDT check in Section VII. The weakest-quench energy deviation from the stated 1% criterion (|Ef−E0|/E0=0.026, Fig. 6) adds a correctness concern but is not itself circularity. On balance, the thermal identification is partially circular, while the dynamical and numerical content remains independent.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the large-N and large-q limits, on the standard Keldysh and Langreth formalism, and on two fit-based characterizations (beta_f and gamma). The thermal-equilibrium verification uses a criterion that partially feeds the numerical solution back into the comparison. No new physical entities are introduced.

free parameters (3)
  • Final inverse temperature beta_f = e.g., Jq beta_f ~ 1e-4 for J2=0.07; up to ~1e-1 for weak quenches
    Obtained by fitting the equilibrium interaction energy V_EQ(beta_f), Eq. (61), to the long-time limit of the non-equilibrium interaction energy V(t1), Eq. (26). The extracted beta_f is used to characterize the final state; no independent estimate, e.g., from KMS or FDT, is reported.
  • Thermalization rate gamma = range roughly 1e-4 to 1e-3 from Fig. 9b
    Obtained by fitting K(t1) to the exponential ansatz A(t1)=B_K + C_K e^{-gamma t1}, Eq. (62). The rate is an output of a fit, not derived from the equations of motion.
  • Initial slope of equilibrium Green's function dg_beta_f/dt at t=0 = taken from late-time numerical solution at Tmax
    Used as the boundary condition to solve the equilibrium ODE Eq. (17) for g_beta_f(t) in Fig. 8. This boundary condition comes from the same numerical solution being tested, which makes the subsequent comparison partially self-referential.
assumptions (7)
  • domain assumption Large-N saddle point: replica-diagonal effective action is exact as N goes to infinity, with no spin-glass instability.
    Used to derive Schwinger-Dyson and KB equations (Appendix A2, Eq. A10). Relies on self-averaging and absence of spin-glass order, cited to Refs. [65-68].
  • domain assumption Large-q expansion: O(1/q^2) terms are negligible in the Green's function ansatz, KB equations, and energy expressions.
    Central to Eq. (11) ansatz and Eq. (15). The justification via Ref. [10] concerns equilibrium transport and is not directly demonstrated for far-from-equilibrium quench dynamics.
  • domain assumption Bogoliubov principle of weakening of correlations: the imaginary-time branch of the Keldysh contour can be discarded for a system prepared in equilibrium in the infinite past.
    Invoked throughout Section III and Appendix B to reduce the contour to C+ + C-, and later to justify the energy calculation on the vertical branch.
  • domain assumption KMS condition Eq. (14) is the correct thermal boundary condition for the large-q Green's function.
    Used to connect g(t) to temperature in Eq. (33) and in the equilibrium analysis; standard but assumed without derivation in this large-q framework.
  • ad hoc to paper Satisfying the stationary equilibrium ODE Eq. (17) with initial slope from the numerical solution is sufficient to identify the late-time state as thermal.
    This validation criterion is constructed in Sec VI A and Eq. (66); the paper does not independently check KMS or FDT for the late-time Green's function.
  • ad hoc to paper Kinetic energy relaxation follows the single-exponential ansatz A(t1)=B_K + C_K e^{-gamma t1}.
    Used in Sec IV D to define the thermalization rate gamma; goodness of fit is shown but no derivation from the KB equations is given.
  • standard math Picard-Lindelof uniqueness of the equilibrium ODE initial value problem Eq. (17).
    Invoked in Sec VI A to argue the KB solution flows to the unique equilibrium solution.

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Pith. "Pith review of Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit." pith.science (2026). https://pith.science/paper/4I4U2W3N

@misc{pith2026241112421,
  author       = {Pith},
  title        = {Pith review of: Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4I4U2W3N}},
  note         = {Machine review of arXiv:2411.12421}
}
abstract

The question of thermalization of a closed quantum system is of central interest in non-equilibrium quantum many-body physics. Here we present one such study analyzing the dynamics of a closed coupled Majorana SYK system. We have a large-$q$ SYK model prepared initially at equilibrium quenched by introducing a random hopping term, thus leading to non-equilibrium dynamics. We find that the final stationary state reaches thermal equilibrium with respect to the Green's functions and energy. Accordingly, the final state is characterized by calculating its final temperature and the thermalization rate. We provide a detailed review of analytical methods and derive the required Kadanoff-Baym equations, which are then solved using the algorithm developed in this work. Our results display rich thermalization dynamics in a closed quantum system in the thermodynamic limit.

Figures

Figures reproduced from arXiv: 2411.12421 by the authors.

Figure 1
Figure 1. Sketch of the Schwinger-Keldysh contour C = C+ + C− + Cimag. The real time forwards and backwards paths go between −∞ to +∞ and are followed by the imaginary time branch which represents the equilibrium state. The branches running from −∞ to ∞ are purely real and are shifted above and below the axis only for visualization purposes. The points on the vertical contour comes later than points lying on both the forward … view at source ↗
Figure 2
Figure 2. Sketch of the fan-like propagation in the two [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Predictor-corrector performance in the limiting cases (as presented in Section [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Predictor-corrector solutions for the case of a [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Visualization of the numerical solution for a mixed SYK quench with [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Additional visualizations of the numerical [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Derivatives along the diagonals T of the real and imaginary components of g(T , t) (namely dg(T ,t) dT [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Stationary limit of the Kadanoff-Baym solution [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Thermalization and approach to the final state of the mixed quench as a function of [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: Thermalization rate plotted against temperature to complement the picture in Fig. 9b where inverse temperature is used. Color coding is the same as in [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Sketch of the deformation of the Keldysh [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: For our quenched case where SYK2 term is switched on at t = 0 starting from an SYKq equilibrium condition, accordingly the forward and the backward contour from time −∞ to 0 in [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]
Figure 14
Figure 14. Figure 14: Comparison of the stationary limit of the [PITH_FULL_IMAGE:figures/full_fig_p033_14.png]
Figure 15
Figure 15. Figure 15: Comparison of the stationary limit of the [PITH_FULL_IMAGE:figures/full_fig_p033_15.png]
Figure 16
Figure 16. Figure 16: Goodness of fit for K(t1) using the exponential ansatz in Eq. (62). The dashed line corresponds K(t1), while the solid one depicts the fitted curve. All four instances shown here have β0 = 18.9 as the initial inverse temperature [PITH_FULL_IMAGE:figures/full_fig_p034…

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