Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems

Gabriel Almeida; Lucas S\'a; Masudul Haque; Pedro Ribeiro

REVIEW 3 major objections 5 minor 1 cited by

Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For a strongly correlated quantum system coupled to a bath whose density of states vanishes as |ω|^ν, the late-time relaxation is set entirely by the bath: correlations decay as t^{-(1+ν)} when ν<1, and chaos-driven exponential decay return

desk verdict The paper's central exponent p=1+ν for ν<1 doesn't follow from its own Eq. (10) — the Fourier transform gives 1−ν — so the headline claim needs major correction, though the qualitative three-regime picture may survive. read the letter →

arxiv 2603.10815 v2 pith:FRCMGA5K submitted 2026-03-11 cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elquant-ph

Gabriel Almeida , Pedro Ribeiro , Masudul Haque , Lucas Sá This is my paper

classification cond-mat.stat-mechcond-mat.dis-nncond-mat.str-elquant-ph

keywords non-MarkoviandissipationSachdev-Ye-KitaevmodelpseudogappedbathalgebraicrelaxationquantumchaosKeldyshformalismSchwinger-Dysonequationslarge-Nlimit

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 an open Sachdev-Ye-Kitaev (SYK) model — a solvable, strongly correlated quantum chaotic system — coupled to a fermionic bath whose density of states vanishes as |ω|^ν at low frequency. Using Keldysh field theory in the large-N limit, it shows that the low-frequency structure of the bath's spectral function completely controls the late-time relaxation: for ν<1 the steady-state spectral function diverges as |ω|^{-ν}, producing algebraic decay t^{-(1+ν)}; for ν>2 the bath is effectively gapped and the system relaxes exponentially through its own chaotic dynamics; between these limits there is a pre-relaxation regime with an exponential-to-power-law crossover. The authors derive this three-regime phase diagram and confirm it numerically, showing that the power-law exponent is universal, fixed only by the bath. This matters because it shows that non-Markovian 'structured' environments do not merely perturb relaxation but can qualitatively select the relaxation mechanism in strongly correlated systems.

What carries the argument

The central object is the low-frequency (infrared) behavior of the steady-state spectral function ρ_-(ω), related to the retarded Green's function by iG_R(t)=Θ(t)∫dω ρ_-(ω)cos(ωt). The paper works with a Keldysh path integral for the open SYK model; after disorder averaging and integrating out the bath, the large-N saddle point yields closed Schwinger-Dyson equations coupling the self-energy σ_-(ω)=J²/4(ρ_-*ρ_-*ρ_-)(ω)+(μ/π)D(ω) to ρ_-(ω). The mechanism is that a pseudogapped D(ω)~|ω|^ν transfers its nonanalyticity into ρ_-(ω): the bath term produces a |ω|^{-ν} divergence for ν<1 and a cusp |ω|^ν for 1<ν<2, and the Fourier transform turns these frequency power laws into power-law decay in ti

What would settle it

A finite-size exact-diagonalization simulation (or a cold-atom experiment with a programmable pseudogapped reservoir) of an open SYK model, measuring the late-time decay of the retarded Green's function for a few values of ν (say 0.5 and 1.5) and checking for t^{-1.5} and t^{-2.5} tails; any deviation—such as a different exponent or a purely exponential decay for ν<1—would falsify the universality claim. Alternatively, solving the Schwinger-Dyson equations without the time-translation-invariance ansatz and finding that non-stationary corrections alter the small-frequency behavior of ρ_-(ω) wou

Watch

Extended reading notes

Core claim

In a large-N open SYK model with linear coupling to a pseudogapped fermionic bath, the steady-state spectral function ρ_-(ω) is either divergent, nonanalytic, or analytic depending on the pseudogap exponent ν. For ν<1, ρ_-(ω)≈[cos²(πν/2)/(πμ)]|ω/Λ|^{-ν}, which by Fourier transform gives a retarded Green's function decaying as t^{-(1+ν)}. For 1<ν<2, the spectral function is finite but has a cusp ρ_-(ω)≈ρ_-(0)-c|ω|^ν, again producing a t^{-(1+ν)} tail; for ν>2 the analytic ω² term dominates and relaxation is exponential, with the gap approaching the isolated SYK value. The boundary between the algebraic and pre-relaxation regimes, ν_c(μ), tends to 1 in the strong-dissipation limit, and the who

Load-bearing premise

The bath is treated as an infinite thermal reservoir that the SYK system cannot affect, and the steady state is assumed to be time-translation invariant, so all relaxation exponents are read off the stationary spectral function; if the bath is back-affected by the strong coupling, or if the transient approach to the steady state is not stationary, the claimed exponents and the phase boundary ν_c(μ) would change.

Editorial extensions

If this is right

For ν<1, the retarded Green's function decays as t^{-(1+ν)} at late times throughout the algebraic regime, so the bath's pseudogap exponent is directly observable in the relaxation law.
For ν>2, the strong depletion of low-energy bath states makes the system relax as if isolated: the relaxation gap Δ and oscillation frequency Ω saturate to their µ=0 values, so internal chaotic dynamics fully govern the decay.
For ν_c(μ)<ν<2, a system will appear to relax exponentially for times shorter than t_*≈(c/d)^{1/(2-ν)}, then cross over to algebraic decay; this crossover time diverges as ν→2⁻, producing an arbitrarily long-lived exponential plateau.
The three-regime structure (bath-driven algebraic, pre-relaxation, chaos-driven exponential) is robust at finite bath temperature, with only quantitative shifts of the boundary ν_c(μ) and the crossover scales.
The results imply that environment engineering—tailoring the low-frequency slope of a reservoir—can deterministically set the relaxation law of a strongly correlated quantum simulator.

Reading between the lines

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

If the exponent p=1+ν is truly universal in the bath-controlled regime, the same t^{-(1+ν)} tail should appear in the decay of observables beyond the retarded Green's function, such as local occupations or entanglement measures once the steady state is reached; this is a direct, testable extension the paper does not compute.
The paper assumes the bath is an infinite reservoir with fixed correlations, but in a finite experimental setup the bath will heat up or be back-affected by the SYK system; a self-consistent treatment of the bath dynamics could renormalize the effective ν and smear the sharp boundary ν_c(μ) into a crossover.
The divergence of ρ_-(ω) for ν<1 implies the steady state is highly sensitive to the precise bath form at the lowest frequencies; if real engineered reservoirs have a small but finite low-frequency cutoff, the power-law tail would terminate at a timescale set by that cutoff, which experiments should look for.
The pre-relaxation crossover near ν=2 resembles critical slowing-down, so a measurement sweeping ν through 2 might be misinterpreted as a dynamical phase transition; distinguishing the divergent timescale t_* from a true gap closure is a clear diagnostic for the predicted mechanism.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: the paper claims a universal power-law relaxation exponent p=1+ν for a SYK system coupled to a pseudogapped bath with spectral density |ω|^{-ν} for ν<1. That doesn't survive contact with their own equations. Eq. (10) gives ρ_-(ω) ~ |ω|^{-ν}; plugging that into their Eq. (6), iG^R(t)=Θ(t)∫ρ_-(ω)cos(ωt)dω, yields a tail t^{-(1-ν)}, not t^{-(1+ν)}. The p=1+ν exponent is what you get from a finite spectral density with a cusp ρ_-(0)-c|ω|^ν, which is the case for 1<ν<2. For ν<1, a genuine |ω|^{-ν} divergence gives a slower decay, not faster. So the central quantitative claim in the bath-driven regime is internally inconsistent.

What's good: the setup is clean — large-N Keldysh solution of an open SYK with a structured bath — and the numerical solution of the Schwinger-Dyson equations is competently done. The ν=0 benchmark against Ref. [81] checks out. The distinction between a divergent spectral function and a cusp, and the resulting three-regime picture, is physically interesting. For 1<ν<2, the pre-relaxation crossover from exponential to algebraic decay seems sound, and the ν>2 regime recovering isolated-SYK exponential relaxation is plausible.

The soft spots: besides the exponent error, the phase boundary ν_c(μ) is defined by an arbitrary threshold δ=0.02 on 1/ρ_-(0+), not an intrinsic criterion. Since ρ_-(0) is actually finite for any ν>0 (the divergence is regularized by interactions), the "divergence" is really just a small value; the phase diagram depends on a numerical cutoff. The finite-temperature analysis in the End Matter imposes the fluctuation-dissipation relation rather than deriving it from the solution, which weakens the robustness claim. And the claimed universality is overstated given the internal error.

Bottom line: this paper deserves a serious referee, but as written it needs major revision. The authors can likely fix the Fourier mapping — the numerics might still support p=1+ν if the true low-frequency behavior is a cusp, not a divergence — but they need to reconcile their Eq. (10) with their data and rewrite the analytical argument. I'd send it to peer review with a request to correct the central derivation, not desk-reject it, but I wouldn't quote the ν<1 exponent in its current form.

Referee Report

3 major / 5 minor

Summary. The manuscript studies an open SYK model coupled to a pseudogapped fermionic bath with density of states D(ω)∼|ω|^ν. Using a Keldysh path integral and large-N saddle point, the authors derive Schwinger-Dyson equations for the steady-state Green's function and analyze the low-frequency structure of the spectral function ρ_-(ω). They claim three dynamical regimes: bath-driven algebraic relaxation with exponent p=1+ν for ν<1, chaos-driven exponential relaxation for ν>2, and an intermediate pre-relaxation regime. Numerical solutions of the SD equations and fits of the time-domain Green's function are presented as confirmation, and finite-temperature robustness is discussed in the End Matter. The central quantitative claim is that the pseudogap exponent ν alone fixes the relaxation exponent.

Significance. If correct, the paper would establish a clean and general mechanism by which a structured environment controls the relaxation law of a strongly correlated many-body system, going beyond the usual Markovian/Lindblad paradigm. The Keldysh framework and the explicit SD equations are a solid technical starting point, and the benchmark against the known Markovian SYK result is a strength. However, the key exponent relation is internally inconsistent with the paper's own Fourier transform, and the phase-diagram interpretation depends on an unstated regularization/crossover issue. Because the central quantitative claim is affected, the paper cannot be accepted in its present form.

major comments (3)

[Main text, Eq. (10) and following paragraph] For 0<ν<1, Eq. (10) gives ρ_-(ω)∼|ω|^{-ν}. Inserting this into the paper's own Eq. (6), iG^R(t)=Θ(t)∫ρ_-(ω)cos(ωt)dω, and substituting x=ωt yields ∫_0^Λ ω^{-ν}cos(ωt)dω ∼ Γ(1−ν)sin(πν/2)t^{-(1−ν)}. Therefore the predicted large-time tail is p=1−ν, not p=1+ν as stated in the text and used in Fig. 5(a). The exponent 1+ν is the Fourier exponent of a cusp |ω|^ν, not of the divergent |ω|^{-ν} form. This is not a minor typo: the central quantitative claim of a universal exponent set by the bath is wrong as stated.
[Eqs. (8)–(10) and Fig. 1(b)] The derivation of Eq. (10) includes a constant interaction self-energy ε=O(J) in σ_- (Eqs. (8),(9)). For an even σ_-, σ_H(0)=0, so Eq. (9) gives ρ_-(0)=1/(π²ε), finite. Thus the |ω|^{-ν} divergence can only be an intermediate-frequency form, valid when μ|ω/Λ|^ν≫ε, with a crossover scale ω_ε∼(εΛ^ν/μ)^{1/ν}. The manuscript states that ρ_-(ω) 'diverges as |ω|^{-ν}' and uses 1/ρ_-(0+)→0 to define the phase boundary in Fig. 1(b). For any finite μ, as ω→0 the spectral function is finite; the 'bath-driven algebraic phase' is therefore a crossover, not a true divergent phase. The relevant limit/order of limits must be specified, and the numerical threshold δ=0.02 cannot define a sharp boundary without a convergence check.
[Eq. (4) and final paragraph] The paper characterizes 'relaxation toward the steady state' via the steady-state retarded Green's function (4), but this object is a steady-state correlation function, not the transient approach of ρ(t) to ρ∞. The Schwinger-Dyson equations are solved under time-translation invariance, and the last paragraph acknowledges that solving without this assumption is an open direction. Hence the claimed relaxation exponents and phase diagram are not directly derived from the actual time evolution of the system from an initial state. The authors should either prove (or argue from linear response) that the GR(t) tail controls the transient relaxation, or reframe the claims as properties of the steady-state response.

minor comments (5)

[Eq. (13)] The effective cusp exponent α_eff is defined through ρ_-(0)−ρ_-(ω), which assumes ρ_-(0) is finite. In the divergent regime ν<1 the definition is not meaningful; clarify that α_eff is extracted only on the non-divergent side.
[Fig. 3 caption] The caption calls the parameters Λ=1000, μ=100 a 'strong dissipation limit', but the text's analytic limit is J≪Λ≪μ; here μ<Λ. Reconcile or reword.
[End Matter, Fig. 6 and Fig. 7 captions] The finite-temperature solutions impose the fluctuation-dissipation relation (7) at every iteration. This should be stated in the main text as an assumption, not as a test of whether the steady state satisfies FDR.
[SM, Eqs. (S26)–(S27)] The bath correlations are taken as fixed equilibrium forms, with no back-action from the SYK system. This is a standard assumption but should be stated explicitly in the main text when interpreting the exponents as universal.
[Eq. (14)] The fit function has six parameters (A, p, B, Δ, Ω, φ). For ν<1, a finite-time window fit may not reliably distinguish t^{-(1−ν)} from t^{-(1+ν)} in the presence of the exponential term. Report residuals or a fixed-window analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular step: the exponents follow from the Schwinger-Dyson equations plus the Fourier relation; self-citations are only benchmarks, and the ν<1 exponent inconsistency is a mathematical correctness issue, not a tautology.

full rationale

The derivation chain is self-contained: the bath exponent ν enters through D(ω)∼|ω|^ν in the Schwinger-Dyson equations (SM Eqs. S30–S32; main-text β=0 limit Eqs. 8–9); the low-frequency forms of ρ_-(ω) are solved from those equations (Eqs. 10–12); and the time exponent p is then obtained from the Fourier relation Eq. (6). No step defines ρ_-(ω) or p in terms of the target p, nor fits a parameter and renames it as a prediction. The α_eff=ν comparison (Eq. 13, Fig. 4) is an asymptotic self-consistency check of an assumed cusp form against the same equations, not a fitted input used to derive the prediction. The numerical solutions are obtained by iterating the same SD equations, so Figs. 4 and 5 are internal-consistency checks rather than independent external tests, but such checks are not circular. The self-cited Ref. [81] is used only as a ν=0 Markovian/isolated benchmark and for µ=0 limiting values, so it is not load-bearing for the new exponent claim. The stated limitation that time-translation invariance is assumed, with the transient problem left open, is a scope restriction, not a circularity. The one serious issue found is mathematical, not circular: for ν<1, Eq. (10) gives ρ_-(ω)∼|ω|^{-ν}; substituting into Eq. (6), ∫_0^Λ ω^{-ν} cos(ωt)dω ∼ Γ(1−ν) sin(πν/2) t^{-(1−ν)}, so the paper's p=1+ν in that regime appears inconsistent with its own spectral function. That is a correctness/consistency error, not a reduction of a prediction to its inputs, so the circularity score is 0.

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

Model inputs (J=1, mu, Lambda=10, nu, beta) are physical parameters scanned or held fixed, not fitted. The three listed free parameters are the hand-chosen or fit-based quantities that enter the central claims. No invented entities are introduced: the pseudogapped bath is a standard reservoir model (Refs. [40-47]), and the 'pre-relaxation phase' is a claimed dynamical regime, not a new entity.

free parameters (3)

Phase-boundary threshold delta = 0.02 = 0.02
Hand-set cutoff defining the 'divergence' region 1/rho_-(0+) < delta in Fig. 1(b); the drawn boundary nu_c(mu) moves with delta and with grid resolution.
Six-parameter fit of Eq. (14): A, p, B, Delta, Omega, phi = p about 1+nu reported in Fig. 5(a); A J^p, Delta, Omega in panels (b)-(d); B, phi not shown
The claimed confirmation of p=1+nu rests on this fit; fits sensitive to the fitting window are manually excluded (shaded points), so the fit is not a closed, reproducible procedure.
Effective cusp exponent alpha_eff = about nu, with deviations near nu -> 2^-
Extracted from the log-log slope of rho_-(0)-rho_-(omega) (Eq. 13) and used as numerical confirmation of alpha=nu; deviation near nu=2 is attributed to discretization, but the grid-degeneracy claim is only self-reported.

assumptions (7)

standard math Large-N saddle-point dominance: Schwinger-Dyson equations (Main Text Eqs. 8-9; SM Eqs. S20-S32) are exact in the N -> infinity limit.
Standard SYK large-N machinery; disorder average and saddle point give closed equations for rho_+/-. All subsequent claims are read off these equations.
domain assumption Bath remains in thermal equilibrium with bare correlations K^>(omega)=-(1-n_F(omega))D(omega) and K^<(omega)=n_F(omega)D(omega) (SM, Eqs. S26-S27).
Neglects any back-action of the SYK system on the bath; the dissipation kernel is the bare thermal one. If strong coupling renormalizes the bath, sigma_+/- (SM Eqs. S24-S25) and hence every quoted exponent change.
domain assumption Time-translation invariance of the steady state: long-time limit tau -> infinity so correlations depend only on relative time (SM-II).
Relaxation exponents are extracted from the stationary spectral function rho_-(omega); the transient approach to the steady state is left for future work (Main Text, final paragraph). Non-TTI transient modes could dominate the actual relaxation.
ad hoc to paper Equilibrium FDR rho_+ = rho_- tanh(beta omega / 2) is imposed at every iteration for the finite-temperature results (End Matter, Figs. 6 and 7 captions).
The finite-beta robustness claim assumes the system equilibrates with the bath at the same temperature rather than demonstrating it; the main text's 'robust against finite beta' statement omits this condition.
domain assumption Asymptotic ansatz rho_-(omega) approx rho_-(0) - c |omega|^alpha (Main Text, around Eq. 13) used to define and 'predict' the cusp exponent alpha_eff = nu.
Inserting the assumed functional form into the Schwinger-Dyson equations returns alpha=nu; the 'prediction' is conditional on the assumed form being the leading singular behavior.
domain assumption Scale hierarchy Lambda >> J, mu and the strong-dissipation ordering J << Lambda << mu used for the analytic low-frequency results (Main Text, Eqs. 10-12).
Eqs. (10)-(12) drop the triple-convolution term; near omega=0 the bath density D(omega) -> 0, so the dropped J^2 term is not obviously negligible there. The paper asserts rather than demonstrates the hierarchy's validity at omega -> 0.
standard math Gaussian disorder average <e^{xJ}> = e^{<J^2> x^2 / 2} (SM, Eq. S9) and quadratic-bath cumulant expansion (SM, Eq. S6).
Standard identities used to reach the effective collective action; exact for Gaussian J and non-interacting baths; approximate at leading order in mu for other baths, as the SM itself notes.

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Cite this review

Pith. "Pith review of Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems." pith.science (2026). https://pith.science/paper/FRCMGA5K

@misc{pith2026260310815,
  author       = {Pith},
  title        = {Pith review of: Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRCMGA5K}},
  note         = {Machine review of arXiv:2603.10815}
}

abstract

In interacting quantum many-body systems, relaxation toward equilibrium reflects a competition between internal chaotic dynamics and environmental dissipation. While conventional Markovian baths typically produce exponential decay, non-Markovian dissipation can give rise to more intricate behavior, including algebraic relaxation. We study an open Sachdev-Ye-Kitaev (SYK) model coupled to a pseudogapped fermionic bath, using the Keldysh formalism to compute steady-state correlations in the large-$N$ limit. Our results uncover a rich dynamical phase diagram, with regimes of bath-driven power-law relaxation, chaos-driven exponential decay, and an intermediate pre-relaxation phase where exponential decay crosses over to algebraic decay. These findings demonstrate that non-Markovian environments can qualitatively reshape relaxation mechanisms in strongly correlated quantum many-body systems.

Figures

Figures reproduced from arXiv: 2603.10815 by the authors.

**Figure 2.** Solution of the Schwinger-Dyson equations for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Self-consistent solutions of the Schwinger-Dyson [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Characterization of the decay of G R(t) for β = 0 and Λ = 10, obtained by fitting the late time behavior of iGR(t) to Eq. (14). (a) Power-law exponent p, which agrees with the analytical prediction p = 1 + ν up to deviations due to the discretization of the frequency g…

**Figure 6.** Figure 6: Finite-temperature relaxation for ν = 0 (i.e., an environment with a constant density of states). (a) Gap ∆ as a function of µ and β. (b) ρ −(ω) behavior in the lowtemperature limit. As µ → 0, ρ −(ω) ∼ ω −1/2 up to ω ≈ 1/β. For this panel only, we impose the fluctuati…

**Figure 7.** Figure 7: Characterization of the low-frequency behavior of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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