Thermalization dynamics of finite-size quantum critical systems

Hao-Tian Sun; Li Li; Yan Liu

REVIEW 2 major objections 4 minor 42 references

Thermalization dynamics of finite-size quantum critical systems

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A holographic simulation shows finite-size quantum critical systems thermalize through three distinct regimes controlled by energy imbalance and scaled size, with wave reflection causing near-complete energy swapping.

desk verdict New finite-size holographic thermalization phenomena, but the no-matter brane constraint for the actual initial data is unproven and needs a direct numerical check. read the letter →

arxiv 2509.06523 v1 pith:CWRCAZEB submitted 2025-09-08 hep-th cond-mat.stat-mechcond-mat.str-elgr-qc

Li Li , Yan Liu , Hao-Tian Sun This is my paper

classification hep-thcond-mat.stat-mechcond-mat.str-elgr-qc MSC 81T4083C5783-08

keywords thermalizationholographicdualityquantumcriticalsystemsnon-equilibriumsteadystatesshockwavesquasi-normalmodesend-of-the-worldbranefinite-size

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 addresses how an isolated, finite-size strongly coupled conformal system approaches equilibrium when two halves at different temperatures are suddenly joined. Using holographic duality, it simulates the full gravitational dynamics in a box and finds that thermalization is not diffusive: it is a wave phenomenon controlled by the energy imbalance Δ and the rescaled size L̄. Depending on those two numbers, the system either cycles through repeated formation and collapse of a non-equilibrium steady state, sustains a long-lived confined shock wave with boundary reflections, or bypasses structure entirely and decays in synchronized oscillations. The paper claims this wave-dominated transport plus boundary reflection can nearly swap the energies of the two halves, and that the late-time decay is quantitatively the linear quasi-normal ringdown of the final equilibrium black hole with nonlinear corrections.

What carries the argument

The central object is a holographic black brane in an asymptotically anti-de Sitter spacetime with a tensionless end-of-the-world brane at each spatial end; the end-of-the-world brane—a bulk hypersurface anchored to the field-theory boundary—encodes the isolated-system condition of zero energy current at the boundary. Real-time evolution uses a characteristic numerical scheme in Eddington-Finkelstein coordinates, with the apparent horizon area supplying entropy production. Late-time decay is analyzed by Fourier-decomposing the energy current and comparing the modes to quasi-normal modes of the final equilibrium black brane, including forced-oscillation corrections when a dominant mode drives

What would settle it

Run the same thermalization with an initial profile that exactly satisfies the odd-derivative mirror-reflection conditions—for example, one constructed from even-parity spatial modes—against the tanh-smoothed step; if the recurrence peak at t=2L/c_s or the extracted quasi-normal frequencies shift measurably, the approximate boundary compatibility is contributing to the claimed dynamics.

Watch

Extended reading notes

Core claim

The paper claims that when two finite-size conformal field theories at different energy densities are joined at a perfectly transmitting interface, thermalization is controlled by two dimensionless parameters: the energy imbalance Δ=(E_R−E_L)/E_L and the scaled size L̄=L((E_R+E_L)/2)^{1/3}. In the holographic dual—a black brane confined between two tensionless end-of-the-world branes—the early contact region forms a non-equilibrium steady state with shock and rarefaction fronts moving at the sound speed. The finite boundaries reflect these fronts. For large L̄ and small Δ, the reflection produces near-mirror recurrences of the initial state at t=2L/c_s, with the NESS re-forming and collapsin

Load-bearing premise

The reliable long-time evolution assumes the boundary stays a perfect, matter-free wall, but that is proven only for specially symmetric starting states; the smoothed-step initial profile used here obeys that condition only approximately.

Editorial extensions

If this is right

For large systems with small energy imbalance, thermalization is not monotonic: the system cycles through repeated formation and collapse of a non-equilibrium steady state, with near-mirror revivals every sound-crossing time 2L/c_s.
For large energy imbalance, shock waves persist through many boundary reflections and dominate late-time energy transport, while rarefaction waves quickly homogenize.
For sufficiently small systems, dissipation dominates: no sustained NESS or shock structure forms, and the system enters rapid synchronized oscillatory decay.
The late-time decay is described by linear quasi-normal modes of the final equilibrium black brane, with higher modes entering a forced-oscillation regime where the fundamental mode acts as the driver.
Wave-propagated energy transfer plus boundary reflection allows near-complete energy swapping between subsystems after separation; total entropy still increases, so the second law is not violated.

Reading between the lines

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

A natural but untested extension: in two spatial dimensions the same phase diagram should persist with curved shock fronts, since the paper states that generalization to other dimensions is straightforward but does not demonstrate it.
If the energy-swap mechanism survives weak external coupling, timed separation of the two subsystems after roughly one sound-crossing time could serve as a cooling or energy-harvesting protocol; the paper mentions cooling applications without detailing an engineered cycle.
The recurrence time t=2L/c_s is a concrete, parameter-free prediction: ultracold-atom or nanowire realizations of conformal systems could look for oscillatory energy exchange at that period, which would support the holographic picture and whose absence would challenge it.
The forced-oscillation crossover suggests that mode amplitudes, not just frequencies, retain memory of the initial energy gap; varying Δ in small systems could expose this memory effect in the late-time ringdown.

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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

Finite-size effects in holographic thermalization are the real new thing here. The three regimes — recurrent NESS, confined shock, oscillatory decay — and the near-complete energy swap do not reduce to the known infinite-size NESS results, and the late-time match to QNMs of the final black brane is a genuine external benchmark with no fitted constants. The numerics look careful: two independent error estimators, energy conservation, entropy monotonicity. Credit where due: this is a serious computation.

The main soft spot is the brane boundary condition. The supplement itself states that the no-matter condition on the tensionless brane (S35) is not guaranteed for general initial data; only the mirror-reflection class (S37) is proven to satisfy it. The tanh-smoothed profile only approximates that class, with odd x-derivatives of s1 exponentially small but not zero. The code imposes the derived conditions (S38) but does not separately enforce or monitor S35. Over t ~ 200L a small residual boundary stress could accumulate and change the revival amplitude. The paper reports no check of S35. This is not a fatal objection by itself — it could well be that the residual is at truncation level — but it is a specific, answerable one. The fix is easy: compute the S35 residual from the numerical data or argue that the tanh profile is effectively in the basin. Until that's shown, the clean mapping to an isolated tensionless-brane BCFT is not established.

The phase diagram rests on few parameter points and the boundaries are schematic; that's acceptable in a letter, but it makes "three regimes" a set of examples rather than a mapped phase diagram. The Fig. 2 caption inconsistency (which run is recurrent NESS) is minor. The absence of released code or data is a negative for this kind of numerical claim.

Who this is for: people working on holographic thermalization, AdS/BCFT, or finite-size quantum critical transport. It deserves a serious referee; the referee should ask for the S35 residual check and a commitment to release data. I would not cite it in my own work until that's resolved.

Referee Report

2 major / 4 minor

Summary. This paper uses holographic duality (AdS/BCFT) to simulate the real-time thermalization of a finite-size strongly coupled CFT prepared with a spatially varying energy density. The setup is an asymptotically AdS_4 spacetime with two dynamical, tensionless end-of-the-world branes, dual to an isolated strip CFT. The initial state is a tanh-smoothed step in energy density (Eq. 4). By numerically solving the bulk Einstein equations, the authors identify three thermalization regimes controlled by two dimensionless parameters: the energy imbalance Δ and the scaled system size L̄. For large L̄ and small Δ, a recurrent NESS forms and collapses, with near-mirror revivals at t=2L/c_s. For large Δ, a long-lived confined shock wave persists, while for small L̄, the system undergoes oscillatory decay without persistent structures. Late-time decay is compared with linear quasi-normal modes of the final black brane, including nonlinear forced-oscillation corrections. The supplementary information contains the metric ansatz, boundary-condition derivation, numerical scheme, perturbation analysis, and error estimates.

Significance. If the results hold, they constitute a useful extension of holographic studies of NESS from infinite systems to finite, isolated systems, with falsifiable predictions: a three-regime phase diagram in (Δ, L̄), a revival time 2L/c_s, and late-time decay governed by black-brane QNMs. The numerical machinery is a notable strength: the paper reports two independent error estimators at the 10^-6–10^-4 level (Fig. S7–S8), energy conservation to 10^-9–10^-6 (Fig. S3), and entropy monotonicity (Fig. S2). The late-time comparison with linear QNMs appears to be an external, parameter-free benchmark, which is valuable. The main risk to the central claim is the consistency of the initial data with the tensionless-brane no-matter condition, as detailed below.

major comments (2)

[SI1, Eqs. (S35)–(S41); main text Sec. II, Eq. (4)] The no-matter brane condition \hat T_uu=0 is not guaranteed for the initial data actually used. The paper itself states that for generic initial conditions (S35) requires a full nonlinear proof, and the only proven class is the mirror-reflection class (S37). The tanh-smoothed step (4) with α=0.07 is not exactly in this class: solving s1 from the energy profile gives odd x-derivatives at x=±L that are exponentially small but nonzero (∼sech^2(1/α)). The evolution scheme imposes (S38) but does not monitor the residual of (S35). Since the recurrent-NESS claim relies on many boundary reflections over t≈200L, a small spurious boundary stress could accumulate and change revival amplitudes. Please report the residual of (S35) on Q during the runs, or repeat the key simulations with initial data constructed to lie exactly in the class (S37), and show that the three regimes persist. This is load-b
[Sec. II, Eq. (4) and Fig. 2] The phase diagram is obtained for a single interface sharpness α=0.07. The statement that this is 'sufficiently small' to not qualitatively alter the dynamics is not supported by a convergence study in α. Because α also controls the magnitude of the boundary odd derivatives of s1, this is intertwined with the consistency issue above. Please provide at least one representative trajectory (e.g., the recurrent-NESS case) for a smaller and a larger α (say 0.03 and 0.1), showing that the revival amplitude and the qualitative phase boundaries are stable.

minor comments (4)

[Sec. IV (Oscillatory decay)] The text says the energy current for oscillatory decay is shown in the 'bottom-right panel of Fig. 2', but the figure caption identifies the bottom-right panel as recurrent NESS and the top-left panel as oscillatory decay. Please correct this cross-reference.
[SI4] Typo: 'preform' should be 'perform'.
[SI1] Typo: 'impost' should be 'impose'.
[Eq. (5)] Eq. (5) introduces constants τ and ω without specifying their origin. If they are taken from the QNM spectrum of the final black brane, state this explicitly in the main text; if they are fits, clarify what is being fitted. This would make the predictive content of the late-time comparison precise.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the three-regime classification and late-time QNM match are independent of the initial data and prior work; the only flagged issue is an unproven brane constraint, which is a correctness gap, not a circular reduction.

full rationale

The paper's derivation chain is: (i) a holographic model with a tensionless EOW brane and the isolated-system condition J=0; (ii) an explicit initial energy profile (Eq. 4); (iii) numerical solution of the bulk Einstein equations; (iv) classification of the observed dynamics into three regimes; (v) a late-time comparison to QNMs of the final black brane. None of these steps uses the target conclusion as an input. The dimensionless parameters Δ and Lbar are fixed by the scaling symmetry (S44)-(S45) from the initial data, not fitted to the phase diagram; the phase diagram is a summary of simulation outputs, not a prediction derived from the inputs. The QNM analysis is a genuine external benchmark: the QNM frequencies are computed from linear perturbations around the final equilibrium black brane determined by conserved energy, and the extracted Fourier modes are compared with those frequencies (Figs. 5, S5, S6); at most amplitudes are matched, which is not circular. The only manuscript-acknowledged weakness is in SI1, where the authors state that the no-matter brane condition (S35) is 'not guaranteed' for generic initial conditions and that a full nonlinear proof is required; the tanh initial profile only asymptotically satisfies the mirror-reflection class (S37). This is a consistency/correctness limitation, not a circular reduction, because the simulation is not assuming the recurrent-NESS outcome. The single self-citation [29] for apparent-horizon entropy is not load-bearing, and independent references [28,30,31] are also cited. Thus the central claims remain self-contained against independent numerical and perturbative benchmarks, and the circularity score is low.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper contributes numerical solutions of known equations in a known framework; its honest inputs are the modeling choices above plus the chosen control parameters (Δ, L̄, α). The main load-bearing extras are the AdS/BCFT boundary-condition setup and the unproven consistency of the brane constraint for the chosen initial data. No new entities (particles, forces, dimensions) are postulated.

free parameters (2)

interface sharpness α = 0.07
Hand-chosen in Eq. (4) of the main text to approximate a step function. The paper asserts this choice does not qualitatively alter the dynamics, but no convergence scan over α is shown.
late-time decay amplitude parameters τ, ω in Eq. (5) = not quoted explicitly; matched to QNM frequencies in Fig. 5, S4-S6
The late-time form E ∝ e^{-t/τ} cos(ωt) sin(πx/2L) in Eq. (5) is read off from the numerical evolution. The subsequent comparison with independently computed quasinormal modes is a validation, but τ and ω themselves are quantities extracted from the simulated data rather than predicted before the run.

assumptions (5)

domain assumption AdS/CFT duality maps the strongly coupled 2+1d CFT to classical gravity in asymptotically AdS4 (action Eq. 1).
Invoked in the Introduction and used throughout as the computational engine.
domain assumption AdS/BCFT with a tensionless end-of-the-world brane and Neumann condition T̂_ab=0 (Eq. 2 of main text, Eq. S22) correctly encodes an isolated finite-size CFT with J=0 at the boundary (Eq. 3).
Central modeling assumption from Takayanagi (ref. 25) and Fujita et al. (ref. 23); the paper builds on this rather than deriving it.
domain assumption The apparent horizon, not the event horizon, gives the correct non-equilibrium entropy used for second-law checks (S = area/4G_N).
Taken from refs. 28-31, including the self-cited ref. 29; standard in the holographic far-from-equilibrium literature but not derived here.
ad hoc to paper Initial data belongs to the mirror-reflection class (Eq. S37) for which the brane no-matter condition (Eq. S35) provably persists in time.
The authors state in SI1 that no general guarantee exists that (S35) is satisfied, and that a full nonlinear analysis is required. The simulated tanh-step profile satisfies the odd-derivative conditions only approximately.
ad hoc to paper α = 0.07 approximates a step function without qualitatively changing the dynamics.
Asserted in the paragraph after Eq. (4); no α-scan is shown.

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Cite this review

Pith. "Pith review of Thermalization dynamics of finite-size quantum critical systems." pith.science (2026). https://pith.science/paper/CWRCAZEB

@misc{pith2026250906523,
  author       = {Pith},
  title        = {Pith review of: Thermalization dynamics of finite-size quantum critical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWRCAZEB}},
  note         = {Machine review of arXiv:2509.06523}
}

read the original abstract

Using holographic duality, we investigate thermalization process when two finite-size quantum critical systems are brought into thermal contact along a perfectly transmitting interface. Through real-time simulations of gravitational dynamics, which are spatially inhomogeneous and anisotropic and are confined within two dynamical bulk branes, we identify three distinct thermalization patterns governed by the energy imbalance (temperature difference) and system size. For systems with large size and small energy imbalance, we observe recurrent cycles of formation and collapse of non-equilibrium steady states (NESS). Under large energy imbalance, shock waves persist for a prolonged period with sustained boundary reflections, while rarefaction waves rapidly homogenize. When the system size is sufficiently small, dissipation dominates and leads to oscillatory decay without sustained NESS or shock structure. In sharp contrast to diffusive systems, we uncover that wave-propagated energy transfer together with boundary reflections enables nearly complete energy swapping between subsystems during thermalization. Our results reveal rich thermalization dynamics in finite-size quantum critical systems across spatial scales and energy gradient regimes.

Figures

Figures reproduced from arXiv: 2509.06523 by the authors.

**Figure 2.** Numerical solutions for the finite size quantum crit [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Far-from-equilibrium dynamics for ∆ = 0.02, L¯ = 100. The system change from the recurrent NESS stage to a phase of oscillatory decay at late times. Top panel: Time evolution of the energy density at two spatial points x/L = −1 (blue) and x/L = −0.5 (red). Bottom panel: Time evolution of the time derivative of the total entropy. While the total energy of the system is conserved, there is entropy production in the sy… view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Temporal evolution of the FM amplitudes of energy [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 4.** Figure 4: Profiles of the energy current J at different times for ∆ = 3, L¯ = 50. The shock wave reflects repeatedly between the boundaries. Note that J = 0 at both boundaries x = ±L, consistent with an isolated system with no energy enters or exits through the boundaries. Oscil…

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Works this paper leans on

42 extracted references · 38 canonical work pages

[1]

J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the Many-Body Localization Transition in Two Dimensions, Science352, 1547 (2016)

work page 2016
[2]

L. V. Delacr´ etaz, A Bound on Thermalization from Dif- fusive Fluctuations, Nature Physics , 1 (2025)

work page 2025
[3]

A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum Thermalization through Entanglement in an Isolated Many-Body System, Science353, 794 (2016)

work page 2016
[4]

Kinoshita, T

T. Kinoshita, T. Wenger, and D. S. Weiss, A Quantum Newton’s Cradle, Nature440, 900 (2006)

work page 2006
[5]

Langen, S

T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schwei- gler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasen- zer, and J. Schmiedmayer, Experimental Observation of a Generalized Gibbs Ensemble, Science348, 207 (2015)

work page 2015
[6]

Y. Le, Y. Zhang, S. Gopalakrishnan, M. Rigol, and D. S. Weiss, Observation of Hydrodynamization and Lo- cal Prethermalization in 1D Bose Gases, Nature618, 494 (2023)

work page 2023
[7]

Neillet al., Ergodic Dynamics and Thermalization in an Isolated Quantum System, Nature Phys.12, 1037 (2016)

C. Neillet al., Ergodic Dynamics and Thermalization in an Isolated Quantum System, Nature Phys.12, 1037 (2016)

work page 2016
[8]

J. J. Pulikkottil, A. Lakshminarayan, S. C. L. Srivas- tava, M. F. I. Kieler, A. B¨ acker, and S. Tomsovic, Quan- tum Coherence Controls the Nature of Equilibration and Thermalization in Coupled Chaotic Systems, Phys. Rev. E107, 024124 (2023)

work page 2023

Show all 42 references

[9]

Rigol, Breakdown of Thermalization in Finite One- Dimensional Systems, Physical Review Letters103, 100403 (2009)

M. Rigol, Breakdown of Thermalization in Finite One- Dimensional Systems, Physical Review Letters103, 100403 (2009)

2009
[10]

Shiraishi and H

N. Shiraishi and H. Tasaki, Nature Abhors a Vacuum: A Simple Rigorous Example of Thermalization in an Iso- lated Macroscopic Quantum System, J. Statist. Phys. 191, 82 (2024)

2024
[11]

Gogolin and J

C. Gogolin and J. Eisert, Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems, Rept. Prog. Phys.79, 056001 (2016)

2016
[12]

Karrasch, R

C. Karrasch, R. Ilan, and J. E. Moore, Nonequilibrium Thermal Transport and Its Relation to Linear Response, Physical Review B88, 195129 (2013)

2013
[13]

Bernard and B

D. Bernard and B. Doyon, Energy Flow in Non- Equilibrium Conformal Field Theory, Journal of Physics A: Mathematical and Theoretical45, 362001 (2012)

2012
[14]

M. J. Bhaseen, B. Doyon, A. Lucas, and K. Schalm, En- ergy Flow in Quantum Critical Systems Far from Equi- librium, Nature Physics11, 509 (2015)

2015
[15]

Lucas, K

A. Lucas, K. Schalm, B. Doyon, and M. J. Bhaseen, Shock Waves, Rarefaction Waves, and Nonequilibrium Steady States in Quantum Critical Systems, Physical Re- view D94, 025004 (2016)

2016
[16]

Chang, A

H.-C. Chang, A. Karch, and A. Yarom, An Ansatz for One Dimensional Steady State Configurations, J. Stat. Mech.1406, P06018 (2014)

2014
[17]

Amado and A

I. Amado and A. Yarom, Black Brane Steady States, JHEP10, 015 (2015)

2015
[18]

Ecker, J

C. Ecker, J. Erdmenger, and W. van der Schee, Non- Equilibrium Steady State Formation in 3+1 Dimensions, SciPost Physics11, 047 (2021), arXiv:2103.10435 [hep- th]

2021 arXiv
[19]

P. M. Chesler and L. G. Yaffe, Numerical Solution of Gravitational Dynamics in Asymptotically Anti-de Sitter Spacetimes, JHEP7, 86
[20]

V. E. Hubeny and M. Rangamani, A Holographic View on Physics out of Equilibrium, Advances in High Energy Physics2010, 1 (2010), arXiv:1006.3675 [gr-qc]

2010 arXiv
[21]

Liu and J

H. Liu and J. Sonner, Holographic Systems Far from Equilibrium: A Review (2018), arXiv:1810.02367 [hep- th]

2018 arXiv
[22]

Cardoso, L

V. Cardoso, L. Gualtieri, C. Herdeiro, U. Sperhake,et al., NR/HEP: Roadmap for the Future, Classical and Quan- tum Gravity29, 244001 (2012), arXiv:1201.5118 [astro- ph]

2012 arXiv
[23]

Fujita, T

M. Fujita, T. Takayanagi, and E. Tonni, Aspects of AdS/BCFT, JHEP11, 043 (2011)

2011
[24]

Nozaki, T

M. Nozaki, T. Takayanagi, and T. Ugajin, Central Charges for BCFTs and Holography, JHEP06, 066 (2012)

2012
[25]

Takayanagi, Holographic Dual of BCFT, Phys

T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett.107, 101602 (2011)

2011
[26]

Andreiet al., Boundary and Defect CFT: Open Prob- lems and Applications, J

N. Andreiet al., Boundary and Defect CFT: Open Prob- lems and Applications, J. Phys. A53, 453002 (2020), arXiv:1810.05697 [hep-th]. 6

2020 arXiv
[27]

Izumi, T

K. Izumi, T. Shiromizu, K. Suzuki, T. Takayanagi, and N. Tanahashi, Brane Dynamics of Holographic BCFTs, JHEP10, 050 (2022)

2022
[28]

Engelhardt and A

N. Engelhardt and A. C. Wall, Decoding the Apparent Horizon: Coarse-Grained Holographic Entropy, Physical Review Letters121, 211301 (2018)

2018
[29]

Baggioli, L

M. Baggioli, L. Li, and H.-T. Sun, Shear Flows in Far- from-Equilibrium Strongly Coupled Fluids, Phys. Rev. Lett.129, 011602 (2022), arXiv:2112.14855 [hep-th]

2022 arXiv
[30]

Hollands, R

S. Hollands, R. M. Wald, and V. G. Zhang, Entropy of Dynamical Black Holes, Physical Review D110, 024070 (2024)

2024
[31]

Rougemont, W

R. Rougemont, W. Barreto, and J. Noronha, Hydrody- namization Times of a Holographic Fluid Far from Equi- librium, Physical Review D105, 046009 (2022)

2022
[32]

Baibhav, M

V. Baibhav, M. H.-Y. Cheung, E. Berti, V. Cardoso, G. Carullo, R. Cotesta, W. Del Pozzo, and F. Duque, Agnostic Black Hole Spectroscopy: Quasinormal Mode Content of Numerical Relativity Waveforms and Limits of Validity of Linear Perturbation Theory, Physical Re- view D108, 104...

2023
[33]

M. H.-Y. Cheunget al., Nonlinear Effects in Black Hole Ringdown, Phys. Rev. Lett.130, 081401 (2023)

2023
[34]

M. H.-Y. Cheung, E. Berti, V. Baibhav, and R. Cotesta, Extracting Linear and Nonlinear Quasinormal Modes from Black Hole Merger Simulations, Phys. Rev. D109, 044069 (2024). 7 SUPPLEMENT AR Y INFORMA TION SI1. The holographic setup We consider a four-dimensional holographic set...

2024
[36]

GivenBands 1, we use (S3) to solve Σ
[37]

GivenB,s 1,Jand Σ, we use (S4) to solveF
[38]

GivenB,s 1, Σ andF, we can use (S5) to solved +S. The boundary condition at the apparent horizon requires vanishing expansion, d+S u=uh = e−B FΣ∂ xB−Σ∂ xF+F 2u2Σ′ 2Σ2 .(S42) By fixing this horizon condition, the energy density can be extracted from the asymptotic solution ofd ...
[39]

GivenB,s 1, Σ,Fandd +S, we determined +Busing (S6)
[40]

The boundary condition at the apparent horizon is obtained by solving (S9) with the boundary condition (S38)

With all previous quantities known,i.e.B,s 1, Σ,F,d +Sandd +B, we solve forAusing (S7). The boundary condition at the apparent horizon is obtained by solving (S9) with the boundary condition (S38). 12
[41]

We then get ˙Busing the definition ofd + andd +B, and get ˙s1 from the asymptotically solution ofA, and get ˙Jfrom energy-momentum tensor conservation
[42]

We initialize our system as a locally boosted black brane

By employing fourth-order Runge-Kutta method for the first three time steps and then the fourth-order Adams- Bashforth method for subsequent steps, we computeB(u, t+δt, x),J(u, t+δt, x) ands1(u, t+δt, x), after which we repeat the entire procedure from step 1. We initialize ou...

2000
[100]

confined shock wave

The system change from the recurrent NESS stage to a phase of oscillatory decay at late times.Top panel:Time evolution of the energy density at two spatial pointsx/L=−1 (blue) andx/L=−0.5 (red).Bottom panel:Time evolution of the time derivative of the total entropy. While the ...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the Many-Body Localization Transition in Two Dimensions, Science352, 1547 (2016)

work page 2016

[2] [2]

L. V. Delacr´ etaz, A Bound on Thermalization from Dif- fusive Fluctuations, Nature Physics , 1 (2025)

work page 2025

[3] [3]

A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum Thermalization through Entanglement in an Isolated Many-Body System, Science353, 794 (2016)

work page 2016

[4] [4]

Kinoshita, T

T. Kinoshita, T. Wenger, and D. S. Weiss, A Quantum Newton’s Cradle, Nature440, 900 (2006)

work page 2006

[5] [5]

Langen, S

T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schwei- gler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasen- zer, and J. Schmiedmayer, Experimental Observation of a Generalized Gibbs Ensemble, Science348, 207 (2015)

work page 2015

[6] [6]

Y. Le, Y. Zhang, S. Gopalakrishnan, M. Rigol, and D. S. Weiss, Observation of Hydrodynamization and Lo- cal Prethermalization in 1D Bose Gases, Nature618, 494 (2023)

work page 2023

[7] [7]

Neillet al., Ergodic Dynamics and Thermalization in an Isolated Quantum System, Nature Phys.12, 1037 (2016)

C. Neillet al., Ergodic Dynamics and Thermalization in an Isolated Quantum System, Nature Phys.12, 1037 (2016)

work page 2016

[8] [8]

J. J. Pulikkottil, A. Lakshminarayan, S. C. L. Srivas- tava, M. F. I. Kieler, A. B¨ acker, and S. Tomsovic, Quan- tum Coherence Controls the Nature of Equilibration and Thermalization in Coupled Chaotic Systems, Phys. Rev. E107, 024124 (2023)

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