Pith. sign in

REVIEW 2 major objections 5 minor 50 references

Chaotic dynamics in a single excitation subspace: deviations from the ETH via long time correlations

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Correlated quenches—one excitation on a system qubit, bath in a single number eigenstate—make local observables deviate from ETH predictions for fluctuations, time evolution, and scrambling.

desk verdict Correct core result in an extremal regime; the robustness claim is not supported by the paper's own Appendix D. read the letter →

arxiv 1908.11773 v2 pith:P66KDF3P submitted 2019-08-30 quant-ph cond-mat.stat-mech

Charlie Nation , Diego Porras This is my paper · ORCID
classification quant-phcond-mat.stat-mech
keywords eigenstatethermalizationhypothesisnon-ergodicsingleexcitationsubspacecorrelatedquenchsurvivalprobabilityout-of-time-orderedcorrelatorquantumscramblinglong-timefluctuations
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 shows that a closed, non-integrable quantum system can thermalize in a way that visibly violates the standard signatures of the eigenstate thermalization hypothesis (ETH), provided the dynamics stays inside a single-excitation sector and the bath starts in a non-degenerate eigenstate of a conserved number. Under this 'correlated quench', the energy-basis matrix elements of a local observable are not random: $O_{\mu\nu} = \Delta O\, c_\mu(\alpha_0)c_\nu(\alpha_0) + O_{\downarrow\downarrow}\delta_{\mu\nu}$, with off-diagonals fixed by the initial-state amplitudes. The paper derives three concrete consequences: equilibrium fluctuations scale as $\delta^2_O(\infty) \approx \Delta O^2\, \mathrm{IPR}(|\psi(0)\rangle)^2$ rather than the ETH's $1/D_{\rm eff}$ scaling; the time-dependent expectation value follows the survival probability $P_0(t)$; and the long-time out-of-time-ordered correlator stays finite, meaning local information does not scramble. These results matter because they give analytic, checkable predictions for when 'thermalization without ETH universality' appears—for example, in rotating-wave approximation settings—and they explain earlier numerical observations of initial-state-dependent scrambling.

What carries the argument

The load-bearing identity is the collapse of the overlap sum $\eta = \sum_k c_\mu(k,\uparrow)c_\nu(k,\uparrow)$ to the single term $c_\mu(\alpha_0)c_\nu(\alpha_0)$. This collapse is forced by the structure of the Hilbert space: with $[H,\hat N]=0$, $\hat N=\hat N_S+\hat N_B$ local, and the initial state $|\psi(0)\rangle=|\!\uparrow\rangle_S|k_0\rangle_B$ with $|k_0\rangle_B$ a non-degenerate eigenstate of $\hat N_B$, the only basis state in which the system qubit is excited has the bath in the fixed state $|k_0\rangle_B$. A local observable $O_S$ that commutes with $\hat N_S$ and $H_S$ is then identified with the survival probability up to a constant, so its matrix elements inherit the initial-state amplitude products. The inverse participation ratio, $\mathrm{IPR}(|\psi(0)\rangle)=\sum_\mu |c_\mu(\alpha_0)|^4$, enters as the natural measure of effective Hilbert-space dimension and controls both the fluctuation scaling and the size of finite-$N$ corrections to the OTOC.

What would settle it

A numerical exact-diagonalization experiment on the NN-XXX chain (Eqs. 25–26) can settle the claim: for initial states $|\!\uparrow\rangle_S|\downarrow\cdots\downarrow\rangle_B$ at increasing chain length $N$, compute the long-time variance $\delta^2_{\sigma_z^{(S)}}(\infty)$ and the IPR, and fit $\log\delta^2$ versus $\log \mathrm{IPR}$. The paper predicts an asymptotic slope of 2 ($\delta^2\propto \mathrm{IPR}^2$), while the ETH/RMT prediction is a slope of 1; observing slope 1 (or a crossover toward it at large $N$) would refute the correlated-quench scaling. The analogous test can be done for the long-time OTOC average $\overline{\langle W(t)VW(t)V\rangle}$: Eq. (29) predicts it saturates to the finite value of Eq. (30) for $W=V=\sigma_z^{(S)}$, whereas scrambling would give a vanishing average.

Watch

Extended reading notes

Core claim

The central claim is Eq. (22): for a correlated quench, observable matrix elements in the energy eigenbasis collapse to $O_{\mu\nu} = \Delta O\, c_\mu(\alpha_0)c_\nu(\alpha_0) + O_{\downarrow\downarrow}\delta_{\mu\nu}$, where $\alpha_0$ indexes the initial state $|\!\uparrow\rangle_S|k_0\rangle_B$. The off-diagonal part is therefore not a random matrix element but a product of the initial state's overlap amplitudes with the two eigenstates; this is what the paper calls an ETH-like relation with non-random off-diagonals. Substituting this form into the diagonal-ensemble formula gives $\delta^2_O(\infty) \approx \Delta O^2\,\mathrm{IPR}(|\psi(0)\rangle)^2$ (Eq. 23), the time dependence $\langle O(t)\rangle = \Delta O\,P_0(t)+O_{\downarrow\downarrow}$ (Eq. 28), and a long-time OTOC average $\overline{F(t)} = \overline{\langle W(t)VW(t)V\rangle}$ that does not vanish when $W=V=\sigma_z^{(S)}$ (Eqs. 29–30). The system still reaches the diagonal-ensemble expectation in the sense of Eq. (3), so it thermalizes, but it fails the ergodicity condition of Eq. (2) and does not scramble. Numerical exact-diagonalization calculations on a non-integrable NN-XXX spin chain and on a spin-boson model under the rotating-wave approximation match the analytic predictions.

Load-bearing premise

The argument depends on the bath starting in exactly one definite configuration with no extra excitations, so that when the system qubit is found excited the bath must be in that single configuration; if the bath starts with several possible configurations, the collapse that produces the non-random matrix elements fails.

Editorial extensions

If this is right

  • Under a correlated quench, equilibrium fluctuations of a local charge-diagonal observable scale as $\mathrm{IPR}^2$, an extra factor of the inverse participation ratio smaller than the ETH scaling $\propto \mathrm{IPR}$; in a Hilbert space of effective dimension $D_{\rm eff}$, fluctuations are suppressed as $1/D_{\rm eff}^2$ rather than $1/D_{\rm eff}$.
  • The time dependence of the local observable after the quench is exactly (up to a constant offset) the survival probability, $\langle O(t)\rangle = \Delta O\,P_0(t)+O_{\downarrow\downarrow}$; hence measuring local spin polarization gives a direct, experimentally accessible readout of $P_0(t)$ and, by Fourier transform, of the local density of states.
  • The long-time average of the out-of-time-ordered correlator does not vanish for charge-diagonal operators: for $W=V=\sigma_z^{(S)}$, $\overline{\langle W(t)VW(t)V\rangle}\to 1$ in the large-system limit, so local information about that observable is not scrambled even though the Hamiltonian is non-integrable.
  • The survival probability itself shows the non-ergodic $\mathrm{IPR}^2$ fluctuation scaling for both correlated and degenerate (e.g., Néel) initial bath states, whereas a local system observable shows it only when the bath state is non-degenerate; this difference is what makes scrambling initial-state dependent.
  • In the rotating-wave approximation, the spin-boson model realizes exactly the correlated-quench condition, so the analytic predictions for fluctuations, dynamics, and OTOC apply to that familiar open-system setting.

Reading between the lines

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

  • Editorial inference: The same mechanism should appear for any conserved quantity that links a local observable to a projection onto the initial state (particle number, magnetization, momentum sectors), so one could look for $\mathrm{IPR}^2$ fluctuation scaling in Hubbard or lattice-gauge models prepared with one particle on a vacuum-like bath.
  • Editorial inference: The OTOC result implies that in single-excitation sectors, late-time OTOC decay is not a reliable chaos diagnostic for charge-diagonal operators; scrambling tests should instead use operators with support transverse to the conserved charge or multi-excitation initial states to avoid false negatives.
  • Editorial inference: Appendix D suggests the non-ergodic scaling survives weak breaking of the conservation law at finite sizes but degrades as the system grows; a systematic finite-size crossover study could map how much symmetry breaking is tolerable before the ETH scaling is restored.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies quantum thermalization in the presence of a single conserved charge, focusing on initial states with exactly one excitation localized on a system qubit while the bath starts in a non-degenerate eigenstate of its local conserved quantity. Under these conditions, the authors derive a rank-one off-diagonal structure for system-local observable matrix elements, Eq. (22), which leads to three deviations from standard ETH expectations: equilibrium fluctuations that scale as IPR^2 (Eq. (23)), a time-dependent decay proportional to the survival probability (Eq. (28)), and a long-time OTOC that need not vanish (Eq. (29)). The analytical results are supported by exact diagonalization in a non-integrable spin chain and in the spin-boson model under the rotating-wave approximation. The paper also discusses the robustness of the effect when the conservation law is only approximate, and outlines a possible extension to larger system Hilbert spaces.

Significance. The central observation is valuable: in a single-excitation sector with an extremal bath state, the standard ETH ansatz for off-diagonal matrix elements is replaced by a deterministic rank-one form, and this yields concrete, testable predictions for fluctuations, correlation functions, and scrambling. The derivations of Eqs. (22), (23), and (28) are straightforward and internally consistent, and the numerical checks in Figs. 2, 3, and 4 and in the spin-boson appendix support the formulas. The identification of the survival probability as governing the observable's time evolution is a clean and useful result. However, the paper's broader claims about robustness and generality go beyond what the evidence establishes, as detailed in the major comments. The exact conservation-law mechanism is a special, although physically relevant, situation rather than a generic feature of non-integrable systems.

major comments (2)
  1. [Appendix D and Abstract] The abstract states that predictions are 'robust to perturbations that break this approximation,' but Appendix D explicitly says that the non-ergodic scaling 'gets worse as the system size increases' (Fig. 7a). This is a finite-size effect whose scaling direction is unfavorable: for a fixed perturbation, the deviation grows with system size and would destroy the effect in the thermodynamic limit. The robustness claim is therefore not supported by the numerical evidence, and the abstract and discussion should be tempered accordingly, or additional evidence should be provided that the deviation remains bounded for a given perturbation strength as N increases.
  2. [Section III.B and Appendix C] The central collapse of the sum in Eq. (20) to a single term, Eq. (21), relies critically on the initial bath state being a unique eigenstate of N_B, which occurs only in extremal charge sectors (e.g., vacuum or fully filled states). The paper's own Appendix C shows that the extension to larger system Hilbert spaces introduces a correction term, Eq. (C2), that can dominate when N_S is large, and the subsequent translational-invariance assumption, Eq. (C5), is not generally justified for a finite system coupled to a bath. The paper should state more precisely the domain of applicability of Eq. (22) and avoid implying that the mechanism is generic for arbitrary system sizes or initial states.
minor comments (5)
  1. [Figure 2 caption] The caption says 'all qubits up' for panel (a), but the text in Section VI describes the correlated initial state as '|↓,↓,...>_B (All down)'. This inconsistency should be corrected.
  2. [Appendix C] There is a typo: 'transnationally invariant' should be 'translationally invariant'.
  3. [Appendix B and Eq. (29)] The derivation of the long-time OTOC uses I_n := sum_mu c_mu^n without complex conjugation, which implicitly assumes real eigenvector coefficients. The paper should state this assumption, since the final expression Eq. (29) is not valid for complex coefficients without modification, although the numerical model has a real Hamiltonian.
  4. [Equation (27)] In the second line, the term O_downarrow downarrow sum_mu c_mu^2(k0,uparrow) should be written as O_downarrow downarrow sum_mu |c_mu|^2 = O_downarrow downarrow if complex coefficients are allowed; the current notation silently assumes real c_mu.
  5. [Section V, Eq. (24)] The penultimate line indicates that <O(t)O> equals O_uparrow uparrow O_downarrow downarrow, but the equality is only approximate after dropping IPR-weighted terms; using 'approximately equals' at that step would be clearer.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the central matrix-element relation is derived from the single-excitation sector, and the paper's self-citations are comparative baselines rather than load-bearing assumptions.

full rationale

The derivation chain is self-contained. Equation (18) is obtained exactly from [O_S, N_S]=0 and the completeness relation; equations (20)-(22) reduce eta = sum_k c_mu(k,up)c_nu(k,up) to c_mu(alpha0)c_nu(alpha0) because the single-excitation initial state leaves only k=k0. This is a stated assumption, not a fit, and equations (23), (28), and (29) are algebraic consequences of (22). The paper explicitly identifies the design feature: 'A following measurement of sigma(S)_z is thus the same as a measurement of the survival probability P0,' so the survival-probability scaling is not concealed as an input. The only self-citations are to the authors' earlier RMT work (Ref. [20] for the ETH form and the ergodic delta^2 proportional to IPR baseline, and Ref. [12] as an independent numerical context); these are used for comparison and motivation, not to force the new result. Appendix D does state that the non-conserving robustness case 'this gets worse as the systems size increases,' which is a limitation of the claimed robustness but is not a circularity in the conserving derivation. On balance, one minor self-citation baseline without load-bearing weight gives score 2.

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

The central claim rests on the stated physical conditions of the correlated quench: a conserved excitation number, a non-degenerate bath eigenstate, a single total excitation, and an observable diagonal in the local number basis. No free parameters are fitted to data; the model parameters in the numerics are chosen, not optimized. No new entities are postulated.

assumptions (6)
  • domain assumption The total Hamiltonian conserves the number of excitations, [H, N] = 0, with N = N_S + N_B, and N_S and N_B are separately conserved by H_S and H_B.
    Used throughout to restrict dynamics to the single-excitation manifold; stated in Section III.B.
  • domain assumption The initial bath state |k0⟩_B is a non-degenerate eigenstate of N_B, and the initial state has exactly one excitation localized to the system.
    This is the correlated-quench condition that makes the sum in Eq. (20) collapse to a single term, yielding Eq. (22).
  • domain assumption The local system observable O_S is diagonal in the basis of the local excitation number and commutes with H_S, so [O_S, N_S] = [O_S, H_S] = 0.
    Required for Eq. (18) to have no O↑↓ terms; stated in Section III.B.
  • domain assumption Eigenstates have a large number of principal components, so max_µ |cµ(α0)| ≪ 1 and Σ_µ |cµ|^8 ≪ (Σ_µ |cµ|^4)^2.
    Used to approximate Eqs. (12), (23), and (24) by dropping Σ|c|^8; standard for non-integrable systems with large effective dimension.
  • standard math No extensive degeneracies, so the time average of e^{-i(Eµ-Eν)t} is δµν.
    Used for the diagonal ensemble result Eq. (5) and for all long-time averages.
  • standard math Completeness of the single-excitation basis {|s⟩_S|k⟩_B} within the conserved sector.
    Used in Eq. (17) to relate the down-component sum to the identity minus the up-component sum.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Chaotic dynamics in a single excitation subspace: deviations from the ETH via long time correlations." pith.science (2026). https://pith.science/paper/P66KDF3P

@misc{pith2026190811773,
  author       = {Pith},
  title        = {Pith review of: Chaotic dynamics in a single excitation subspace: deviations from the ETH via long time correlations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P66KDF3P}},
  note         = {Machine review of arXiv:1908.11773}
}
read the original abstract

In this work we study a scenario where dynamics is restricted to a single excitation manifold, for particular physical observables with support in the manifold, which we label a `correlated quench'. We ask how such dynamics may in general differ from predictions of the eigenstate thermalization hypothesis (ETH). We show that if thermalization occurs, it will not fulfil other key predictions of the ETH; instead following differing generic behaviours. We show this by analysing long-time fluctuations, two-point correlation functions, and the out-of-time-ordered correlator; analytically detailing deviation from ETH predictions. We derive instead an ETH-like relation, with non-random off-diagonals, for matrix elements of observables, with correlations that alter long-time behaviour and constrain dynamics. Further, we analytically compute the time-dependence of the decay to equilibrium, showing that it is proportional to the survival probability of the initial state. We finally note that the conditions studied are common in many physical scenarios, such as under the rotating-wave approximation. We show numerically that predictions are robust to perturbations that break this approximation.

Figures

Figures reproduced from arXiv: 1908.11773 by the authors.

Figure 1
Figure 1. FIG. 1. Idealised scenario of a correlated quench where [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. DE Fluctuations versus IPR for Hamiltonian (25), (26). [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Dynamics of observable [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. DE Fluctuations versus IPR for Spin-Boson model [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. DE Fluctuations versus IPR for Hamiltonian Eqs. (D1), (D2), and (D3). [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Dynamics of observable [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 36 canonical work pages

  1. [1]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008)

    2008

  2. [2]

    Eisert, M

    J. Eisert, M. Friesdorf, and C. Gogolin, Nature Physics 11, 124 (2014)

    2014

  3. [3]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics 65, 239 (2016)

    2016

  4. [4]

    Gogolin and J

    C. Gogolin and J. Eisert, Reports on Progress in Physics 79, 056001 (2016)

    2016

  5. [5]

    Borgonovi, F

    F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky, Physics Reports 626, 1 (2016)

    work page 2016

  6. [6]

    Gring, M

    M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. Adu Smith, E. Dem- ler, and J. Schmiedmayer, Science 337, 1318 (2012)

    work page 2012

  7. [7]

    Schneider, D

    C. Schneider, D. Porras, and T. Schaetz, Rep. Prog. Phys. 75, 24401 (2012)

    work page 2012

  8. [8]

    I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153 (2014)

    2014

Show all 50 references
  1. [9]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science 349, 842 (2015)

    2015

  2. [10]

    Neill, P

    C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodru- betz, Z. Chen, A. Megrant, R. Barends, B. Campbell, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, J. Mutus, P. J. O’Malley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. Polkovnikov, and J. M. Martinis, Nat...

    2016

  3. [11]

    A. M. Kaufman, M. Eric Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 790 (2016)

    2016

  4. [12]

    G. Clos, D. Porras, U. Warring, and T. Schaetz, Phys. Rev. Lett. 117, 170401 (2016)

    2016

  5. [13]

    J. M. Deutsch, Physical Review A 43, 2046 (1991)

    1991

  6. [14]

    Srednicki, Journal of Physics A: Mathematical and General 29, 75 (1996), 9511001 [chao-dyn]

    M. Srednicki, Journal of Physics A: Mathematical and General 29, 75 (1996), 9511001 [chao-dyn]

    1996

  7. [15]

    H. Kim, T. N. Ikeda, and D. A. Huse, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 90 (2014)

    2014

  8. [16]

    Mondaini, K

    R. Mondaini, K. R. Fratus, M. Srednicki, and M. Rigol, Phys. Rev. E 93, 032104 (2016)

    2016

  9. [17]

    Mondaini and M

    R. Mondaini and M. Rigol, Phys. Rev. E 96, 012157 (2017)

    2017

  10. [18]

    Srednicki, J

    M. Srednicki, J. Phys. A: Math. Gen. 32, 1163 (1999)

    1999

  11. [19]

    Reimann, Physical Review Letters 101 (2008)

    P. Reimann, Physical Review Letters 101 (2008)

    2008

  12. [20]

    Nation and D

    C. Nation and D. Porras, New J. Phys.20, 103003 (2018)

    2018

  13. [21]

    Nation and D

    C. Nation and D. Porras, Physical Review E 99, 052139 (2019)

    2019

  14. [22]

    Dabelow and P

    L. Dabelow and P. Reimann, (2019), arXiv:1903.11881

    2019 arXiv

  15. [23]

    A. C. Cassidy, C. W. Clark, and M. Rigol, Physical Review Letters 106, 1 (2011), arXiv:1008.4794

    2011 arXiv

  16. [24]

    A. S. L. Malabarba, L. P. Garcia-Pintos, N. Linden, T. C. Farrelly, and A. J. Short, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 90 (2014)

    2014

  17. [25]

    L. P. Garc´ ıa-Pintos, N. Linden, A. S. Malabarba, A. J. Short, and A. Winter, Physical Review X 7 (2017)

    2017

  18. [26]

    Reimann, Nature Communications 7, 10821 (2016)

    P. Reimann, Nature Communications 7, 10821 (2016)

    2016

  19. [27]

    Borgonovi, F

    F. Borgonovi, F. M. Izrailev, and L. F. Santos, Phys. Rev. E 99, 010101 (2019)

    2019

  20. [28]

    ´A. M. Alhambra, J. Riddell, and L. P. Garc´ ıa-Pintos, (2019), arXiv:1906.11280

    2019 arXiv

  21. [29]

    Berkovitz, R

    J. Berkovitz, R. Frigg, and F. Kronz, Studies in History and Philosophy of Science Part B - Studies in History and Philosophy of Modern Physics 37, 661 (2006)

    2006

  22. [30]

    Peres, Physical Review A 30 (1984)

    A. Peres, Physical Review A 30 (1984)

    1984

  23. [31]

    Feingold, N

    M. Feingold, N. Moiseyev, and A. Peres, Physical Review A 30 (1984)

    1984

  24. [32]

    Zhang, H

    D. Zhang, H. T. Quan, and B. Wu, Physical Review E 94 (2016)

    2016

  25. [33]

    Mazur, Physica 43, 533 (1969)

    P. Mazur, Physica 43, 533 (1969)

    1969

  26. [34]

    Suzuki, Physica 51, 277 (1971)

    M. Suzuki, Physica 51, 277 (1971)

    1971

  27. [35]

    L. C. Venuti and L. Liu, (2019), arXiv:1904.02336

    2019 arXiv

  28. [36]

    Vidmar and M

    L. Vidmar and M. Rigol, Journal of Statistical Mechan- ics: Theory and Experiment 2016 (2016)

    2016

  29. [37]

    Nation and D

    C. Nation and D. Porras, (2019), arXiv:1906.06206

    2019 arXiv

  30. [38]

    Borgonovi, F

    F. Borgonovi, F. Mattiotti, and F. M. Izrailev, Physical Review E 95, 42135 (2017)

    2017

  31. [39]

    A. J. Short, New Journal of Physics 13, 053009 (2011)

    2011

  32. [40]

    Borgonovi and F

    F. Borgonovi and F. M. Izrailev, Physical Review E 99, 12115 (2019)

    2019

  33. [41]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, Journal of High Energy Physics 2016, 106 (2016)

    2016

  34. [42]

    Hashimoto, K

    K. Hashimoto, K. Murata, and R. Yoshii, Journal of High Energy Physics 2017 (2017)

    2017

  35. [43]

    B. Yan, L. Cincio, and W. H. Zurek, (2019), arXiv:1903.02651

    2019 arXiv

  36. [44]

    Swingle, Nature Physics 14, 988 (2018)

    B. Swingle, Nature Physics 14, 988 (2018)

    2018

  37. [45]

    Iyoda and T

    E. Iyoda and T. Sagawa, Physical Review A 97 (2018)

    2018

  38. [46]

    Huang, F

    Y. Huang, F. G. S. L. Brandao, and Y.-L. Zhang, (2017), arXiv:1705.07597

    2017 arXiv

  39. [47]

    Foini and J

    L. Foini and J. Kurchan, Physical Review E 99 (2019)

    2019

  40. [48]

    Schiulaz, E

    M. Schiulaz, E. J. Torres-Herrera, F. P´ erez-Bernal, and L. F. Santos, (2019), arXiv:1906.11856

    2019 arXiv

  41. [49]

    Weisskopf and E

    V. Weisskopf and E. Wigner, Zeitschrift f¨ ur Physik 63, 54 (1930)

    1930

  42. [50]

    Cohen-Tannoudji, J

    C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applica- tions, by Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Grynberg, pp. 678. ISBN 0-471-29336-9. Wiley- VCH , March 1998. -1 (1998), 10.1063/1.2809840

    1998 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.