Pith. sign in

REVIEW 3 major objections 4 minor 52 references

Mean-field theory is exact for long-range open quantum systems starting from strongly correlated states, provided the initial state is first decomposed into its clustering components and each component is evolved independently by the mean-f

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review

2026-08-01 02:27 UTC pith:KCLEKULV

load-bearing objection The core decomposition theorem for finite mixtures of clustering states is correct and gives the field a genuinely useful tool; the time-crystal continuum constructions and the power-law extension outrun the proof and should be flagged or closed. the 3 major comments →

arxiv 2607.25434 v1 pith:KCLEKULV submitted 2026-07-28 cond-mat.stat-mech quant-ph

Don't truncate, decompose: mean-field dynamics of long-range quantum systems from strongly correlated states

Federico Carollo This is my paper
classification cond-mat.stat-mech quant-ph
keywords mean-field theorylong-range quantum systemsstrongly correlated statescumulant expansionsopen quantum systemsmoment generating functionsymmetry breakingtime crystals
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper challenges the standard view that mean-field theory breaks down when a long-range open quantum system starts from a strongly correlated state, such as a macroscopic superposition of distinct phases. The proposed fix is not to add higher-order cumulants but to decompose the initial state into its clustering components—states whose correlations decay rapidly—evolve each component separately under the nonlinear mean-field equations, and recombine the results with the original weights. The central theorem states that, in the thermodynamic limit and for infinite-range interactions, this weighted recombination reproduces the entire moment generating function of the macroscopic observables, so all moments and cumulants are obtained exactly. A corollary shows that for equal-weight binary superpositions or mixtures, all third-order cumulants vanish, making second-order cumulant expansions exact—and therefore conceptually redundant. If the construction holds, strongly correlated many-body open systems can be simulated with the cost set by the number of components rather than the system size.

Core claim

On the paper's own terms, the discovery is a decomposition theorem. For a collective open quantum system with infinite-range interactions, any initial state that is a finite statistical mixture—or a macroscopic superposition whose components are asymptotically orthogonal—of clustering states is dynamically indistinguishable from the corresponding mixture. The time-evolved moment generating function of the macroscopic observables satisfies lim_{N→∞} Tr[ρ e^{tL*}(e^{-Σ_μ s_μ m^N_μ})] = Σ_k P_k e^{-Σ_μ s_μ x^{ρ_k}_μ(t)}, where x^{ρ_k}_μ(t) is the solution of the mean-field equations initialized from component ρ_k. This identity is the exact replacement for cumulant truncation: it delivers the f

What carries the argument

The load-bearing object is the decomposed mean-field Ansatz, expressed in the identity Eq. (6). The linearity of the quantum master equation makes the evolution of a mixture the mixture of evolutions; the thermodynamic-limit indistinguishability of a superposition from a mixture is supplied by the vanishing overlap of asymptotically orthogonal clustering components; and the previously established exactness of mean-field equations for each clustering component supplies the trajectory x^{ρ_k}_μ(t). Together these three facts convert a strongly correlated many-body state into a weighted ensemble of independent, low-dimensional classical trajectories. The corollary's mechanism is equally simple:

Load-bearing premise

The theorem is proven only for infinite-range interactions and for finite mixtures or superpositions of clustering components; the broader claim that the decomposition is exact for all strong long-range systems, and the pure-state time-crystal construction with a continuum of components, are asserted extensions without proof.

What would settle it

Take an equal-weight binary statistical mixture of two clustering states in the infinite-range open collective spin model with Hamiltonian ωS3 − (Ω/N)S1S1 and collective decay, and compute the third-order cumulant of the macroscopic component m1 at increasing N. The theorem predicts a limit of zero at all times; a nonzero N→∞ limit would falsify the decomposition. A second check targets the extension: in a one-dimensional power-law spin chain with interaction exponent 1, evolve a binary superposition of clustering states and compare the moment generating function with the weighted sum of indep

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Full statistics, not just averages: the moment generating function—and hence all cumulants—of macroscopic observables from strongly correlated states is computable from independent mean-field trajectories, so simulating such states costs the number of components, not N.
  • A benchmark for cumulant expansions: for equal-weight binary mixtures or superpositions, third-order cumulants vanish at all times, so second-order cumulant expansions are exactly correct for second-order cumulants and add no information beyond the decomposition.
  • Symmetry-preserving mean-field dynamics: symmetric strongly correlated initial states (e.g., an equal superposition of the two symmetry-broken branches) are captured exactly by the weighted independent evolutions, giving the bimodal non-Gaussian stationary state that a single mean-field trajectory misses.
  • Time-crystalline symmetry restoration: in the boundary time-crystal, averaging the initial condition over a full period of the limit cycle produces a stationary state that restores time-translation symmetry within the time-crystal phase.
  • Wider long-range scope: the paper argues the same decomposition applies to generic strong long-range systems with power-law interactions decaying as an inverse power of the distance with exponent smaller than or equal to the spatial dimension, extending the exactness beyond infinite-range models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This suggests a practical algorithm for arbitrary initial states: find the smallest finite decomposition into approximately clustering components and evolve each under mean field; the error should be controlled by the discarded overlaps and component correlations.
  • The exact vanishing of third-order cumulants for equal-weight binary states is a sharp, parameter-free test of any numerical method: in the thermodynamic limit, a nonzero third-order cumulant for such an initial condition signals a violation of the theorem's assumptions, not physical fluctuation.
  • The orbit-averaged stationary state in the time crystal points to a general symmetry-restoration mechanism—averaging over the symmetry orbit of mean-field initial conditions—that could be adapted to restore other broken symmetries (translation, rotation, gauge) in long-range open systems.
  • The unproven extension to power-law interactions is the natural place to test the boundary of the claim: if a binary superposition in a power-law model with exponent equal to dimension deviates from the weighted mean-field prediction in the thermodynamic limit, the exactness is confined to infinite-range interactions.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims that for long-range open quantum systems in the strong long-range regime, starting from strongly correlated states (macroscopic superpositions or mixtures) does not require going beyond mean-field theory. The main result, Theorem (Eq. 6), states that if the initial state is a finite mixture or asymptotically orthogonal superposition of clustering states, then the time-evolved moment generating function of the collective observables is, in the thermodynamic limit, the weighted sum of exponential mean-field predictions for each component. The proof in the Supplemental Material is based on prior exactness results for clustering states and a Cauchy-Schwarz / CP-Schwarz bound. From this theorem the author derives a corollary: for equal-weight binary mixtures or superpositions, third-order cumulants vanish, so a second-order cumulant expansion is exact for second-order cumulants. The paper applies the decomposition to the open LMG model (nonequilibrium Z2 symmetry breaking) and to the boundary time-crystal, where it constructs a time-translation-invariant state by averaging over a cycle of the mean-field dynamics, and also considers a pure-state analogue.

Significance. If the theorem and its advertised extensions hold, this is a significant conceptual and practical result: it replaces cumulant truncations, which lack controlled errors, with an exact decomposition into independent mean-field trajectories. The central theorem for finite sums of clustering states and infinite-range interactions is convincingly proved in the Supplemental Material, using established exactness results for clustering states; the corollary is a simple algebraic consequence. The numerical figures for N=16..512 support the claimed convergence. The main weaknesses are that several load-bearing claims go beyond the proved theorem: the continuum-mixture stationary state of Eq. (8), the pure-state superposition of Eq. (S18), and the power-law extension in the Discussion are asserted without proof. These gaps affect the breadth advertised in the title and abstract, though they do not invalidate the core theorem.

major comments (3)
  1. [Application 2, Eq. (8)] Equation (8) defines the time-crystal stationary state as a continuum mixture, ρ = (1/T)∫ dτ0 ρ_τ0, while the theorem in Eq. (6) is stated and proved only for finite sums ρ = Σ_k P_k ρ_k. The subsequent evaluation of lim_{N→∞} Tr[ρ F] uses the theorem as though it applied directly to the continuum mixture. No argument is given for exchanging the N→∞ limit with the continuum (Riemann-sum) limit. This can likely be justified because the expectation of a bounded function of m^N is continuous in the initial state uniformly in N, but the paper should supply that argument. As written, the exactness claim for the state in Eq. (8) is not proven.
  2. [Supplemental Eq. (S18), Section V] The pure-state construction |Ψ⟩ ∝ ∫ dτ0 |φ_τ0⟩ is outside the theorem's hypotheses. The proof of point (i) requires lim_{N→∞} ⟨φ_h|φ_k⟩ = 0 for distinct components; for nearby values of τ0 in a continuum this condition fails. A Riemann-sum discretization with spacing Δτ0 gives a finite superposition to which the theorem could apply, but no scaling of Δτ0 with N is provided. The Supplemental Material acknowledges the difficulty and reports numerical observation, but this is not a proof. The main text's statement that 'analogous results hold' for this pure state is therefore unsupported. Please either prove the limit under an explicit Δτ0(N) scaling or reclassify this as a numerical observation/conjecture.
  3. [Discussion (power-law extension)] The Discussion asserts that the decomposition extends to generic strong long-range systems with power-law interactions (exponent α ≤ d), where the time-evolved reduced density matrix for local observables coincides with the weighted sum of mean-field evolutions from clustering components. This is a different statement from the theorem, which is proved for infinite-range collective observables and relies on exactness results specific to that setting. No proof or precise reference is given for the power-law case. Since the title and abstract advertise 'long-range quantum systems' broadly, this is a load-bearing point. Either provide a proof or restrict the claims to infinite-range interactions and mark the power-law case as a conjecture.
minor comments (4)
  1. [Supplemental Section I (generic functions)] The proof that an analogous relation holds for regular functions of m^N exchanges an infinite Taylor series with the large-N limit. The exchange is not justified in detail; a dominated convergence or uniform convergence argument should be added, rather than simply stating that the series converges on the relevant domain.
  2. [Equation (S3) / notation] In the statement ρ = Σ_k P_k ρ_k, the subscript k is used both for the mixture components and for the individual clustering states in Eq. (S3). This becomes confusing when ρ_k may be pure or mixed; a clearer notation distinguishing the component index from the state label would help.
  3. [Figure 2 caption] In panel (b), the dotted line is described as the 'Gaussian prediction of the second-order cumulant expansion'. It would be clearer to state explicitly that this is the second-order truncated cumulant result, which effectively approximates the stationary measure as Gaussian, and that the full cumulant generating function is not reproduced.
  4. [Data availability] The data availability statement cites Zenodo (2026) without a DOI or identifier. If a permanent record is intended, a DOI or accession number should be provided.

Circularity Check

0 steps flagged

No significant circularity: Theorem Eq. (6) is a genuine algebraic extension of the cited clustering-state mean-field results, not a restatement of them.

full rationale

The paper's central claim, Eq. (6), states that for a finite mixture or superposition of clustering states, the large-N moment generating function is the weighted sum of independent mean-field solutions. The proof in Supplemental §I uses two ingredients: (a) the previously established mean-field convergence for each individual clustering component, cited from Refs. [32–34], and (b) linearity of the generator and boundedness of the finite sum, which allow the thermodynamic limit to be interchanged with the sum. This is not circular: the cited prior results concern single clustering states, while the new statement concerns mixtures and superpositions of such states. The mixture MGF is not assumed; it is derived by bounding each term |I_k^N| via Cauchy–Schwarz and invoking the cited component-level convergence. No parameter is fitted to the predicted MGF, no uniqueness theorem from the author's prior work is used to force the conclusion, and no ansatz is smuggled in via citation. The corollary about vanishing third-order cumulants for equal-weight binary states is an explicit algebraic calculation (Supplemental §II) from the derived moments, not an input. The applications to the open LMG model and boundary time-crystal are numerical demonstrations of the theorem, not fits. The continuum mixture in Eq. (8), the pure-state construction in Eq. (S18), and the extension to power-law interactions are asserted beyond the theorem's stated finite-sum/infinite-range assumptions, and the Supplemental Material itself acknowledges slower convergence for the pure state. These are limitations in scope or open correctness questions, but they do not make the derivation circular. Therefore no circular step is identified.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters: all constants are physical model inputs or numerical discretization choices. The main external input is the prior exactness theorem for clustering states, which is a mathematical result rather than an entity. No new particles, forces, or dimensions are postulated.

axioms (5)
  • domain assumption Mean-field exactness for clustering initial states: lim_N <φ_k|e^{tL*}[(m^N_ν - x^{φ_k}_ν(t))^2]|φ_k> = 0
    Imported from Refs [32-34]; used to control every diagonal and off-diagonal term in the proof of the Theorem (Supplemental Eqs. S4-S6).
  • domain assumption Component states |φ_k> are clustering and mutually asymptotically orthogonal: lim_N <φ_h|φ_k> = 0
    Assumed in the Theorem; needed to make off-diagonal coherences vanish in Eq. (S1).
  • domain assumption Initial state is a finite superposition or mixture of such components; the time-crystal pure state replaces the finite sum by a continuum integral over τ0
    The finite case is proven; the continuum case in Supplemental Eq. (S18) is used with only numerical evidence and slower convergence.
  • standard math Time-evolved generator e^{tL*} is a completely positive unital map, so the operator Schwarz inequality applies
    Used implicitly in the Cauchy-Schwarz bounds in the Supplement; standard but not stated.
  • standard math Taylor expansion of functions F of macroscopic observables converges on [-M,M]^{d^2}
    Supplemental Section I assumes this to extend the MGF statement to generic functions of macroscopic observables.

reviewed 2026-08-01 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Don't truncate, decompose: mean-field dynamics of long-range quantum systems from strongly correlated states." pith.science (2026). https://pith.science/paper/KCLEKULV

@misc{pith2026260725434,
  author       = {Pith},
  title        = {Pith review of: Don't truncate, decompose: mean-field dynamics of long-range quantum systems from strongly correlated states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCLEKULV}},
  note         = {Machine review of arXiv:2607.25434}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We challenge the widespread consensus that mean-field theory fails to describe long-range open quantum systems in the presence of symmetry breaking and/or when starting from strongly correlated states (e.g., macroscopic superpositions). While recent literature relies on cumulant expansions to capture such systems, this approach rests on truncations with no clear justification. Here, we show that it is, at best, conceptually redundant in the strong long-range regime. We show that the evolution can be decomposed into, and fully reconstructed from, independent mean-field dynamics. This decomposition generates the entire hierarchy of cumulants and, as a byproduct, identifies---to our knowledge, for the first time---a regime in which cumulant expansions exactly predict low-order cumulants. We illustrate the power of our findings with two applications: we compute the moment generating function for nonequilibrium $\mathcal{Z}_2$ symmetry breaking, and construct states restoring time-translation symmetry in time crystals. In both cases, our method fully reproduces the exact many-body dynamics, which is out of reach of cumulant expansions. Our results reclaim the exactness of mean-field theory, offering a transparent framework for large-scale open quantum systems.

Figures

Figures reproduced from arXiv: 2607.25434 by Federico Carollo.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 3 linked inside Pith

  1. [1]

    Defenu, T

    N. Defenu, T. Donner, T. Macr ` ı, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys.95, 035002 (2023)

    2023

  2. [2]

    Defenu, A

    N. Defenu, A. Lerose, and S. Pappalardi, Out-of- equilibrium dynamics of quantum many-body systems with long-range interactions, Phys. Rep.1074, 1 (2024)

    2024

  3. [3]

    Kastner, Diverging equilibration times in long-range quantum spin models, Phys

    M. Kastner, Diverging equilibration times in long-range quantum spin models, Phys. Rev. Lett.106, 130601 (2011)

    2011

  4. [4]

    Defenu, Metastability and discrete spectrum of long- range systems, Proc

    N. Defenu, Metastability and discrete spectrum of long- range systems, Proc. Natl. Acad. Sci. U.S.A.118, e2101785118 (2021)

    2021

  5. [5]

    Eisert, M

    J. Eisert, M. van den Worm, S. R. Manmana, and M. Kastner, Breakdown of quasilocality in long-range quantum lattice models, Phys. Rev. Lett.111, 260401 (2013)

    2013

  6. [6]

    Richerme, Z.-X

    P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions, Nature 511, 198 (2014)

    2014

  7. [7]

    Jurcevic, B

    P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body sys- tem, Nature511, 202 (2014)

    2014

  8. [8]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys.82, 2313 (2010)

    2010

  9. [9]

    Browaeys and T

    A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys.16, 132 (2020)

    2020

  10. [10]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys.93, 025001 (2021)

    2021

  11. [11]

    Steinert, P

    L.-M. Steinert, P. Osterholz, R. Eberhard, L. Festa, N. Lorenz, Z. Chen, A. Trautmann, and C. Gross, Spa- tially tunable spin interactions in neutral atom arrays, Phys. Rev. Lett.130, 243001 (2023)

    2023

  12. [12]

    C. Chen, G. Bornet, M. Bintz, G. Emperauger, L. Leclerc, V. S. Liu, P. Scholl, D. Barredo, J. Hauschild, S. Chatterjee, M. Schuler, A. M. L¨ auchli, M. P. Zale- tel, T. Lahaye, N. Y. Yao, and A. Browaeys, Continuous symmetry breaking in a two-dimensional Rydberg array, Nature616, 691 (2023)

    2023

  13. [13]

    Foss-Feig, G

    M. Foss-Feig, G. Pagano, A. C. Potter, and N. Y. Yao, Progress in trapped-ion quantum simulation, Annu. Rev. Condens. Matter Phys.16, 145 (2025)

    2025

  14. [14]

    Z. Bai, C. Chen, F. Yang, and W. Li, Many-Body Physics with Rydberg Atoms: Quantum Simulation and Non- equilibrium Dynamics (2026), arXiv:2607.11038 [quant- ph]

    Pith/arXiv arXiv 2026

  15. [15]

    V. D. Vaidya, Y. Guo, R. M. Kroeze, K. E. Ballan- tine, A. J. Koll´ ar, J. Keeling, and B. L. Lev, Tunable- Range, Photon-Mediated Atomic Interactions in Multi- mode Cavity QED, Phys. Rev. X8, 011002 (2018)

    2018

  16. [16]

    Mivehvar, F

    F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Cavity QED with quantum gases: new paradigms in many-body physics, Adv. Phys.70, 1 (2021)

    2021

  17. [17]

    Ferioli, A

    G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, A non-equilibrium superradiant phase tran- sition in free space, Nat. Phys.19, 1345 (2023)

    2023

  18. [18]

    Bonifacio, F

    M. Bonifacio, F. Piazza, and T. Donner, Laser-painted cavity-mediated interactions in a quantum gas, PRX Quantum5, 040332 (2024)

    2024

  19. [19]

    E. S. Cooper, P. Kunkel, A. Periwal, and M. Schleier- Smith, Graph states of atomic ensembles engineered by photon-mediated entanglement, Nat. Phys.20, 770 (2024)

    2024

  20. [20]

    E. A. Bohr, S. L. Kristensen, C. Hotter, S. A. Sch¨ affer, J. Robinson-Tait, J. W. Thomsen, T. Zelevin- sky, H. Ritsch, and J. H. M¨ uller, Collectively enhanced ramsey readout by cavity sub- to superradiant transition, Nat. Commun.15, 1084 (2024)

    2024

  21. [21]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The theory of open quantum systems(Oxford University Press on Demand, New York, 2002). 6

    2002

  22. [22]

    Gardiner and P

    C. Gardiner and P. Zoller,Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic meth- ods with applications to quantum optics(Springer Science & Business Media, New York, 2004)

    2004

  23. [23]

    Kirton and J

    P. Kirton and J. Keeling, Suppressing and Restoring the Dicke Superradiance Transition by Dephasing and Decay, Phys. Rev. Lett.118, 123602 (2017)

    2017

  24. [24]

    Iemini, A

    F. Iemini, A. Russomanno, J. Keeling, M. Schir` o, M. Dal- monte, and R. Fazio, Boundary time crystals, Phys. Rev. Lett.121, 035301 (2018)

    2018

  25. [25]

    Buˇ ca, J

    B. Buˇ ca, J. Tindall, and D. Jaksch, Non-stationary coher- ent quantum many-body dynamics through dissipation, Nat. Commun.10, 1730 (2019)

    2019

  26. [26]

    Russo and T

    F. Russo and T. Pohl, Quantum dissipative continuous time crystals, Phys. Rev. Lett.135, 110404 (2025)

    2025

  27. [27]

    Z. Wang, R. Gao, X. Wu, B. Buˇ ca, K. Mølmer, L. You, and F. Yang, Boundary time crystals induced by local dissipation and long-range interactions, Phys. Rev. Lett. 135, 230401 (2025)

    2025

  28. [28]

    Spohn, Kinetic equations from hamiltonian dynamics: Markovian limits, Rev

    H. Spohn, Kinetic equations from hamiltonian dynamics: Markovian limits, Rev. Mod. Phys.52, 569 (1980)

    1980

  29. [29]

    Mattes, I

    R. Mattes, I. Lesanovsky, and F. Carollo, Long-range in- teracting systems are locally noninteracting, Phys. Rev. Lett.134, 070402 (2025)

    2025

  30. [30]

    Winter, P

    L. Winter, P. Brighi, and A. Nunnenkamp, Restor- ing thermalization in long-range quantum magnets with staggered magnetic fields, Phys. Rev. Res.8, 023112 (2026)

    2026

  31. [31]

    Alicki and J

    R. Alicki and J. Messer, Nonlinear quantum dynamical semigroups for many-body open systems, J. Stat. Phys. 32, 299 (1983)

    1983

  32. [32]

    Carollo and I

    F. Carollo and I. Lesanovsky, Exactness of Mean-Field Equations for Open Dicke Models with an Application to Pattern Retrieval Dynamics, Phys. Rev. Lett.126, 230601 (2021)

    2021

  33. [33]

    Carollo and I

    F. Carollo and I. Lesanovsky, Applicability of Mean- Field Theory for Time-Dependent Open Quantum Sys- tems with Infinite-Range Interactions, Phys. Rev. Lett. 133, 150401 (2024)

    2024

  34. [34]

    Fiorelli, M

    E. Fiorelli, M. M¨ uller, I. Lesanovsky, and F. Carollo, Mean-field dynamics of open quantum systems with col- lective operator-valued rates: validity and application, New J. Phys.25, 083010 (2023)

    2023

  35. [35]

    Strocchi,Symmetry breaking(Springer, Berlin, 2020)

    F. Strocchi,Symmetry breaking(Springer, Berlin, 2020)

    2020

  36. [36]

    Mukherjee, Y

    A. Mukherjee, Y. Ibrahim, M. Hajduˇ sek, and S. Vin- janampathy, Symmetries and correlations in continuous time crystals, Phys. Rev. A110, 012220 (2024)

    2024

  37. [37]

    Plankensteiner, C

    D. Plankensteiner, C. Hotter, and H. Ritsch, Quantum- Cumulants.jl: A Julia framework for generalized mean- field equations in open quantum systems, Quantum6, 617 (2022)

    2022

  38. [38]

    Kerber, H

    J. Kerber, H. Ritsch, and L. Ostermann, The cumulants expansion approach: The good, the bad and the ugly (2025), arXiv:2511.20115 [quant-ph]

    arXiv 2025

  39. [39]

    Fowler-Wright, K

    P. Fowler-Wright, K. B. Arnard´ ottir, P. Kirton, B. W. Lovett, and J. Keeling, Determining the validity of cu- mulant expansions for central spin models, Phys. Rev. Res.5, 033148 (2023)

    2023

  40. [40]

    Lipkin, N

    H. Lipkin, N. Meshkov, and A. Glick, Validity of many- body approximation methods for a solvable model: (i). exact solutions and perturbation theory, Nucl. Phys.62, 188 (1965)

    1965

  41. [41]

    Morrison and A

    S. Morrison and A. S. Parkins, Dynamical Quantum Phase Transitions in the Dissipative Lipkin-Meshkov- Glick Model with Proposed Realization in Optical Cavity QED, Phys. Rev. Lett.100, 040403 (2008)

    2008

  42. [42]

    T. E. Lee, C.-K. Chan, and S. F. Yelin, Dissipative phase transitions: Independent versus collective decay and spin squeezing, Phys. Rev. A90, 052109 (2014)

    2014

  43. [43]

    Kothe, C

    S. Kothe, C. Oliver, and P. Kirton, PT-symmetry breaking phase transitions in an LMG dimer (2026), arXiv:2504.18426 [quant-ph]

    Pith/arXiv arXiv 2026

  44. [44]

    Carollo and I

    F. Carollo and I. Lesanovsky, Exact solution of a bound- ary time-crystal phase transition: Time-translation sym- metry breaking and non-Markovian dynamics of correla- tions, Phys. Rev. A105, L040202 (2022)

    2022

  45. [45]

    Thirring, On the mathematical structure of the B.C.S.-model

    W. Thirring, On the mathematical structure of the B.C.S.-model. II, Commun. Math. Phys.7, 181 (1968)

    1968

  46. [46]

    J. A. Sheikh, J. Dobaczewski, P. Ring, L. M. Robledo, and C. Yannouleas, Symmetry restoration in mean-field approaches, J. Phys. G: Nucl. Part. Phys.48, 123001 (2021)

    2021

  47. [47]

    M. Kac, G. E. Uhlenbeck, and P. C. Hemmer, On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model, J. Math. Phys. 4, 216 (1963)

    1963

  48. [48]

    Benatti, F

    F. Benatti, F. Carollo, R. Floreanini, and H. Narnhofer, Quantum spin chain dissipative mean-field dynamics, J. Phys. A: Math. Theor.51, 325001 (2018)

    2018

  49. [49]

    See Supplemental Material for details

  50. [50]

    Hosseinabadi, O

    H. Hosseinabadi, O. Chelpanova, and J. Marino, User- Friendly Truncated Wigner Approximation for Dissipa- tive Spin Dynamics, PRX Quantum6, 030344 (2025)

    2025

  51. [51]

    Noel and I

    V. Noel and I. Lesanovsky, Quantum to classical relax- ation dynamics of the dissipative Rydberg gas (2026), arXiv:2604.10538 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2026

  52. [52]

    Don’t truncate, decompose: mean- field dynamics of long-range quantum systems from strongly correlated states

    F. Carollo, Data for “Don’t truncate, decompose: mean- field dynamics of long-range quantum systems from strongly correlated states”, Zenodo (2026). 1 SUPPLEMENTAL MATERIAL Don’t truncate, decompose: mean-field dynamics of long-range quantum systems from strongly correlated states Federico Carollo1 1Dipartimento di Fisica, Sapienza Universit` a di Roma, P...

    2026

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.