Lindbladian PT phase transitions

Tomohiro Sasamoto; Yuma Nakanishi

REVIEW 2 major objections 5 minor 1 cited by

Lindbladian PT phase transitions

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For dissipative many-body systems whose Lindbladian is parity-time symmetric, the mean-field equations inherit a nonlinear PT symmetry, and that symmetry alone forces permanent oscillations on one side of the transition and a stable/unstabl

desk verdict Honest consolidation of the authors' own L-PT program, with a clean analytic core but two verifiability gaps (Theorem 2's deferred proof, and the linear-center oscillation claim) that should be tightened before publication. read the letter →

arxiv 2512.24981 v2 pith:R2NW4WB2 submitted 2025-12-31 quant-ph

Yuma Nakanishi , Tomohiro Sasamoto This is my paper · ORCID

classification quant-ph

keywords LindbladianPTsymmetrydissipativephasetransitioncriticalexceptionalpointcontinuous-timecrystalcollectivespinsystemsnon-reciprocaldrivenDickemodelmean-fieldtheory

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

Markovian open quantum systems are usually expected to relax to a time-independent steady state, but the authors show that when the Lindbladian has a parity-time (L-PT) symmetry and the dissipation is collective, the thermodynamic-limit dynamics is forced to be periodic: the mean-field equations inherit a nonlinear PT symmetry, PT-symmetric fixed points are centers surrounded by closed orbits, and PT-broken fixed points appear as stable/unstable pairs. The transition between these phases occurs at a critical exceptional point where collective excitation modes coalesce, with critical exponents ν_t = x_Z = q/2. This makes L-PT symmetry a single microscopic origin for dissipative continuous-time-crystal behaviour and connects it to non-reciprocal phase transitions; the driven Dicke model realizes the scenario at κ=g, where a zero-mode Lindbladian exceptional point appears. Beyond mean field, the time-independent steady state becomes PT-symmetric in the PT phase and PT-broken in the ordered phase in the thermodynamic limit, with purity dropping and spin squeezing and entanglement growing near criticality.

What carries the argument

The carrier of the argument is the nonlinear PT (n-PT) symmetry of the mean-field vector field, f=i g satisfying P̃T̃ f(q)=f(P̃T̃ q) with P̃=diag(1,1,-1) (Eq. 2.8), and its bridge from the microscopic scale: the lemma's vanishing commutator-in-expectation, ⟨[iL†, P̂T̂]m_α⟩→0 (Eq. 5.8), which holds because the scaling H=S h, L_μ=√S l_μ (Eq. 5.7) makes each double commutator of normalized spin operators an O(1/S) contribution. The n-PT symmetry then forces the off-diagonal Jacobian form (5.23) at PT-symmetric fixed points — so spectra are λ=±√(αβ), centers occur when αβ<0 with period 2π/√(-αβ), and αβ=0 is a CEP — and the stable/unstable pair structure (5.26)–(5.28) at PT-broken fixed points.

What would settle it

Compute the mean-field flow of an L-PT-symmetric driven-Dicke-type model with local dissipators Σ_i D[σ^-_i] instead of collective decay κ/S D[S^-]. The paper predicts no PT-protected center oscillations in the thermodynamic limit for any parameter choice; observation of persistent periodic orbits in that setting would falsify the double-commutator scaling argument behind Theorem 1.

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Extended reading notes

Core claim

In the paper's own terms: an L-PT-symmetric Lindbladian is one that is invariant when every Hamiltonian and jump operator is replaced by its PT transform, PT(O)=PT O†(PT)^{-1}; this symmetry cannot be written as a commutation relation with a superoperator, so it does not block-diagonalize the Lindbladian. The paper proves (Lemma and Theorems 1 and 2) that for single-collective-spin models (and U(1)-conserving bipartite bosonic systems) satisfying the scaling H=S h, L_μ=√S l_μ, the microscopic symmetry is transferred to the mean-field flow in the thermodynamic limit: ⟨[iL†, P̂T̂]m_α⟩ → 0, so the flow obeys the n-PT symmetry (2.8) with P̃=diag(1,1,-1). The forced structure: PT-symmetric fixed

Load-bearing premise

The scaling H=S h and L_μ=√S l_μ, i.e., that the dissipation is collective or long-range, is the load-bearing premise: for local dissipation (η>1) the double-commutator contributions stay O(1), so the n-PT symmetry of the mean-field flow is destroyed even though the microscopic L-PT symmetry remains.

Editorial extensions

If this is right

For any single-collective-spin model satisfying L-PT symmetry and the scaling H=S h, L_μ=√S l_μ, the PT-symmetric mean-field fixed point is a center when αβ<0, so persistent periodic oscillations are generic rather than fine-tuned (Sec. 5.3.1).
At the L-PT transition the collective excitation spectrum closes as a critical exceptional point; in the driven Dicke model this appears as a zero-mode Lindbladian exceptional point at κ=g with finite-size scalings |κ-κ_c|~S^{-0.62} and Re[λ_EP]~S^{-0.35} (Sec. 5.5).
The mean-field results carry over to U(1)-conserving bipartite bosonic systems and to long-range dissipation with η≤1 (Sec. 5.7); local dissipation (η>1) destroys the n-PT symmetry of the mean-field flow and with it the protected oscillations.
L-PT-induced oscillations satisfy the persistence and dissipation-induced-origin criteria for dissipative continuous-time crystals, but only the weak versions of robustness: they are destroyed by a small longitudinal field or by local dissipation (Sec. 6.4).
Beyond mean field, the time-independent steady state becomes PT-symmetric in the PT phase and PT-broken in the ordered phase in the thermodynamic limit, and purity drops toward zero while spin squeezing and quantum Fisher information are enhanced near criticality (Sec. 7).

Reading between the lines

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

Editorial inference: because the n-PT structure depends only on the L-PT symmetry and the collective scaling, the mechanism should transfer to other platforms with engineered long-range dissipation — cavity-mediated spin ensembles, Rydberg arrays, or trapped ions — where the critical exceptional point could be detected as a divergent relaxation time and a gap closing with a definite power law.
Editorial inference: the stable/unstable pair structure means the PT-broken phase has no steady-state degeneracy, so the eventual state is selected by the basin of attraction rather than by symmetry breaking; a testable consequence is hysteresis when sweeping the dissipation rate across the transition, which would distinguish L-PT transitions from conventional Z_2-breaking transitions.
Editorial inference: the oscillating frequency's dependence on dissipative rates, T=2π/√(-αβ), suggests dissipation can tune the time-crystal frequency, pointing toward AC-sensing applications of L-PT systems — a direction the review mentions only for the driven Dicke model and does not derive from the n-PT structure.
Editorial inference: the review leaves open a spectral theory that derives purely imaginary Lindbladian eigenvalues directly from L-PT symmetry; a natural next step is to use the n-PT Jacobian structure to predict the number of PIE modes in the PT phase, extending the mean-field result to the full Lindbladian spectrum.

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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

Short version: this is a readable, honest consolidation of the L-PT program the authors have been building in their own preprints. The core analytic machinery — the lemma that the PT-commutator vanishes in the mean-field limit under the √S Lindblad scaling, and the consequent n-PT symmetry of the mean-field dynamics — is clean, and the specific model calculations (DDM, LMG, two-spin) check out. But the paper is not a self-contained proof of the broad claims: Theorem 2's proof lives in an unpublished arXiv preprint, and the 'generic' persistent-oscillation claim rests on linear centers without nonlinear control. If you treat it as a review with some new fragments, it's a useful contribution.

What's actually new: the critical-exponent equality ν_t = x_Z = q/2, the η = 1 threshold for long-range dissipation that breaks n-PT symmetry, the DCTC taxonomy, and extra numerics. The rest is re-derivation from Refs. [59–63]. The authors are upfront about this; they call it a review.

Soft spots. The stress-test note is correct: Sec. 5.7.2 says the detailed proof of Theorem 2 is in Ref. [61], a same-group arXiv preprint. The spatially extended bosonic leg is therefore not verifiable from this document alone. That should be fixed by either including the proof, citing a published version, or softening the claim. Second, the claim that PT-symmetric fixed points 'generically' support persistent oscillations is only established at the level of linear stability; for the general class there is no nonlinear argument that the center persists as a family of closed orbits. The specific models with exact closed orbits are fine, but the general statement is over-broad. Third, the quantum properties (purity, squeezing, negativity) are computed for specific models, and the authors themselves note these are not derived from L-PT symmetry — so the generalization is weak, but honestly flagged. The √S scaling is a real restriction, but it's stated and its failure mode is analyzed, so I don't count it as a hidden flaw.

Who it's for: people working on dissipative phase transitions, PT symmetry, and time crystals who want a guided tour of the L-PT program. It deserves a serious referee, not a desk reject: the analytic core is sound, the limitations are disclosed, and the consolidation itself has value. With Theorem 2 made verifiable and the oscillation claim tightened, I'd be happy to see it published.

Send it to peer review.

Referee Report

2 major / 5 minor

Summary. The manuscript is a review of Lindbladian PT (L-PT) phase transitions in Markovian open quantum systems. It defines L-PT symmetry at the level of the GKSL generator, reviews non-Hermitian and nonlinear PT symmetry, and then develops mean-field theory for two broad classes: single-collective-spin models and bipartite bosonic systems with conserved particle number. The central theoretical claim is that, under the scaling H = S h, L_μ = √S l_μ (Eq. 5.7), microscopic L-PT symmetry (4.1) implies that the mean-field flow inherits a nonlinear PT symmetry (Theorem 1 and Theorem 2). Linear-stability analysis then shows that PT-symmetric fixed points are centers, PT-broken fixed points appear as stable/unstable pairs, and the transition is a critical exceptional point. These results are applied to the driven Dicke model and the dissipative Lipkin-Meshkov-Glick model, and connected to dissipative continuous-time crystals. The paper also presents exact and numerical results for steady-state purity, spin squeezing, and entanglement in specific models, and discusses generalizations to long-range dissipation and spatially extended bosonic lattices.

Significance. If the central theorems hold, the paper establishes a valuable unifying principle: a single microscopic anti-unitary symmetry of the Lindbladian can explain persistent periodic dynamics, continuous-time-crystal behavior, and critical exceptional points in a broad class of dissipative many-body systems. The manuscript includes several genuinely parameter-free derivations: the G_η^(N) threshold computation in Sec. 5.7.1, the exact third-quantization spectra (6.13)–(6.17) for the two-collective-spin model, and the exact TISS solution (7.1) for the DDM. The main caveat is that the bosonic generalization is not self-contained: Theorem 2 is deferred to an unpublished same-group preprint, and this is a load-bearing part of the abstract's claim. Nevertheless, the analytic core for collective-spin systems appears sound and the review is a useful synthesis of recent developments.

major comments (2)

[§5.7.2] Theorem 2 is a central component of the paper's main claim for spatially extended bipartite bosonic systems, yet its proof is not contained in the manuscript; the text states only 'A detailed proof of Theorem 2 is given in Ref. [61]', an arXiv preprint from the same group. The Schwinger-boson mapping (5.63) provides intuition but does not by itself establish the theorem for general U(1)-conserving bipartite bosonic Lindbladians. Since the abstract and conclusions explicitly claim this class of systems, this is a verifiability gap. The authors should either include a self-contained proof, or clearly mark Theorem 2 as a result quoted from [61] and restrict the abstract/conclusions accordingly.
[§5.2, Eq. (5.19)–(5.20)] The step from the expectation-value relation (5.19) to the function identity (5.20) uses the mean-field factorization ⟨G(m_x,m_y,±m_z)⟩ ≃ G(⟨m_x⟩,⟨m_y⟩,±⟨m_z⟩). This factorization is stated in (5.4) and is known to become exact for initial states with vanishing higher cumulants, but it is not generally valid for arbitrary states, including the TISS in the time-crystal phase (as the paper itself notes in Sec. 7). The theorem is therefore best phrased as a property of the mean-field equation under the explicit mean-field closure, not as a rigorous consequence of L-PT symmetry in the thermodynamic limit. The authors should state this limitation directly in the theorem or its proof.

minor comments (5)

[Abstract / §1.4] The phrase 'a broad class of systems' should be qualified by the scaling assumption (5.7) and by the result of Sec. 5.7.1 that local dissipation (η > 1) destroys the n-PT symmetry of the mean-field flow even though microscopic L-PT symmetry is preserved. The paper is internally consistent on this point, but the abstract currently risks overstating the scope.
[§5.7.2, Eq. (5.63)] The Schwinger-boson representation is used to motivate Theorem 2, but the non-compact bosonic phase space and the large-N limit require care. A brief discussion of how the strong U(1) symmetry controls the mean-field factorization for bosonic amplitudes would improve clarity.
[§5.5.2, Fig. 5(i)] The finite-size scaling exponents |κ−κ_c| ∼ S^{−0.62} and Re[λ_EP]/g ∼ S^{−0.35} are presented as numerical observations without error bars or an analytic estimate. The text labels these as 'possible' scaling; please state whether these exponents are expected to be universal or model-dependent.
[§6.3.1] The three-way distinction among strict, mean-field, and symmetry-protected DCTCs is useful, but the wording of condition (B-4) ('Dissipation-induced origin') could be sharpened: the paper excludes oscillations that become dissipation-free only in a limit, yet the mean-field DCTC class itself relies on the thermodynamic limit. Please clarify how the thermodynamic limit interacts with condition (B-4).
[Appendix G] The measure R_opt^{(n)}(S) and the claim of algebraic decay to zero would be more convincing with a stated power-law fit and a discussion of the choice of eigenmodes used in the plot.

Circularity Check

1 steps flagged · score 4.0 of 10

Central bipartite-bosonic theorem is deferred to a same-group preprint; the collective-spin derivation is self-contained.

self citation load bearing [Sec. 5.7.2 (Theorem 2, around Eqs. (5.60)-(5.65))]
"Above the consideration, the following theorem holds [61]. Theorem 2 The mean-field equation possesses an n-PT symmetry [Eq.(2.8)], where the parity operator \tilde{P} is a corresponding permutation in a nonlinear dynamical system. ... A detailed proof of Theorem 2 is given in Ref. [61]."

The abstract and conclusion claim persistent periodic dynamics for a broad class including spatially extended bipartite bosonic lattices with conserved U(1) charge. For that class the central statement is Theorem 2, but the proof is not in this manuscript; it is referred to Ref. [61], a preprint by the same authors. The derivation chain for the bosonic leg therefore terminates in a self-citation rather than a self-contained proof. This is load-bearing because the paper does not reproduce the proof or state the precise conditions under which the mean-field/factorization step is valid for the bosonic amplitudes. The collective-spin half (Lemma + Theorem 1) is proved in the text, so the circularity is partial rather than total.

full rationale

The collective-spin part of the paper is self-contained: the Lemma proves Eq. (5.8) from L-PT symmetry (4.1) and the stated scaling (5.7), and Theorem 1 derives the n-PT symmetry (2.8) explicitly in Eqs. (5.16)-(5.21). The linear-stability classification (centers, stable/unstable pairs, CEP) follows algebraically from the Jacobian forms (5.23)/(5.26) and is presented in the text; the DDM and LMG applications are worked out and compared with external known results (Puri-Lawande exact TISS, Walls/Carmichael DDM behavior, etc.). No fitted parameter is renamed as a prediction, and no derived quantity is fed back as its own input. The only load-bearing step that is not self-contained is Theorem 2 for bipartite bosonic systems, whose proof is deferred to Ref. [61], a same-group preprint. This is a citation-based gap in the bosonic half of the central claim, not a reduction of the collective-spin derivation to its assumptions. Because the paper acknowledges the scaling assumption and its failure mode in Sec. 5.7.1, and because the collective-spin half remains independently derived, I score this as moderate partial circularity (4) rather than a full by-construction reduction.

Assumptions & free parameters 2 free parameters · 10 assumptions · 0 invented entities

The ledger is dominated by domain assumptions of the mean-field/thermodynamic-limit program rather than fitted numbers: scaling (5.7), order of limits, mean-field factorization, and the concrete PT form. Only two descriptive fits enter (Fig. 5(i) exponents) and they are not load-bearing. No new physical entities are postulated: 'zero-mode LEP', 'n-PT symmetry', and 'critical exceptional point' reorganize existing concepts (Prosen's LEP, Konotop et al.'s n-PT, Fruchart et al.'s CEP).

free parameters (2)

DDM finite-size shift exponent = ≈ 0.62
Fit of |κ − κ_c| versus S in Fig. 5(i); presented as a descriptive finite-size scaling observation, not load-bearing for the central argument and without error bars.
DDM zero-mode LEP decay exponent = ≈ 0.35
Fit of Re[λ_EP]/g versus S in Fig. 5(i); descriptive observation, no error bars or fit range given.

assumptions (10)

domain assumption Markovian GKSL master equation (1.8) with CPTP semigroup
The entire L-PT framework is formulated for GKSL generators; non-Markovian extensions are cited but excluded.
domain assumption Thermodynamic-limit scaling H = S h, L_μ = √S l_μ (Eq. (5.7))
Load-bearing for the Lemma: it makes double-commutator contributions O(1/S) and yields n-PT symmetry in the mean-field limit. Dissipators with different scaling (local dissipation, η > 1) break the mean-field n-PT symmetry even under L-PT symmetry (Sec. 5.7.1).
domain assumption Mean-field factorization exact for initial states with vanishing cumulants (Sec. 5.2, Refs. [176,177])
Links quantum expectation values to the classical flow (5.5); the paper states but does not prove this.
domain assumption Order of limits: S → ∞ before t → ∞ for CTCs; t → ∞ before S → ∞ for TISS (Sec. 5.1.3)
The non-commuting limits define which object (TISS or oscillating state) is discussed; the whole phase-diagram reading depends on this.
domain assumption PT operators P = ∏σ^x_i (or site exchange for bipartite models) and T = K with P² = T² = 1, [P,T] = 0 (Eq. (2.2))
The concrete form of P fixes the n-PT structure diag(1,1,−1) in Theorem 1; other PT implementations would give different conclusions.
domain assumption n-PT definition (2.8) adopted over factorization-based (2.9)
The choice of definition affects which nonlinear flows count as n-PT-symmetric; the paper explicitly rejects (2.9) as factorization-dependent.
domain assumption Holstein–Primakoff validity ⟨a†a⟩/(2S) ≪ 1 for the two-collective-spin model (Sec. 6.5.1)
Underlies the exact third-quantization spectrum and the closed-form purity/negativity; fails near spin-up saturation.
standard math Third-quantization machinery: quadratic H, linear L_μ, Re β_j > 0 (Appendix F)
Standard exact method (Prosen 2008) invoked in Appendix F; the assumptions are stated for the two-spin model.
domain assumption DCTC criteria (B-1)–(B-4) as adopted classification
The taxonomy of strict/mean-field/symmetry-protected DCTCs is a definitional choice against which the L-PT–CTC connection is measured.
domain assumption Genericity c_0 ~ (κ − κ_c)^q with q = 1 (footnote, Sec. 5.3.3)
The critical-exponent relation x_Z = q/2 assumes a simple zero; the paper notes extra symmetry or fine-tuning would change q.

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Cite this review

Pith. "Pith review of Lindbladian PT phase transitions." pith.science (2026). https://pith.science/paper/R2NW4WB2

@misc{pith2026251224981,
  author       = {Pith},
  title        = {Pith review of: Lindbladian PT phase transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R2NW4WB2}},
  note         = {Machine review of arXiv:2512.24981}
}

read the original abstract

A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.

Figures

Figures reproduced from arXiv: 2512.24981 by the authors.

**Figure 1.** Connections between L-PT phase transitions and other physical phenomena, symmetry and applications. microscopic and macroscopic dynamics developed in different contexts. Specifically, it has been proven that when the Lindbladian exhibits an L-PT symmetry at the microscopic level, its corresponding mean-field equation has a nonlinear PT (n-PT) symmetry, and a DPT associated with spontaneous n-PT symmetry breaking occ… view at source ↗

**Figure 2.** A basic PT-symmetric example with balanced gain and loss: the parity operator exchanges two particles, while the time-reversal operator swaps gain and loss. [Inspired by Ref. [98]] 2.2. Nonlinear dynamical system We consider a finite-dimensional nonlinear dynamical system on C n of the form i∂tq = f(q), (2.7) with q := (q1,q2,..,qn) ∈ C n and f := (f1, f2,..., fn) with component functions fj : C n → C that are in g… view at source ↗

**Figure 3.** (Top) Lindbladian eigenvalues and (Bottom) their corresponding dynamics of an observable ⟨O⟩ for (a) typical case and (b) persistent oscillations including CTCs. of observables in time [18]. Therefore, a special Lindbladian exceptional point (LEP), where multieigenmodes with zero eigenvalue λ = 0 coalesce, exists only in the thermodynamic limit. We call it zero-mode LEP. 3.1.4. Choi–Jamiolkowski isomorphism We now … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Dissipative continuous phase transition with spontaneous breaking of a weak unitary symmetry: Beyond the critical point, multiple symmetry-broken steady states emerge. This indicates that the Lindbladian gap ∆L vanishes and an order parameter | ⟨O⟩ | takes a finite val…

**Figure 5.** Figure 5: Numerical calculation for the DDM. (a) Illustration of the DDM. (b) The magnetization dynamics for S = 20,40,80 and mean-field analysis (S = ∞). (c) Top: The Lindbladian gap in the TISS. Middle: The normalized magnetization ⟨mz⟩ in the TISS. Viewing ⟨mz⟩ as the order p…

**Figure 6.** Figure 6: Illustrative explanation of (a) L-PT phase transition, (b) nonequilibrium phase transition with Z2 symmetry breaking, and (c) (standard continuous) nonreciprocal phase transition. For case (b), the transition from a unique stable fixed point to two degenerated stable …

**Figure 7.** Figure 7: The phase diagram and numerical calculation for the dissipative LMG model (5.46). (a) The normalized magnetization in the TISS for finite S and mean field solution (S = ∞) with χy = −4. (b) Phase diagram: In the yellow (blue) region, there are only PT-symmetric (PTbro…

**Figure 8.** Figure 8: Illustration of the Schwinger boson transformation (SBT). For example, the collective lowering operator S − maps onto a bilinear product of two bosonic modes, yielding a nonlinear gain–loss dissipator ab† . In this SBT representation, the resulting model is physically …

**Figure 9.** Figure 9: Difference between continuous- and discretetime crystals. The left column sketches a continuoustime crystal: the generator is time-independent (flat blue line), yet a correlation function or macroscopic observable (green) exhibits self-sustained oscillations with an …

**Figure 10.** Figure 10: (a) Schematic picture of the two-collective spin model with gain and loss (6.10). (b) Phase diagram: Red, blue, and green regions indicate the ferromagnetic (FM) phase with ZA = ZB = 1, the FM phase with ZA = ZB = −1, the anti-ferromagnetic (AM) phase with ZA = 1, ZB …

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This state in the time crystal phase is known not to be captured by the mean-field approximation, even in the large-spin limit, as evidenced by inequality⟨S 2 α⟩,⟨S α⟩2 [162, 176]

Quantum and Statistical properties for LindbladianPTphase transitions We here investigate the quantum and statistical properties beyond the mean field theory, most notably the PT symmetry breaking, purity, and quantum entanglement indicators in the TISSρ ss(S)= limt→∞ρ(t,S). This state in the time crystal phase is known not to be captured by the mean-fiel...