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REVIEW 4 major objections 4 minor 72 references

Exceptional stationary state in a dephasing many-body open quantum system

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that a one-dimensional dephasing Lindblad model has exactly two stationary states — the infinite-temperature state and the pure fully polarized state…

desk verdict A solid, honest paper with a genuinely interesting mechanism—finite gap yet size-dependent slow relaxation—but the quantitative membrane prediction is backed by fits rather than by testing the analytic values. read the letter →

arxiv 2412.13820 v3 pith:RCLX3NFA submitted 2024-12-18 quant-ph cond-mat.quant-gascond-mat.stat-mechcond-mat.str-el

Alice Marché , Gianluca Morettini , Leonardo Mazza , Lorenzo Gotta , Luca Capizzi This is my paper · ORCID
classification quant-phcond-mat.quant-gascond-mat.stat-mechcond-mat.str-el
keywords openquantumsystemsLindbladmasterequationmany-bodyscarsdephasingnoisedarkstatesstrongsymmetriescommutantalgebrasnon-Hermitianskineffect
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a one-dimensional spin chain with dephasing noise and shows that, besides the usual infinite-temperature state, it has a second stationary state: the pure fully polarized state with all spins up. That pure state is exceptional because it keeps memory of the initial condition, while every orthogonal state relaxes to the infinite-temperature state and loses all information. The authors argue that this pure dark state is an open-system analog of a quantum many-body scar, since it is not protected by any extensive conserved operator and no hydrodynamic slow modes exist. They then show that the interface between the two stationary states melts by diffusive broadening with a drift, on timescales that grow with system size even though the Lindbladian spectral gap is finite, calling the standard gap-to-relaxation-timescale identification into question.

What carries the argument

The argument runs on the commutant algebra $\mathcal{C} = \mathrm{Span}\{1, |\!\uparrow\uparrow\cdots\uparrow\rangle\langle\uparrow\uparrow\cdots\uparrow\!|\}$, which fixes the stationary manifold, and on an effective Markov generator obtained by second-order perturbation theory in $g/\gamma$ and $J/\gamma$, acting on diagonal density matrices in the $z$ basis. In the Doi-Peliti representation this generator becomes a classical Markov process $W$. The paper introduces a simplified generator $\widetilde W = W - (2g^2/\gamma)\sum_j \pi_{j-1}^z\pi_j^x\pi_{j+1}^z$ with rates chosen so that $J^2/(4\gamma) = 2g^2/\gamma = 1/6$; in $\widetilde W$ the single-domain-wall subspace is closed and the interface performs a biased random walk with drift $v = 1/3$ and diffusion constant $D = 1$, giving the error-function profile. The spectral decomposition of $\widetilde W$, together with the non-Hermitian skin effect in its one-domain-wall sector, explains why the finite gap does not control the relaxation of local observables.

What would settle it

Run the full Lindblad dynamics (or the full Markov generator $W$) from a sharp interface at small but non-special $g/\gamma$ and $J/\gamma$, and measure the weight that leaves the single-domain-wall subspace: if that weight grows with $L$ or the rescaled profile fails to collapse to the error function at large $t$, the membrane picture is not the correct mechanism. Alternatively, compute the Lindbladian gap for $L \to \infty$; if $\Delta_0$ extrapolates to zero, the finite-gap premise fails.

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Extended reading notes

Core claim

The central claim is that the Lindblad dynamics of Eq. (1), with Hamiltonian $H = \sum_j [J(\sigma_j^+\sigma_{j+1}^- + \mathrm{H.c.}) + g(\sigma_j^x\pi_{j+1}^z + \pi_j^z\sigma_{j+1}^x)]$ and dephasing jump operators $L_j = \sqrt{\gamma}\,\sigma_j^z$, has exactly two stationary states: the infinite-temperature state and the pure fully polarized state $|\!\uparrow\uparrow\cdots\uparrow\rangle\langle\uparrow\uparrow\cdots\uparrow\!|$. The commutant algebra of the local terms is generated by these two operators, so no extensive conserved quantity protects the polarized state; the paper therefore calls it an open-system quantum many-body scar. Starting from an interface that joins the two stationary states, the magnetization profile evolves under an error-function scaling in $(x-vt)/\sqrt{Dt}$ with nonzero drift and diffusion, even though no conserved densities exist and the Lindbladian gap is finite. The paper reconciles this apparent contradiction by showing that the overlaps of the initial state onto the Lindbladian eigenvectors grow exponentially with system size, a phenomenon tied to the non-Hermitian skin effect.

Load-bearing premise

The quantitative membrane prediction is derived from a simplified generator $\widetilde W$ that drops the three-site coupling and pins rates to a special value; the paper assumes this simplified generator faithfully describes the interface dynamics of the full model at the parameters actually simulated, where the dropped terms are not small.

Editorial extensions

If this is right

  • Any initial state orthogonal to $|\!\uparrow\uparrow\cdots\uparrow\rangle$ eventually becomes locally indistinguishable from the infinite-temperature state, so all memory of the initial condition is erased except for the component along the polarized state.
  • The interface magnetization satisfies $\langle\sigma^z(x,t)\rangle \simeq \mathrm{erf}((x-vt)/\sqrt{Dt})$ with finite $v$ and $D$, so local probes see relaxation times that grow with system size rather than saturating at $1/\Delta_0$.
  • Coherence between the scar subspace and its complement decays at least as $e^{-4\gamma t}$, so the two sectors decouple exponentially fast.
  • A finite Lindbladian gap is not sufficient to conclude fast, size-independent thermalization; the size dependence of initial-state overlaps onto Lindbladian eigenvectors must be checked.
  • The same membrane-interface collapse appears in two further dephasing models (PXP-like and XPX-like), indicating the mechanism is not an accident of the East-West Hamiltonian.

Reading between the lines

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

  • A direct test of the membrane mechanism would be to tune $g/\gamma$ and $J/\gamma$ to the perturbative regime and check whether fitted $v$ and $D$ approach $1/3$ and $1$; the paper's simulations sit at moderate couplings where the simplified generator is not guaranteed to dominate.
  • The non-Hermitian skin-effect explanation suggests boundary conditions are a switch: periodic boundary conditions force an even number of domain walls, so the slowest interface relaxation may look qualitatively different from the open-chain case at the same parameters.
  • If the scar classification by absence of extensive strong symmetries is adopted, other dephasing models with non-extensive symmetries should also display finite-gap, slow-interface relaxation, making the membrane scaling a generic experimental benchmark for weak ergodicity breaking in open systems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the Lindblad master equation (1) for a spin-1/2 chain with dephasing jump operators and a Hamiltonian consisting of hopping plus an East-West term. The main claims are: (i) for generic g,J≠0 the only stationary states are the infinite-temperature state and the pure fully-polarized state |⇑⟩⟨⇑|, with the latter a dark state protected by a non-extensive strong symmetry; (ii) coherences between the sector of |⇑⟩ and its orthogonal complement decay exponentially at a rate bounded by 4γ; (iii) the Lindbladian gap is finite in the thermodynamic limit but, despite this, an interface prepared between the two stationary states melts diffusively with drift on timescales growing with L; (iv) this apparent contradiction is reconciled through eigenvector localization, i.e., a non-Hermitian skin effect, of the effective Markov generator; and (v) |⇑⟩ should be regarded as an open-system quantum many-body scar. The analytic membrane solution with v=1/3 and D=1 is obtained for a simplified Markov generator W-tilde, while the full-model comparison in Figs. 3 and 4 uses fitted values of v and D.

Significance. If the claims hold, the paper provides a concrete dephasing model in which a finite Lindbladian gap coexists with system-size-dependent relaxation of a local observable, and it formulates a criterion for open-system many-body scars based on the absence of extensive strong symmetries. The strengths include the algebraic commutant calculation, the rigorous exponential bound on inter-sector coherence, the exact solvability of the auxiliary membrane generator, and the transparent tensor-network and exact-diagonalization numerics. These elements make the central existence claim credible and the proposed scar terminology worth discussing. However, the quantitative membrane prediction is not yet connected to the full model in a controlled way, so the broad significance of the paper depends on a reframing or on additional controlled numerics.

major comments (4)
  1. [Membrane diffusion of the interface, Eqs. (5)-(7)] The analytic values v=1/3 and D=1 in Eq. (6) are derived for W-tilde, not for the model in Eq. (1). W-tilde is obtained from the effective generator W of Eq. (5) by dropping the three-site term 2(g^2/γ)∑_j π^z_{j-1}π^x_jπ^z_{j+1} and by fixing J^2/(4γ)=2g^2/γ=1/6. At that point the dropped term has the same coefficient as the retained two-site terms, so W-tilde is not a controlled approximation to W; its single-domain-wall closure is a property of W-tilde alone. Since the full-model comparison in Fig. 4 fits both v and D, the predicted constants v=1/3 and D=1 are never tested against the original Lindbladian. The paper should either present W-tilde as an auxiliary model that motivates the scaling form, or provide a controllability argument that connects it to Eq. (1).
  2. [Membrane diffusion of the interface, Figs. 3 and 4] The full-model simulations in Figs. 3 and 4 use J=g=0.5 and γ=1, i.e., g/γ=J/γ=0.5, outside the large-γ regime (g/γ, J/γ≪1) in which the effective generator W in Eq. (5) was derived; the authors acknowledge this in the text, but the consequence is that the error-function collapse with fitted v≃0.453 and D≃1.33 does not quantitatively test the membrane mechanism. A test in the controlled regime, or an analysis of the dependence of v and D on γ, would be needed to support the claim that the interface of the original model obeys the membrane dynamics rather than merely the diffusive scaling form.
  3. [Late-time asymptotics and spectral gap, End Matter] The skin-effect reconciliation in the section 'Late-time asymptotics and spectral gap' and in the End Matter is carried out for the projected single-domain-wall matrix of W-tilde. The estimate in Eq. (14) shows that the matrix elements of W-tilde grow exponentially in L; no analogous statement is demonstrated for the low-lying eigenvectors of the actual Lindbladian L. Since the finite gap and the slow interface relaxation are both properties of Eq. (1), the reconciliation currently applies to the auxiliary model. The authors should state this limitation explicitly or provide numerical evidence for the corresponding eigenvector localization in the full Lindbladian.
  4. [Stationary states, End Matter] The claim that there are exactly two stationary states for all L is stronger than what is proven in the main text. The End Matter proves C=Span{1, |⇑⟩⟨⇑|} and notes the inclusion C⊆kerL, while the numerical double degeneracy of the zero eigenvalue is shown only for L=6. Unless a theorem ensures ker L=C for this class of dephasing Lindbladians (which the paper does not state), the uniqueness of the stationary states is not established in the thermodynamic limit. Please state the relevant theorem or extend the argument.
minor comments (4)
  1. [End Matter, Commutant algebra] The sentence 'since C ⊆ ker L ... one safely concludes that C = Span{1, |⇑⟩⟨⇑|}' is logically circular and should be rewritten; the conclusion follows from the induction in the Supplemental Material, not from the observed degeneracy of the kernel.
  2. [Throughout] There are several typos and minor notational slips: 'reconcyling' in the heading of the last section, 'isporphic' and 'ahereas' in the Supplemental Material, and '(σ^z_j(t))' in Eq. (11) should be 'σ^z_j'.
  3. [Fig. 1(b)] The fit Δ(L)=Δ0+Δ1/L^2 is reported without fitted values or residuals; please provide the parameters and error bars, since the finite Δ0 is a central assertion.
  4. [Membrane diffusion of the interface, Fig. 4] The fitted values v≃0.453 and D≃1.33 differ from the analytic v=1/3 and D=1 by about 36% and 33%, respectively; the paper should comment on this discrepancy even under the interpretation that W-tilde is only an auxiliary model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stationary-state and exact W-tilde membrane calculations are self-contained, and the full-model collapse is an openly fitted comparison, not a disguised prediction.

full rationale

The stationary-state structure is proved independently of the paper's own conclusions: the commutant algebra is derived by induction in the End Matter, the two-dimensional kernel of the Lindbladian is checked by exact diagonalization, and the exponential decay of the coherence between the scar sector and its orthogonal complement follows from an algebraic inequality (End Matter, 'On the exponential loss of coherence'). The membrane result is obtained for the explicitly defined simplified generator W-tilde, with the parameter choice J^2/(4γ)=2g^2/γ=1/6 disclosed before the computation; the paper does not present this as a first-principles derivation for the full Lindbladian at J=0.5, g=0.5, γ=1. The only step that could be mistaken for a fitted prediction is Fig. 4, where the magnetization profiles are collapsed onto Eq. (7) using v and D as fit parameters. However, the text explicitly states 'where v and D have been fitted', and the analytic values v=1/3, D=1 for W-tilde are not claimed to be the fitted full-model values (v≃0.453, D≃1.33). Fitting the two parameters of a Gaussian/error-function scaling form weakens the comparison as a test of the membrane picture, but it is a disclosed empirical calibration rather than a circular derivation: the existence of the scaling form, the drift, and the diffusive broadening are not guaranteed by the construction of the full Lindbladian. Citations such as Refs. [18], [29], [62], and [63] are external or rigorous benchmarks, and no load-bearing self-citation chain is present. Therefore no step in the claimed derivation reduces by construction to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

No new physical particles or forces are introduced. The main free parameters are the fitted membrane constants v and D and the hand-chosen rate ratio for the simplified generator; the key axioms are the Lindblad framework, the large-gamma perturbative reduction, and the representativeness of W-tilde. The only invented entity is the proposed scar label, which the authors explicitly mark as non-rigorous.

free parameters (4)
  • Drift velocity v in membrane fit = v ~ 0.453 (Fig. 4, full Lindbladian); v = 1/3 analytically for W-tilde
    Fit to the numerical interface profile used to demonstrate the scaling collapse of Eq. (7); the analytic value 1/3 applies only to the simplified generator at special rates, not to the simulated model.
  • Diffusion constant D in membrane fit = D ~ 1.33 (Fig. 4, full Lindbladian); D = 1 analytically for W-tilde
    Same as v: fitted from the same data against which the membrane prediction is displayed; no independent prediction of D for the full model is provided.
  • Simplified-model rate ratio = J^2/(4 gamma) = 2 g^2/gamma = 1/6
    Chosen by hand so that W-tilde leaves the single-domain-wall subspace closed and the interface problem is exactly solvable; not a measured parameter of the full model.
  • Lindbladian gap intercept Delta_0 = non-zero for g != 0 (extrapolated from L = 6 to 12)
    Extracted from a fit of Delta(L) = Delta_0 + Delta_1 / L^2 or similar; it is the numerical evidence for the finite-gap claim, but is itself a fitted quantity.
assumptions (5)
  • domain assumption The Lindblad master equation with Hermitian dephasing jump operators L_j = sqrt(gamma) sigma^z_j is an accurate description of the dynamics.
    The entire analysis is built on Eq. (1); all spectral and dynamical statements refer to this generator.
  • standard math Second-order perturbation theory in g/gamma and J/gamma produces the effective Markov generator W of Eq. (5).
    Used in the End Matter to reduce the Lindbladian to a classical Markov process on diagonal density matrices; the paper notes this holds for g and J at fixed L.
  • ad hoc to paper The simplified generator W-tilde is representative of W for interface dynamics.
    Dropping the three-site term and setting coupling ratios to 1/6 is needed for exact single-domain-wall closure; the paper validates it only indirectly through fitted scaling collapses.
  • domain assumption The initial state in the bipartition protocol can be prepared as a product state joining the infinite-temperature state and the fully polarized state.
    Used for Figs. 3 and 4; the paper calls the notation rho_inf tensor rho_up a slight abuse, and deviations at the junction are not analyzed.
  • standard math Knabe's method gives a rigorous lower bound on the spectral gap of W-tilde.
    Invoked in the Supplemental Material to prove the finite gap of the Markov generator; it is a standard technique for frustration-free local Hamiltonians.
invented entities (1)
  • Open-system quantum many-body scar (proposed label for |up><up|)
    purpose: Provides a conceptual classification of pure dark stationary states not protected by extensive strong symmetries in dephasing Lindblad systems.
    The paper itself states that no rigorous widely accepted definition exists and that alternative definitions appear in Refs. [47-51]; the label is a proposal, not an independently evidenced entity.

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Cite this review

Pith. "Pith review of Exceptional stationary state in a dephasing many-body open quantum system." pith.science (2026). https://pith.science/paper/RCLX3NFA

@misc{pith2026241213820,
  author       = {Pith},
  title        = {Pith review of: Exceptional stationary state in a dephasing many-body open quantum system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCLX3NFA}},
  note         = {Machine review of arXiv:2412.13820}
}
read the original abstract

We study a dephasing many-body open quantum system that hosts, together with the infinite-temperature state, another additional stationary state. The latter is exceptional in many respects, as it is pure and retains memory of the initial condition, whereas any orthogonal state evolves towards the infinite-temperature state erasing any information on the initial state. We discuss the approach to stationarity of the model focusing in particular on the fate of interfaces between the two states; a simple classical model based on a membrane picture helps developing an effective hydrodynamic theory even if the dynamics does not feature any conserved quantity. The fact that the model reaches stationary properties on timescales that depend on the system size while the asymptotic decay rate is finite is duly highlighted. We point out the reasons for considering these exceptional stationary states as quantum many-body scars in the open system framework.

Figures

Figures reproduced from arXiv: 2412.13820 by the authors.

Figure 1
Figure 1. (a) Lindbladian spectrum for the following paramet [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Magnetization profile ⟨σ z j (t)⟩ as a function of j and t. The initial state is obtained by joining the infinite￾temperature state, with support j ∈ [−L/2 + 1, 0], and the full-aligned up state along the z-axis elsewhere. The evolu￾tion is performed for the model in Eqs. (1) with the following parameters: [J, g, γ, L] = [0.5, 0.5, 1, 40]. possible stationary states [3, 4]; in particular, when only H is conserved, t… view at source ↗
Figure 4
Figure 4. Profile of the magnetization shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: Comparison between the slow decaying modes of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Lindbladian gap ∆ as function of 1/L2 for L ∈ {6, . . . , 12} and periodic boundary conditions. gives d dt ⟨σ| ρ(t)|⇑⟩ = ⟨σ| − i[H, ρ(t)] + γ X j (σ z j ρ(t)σ z j (t) − ρ(t))|⇑⟩. (11) Using H |⇑⟩ = 0 and σ z j |⇑⟩ = |⇑⟩, we express d dt ⟨σ| ρ(t)|⇑⟩ = − X |σ′⟩̸=|⇑⟩ ⟨σ| …
Figure 7
Figure 7. Figure 7: (Left) Profile of the magnetization ⟨σ z j (t)⟩ for the model in Eq. (36) with parameters [J, g, γ, L] = [0.5, 0.5, 1, 20] and open boundary conditions. (Right) Scaling collapse of the magnetization against the prediction (7), with the parameters v = 0.023, D = 1.72 be…
Figure 8
Figure 8. Figure 8: (Left) Profile of the magnetization ⟨σ z j (t)⟩ for the model in Eq. (37) with parameters [J, g, γ, L] = [0.5, 0.5, 1, 40] and open boundary conditions. (Right) Scaling collapse of the magnetization against the prediction (7), with the parameters v = 0.247, D = 0.804 b…
Figure 9
Figure 9. Figure 9: We plot the spectral gap ∆, choosing the parameters [J, g, γ] = [0.5, 0.5, 1] for the PXP and XPX with hopping and dephasing. (Left) Model (36): We fit ∆ = ∆0 + ∆1/L3 , extrapolating ∆0 ≃ 5.1314 · 10−3 (Right) Model (37): We fit ∆ = ∆0 + ∆1/L2 and we obtain ∆0 = 7.596 …

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Reviewed August 11, 2026 · model on record in the stance chip above.