Dissipation engineering of fermionic long-range order beyond Lindblad

Andrew J. Daley; Fran\c{c}ois Damanet; Jorge Yago Malo; Maria Luisa Chiofalo; Silvia Neri

REVIEW 3 major objections 4 minor 65 references

Dissipation engineering of fermionic long-range order beyond Lindblad

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Dissipative η-pairing long-range order survives spin exchange and requires a bath spectral density that is symmetric about zero energy and wide enough to cover the model's transitions.

desk verdict Useful extension of dissipative eta-pairing to the tUJ model, but the Redfield symmetry criterion is not physically established—it is baked into the symmetrized bath spectrum of Eq. (24). read the letter →

arxiv 2507.00553 v2 pith:DWG4GZ5M submitted 2025-07-01 cond-mat.supr-con quant-ph

Silvia Neri , François Damanet , Andrew J. Daley , Maria Luisa Chiofalo , Jorge Yago Malo This is my paper · ORCID

classification cond-mat.supr-conquant-ph

keywords dissipationengineeringeta-pairingoff-diagonallong-rangeorderRedfieldmasterequationtUJmodelHubbardspectraldensitysymmetrycavityQED

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 claims that a dissipative protocol for preparing η-pairing off-diagonal long-range order, previously established for the Hubbard model in the Markovian limit, still works in the tUJ model (a Hubbard model plus planar spin-exchange coupling) and continues to work for structured baths, provided the bath spectral density is symmetric about zero frequency and has a cutoff covering the system's relevant transition energies. This matters because it turns a symmetry-based preparation scheme into a concrete spectral-density criterion for experiments such as ultracold atoms in cavity QED, where the environment is rarely perfectly flat. The authors find that the strong η symmetry can survive beyond the Lindblad limit while the steady state still loses translationally invariant correlations, so the engineered order is tied to the Lindblad-type structure of the dissipation rather than to the symmetry alone.

What carries the argument

The central object is Yang's η algebra, $\eta^+=\sum_i(-1)^i c^{\dagger}_{i\uparrow}c^{\dagger}_{i\downarrow}$, $\eta^-=(\eta^+)^\dagger$, $\eta^z$, which commutes with the tUJ Hamiltonian at half filling while the dephasing jump operators $L_j=s^z_j=n_{\uparrow,j}-n_{\downarrow,j}$ commute with $\eta^\pm$. Under Lindblad dephasing the spin sector is driven to infinite temperature while charge-sector correlations survive. In the Redfield treatment the rotated jump operator $L_j(\tau)=\int_0^\tau \alpha(s)e^{-iHs}L_j e^{iHs}ds$ becomes non-Hermitian unless the spectral density $J(\omega)$ is symmetric, and a narrow cutoff leaves some dissipative channels closed; both effects suppress the distance-invariant η-correlations.

What would settle it

Simulate the same $tUJ$ chain coupled to a structured bath with a numerically exact non-perturbative method at $\gamma=2t$ and check whether distance-independent $\langle\eta^+_i\eta^-_{i+j}\rangle$ still appears only for symmetric, wide-enough spectral densities; if it appears for asymmetric or narrow baths, the criterion is an artifact of the Redfield approximation.

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Extended reading notes

Core claim

The paper shows that the Lindblad mechanism for stabilizing η-pairing off-diagonal long-range order survives in the tUJ model and, for structured baths, the steady state exhibits distance-independent correlations $\langle\eta^+_i\eta^-_{i+j}\rangle$ only when the bath spectral density is even around $\omega=0$ and its cutoff covers the relevant transition-energy range of the Hamiltonian. It further finds that going beyond Lindblad does not break the strong symmetry protecting η-pairing, yet the steady state can still lose the translational invariance of the correlations, so the ODLRO is genuinely tied to the Lindblad-type structure of the dissipation rather than to the symmetry alone.

Load-bearing premise

The Bloch-Redfield master equation with a zero-temperature bath and a manually symmetrized spectral density remains accurate at $\gamma=2t$, even though the dissipation rate is not small compared with the energy scales $t$ and $U$.

Editorial extensions

If this is right

The symmetry-based dissipative preparation of η-pairing superfluidity, originally formulated for the Hubbard model, works unchanged when a planar spin-exchange term $J$ is added, so the tUJ model is a viable target for cavity-QED emulators.
For any realistic structured bath, the protocol succeeds only if $J(\omega)$ is approximately even about $\omega=0$ over the transition-energy range of $H$; a bath that is flat only on one side will not produce ODLRO.
The frequency cutoff $\omega_c$ must be at least as large as the largest relevant transition energy; below that, the steady-state η-correlations are not translationally invariant.
Even with the strong η symmetry intact, non-Lindblad dissipation can change the steady state, so the preparation scheme is robust only around a Lindblad-type description.
Moving away from the Markovian limit generally lowers the condensate fraction and lengthens the convergence time, with an Ohmic linear spectral density staying closest to the Markovian result.

Reading between the lines

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

The symmetry-plus-cutoff criterion is plausibly general: any dissipative protocol that relies on a strong symmetry to protect a target sector should require the bath to open all symmetry-preserving dissipative channels across the relevant spectral range, not just to preserve the algebra.
A direct experimental test would be to tune a cavity or bosonic reservoir so that its coupling strength varies asymmetrically in frequency and to measure the distance dependence of the η-correlations; the paper predicts ODLRO will disappear when the asymmetry enters the Hamiltonian's transition range.
The Lorentzian results in the appendix suggest that a centered, sufficiently broad Lorentzian behaves like the Markovian limit whereas a shifted one slows convergence; this could be used as a finite-size diagnostic, with the optimal Lorentzian width tracking the system's largest transition energy.

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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

Jake, quick take on 2507.00553. The paper does two things: it shows the Tindall et al. Lindblad mechanism for eta-pairing ODLRO survives adding a spin-exchange J term (the tUJ model), and it starts to go beyond Lindblad with Bloch-Redfield, claiming that ODLRO requires the bath spectral density to be symmetric about zero frequency and to have a cutoff covering the system's transition energies.

The tUJ extension is genuine and the numerics back it up: Fig. 1 shows distance-invariant eta correlations at M=8, and the symmetry argument is clean. The Redfield part is more exploratory. The cutoff dependence is clearly illustrated for M=4 with a linear spectral density, and the Lorentzian appendix supports the general trend. Credit where due: the paper is a reasonable step in an established program, and the experimental discussion is grounded.

Now the soft spots. The main one is not the absence of finite-size scaling or error bars, though those matter; it's the bath model. Eq. (22) defines a zero-temperature bath with a one-sided spectrum. Eq. (24) then extends the integral to -infinity with an even J(omega). That is not the zero-temperature quantum bath—an even spectral density gives a real, even correlation function, which is the noise of a classical or infinite-temperature bath. So the paper's central criterion that J must be symmetric about zero is partly built in by construction. The observed failures for asymmetric or low-cutoff J may be artifacts of that symmetrization, not generic properties of structured quantum baths. A stress-test note makes exactly this point, and I think it lands. The authors should either re-frame the claim as conditional on an engineered symmetric (infinite-temperature-like) bath or re-derive with a one-sided spectrum.

Second, the Redfield equation is used at gamma = 2t with U = 4t and t = 1. That is not a controlled weak-coupling regime, so the quantitative conclusions are shaky. Third, there is no code or data release, and the sigma quantifier is presented without uncertainty propagation—minor but worth mentioning.

Overall, the Lindblad tUJ result is solid, and the paper is a plausible first exploration of structured baths. But the 'symmetric spectral density' criterion is not established as a physical statement. A serious referee should send it back for major revision: fix the bath model, run finite-size scaling at least to M=6 or M=8 for the Redfield part, and temper the conclusions accordingly. If they do that, it could be a useful contribution. If not, the central claim remains a heuristic tied to a specific, somewhat unphysical master equation.

My honest verdict: worth a referee, but not close to acceptance as is. The tUJ Lindblad result alone would be a modest but solid paper.

Referee Report

3 major / 4 minor

Summary. The manuscript studies dissipative preparation of η-pairing off-diagonal long-range order (ODLRO) in the tUJ model, extending the mechanism of Tindall et al. (PRL 123, 030603). In the Lindblad part (Sec. III), spin dephasing with jump operators L_j = s^z_j is shown numerically, for M=8 sites, to produce a steady state with distance-independent ⟨η_i^+ η_{i+j}^-⟩, indicating that the mechanism survives the spin-exchange term J. In the Redfield part (Secs. IV–V), the flat bath is replaced by structured spectral densities, using Eq. (24) with an extended lower integration limit; numerical results show that for J(ω)=|ω| a sufficiently large cutoff is needed, and for a step-like J(ω)=θ(ω−ω_c) a sufficiently negative step position is needed to approach ODLRO. Section V condenses these observations into two criteria: the spectral density should be symmetric about zero and should symmetrically cover the relevant transition-energy range. Appendices A and B add Lorentzian and multi-peak spectral densities and numerical details.

Significance. If correct, the paper would generalize a known symmetry-based dissipative preparation to the tUJ model and would provide a simple frequency-domain design rule for structured baths, which is potentially useful for cavity-QED implementations. The Lindblad extension is a nontrivial numerical result, and the paper is honest about the sufficient-condition character of its ODLRO criterion and about the parameter-specificity of some Appendix A results. However, the Redfield part is presently a heuristic numerical criterion rather than a demonstrated physical law: the master-equation model in Eq. (24) uses a symmetrized bath correlation function, the simulated coupling strength is outside the weak-coupling regime, and no finite-size or sampling-error analysis supports the σ(ω_c)→0 claim. With those gaps closed, or with the claims suitably restated as conditional on an engineered symmetric bath, the criterion could become a practical guideline.

major comments (3)

[Sec. IV, Eq. (24)] The replacement of the one-sided zero-temperature bath correlation function of Eq. (22) by the two-sided integral with an even J(ω) is not a harmless extension. For an even, positive J, α(s) is real and even, which is the correlation function of classical (infinite-temperature) noise, not of the zero-temperature Bose bath stated in Eq. (22); a physical quantum bath obeys α(−s)=α(s)^* with the negative-frequency weight fixed by detailed balance. Since every Redfield simulation in Figs. 2–4 uses this symmetrized α(s), the conclusion in Sec. V that J must be symmetric about zero is at least partly imposed by construction. Please either re-derive the analysis with the one-sided spectrum J(ω≥0) of Eq. (22) and test the same cutoff/asymmetry questions, or restate the central claim as applying to an engineered symmetric (effective-infinite-temperature) bath.
[Sec. IV and Appendix B] The Bloch-Redfield master equation is used with γ=2 (in units of t) while U=4t and t=1, so the dissipation rate is not small compared with the system energy scales. The paper gives no check of the Born-Markov or secular assumptions and no check of complete positivity of the resulting Redfield dynamics, which is known to be violated in general. The cutoff and symmetry dependences in Figs. 2–4 and the Liouvillian spectra in Fig. 3 could therefore be artifacts of the Redfield approximation rather than properties of the dissipative preparation scheme. Please add a validity test such as a small-γ extrapolation, a comparison with a non-perturbative method, or a positivity check on ρ(t), and restrict the conclusions to the regime in which Redfield is controlled.
[Sec. IV, Eq. (25) and Fig. 2(d)] The central quantifier σ(ω_c) is evaluated at M=4 (with M=8 in Fig. 1) without finite-size scaling, and no error bars or uncertainty propagation are given for σ, μ, or σ_L even though the data come from quantum-jump Monte Carlo with N=2000 trajectories (Fig. 1) or 1000 trajectories (Appendix B). Because ODLRO is a thermodynamic-limit statement and the claim is that σ(ω_c)→0 as ω_c grows, the paper should demonstrate that the cutoff threshold and the j-independence of the correlations are stable under system-size and sampling-error variation. As written, the criterion is a heuristic numerical observation rather than a demonstrated property.

minor comments (4)

[Throughout] There are several typos and wording errors that should be corrected: 'particule-hole' (Sec. II), 'implentations' and 'impassibility of realizing a SF state' (Secs. IV–V), 'without loose of generality' near Eq. (25), and 'ORDLO' in the caption of Fig. 3.
[Fig. 1 caption] The caption says 'M = 6 particles at half-filling' for the projections in panel (c), while elsewhere M denotes the number of lattice sites; please use a consistent notation such as 'M=6 sites at half-filling (N=M particles)'.
[Fig. 4 and Sec. IV.B] The step-like spectral density J(ω)=θ(ω−ω_c) is described as a 'cutoff', but it is a one-sided step whose support is [ω_c,∞). The text should clarify how varying ω_c from 0 to large negative values changes the symmetry of the spectral density over the relevant transition-energy range, since the terminology 'low cutoff' and 'large cutoff' is confusing for this functional form.
[Appendix A, Fig. 7] The note in Appendix A states that in Fig. 7 the super-Ohmic, sub-Ohmic, and Lorentzian values are specific and do not represent a general trend. This caveat should be carried into the main text where the Ohmic-like bath is described as performing best, to avoid overgeneralization from a single parameter choice.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: numerical Redfield results are independent; the bath-symmetry criterion is a modeling conclusion, not an input.

full rationale

The paper's central claims are supported by direct computation rather than by importing the target result. In Sec. III, Eq. (17) is solved numerically for the tUJ model, with η-pairing ODLRO diagnosed from the simulated two-point correlators in Fig. 1(d); Eq. (19) verifies [L_j, η±] = 0 in the text. Ref. [33] supplies the external Lindblad benchmark and is not by any of the present authors; it is used as a comparison, not as a proof that the tUJ version works. The Redfield analysis in Sec. IV defines the spectral-density dependence through Eq. (24) and measures convergence through σ(ωc) and σL(ωc) in Eqs. (25)-(26). No parameter is fitted to the target ODLRO, and symmetric J is not used in the definition of ODLRO; the Sec. V conditions ('the spectral density has to be symmetrical respect to the zero-energy axis' and the cutoff must cover the relevant transition range) are interpretations of the observed steady-state behavior. The main caveat, that extending Eq. (22) to Eq. (24) may impose a symmetric or infinite-temperature-like bath and that γ = 2t may lie outside the Bloch-Redfield weak-coupling regime, is a physical-modeling and validity limitation rather than a circular reduction; it does not make the derivation equivalent to its inputs. Self-citations by the authors appear only in background and experimental contexts and are not load-bearing for the central derivation.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper adds no new particles or forces. Its central result depends on chosen model parameters (U, J, gamma), trust in Redfield at moderate coupling, and an assumed finite-size-to-thermodynamic-limit correspondence. No parameters are fitted to reproduce the target outcome, so the circularity burden is low.

free parameters (5)

Dissipative rate gamma = 2t (t=1)
Sets the coupling to the bath in all Lindblad and Redfield simulations; chosen by hand, not fitted to external data.
On-site interaction U = 4t
Defines the model's interaction scale; chosen as an illustrative parameter, as in Ref. [33].
Spin-exchange J = -0.1U
Selected to introduce tUJ spin-exchange while preserving the eta symmetry; sign and magnitude chosen by hand.
Spectral density cutoffs omega_c = 3/t and 10/t (linear); -1, -5, -10 (step)
Scanning these cutoffs is how the paper establishes its central criterion; they are chosen limits, not data-derived constants.
Spectral density amplitude = Not explicitly stated for J(w) proportional to |w|; implied by gamma=2t in the Markovian limit
The dissipation strength of the colored bath is not fully specified, which affects reproducibility.

assumptions (4)

domain assumption The Redfield/Bloch-Redfield master equation (20)-(24) with zero-temperature bath and negative-frequency extension describes the open-system dynamics to sufficient accuracy.
Invoked in Sec. IV; weak-coupling and Born-Markov assumptions are only mentioned, not verified at gamma=2t.
domain assumption A finite chain of M=4 or 8 sites at half filling with distance-invariant eta correlations indicates ODLRO in the thermodynamic limit.
The paper notes that equal correlations are a sufficient condition for ODLRO, but finite-size extrapolation is not performed (Sec. III, Figs. 1-4).
standard math The Lindblad result of Tindall et al. [33] regarding unital dephasing and eta-symmetry protection is correct.
Basis of the whole protocol, cited in Sec. III.
standard math Yang's criterion and the eta-SU(2) algebra at half filling.
Used in Sec. II to identify eta-pairing as the target order.

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Pith. "Pith review of Dissipation engineering of fermionic long-range order beyond Lindblad." pith.science (2026). https://pith.science/paper/DWG4GZ5M

@misc{pith2026250700553,
  author       = {Pith},
  title        = {Pith review of: Dissipation engineering of fermionic long-range order beyond Lindblad},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DWG4GZ5M}},
  note         = {Machine review of arXiv:2507.00553}
}

read the original abstract

We investigate the possibility of engineering dissipatively long-range order that is robust against heating in strongly interacting fermionic systems, relevant for atoms in cavity QED. It was previously shown [Tindall et al. Phys.Rev.Lett. 123, 030603 (2019)] that it is possible to stabilize long-range order in a Hubbard model by exploiting a dissipative mechanism in the Lindblad limit, this latter being valid for spectrally unstructured baths. Here, we first show that this mechanism still holds when including additional spin-exchange interactions in the model, that is for the tUJ model. Moreover, by means of a Redfield approach that goes beyond the Lindblad case, we show how the stability of the engineered state depends crucially on properties of the bath spectral density and discuss the feasibility of those properties in an experiment.

Figures

Figures reproduced from arXiv: 2507.00553 by the authors.

**Figure 3.** FIG. 3. (a) Eigenspace spectrum of the Redfield tensor [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. In Fig. 4(a), we present the spectral density used to [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Summary of the beyond-GSKL analysis using differ [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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