Kibble-Zurek scaling immune to anti-Kibble-Zurek behavior in driven open systems at the limit of loss difference

Han-Chuan Kou; Peng Li; Zhi-Han Zhang

REVIEW 3 major objections 4 minor 61 references

Kibble-Zurek scaling immune to anti-Kibble-Zurek behavior in driven open systems at the limit of loss difference

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper shows that at the limit of loss difference, the residual fermion density obeys Kibble-Zurek scaling $N \propto \tau_Q^{-\beta}$, immune to the anti-Kibble-Zurek behavior that saturates the standard excitation signal.

desk verdict A useful result on KZ scaling in dissipative quenches, with a real proof gap behind the word 'rigorous'. read the letter →

arxiv 2411.16406 v3 pith:7E7DOQKH submitted 2024-11-25 quant-ph

Han-Chuan Kou , Zhi-Han Zhang , Peng Li This is my paper · ORCID

classification quant-ph

keywords Kibble-Zurekmechanismanti-Kibble-ZurekbehaviorLindbladmasterequationlossdifferencedissipativequenchdynamicstwo-bandfermionicsystemsresidualfermiondensityopenquantum

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 asks whether the Kibble-Zurek power law survives in an open two-band fermionic system that loses particles while its staggered on-site energy is ramped linearly. It finds that ordinary uniform loss exponentially suppresses the usual defect-density signal and adds an anti-Kibble-Zurek background. At the 'limit of loss difference', where one sublattice is dissipated and the other is not, the density of residual fermions obeys $N \propto \tau_Q^{-\beta}$, recovering the universal Kibble-Zurek scaling even though the standard excitation signal has saturated. A quench that avoids the critical point instead produces a dissipation-dependent 'pseudo-Kibble-Zurek' scaling $N \propto [(1/u_f - 1/u_i)\gamma\tau_Q]^{-\beta}$. The authors show both behaviors in the Rice-Mele, Shockley, and Haldane models, and propose counting residual particles as an observable that is immune to anti-Kibble-Zurek contamination.

What carries the argument

The supporting object is the momentum-sector Liouvillian superoperator $\mathcal{L}_q$ and its lowest spectral gap $\Delta\lambda = \lambda_0 - \lambda_{1,+}$. At the limit of loss difference this gap closes, and its dependence on the mode momentum is set by the instantaneous parameter $d_z(t)$: gapless at the critical point, $\Delta\lambda \propto |\Delta_q|^2/\gamma$, and finite away from it, $\Delta\lambda \propto \gamma|\Delta_q|^2/(4d_z^2)$. The calculation uses two approximations: the near-commutativity of Liouvillians at different times for $\Delta_q \sim 0$ (Eq. 16), and a conjectural ansatz for the loss-difference solution (Eq. C4a) in which the correction factor is $e^{-\pi\tau_Q q^2}(e^{\bar u \delta \tau_Q}-1)$. The observable that carries the result is the residual fermion density $N = \frac{1}{N}\sum_q \mathrm{Tr}[\rho_q(c^\dagger_{a,q}c_{a,q}+c^\dagger_{b,q}c_{b,q})]$, which is free of the quantum-jump contribution responsible for anti-Kibble-Zurek behavior.

What would settle it

Solve the full Lindblad equations numerically for a finite Rice-Mele chain at $\delta=\gamma$ with the quench crossing the critical point, and check whether $N \tau_Q^{1/2}$ saturates to a $\gamma$-independent constant for large $\tau_Q$; or directly compute the function $g(q,t_f)$ from Appendix C for parameters outside those of Fig. 5 and check whether it remains $e^{-\pi\tau_Q q^2}$, since any deviation breaks the derivation of Eqs. (30)-(33).

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Extended reading notes

Core claim

The central claim is that the Liouvillian spectral gap closes at the limit of loss difference $|\delta|=\gamma$, and that the way it closes depends on whether the instantaneous Hamiltonian is at its critical point: $\Delta\lambda \propto (4/\gamma)|\Delta_q|^2$ at criticality, versus $\Delta\lambda \propto \gamma |\Delta_q|^2/(4 d_z^2)$ away from it. This dichotomy is the origin of two distinct power laws in the final residual-fermion density $N$: a true Kibble-Zurek law $N \propto \tau_Q^{-\beta}$ for quenches that cross the critical point, where the criticality supplies the impulse stage, and a dissipation-controlled pseudo-Kibble-Zurek law $N \propto [(1/u_f - 1/u_i)\gamma\tau_Q]^{-\beta}$ for quenches that do not. Because $N$ counts the total fermion number and contains no contribution from the quantum-jump terms, it avoids the anti-Kibble-Zurek saturation that masks the standard excitation density, so the universal scaling can be observed by counting leftover particles.

Load-bearing premise

The load-bearing premise is the unproven conjectural ansatz for the loss-difference solution (Eq. C4a), that the correction factor factorizes as $e^{-\pi\tau_Q q^2}(e^{\bar u \delta \tau_Q}-1)$, together with the approximation that Liouvillians at different times nearly commute for small $\Delta_q$; if either fails outside the tested parameter range, the derived scaling laws do not follow.

Editorial extensions

If this is right

For any two-band bipartite model, a quench crossing the critical point at the limit of loss difference should show $N \propto \tau_Q^{-\beta}$ for long quench times, independent of the loss strength once $\bar u \gamma \tau_Q \gg 1$.
A quench staying on one side of the critical point should show $N \propto [(1/u_f - 1/u_i)\gamma \tau_Q]^{-\beta}$, so the apparent exponent can be tuned by changing the dissipation strength $\gamma$.
The two scalings can be made to appear together or separately by choosing the start and end points relative to the critical modes, as demonstrated in the Shockley and Haldane models.
Counting residual fermions (or holes, under gain) provides an experimentally accessible signal that bypasses the $1/2$ saturation produced by anti-Kibble-Zurek behavior.
Dropping the quantum-jump terms reduces the Lindblad equation to an effective non-Hermitian Hamiltonian that reproduces Kibble-Zurek scaling qualitatively but misses the correct prefactor, so the full master equation is needed for quantitative predictions.

Reading between the lines

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

If the conjectural ansatz (Eq. C4a) can be proven, the same technique would give exact finite-quench-time results for other observables, such as two-point correlation functions or entanglement measures, in driven open systems.
The contrast between KZ and pKZ suggests a practical diagnostic: measuring the dependence of the residual-particle scaling on $\gamma$ reveals whether a quench crossed a critical point, since the true KZ exponent is $\gamma$-independent while pKZ is not.
A testable extension is to ramp the dissipation strength itself rather than the on-site energy; the Liouvillian-gap analysis would then predict a scaling crossover as the system approaches the loss-difference limit.
In cold-atom or photonic-lattice setups with engineered loss difference, the predicted crossover from pKZ to KZ as $u_f$ crosses the critical point could be observed by time-resolved counting of remaining atoms.

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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 paper deserves a serious referee, but the referee should push on the analytical derivation. The central claim—that counting residual fermions reveals KZ scaling at the limit of loss difference, immune to the anti-KZ saturation—is plausible and well supported numerically, but it is not proven.

What's actually new: the loss-difference KZ scaling for the residual fermion number N, the pseudo-KZ scaling for quenches that avoid the critical point, and the identification of the quantum jump term as the source of the anti-KZ behavior. The uniform-loss part is exact and clean: it reduces to the same integral equation as the isolated case, so there is no controversy there. The N observable is a nice diagnostic because it sidesteps the 1/2 saturation that buries the usual excitation density.

Soft spots, in order. The loss-difference solution in Appendix C is introduced as a speculation—the text literally says 'we speculate the solution'—then verified numerically at tau_Q = 30. That is legitimate evidence, but the appendix is titled 'Rigorous Solution' and the main text claims Eq. (30) is rigorously demonstrated through Eq. (17). Eq. (17) rests on the Liouvillian commutator being small in Delta_q, with no error estimate; the scaling-relevant modes have |Delta_q| ~ tau_Q^{-1/2}, so the approximation is not controlled. So the proof chain is incomplete. I don't think this sinks the paper—the numerical checks on Rice-Mele, Shockley, and Haldane models are genuine full-Lindblad solutions—but the gap should be stated honestly, and either closed or explicitly left as a conjecture.

Minor: the Haldane section is purely numerical, which is fine. The phrase 'universal KZ scaling' is used loosely; the exponent is the standard one, and the prefactor is model-dependent.

Who should read it: people working on dissipative quench dynamics, KZ physics in open systems, and experimentalists engineering loss on bipartite lattices. It deserves a peer review slot. My main revision request would be to sort out the proof of the ansatz or at least provide an error bound for Eq. (17).

Referee Report

3 major / 4 minor

Summary. The paper studies adiabatic quenches across quantum critical points in two-band fermion systems subject to Lindblad loss, with particular focus on the limit of loss difference (|δ|=γ). For uniform loss, the authors derive an exact solution for the excitation density, showing exponential suppression of Kibble-Zurek scaling and attributing the anti-Kibble-Zurek (AKZ) contribution to the quantum-jump part of the dissipator. For loss difference, they propose that the final residual-fermion number obeys N ∝ τ_Q^{-β} when the quench crosses the critical point, and a dissipation-strength-dependent pseudo-KZ scaling N ∝ [(1/u_f−1/u_i) γ τ_Q]^{-β} when it does not. These claims are illustrated in the Rice-Mele, Shockley, and Haldane models, with analytical formulas checked against numerical solutions of the Lindblad master equation.

Significance. If correct, the paper makes several useful contributions: it provides an exact treatment of uniform loss in this class of driven bipartite systems, cleanly separates the roles of the non-Hermitian and quantum-jump terms in producing AKZ behavior, and proposes a residual-fermion observable that is immune to the AKZ saturation. The predicted KZ and pseudo-KZ scalings at the loss-difference limit are specific and falsifiable, and the numerical checks in the Rice-Mele model collapse convincingly onto the advertised power laws. The scaling exponents are not fitted to the target results but come from Gaussian integrals, so the numerics provide genuine evidence. The main weakness is that the central loss-difference solution is introduced as a conjecture rather than proved, and the Liouvillian approximation on which it is said to rest has no controlled error estimate; therefore the paper is not yet at the standard of rigor claimed in the text.

major comments (3)

[Appendix C, Eqs. (C3)-(C4a)] The central loss-difference result used in Eqs. (28)-(32) rests on the conjectural factorization tilde R_q(t_f) = [tilde R_q(t_f)]_{δ=0} + e^{-π τ_Q q^2}(e^{bar u δ τ_Q} − 1). The main text and appendix introduce this with 'we speculate the solution', yet the section is titled 'Rigorous Solution'. The factorization is verified numerically only for the Rice-Mele model: Fig. 5 uses τ_Q=30 and two values of total loss, and Fig. 1 shows agreement of Eq. (31) with numerics over a range of τ_Q. No derivation from Eq. (A15c) is given, and no argument explains why g(q,t_f) should equal e^{-π τ_Q q^2} independently of model parameters. If this ansatz fails outside the tested Rice-Mele parameters, Eqs. (28)-(32) and the headline N ∝ τ_Q^{-β} at the loss-difference limit collapse. This needs either a proof, a controlled asymptotic estimate, or a clear downgrade of the claim to a numerically supported conjecture.
[Section II.B.1, Eqs. (16)-(17)] Eq. (17) replaces the time-ordered Liouvillian evolution with ρ_q(t) ≈ exp(∫_0^t L_q(t') dt'), justified only by the statement in Eq. (16) that [L_q(t1), L_q(t2)] ∝ Δ_q. No explicit commutator is displayed and no error estimate is given for the truncated Magnus series. The modes responsible for the KZ scaling have |Δ_q| ~ τ_Q^{-1/2}, so the instantaneous smallness of the commutator in Δ_q does not imply that the integrated correction over the full quench is small; in fact the correction can be O(1) in the scaling window. Since Eq. (30) is asserted to be 'rigorously demonstrated through the Liouvillian dynamics' via Eq. (17), this missing estimate is load-bearing for the central claim.
[Section V, Eqs. (41)-(42) and Fig. 4] The Haldane-model formulas are presented by analogy with the one-dimensional Rice-Mele solution, but no derivation from Eq. (A15c) is supplied and the only evidence is the finite-lattice numerics in Fig. 4. In particular, the pKZ prefactor 2/(√3 π γ τ_Q) and the decomposition into the two sets of Dirac points are asserted rather than derived. This does not invalidate the Rice-Mele results, but it weakens the general claim that the two scaling behaviors appear together or separately in the Haldane model.

minor comments (4)

[Appendix C, Eq. (C2)] The stated exact solution tilde R'_0(t) = e^{γ t/2} + e^{δ t/2} − 2 appears inconsistent with Eq. (A15a) and the initial condition tilde R'_0(0)=0; solving A15a with tilde R_0(t)=e^{δ t/2} gives e^{γ t/2} − e^{δ t/2}. The later formulas appear to use the latter form, but the printed equation should be corrected.
[Figure 5 caption] The caption states that the analytical expressions 'match the numerical results rigorously', but the figure covers only a single quench time τ_Q=30 and two values of the total loss. Please soften this wording and state the tested parameter range explicitly.
[Eq. (17)] The notation e^{∫ L_q(t') dt'} is ambiguous for time-dependent noncommuting superoperators; a time-ordering symbol or an explicit Magnus-series expression would clarify the approximation being made.
[Abstract and Section II.B] The abstract and the main text describe the loss-difference solution as an 'analytical solution' without qualification. Given that the solution is introduced as a conjecture and verified numerically, the wording should be adjusted to distinguish the uniform-loss exact solution from the conjectured loss-difference ansatz.

Circularity Check

1 steps flagged · score 2.0 of 10

The central LLD scalings rest on the explicitly conjectural factorization of Appendix C, but the conjecture is benchmarked against untuned numerical Lindblad solutions, so the derivation is not circular; the main caveats are incompleteness and approximation error rather than self-referentiality.

ansatz smuggled in via citation [Section III.A, Eq. (28), and Appendix C, Eqs. (C3)-(C4); main-text statement at Eq. (28) 'It is hard to infer exact solution from Eq. (6).]
"It is hard to infer exact solution from Eq. (6). Nonetheless, the exact solution presented above provides a good starting point for us to conjecture the full solution, which is given by Rq ≈ ... (28). ... In Appendix C: 'We speculate the solution in two steps. ... Next, we consider q ̸= 0 and speculate the solution of ˜Rq(t) as, ˜Rq(t) = [˜Rq(t)]δ=0 + g(q, t)(eδt/2 − 1) ... (C3) ... ˜Rq = [˜R′ q(tf)]δ=0 + e−πτQ q2 (e¯uδτQ − 1) (C4a)'."

This is not circularity in the sense of a result being identical to its input: the Gaussian factor g(q,tf)=e^{-πτQ q^2} is an ansatz, not derived, and the δ-dependent exponential is inserted by hand. However, the headline scaling N ∝ τ_Q^{-β} at LLD is obtained by integrating exactly this conjectured Gaussian form (Eqs. 30-33), so the predicted scaling exponent is contained in the ansatz rather than independently derived. The authors are explicit that they 'conjecture' and 'speculate' the solution, and they verify it numerically against the full Lindblad equations for Rice-Mele parameters, so the burden is reduced. I flag it as an ansatz-within-the-analysis rather than a self-citation smuggling.

full rationale

I checked the derivation chain end to end. The isolated-system KZ result (Eq. 7, Appendix A) is derived by standard iterative solution of the integral-differential equation, giving the Gaussian e^{-πτQ|Δq|^2}; the uniform-loss result (Eq. 8) follows because setting δ=0 leaves the equation formally unchanged, so tilde-R_q equals the lossless R_q. At LLD, the paper does not fit scaling exponents to numerical data: the exponent β=1/2 comes from Gaussian integration of |Δq|∝|q-qc|, i.e. from the known KZM exponent, not from a fit. The pKZ result (Eqs. 26, 33) also emerges from integrating N_q=e^{-f τQ |Δq|^2} with f = arctan(4ui/γ)-arctan(4uf/γ), and in the limit |uf|,|ui| large this gives the quoted ∝[(1/uf-1/ui)γτQ]^{-β} form. No fitted parameter is later relabeled as a prediction, and no uniqueness theorem is imported from the authors' prior work; the self-citations [11,12] are background KZM references and are not load-bearing for the LLD claim. The genuine weakness is that Eqs. (C3)-(C4a) are an unproven factorization of the loss-difference correction: the function g(q,t_f) is posited as e^{-πτQ q^2}, and Eq. (17) is justified only by the small-Δ_q commutator statement (Eq. 16) with no controlled Magnus error estimate for modes |Δ_q| ~ τ_Q^{-1/2}. These are correctness/completeness risks, not circularity: the numerical checks in Fig. 5 are full Lindblad solutions independent of the ansatz and are not tuned to reproduce the scaling law. Because the predicted scaling form is admittedly inserted through the conjectured solution rather than deduced from the equation of motion, I assign a modest score of 2 rather than 0, but this is not circular in the reduction-by-construction sense. No step reduces, by definition or by fitting, to its own output.

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

The paper introduces no fitted free parameters; prefactors such as A=1/(2π) are computed from Gaussian integrals. The scaling claims rest on the Lindblad master equation, the small-Δ_q commutator approximation, and a conjectured loss-difference solution verified numerically. No new particles, forces, or conserved quantities are postulated.

assumptions (4)

domain assumption Lindblad master equation with local loss jump operators L_{a,j}=√γ_a c_{a,j}, L_{b,j}=√γ_b c_{b,j} accurately models the dissipative quench dynamics.
Used in Sec. II, Eq. (1). The entire analysis lives in this Markovian open-quantum-system framework.
domain assumption At the loss-difference limit and for small |Δ_q|, the Liouvillian superoperators at different times approximately commute, [L_q(t1), L_q(t2)] ∝ Δ_q, so time evolution is the ordinary exponential of the integrated Liouvillian.
Eq. (16) and Eq. (17). This is the key enabling approximation for the Gaussian decay form of N_q.
ad hoc to paper The conjectured loss-difference solution, tilde R_q(t) = [tilde R_q(t)]_{δ=0} + g(q,t)(e^{δt/2}-1) with g(q,t_f)=e^{-πτ_Q q^2}, is valid.
Appendix C, Eqs. (C3)-(C4). This ansatz directly produces the KZ and pKZ scaling laws and is verified numerically for the Rice-Mele model but not proven.
domain assumption The initial state is the ground state at half filling far from the critical point, and only modes near the critical mode q_c contribute to the scaling.
Sec. II and Appendix A. This is the standard KZM assumption, used to replace |Δ_q| by a linear function of |q-q_c|.

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Cite this review

Pith. "Pith review of Kibble-Zurek scaling immune to anti-Kibble-Zurek behavior in driven open systems at the limit of loss difference." pith.science (2026). https://pith.science/paper/7E7DOQKH

@misc{pith2026241116406,
  author       = {Pith},
  title        = {Pith review of: Kibble-Zurek scaling immune to anti-Kibble-Zurek behavior in driven open systems at the limit of loss difference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7E7DOQKH}},
  note         = {Machine review of arXiv:2411.16406}
}

read the original abstract

We investigate the dissipative quench dynamics in a family of two-band fermionic systems by linearly ramping the staggered on-site energy. In the Lindblad formalism, we present an analytical solution in the presence of uniform loss or loss difference on bipartite lattices, which tells that dissipation exponentially suppresses the Kibble-Zurek (KZ) scaling behavior and the quantum jump term of the dissipation is responsible for the anti-KZ (AKZ) behavior. Interestingly, we find two different scaling behaviors at the limit of loss difference. Both scaling behaviors arise from the gapless Liouvillian. But one is accompanied by impulse stage rendered by the criticality of the system, so that it is ascribed to the universal KZ scaling law. Another depends on the dissipation strength and there is no impulse stage in it. We also point out a convenient way to observe the two new scaling behaviors by counting the number of residual particles in the end, since it is immune to the influence of AKZ behavior. We illustrate our findings through the prototypical one-dimensional Rice-Mele model first. Then, in the one-dimensional Shockley model and the two-dimensional Haldane model for Chern insulators, we show that the two scaling behaviors can appear together or separately with appropriate quench protocols.

Figures

Figures reproduced from arXiv: 2411.16406 by the authors.

**Figure 3.** FIG. 3. Dissipative quench dynamics of the Shockley model. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** FIG. 4. (a) Phase diagram of the Haldane model and the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5. The functions [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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The solution of this equation is the key to all the 6 nonzero elements

After quench, the final on-site energy parameter meets uf ≪ 0. The solution of this equation is the key to all the 6 nonzero elements. The Lindblad equation can be solved numerically. However, we seek for an analytical solution. Firstly, if there is no dissipation at all, the first two terms in Eq. (6) disappear. So we get a reduced equation that provides...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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The solution of this equation is the key to all the 6 nonzero elements

After quench, the final on-site energy parameter meets uf ≪ 0. The solution of this equation is the key to all the 6 nonzero elements. The Lindblad equation can be solved numerically. However, we seek for an analytical solution. Firstly, if there is no dissipation at all, the first two terms in Eq. (6) disappear. So we get a reduced equation that provides...

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(4), can be reduced to the two- level system with different modes, Hq = ∆qσ+ + ∆∗ qσ− + dz qσz

Liouvillian quench dynamics in the dissipative two-level system The Hamiltonian, Eq. (4), can be reduced to the two- level system with different modes, Hq = ∆qσ+ + ∆∗ qσ− + dz qσz. (10) Then, the dissipative quench dynamics of the two-level system can be entirely specified by looking at the Liou- villian eigenvalue problem [37]. The Lindblad equation can ...

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KZ and pseudo-KZ scaling behaviors at LLD We return to the discussion of the many-body system and assume, for simplicity, that the Hamiltonian has only one critical point. To avoid the influence of AKZ behav- ior, we count the density of residual fermion numbers, N = 1 N X q Nq, (18) where Nq = Na,q + Nb,q = Tr[ρq(c† a,qca,q + c† b,qcb,q)] (19) represents...

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Lindblad Master Equation The two-band model on a bipartite lattice in momen- tum space reads H = X q c† a,q c† b,q dx q σx + dy q σy + dz qσz ca,q cb,q , (A1) where σµ’s represent a Pauli matrix, ca,q and cb,q denote the fermion operators on a and b sublattices respectively. By the canonical Bogoliubov transformation, η1,q = uqca,q + vqcb,q, η 2,q = −v∗ q...

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Exact Solution in the Absence of Loss, γa = γb = 0 Now, we revisit the conventional Kibbke-Zurek mecha- nism (KZM) by the Lindblad formalism for ∀γi = 0. Eq. (A15c) becomes dRq(s) ds = −4|∆q|2τQ Z s si Rq(s′) cos s2 − s′2 ds′ (A16) where s = [t − dz q(ti)τQ]/√τQ, si = −√τQdz q(ti), and the initial condition Rq(0) = 1. According to KZM, only the modes near...

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Exact Solution in the Presence of Uniform Loss, γa = γb ̸= 0 For the situation of uniform loss, γa = γb ̸= 0, the equation in Eq. (A15c) can be simplified to d ˜Rq(s) ds = −4|∆q|2τQ R s si ˜Rq(s′) cos s2 − s′2 ds′, (A25) where s = [ t − dz q(ti)τQ]/√τQ and si = −dz q(ti)√τQ, which has the same form as that in Eq. (A16) and shares the same initial conditio...

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Sachdev, Quantum Phase Transitions, 2nd ed(Cam- bridge University Press, 2011.)

S. Sachdev, Quantum Phase Transitions, 2nd ed(Cam- bridge University Press, 2011.)

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1976