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REVIEW 2 major objections 4 minor 68 references

Entanglement transition in unitary system-bath dynamics

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A unitary, unmonitored system-bath evolution exhibits an entanglement phase transition from logarithmic to area-law scaling as system-bath coupling increases, with the transition visible only in bath-bath correlations.

desk verdict A genuinely new unitary system-bath route to an entanglement transition, but the 'genuine phase transition' claim needs a more careful finite-size scaling before I'd buy it. read the letter →

arxiv 2512.06081 v3 pith:X4MZTWGI submitted 2025-12-05 quant-ph cond-mat.quant-gascond-mat.stat-mech

Bo Xing , Giuliano Chiriacò , Paola Cappellaro , Rosario Fazio , Dario Poletti This is my paper
classification quant-phcond-mat.quant-gascond-mat.stat-mech
keywords entanglementphasetransitionunitarysystem-bathdynamicsfreefermionslogarithmicnegativityarealawscalingmeasurement-inducedbath-bathcorrelationsfinite-size
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 an entanglement phase transition—previously seen only when a dissipative system is monitored along quantum trajectories—can appear in a single deterministic unitary evolution of a system together with its baths. It shows, for a 2D lattice of free fermions coupled at every site to a fermionic bath, that the steady-state bipartite entanglement of the combined system-bath state changes from logarithmic-law scaling to area-law scaling as the system-bath coupling γ is increased. The transition is witnessed by the logarithmic fermionic negativity, mutual information, and a connected correlation weight, and it occurs while the reduced state of the system remains a trivial infinite-temperature state, so the signature lives entirely in bath-bath correlations. Finite-size scaling of the correlation weight places the critical point at γ_c = (0.13 ± 0.01)J with a correlation-length exponent ν ≈ 1.26. This matters because it suggests entanglement transitions can be studied without post-selection, using a single unitary evolution.

What carries the argument

The central object is the full correlation matrix C0 of the system and its baths, which evolves unitarily under H_tot = H_S + H_B + H_SB. From it, the paper defines three bipartite information quantifiers—logarithmic fermionic negativity E (a fermionic analogue of logarithmic negativity for mixed Gaussian states), mutual information I, and the connected correlation weight C = Σ |C_LR|², the sum of squared correlations between left and right partitions. The connected correlation weight is the key diagnostic because it is numerically cheap and supports the finite-size scaling ansatz C(γ,x) = c(γ,L) x ln x + b(γ,L) x; the transition is located by the crossing where c and b become size-independe

What would settle it

Compile high-precision data for L up to 100 (e.g., using optimized free-fermion codes) and test whether c(γ,L) and b(γ,L) become size-independent at a single γ_c; if the crossing point shifts with L or the collapse degrades, the apparent transition is a crossover. Alternatively, measure the bath-bath correlation weight in a small quantum simulator as a function of γ and L, and check whether the size-invariance point matches γ_c ≈ 0.13 J.

Watch

Extended reading notes

Core claim

For a 2D free-fermion lattice (N = L × L sites) coupled to M = 100 local fermionic bath modes per site and evolving under the full unitary Hamiltonian H_tot, the steady-state bipartite entanglement between left and right halves of the total system-bath setup scales as L log L at weak coupling and as L at strong coupling. The logarithmic fermionic negativity, mutual information, and connected correlation weight all show this crossover, with the connected correlation weight enabling finite-size scaling up to L = 30. Fitting C(γ,x) = c(γ,L) x ln x + b(γ,L) x, the coefficients become size-independent at γ_c = (0.13 ± 0.01)J, giving a data collapse with ν = 1.26 ± 0.48 and ζ = 0.00 ± 0.01. The ba

Load-bearing premise

The identification of a genuine phase transition rests on the finite-size scaling ansatz C(γ,x) = c(γ,L) x ln x + b(γ,L) x and the assumption that the coefficients become size-independent at the critical point; with sizes only up to L = 30 and no reported error bars on the scaling data, the logarithmic phase could instead be a finite-size crossover.

Editorial extensions

If this is right

  • Entanglement transitions can be probed without measurement or post-selection: a single unitary evolution of the system-bath setup suffices, with the transition visible in bath-bath correlations.
  • Because the reduced system state stays at infinite temperature, any experimental probe of the transition must access bath observables, such as the connected correlation weight.
  • The connected correlation weight provides a scalable witness for the transition, requiring only correlation measurements rather than full state tomography.
  • The critical coupling in the unitary setup differs from the trajectory-unraveled GKSL value, implying that the master equation misses system-bath correlations that the unitary description retains.
  • The setup is compatible with quantum simulators, where a deterministic evolution avoids the post-selection bottleneck that limits measurement-induced-transition experiments.

Reading between the lines

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

  • The transition may be a generic feature of coherent system-bath hybridizations: any bath that can store correlations while the system thermalizes could show an analogous entanglement transition with a tuning parameter like γ.
  • The critical point γ_c ≈ 0.13 J may coincide with a localization or delocalization threshold for the effective single-particle spectrum of the system-bath Hamiltonian; examining eigenstate entanglement across γ could give an independent check.
  • An experiment on a programmable fermionic simulator could directly measure the connected correlation weight at fixed γ and increasing L, testing the predicted size invariance at γ_c without needing entropy measurements.
  • If the mechanism extends to interacting systems, the same transition might appear in the mutual information or coherence of bath degrees of freedom, which are easier to measure than negativity.
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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

2 major / 4 minor

Summary. The paper studies a 2D free-fermion lattice coupled to local fermionic baths. In the GKSL trajectory-unraveled description, it reports an entanglement transition as the system–bath coupling γ is varied. The main new claim is that the same transition appears in a single deterministic unitary evolution of the full system–bath Hamiltonian: the steady-state bipartite logarithmic fermionic negativity, mutual information, and connected correlation weight switch from ∼L log L to ∼L scaling as γ increases, with a critical point γ_c=(0.13±0.01)J inferred from finite-size scaling of the connected correlation weight, while the reduced system density matrix remains featureless. The transition is witnessed by bath–bath correlations, not by system observables.

Significance. If correct, the result is conceptually significant: it shows that entanglement-scaling transitions need not rely on postselected monitored trajectories, and can be observed in a deterministic unitary evolution with the bath degrees of freedom retaining the relevant information. This is experimentally relevant because it avoids postselection. The paper also makes good use of free-fermion techniques, computes three independent-looking observables (E, I, C), validates that the system reduced state matches the GKSL prediction, and honestly notes the discrepancy between the unitary and trajectory critical couplings. The main weakness is that the central phase-transition identification rests on a single fitting ansatz at modest system sizes and without reported error bars.

major comments (2)
  1. [§4, Fig. 4(b,c)] The central claim rests on fitting C(γ,x)=c(γ,L)x ln x + b(γ,L)x for x∈[8,L] with L≤30 and on the crossing of c(γ,L). Because no error bars are reported, because the range in L is only a factor ~3–4, and because the text states c(γ,L) still grows with L for γ<0.14J, the asymptotic log-law coefficient is inferred, not directly observed. The γ→0 limit is exactly decoupled, so apparent L log L could be a finite-size crossover to area law. Consistency among E, I, C does not remove this ambiguity: all three are derived from the same correlation matrix, and E and I are shown at only one log-phase γ value. Please add bootstrap/jackknife uncertainties for c and b, compare competing fits (C∼a L^p, C∼L(log L)^κ) with a goodness-of-fit metric, extrapolate c(γ,L) for γ<γ_c, and test sensitivity of the crossing to the fit range x∈[8,L].
  2. [Eqs. (3)–(5) and Fig. S1] The unitary model is validated only through C_SS matching C_ME at t_s. The transition is carried by bath–bath correlations C_BB, and no convergence with M or ω_max is shown for E, I, or C. The statement 'Increasing M further does not quantitatively change the results' is not supported by displayed data. Please show at least a representative convergence check (e.g., M=50,100,200 and/or ω_max=5,10,20 J) for the three observables at a few γ values.
minor comments (4)
  1. [§2, Fig. 2] The trajectory-unraveling section claims an MIPT, but the evidence is limited to S and E at two small-γ values (0.1J, 0.5J) and area-law behavior at γ≥2.0J, with no intermediate couplings and no finite-size collapse. If this section is intended only as motivation, please state so explicitly; if it is part of the paper's claims, additional data are needed.
  2. [Main text, first use of E(C)] The logarithmic fermionic negativity E(C) is used in the main text before being defined; the definition is relegated to the SM. Please define it in the main text or refer explicitly to Eq. (S12).
  3. [Eq. (8)–(9)] Please define N_pairs(d) and clarify the distance convention in the figure/equation; 'city block distance' is mentioned in the text but not in the figure caption.
  4. [Throughout] There is a typo: 'betwen' should be 'between' in the sentence introducing the connected correlation weight. Also, the conclusion states that 'all three quantities exhibit a transition from logarithmic-law to area-law', but for E and I only one log-phase coupling value is shown; this should be softened or supported.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the unitary system-bath transition is inferred from independent numerical scaling and finite-size collapse, without fitting the target conclusion into the inputs.

full rationale

The paper's central claim is an entanglement-scaling transition in a unitary system-bath model, supported by three independent quantities (logarithmic fermionic negativity E, mutual information I, and connected correlation weight C) computed from direct unitary evolution of the total correlation matrix. The critical point gamma_c = (0.13 +/- 0.01)J is obtained by finite-size scaling of the fitted coefficient c(gamma, L) from the ansatz C(gamma,x) = c(gamma,L) x ln(x) + b(gamma,L) x, a standard FSS procedure (pyfssa) rather than a quantity that is defined to equal the desired conclusion. The unitary model's system reduced dynamics is explicitly engineered to reproduce the GKSL master equation via the spectral density V(omega_m) = sqrt(gamma omega_m M omega_max/(pi h_s)), but this construction constrains only C_SS, not the bath-bath correlations in which the transition is witnessed; the paper shows quantitatively that C_SS approaches the GKSL steady state while the transition appears in bath-bath correlations (Supplemental Fig. S3). The citations to the authors' prior work are used as methodological tools (e.g., trajectory unraveling, FSS for monitored free fermions) and are not load-bearing as unverified uniqueness or ansatz-justifying claims. The skeptical concern that L <= 30 and the singular gamma=0 limit may mimic L log L behavior is a finite-size/correctness ambiguity, not a circular reduction of the derivation. No step reduces, by the paper's own equations or by self-citation, to its own inputs.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

No new particles, forces, or entities are introduced; the bath modes are a standard discretized environment model. The central claim rests on Gaussian free-fermion simulation, the chosen spectral density mapping to GKSL, and the FSS ansatz for the correlation weight.

free parameters (6)
  • γ_c = 0.13 ± 0.01 J
    Critical system-bath coupling determined by finite-size scaling collapse of c(γ, L); not derived from first principles.
  • ν = 1.26 ± 0.48
    Correlation-length exponent obtained from the FSS collapse in Fig. 4(c).
  • ζ = 0.00 ± 0.01
    Exponent governing the scaling of c(γ, L) at criticality, obtained from the same collapse.
  • h_s = 5J
    On-site energy hand-chosen; enters the spectral density V(ω_m) and therefore the mapping to the GKSL description.
  • ω_max = 10J
    Bath bandwidth hand-chosen; the authors state increasing M does not change results, but the value sets the Markovian approximation.
  • M = 100
    Number of bath modes per site, hand-chosen; convergence checked.
assumptions (4)
  • standard math Gaussian state / Wick's theorem validity
    The entire simulation uses correlation matrices; this is valid because H_tot is quadratic and the initial states (checkerboard system, infinite-temperature baths) are Gaussian. Wick's theorem is invoked for Eq. (8) (footnote [60]).
  • domain assumption Born–Markov–secular mapping to GKSL
    The unitary system-bath model with the chosen V(ω_m) is assumed to reproduce the GKSL master-equation dynamics after tracing the baths; stated around Eq. (5), with ref [55] as support.
  • domain assumption Finite-size scaling ansatz
    The transition analysis assumes C(γ,x)=c x ln x + b x and c(γ,L)=L^{ζ/ν} f(L^{1/ν}(γ−γ_c)); this is the standard FSS hypothesis, not derived in the paper.
  • domain assumption Infinite-temperature bath initial condition
    Baths are initialized in an infinite-temperature mixed state (with random pure-state initializations tested in the supplement); the central claim is framed for this class of initial states.

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Cite this review

Pith. "Pith review of Entanglement transition in unitary system-bath dynamics." pith.science (2026). https://pith.science/paper/X4MZTWGI

@misc{pith2026251206081,
  author       = {Pith},
  title        = {Pith review of: Entanglement transition in unitary system-bath dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4MZTWGI}},
  note         = {Machine review of arXiv:2512.06081}
}
read the original abstract

The evolution of a system coupled to baths is commonly described by a master equation that, in the long-time limit, yields a steady-state density matrix. However, when the same evolution is unraveled into quantum trajectories, it is possible to observe a transition in the scaling of entanglement within the system as the system-bath coupling increases - a phenomenon that is invisible in the trajectory-averaged reduced density matrix of the system. Here, we go beyond the paradigm of trajectories from master equations and explore whether a qualitatively analogous entanglement-scaling transition emerges in a single unitary evolution of the combined system-bath setup, without monitoring the dynamics of the system. We investigate the scaling of entanglement in a unitary quantum setup composed of a two-dimensional lattice of free fermions, where each site is coupled to a fermionic bath. As the system-bath coupling increases, the logarithmic fermionic negativity reveals an entanglement transition from logarithmic-law to area-law scaling. This occurs while the system's steady-state properties are trivial, highlighting that the signatures of these different scalings are within the bath-bath correlations. Evidence of the transition is also found in the mutual information and the correlations of the full system-bath setup, suggesting that the entanglement transition is underpinned by a change in the spatial structure of quantum information.

Figures

Figures reproduced from arXiv: 2512.06081 by the authors.

Figure 1
Figure 1. (a) A 2D lattice of free fermions with open bound￾ary conditions is coupled to external baths. After tracing out the baths, the dissipative dynamics of the system can be ap￾proximated by the GKSL master equation. For further ana￾lysis, we partition the system into the left and right partitions, each consisting of N/2 system sites. (b) We can also consider a unitary evolution, in which the environment is modeled as M… view at source ↗
Figure 2
Figure 2. The trajectory-averaged steady-state (a) bipartite [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The unitary-evolved steady-state (a) bipartite log [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (a) The steady-state density-density correlation as [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Works this paper leans on

68 extracted references · 1 canonical work pages

  1. [1]

    Lindblad, Communications in mathematical physics 48, 119 (1976)

    G. Lindblad, Communications in mathematical physics 48, 119 (1976)

    1976

  2. [2]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics17, 821 (1976)

    1976

  3. [3]

    Manzano, AIP Advances10, 025106 (2020)

    D. Manzano, AIP Advances10, 025106 (2020)

    2020

  4. [4]

    Stefanini, A

    M. Stefanini, A. A. Ziolkowska, D. Budker, U. Poschinger, F. Schmidt-Kaler, A. Browaeys, A. Im- amoglu, D. Chang, and J. Marino, arXiv.2506.22436 10.48550/arXiv.2506.22436 (2025)

    arXiv doi:10.48550/arxiv.2506.22436 2025

  5. [5]

    A. J. Daley, Advances in Physics63, 77 (2014)

    2014

  6. [6]

    Fazio, J

    R. Fazio, J. Keeling, L. Mazza, and M. Schirò, SciPost Phys. Lect. Notes , 99 (2025)

    2025

  7. [7]

    Dalibard, Y

    J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992)

    1992

  8. [8]

    Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B98, 205136 (2018)

    2018

Show all 68 references
  1. [9]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X9, 031009 (2019)

    2019

  2. [10]

    Szyniszewski, A

    M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. B100, 064204 (2019)

    2019

  3. [11]

    Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B100, 134306 (2019)

    2019

  4. [12]

    Szyniszewski, A

    M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. Lett.125, 210602 (2020)

    2020

  5. [13]

    A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, inEntanglement in Spin Chains: From Theory to Quantum Technology Applic- ations, edited by A. Bayat, S. Bose, and H. Johan- nesson (Springer International Publishing, Cham, 2022) pp. 211–249

    2022

  6. [14]

    O. Lunt, J. Richter, and A. Pal, Quantum simulation using noisy unitary circuits and measurements, inEn- tanglement in Spin Chains: From Theory to Quantum Technology Applications, edited by A. Bayat, S. Bose, and H. Johannesson (Springer International Publishing, Cham, 2022) p...

    2022

  7. [15]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

    2023

  8. [16]

    S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Phys. Rev. Lett.125, 030505 (2020)

    2020

  9. [17]

    Fidkowski, J

    L. Fidkowski, J. Haah, and M. B. Hastings, Quantum5, 382 (2021)

    2021

  10. [18]

    X. Cao, A. Tilloy, and A. D. Luca, SciPost Phys.7, 024 (2019)

    2019

  11. [19]

    Y. Bao, S. Choi, and E. Altman, Phys. Rev. B101, 104301 (2020)

    2020

  12. [20]

    M. J. Gullans and D. A. Huse, Phys. Rev. Lett.125, 070606 (2020)

    2020

  13. [21]

    Fuji and Y

    Y. Fuji and Y. Ashida, Phys. Rev. B102, 054302 (2020)

    2020

  14. [22]

    M. J. Gullans and D. A. Huse, Phys. Rev. X10, 041020 (2020)

    2020

  15. [23]

    Alberton, M

    O. Alberton, M. Buchhold, and S. Diehl, Phys. Rev. Lett. 126, 170602 (2021)

    2021

  16. [24]

    Y. Bao, S. Choi, and E. Altman, Annals of Physics435, 168618 (2021), special issue on Philip W. Anderson

    2021

  17. [25]

    Turkeshi, A

    X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schiró, Phys. Rev. B103, 224210 (2021)

    2021

  18. [26]

    Sierant, G

    P. Sierant, G. Chiriacò, F. M. Surace, S. Sharma, X. Turkeshi, M. Dalmonte, R. Fazio, and G. Pagano, 6 Quantum6, 638 (2022)

    2022

  19. [27]

    Coppola, E

    M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Phys. Rev. B105, 094303 (2022)

    2022

  20. [28]

    Y. L. Gal, X. Turkeshi, and M. Schirò, SciPost Phys.14, 138 (2023)

    2023

  21. [29]

    Granet, C

    E. Granet, C. Zhang, and H. Dreyer, Phys. Rev. Lett. 130, 230401 (2023)

    2023

  22. [30]

    M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Phys. Rev. X13, 041045 (2023)

    2023

  23. [31]

    H. Lóio, A. De Luca, J. De Nardis, and X. Turkeshi, Phys. Rev. B108, L020306 (2023)

    2023

  24. [32]

    B. Xing, X. Turkeshi, M. Schiró, R. Fazio, and D. Poletti, Phys. Rev. B109, L060302 (2024)

    2024

  25. [33]

    Poboiko, P

    I. Poboiko, P. Pöpperl, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B111, 024204 (2025)

    2025

  26. [34]

    C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse,et al., Nature Physics18, 760 (2022)

    2022

  27. [35]

    Nature622, 481 (2023)

    2023

  28. [36]

    J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Nature Physics19, 1314 (2023)

    2023

  29. [37]

    Turkeshi, R

    X. Turkeshi, R. Fazio, and M. Dalmonte, Phys. Rev. B 102, 014315 (2020)

    2020

  30. [38]

    O. Lunt, M. Szyniszewski, and A. Pal, Phys. Rev. B104, 155111 (2021)

    2021

  31. [39]

    C.-M. Jian, B. Bauer, A. Keselman, and A. W. W. Lud- wig, Phys. Rev. B106, 134206 (2022)

    2022

  32. [40]

    H. Liu, T. Zhou, and X. Chen, Phys. Rev. B106, 144311 (2022)

    2022

  33. [41]

    Sierant, M

    P. Sierant, M. Schirò, M. Lewenstein, and X. Turkeshi, Phys. Rev. B106, 214316 (2022)

    2022

  34. [42]

    Poboiko, I

    I. Poboiko, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. Lett.132, 110403 (2024)

    2024

  35. [43]

    Chiriacò, M

    G. Chiriacò, M. Tsitsishvili, D. Poletti, R. Fazio, and M. Dalmonte, Phys. Rev. B108, 075151 (2023)

    2023

  36. [44]

    Core7, 011 (2024)

    M.Tsitsishvili, D.Poletti, M.Dalmonte,andG.Chiriacò, SciPost Phys. Core7, 011 (2024)

    2024

  37. [45]

    Muzzi, M

    C. Muzzi, M. Tsitsishvili, and G. Chiriacò, Phys. Rev. B 111, 014312 (2025)

    2025

  38. [46]

    S. P. Kelly and J. Marino, Phys. Rev. A111, L010402 (2025)

    2025

  39. [47]

    S. P. Kelly and J. Marino, Phys. Rev. A111, 012425 (2025)

    2025

  40. [48]

    Gardiner and P

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

    2004

  41. [49]

    H. M. Wiseman and G. J. Milburn,Quantum Measure- ment and Control(Cambridge University Press, Cam- bridge, England, 2009)

    2009

  42. [50]

    K.Jacobs,Quantum Measurement Theory and its Applic- ations(Cambridge University Press, Cambridge, Eng- land, 2014)

    2014

  43. [51]

    Breuer, F

    H.-P. Breuer, F. Petruccione,et al.,The theory of open quantum systems(Oxford University Press on Demand, 2002)

    2002

  44. [52]

    See supplemental material

  45. [53]

    Audenaert, M

    K. Audenaert, M. B. Plenio, and J. Eisert, Phys. Rev. Lett.90, 027901 (2003)

    2003

  46. [54]

    M. B. Plenio, Phys. Rev. Lett.95, 090503 (2005)

    2005

  47. [55]

    de Vega, U

    I. de Vega, U. Schollwöck, and F. A. Wolf, Phys. Rev. B 92, 155126 (2015)

    2015

  48. [56]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of statistical mech- anics: theory and experiment2004, P06002 (2004)

    2004

  49. [57]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys.80, 517 (2008)

    2008

  50. [58]

    M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Physical review letters100, 070502 (2008)

    2008

  51. [59]

    Lepori, S

    L. Lepori, S. Paganelli, F. Franchini, and A. Trombettoni, Phys. Rev. Res.4, 033212 (2022)

    2022

  52. [60]

    The simplification of Eq

    Forn=n ′, we neglect contributions fromm=m ′ to exclude trivial self-correlations of the bath modes. The simplification of Eq. (8) is done using Wick’s theorem

  53. [61]

    M. E. Fisher and M. N. Barber, Phys. Rev. Lett.28, 1516 (1972)

    1972

  54. [62]

    Campostrini, A

    M. Campostrini, A. Pelissetto, and E. Vicari, Phys. Rev. B89, 094516 (2014)

    2014

  55. [63]

    Chahine and M

    K. Chahine and M. Buchhold, Phys. Rev. B110, 054313 (2024)

    2024

  56. [64]

    Melchert, arXiv preprint arXiv:0910.5403 10.48550/arXiv.0910.5403 (2009)

    O. Melchert, arXiv preprint arXiv:0910.5403 10.48550/arXiv.0910.5403 (2009)

    2009 doi arXiv

  57. [65]

    Sorge, Zenodo 10.5281/zenodo.592722 (2015)

    A. Sorge, Zenodo 10.5281/zenodo.592722 (2015)

    2015 doi

  58. [66]

    X. Xu, C. Guo, and D. Poletti, Phys. Rev. A105, L040203 (2022)

    2022

  59. [67]

    X. Xu, C. Guo, and D. Poletti, Phys. Rev. A107, 022220 (2023)

    2023

  60. [68]

    Zhang, Y

    P. Zhang, Y. Gao, X. Xu, N. Wang, H. Dong, C. Guo, J. Deng, X. Zhang, J. Chen, S. Xu,et al., Nature Com- munications15, 10115 (2024). [69]https://www.nscc.sg/. 7 Supplemental Material UNRA VELING PROTOCOL Our system with Hamiltonian HS =− X ⟨n,n′⟩ J(a † nan′ +h.c.) + X n hsa† ...

    2024

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.