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Robustness of topological edge states in alternating spin chains against environment

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Pith's one-line read Dimerized chains keep topological edge-state signatures longer than Haldane-like chains when coupled to a surface.

desk verdict Careful Lindblad study with a solid steady-state result, but the headline SSH-vs-Haldane ordering rests on a heuristic threshold the authors themselves call debatable. read the letter →

arxiv 2505.22420 v2 pith:GSJF5W34 submitted 2025-05-28 cond-mat.str-el

Alexander Sattler , Maria Daghofer This is my paper
classification cond-mat.str-el
keywords topologicaledgestatesopenquantumsystemsLindbladmasterequationalternatingHeisenbergchainSu-Schrieffer-HeegermodelHaldanephaseground-statedegeneracyZenoeffect
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 topological edge states of spin chains survive when the chain is placed on a surface and coupled to it as an environment—the setting of recent experiments that assemble such chains atom by atom on substrates. Using a Lindblad master equation for an alternating spin-1/2 Heisenberg chain with local dephasing and spin-flip dissipation, the authors track how long the quasi-fourfold ground-state degeneracy, the edge magnetization, and spin correlations persist for chains of 4 to 12 spins. The central finding is an ordering: signatures of topological edge states are more robust in dimerized, Su-Schrieffer-Heeger-like chains than in Haldane-like chains, for both dissipation types and across coupling strengths. In the steady state every topological signature vanishes, so the protection is transient; the paper also identifies which observables distinguish topological from trivial chains and which do not.

What carries the argument

The load-bearing diagnostic is the ratio $\Theta_{\mathrm{PT}}(t)$ (Eq. 4), a time-dependent generalization of the authors' closed-chain gap criterion: the maximum of the three lowest energy gaps above the ground state divided by the gap $\Delta E_{4\text{--}5}$ that separates the quasi-fourfold ground-state manifold from excited states, with a flag that sets the ratio to 1 if level crossings destroy the four-state ordering. Values below the threshold $\Theta_{\mathrm{PT}}^{\mathrm{tran}}=0.5$ mark parameters where observing edge states is judged viable. The second essential element is the Lindblad master equation (Eq. 3), solved exactly by vectorizing the density matrix into a superoperator eigenvalue problem via the Choi–Jamiolkowski isomorphism, with jump operators $L_i=S_i^z$ (dephasing) and $L_i=S_i^x$ (spin flips) summed over all sites.

What would settle it

Measure time-resolved edge spin correlations on an on-surface alternating spin-1/2 chain, comparing a dimerized SSH-like coupling ($|J_1| < 1$) with a Haldane-like coupling ($J_1 < -1$) under identical surface-induced dissipation: if the Haldane-like chain retains edge correlations with aligned spin and jump operators for as long as—or longer than—the SSH-like chain, the claimed ordering fails. A cheaper check is to recompute the $\Theta_{\mathrm{PT}}$ diagrams with thresholds of 0.25 and 0.75 and across several times $t$; the SSH-like region must remain the larger one for the paper's conclusion to hold.

Watch

Extended reading notes

Core claim

For an alternating spin-1/2 Heisenberg chain with open boundary conditions, coupled to a Markovian environment through local jump operators—dephasing ($L_i = S_i^z$) or spin flips ($L_i = S_i^x$)—the lifetime of topological edge-state signatures is longer when the intra-dimer coupling is antiferromagnetic and the chain resembles an interacting Su-Schrieffer-Heeger model ($|J_1| < 1$) than when it resembles a Haldane spin-1 chain ($J_1 < -1$). The authors establish this by exact diagonalization of the Lindblad master equation for chains up to 12 spins, using a time-dependent energy-gap ratio $\Theta_{\mathrm{PT}}(t)$ that tracks the persistence of the quasi-fourfold degenerate ground-state manifold. They further show that in the steady state all topological signatures vanish; that edge magnetization under spin-flip dissipation decays approximately as a single spin's $e^{-\gamma t/2}$ regardless of topology; and that correlation functions only discriminate the two chain types when the spin operator and the jump operator point in the same direction.

Load-bearing premise

The comparison rests on the assumption that the gap ratio $\Theta_{\mathrm{PT}}$ with threshold 0.5 correctly tracks when topological edge states would actually be observable in an experiment; if that heuristic misjudges observability during time evolution, the claimed ordering of SSH-like over Haldane-like chains does not follow from the $\Theta_{\mathrm{PT}}$ diagrams alone, and the correlation and magnetization data give only partial independent support.

Editorial extensions

If this is right

  • For assembling spin chains on surfaces, dimerized SSH-like chains are the more forgiving platform: they keep a near-fourfold degenerate ground state and matching spin correlations for longer times under Markovian dissipation.
  • Topological edge states in these chains are transient: in the steady state all topological signatures vanish, so any application that relies on the edge states must operate on timescales short compared with the dissipation-induced decay.
  • The quantum Zeno effect can prolong some signatures at strong dissipation, with minimum robustness around $\gamma \sim J_{\mathrm{NN}}$; this behavior already appears for two coupled spins and is not a topological property.
  • Observables differ sharply in diagnostic power: edge magnetization under $L_x$ and the entropy, fidelity, and purity do not distinguish topological from trivial chains, while the gap ratio and correlation functions with aligned spin and jump orientation do.
  • Robustness generally increases with chain length $N$, so the ordering found for $N = 4$ to $12$ should persist—and strengthen—for the longer chains achievable in experiments.

Reading between the lines

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

  • The robustness ordering probably tracks the size of the protecting energy gap (dimer gap versus Haldane gap) more than topology itself: the paper finds that topologically trivial chains with $J_1 = 1$ also decay quickly, and that $L_x$ dynamics is universal. A testable extension would be to compare two chains with identical gap sizes but different topological classification.
  • The $\Theta_{\mathrm{PT}}$ lifetime diagnostic could be exported to other one-dimensional topological wires, such as Kitaev chains, as a practical pre-experiment screen for dissipation tolerance, even though it is explicitly not a topological phase criterion.
  • Because the paper treats Markovian, local, site-independent dissipation, a natural open question is whether non-Markovian or spatially correlated surface noise changes the ordering; the Lindblad framework used here cannot capture memory effects.
  • The finding that entropy, fidelity, and purity evolve nearly identically for topological and trivial chains suggests that bulk decoherence, not edge physics, dominates these quantities; confirming this with bulk-only observables would separate the two contributions.
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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

3 major / 5 minor

Summary. The paper studies alternating spin-1/2 Heisenberg chains with nearest-neighbor and next-nearest-neighbor couplings, focusing on SSH-like (dimerized) and Haldane-like parameter regimes, and examines the robustness of their topological edge states under Markovian dissipation described by a Lindblad master equation with S^z (dephasing) and S^x (noise) jump operators. Using exact diagonalization of the Lindblad superoperator for short chains (N=4 to 12), the authors track the time evolution of the quasi-fourfold ground-state degeneracy via the Θ_PT criterion, edge magnetization, spin-spin correlations, entropy, fidelity, purity, and the spectral gap. The central claim, stated in the abstract and Section IV, is that signatures of topological edge states are generally more robust in SSH-like chains than in Haldane-like chains under the investigated dissipative dynamics, while the steady state is topologically trivial. The paper also reports quantum Zeno effects, the absence of topological protection in several observables, and the universality of a fitted decay form for entropy/fidelity/purity.

Significance. If the central claim holds, the paper provides a useful extension of the authors' earlier closed-chain analysis to open systems, with direct relevance to recent experiments assembling spin chains on surfaces. The work is commendable for investigating several complementary observables, for comparing the full chain dynamics with analytically solvable single- and two-spin limits (Appendix B), and for clearly identifying observables that do not distinguish topological from trivial chains. The finding that the steady state is topologically trivial, and that L^x-driven edge magnetization decays like a single spin, is an important negative result. However, the main quantitative support for the SSH-versus-Haldane ordering comes from the Θ_PT diagnostic, which is a heuristic from the authors' previous work and whose open-system validity is not yet established. The paper would be significantly strengthened by validating Θ_PT against direct edge-state observables and by documenting numerical convergence.

major comments (3)
  1. [Section III A, Eq. (4)] The headline ordering between SSH-like and Haldane-like chains is carried by the Θ_PT(t) criterion with the threshold Θ_tran_PT=0.5 imported from the authors' closed-chain work Ref. [86]. The text itself calls the threshold 'debatable', and the open-system evaluation retains only the 70 lowest states without a documented convergence check in the number of retained states or in the expansion order and time step of Eq. (A2b). Because the definition of Θ_PT(t) must handle time-dependent level crossings, a miscalibrated or underconverged Θ_PT would remove the main evidence for the central claim in Figs. 1-4. Please add a convergence study (e.g., retaining 100 and 150 states) and compare the Θ_PT-based lifetimes with a direct edge-state survival measure, such as the weight of the state on the edge sites or the decay of |C(t,L^β,S^α_r)| for α=β, for representative parameter sets.
  2. [Section III A, Figs. 1(b) and 3(b)] The L_z results in Figs. 1(b) and 3 are obtained with an ad hoc z-anisotropy of Δ_z=1.001 instead of the isotropic value Δ_z=1, introduced to remove 'rather chaotic behavior' that 'might be due to numerics or finite-size effects'. Since the parameter scan is performed at a slightly different Hamiltonian, these figures do not directly demonstrate robustness of the isotropic model under L_z dissipation. Please either demonstrate convergence of the Δ_z=1 calculation by varying the numerical parameters (Δt, m_max, number of retained states) or present the Δ_z=1.001 results explicitly as an exploratory check with a clear caveat that the isotropic L_z case is not fully resolved. Without this, the L_z columns of Figs. 1-3 should not be used as quantitative support for the SSH-versus-Haldane ordering.
  3. [Sections III B and IV] The paper's abstract and Section IV state a general robustness ordering, but the independent support from other observables is partial: under L_x, the edge magnetization follows the single-spin exponential decay e^{-γt/2} for both topological and trivial chains (Eq. (9)), and the entropy, fidelity, and purity show no topological distinction (Section III D). The correlation function in Section III C distinguishes the scenarios only for α=β and for the singlet initial state. Thus the only parameter-resolved quantitative diagnostic supporting the ordering remains Θ_PT. Please make the scope of the claim explicit in the abstract and conclusions: the ordering is established by the ground-state-degeneracy diagnostic, with supporting evidence from α=β correlations and L_z edge magnetization, and not by a generic measure of topological protection. This is a framing issue, but it is load-bearing because the current wording overstates the breadth of the evidence.
minor comments (5)
  1. [Introduction] The phrase 'expensively studied example' should be 'extensively studied example'.
  2. [Title page] The author affiliation contains a LaTeX artifact 'f¨ ur' that should be rendered as 'für'.
  3. [Eq. (4)] In the definition of Θ_PT(t), the condition for setting Θ_PT=1 is written compactly; please specify explicitly the index ranges for the max and min operations in the displayed formula to avoid ambiguity about which energy gaps are compared.
  4. [Figure 5 and Figure 6 captions] The legend entries such as '(a,c)' and '(b,d)' are confusing because they refer to panels in other subfigures; please spell out which panels share which parameters.
  5. [Appendix A] The iterative scheme with m_max=10 and Δt=0.1 is stated as a 'good parameter selection', but no convergence check or error estimate is reported; a brief statement on how these values were validated for the longest chains would be helpful.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the SSH-versus-Haldane robustness ordering is a new open-system result supported by independent diagnostics; only the Theta_PT threshold is a minor self-citation.

full rationale

The paper's central claim is that signatures of topological edge states are more robust in SSH-like alternating chains than in Haldane-like chains under Markovian dissipation. The claim is supported by exact-diagonalization solutions of the Lindblad master equation (Eq. 3) and by several diagnostics: the Theta_PT gap-ratio diagrams (Eq. 4, Figs. 1-4), correlation functions with alpha=beta (Sec. III C), and edge-state magnetization (Sec. III B). No fitted parameter is renamed as a prediction: Eq. (12) is explicitly called a fit-function with fit parameters, and its role is to summarize entropy, fidelity, and purity curves, not to generate the ordering. The single-spin comparison in Eq. (9) and Appendix B provides a parameter-free external benchmark. The only self-citation that enters the analysis is the threshold Theta_tran_PT = 0.5 taken from the authors' closed-chain work [86]; the paper itself flags this value as debatable and asserts robustness to variations. Because the SSH-versus-Haldane ordering also appears in the alpha=beta correlation data, which do not use that threshold, the self-citation is not load-bearing. Numerical caveats, such as the 70-state truncation without a documented convergence check and the ad hoc Delta_z = 1.001 shift for L_z in Figs. 1(b) and 3, are correctness risks rather than circular steps. No uniqueness theorem is invoked, and no step reduces by construction to an input. The derivation chain is therefore self-contained in the sense relevant to circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim relies on one heuristic diagnostic imported from prior work, several hand-chosen parameters, and the modeling assumption that a Markovian Lindblad equation captures the surface. No new particles, forces, or conserved quantities are introduced. The absence of invented entities is a positive feature, but the free parameters and the ad hoc Delta_z shift add to the circularity and soundness burden.

free parameters (3)
  • Theta_PT threshold = 0.5
    Defines edge-state observability regions in all Theta_PT diagrams. The value is taken from the authors' closed-chain study [86] and described as debatable.
  • Fit parameters a, b, c, d in Eq. (12) = not given
    Fit to numerical entropy, fidelity, and purity curves. Not derived from the model, and claimed universal despite exceptions for large gamma.
  • Delta_z = 1.001 for L_z runs = 1.001
    Chosen instead of Delta_z=1 to remove chaotic behavior in specific L_z diagrams. A hand-chosen anisotropy that affects the L_z robustness maps.
assumptions (4)
  • domain assumption Lindblad master equation with local jump operators S^z_i or S^x_i models the surface environment
    The surface is treated as a Markovian bath; no microscopic derivation from electron tunneling or spin-phonon coupling is given (Sec. II, Eq. 3).
  • ad hoc to paper Theta_PT criterion with threshold 0.5 is a valid proxy for topological edge-state observability
    Criterion imported from the authors' own Ref. [86] and used to label parameter regions as robust (Sec. III A, Eq. 4).
  • domain assumption Short chains (N=4 to 12) are representative for on-surface experiments
    Motivated by STM measurements of short spin chains; finite-size effects are acknowledged but not systematically extrapolated (Sec. IV).
  • ad hoc to paper The chaotic behavior at Delta_z=1 for L_z is a numerical artifact, not physics
    Authors replace Delta_z=1 with 1.001 or 0.999 to suppress it and assume the deviation is irrelevant (Sec. III A).

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Pith. "Pith review of Robustness of topological edge states in alternating spin chains against environment." pith.science (2026). https://pith.science/paper/GSJF5W34

@misc{pith2026250522420,
  author       = {Pith},
  title        = {Pith review of: Robustness of topological edge states in alternating spin chains against environment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GSJF5W34}},
  note         = {Machine review of arXiv:2505.22420}
}
abstract

Both the Haldane spin-$1$ chain and dimerized chains of spin-$1/2$ exhibit topologically protected edge states that are robust against specific perturbations. Recently, such spin chains have been specifically assembled on surfaces and we investigate here the robustness of these edge states against coupling to the surface. Since no physical system can be considered perfectly isolated, it is crucial to examine whether topological robustness is maintained in the presence of environmental coupling. We apply exact diagonalization to a Lindblad master equation that couples an alternating Heisenberg spin chain based on spins $1/2$ to a surface via various jump operators. The robustness of topological states is assessed via the time evolution of quantities such as the ground-state degeneracy, correlation function, entropy, and magnetization of edge states. We investigate chains built from dimers with antiferromagnetic and ferromagnetic intra-dimer coupling, which resemble Su-Schrieffer-Heeger and the Haldane models, resp., and assess the impact of $z$-axis anisotropy and longer-ranged couplings. Generally, we find that signatures of topological properties are more robust in Su-Schrieffer-Heeger-like chains than in Haldane-like chains.

Figures

Figures reproduced from arXiv: 2505.22420 by the authors.

Figure 2
Figure 2. FIG. 2. Θ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Θ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Θ [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: shows the time evolution ⟨S z edge⟩(t) for Lz and Lx. Longer chains do not have a strong effect on the decay time in the case of Lx, while they slow it down for Lz, see [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Entropy ( [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 11
Figure 11. Figure 11: (d) reveals that larger spin chains exhibit greater τ . Apart from this, τ remains quite similar for different chain lengths. τ is further identical for both jump opera￾tors Lx and Lz in isotropic chains. Therefore, the results depicted in Figs. 11(a), 11(d) and 11(e)…
Figure 12
Figure 12. Figure 12: The figure clearly illustrates that the results for [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The inverse spectral gap [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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Works this paper leans on

159 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [86]

    Sattler and M

    A. Sattler and M. Daghofer, Promising regimes for the observation of topological degeneracy in spin chains, Phys. Rev. B110, 024404 (2024)

    work page 2024

  2. [1]

    F. D. M. Haldane, Nobel Lecture: Topological quantum matter, Rev. Mod. Phys.89, 040502 (2017)

    2017

  3. [2]

    Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev

    X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys.89, 041004 (2017)

    2017

  4. [3]

    C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with sym- metries, Rev. Mod. Phys.88, 035005 (2016)

    2016

  5. [4]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83, 1057 (2011)

    2011

  6. [5]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

    2010

  7. [6]

    Hornberg,Entanglement and Decoherence: Foun- dations and Modern Trends, edited by A

    K. Hornberg,Entanglement and Decoherence: Foun- dations and Modern Trends, edited by A. Buchleitner, C. Viviescas, and M. Tiersch (Springer Berlin, Heidel- berg, 2009) Chap. 5 Introduction to Decoherence The- ory, Lecture Notes in Physics 768

    2009

  8. [7]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2007)

    2007

Show all 159 references
  1. [8]

    Suter and G

    D. Suter and G. A. ´Alvarez, Colloquium: Protecting quantum information against environmental noise, Rev. Mod. Phys.88, 041001 (2016)

    2016

  2. [9]

    I. L. Chuang, R. Laflamme, P. W. Shor, and W. H. Zurek, Quantum computers, factoring, and decoher- ence, Science270, 1633 (1995)

    1995

  3. [10]

    Beige, D

    A. Beige, D. Braun, B. Tregenna, and P. L. Knight, Quantum computing using dissipation to remain in a decoherence-free subspace, Phys. Rev. Lett.85, 1762 (2000)

    2000

  4. [11]

    D. P. DiVincenzo, The physical implementation of quantum computation, Fortschritte der Physik48, 771 (2000)

    2000

  5. [12]

    T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Na- ture464, 45 (2010)

    2010

  6. [13]

    D. P. DiVincenzo, Two-bit gates are universal for quan- tum computation, Phys. Rev. A51, 1015 (1995)

    1995

  7. [14]

    Lahtinen and J

    V. Lahtinen and J. K. Pachos, A short introduction to topological quantum computation, SciPost Phys.3, 021 (2017)

    2017

  8. [15]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topologi- cal quantum computation, Rev. Mod. Phys.80, 1083 (2008)

    2008

  9. [16]

    Steane, Quantum computing, Reports on Progress in Physics61, 117 (1998)

    A. Steane, Quantum computing, Reports on Progress in Physics61, 117 (1998)

    1998

  10. [17]

    Lang and H

    N. Lang and H. P. B¨ uchler, Topological networks for quantum communication between distant qubits, Quan- tum Information3, 47 (2017)

    2017

  11. [18]

    Dlaska, B

    C. Dlaska, B. Vermersch, and P. Zoller, Robust quan- tum state transfer via topologically protected edge chan- nels in dipolar arrays, Quantum Sci. Technol.2, 015001 (2017)

    2017

  12. [19]

    Jaworowski and P

    B. Jaworowski and P. Hawrylak, Quantum bits with macroscopic topologically protected states in semicon- ductor devices, Applied Sciences9, 474 (2019)

    2019

  13. [20]

    Jaworowski, N

    B. Jaworowski, N. Rogers, M. Grabowski, and P. Hawrylak, Macroscopic singlet-triplet qubit in syn- thetic spin-one chain in semiconductor nanowires, Sci- entific Reports7, 5529 (2017)

    2017

  14. [21]

    S. D. Bartlett, G. K. Brennen, A. Miyake, and J. M. Renes, Quantum computational renormalization in the Haldane phase, Phys. Rev. Lett.105, 110502 (2010)

    2010

  15. [22]

    Miyake, Quantum computation on the edge of a symmetry-protected topological order, Phys

    A. Miyake, Quantum computation on the edge of a symmetry-protected topological order, Phys. Rev. Lett. 105, 040501 (2010)

    2010

  16. [23]

    T.-C. Wei, I. Affleck, and R. Raussendorf, Two- dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation, Phys. Rev. A86, 032328 (2012)

    2012

  17. [24]

    Hsieh, Y.-P

    C.-Y. Hsieh, Y.-P. Shim, M. Korkusinski, and P. Hawry- lak, Physics of lateral triple quantum-dot molecules with controlled electron numbers, Reports on Progress in Physics75, 114501 (2012)

    2012

  18. [25]

    Manalo, M

    J. Manalo, M. Cygorek, A. Altintas, and P. Hawrylak, Electronic and magnetic properties of many-electron complexes in charged InAs xP1−x quantum dots in InP nanowires, Phys. Rev. B104, 125402 (2021)

    2021

  19. [26]

    S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln´ ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Spintronics: A spin- based electronics vision for the future, Science294, 1488 (2001)

    2001

  20. [27]

    Hirohata, K

    A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Di´ eny, P. Pirro, and B. Hillebrands, Review on spin- tronics: Principles and device applications, Journal of Magnetism and Magnetic Materials509, 166711 (2020)

    2020

  21. [28]

    ˇZuti´ c, J

    I. ˇZuti´ c, J. Fabian, and S. Das Sarma, Spintronics: Fun- damentals and applications, Rev. Mod. Phys.76, 323 (2004)

    2004

  22. [29]

    S. D. Bader and S. S. P. Parkin, Spintronics, Annual Review of Condensed Matter Physics1, 71 (2010)

    2010

  23. [30]

    Klett, H

    M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wun- ner, Topological edge states in the Su-Schrieffer-Heeger model subject to balanced particle gain and loss, The European Physical Journal D72, 214 (2018)

    2018

  24. [31]

    Dangel, M

    F. Dangel, M. Wagner, H. Cartarius, J. Main, and G. Wunner, Topological invariants in dissipative exten- sions of the Su-Schrieffer-Heeger model, Phys. Rev. A 98, 013628 (2018)

    2018

  25. [32]

    E. G. Cinnirella, A. Nava, G. Campagnano, and D. Giu- 11 liano, Fate of high winding number topological phases in the disordered extended Su-Schrieffer-Heeger model, Phys. Rev. B109, 035114 (2024)

    2024

  26. [33]

    J. C. Budich, S. Walter, and B. Trauzettel, Failure of protection of Majorana based qubits against decoher- ence, Phys. Rev. B85, 121405 (2012)

    2012

  27. [34]

    Viyuela, A

    O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Thermal instability of protected end states in a one- dimensional topological insulator, Phys. Rev. B86, 155140 (2012)

    2012

  28. [35]

    Deng and L

    T.-S. Deng and L. Pan, Fate of symmetry protected co- herence in open quantum system, Phys. Rev. B104, 094306 (2021)

    2021

  29. [36]

    Zhang, K

    J.-H. Zhang, K. Ding, S. Yang, and Z. Bi, Fractonic higher-order topological phases in open quantum sys- tems, Phys. Rev. B108, 155123 (2023)

    2023

  30. [37]

    T.-S. Deng, L. Pan, Y. Chen, and H. Zhai, Stability of time-reversal symmetry protected topological phases, Phys. Rev. Lett.127, 086801 (2021)

    2021

  31. [38]

    Z. Wang, Q. Li, W. Li, and Z. Cai, Symmetry-protected topological edge modes and emergent partial time- reversal symmetry breaking in open quantum many- body systems, Phys. Rev. Lett.126, 237201 (2021)

    2021

  32. [39]

    Linzner, L

    D. Linzner, L. Wawer, F. Grusdt, and M. Fleischhauer, Reservoir-induced Thouless pumping and symmetry- protected topological order in open quantum chains, Phys. Rev. B94, 201105 (2016)

    2016

  33. [40]

    Rivas, O

    A. Rivas, O. Viyuela, and M. A. Martin-Delgado, Density-matrix Chern insulators: Finite-temperature generalization of topological insulators, Phys. Rev. B 88, 155141 (2013)

    2013

  34. [41]

    Diehl, E

    S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topol- ogy by dissipation in atomic quantum wires, Nature Physics7, 971 (2011)

    2011

  35. [42]

    Bardyn, M

    C.-E. Bardyn, M. A. Baranov, E. Rico, A. ˙Imamo˘ glu, P. Zoller, and S. Diehl, Majorana modes in driven- dissipative atomic superfluids with a zero Chern num- ber, Phys. Rev. Lett.109, 130402 (2012)

    2012

  36. [43]

    Bardyn, M

    C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. ˙Imamo˘ glu, P. Zoller, and S. Diehl, Topology by dis- sipation, New Journal of Physics15, 085001 (2013)

    2013

  37. [44]

    J. C. Budich, P. Zoller, and S. Diehl, Dissipative prepa- ration of Chern insulators, Phys. Rev. A91, 042117 (2015)

    2015

  38. [45]

    Verstraete, M

    F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quan- tum computation and quantum-state engineering driven by dissipation, Nature Physics5, 633 (2009)

    2009

  39. [46]

    X.-D. Dai, Z. Wang, H.-R. Wang, and Z. Wang, Steady- state topological order, Phys. Rev. B111, 115142 (2025)

    2025

  40. [47]

    X. Mi, M. Sonner, M. Y. Niu, K. W. Lee, B. Foxen, R. Acharya, I. Aleiner, T. I. Andersen, F. Arute, and K. A. et al., Noise-resilient edge modes on a chain of superconducting qubits, Science378, 785 (2022)

    2022

  41. [48]

    S. R. White and D. A. Huse, Numerical renormalization- group study of low-lying eigenstates of the antiferromag- neticS=1 Heisenberg chain, Phys. Rev. B48, 3844 (1993)

    1993

  42. [49]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B81, 064439 (2010)

    2010

  43. [50]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Phys. Rev. B80, 155131 (2009)

    2009

  44. [51]

    Affleck, Quantum spin chains and the Haldane gap, Journal of Physics: Condensed Matter1, 3047 (1989)

    I. Affleck, Quantum spin chains and the Haldane gap, Journal of Physics: Condensed Matter1, 3047 (1989)

    1989

  45. [52]

    Renard, L.-P

    J.-P. Renard, L.-P. Regnault, and M. Verdaguer, Hal- dane quantum spin chains, inMagnetism: Molecules to Materials I(John Wiley & Sons, Ltd, 2001) Chap. 2, pp. 49–93

    2001

  46. [53]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Os- hikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012)

    2012

  47. [54]

    F. D. M. Haldane, Continuum dynamics of the 1- D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys. Rev. A93, 464 (1983)

    1983

  48. [55]

    F. D. M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis N´ eel state, Phys. Rev. Lett.50, 1153 (1983)

    1983

  49. [56]

    Yamanaka, Y

    M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Phase diagram of theS=1/2 quantum spin chain with bond alternation, Phys. Rev. B48, 9555 (1993)

    1993

  50. [57]

    Hida, Ground-state phase diagram of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with anisotropy, Phys

    K. Hida, Ground-state phase diagram of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with anisotropy, Phys. Rev. B46, 8268 (1992)

    1992

  51. [58]

    Hida, Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain, Phys

    K. Hida, Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain, Phys. Rev. B45, 2207 (1992)

    1992

  52. [59]

    Kohmoto and H

    M. Kohmoto and H. Tasaki, HiddenZ 2×Z2 symmetry breaking and the Haldane phase in theS=1/2 quantum spin chain with bond alternation, Phys. Rev. B46, 3486 (1992)

    1992

  53. [60]

    Hagiwara, K

    M. Hagiwara, K. Katsumata, I. Affleck, B. I. Halperin, and J. P. Renard, Observation ofS=1/2 degrees of freedom in anS=1 linear-chain Heisenberg antiferro- magnet, Phys. Rev. Lett.65, 3181 (1990)

    1990

  54. [61]

    Sahoo, D

    S. Sahoo, D. Dey, S. K. Saha, and M. Kumar, Haldane and dimer phases in a frustrated spin chain: An ex- act groundstate and associated topological phase transi- tion, Journal of Physics: Condensed Matter32, 335601 (2020)

    2020

  55. [62]

    G. Xu, J. F. DiTusa, T. Ito, K. Oka, H. Takagi, C. Bro- holm, and G. Aeppli, Y 2BaNiO5: A nearly ideal real- ization of theS=1 Heisenberg chain with antiferromag- netic interactions, Phys. Rev. B54, R6827 (1996)

    1996

  56. [63]

    Mutka, J

    H. Mutka, J. L. Soubeyroux, G. Bourleaux, and P. Colombet, Support for the Haldane conjecture: Gap for magnetic excitations in the quasi-one-dimensional S=1 Heisenberg antiferromagnet AgVP 2S6, Phys. Rev. B39, 4820 (1989)

    1989

  57. [64]

    J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. Erkelens, J. Rossat-Mignod, J. Ribas, W. G. Stirling, and C. Vettier, Quantum energy gap in two quasi- one-dimensionalS=1 Heisenberg antiferromagnets (in- vited), Journal of Applied Physics63, 3538 (1988)

    1988

  58. [65]

    W. J. L. Buyers, R. M. Morra, R. L. Armstrong, M. J. Hogan, P. Gerlach, and K. Hirakawa, Experi- mental evidence for the Haldane gap in a spin-1 nearly isotropic, antiferromagnetic chain, Phys. Rev. Lett.56, 371 (1986)

    1986

  59. [66]

    J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. Erkelens, J. Rossat-Mignod, and W. G. Stirling, Pre- sumption for a quantum energy gap in the quasi- one-dimensionalS=1 Heisenberg antiferromagnet 12 Ni(C2H8N2)2NO2(ClO4), Europhysics Letters3, 945 (1987)

    1987

  60. [67]

    Katsumata, H

    K. Katsumata, H. Hori, T. Takeuchi, M. Date, A. Ya- magishi, and J. P. Renard, Magnetization process of an S=1 linear-chain Heisenberg antiferromagnet, Phys. Rev. Lett.63, 86 (1989)

    1989

  61. [68]

    R. M. Morra, W. J. L. Buyers, R. L. Armstrong, and K. Hirakawa, Spin dynamics and the Haldane gap in the spin-1 quasi-one-dimensional antiferromagnet CsNiCl 3, Phys. Rev. B38, 543 (1988)

    1988

  62. [69]

    L. P. Regnault and J. P. Renard, Spin dynamics in the Haldane-gap system NENP, Physica B: Condensed Matter215, 71 (1995)

    1995

  63. [70]

    Darriet and L

    J. Darriet and L. P. Regnault, The compound Y2BaNiO5: A new example of a Haldane gap in aS=1 magnetic chain, Solid State Communications86, 409 (1993)

    1993

  64. [71]

    Ajiro, T

    Y. Ajiro, T. Goto, H. Kikuchi, T. Sakakibara, and T. Inami, High-field magnetization of a quasi-one-dimensionalS=1 antiferromagnet Ni(C2H8N2)2NO2(ClO4): Observation of the Hal- dane gap, Phys. Rev. Lett.63, 1424 (1989)

    1989

  65. [72]

    ˇCiˇ zm´ ar, M

    E. ˇCiˇ zm´ ar, M. Ozerov, O. Ignatchik, T. P. Papageor- giou, J. Wosnitza, S. A. Zvyagin, J. Krzystek, Z. Zhou, C. P. Landee, B. R. Landry, M. M. Turnbull, and J. L. Wikaira, Magnetic properties of the Haldane-gap mate- rial [Ni(C 2H8N2)2NO2](BF4), New Journal of Physics 10, ...

    2008

  66. [73]

    Kenzelmann, G

    M. Kenzelmann, G. Xu, I. A. Zaliznyak, C. Broholm, J. F. DiTusa, G. Aeppli, T. Ito, K. Oka, and H. Takagi, Structure of end states for a Haldane spin chain, Phys. Rev. Lett.90, 087202 (2003)

    2003

  67. [74]

    S. H. Glarum, S. Geschwind, K. M. Lee, M. L. Kaplan, and J. Michel, Observation of fractional spinS=1/2 on open ends ofS=1 linear antiferromagnetic chains: Nonmagnetic doping, Phys. Rev. Lett.67, 1614 (1991)

    1991

  68. [75]

    Kimura, H

    S. Kimura, H. Ohta, M. Motokawa, T. Yokoo, and J. Akimitsu,S=1/2 degrees of freedom in Haldane sys- tem Y 2BaNiO5 observed by submillimeter wave ESR, Journal of the Physical Society of Japan67, 2514–2521 (1998)

    1998

  69. [76]

    Yoshida, K

    M. Yoshida, K. Shiraki, S. Okubo, H. Ohta, T. Ito, H. Takagi, M. Kaburagi, and Y. Ajiro, Energy structure of a finite Haldane chain in Y2BaNi0.96Mg0.04O5 studied by high field electron spin resonance, Phys. Rev. Lett. 95, 117202 (2005)

    2005

  70. [77]

    Tedoldi, R

    F. Tedoldi, R. Santachiara, and M. Horvati´ c,89YNMR imaging of the staggered magnetization in the doped Haldane chain Y 2BaNi1 –xMgx O5, Phys. Rev. Lett.83, 412 (1999)

    1999

  71. [78]

    Delgado, C

    F. Delgado, C. D. Batista, and J. Fern´ andez-Rossier, Local probe of fractional edge states ofS=1 Heisenberg spin chains, Phys. Rev. Lett.111, 167201 (2013)

    2013

  72. [79]

    Mishra, G

    S. Mishra, G. Catarina, F. Wu, R. Ortiz, D. Jacob, K. Eimre, J. Ma, C. A. Pignedoli, X. Feng, P. Ruffieux, J. Fern´ andez-Rossier, and R. Fasel, Observation of frac- tional edge excitations in nanographene spin chains, Na- ture598, 287 (2021)

    2021

  73. [80]

    Y. Zhao, K. Jiang, C. Li, Y. Liu, G. Zhu, M. Pizzochero, E. Kaxiras, D. Guan, Y. Li, H. Zheng, C. Liu, J. Jia, M. Qin, X. Zhuang, and S. Wang, Quantum nanomag- nets in on-surface metal-free porphyrin chains, Nature Chemistry15, 53 (2023)

    2023

  74. [81]

    H. Wang, P. Fan, J. Chen, L. Jiang, H.-J. Gao, J. L. Lado, and K. Yang, Construction of topological quan- tum magnets from atomic spins on surfaces, Nature Nanotechnology19, 1782 (2024)

    2024

  75. [82]

    C. Zhao, G. Catarina, J.-J. Zhang, J. C. G. Henriques, L. Yang, J. Ma, X. Feng, O. Gr¨ oning, P. Ruffieux, J. Fern´ andez-Rossier, and R. Fasel, Tunable topologi- cal phases in nanographene-based spin-1/2 alternating- exchange Heisenberg chains, Nature Nanotechnology 19, 1789 (2024)

    2024

  76. [83]

    V. V. Baran and J. Paaske, Spin-1 Haldane chains of superconductor-semiconductor hybrids, Phys. Rev. B 110, 064503 (2024)

    2024

  77. [84]

    Y.-P. Shim, A. Sharma, C.-Y. Hsieh, and P. Hawrylak, Artificial Haldane gap material on a semiconductor chip, Solid State Communications150, 2065 (2010)

    2010

  78. [85]

    Manalo, D

    J. Manalo, D. Miravet, and P. Hawrylak, Microscopic design of a synthetic spin-1 chain in an InAsP quantum dot array, Phys. Rev. B109, 085112 (2024)

    2024

  79. [87]

    W.-X. Chen, J. Ren, W.-L. You, X. Hao, and Y.-Z. Wu, Phase diagram of the spin-1/2 Heisenberg alternating chain in a magnetic field, Communications in Theoreti- cal Physics71, 1029 (2019)

    2019

  80. [88]

    K. Hida, Ground-state phase diagram of the spin 1/2 alternating Heisenberg chain with anisotropy on the an- tiferromagnetic bond, Journal of the Physical Society of Japan62, 1463 (1993)

    1993

  81. [89]

    Barnes, J

    T. Barnes, J. Riera, and D. A. Tennant,S= 1 2 alternat- ing chain using multiprecision methods, Phys. Rev. B 59, 11384 (1999)

    1999

  82. [90]

    Nakamura and S

    M. Nakamura and S. Todo, Order parameter to charac- terize valence-bond-solid states in quantum spin chains, Phys. Rev. Lett.89, 077204 (2002)

    2002

  83. [91]

    Collins, C

    A. Collins, C. J. Hamer, and Z. Weihong, Modified triplet-wave expansion method applied to the alternat- ing Heisenberg chain, Phys. Rev. B74, 144414 (2006)

    2006

  84. [92]

    Hida, XY Phase in the ground state of the spin-1/2 anisotropic alternating Heisenberg chain, Journal of the Physical Society of Japan62, 3357 (1993)

    K. Hida, XY Phase in the ground state of the spin-1/2 anisotropic alternating Heisenberg chain, Journal of the Physical Society of Japan62, 3357 (1993)

    1993

  85. [93]

    M. S. Bahovadinov, O. G¨ ulseren, and J. Schnack, Lo- cal entanglement and string order parameter in dimer- ized models, Journal of Physics: Condensed Matter31, 505602 (2019)

    2019

  86. [94]

    Haghshenas, A

    R. Haghshenas, A. Langari, and A. T. Rezakhani, Sym- metry fractionalization: Symmetry-protected topolog- ical phases of the bond-alternating spin-1/2 Heisen- berg chain, Journal of Physics: Condensed Matter26, 456001 (2014)

    2014

  87. [95]

    Sahoo, V

    S. Sahoo, V. M. L. Durga Prasad Goli, D. Sen, and S. Ramasesha, Studies on a frustrated Heisenberg spin chain with alternating ferromagnetic and antiferromag- netic exchanges, Journal of Physics Condensed Matter 26, 276002 (2014)

    2014

  88. [96]

    Yoshida and K

    S. Yoshida and K. Okamoto, Phase diagram of spin-1/2 alternating ferromagnetic chain with XY-like anisotropy, Journal of the Physical Society of Japan58, 4367 (1989)

    1989

  89. [97]

    K. Hida, Topological phases of spin-1/2 ferromagnetic- antiferromagnetic alternating Heisenberg chains with al- ternating next-nearest-neighbour interaction, Journal of the Physical Society of Japan85, 124712 (2016)

    2016

  90. [98]

    K. Hida, Characterization of topological phases of spin- 13 1/2 frustrated ferromagnetic-antiferromagnetic alter- nating Heisenberg chains by entanglement spectrum, Journal of the Physical Society of Japan85, 024705 (2016)

    2016

  91. [99]

    Liu, W.-L

    G.-H. Liu, W.-L. You, W. Li, and G. Su, Quantum phase transitions and string orders in the spin-1/2 Heisenberg- Ising alternating chain with Dzyaloshinskii-Moriya in- teraction, Journal of Physics Condensed Matter27, 165602 (2015)

    2015

  92. [100]

    H. T. Wang, B. Li, and S. Y. Cho, Topological quantum phase transition in bond-alternating spin-1/2 Heisen- berg chains, Phys. Rev. B87, 054402 (2013)

    2013

  93. [101]

    K. Hida, K. Takano, and H. Suzuki, Topological phases of the spin-1/2 ferromagnetic-antiferromagnetic alter- nating Heisenberg chain with frustrated next-nearest- neighbour iInteraction, Journal of the Physical Society of Japan82, 064703 (2013)

    2013

  94. [102]

    Duffy and K

    W. Duffy and K. P. Barr, Theory of alternating antifer- romagnetic Heisenberg linear chains, Phys. Rev.165, 647 (1968)

    1968

  95. [103]

    Manaka, I

    H. Manaka, I. Yamada, M. Hagiwara, and M. Toku- naga, High-field and high-frequency ESR study of the Haldane state formed in the ferromagnetic and an- tiferromagnetic alternating Heisenberg chain system (CH3)2CHNH3CuCl3, Phys. Rev. B63, 144428 (2001)

    2001

  96. [104]

    Hida and S

    K. Hida and S. Takada, String order parameters in the Haldane gap phase of the spin-1/2 alternating Heisen- berg chain – numerical diagonalization and variational study–, Journal of the Physical Society of Japan61, 1879 (1992)

    1992

  97. [105]

    Chatterjee, M

    M. Chatterjee, M. Kumar, and Z. G. Soos, Spin- 1 2 string correlations and singlet-triplet gaps of frustrated lad- ders with ferromagnetic legs and alternate ferromag- netic and antiferromagnetic rungs, Phys. Rev. B109, 094439 (2024)

    2024

  98. [106]

    Chatterjee, M

    M. Chatterjee, M. Kumar, and Z. G. Soos, Singlet quan- tum phases of the frustrated spin-1/2 ladder with fer- romagnetic (F) exchange in legs and alternating F-AF exchange in rungs, Physica Scripta99, 025973 (2024)

    2024

  99. [107]

    Lindblad, On the generators of quantum dynamical semigroups, Commun.Math.Phys48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Commun.Math.Phys48, 119 (1976)

    1976

  100. [108]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, Journal of Mathematical Physics17, 821 (1976)

    1976

  101. [109]

    C. A. Brasil, F. F. Fanchini, and R. d. J. Napolitano, A simple derivation of the Lindblad equation, Revista Brasileira de Ensino de Fisica35(2013)

    2013

  102. [110]

    D. A. Lidar, Lecture notes on the theory of open quan- tum systems (2020), arXiv:1902.00967 [quant-ph]

    2020 arXiv

  103. [111]

    Manzano, A short introduction to the Lindblad mas- ter equation, AIP Advances10, 025106 (2020)

    D. Manzano, A short introduction to the Lindblad mas- ter equation, AIP Advances10, 025106 (2020)

    2020

  104. [112]

    ˇZnidariˇ c, Relaxation times of dissipative many-body quantum systems, Phys

    M. ˇZnidariˇ c, Relaxation times of dissipative many-body quantum systems, Phys. Rev. E92, 042143 (2015)

    2015

  105. [113]

    Cai and T

    Z. Cai and T. Barthel, Algebraic versus exponential de- coherence in dissipative many-particle systems, Phys. Rev. Lett.111, 150403 (2013)

    2013

  106. [114]

    van Caspel and V

    M. van Caspel and V. Gritsev, Symmetry-protected co- herent relaxation of open quantum systems, Phys. Rev. A97, 052106 (2018)

    2018

  107. [115]

    Shibata and H

    N. Shibata and H. Katsura, Dissipative spin chain as a non-Hermitian Kitaev ladder, Phys. Rev. B99, 174303 (2019)

    2019

  108. [116]

    ˇZnidariˇ c, Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product rep- resentation, Phys

    M. ˇZnidariˇ c, Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product rep- resentation, Phys. Rev. E83, 011108 (2011)

    2011

  109. [117]

    Ajisaka and Y

    S. Ajisaka and Y. B. Band, Decoherence of three-level systems: Application to nitrogen-vacancy centers in di- amond near a surface, Phys. Rev. B94, 134107 (2016)

    2016

  110. [118]

    X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B41, 9377 (1990)

    1990

  111. [119]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi,A Short Course on Topological Insulators(Springer Cham, 2016) Lecture Notes in Physics 919

    2016

  112. [120]

    Chen and D.-W

    B.-H. Chen and D.-W. Chiou, An elementary rigorous proof of bulk-boundary correspondence in the general- ized Su-Schrieffer-Heeger model, Physics Letters A384, 126168 (2020)

    2020

  113. [121]

    Hung and X.-G

    L.-Y. Hung and X.-G. Wen, Universal symmetry- protected topological invariants for symmetry-protected topological states, Phys. Rev. B89, 075121 (2014)

    2014

  114. [122]

    den Nijs and K

    M. den Nijs and K. Rommelse, Preroughening tran- sitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B40, 4709 (1989)

    1989

  115. [123]

    Tasaki, Quantum liquid in antiferromagnetic chains: A stochastic geometric approach to the Haldane gap, Phys

    H. Tasaki, Quantum liquid in antiferromagnetic chains: A stochastic geometric approach to the Haldane gap, Phys. Rev. Lett.66, 798 (1991)

    1991

  116. [124]

    Misra and E

    B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quantum theory, Journal of Mathematical Physics18, 756 (1977)

    1977

  117. [125]

    Tonielli, R

    F. Tonielli, R. Fazio, S. Diehl, and J. Marino, Orthog- onality catastrophe in dissipative quantum many-body systems, Phys. Rev. Lett.122, 040604 (2019)

    2019

  118. [126]

    Roberts, M

    W. Roberts, M. Vogl, and G. A. Fiete, Fidelity of the Kitaev honeycomb model under a quench, Phys. Rev. B 109, L220406 (2024)

    2024

  119. [127]

    V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014)

    2014

  120. [128]

    Kawabata, R

    K. Kawabata, R. Sohal, and S. Ryu, Lieb-Schultz- Mattis theorem in open quantum systems, Phys. Rev. Lett.132, 070402 (2024)

    2024

  121. [129]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009)

    2009

  122. [130]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rig- orous results on valence-bond ground states in antifer- romagnets, Phys. Rev. Lett.59, 799 (1987)

    1987

  123. [131]

    A. Nava, C. A. Perroni, R. Egger, L. Lepori, and D. Giu- liano, Dissipation-driven dynamical topological phase transitions in two-dimensional superconductors, Phys. Rev. B109, L041107 (2024)

    2024

  124. [132]

    Wolff, A

    S. Wolff, A. Sheikhan, S. Diehl, and C. Kollath, Nonequilibrium metastable state in a chain of interact- ing spinless fermions with localized loss, Phys. Rev. B 101, 075139 (2020)

    2020

  125. [133]

    Fr¨ oml, A

    H. Fr¨ oml, A. Chiocchetta, C. Kollath, and S. Diehl, Fluctuation-induced quantum Zeno effect, Phys. Rev. Lett.122, 040402 (2019)

    2019

  126. [134]

    Shackleton and M

    H. Shackleton and M. S. Scheurer, Exactly solvable dis- sipative spin liquid, Phys. Rev. B109, 085115 (2024)

    2024

  127. [135]

    Wolff, A

    S. Wolff, A. Sheikhan, and C. Kollath, Dissipative time evolution of a chiral state after a quantum quench, Phys. Rev. A94, 043609 (2016)

    2016

  128. [136]

    Secli, M

    M. Secli, M. Capone, and M. Schiro, Steady-state quan- tum Zeno effect of driven-dissipative bosons with dy- 14 namical mean-field theory, Phys. Rev. A106, 013707 (2022)

    2022

  129. [137]

    Nakagawa, N

    M. Nakagawa, N. Kawakami, and M. Ueda, Exact Liou- villian spectrum of a one-dimensional dissipative Hub- bard model, Phys. Rev. Lett.126, 110404 (2021)

    2021

  130. [138]

    Viciani, M

    S. Viciani, M. Lima, M. Bellini, and F. Caruso, Observa- tion of noise-assisted transport in an all-optical cavity- based network, Phys. Rev. Lett.115, 083601 (2015)

    2015

  131. [139]

    M. C. Fischer, B. Guti´ errez-Medina, and M. G. Raizen, Observation of the quantum Zeno and anti-Zeno effects in an unstable system, Phys. Rev. Lett.87, 040402 (2001)

    2001

  132. [140]

    Syassen, D

    N. Syassen, D. M. Bauer, M. Lettner, T. Volz, D. Di- etze, J. J. Garcia-Ripoll, J. I. Cirac, G. Rempe, and S. D¨ urr, Strong dissipation inhibits losses and induces correlations in cold molecular gases, Science320, 1329 (2008)

    2008

  133. [141]

    W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Quantum Zeno effect, Phys. Rev. A41, 2295 (1990)

    1990

  134. [142]

    Y. S. Patil, S. Chakram, and M. Vengalattore, Measurement-induced localization of an ultracold lat- tice gas, Phys. Rev. Lett.115, 140402 (2015)

    2015

  135. [143]

    Wiesendanger, Spin mapping at the nanoscale and atomic scale, Rev

    R. Wiesendanger, Spin mapping at the nanoscale and atomic scale, Rev. Mod. Phys.81, 1495 (2009)

    2009

  136. [144]

    D.-J. Choi, N. Lorente, J. Wiebe, K. von Bergmann, A. F. Otte, and A. J. Heinrich, Colloquium: Atomic spin chains on surfaces, Rev. Mod. Phys.91, 041001 (2019)

    2019

  137. [145]

    Gauyacq, N

    J.-P. Gauyacq, N. Lorente, and F. D. Novaes, Excita- tion of local magnetic moments by tunneling electrons, Progress in Surface Science87, 63 (2012)

    2012

  138. [146]

    J¨ ack, Y

    B. J¨ ack, Y. Xie, and A. Yazdani, Detecting and dis- tinguishing Majorana zero modes with the scanning tunnelling microscope, Nature Reviews Physics3, 541 (2021)

    2021

  139. [147]

    Feldmeier, W

    J. Feldmeier, W. Natori, M. Knap, and J. Knolle, Local probes for charge-neutral edge states in two-dimensional quantum magnets, Phys. Rev. B102, 134423 (2020)

    2020

  140. [148]

    Udagawa, S

    M. Udagawa, S. Takayoshi, and T. Oka, Scanning tun- neling microscopy as a single Majorana detector of Ki- taev’s chiral spin liquid, Phys. Rev. Lett.126, 127201 (2021)

    2021

  141. [149]

    E. J. K¨ onig, M. T. Randeria, and B. J¨ ack, Tunneling spectroscopy of quantum spin liquids, Phys. Rev. Lett. 125, 267206 (2020)

    2020

  142. [150]

    Z. Wang, L. Liu, H. Zheng, M. Zhao, K. Yang, C. Wang, F. Yang, H. Wu, and C. Gao, Direct observation of the mottness and p–d orbital hybridization in the epitaxial monolayerα-RuCl 3, Nanoscale14, 11745 (2022)

    2022

  143. [151]

    Kohsaka, S

    Y. Kohsaka, S. Akutagawa, S. Omachi, Y. Iwamichi, T. Ono, I. Tanaka, S. Tateishi, H. Murayama, S. Suet- sugu, K. Hashimoto, T. Shibauchi, M. O. Takahashi, S. Nikolaev, T. Mizushima, S. Fujimoto, T. Terashima, T. Asaba, Y. Kasahara, and Y. Matsuda, Imaging quan- tum interferenc...

    2024

  144. [152]

    Cookmeyer and S

    T. Cookmeyer and S. Das Sarma, Engineering the Ki- taev spin liquid in a quantum dot system, Phys. Rev. Lett.132, 186501 (2024)

    2024

  145. [153]

    Choi, Completely positive linear maps on com- plex matrices, Linear Algebra and its Applications10, 285 (1975)

    M.-D. Choi, Completely positive linear maps on com- plex matrices, Linear Algebra and its Applications10, 285 (1975)

    1975

  146. [154]

    Jamio lkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Reports on Mathematical Physics3, 275 (1972)

    A. Jamio lkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Reports on Mathematical Physics3, 275 (1972)

    1972

  147. [155]

    Jaschke and L

    D. Jaschke and L. D. Carr, Open source matrix product states: Exact diagonalization and other entanglement- accurate methods revisited in quantum systems, Journal of Physics A Mathematical General51, 465302 (2018)

    2018

  148. [156]

    Am-Shallem, A

    M. Am-Shallem, A. Levy, I. Schaefer, and R. Kosloff, Three approaches for representing Lindblad dynamics by a matrix-vector notation (2015), arXiv:1510.08634 [quant-ph]. 15 Appendix A: Numerics The Lindblad operatorL, see Eq. (A1), is a so- called superoperator, since it acts ...

    2015 arXiv

  149. [157]

    (A1), for the diagonal Hamilto- nian Eq

    Jump operatorL=S z Following a straightforward calculation of the LME in vectorized form, see Eq. (A1), for the diagonal Hamilto- nian Eq. (B1) and the jump operatorL=S z, we obtain ρ(t)=( ρ11(0)ρ 12(0)e−it∆Ee−t γ 2 ρ21(0)eit∆Ee−t γ 2 ρ22(0) ).(B2) The diagonal elements remain...

  150. [158]

    B 1, we now discuss the jump operatorL=S x for the Hamiltonian Eq

    Jump operatorL=S x Similar to the discussion of the jump operatorL=S z in Sec. B 1, we now discuss the jump operatorL=S x for the Hamiltonian Eq. (B1). The diagonal elements ofρ(t) are ρ11(t)= 1 2 − 1 2 (1−2ρ 11(0))e −γt 2 (B4) ρ22(t)= 1 2 − 1 2 (1−2ρ 22(0))e −γt 2 (B5) and th...

  151. [159]

    In contrast toL=S x, the combination of these two jump operators does not mix the off-diagonal elements ofρ

    Jump operatorsL 1 =S + andL 2 =S − Instead of employingL=S x as a jump operator, an alternative approach involves utilizing two distinct jump operatorsL 1 =S + andL 2 =S −, each with the dissipation strengthsγ + andγ −, respectively. In contrast toL=S x, the combination of the...

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