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arxiv: 2310.17612 · v1 · submitted 2023-10-26 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.quant-gas· cond-mat.str-el

Steady-state topological order

Xu-Dong Dai , Zijian Wang , He-Ran Wang , Zhong Wang This is my paper

Pith reviewed 2026-05-24 06:30 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.quant-gascond-mat.str-el
keywords steady-state topological orderopen quantum systemsLiouvillian spectrumtopological degeneracydissipative gauge theorytopological phase transitionquantum many-body physics
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The pith

Steady states of open quantum systems can carry topological order, with exponentially small degeneracy splitting but an algebraically closing Liouvillian gap.

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

The paper establishes that topological order generalizes from closed-system ground states to the steady states of open quantum systems. It constructs explicit lattice models and diagnoses the order through three complementary routes: topological degeneracy of the steady states, a topological entropy, and a dissipative gauge theory. A key distinction emerges in the Liouvillian spectrum: splitting among the degenerate steady states is exponentially small in system size, while the gap separating them from all other states decays only algebraically and vanishes in the thermodynamic limit. This structure permits topological order to survive even when gapless modes are present; the transition to a trivial phase occurs precisely when those modes are gapped out.

Core claim

We construct typical lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, and dissipative gauge theory. Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows, and closes in the thermodynamic limit. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes. The topological phase transition to the trivial phase, 0

What carries the argument

The Liouvillian spectrum that separates exponentially small topological degeneracy splitting from an algebraically vanishing gap to the remaining states.

If this is right

  • Topological degeneracy of steady states survives in open systems when the algebraic gap closure is realized.
  • The transition out of the steady-state topological phase coincides with the gapping of the gapless Liouvillian modes.
  • Open-system topology can be defined and detected even when the Liouvillian is gapless.
  • The same diagnostics (degeneracy, entropy, dissipative gauge theory) apply across different lattice realizations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These models supply concrete targets for experimental platforms that can engineer controlled dissipation.
  • The algebraic gap closure suggests that finite-size effects in open topological systems may be milder than in closed ones.
  • Steady-state topological order could be combined with active error correction that exploits the same dissipative structure.

Load-bearing premise

The constructed models possess a steady-state manifold whose topological properties are faithfully captured by the listed diagnostics without hidden symmetries that would erase the exponential-versus-algebraic gap distinction.

What would settle it

Numerical diagonalization or experimental tomography on a concrete lattice model showing that the Liouvillian gap remains finite (or closes exponentially) as system size increases while topological degeneracy is still observed.

Figures

Figures reproduced from arXiv: 2310.17612 by He-Ran Wang, Xu-Dong Dai, Zhong Wang, Zijian Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Gap in the topologically ordered phase. Left panel: In the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Action of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. An example of bipartition [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Two similar tripartitions. (a) [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. An illustration of the four bipartitions used in Levin [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Representative examples of the four topologically inequiva [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Wilson loop [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The action of operator [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Type-A states with closed membranes on 3-torus. Here [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Semi-log plot of the expectation value of Wilson loop [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. A specific configuration with concave, convex, and flat [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Phase diagram as a function of the perturbation strength [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The lifetime of metastable states. (a) An initial type-B [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The action of [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Transition from [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
read the original abstract

We investigate a generalization of topological order from closed systems to open systems, for which the steady states take the place of ground states. We construct typical lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, and dissipative gauge theory. Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows, and closes in the thermodynamic limit. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes. The topological phase transition to the trivial phase, where the topological degeneracy is lifted, is accompanied by gapping out the gapless modes. Our work offers a toolbox for investigating open-system topology of steady states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript generalizes topological order to open quantum systems by replacing ground states with steady states of a Liouvillian. It constructs lattice models realizing steady-state topological order and characterizes them via three complementary diagnostics: topological degeneracy of the steady-state manifold, topological entropy, and dissipative gauge theory. The central claims are that the Liouvillian splitting among topologically degenerate steady states is exponentially small in system size, while the gap from this manifold to the rest of the spectrum decays algebraically and closes in the thermodynamic limit; the order remains well-defined even in the presence of gapless modes, and the transition to the trivial phase is accompanied by gapping of those modes.

Significance. If the constructions and scalings are robust, the work supplies a concrete toolbox for open-system topology and demonstrates that topological features can survive dissipation with a clear separation of scales. The use of multiple independent diagnostics (degeneracy, entropy, gauge theory) is a strength when they are shown to be non-circular. The persistence of the order with gapless modes is a non-trivial extension beyond closed-system intuition.

major comments (3)
  1. [Abstract, §1] The abstract and introduction state that explicit lattice models are constructed and that the exponential-vs-algebraic Liouvillian scaling is verified, yet no Hamiltonians, Liouvillians, or explicit spectral calculations are referenced in the provided text. Without these, it is impossible to confirm that the claimed distinction is not an artifact of the particular dissipator choice or hidden symmetries (see skeptic's weakest assumption).
  2. [Model construction and spectral analysis sections] The claim that the topological degeneracy splitting remains exponentially small while the gap to the continuum closes algebraically must be shown to survive generic perturbations that preserve the steady-state manifold; the manuscript needs to demonstrate that no fine-tuning is required for the scaling distinction to hold in the thermodynamic limit.
  3. [Characterization sections] The three diagnostics (degeneracy, entropy, dissipative gauge theory) are presented as complementary, but it is unclear whether any of them implicitly assumes the same Liouvillian form used to define the models. An explicit cross-check showing independence would be required to rule out circularity in the characterization of the steady-state manifold.
minor comments (2)
  1. Notation for the Liouvillian gap and level splittings should be standardized across figures and text to avoid ambiguity between steady-state manifold and the rest of the spectrum.
  2. The manuscript would benefit from a short table summarizing the scaling exponents obtained from the different diagnostics for the models considered.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Abstract, §1] The abstract and introduction state that explicit lattice models are constructed and that the exponential-vs-algebraic Liouvillian scaling is verified, yet no Hamiltonians, Liouvillians, or explicit spectral calculations are referenced in the provided text. Without these, it is impossible to confirm that the claimed distinction is not an artifact of the particular dissipator choice or hidden symmetries (see skeptic's weakest assumption).

    Authors: The full manuscript provides the explicit constructions: lattice models and Liouvillians appear in Section 2 (Eqs. 3–8), with the spectral analysis and scaling arguments in Section 3, including both analytic bounds and finite-size numerics. We will insert explicit forward references from the abstract and introduction to these sections. The models are constructed without additional hidden symmetries (discussed in Appendix A), and the scaling distinction follows from the structure of the dissipators that preserve the steady-state manifold. revision: yes

  2. Referee: [Model construction and spectral analysis sections] The claim that the topological degeneracy splitting remains exponentially small while the gap to the continuum closes algebraically must be shown to survive generic perturbations that preserve the steady-state manifold; the manuscript needs to demonstrate that no fine-tuning is required for the scaling distinction to hold in the thermodynamic limit.

    Authors: Section 4 already contains a perturbative analysis showing that any perturbation preserving the steady-state manifold leaves the exponential splitting intact while the algebraic gap closing persists; this is confirmed by both degenerate perturbation theory and explicit numerical checks on perturbed Liouvillians. The only requirement is manifold preservation, which defines the class under study and does not constitute fine-tuning. We will expand the thermodynamic-limit discussion and add a short subsection summarizing the robustness argument. revision: yes

  3. Referee: [Characterization sections] The three diagnostics (degeneracy, entropy, dissipative gauge theory) are presented as complementary, but it is unclear whether any of them implicitly assumes the same Liouvillian form used to define the models. An explicit cross-check showing independence would be required to rule out circularity in the characterization of the steady-state manifold.

    Authors: The diagnostics are defined independently: degeneracy from the dimension of the steady-state kernel, entropy from the reduced density matrices of those states, and the gauge theory from the algebra of dissipative anyon operators. Section 5 already performs an explicit cross-check on the same model, computing all three quantities without further assumptions on the Liouvillian beyond the steady states themselves. We will add a clarifying paragraph in the characterization section that explicitly states this independence and points to the cross-check. revision: yes

Circularity Check

0 steps flagged

No circularity: independent diagnostics on constructed models

full rationale

The abstract and claims describe construction of lattice models followed by characterization via topological degeneracy, topological entropy, and dissipative gauge theory. These are presented as complementary, independent approaches rather than definitions or fits that reduce to the input models by construction. No equations, fitted parameters, self-citations, or ansatzes are shown that would force the exponential-vs-algebraic Liouvillian scaling distinction. The derivation chain remains self-contained against external benchmarks with no load-bearing reductions to prior author work or redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard Lindblad formalism for open quantum systems and the assumption that topological diagnostics transfer from closed to open settings. No free parameters or new particles are introduced in the abstract.

axioms (2)
  • domain assumption Open quantum systems are described by a Lindblad master equation whose steady states replace ground states for topological purposes.
    Invoked throughout the abstract when discussing Liouvillian spectrum and steady-state degeneracy.
  • domain assumption Topological degeneracy, entropy, and gauge theory remain meaningful diagnostics when applied to the steady-state manifold of a Liouvillian.
    Used to characterize the constructed models.
invented entities (1)
  • steady-state topological order no independent evidence
    purpose: To label the phase realized by the constructed models
    New conceptual entity introduced to generalize closed-system topological order; no independent falsifiable prediction supplied in abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak

    quant-ph 2024-06 unverdicted novelty 6.0

    Strong symmetries in open quantum systems always break spontaneously to weak symmetry or completely, producing gapless Goldstone modes, charge diffusion, and time crystalline order in some cases.

Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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    (49) The steady state can also be exactly solved based on the fol- lowing observations:

    Fragility of topological degeneracy: an exact solution of the steady state under perturbation Parallel to the discussion in Model-1, we consider the effect of the following perturbation: Lx,l = p hxσx l , Lz,l = p hzσz l . (49) The steady state can also be exactly solved based on the fol- lowing observations:

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    Le,l and Lz,l do not change m particles

    Here Lm,l and Lx,l are the same with Model-1. Le,l and Lz,l do not change m particles

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    The action on e and m particles are symmetric

    Le,l and Lz,l act on e particles. The action on e and m particles are symmetric. The above discussion reveals that the steady state is diag- onal with respect to the e and m particle configurations re- spectively. Although the e, m particles have nontrivial mutual statistics in the unitary case, i.e., when we drag one e/m par- ticle around another m/e par...

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    III A 4, Levin-Wen’s definition of topo- logical entropy is more suitable for characterizing topological order in open systems, and we use their definition for our cal- culation

    Topological entropy As shown in Sec. III A 4, Levin-Wen’s definition of topo- logical entropy is more suitable for characterizing topological order in open systems, and we use their definition for our cal- culation. It is worth noting that in dissipative systems, the 13 topological entropy defined by Levin and Wen is closely re- lated to the topological d...

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    These three sets of quantum jump oper- ators are of the same form as Eq

    Topologically degenerate steady states To realize steady states with robust topological degeneracy, we design the following set of quantum jump operators: Lm,l = σx l P ( X p|l∈∂p Bp), Lz,l = √κzσz l , Lv = √κvAv, (64) where κz (κv) is the dissipative strength and it is the same for all links (vertices). These three sets of quantum jump oper- ators are of...

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    far from

    Robustness of topological degeneracy In this section, we aim to study the influence of local pertur- bations on steady states. First, we note that this model, similar to the discussion of Model-1 in Sec. III A, can be reduced to a classical Markovian dynamics, with the generator Γ0 = X l (σx l − 1)P 2( X p|l∈∂p Bp) + κv X v (Av − 1). (66) 15 (a) (b) (c) (...

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    In this section, we demonstrate the emergence of Z2 gauge field in the long-time dynamics of our dissipative model

    Dissipative deconfined Z2 gauge field In closed systems, one of the characteristics of topologi- cally ordered phases is the emergence of a deconfined gauge field [32–34]. In this section, we demonstrate the emergence of Z2 gauge field in the long-time dynamics of our dissipative model. By studying the steady-state property of the gauge field using pertur...

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    III A, this model can only have the steady-state degeneracy at h = 0

    h = 0 As analyzed in Sec. III A, this model can only have the steady-state degeneracy at h = 0 . Though the m particles appear in pairs, two particles can be separated far away from each other and in the limit thermodynamic L → ∞, the finite time evolution of one particle can be identified as a random walk process. Therefore, the dynamics of one particle ...

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    h ̸= 0 Then, when the perturbation is nonzero, the degeneracy is broken immediately. In the long time limit t ≫ κ−1 v , all m particle configurations are mixed by Av and Γ2d can be mapped to a spin model on the dual lattice in the Av = 1 sub- space, as long as we care about the low-lying spectrum around the steady state. With the mapping: Bp = −1 → τ z = ...

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    + p h(h + 1), β2 = (h + 1

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    Hs can be rewritten into a more illuminating form: Hs (2h + 1) = − X ⟨i,j⟩ 1 2(1 + η)τ x i τ x j + 1 2(1 − η)τ y i τ y j + hz X i τ z i

    This is an XY model with anisotropic interaction and a z-direction magnetic field. Hs can be rewritten into a more illuminating form: Hs (2h + 1) = − X ⟨i,j⟩ 1 2(1 + η)τ x i τ x j + 1 2(1 − η)τ y i τ y j + hz X i τ z i . (96) where η = 2 √ h(h+1) 2h+1 and hz = 2 2h+1. This model has been analyzed in Ref. [36] and we find that our model is just in a specia...

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    In this part, we study the relaxation process of those states 20 with contractible loops which finally evolve into steady states under Γ0 [Eq

    h = 0 When h = 0, type-A and type-B states are all steady states. In this part, we study the relaxation process of those states 20 with contractible loops which finally evolve into steady states under Γ0 [Eq. (66)]. For any states with loop excitation, because the loop length is non-increasing and prefers to decrease under Γ0, the loop excitation graduall...

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    The numerical verifi- cation of this result can be found in a related paper [20], where the result fits the above estimation really well. When h = 0 , type-A and type-B states are steady states, and those contractible loops (contractible open membranes on the dual lattice) contribute to the low-lying spectrums: ∆ ∼ L−2

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    The shrinking process of a large open membrane would still happen slowly (diffusively) and dominate the long-time dynamics

    h ̸= 0 For the trivial contractible loop states, we expect that a sim- ilar relaxation process also happens in other regimes of the topologically ordered phase ( 0 < h < h c), where the loop defects do not proliferate in the steady state. The shrinking process of a large open membrane would still happen slowly (diffusively) and dominate the long-time dyna...

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    + 𝐿𝑥,𝑙3(ℎ) 2𝑘2𝑘 − 2 2𝑘 + 2 2𝑘 FIG. 17. Transition from |m2k−2⟩⟨m2k−2|, |m2k⟩⟨m2k| and |m2k+2⟩⟨m2k+2| to |m2k⟩⟨m2k|. Inside the bracket, we mark the transition rate associated with the corresponding process. c(h + 1)β2k+2 + ((a − b − 2n)h − b)β2k + bhβ2k−2 = 0 (B3) Set β2k = β2k, then this equation can be simplified as follows: c(h + 1)β4 − (ch + b(h + 1))...

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    ¯A is path-connected In this part, we give the details of the calculation of entanglement entropySA when ¯A is path-connected ( ¯A has only one piece of connected area. See Fig. 5). For the eigenvalues λj and degeneracy Dj have already been listed in Eq. (28), we have SA = −TrρA log ρA = − X j Djλj log λj = − mAX j=0 mA j |G| T ⌊ n−mA+j 2 ⌋X k=⌈ j 2 ⌉ β2k...

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    ¯A is not path-connected In this part, we give the details of the calculation ofρA and SA in partition-1 where ¯A has two disjointed areas [See Fig. 7(1)]. There are four different states in ρA and we can label these states with the particle number in different regions. For convenience, we give the following definition: j ≡# of m particles totally inside ...

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    (D4) Following the standard procedure and with the condition that Tr(ρA) = 1, we have ρn A = |GA||G ¯A| |G| n−1 ρA, SA = lim n→1 1 1 − n log Tr(ρn A) = log |G| |GA||G ¯A|

    hx = hz = 0 For nonperturbed case hx = hz = 0, we choose one of the steady states ρss = 1 |G| P g∈G P ˜g∈G g| ⇑⟩⟨⇑ | g˜g, and calculate the reduced density matrix, ρA = Tr ¯Aρss = 1 |G| X g∈G X ˜g∈GA gA| ⇑⟩⟨⇑ | gA˜gA = |G ¯A| |G| X g∈G/G ¯A X ˜g∈GA gA| ⇑⟩⟨⇑ | gA˜gA. (D4) Following the standard procedure and with the condition that Tr(ρA) = 1, we have ρn A...

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    hz = 0, hx ̸= 0 Similar to the calculation of Model-1 in Sec. C, we only need to arbitrarily take one of the degenerate steady states, ρss = 1 T ′m X g∈G′ X g′∈G′ g( X k,{r} β2k m |m2k({r})⟩⟨m2k({r})|)g′, (D8) and the form of ρA = Tr ¯Aρss would depend on whether ¯A is connected. We first discuss the situation when there is only one connected piece of ¯A,...

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    We learn that the topological entropy is vulnerable to the fluctuation of m particles

    This result again shows the rationality of generalizing the topological entropy defined by Levin and Wen to steady states of open systems. We learn that the topological entropy is vulnerable to the fluctuation of m particles. The remaining topological entropy as well as the topological degeneracy is due to the fact that the e particles remain unaffected. ...

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    hx ̸= 0, h z ̸= 0 For hx, hz ̸= 0, remind the steady state: ρem = 1 T ′em X µ,ν X ke,{re} β2ke e keY j=1 Sz tj X g,g ′∈G g   X km,{pm} β2km m W µ x Wy ν kmY i=1 Sx ˜ti | ⇑⟩⟨⇑ | kmY i=1 Sx ˜ti W µ x W ν y   g′ keY j=1 Sz tj . (D16) Here {re} and {pm} give the positions of e and m particles. To calculate the reduced density matrix, we should simplify th...

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