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arxiv: 2604.02705 · v1 · submitted 2026-04-03 · ❄️ cond-mat.stat-mech · cond-mat.soft

The unique control features of topological stochastic and quantum systems

Ziyin Xiong , Aleksandra Nelson , Evelyn Tang This is my paper

Pith reviewed 2026-05-13 18:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft
keywords topological phasesstochastic systemsquantum systemsspectral propertiesnon-reciprocitytopologically emerging stateedge statesnon-equilibrium currents
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The pith

Topological features create a unique persisting state in stochastic systems that quantum systems lack.

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

The paper compares how topology and non-reciprocity shape spectral properties in stochastic versus quantum lattice models. Increasing topology clusters states around the steady state in stochastic systems but disperses them from zero energy in quantum ones, while non-reciprocity does the reverse in each case. These opposite responses supply separate control knobs for tuning the longest-lived states and the protecting gap. The work identifies a mode called the topologically emerging state that appears only in stochastic systems and survives across models, dimensions, and non-equilibrium currents.

Core claim

We derive analytical expressions for the spectral properties of simple quantum and stochastic models on the same lattice. Non-reciprocity moves states away from the steady-state in stochastic systems while clustering states at zero-energy in quantum systems. Making the system more topological clusters more states around the steady-state in stochastic systems but moves states away from the zero-energy state in quantum systems. We discover a mode unique to stochastic systems that we dub the topologically emerging state, which persists across different models and dimensions, including in the presence of non-equilibrium currents. These results provide control parameters for selection and modulat

What carries the argument

The topologically emerging state, a spectral mode that arises only in stochastic topological systems and remains robust under non-reciprocity and dimensional changes.

Load-bearing premise

Simple lattice models and their derived analytical spectral properties capture the essential behavior of more complex real-world quantum and stochastic topological systems.

What would settle it

Direct observation that the topologically emerging state fails to appear or loses its persistence in a higher-dimensional stochastic model with explicit non-equilibrium currents would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.02705 by Aleksandra Nelson, Evelyn Tang, Ziyin Xiong.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

Topological phases support edge states that can be robust to material deformations and other perturbations. While well-studied in quantum systems, topological phases have also been observed in stochastic and biochemical systems, yet it remains unclear which of their properties remain similar or different from those in quantum systems. In this paper, we derive analytical expressions for the spectral properties of simple quantum and stochastic models on the same lattice to rigorously characterize these complex systems. Intriguingly, we find that non-reciprocity moves states away from the steady-state in stochastic systems while clustering states at zero-energy in quantum systems. In contrast, making the system more topological does the opposite: it clusters more states around the steady-state in stochastic systems but moves states away from the zero-energy state in quantum systems. These results provide control parameters for selection and modulation of different purposes while quantifying the size of gap which protects the longest-lived states. Lastly, we discover a mode unique to stochastic systems that we dub the topologically emerging state, which persists across different models and dimensions, including in the presence of non-equilibrium currents.

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

2 major / 2 minor

Summary. The paper derives analytical expressions for the spectral properties of simple quantum and stochastic topological models defined on identical lattices. It reports that non-reciprocity shifts states away from the steady state in stochastic systems while clustering them at zero energy in quantum systems, whereas increasing topological character produces the opposite clustering behavior. The authors identify a mode unique to the stochastic case, termed the topologically emerging state, and claim it persists across models, dimensions, and in the presence of non-equilibrium currents, providing control parameters and gap sizes for longest-lived states.

Significance. If the analytical spectra are correctly derived and the emerging state is shown to be absent from quantum analogs while robust to model variations, the work supplies concrete, tunable distinctions between stochastic and quantum topological control. This could inform design of non-equilibrium systems in statistical mechanics and biochemical networks by quantifying how topology and drive separately modulate state clustering and lifetimes.

major comments (2)
  1. [Abstract / spectral analysis] Abstract and results on spectral properties: the uniqueness of the topologically emerging state is asserted to hold across models and dimensions, yet the derivations are performed only on minimal translation-invariant lattices with nearest-neighbor couplings; no model-independent argument is supplied showing that zero-mode clustering or the emerging state is forced by stochasticity plus topology rather than by the specific rate-matrix choice, which is load-bearing for the central distinction from quantum systems.
  2. [Results / spectral derivations] Comparison of stochastic and quantum spectra: the opposite effects of non-reciprocity and topology on state clustering are described qualitatively, but explicit side-by-side eigenvalue expressions or tables for the same lattice are needed to confirm the emerging state is absent from the quantum spectrum and survives non-equilibrium currents; without these, the claimed control features rest on unverified generality.
minor comments (2)
  1. [Methods] Notation for the steady-state eigenvalue and the emerging mode should be standardized across equations to avoid ambiguity when comparing stochastic and quantum cases.
  2. [Figures] Figure captions should explicitly label which curves correspond to the topologically emerging state and indicate the lattice size and boundary conditions used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which have helped us improve the clarity and rigor of our manuscript. We address each major comment below and have made revisions accordingly.

read point-by-point responses

  1. Referee: [Abstract / spectral analysis] Abstract and results on spectral properties: the uniqueness of the topologically emerging state is asserted to hold across models and dimensions, yet the derivations are performed only on minimal translation-invariant lattices with nearest-neighbor couplings; no model-independent argument is supplied showing that zero-mode clustering or the emerging state is forced by stochasticity plus topology rather than by the specific rate-matrix choice, which is load-bearing for the central distinction from quantum systems.

    Authors: We thank the referee for pointing this out. Our derivations focus on minimal models to derive exact analytical expressions, which we then use to illustrate the general behavior. We have checked the emerging state in additional models beyond the minimal lattice, including in 2D and with longer-range couplings (as mentioned in the manuscript). However, we agree that a model-independent argument would be ideal. In the revised manuscript, we will expand the discussion to explicitly state the assumptions and the representative character of the models, and note that the distinction stems from the non-Hermitian nature of the stochastic generator versus the Hermitian quantum Hamiltonian. We will also add references to related works on general stochastic topological systems to support the broader applicability. revision: partial

  2. Referee: [Results / spectral derivations] Comparison of stochastic and quantum spectra: the opposite effects of non-reciprocity and topology on state clustering are described qualitatively, but explicit side-by-side eigenvalue expressions or tables for the same lattice are needed to confirm the emerging state is absent from the quantum spectrum and survives non-equilibrium currents; without these, the claimed control features rest on unverified generality.

    Authors: We agree that side-by-side comparisons would make the distinction clearer. The analytical expressions for both cases are derived in the main text (see Eqs. for stochastic and quantum spectra). In the revision, we will add a new table that lists the eigenvalues for the stochastic and quantum versions on the same 1D lattice, explicitly showing the topologically emerging state (which has no counterpart in the quantum spectrum) and confirming its survival under non-equilibrium driving. This will provide direct verification of the claimed control features. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations are direct from lattice model equations

full rationale

The paper states it derives analytical expressions for spectral properties directly from the rate matrices and Hamiltonians of minimal lattice models. No parameters are fitted to data subsets and then relabeled as predictions; the topologically emerging state is identified by explicit spectral calculation rather than by self-definition or renaming. No load-bearing self-citations or uniqueness theorems imported from prior author work appear in the derivation chain. The analysis remains self-contained within the chosen models and does not reduce any central claim to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on analytical derivations from simple lattice models; no free parameters are mentioned, and the only domain assumption is that these minimal models suffice to characterize topological features in both quantum and stochastic regimes.

axioms (1)
  • domain assumption Simple lattice models capture the essential topological and spectral features of both quantum and stochastic systems.
    Invoked to justify deriving analytical expressions that generalize to the claimed control features and unique state.
invented entities (1)
  • topologically emerging state no independent evidence
    purpose: A long-lived mode that appears uniquely in stochastic systems due to topology and persists across models and dimensions.
    Discovered within the stochastic models but presented without external falsifiable evidence outside the derivations.

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Reference graph

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