{"id":"30937334-dd2c-47bd-91b8-79e65e724f2c","arxiv_id":"2509.05411","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-Hermitian many-body Hamiltonians are mapped to Markov-chain generators, yielding new classical steady states: a Fermi-Dirac-like exclusion profile and exactly staggered, sector-dependent spin densities.","lead":"The authors reinterpret interacting non-Hermitian quantum many-body models as classical Markov chains, so wavefunction amplitudes become probabilities and steady states become exactly computable. Two worked models give a Fermi-Dirac-like density profile and exactly staggered spin profiles that differ from the quantum dynamics despite identical transitions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (8)'s Fermi-Dirac profile is a mean-field approximation whose own supplement restricts to weak asymmetry, yet the main-text figure uses λ+/λ-=0.1; this unstated limitation is the main load-bearing flaw.","rationale":"I read the manuscript and the reader's verdict in good faith. The central construction—L=D-H with probability-conserving diagonal terms—is explicit and coherent for the presented models, and the correlated spin-flip steady states are exact: the weights ψ∝γ^{n_tot/2} satisfy detailed balance on every pair-flip edge, and the combinatorics in S5/S6 reproduce the stated staggered profiles and Δρ_+^ss=2m/N for even N. I also checked small sectors by hand (e.g., N=4, m=1) and found no disconnectivity that would break those formulas. The anti-correlated spin-flip reduction to the n_max=1 interacting HN model is algebraically sound. The weak point is unambiguously the interacting-HN Fermi-Dirac formula. The supplement itself identifies the mean-field factorization as the source and restricts it to weak asymmetry; the main text both omits that provenance and displays the formula at λ+/λ-=0.1, which is strongly asymmetric and where the supplement's own numerics show disagreement. This is not an external consensus objection but an internal limitation that the authors themselves acknowledge, so it is precisely the kind of missing support that should be flagged. The concrete test of computing the exact steady state for the Fig. 1 parameters would settle whether Eq. (8) is quantitatively valid there; until then the quantitative Fermi-Dirac claim should be conditional. The central mapping claim and the exact spin-model results are not affected, so I do not see grounds for rejection.","tokens_in":35823,"tokens_out":27216,"duration_ms":291775,"concrete_test":"Compute the exact stationary distribution of the K-exclusion generator -Lλ of Eq. (7) for the Fig. 1 parameters (N=10, nmax=2, n=10, λ+=0.1, λ-=1) by numerically finding the right null vector of the Lλ matrix (or by long-time Gillespie simulation to convergence), then compare ρss(x) to Eq. (8). If max_x |ρ_num(x)-ρ_FD(x)| exceeds about 0.2 (10% of nmax), the displayed quantitative FD claim fails in the plotted regime. As a control, repeat at λ+/λ-=0.9 to confirm the mean-field formula is accurate in the weak-asymmetry regime; this isolates the asymmetry breakdown.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equation (8) of the main text is presented as the steady-state density of the interacting Hatano-Nelson Markov chain, with equality and an effective temperature set by 1/ln(λ-/λ+). The derivation in Supplemental Sec. S1 relies on the mean-field factorization Eq. (S5), and S1.2 explicitly states that the approximation is valid only when the hopping asymmetry is weak, because the error scales as (λ--λ+)(ε_{x,x+1}-ε_{x,x-1}). The main-text Fig. 1 and the text around Eq. (8) use λ+=0.1, λ-=1, nmax=2—a strongly asymmetric regime. The supplement's own Fig. S3, at nmax=1, shows visible deviations between the exact numerical steady state and the mean-field/Fermi-Dirac forms at exactly this ratio. No nmax=2 benchmark is shown, and the main text does not state that Eq. (8) is a mean-field result. Thus the first worked example's quantitative headline is unsupported in its own displayed parameter regime; at best it is a qualitative step-like profile. This does not invalidate the central L=D-H mapping or the exact correlated-spin steady states, but it is a load-bearing gap for the paper's 'Fermi-Dirac-like' claim. A related minor inconsistency is that Eq. (8) uses x0=n/nmax, whereas Supplement S1.3 derives x0≈n/nmax+1/2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a mapping between non-Hermitian many-body quantum Hamiltonians and continuous-time Markov chains. Given a Hamiltonian H whose off-diagonal matrix elements are nonnegative, one defines L=D-H with D chosen so that columns sum to zero; the state amplitudes become probabilities and evolve by exp(-Lt). Two models are studied: (i) the interacting Hatano-Nelson model with exclusion, mapped to a K-exclusion process, for which a Fermi-Dirac-like steady-state profile Eq. (8) is derived; (ii) a correlated spin-flip model, whose state space fragments into sectors labelled by a Néel imbalance m (even N, OBC/PBC) or parity Π (odd N, PBC), with exact staggered steady states Eqs. (12)-(15). The paper emphasizes that quantum and stochastic interpretations share transitions but differ in steady-state and dynamical behavior.","tokens_in":36070,"tokens_out":6760,"duration_ms":71074,"significance":"If the Fermi-Dirac claim is properly qualified, the paper is a valuable contribution: it gives a clean dictionary (L=D-H), and the correlated spin-flip steady states are exact and nontrivial. I verified the detailed-balance structure: the weight ψ∝γ^{n_tot/2} balances every single pair-flip edge, so Eqs. (13)-(15) and S53/S64 follow. The state-space fragmentation and boundary-condition dependence are clearly demonstrated. The main weakness is the unsupported quantitative status of Eq. (8) in the strong-asymmetry regime used in the main text.","major_comments":[{"comment":"Equation (8) is stated as an equality, but its derivation uses the mean-field factorization Eq. (S5). Supplement S1.2 itself concludes the approximation is valid when hopping asymmetry is weak, because the error is proportional to (λ_- - λ_+)(ε_{x,x+1} - ε_{x,x-1}). The main text's Fig. 1 uses λ_+=0.1, λ_-=1, nmax=2, a strongly asymmetric regime; the supplement's Fig. S3, at nmax=1, shows visible deviations between exact numerics and the analytic FD form at this ratio. No nmax=2 benchmark is given. Thus the quantitative FD profile (including the value of k_B T_eff) is not supported in the displayed parameter regime and should be presented as a mean-field approximation with its validity condition, or verified numerically in that regime.","section":"HN model with exclusion interactions, Eq. (8) and Supplement S1.2, Fig. S3"},{"comment":"The construction L=D-H is only a Markov generator if all off-diagonal matrix elements of H are nonnegative (real) in the chosen basis; otherwise L's off-diagonal entries are positive, corresponding to negative transition rates. The text says 'any given quantum Hamiltonian H' can be mapped. This is too broad; many non-Hermitian many-body Hamiltonians have negative or complex hoppings. The authors should state the positivity condition and clarify whether their formalism is intended for the class of stochastic-rate-compatible Hamiltonians. This does not affect the two examples (whose couplings are nonnegative rates), but it is load-bearing for the claimed generality.","section":"Markov chain formalism, Eqs. (4)-(7)"}],"minor_comments":[{"comment":"The main text uses x0=n/nmax in Eq. (8), while S1.3 derives x0≈n/nmax+1/2. This shift affects the quantitative profile and should be reconciled.","section":"Eq. (8) and Supplement S1.3"},{"comment":"Fig. S3 benchmarks only nmax=1. Since main-text Fig. 1 uses nmax=2, an nmax=2 comparison between exact numerics and the analytic form would be helpful, even if only as a supplement.","section":"Supplement Fig. S3"},{"comment":"Two effective temperatures are defined: k_B T^(site)_eff=(λ_-+λ_+)/(2(λ_- - λ_+)) and k_B T^(current)_eff=1/ln(λ_-/λ_+). The main text uses the latter without noting that the two coincide only in the weak-asymmetry limit.","section":"Supplement Eqs. (S15) and (S24)"},{"comment":"The boson normalization is nonstandard: \\hat b_x|n_x⟩=n_x|n_x-1⟩ without the usual sqrt factor. This convention should be stated explicitly at first use, since it affects all matrix elements.","section":"Eq. (1)"},{"comment":"The phrase 'considerable freedom' in choosing D should be sharpened: D must be a diagonal matrix with nonnegative entries such that each column of L=D-H sums to zero, so that L is a valid Markov-chain Laplacian.","section":"Eq. (4)"}],"recommendation":"major_revision","confidential_remarks":"The exact correlated-spin analysis is the strongest part of the paper. The main revision needed is to qualify the Fermi-Dirac result (Eq. 8) as a mean-field approximation with a stated validity regime, or to provide numerical verification in the parameter regime shown in Fig. 1. The generality claim of the L=D-H mapping should also be narrowed to Hamiltonians with nonnegative off-diagonal elements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely useful paper with a clean dictionary and two worked examples, one of which (correlated spin-flip) is exact and very nice. The other (interacting Hatano-Nelson) is oversold in the main text: the Fermi-Dirac formula is a mean-field result, and the supplement itself shows it fails quantitatively at the parameter point the main text puts front and center. That's a fixable presentation issue, not a fatal flaw.\n\nWhat's actually new: the L = D - H construction is textbook Markov-chain material (Merris; Mirzaev-Gunawardena), and the authors say so. The value is in the application: they spell out how to take a many-body non-Hermitian Hamiltonian, keep the transitions, and choose the diagonal terms to make a probability-conserving Laplacian. That's a helpful reference for anyone trying to connect quantum skin effects to classical processes. The second model is the real gem: the correlated spin-flip chain has exact steady states with a staggered profile, the m-invariant fragmentation, and the parity-indexed uniform states for odd N under PBC. I checked the detailed balance argument and the N=4 m=0 sector by hand and it holds. The linear contrast Δρ = 2m/N is clean.\n\nSoft spots: Eq. (8)'s Fermi-Dirac profile is derived via mean-field factorization in S1, and S1.2 explicitly says it's valid for weak asymmetry. The main text uses λ+/λ-=0.1, which is not weak; the supplement's own Fig S3 shows visible deviation for nmax=1 at that ratio. No nmax=2 benchmark is shown. The formula also has an x0 inconsistency (n/nmax vs n/nmax + 1/2) between Eq. (8) and S1.3. These are bounded: the mapping is fine, the qualitative step profile is likely right, and the exact spin results stand. The triple-spin Laplacian in S32 appears without derivation, and the sector-connectivity claims would benefit from a proof or at least a clearer argument. Minor.\n\nWho should read it: people working on non-Hermitian many-body physics, stochastic processes, or social/opinion dynamics looking for new exact tractable models. It deserves a serious referee; the authors just need to be honest about the mean-field provenance and show a benchmark in the displayed regime. I'd send it to review with the expectation of a revision.","headline":"Useful dictionary, exact spin-flip steady states, but the Fermi-Dirac headline is a mean-field approximation displayed in a regime where its own supplement shows it fails quantitatively.","tokens_in":36665,"tokens_out":2974,"would_cite":true,"duration_ms":34164,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a general dictionary that recasts interacting non-Hermitian quantum many-body Hamiltonians as continuous-time Markov chains, and shows that the stochastic steady states exhibit Fermi-Dirac-like spatial profiles and parit","keywords":["non-Hermitian many-body systems","Markov chain Laplacian","Hatano-Nelson model","asymmetric exclusion process","state-space fragmentation","Fermi-Dirac steady state","spin-flip dynamics","Néel order"],"falsifier":"Exact numerical steady state of the K-exclusion process (e.g., N=10, n=5, nmax=2, λ+=0.1, λ-=1) compared against Eq. (8) would settle the quantitative validity of the Fermi-Dirac form; the supplement's own Fig. S3 indicates visible deviations in this regime. Separately, computing the exact steady state of the correlated spin-flip model for N=8 and fixed m directly tests the linear relation Δρ=2m/N.","tokens_in":35575,"feed_emoji":"🎲","tokens_out":5376,"duration_ms":56970,"temperature":0.7,"pith_summary":"The paper proposes a general dictionary that turns interacting non-Hermitian quantum Hamiltonians into continuous-time Markov chains: the off-diagonal hopping terms are kept exactly the same, while diagonal terms are replaced by a probability-conserving Laplacian, so wavefunction amplitudes become actual probabilities. With this dictionary, the paper derives two concrete steady-state results. The first maps an interacting Hatano-Nelson chain to a biased exclusion process whose steady-state density is a real-space Fermi-Dirac distribution with an effective temperature set by the hopping asymmetry. The second maps a correlated spin-flip model to a stochastic process whose state space fragments into sectors labeled by a Néel invariant, and the steady state is a staggered profile with contrast proportional to that invariant (or, for odd rings under periodic boundary conditions, determined by particle-number parity). The paper argues these steady states are genuinely new: the quantum analogs do not relax to them, because quantum evolution is governed by oscillation and interference rather than by probability conservation.","feed_headline":"Quantum non-Hermitian models become Markov chains","feed_subtitle":"Same transitions as the quantum model, but steady states follow Fermi-Dirac and parity rules instead of oscillating.","key_machinery":"The central object is the Markov-chain Laplacian L = D - H, where H is the original non-Hermitian many-body Hamiltonian and D is a diagonal operator chosen so that every column of L sums to zero. This converts the transition amplitudes of H into probability-conserving rates; the zero-eigenvalue eigenvector of L is the steady state. For the first model the machinery reduces to a K-exclusion (partial asymmetric exclusion) process; for the second, to a kinetically constrained pair-flipping process whose conserved quantities are the Néel invariant m (even N or odd-N OBC) or the parity Π (odd-N PBC).","core_discovery":"The central claim is that any many-body non-Hermitian Hamiltonian can be reinterpreted as a Markov-chain generator L = D - H, where D is a diagonal operator that enforces probability conservation. This preserves all state transitions while changing the equation of motion from Schrödinger to a master equation, so real non-negative amplitudes evolve as probabilities. In the interacting Hatano-Nelson chain (with site-occupancy cap nmax), the resulting stochastic process is the K-exclusion process, whose open-boundary steady state is shown to be the Fermi-Dirac form ρ(x)=nmax/(1+e^{(x-x0)/k_B T_eff}) with k_B T_eff=1/ln(λ-/λ+). In the correlated spin-flip model, the Laplacian fragments the state","pith_inferences":["The mapping likely extends beyond the two models: any non-Hermitian many-body Hamiltonian with non-negative off-diagonal transition amplitudes admits a Markov-chain counterpart, so the framework could be used to engineer classical stochastic simulators of many-body non-Hermitian skin effects and related phenomena.","The Fermi-Dirac steady state implies a dictionary between nonreciprocity strength and an effective temperature in real space; one could test in a simple exclusion experiment whether tuning λ+/λ- across 1 produces negative-temperature distributions.","For the correlated spin-flip model, the linear relation Δρ=2m/N suggests a robust measurable probe: in an opinion-dynamics or agent-based setting, a persistent even-odd bias encodes the initial imbalance m, providing a form of initial-state memory without energetic barriers."],"forward_implications":["The interacting Hatano-Nelson chain, viewed stochastically, has a unique zero-mode steady state that all initial conditions converge to, so the Fermi-Dirac-like density profile is a universal attractor for the stochastic process—unlike the quantum case where no such attractor exists.","The anti-correlated spin-flip model reduces exactly to the nmax=1 interacting Hatano-Nelson model, so its steady state follows the same Fermi-Dirac-like profile with u+/u- playing the role of the asymmetry.","The correlated spin-flip model exhibits state-space fragmentation: the steady-state profile depends on the initial Néel order m, and the odd/even density contrast grows linearly as 2m/N, so an initial imbalance imprints itself permanently on the final spatial profile.","Under periodic boundary conditions with odd N, only the parity of the number of + particles survives as a conserved quantity; this yields uniform steady states, one of which cannot reach full polarization no matter how strong the asymmetry.","The paper argues these results reveal robust signatures of non-Hermitian phenomena in classical stochastic settings such as ecological networks, traffic flow, and social opinion dynamics."],"supporting_citations":[{"why":"Supplies the non-Hermitian Hatano-Nelson model that is the starting point for the first example.","marker":"[102]"},{"why":"Establishes that the interacting Hatano-Nelson Laplacian is mathematically equivalent to the K-exclusion process, whose steady-state properties support the Fermi-Dirac-like claim.","marker":"[103–106]"},{"why":"Provides the graph-Laplacian formalism used to construct L = D - H with probability conservation.","marker":"[70, 71, 100]"},{"why":"Supplies the concept of Hilbert-space fragmentation that underlies the sector structure of the correlated spin-flip model.","marker":"[117, 118]"},{"why":"Defines the even-odd imbalance (Néel order parameter) m that is the conserved quantity governing the staggered steady states.","marker":"[116]"}],"fun_headline_variants":["Non-Hermitian dynamics become classical Markov processes","Interacting non-Hermitian models yield Markov-chain steady states","Steady states from non-Hermitian Hamiltonians in Markov chains","Non-Hermitian many-body systems map onto Markov-chain generators"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The Fermi-Dirac-like steady-state formula for the interacting Hatano-Nelson chain rests on a mean-field factorization that ignores correlations between neighboring occupations; the supplement itself limits the approximation to weak hopping asymmetry.","fun_headline_variants_meta":{"raw":{"variants":["Non-Hermitian dynamics become classical Markov processes","Interacting non-Hermitian models yield Markov-chain steady states","Steady states from non-Hermitian Hamiltonians in Markov chains","Non-Hermitian many-body systems map onto Markov-chain generators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00177,"raw_usage":{"total_tokens":6800,"prompt_tokens":704,"completion_tokens":6096,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":6027}},"tokens_in":448,"tokens_out":6096,"duration_ms":46409,"temperature":1.0,"reasoning_tokens":6027,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:27:35.796684+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exact numerical steady state of the K-exclusion process (e.g., N=10, n=5, nmax=2, λ+=0.1, λ-=1) compared against Eq. (8) would settle the quantitative validity of the Fermi-Dirac form; the supplement's own Fig. S3 indicates visible deviations in this regime. Separately, computing the exact steady state of the correlated spin-flip model for N=8 and fixed m directly tests the linear relation Δρ=2m/N.","supporting_citations":[],"review_version":1}