{"id":"966bc98e-df99-434e-90be-13ac8cc67ae5","arxiv_id":"2507.22583","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-Hermitian driving can stabilize quantum many-body scar states as non-equilibrium steady states, with a predicted first-order phase transition from a thermal phase.","lead":"This paper shows that non-Hermitian driving, such as repeated measurements, can keep special high-energy quantum states (scars) alive and turn them into long-lived steady states. It predicts a sharp phase transition between this new 'scar phase' and the usual thermal phase.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The variational step in Eq. (44) maximizes <Φ|U†U|Φ>, the largest singular value of non-normal U, not its largest eigenvalue, so the RQC phase diagrams in Fig. 5 may not describe the stated steady state of L.","rationale":"The reader's weakest assumption is the same concern identified here: the variational replacement of the dominant eigenvector of a non-normal transfer matrix by the dominant singular vector of U†U. The paper's only rigorous pieces are the exact solvable limits in Sec. 3.1.2; the RQC mapping is transparent and plausible, but the variational step in Eq. (44) changes the object being computed. The manuscript itself supplies a red flag by admitting that the exact steady states do not satisfy Eq. (44) and that the variational/MPS methods underestimate the QMBS phase relative to the solvable limit. The remaining evidence is not independent enough to rescue the central claim: the real-space RG is an uncontrolled approximation of the same mapped partition function; the SU(1,1) effective field theory assumes the physically relevant irrep rather than deriving it; and the XY-model exact diagonalization is restricted to L ≤ 10 with no clear finite-size scaling and no code or data. Since the reader already recommends REJECT and my analysis supports that verdict, no verdict change is needed. The concrete check above would be decisive: if the actual dominant eigenvector of U/L reproduces the variational phase diagram, the manuscript should be revisited as CONDITIONAL or ACCEPT.","tokens_in":31855,"tokens_out":4521,"duration_ms":52567,"concrete_test":"For small L (e.g., L=8) and several (a,c) values spanning the Fig. 5 phase boundary, construct the exact transfer matrix U from Eq. (42) (or the channel L from Eq. (34)) and compute its actual dominant eigenvector by power iteration on U/L directly, not on U†U. Evaluate the unrotated magnetization ⟨M_eff⟩ of that eigenvector and compare with the mean-field/DMRG curves in Fig. 5. Also check whether U†U ≠ UU† to confirm non-normality. If the true dominant-eigenvector magnetization reproduces Fig. 5, the concern is mitigated; if it disagrees, especially at small c or near the reported boundary, the phase diagram is not for the stated steady state.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Sec. 3.3, the paper replaces the dominant eigenvalue problem of the transfer matrix U (Eq. 42) with the Rayleigh quotient |λ|^2 = max_Φ <Φ|U†U|Φ>/<Φ|Φ> (Eq. 44). This is exact only for the largest eigenvalue of U†U, i.e. the largest singular value of U, and its maximizer is a right singular vector. To identify this with the dominant eigenvector of U, one must show U is normal (U†U = UU†); the paper never does, and for a non-symmetric, non-Hermitian channel normality is not expected. The distinction is load-bearing because the non-equilibrium steady state is defined as the eigenvector of L (equivalently U) with largest real part, not the singular vector of U. The paper gives direct evidence that the variational object is not the true steady state: in Sec. 3.4.2 it states that the exact steady states |↑⋯↑⟩ and |↓⋯↓⟩ of the solvable limits do not solve Eq. (44), and that the variational calculation puts the QMBS phase only at large c whereas the exact solution puts the whole line except (a,c)=(1,0) in the QMBS phase. Thus the mean-field and DMRG phase diagrams in Fig. 5, and the magnetization used as an order parameter, are not shown to describe the steady state of the original channel. The real-space RG and effective field theory do not repair this: RG is an uncontrolled approximation of the same partition function, and the SU(1,1) analysis rests on an assumed irrep (Eq. 77). The central claim of a QMBS phase with a sharp first-order transition therefore rests on a variational problem that may have a different fixed point from the true NESS.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new non-equilibrium phase, the quantum many-body scar (QMBS) phase, in non-Hermitian many-body dynamics, in which projective measurements or non-Hermitian driving preferentially stabilize scarred wavefunctions against thermalization. The authors analyze three models: a random quantum circuit (RQC) model with a Shiraishi-Mori scar subspace, a large- q SU(q) spin model, and the spin-1 XY model with a non-Hermitian perturbation. For the RQC model, the dynamics is mapped to a two-dimensional classical statistical mechanics problem; the paper presents exact solvable limits (b = 0 and c = 0), a variational mean-field/DMRG treatment of a rotated transfer matrix, and a real-space RG analysis, all leading to a claimed first-order transition between a thermal phase and a QMBS phase. The SU(q) model is treated by an effective field theory built on an assumed SU(1,1) irrep, and the spin-1 XY model is studied by exact diagonalization. The central claim is that the QMBS phase exists as a sharp phase of non-Hermitian dynamics, distinct from both thermalization and measurement-induced entanglement transitions.","tokens_in":32192,"tokens_out":7094,"duration_ms":89943,"significance":"If the central claim were established, the paper would introduce a genuinely new non-equilibrium phase and a general mechanism for stabilizing otherwise fragile quantum many-body scars, with potential experimental relevance to monitored or post-selected quantum dynamics. The paper has clear strengths: the exact solvable limits of Sec. 3.1.2 are correct and already show that infinitesimal measurement stabilizes the scarred steady state in the absence of scar-breaking perturbation, and the exact-diagonalization study of Sec. 5 directly accesses the steady state and exhibits a first-order-looking jump in the order parameter. These are valuable, falsifiable results. However, the main RQC-based evidence and the analytical field theory both rest on approximations that are not shown to describe the true steady state of the original channel, so the existence of a sharp QMBS phase is not established at the level claimed in the abstract.","major_comments":[{"comment":"The variational step that defines the phase diagram identifies the largest eigenvalue of the transfer matrix U with the maximum of the Rayleigh quotient <Φ|U†U|Φ>/<Φ|Φ>. For a non-normal U, this maximum gives the largest singular value and a right singular vector, not the dominant eigenvalue and eigenvector of U. The non-equilibrium steady state is defined in Sec. 3 as the eigenvector of the channel L (equivalently of U after the rotation) with largest real part, so the variational object is not obviously the steady state. The paper never proves that U is normal, and for the present non-Hermitian, non-symmetric channel normality is not expected. This is not a technicality: Sec. 3.4.2 explicitly states that the exact steady states |↑⋯↑> and |↓⋯↓> of the solvable limits do not solve Eq. (44), and that the variational calculation places the QMBS phase only at large c whereas the exact solution places the whole b = 0 line except (a,c) = (1,0) in the QMBS phase. The attribution of this discrepancy to the rotated quantization axis does not repair the mismatch; it confirms that the variational target is not the steady state. Consequently, the mean-field and DMRG phase diagrams in Fig. 5, and the magnetization used as an order parameter, are not shown to describe the steady state of the original channel. This is a load-bearing issue for the RQC-based claim of a sharp first-order QMBS transition.","section":"Sec. 3.3, Eq. (44)"},{"comment":"The effective field theory restricts the dynamics to the single irrep D_{|s|/2} ⊗ D_{(q-|s|)/2} of the emergent SU(1,1) algebra. The paper itself states that the initial state |1_{SU(q)}> does not belong to a single irrep and that a rigorous treatment requires all irreps. The argument in Appendix D that non-symmetric irreps correspond to traceless density matrices is not sufficient to exclude them from the linear, trace-non-preserving evolution: a traceless component can affect the transient dynamics and the normalization of the state, and the dominant eigenvector of the unnormalized channel need not be a positive trace-one density matrix. Therefore the SU(1,1) phase diagram in Fig. 8 and the spin-wave stability analysis in Sec. 4.4 are conditional on an unproven representation-theoretic assumption. The mean-field justification in Appendix E is a heuristic, not a derivation. Since this analytical argument is one of the three pillars supporting the claimed existence of the QMBS phase, this assumption needs either proof or a clear statement that the field-theoretic analysis is an uncontrolled approximation.","section":"Sec. 4.2, Eq. (77)"},{"comment":"The paper's own benchmark against the exactly solvable limit shows that the variational method does not reproduce the known steady state even qualitatively: the exact b = 0 solution says that any c > 0 drives the system into the QMBS phase, while Fig. 5 shows the QMBS phase only for sufficiently large c. This is a direct falsification of the numerical method as a probe of the steady state in a limit where the answer is known. The text acknowledges this and attributes it to the rotated quantization axis, but no independent evidence is provided that the variational state becomes correct away from the solvable limit. A revised manuscript should either replace Eq. (44) by a correct dominant-eigenvector computation (for example, a power method acting directly on U, or a method for non-normal transfer matrices) and benchmark it against the b = 0 exact line, or substantially weaken the claims drawn from the RQC model.","section":"Sec. 3.4.2"}],"minor_comments":[{"comment":"The inequality |s|^l ≤ S_l(t) cannot be correct as written: |s|^l is the dimension of the l-site scar subspace, while S_l(t) is an entanglement entropy. The correct statement is the upper bound S_l(t) ≤ l log |s| (or equivalently S_l(t) ≤ log of the subspace dimension). Please clarify.","section":"Sec. 3.1.2, Eq. (37)"},{"comment":"The spin-1 XY Hamiltonian is written with the interaction term J( S^x_i S^y_{i+1} + S^y_i S^y_{i+1} ), which appears to be a typo; the standard form (and the one used in the cited Ref. [27]) is J( S^x_i S^x_{i+1} + S^y_i S^y_{i+1} ). The same typo appears in Eq. (19).","section":"Sec. 2.3, Eqs. (9) and (19)"},{"comment":"The sentence attributing the discrepancy between the variational result and the exact solvable limit to the rotated quantization axis is too terse; since the exact steady states do not solve Eq. (44), the reader needs an explicit discussion of why the rotated transfer matrix should nevertheless be expected to share the dominant eigenvector with the original channel, or why the variational approximation is still predictive.","section":"Sec. 3.4.2"}],"recommendation":"major_revision","confidential_remarks":"The central idea is interesting and the exact solvable limits and the spin-1 XY numerical study are valuable, but the RQC phase diagram rests on the invalid variational replacement in Eq. (44) and the field-theoretic argument rests on an unproven irrep assumption. These are load-bearing but potentially fixable: the authors could compute the true dominant eigenvector of U directly, benchmark against the b = 0 exact solution, and either prove or relax the irrep assumption. I therefore recommend major revision rather than rejection, but the authors should be told that the current version does not establish the abstract's central claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the stress-test note is right. Eq. (44) maximizes <Phi|U†U|Phi>, i.e. the largest singular value of a non-normal transfer matrix, not its largest eigenvalue. The paper never shows U is normal, and Sec. 3.4.2 concedes that the exact steady states of the solvable limits do not solve Eq. (44). That is a load-bearing mismatch between the variational object and the actual non-equilibrium steady state of L.\n\nWhat is genuinely new: the notion of a distinct QMBS phase selected by non-Hermitian driving, the RQC-to-classical mapping, and the SU(1,1) effective theory. The exact solvable limits in Sec. 3.1.2 are correct and give a sharp result: infinitesimal measurement stabilizes the scar subspace when b=0. The spin-1 XY exact diagonalization (L up to 10) directly targets the NESS (largest imaginary part) and shows a clear jump for complex V and near-instant stabilization for real V. That is the strongest evidence in the paper and is not affected by the transfer-matrix issue.\n\nSoft spots beyond Eq. (44): the real-space RG is an uncontrolled approximation of the same partition function, and the SU(1,1) analysis rests on an assumed irrep (Eq. 77). No code or data is provided, and ED sizes are small. Still, the XY numerics suggest the qualitative claim may survive; the problem is that the Fig. 5 phase diagram is not currently tied to the actual steady state.\n\nWho this is for: people working on QMBS stability, non-Hermitian dynamics, and measurement-induced phases. A serious referee should engage: the exact limits and XY ED are worth publishing even if the RQC derivation needs major rework. Recommendation: send to peer review, require the authors to either prove normality of U or replace the variational step with a direct NESS computation for small systems, and provide data/code.","headline":"The paper has a genuinely interesting idea and one clean exact result, but the main analytical engine for the RQC phase diagram is solving a variational problem that is not the steady-state problem.","tokens_in":32761,"tokens_out":1957,"would_cite":false,"duration_ms":23340,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Non-Hermitian driving turns quantum many-body scars into a stable non-equilibrium phase, with a sharp first-order transition from thermalization.","keywords":["quantum many-body scars","non-Hermitian dynamics","non-equilibrium steady states","first-order phase transition","random quantum circuit","spin-1 XY model","Shiraishi-Mori construction","measurement-induced stabilization"],"falsifier":"Take the random circuit model at small system size (for example $L=6$ to $8$), construct the exact transfer matrix $\\hat U$ or the original channel $\\mathcal{L}$, and compute its complete spectrum; then compare the eigenvector with the largest eigenvalue against the mean-field and DMRG variational states. If their scar weights differ substantially in the parameter region where the paper predicts the QMBS phase, the variational identification fails and the phase boundary must be redrawn.","tokens_in":31604,"feed_emoji":"⚛️","tokens_out":4837,"duration_ms":57079,"temperature":0.7,"pith_summary":"Quantum many-body scars are atypical high-energy eigenstates that evade thermalization in closed systems, but in ordinary unitary dynamics their weight is negligible and perturbations destroy them. This paper argues that the situation reverses under non-Hermitian driving: by preferentially damping everything outside the scar subspace, the driving converts scarred wavefunctions into true non-equilibrium steady states that are robust against scar-breaking perturbations. The authors establish this QMBS phase in three models—a random quantum circuit, a large-$q$ $\\mathrm{SU}(q)$ spin chain, and the spin-1 XY model—and find a sharp first-order phase transition separating it from an ergodic thermal phase. This makes quantum many-body scars potentially observable and manipulable in open quantum systems rather than fragile exceptions to eigenstate thermalization.","feed_headline":"Measurements turn fragile quantum scars into a stable phase","feed_subtitle":"Non-Hermitian driving makes scar states robust steady states, with a sharp first-order transition from thermalization.","key_machinery":"The argument runs through several linked devices. The Shiraishi–Mori construction provides the scar subspace $\\mathcal{S}$: a common kernel of local projectors that is invariant under the Hamiltonian, so exact scar eigenstates exist by construction. The random circuit model averages the two-site gates and the scar-preserving projector into a quantum channel whose density-matrix dynamics closes on two local states, the maximally mixed thermal state and the scar-subspace state; this maps to an effective spin-1/2 chain and then, via a rotation of the quantization axis, to a two-dimensional classical lattice model with transfer matrix $\\hat U$. The variational estimate of the largest eigenvalue of $\\hat U^\\dagger \\hat U$ supplies the phase diagram. In the large-$q$ limit the same physics is captured by an emergent $\\mathrm{SU}(1,1)$ spin chain, whose coherent-state path integral yields a Ginzburg–Landau-type free energy whose maximum determines the steady state. The first-order nature of the transition comes from a discontinuous jump in the scar magnetization between the two phases.","core_discovery":"The central claim is that selective non-Hermitian driving—the continuous-time equivalent of repeatedly projecting onto configurations that avoid the scar subspace—stabilizes quantum many-body scar states as the non-equilibrium steady states of an open system. Because the scar subspace is invariant under the unperturbed dynamics and is nearly annihilated by the non-Hermitian term, scarred wavefunctions acquire a large imaginary-energy advantage over thermal ones; when that advantage exceeds the entropy gain of the thermal phase, the steady state jumps discontinuously into the scarred phase. The paper presents this as a genuine phase of non-Hermitian matter, distinct from the measurement-induced entanglement transition, and supports it with analytic transfer-matrix, mean-field, DMRG, real-space RG, and exact-diagonalization results.","pith_inferences":["Beyond the paper's three models, the same selective-decay mechanism could stabilize other fragile states, such as topological or chiral wavefunctions, provided they lie in the kernel of the non-Hermitian drive; the paper raises this as a question rather than a claim.","The variational transfer-matrix step (Eq. 44) assumes that maximizing $\\langle\\Phi|\\hat U^\\dagger \\hat U|\\Phi\\rangle$ finds the steady state; if $\\hat U$ is non-normal, that maximum gives the largest singular value rather than the largest eigenvalue, so an exact diagonalization check on small circuits would directly test whether the reported phase boundaries survive.","The result suggests a concrete experimental signature in Rydberg or trapped-ion chains: after repeated postselected projections, the scarred order parameter should show a discontinuous jump as the driving strength crosses a threshold, analogous to the jump seen in the spin-1 XY exact diagonalization."],"forward_implications":["If the QMBS phase exists, scar states cease to be fragile curiosities of closed systems; they become the attractors of non-Hermitian dynamics and can persist indefinitely rather than decaying at late times.","The sharp first-order transition gives a quantitative order parameter, such as the expectation value of the scar-projector order parameter $\\hat O$, that experiments can measure to locate the phase boundary.","The random circuit mapping means the phase transition can be studied with classical statistical mechanics tools, including transfer-matrix and renormalization-group methods, allowing predictions beyond the specific models treated here.","In the spin-1 XY model, a Hermitian perturbation is compatible with a QMBS phase for any nonzero non-Hermitian driving strength $g$, while a genuinely non-Hermitian perturbation instead exhibits a finite critical $g$; the nature of the perturbation changes the phase diagram.","Both phases carry volume-law entanglement, so the transition is distinguishable from a measurement-induced entanglement transition: it is a transition in the steady-state density matrix rather than in entanglement scaling."],"supporting_citations":[{"why":"Supplies the Shiraishi–Mori construction that defines the scar subspace and guarantees exact non-thermal eigenstates, the algebraic foundation of every model in the paper.","marker":"[10]"},{"why":"Introduces and names quantum many-body scars as weak ergodicity breaking, the target phenomenon that the paper aims to stabilize.","marker":"[12]"},{"why":"Proposes that non-Hermitian 'Fock skin' effects stabilize scarred dynamics, the starting point for the paper's non-Hermitian stabilization mechanism.","marker":"[21]"},{"why":"Provides the spin-1 XY model whose scar states and order parameter are used for the exact diagonalization checks of the phase transition.","marker":"[27]"},{"why":"Shows how adding a non-Hermitian term makes the PXP model exactly solvable and reveals hidden Shiraishi–Mori structure, motivating the algebraic connection between QMBS and non-Hermitian physics.","marker":"[22]"},{"why":"Extends the hidden local-projector embedding to other scar models, supporting the claim that non-Hermitian terms arise naturally in QMBS settings.","marker":"[23]"},{"why":"Reports the Rydberg experiment that first observed dynamical signatures of scars, the empirical motivation for asking whether scarred dynamics can be stabilized.","marker":"[11]"}],"fun_headline_variants":["Non-Hermitian driving turns quantum scars into a phase","Scars become a steady phase via non-Hermitian driving","First-order transition to a scarred non-equilibrium phase","Non-Hermitian driving stabilizes a scar phase","Quantum scars become steady under non-Hermitian driving"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the state maximizing $\\langle\\Phi|\\hat U^\\dagger \\hat U|\\Phi\\rangle$ over matrix-product states is the true steady state of the averaged channel; if the transfer matrix $\\hat U$ is not normal, this variational maximum gives the largest singular value, and the resulting 'steady state' need not be the eigenstate with the largest eigenvalue of the original non-Hermitian dynamics.","fun_headline_variants_meta":{"raw":{"variants":["Non-Hermitian driving turns quantum scars into a phase","Scars become a steady phase via non-Hermitian driving","First-order transition to a scarred non-equilibrium phase","Non-Hermitian driving stabilizes a scar phase","Quantum scars become steady under non-Hermitian driving"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000741,"raw_usage":{"total_tokens":3250,"prompt_tokens":830,"completion_tokens":2420,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":2340}},"tokens_in":446,"tokens_out":2420,"duration_ms":19300,"temperature":1.0,"reasoning_tokens":2340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:33:57.934968+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the random circuit model at small system size (for example $L=6$ to $8$), construct the exact transfer matrix $\\hat U$ or the original channel $\\mathcal{L}$, and compute its complete spectrum; then compare the eigenvector with the largest eigenvalue against the mean-field and DMRG variational states. If their scar weights differ substantially in the parameter region where the paper predicts the QMBS phase, the variational identification fails and the phase boundary must be redrawn.","supporting_citations":[{"cited_title":"Systematic construction of counterexamples to the eigenstate thermalization hypothesis.Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the Shiraishi–Mori construction that defines the scar subspace and guarantees exact non-thermal eigenstates, the algebraic foundation of every model in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces and names quantum many-body scars as weak ergodicity breaking, the target phenomenon that the paper aims to stabilize."},{"cited_title":"En- hancedmany-bodyquantumscarsfromthenon-hermitianfockskineffect","cited_arxiv_id":null,"evidence_quote":"Proposes that non-Hermitian 'Fock skin' effects stabilize scarred dynamics, the starting point for the paper's non-Hermitian stabilization mechanism."},{"cited_title":"Weakergodicitybreakingandquantummany-body scars in spin-1xy magnets","cited_arxiv_id":null,"evidence_quote":"Provides the spin-1 XY model whose scar states and order parameter are used for the exact diagonalization checks of the phase transition."},{"cited_title":"Quantum many-body scars in bipartite rydberg arrays originating from hidden projector embedding.Phys","cited_arxiv_id":null,"evidence_quote":"Shows how adding a non-Hermitian term makes the PXP model exactly solvable and reveals hidden Shiraishi–Mori structure, motivating the algebraic connection between QMBS and non-Hermitian physics."},{"cited_title":"Fractionalizationpavesthewaytolocalprojectorembed- dings of quantum many-body scars.Phys","cited_arxiv_id":null,"evidence_quote":"Extends the hidden local-projector embedding to other scar models, supporting the claim that non-Hermitian terms arise naturally in QMBS settings."},{"cited_title":"Zibrov, Manuel Endres, Markus Greiner, VladanVuletić,andMikhailD.Lukin","cited_arxiv_id":null,"evidence_quote":"Reports the Rydberg experiment that first observed dynamical signatures of scars, the empirical motivation for asking whether scarred dynamics can be stabilized."}],"review_version":1}