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REVIEW 3 major objections 3 minor 33 references

Non-Hermitian Quantum Many-Body Scar Phase

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Non-Hermitian driving turns quantum many-body scars into a stable non-equilibrium phase, with a sharp first-order transition from thermalization.

desk verdict 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. read the letter →

arxiv 2507.22583 v1 pith:LRHFGDXP submitted 2025-07-30 quant-ph cond-mat.stat-mechcond-mat.str-el

Keita Omiya , Yuya O Nakagawa This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-el
keywords quantummany-bodyscarsnon-Hermitiandynamicsnon-equilibriumsteadystatesfirst-orderphasetransitionrandomcircuitspin-1XYmodelShiraishi-Moriconstructionmeasurement-inducedstabilization
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 3 minor

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.

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 (3)
  1. [Sec. 3.3, Eq. (44)] 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.
  2. [Sec. 4.2, Eq. (77)] 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.
  3. [Sec. 3.4.2] 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.
minor comments (3)
  1. [Sec. 3.1.2, Eq. (37)] 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.
  2. [Sec. 2.3, Eqs. (9) and (19)] 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).
  3. [Sec. 3.4.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is derived from the stated model parameters without fitting the target phase diagram.

full rationale

The paper's inputs are the model parameters themselves: the circuit probabilities a, b, c, the coupling constants J, V, g, and the system sizes. No parameter is fitted to reproduce the phase diagram, so the phase diagrams in Figs. 5, 6, 8, and 10 are honest outputs of the stated models. The RQC construction is mapped exactly to a classical statistical model and then approximated by mean-field, DMRG, and real-space RG, each of which is an independent numerical/analytical procedure. The effective field theory is built from the explicitly stated SU(q) spin model and the Schwinger-boson representation, with the ansatz for the SU(1,1) irrep justified in Appendices D and E rather than imported as an unexamined external result. The spin-1 XY exact diagonalization directly diagonalizes the non-Hermitian Hamiltonian and identifies the steady state as the eigenstate with largest imaginary part. The known variational limitation in Sec. 3.4.2—that the exact steady states in the solvable limits do not solve Eq. (44)—is a correctness/approximation concern, not a circularity: it shows the variational object may not be the true dominant eigenvector of the non-normal transfer matrix, but it does not show that the paper's equations reduce to their own inputs. The paper is transparent about this limitation, and the central claim does not depend on treating Eq. (44) as exact. The self-citations, Refs. [22,23], provide background examples of non-Hermitian emergence in QMBS models and are not load-bearing for the main results; Ref. [29] supports an auxiliary classification statement but the main derivation chain does not reduce to it. Overall, no step in the derivation is equivalent to its input by construction, and no fitted quantity is renamed as a prediction.

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

The central claim rests on standard quantum mechanics and the Shiraishi-Mori construction. The key assumptions are the uniqueness of the steady state, the validity of the non-Hermitian Hamiltonian as a model of stochastic measurements, the restriction to a particular SU(1,1) irrep, and the equivalence of the largest eigenvector of the transfer matrix with the steady state (which is flawed). No parameters are fitted to data.

assumptions (4)
  • domain assumption The steady state of the non-Hermitian channel is unique and independent of the initial state.
    Assumed in Sec. 4 (before Eq. (63): 'assuming that the steady state is independent of the initial state'); not proven for the RQC channel at generic parameters.
  • ad hoc to paper The physically relevant irrep of the emergent su(1,1) algebra is D_{|s|/2} ⊗ D_{(q-|s|)/2}.
    Eq. (77) restricts to this irrep; Appendix D and E provide a sketch and a mean-field argument, not a rigorous derivation.
  • domain assumption The non-Hermitian Hamiltonian −igΣP (Eq. 16) is equivalent to stochastic projective measurements.
    Standard for quantum trajectories when the measurement rate is constant; used throughout Secs. 2 and 5.
  • domain assumption The eigenvector of the non-Hermitian Hamiltonian with the largest imaginary part represents the steady state of the open system (Sec. 5).
    The authors note that a rigorous treatment would require a Lindblad operator; the simplified non-Hermitian Hamiltonian is an approximation.

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Pith. "Pith review of Non-Hermitian Quantum Many-Body Scar Phase." pith.science (2026). https://pith.science/paper/LRHFGDXP

@misc{pith2026250722583,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian Quantum Many-Body Scar Phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LRHFGDXP}},
  note         = {Machine review of arXiv:2507.22583}
}
abstract

We introduce a novel non-equilibrium phase -- the quantum many-body scar (QMBS) phase -- that emerges in non-Hermitian many-body dynamics when scarred wavefunctions are selectively stabilized via non-Hermitian driving. Projective measurements, or non-Hermitian counterparts, preferentially reinforce QMBS, counteracting the entropy growth that drives thermalization. As a result, atypical, high-energy scarred wavefunctions that are negligible in the long-time dynamics of closed systems become non-equilibrium steady states. We establish the existence of the QMBS phase and its sharp, first-order phase transition from an ergodic thermal phase, through both analytical arguments and numerical simulations of three representative models: a random quantum circuit model, the $SU(q)$ spin model, and the paradigmatic spin-1 XY model.

Figures

Figures reproduced from arXiv: 2507.22583 by the authors.

Figure 1
Figure 1. Simulation of the non-unitary dynamics of the spin-1 XY model ensuing from the [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The circuit architecture (left panel) is mapped to the two dimensional “spacetime” [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. The sites on the red line are identified as the time slice with the corresponding quantum [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Left panel: the dynamics can be expressed as the MPO acting on the MPS. Note [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: The expectation value of the magnetization [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Left panel: a tensor network representation of the RG procedure, where the circles [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Classification of the optimal configuration in our theory: the left panel describes Case [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: Left panel: the plot of mz in Eq. (88) for the steady state configuration as a function of g/J and V/J. We fix the parameters as κ = 1 and r = 1/10. The yellow region corresponds to the QMBS phase, and the purple region to the thermal phase. Right panel: the expectatio…
Figure 9
Figure 9. Figure 9: The minimum value of the factor mink tϑ cos k + mϑ. The positivity of this factor indicates the stability of the mean-field configuration in Case 1. Squad is diagonal in the momentum space, and for each k we can define the simple harmonic oscillator with the mass mk an…
Figure 10
Figure 10. Figure 10: The expectation value of the order parameter [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: Top panels: The imaginary part of 10 energy eigenvalues as a function of the strength [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]
Figure 12
Figure 12. Figure 12: The expectation value of PˆXY i,i+1 in Eq. (11). A number of follow-up questions should be discussed: in the simplest QMBS phase discussed in this paper, the transition is discontinuous. It is an interesting question whether this transition must be first-order or cont…

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