{"id":"9d120055-e4a4-460e-90ea-214f5fe6fb0f","arxiv_id":"2507.20155","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For PT-broken quenches, the biorthogonal density matrix drives exponential growth of observables and entanglement, but free-fermion TTC entropy decays linearly due to approximate spectral symmetry.","lead":"This paper studies what happens to observables, quantum geometry, and entanglement measures when a non-Hermitian PT-symmetric system is suddenly quenched into a PT-broken phase, using a biorthogonal formalism. The authors find exponential growth in most quantities, but a surprising linear decay in the Tu-Tzeng-Chang entropy for non-interacting fermions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (34) under-specifies the multiplicity of dominant correlation eigenvalues; without M=1 the linear-decay slope is not -2EIt, and Table I's 'linear growth' contradicts the text.","rationale":"The reader's conditional verdict is appropriate. The paper's numerics for the XXZ chain at Jz=0 and the Yang-Lee model provide direct evidence for linear decay, and the interacting crossover in Fig. 5(b) supports the qualitative interacting-versus-free-fermion distinction. However, the analytical derivation of the linear decay, Eq. (34), is the weakest load-bearing step: it implicitly assumes a single dominant eigenvalue without proof, and the spectral-symmetry explanation is qualitative rather than derived. This is a concrete, checkable gap, not a disagreement with consensus. The Table I 'linear growth' entry is an internal inconsistency that should be corrected. The reader's own concern about the exceptional-point limit and nonzero-overlap assumption is real but secondary: away from exceptional points and for generic initial states the expansion in Eq. (15) should hold, and the numerics operate in that regime. Since the central claim is plausible and numerically supported, the verdict should remain conditional rather than being upgraded or rejected, but the conditions should include an explicit verification of the multiplicity and slope in Eq. (34).","tokens_in":15186,"tokens_out":14173,"duration_ms":147942,"concrete_test":"Compute the single-particle correlation matrix C_A(t) for the Jz=0 non-Hermitian XXZ chain at several late times (e.g., t=20, 50, 100), diagonalize it, and extract the eigenvalues νδ(t). Determine how many eigenvalues grow as e^{2EIt} and whether all others grow with smaller exponents. Then compute S_TTC(t) from Eq. (33) and fit the slope; compare with -2EI. Repeat for a generic free-fermion PT-broken chain, e.g., the non-Hermitian SSH model, to test universality. If the slope is -M·2EI with M>1, or if any dominant eigenvalue branching changes the logarithm sign, Eq. (34) is incorrect and the central claim needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central free-fermion claim rests on Eq. (34), where S_TTC(t) is taken to approach -2EIt because a single correlation-matrix eigenvalue νδ0' grows as e^{2EIt}. This is not justified. For a Gaussian state, every eigenvalue of the reduced correlation matrix that grows as e^{2EIt} contributes approximately -2EIt to S_TTC, so the asymptotic slope is -M·2EIt, where M is the rank of the leading-order projector P_A C(0) P_A. The paper never proves M=1 for the XXZ chain or for generic free-fermion PT-broken systems, nor does it examine the branch structure of ln|1-ν| for complex ν. The 'approximate spectral symmetry' argument in §V.A is qualitative: no precise statement of the symmetry or proof that exponential terms cancel is given. If subleading eigenvalues have positive growth exponents, they too contribute linear terms, and if M>1 the quoted slope changes. Additionally, Table I states 'linear growth' for free-fermion TTC entropy while the abstract, Sec. V, and Fig. 5 all say 'linear decay,' an internal inconsistency that suggests the sign and slope of Eq. (34) have not been carefully cross-checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies sudden quenches into PT-symmetric non-Hermitian Hamiltonians using the biorthogonal density matrix rho_RL. It derives asymptotic forms for observables, the non-Hermitian quantum metric, and three entanglement measures in PT-unbroken and PT-broken cases. The central claim is that a PT-broken quench produces exponential growth in generic interacting systems, while the TTC entropy of non-interacting fermions decays linearly as S_TTC ~ -2 E_I t, attributed to an approximate spectral symmetry of the biorthogonal reduced density matrix. The claims are benchmarked numerically on the Yang-Lee model and the non-Hermitian XXZ chain.","tokens_in":15364,"tokens_out":12293,"duration_ms":126943,"significance":"If correct, the distinction between exponential TTC growth in interacting systems and linear decay in free-fermion systems is a clean and falsifiable signature of interaction effects in non-Hermitian quench dynamics, and the proposed spectral-symmetry mechanism is conceptually appealing. The paper also gives a useful comparison among TTC, SVD, and right-right entanglement entropies and provides explicit asymptotic formulas, Eqs. (15), (28), and (34), against which numerics can be checked. The use of two distinct models and the clear Table I summary are strengths. However, several load-bearing derivations are incomplete or currently incorrect, so the central results are not yet established at the level required for publication.","major_comments":[{"comment":"The claimed analytical solution for the quantum metric is not the correct derivative of the time-evolution operator. For a time-independent, parameter-dependent Hamiltonian, ∂_{hz} e^{-iHt} is not -i(∂H/∂hz)t e^{-iHt} unless [H, ∂H/∂hz]=0; the correct expression is a time-ordered integral. Since the paper later invokes [H, ∂H/∂hz]≠0 as the mechanism for exponential growth, the formulas for χ^RL_hzhz, χ^RL_hzt, and χ^RL_thz need to be re-derived before the quantum-geometry claims in Fig. 3 are supported.","section":"Sec. III.C, Eq. (22)"},{"comment":"The limit S_TTC(t)→-2E_I t replaces the sum in Eq. (33) by a single correlation-matrix eigenvalue without proving that only one eigenvalue grows at the maximal rate. If C(t)∼e^{2E_I t}C_∞ with rank M, each nonzero eigenvalue of C_∞ contributes approximately -2E_I t, giving an asymptotic slope -2M E_I t; the abstract and Fig. 5 specifically assert -2E_I t. The paper neither computes M for the non-interacting XXZ chain nor excludes other eigenvalues with positive growth exponents, and it does not address the branch structure of ln|1-ν| for complex ν. This is a load-bearing step for the central free-fermion claim.","section":"Sec. V.A, Eq. (34)"},{"comment":"The conclusion that the TTC entropy grows exponentially in the interacting PT-broken case depends on the sign of -Tr(P_A^2 ln|P_A^2|). Since Tr P_A^2 = 0 and P_A^2 is non-Hermitian, this quantity is not manifestly positive; for example, a pair of eigenvalues ±a gives zero, and other spectra can give either sign. The derivation in Eqs. (26)-(28) therefore does not by itself establish exponential growth, and the sign should be proved or verified for the Yang-Lee model.","section":"Sec. IV.A, Eq. (28)"},{"comment":"The long-time expansion assumes that the largest imaginary part E_I is non-degenerate and that the overlaps c_max and c_min of the initial biorthogonal state with |R_max⟩ and ⟨L_min| are nonzero. The text states that the initial state is composed of contributions from all eigenmodes, but this is an assumption rather than a demonstrated property, and the exceptional-point limit is not discussed. Special initial states or a non-diagonalizable Hamiltonian invalidate the single-mode dominance used in all subsequent asymptotic formulas.","section":"Sec. II.B, Eq. (15)"},{"comment":"The approximate spectral symmetry is invoked as the origin of the distinct free-fermion behavior, but it is only described qualitatively. The statement that the RDM eigenvalues form a set {α_i e^{2E_I t}, -α_i e^{2E_I t}+ϵ} needs a precise bound on ϵ and a proof that the error terms do not produce exponential contributions to S_TTC. Without this, the claim that the symmetry is broken for interacting systems remains an observation rather than a demonstrated mechanism.","section":"Sec. V.A, spectral symmetry discussion"}],"minor_comments":[{"comment":"The TTC entropy row states 'linear growth' for free-fermion systems, while the abstract, Sec. V, and Fig. 5 all report linear decay; this inconsistency should be corrected.","section":"Table I"},{"comment":"The sentence 'If the post-quench Hamiltonian is in PT-unbroken phase, some of the eigen-energies come in complex-conjugation pairs' should refer to the PT-broken phase, since the immediately preceding sentence correctly states that all eigen-energies are real in the PT-unbroken phase.","section":"Sec. II.A, first paragraph"},{"comment":"The notation for the imaginary part of the energy is inconsistent: E_1 and E_I are used interchangeably, and the symbol ν_{δ0}' in Eq. (34) is not defined clearly.","section":"Eqs. (27)-(28) and (34)"},{"comment":"There are several typographical errors, including 'biothogonal' in the Table I caption, 'paramter' in the Fig. 5 caption, and 'Y ANG-LEE' in the Sec. III heading.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the central idea is interesting, but the analytic gaps in Eqs. (22), (28), and (34) are substantial enough that I cannot recommend acceptance in the present form. A revision that fixes the quantum-metric derivative, proves or sharpens the multiplicity and sign statements behind the TTC asymptotics, and reconciles Table I with the text would make the paper publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing you should know first: this is not just another non-Hermitian quench paper. The central claim — that in a PT-broken post-quench Hamiltonian the TTC entropy decays linearly in time for free fermions, while it grows exponentially once interactions are added — is new and interesting. The exponential growth of observables and standard entropies follows from the biorthogonal formalism, but the linear decay is a real deviation, and the 'approximate spectral symmetry' explanation is plausible.\n\nWhat the paper does well: it works out the quench structure of the biorthogonal density matrix in the PT-broken phase, connects it to the quantum geometric tensor, and contrasts three entropy measures (TTC, SVD, and right-right). The numerical check on the non-Hermitian XXZ chain in Fig. 5 shows the crossover from linear decay to exponential growth as Jz is turned on, which is a nice demonstration. The authors also correctly benchmark against the Yang-Lee model.\n\nNow the soft spots. The derivation of Eq. (34) is under-specified. The step from the sum over correlation eigenvalues to a single dominant eigenvalue needs justification. In a generic PT-broken free-fermion quench the long-time correlation matrix is actually rank-one (the leading right and left modes are unique), so the slope -2EIt is right, but the paper never says that. This is a fixable gap, not a fatal flaw. The approximate spectral symmetry is shown numerically, not derived; that is fine for a claim of this scope, but the authors should state it more carefully. There is an unambiguous internal inconsistency: Table I (and the conclusion) say 'linear growth' for the free-fermion TTC entropy, while the abstract, Eq. (34), Section V, and Figure 5 all say 'linear decay.' That is a sign error somewhere, and it needs to be corrected before the paper is cited. There is also no discussion of the non-diagonalizable exceptional-point limit, which is a natural edge case for this expansion. No code or data is provided, though the numerics are simple enough to reproduce.\n\nBottom line: this is a serious contribution with one new result and a mostly sound argument. A good referee would ask for the derivation gap and the sign inconsistency to be fixed, not reject the work. I would bring it to a reading group if you care about non-Hermitian many-body dynamics.","headline":"A genuinely new result on linear decay of TTC entropy in free-fermion PT-broken quenches, wrapped in a paper with a sign inconsistency and a sketchy derivation.","tokens_in":15952,"tokens_out":6079,"would_cite":true,"duration_ms":61046,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.-w","03.65.Ud","05.30.-d","75.10.Jm"],"model":"deepseek-v4-flash","headline":"The central claim is that a quench into the PT-broken phase of a PT-symmetric non-Hermitian Hamiltonian produces exponentially growing off-diagonal terms in the biorthogonal density matrix, making observables, quantum geometry, and TTC…","keywords":["non-Hermitian quantum systems","PT symmetry","biorthogonal formalism","quench dynamics","entanglement entropy","Tu-Tzeng-Chang entropy","quantum metric tensor","Yang-Lee model"],"falsifier":"Compute the TTC entropy after a PT-broken quench from an initial state chosen to have zero overlap with the dominant biorthogonal pair, or tune the post-quench Hamiltonian exactly to an exceptional point where the two dominant eigenvectors coalesce; if the paper's mechanism is correct, the exponential or linear asymptotics in Eqs. (28) and (34) should break down because the expansion in Eq. (15) is no longer valid.","tokens_in":14921,"feed_emoji":"⚛️","tokens_out":10525,"duration_ms":90448,"temperature":0.7,"pith_summary":"This paper studies what happens after a sudden quench into the PT-broken phase of a parity-time-symmetric non-Hermitian Hamiltonian, the regime where energy levels come in complex-conjugate pairs; it uses the biorthogonal formalism, which keeps both left and right eigenstates and preserves the trace of the density matrix. It claims that the biorthogonal density matrix develops exponentially growing off-diagonal components, so physical observables, the quantum metric tensor, and the Tu-Tzeng-Chang (TTC) entanglement entropy generically grow exponentially, at a rate set by the imaginary part of the energy spectrum. The exception is non-interacting fermions, where the TTC entropy decays linearly in time because the biorthogonal reduced density matrix has an approximate plus-minus spectral symmetry. The significance is that this yields a clean dynamical distinction between interacting and free-fermion non-Hermitian systems and ties entanglement growth to PT-symmetry breaking after a quench.","feed_headline":"Quenching into a PT-broken phase makes entanglement grow exponentially","feed_subtitle":"Free-fermion systems instead show linear entropy decay, a signature of reduced-density-matrix symmetry","key_machinery":"The biorthogonal density matrix $\\rho_{RL}=|\\psi_R\\rangle\\langle\\psi_L|$ is the central object: built from right and left eigenstates of the non-Hermitian Hamiltonian, it is non-Hermitian and can have negative eigenvalues, yet it stays trace-preserving. Its long-time expansion in Eq. (15), controlled by the pair of eigenmodes with largest and smallest imaginary energies, is what produces the exponentially growing off-diagonal terms. The second load-bearing mechanism is the approximate spectral symmetry of the biorthogonal reduced density matrix in free-fermion systems—its eigenvalues come in near $\\pm\\lambda$ pairs—which is what converts the generic exponential growth of the TTC entropy into linear decay.","core_discovery":"In the PT-broken case, the paper's central claim is that the long-time biorthogonal density matrix has the structure $\\rho_{RL}(t)\\to P_0+e^{E_I t}P_1+e^{2E_I t}P_2$ with $\\mathrm{Tr}P_0=1$ and $\\mathrm{Tr}P_1=\\mathrm{Tr}P_2=0$: the trace is preserved while individual off-diagonal components grow. This happens because the right state is dominated by the eigenmode with the largest imaginary energy, $|R_{\\max}\\rangle$, while the left state is dominated by the eigenmode with the smallest imaginary energy, $\\langle L_{\\min}|$; their biorthogonal inner product is zero, so their outer product contributes an off-diagonal term proportional to $e^{2E_I t}$ without affecting the trace. For generic interacting systems the TTC entropy consequently grows as $S_A^{TTC}\\sim -2E_I t - e^{2E_I t}\\mathrm{Tr}(P_A^2\\ln|P_A^2|)$. For non-interacting fermions the reduced density matrix can be built from the two-point correlation matrix, and its eigenvalues come in approximate $\\pm\\lambda$ pairs; that approximate spectral symmetry turns the exponential growth into linear decay, $S_A^{TTC}\\sim -2E_I t$. The paper confirms both behaviors in the Yang-Lee model and the non-Hermitian XXZ chain.","pith_inferences":["The paper leaves implicit that the TTC entropy itself, because its growth rate is fixed by $E_I$, could serve as a dynamical order parameter for PT-symmetry breaking in experiments where the spectrum is not directly measurable.","The approximate plus-minus spectral symmetry responsible for linear decay is a property of Gaussian free-fermion states; by extension, analogous linear decay may appear in other free-particle or integrable settings, a testable prediction beyond the two models studied here.","Because the biorthogonal and right-right conventions give such different late-time entropies, experimental comparisons of the two could discriminate which density-matrix convention describes a given open-system setup, a question the paper raises but does not settle."],"forward_implications":["In the PT-broken phase, observables and quantum metric tensor components grow as $e^{2E_I t}$, so the imaginary part of the energy spectrum controls the rate of dynamical amplification.","For generic interacting systems the TTC entropy grows exponentially in time, while the usual right-right entanglement entropy and the SVD entropy saturate to constants; both saturating entropies show a volume-law to area-law transition at the level crossing.","For non-interacting fermions the TTC entropy instead decays linearly as $-2E_I t$, making late-time entropy dynamics a sharp diagnostic that distinguishes free from interacting post-quench Hamiltonians.","Quenches into the PT-unbroken phase remain oscillatory, so the onset of exponential growth in any of these quantities acts as a dynamical detector of the PT transition."],"supporting_citations":[{"why":"Supplies the biorthogonal quantum-mechanics framework: right/left eigenstates, trace-preserving biorthogonal density matrix, and conserved inner product.","marker":"[18]"},{"why":"Provides the long-time dominant-eigenmode analysis and the right-right density-matrix normalization that the biorthogonal result is contrasted against.","marker":"[33]"},{"why":"Defines the TTC entropy $S=-\\mathrm{Tr}(\\rho_A\\ln|\\rho_A|)$ that is the paper's main entanglement measure.","marker":"[29]"},{"why":"Defines the SVD reduced density matrix and SVD entropy whose saturation and level-crossing transition the paper analyzes.","marker":"[28]"},{"why":"Gives the correlation-matrix/Gaussian form of the reduced density matrix used to obtain the linear TTC decay in free-fermion systems.","marker":"[19]"},{"why":"Supplies the Yang-Lee model context and the level-crossing entanglement transition against which the biorthogonal results are compared.","marker":"[11]"},{"why":"Introduces the PT-symmetric non-Hermitian XXZ chain used to compare non-interacting and interacting TTC dynamics.","marker":"[20]"},{"why":"Defines the non-Hermitian quantum metric tensor whose exponential PT-broken growth is one of the paper's main results.","marker":"[21]"}],"fun_headline_variants":["PT-broken quench drives exponential entanglement growth","Free fermions show linear entropy decay after PT-broken quench","Biorthogonal quench: exponential entanglement in PT-broken phase","PT-symmetric quenches: exponential growth, except free fermions","Non-Hermitian quench: entanglement grows, free fermions decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the post-quench Hamiltonian is diagonalizable and that the initial state has nonzero overlap with the eigenmode pair whose imaginary energies are largest and smallest, so that the single-pair asymptotic expansion really dominates; the non-diagonalizable exceptional-point limit is not treated.","fun_headline_variants_meta":{"raw":{"variants":["PT-broken quench drives exponential entanglement growth","Free fermions show linear entropy decay after PT-broken quench","Biorthogonal quench: exponential entanglement in PT-broken phase","PT-symmetric quenches: exponential growth, except free fermions","Non-Hermitian quench: entanglement grows, free fermions decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1471,"prompt_tokens":994,"completion_tokens":477,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":384}},"tokens_in":610,"tokens_out":477,"duration_ms":4776,"temperature":1.0,"reasoning_tokens":384,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:48:10.029883+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the TTC entropy after a PT-broken quench from an initial state chosen to have zero overlap with the dominant biorthogonal pair, or tune the post-quench Hamiltonian exactly to an exceptional point where the two dominant eigenvectors coalesce; if the paper's mechanism is correct, the exponential or linear asymptotics in Eqs. (28) and (34) should break down because the expansion in Eq. (15) is no longer valid.","supporting_citations":[{"cited_title":"Dissipative Dynamical Phase Transition as a Complex Ising Model","cited_arxiv_id":"2412.09591","evidence_quote":"Provides the long-time dominant-eigenmode analysis and the right-right density-matrix normalization that the biorthogonal result is contrasted against."},{"cited_title":"PT -broken case: saturate to a constant and has a volume to area law transition across the level crossing point","cited_arxiv_id":null,"evidence_quote":"Defines the SVD reduced density matrix and SVD entropy whose saturation and level-crossing transition the paper analyzes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the PT-symmetric non-Hermitian XXZ chain used to compare non-interacting and interacting TTC dynamics."},{"cited_title":"Chang, J.-S","cited_arxiv_id":null,"evidence_quote":"Defines the non-Hermitian quantum metric tensor whose exponential PT-broken growth is one of the paper's main results."}],"review_version":2}