{"id":"e2bff9d6-b8d4-4ad8-8517-5248a3f42415","arxiv_id":"2412.08141","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A first-order perturbative derivation shows that both the right-right and left-right quantum geometric tensors contribute to second-order field-induced corrections in non-Hermitian two-band wavepacket dynamics.","lead":"This paper extends the quantum geometric tensor formalism to non-Hermitian systems, deriving corrected wavepacket equations that show both the right-right and left-right geometric tensors affect the motion. The authors validate the theory against simulations of microcavity exciton polaritons, a platform where the geometric quantities can be measured experimentally.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SI Fig. S1 shows a linear-in-F deviation in the y-direction that neither Eq. (6) nor the new Eq. (11) captures; the SI admits a new theory is needed for the non-Hermitian anomalous Berry connection, so Eq. (11) is not demonstrated as the complete second-order generalization.","rationale":"The paper's analytical derivation of Eq. (11) is internally consistent, and the algebraic reduction from the SI to the main-text form checks out. The RR QGT and LR QMT do appear in the derived corrections, supporting the qualitative claim that both formalisms contribute to the dynamics. However, the SI's own numerical analysis at high force reveals a deviation in the y-direction that scales linearly with F for both the old and new theories, and the authors explicitly state that a new theory would be needed to capture the correction to the non-Hermitian anomalous Berry connection. This is a direct admission that Eq. (11) is incomplete at the claimed order, which is a more specific threat to the central claim than the single-band limitation identified by the reader. The reader's CONDITIONAL verdict remains appropriate: the paper makes a substantial contribution to the QGT formalism, but the claim that Eq. (11) is the full second-order-in-F generalization is not fully established by the presented evidence. My concern does not change the verdict, but it sharpens the condition: the authors should either identify and include the missing correction or explicitly temper the conclusion about completeness. I did not find grounds for REJECT because the analytical framework and the x-direction validation are sound, and the missing term may be a known but omitted contribution that can be incorporated in a revised version.","tokens_in":21728,"tokens_out":26473,"duration_ms":229956,"concrete_test":"Extract the instantaneous center-of-mass velocity v_y(t) from the split-step simulation for the same parameters at several force magnitudes F. Compare v_y(t) with the prediction of Eq. (11) at each time; compute the time-averaged absolute error. If the error scales linearly with F (rather than quadratically) after the initial transient, Eq. (11) is missing a first-order-in-F term in the y-velocity. Alternatively, rerun the y-deviation analysis at later times (e.g., t = 2.5 and 5 ps) to confirm the linear-in-F scaling persists and is not a transient artifact; if it persists, the missing anomalous-Berry-connection correction is real and must be identified before the central claim can stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is an explicit limitation in SI Section V ('Additional data from simulation'). The deviation in the y-position between the numerical simulation and the analytical theory is fitted by a linear function in F for both the first-order theory (Eq. 6) and the new second-order theory (Eq. 11). If the analytic velocity error were truly second-order in F, the position deviation at fixed time would scale as F^2; a linear scaling indicates a missing first-order-in-F contribution to the y-velocity. The SI states: 'It would require a new theory to capture the correction to the non-Hermitian anomalous Berry connection, which is beyond the scope of this work.' This is an explicit admission that Eq. (11) does not fully capture the dynamics even at the claimed order. The central claim that Eq. (11) is 'the non-Hermitian generalization of Eq. (5)' and that the derived corrections completely describe the second-order-in-F wavepacket dynamics is therefore not fully supported by the paper's own numerical evidence. The x-direction deviation is captured (quadratic-to-cubic scaling as expected), but the y-direction failure suggests an incomplete correction to the anomalous Berry connection, not merely a finite-width artifact, because the authors themselves attribute it to a missing term. This concern is distinct from, and more specific than, the single-band assumption identified by the reader.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives a semiclassical equation of motion for wave-packet center-of-mass dynamics in two-band non-Hermitian systems, including first-order perturbative corrections that are second order in the external force. The central result, Eq. (11), generalizes the Hermitian equation of motion and incorporates both right-right (RR) and left-right (LR) quantum geometric tensors. The authors verify the theory against split-step simulations of a two-dimensional exciton-polariton model with experimentally motivated parameters. The derivation is presented in the Supplemental Material, and the simulation data are deposited in a public repository.","tokens_in":21990,"tokens_out":8195,"duration_ms":76648,"significance":"If the derived equations are complete, the work would help resolve the ongoing debate about which quantum-geometric-tensor generalization governs non-Hermitian wave-packet dynamics, showing distinct roles for the RR and LR tensors. The use of an experimentally relevant exciton-polariton model and the public data deposit are strengths. However, the numerical verification in the SI reveals a missing linear-in-force contribution to the y-velocity, which leaves the central claim that Eq. (11) is the complete non-Hermitian generalization of Eq. (5) only partially supported.","major_comments":[{"comment":"The deviation in the y-component of the center-of-mass position between the numerical simulation and both the first-order equation (Eq. (6)) and the new equation (Eq. (11)) is linear in the external force F, as shown in Fig. S1(b). At fixed time, a position deviation proportional to F implies a missing velocity contribution of order F, i.e., a first-order-in-F term along the force direction. The SI explicitly states that \"It would require a new theory to capture the correction to the non-Hermitian anomalous Berry connection, which is beyond the scope of this work.\" This directly contradicts the main-text claim that Eq. (11) is the non-Hermitian generalization of Eq. (5) and that Eqs. (12)-(14) completely capture the second-order-in-F dynamics. The central claim is therefore not fully supported by the paper's own numerical evidence.","section":"SI Section V (Additional data from simulation), Fig. S1(b)"},{"comment":"The concluding statement that the results \"settle the dispute\" on which QGT generalization governs non-Hermitian dynamics is too strong given the missing linear-in-F contribution identified in Fig. S1(b). The incompleteness in the y-direction means that the roles of the LR QMT in the Berry-phase correction (Eq. (19)) and of the anomalous Berry connection in the positional shift (Eq. (18)) may not be fully captured. The conclusions should be moderated to state that both tensors contribute to the captured dynamics, while explicitly acknowledging the unaccounted term.","section":"Conclusion"}],"minor_comments":[{"comment":"The main text states that the solution of Eq. (11) \"gives a better fit\" to the numerical results, but it does not mention that the y-deviation is essentially unchanged and linear in F, as shown in SI Fig. S1(b). The reader should be alerted to this limitation in the main text.","section":"Results, Fig. 2"},{"comment":"There are several typographical errors, including \"assopciated\" in the first paragraph and \"Brillouine\" in the second paragraph.","section":"Introduction"},{"comment":"The caption reads \"(a, d) The real-space trajectory\" but the figure contains four panels (a)-(d); it should read \"(a,b)\" and \"(c,d)\" respectively.","section":"Figure 2 caption"},{"comment":"Reference [89] contains a LaTeX artifact \"/suppress\" in the author list.","section":"References"},{"comment":"The phrase \"non-Hermitian contribution\" to describe the term F·(A_RR_00 - A_LR_00) is imprecise, as this term also affects the group velocity even in the presence of Hermitian-like dynamics; consider rephrasing.","section":"Eq. (6)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a deposited dataset (Zenodo DOI) and a systematic SI derivation, which are strengths. The SI's admission of a missing term, however, is not reflected in the main text's strong claims, and the y-direction discrepancy is not disclosed in the main text. I recommend the authors derive the missing term or substantially qualify the central claims. With these changes, the paper could be suitable for publication in a condensed-matter journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the headline: this is a serious derivation that probably extends the non-Hermitian semiclassical EOM to second order in the force, but the authors' own SI shows the result is not as complete as claimed, so it needs revision rather than a desk reject.\n\nWhat's genuinely new: prior non-Hermitian results (Refs. 71-73) had only first-order-in-F corrections. The paper derives explicit second-order corrections—Eqs. (11)-(14)—showing the RR QGT controls the field-induced positional shift and Berry-curvature correction, while the LR QMT enters the Berry phase. The SI derivation is systematic: biorthogonal completeness, non-orthogonality corrections, and a phase from the time-dependent Schrödinger equation. It reduces to the Hermitian limit and to the first-order non-Hermitian results. That is real work, and the new terms (Im Δε Berry curvature, imaginary part of g^LR) are plausible and testable in polariton systems. Data are archived on Zenodo.\n\nThe soft spot is in the paper's own SI Section V. Figure S1 shows that the deviation in the y-position at fixed time is linear in F for both the old and new theories. That means a missing first-order-in-F contribution to the y-velocity, not a second-order effect. The SI explicitly says: 'It would require a new theory to capture the correction to the non-Hermitian anomalous Berry connection, which is beyond the scope of this work.' So Eq. (11) is not demonstrated as the complete second-order generalization; the x-deviation is captured but the y-deviation is not. That is more specific than the usual finite-width artifact, and the authors themselves flag it. The conclusion that the paper 'settles the dispute' over which QGT to use is stronger than the evidence supports, given a single two-band model, one gapped phase, one force direction, and one initial momentum. The single-band assumption also restricts the domain: crossing the imaginary Fermi arc breaks the two-level truncation, and that is stated in the main text.\n\nNone of this makes the paper wrong in the derivational sense; it makes the central claim overbroad. The right fix is to temper the conclusion, discuss the y-deviation explicitly, and either extend the theory to the anomalous Berry connection correction or delimit the regime more carefully.\n\nWho should read it: anyone working on non-Hermitian wavepacket dynamics, quantum geometry, or polariton transport. It deserves a serious referee—send it out—but the referee should insist on the y-correction being addressed or clearly scoped. I'd bring it to a reading group as a good example of a careful derivation with an honest numerical limitation.","headline":"A careful derivation of second-order non-Hermitian QGT corrections whose own numerics reveal a missing first-order term in the anomalous Berry connection, so the central claim outruns the evidence.","tokens_in":22569,"tokens_out":2454,"would_cite":true,"duration_ms":24066,"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":"This paper derives the non-Hermitian generalization of the semiclassical wave-packet equation of motion and shows that the right-right quantum geometric tensor controls field-induced positional shifts and Berry-curvature corrections…","keywords":["quantum geometric tensor","non-Hermitian systems","wave-packet dynamics","Berry curvature","quantum metric","perturbation theory","exciton polaritons","anomalous Hall drift"],"falsifier":"Track a polariton wave packet under a controlled external force and measure its centre-of-mass $x$-displacement after a fixed time for several force magnitudes. The old equation (6) predicts a deviation from simulation that grows as $F^2$, while Eq. (11) predicts a residual growing as $F^3$; observing no $F^3$ term, or observing that the trajectory still follows Eq. (11) after the packet crosses the imaginary Fermi arc, would falsify the corrected equation.","tokens_in":21518,"feed_emoji":"🌀","tokens_out":8339,"duration_ms":70075,"temperature":0.7,"pith_summary":"The paper asks which of the two proposed generalizations of the quantum geometric tensor governs wave-packet motion in two-band systems with gain and loss. It derives the non-Hermitian counterpart of the standard semiclassical equation of motion for a narrow wave packet, including first-order perturbative corrections from interband mixing. The result shows that both generalizations are physical: the right-right (RR) tensor produces the field-induced positional shift and corrections to the anomalous Hall velocity, while the left-right (LR) quantum metric enters the geometric phase. The authors confirm the improved accuracy of the equation by simulating exciton-polariton wave packets in a semiconductor microcavity, where the relevant geometric quantities can be measured. If correct, this settles the earlier dispute over which QGT definition is the right one: each appears in a different dynamical role.","feed_headline":"Wave packets in lossy systems move by two quantum geometries","feed_subtitle":"A corrected semiclassical equation matches polariton wave-packet simulations where the old first-order theory does not.","key_machinery":"The load-bearing object is the first-order perturbative expansion of the perturbed right eigenstate, Eq. (10): $|\\tilde{u}^R_0\\rangle = |u^R_0\\rangle - \\frac{F\\cdot A^{LR}_{10}}{\\epsilon_0-\\epsilon_1}|u^R_1\\rangle$, with $A^{LR}_{10}=\\langle u^L_1|i\\partial_k u^R_0\\rangle$ the left-right inter-band Berry connection. This expansion is inserted into the centre-of-mass position and the Schrödinger equation under the narrow-wave-packet approximation, producing the corrected connections (12), (13) and the corrected curvature (14). The complex denominator $\\epsilon_1-\\epsilon_0$ is the mechanism that distributes the corrections: its real part weights the RR quantum metric, its imaginary part couples the RR Berry curvature and the imaginary parts of the LR quantities, and both parts vanish in the Hermitian limit or on the imaginary Fermi arc.","core_discovery":"Equation (11) is the central result: it is the non-Hermitian generalization of the Hermitian semiclassical equation of motion (5), and Eqs. (12)-(14) give the first-order corrections to the Berry connections and Berry curvature. The paper establishes that, for a wave packet occupying the slower-decaying band, the field-induced positional shift and the Berry-curvature correction are controlled by the RR QGT $Q^{\\mathrm{RR}}_0$, while the correction to the Berry phase is controlled by the LR QMT $g^{\\mathrm{LR}}_0$. Because the corrections involve the complex energy gap $\\epsilon_1-\\epsilon_0$, new terms appear that have no Hermitian counterpart: for example, the imaginary part of the gap couples the RR Berry curvature to the motion, and both real and imaginary parts of the LR QMT enter. In the Hermitian limit the equation and its corrections reduce to the known result (5), and the two QGT formalisms are seen not as rivals but as descriptions of different physical observables.","pith_inferences":["One could design a quantitative experiment that measures the wave-packet $x$-displacement after a fixed time for several force strengths; the predicted $F^3$ residual over Eq. (6) is a sharper signature than the trajectory plots alone.","The same perturbative machinery should carry over to multiband non-Hermitian systems, with sums over all other bands replacing the two-band gap; the separation of roles (RR for shifts and curvature, LR for phase) is likely to persist.","The dispute over whether the RR or LR QGT is 'ill-defined' may be recast: the useful question is not which tensor is fundamental but which observable each tensor couples to, and future definitions should specify the dynamical context.","Near but not across the imaginary Fermi arc, the complex gap suppresses high-order corrections, so the theory predicts a sharp crossover in validity as a trajectory crosses the arc; this could be probed in existing polariton experiments by steering the wave packet through the crossing point."],"forward_implications":["The previously used first-order equation (6) is incomplete in the presence of finite force; deviations from it grow as $F^2$, whereas the new equation (11) leaves only $F^3$ deviations, so experiments with stronger acceleration or smaller gaps need the corrected terms.","Both QGT formalisms carry measurable content: the RR QGT can be extracted from field-induced positional shifts and anomalous Hall drifts, and the LR QMT from the field-induced correction to the Berry phase.","Corrections proportional to $\\mathrm{Im}[\\Delta\\epsilon]$ mean the non-Hermitian geometric response changes strength away from the imaginary Fermi arc and vanishes on it, giving a tunable knob through the complex band structure.","In the exciton-polariton platform used for the numerical test, the same quantities—complex eigenenergies, QGT components, and anomalous Hall drift—are in principle experimentally accessible, so the improved equation is directly testable."],"supporting_citations":[{"why":"supplies the Hermitian field-induced positional shift that the RR QGT generalizes","marker":"[12]"},{"why":"provides the Hermitian QGT-based nonadiabatic corrections, Eq. (5), being generalized","marker":"[14]"},{"why":"supplies the exciton-polariton model and the imaginary Fermi arc limiting the single-band approximation","marker":"[58]"},{"why":"is the counter-position that the RR quantum metric is ill-defined, the dispute this work addresses","marker":"[70]"},{"why":"gives the previous first-order non-Hermitian equation of motion involving RR Berry curvature and anomalous Berry connection","marker":"[71]"},{"why":"derives the anomalous Berry connection dynamics that Eq. (11) corrects to second order in force","marker":"[72]"},{"why":"contributes the wave-packet formalism in which the corrected equation of motion is derived","marker":"[73]"},{"why":"shows that QGT components of a non-Hermitian exciton-polariton system can be measured, making the predictions testable","marker":"[75]"},{"why":"establishes the single-band validity condition and the role of the imaginary Fermi arc in wave-packet dynamics","marker":"[78]"},{"why":"supplies the non-Hermitian perturbation theory used to expand the eigenstate in Eq. (10)","marker":"[79]"}],"fun_headline_variants":["Two quantum geometries steer wave packets in lossy systems","How two quantum geometries shape wave packet motion in loss","Non-Hermitian wave packets split geometry into two roles","Wavepackets in loss obey a new semiclassical rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the wave packet can be treated as a single band throughout its motion: it must remain in the eigenstate with the larger imaginary energy and must not cross the imaginary Fermi arc, where the loss rates of the two bands switch. If that condition fails, the two-level truncation and the first-order expansion in $F$ break down, and Eq. (11) no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["Two quantum geometries steer wave packets in lossy systems","How two quantum geometries shape wave packet motion in loss","Non-Hermitian wave packets split geometry into two roles","Wavepackets in loss obey a new semiclassical rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000632,"raw_usage":{"total_tokens":2918,"prompt_tokens":947,"completion_tokens":1971,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":1905}},"tokens_in":563,"tokens_out":1971,"duration_ms":13074,"temperature":1.0,"reasoning_tokens":1905,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:10:09.589533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Track a polariton wave packet under a controlled external force and measure its centre-of-mass $x$-displacement after a fixed time for several force magnitudes. The old equation (6) predicts a deviation from simulation that grows as $F^2$, while Eq. (11) predicts a residual growing as $F^3$; observing no $F^3$ term, or observing that the trajectory still follows Eq. (11) after the packet crosses the imaginary Fermi arc, would falsify the corrected equation.","supporting_citations":[{"cited_title":"Chen Ye, W","cited_arxiv_id":null,"evidence_quote":"is the counter-position that the RR quantum metric is ill-defined, the dispute this work addresses"},{"cited_title":"Silberstein, J","cited_arxiv_id":null,"evidence_quote":"derives the anomalous Berry connection dynamics that Eq. (11) corrects to second order in force"},{"cited_title":"Wang, Y.-L","cited_arxiv_id":null,"evidence_quote":"contributes the wave-packet formalism in which the corrected equation of motion is derived"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"shows that QGT components of a non-Hermitian exciton-polariton system can be measured, making the predictions testable"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the single-band validity condition and the role of the imaginary Fermi arc in wave-packet dynamics"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the non-Hermitian perturbation theory used to expand the eigenstate in Eq. (10)"}],"review_version":1}