{"id":"c8aa93ae-d0f4-434b-b966-8cae08e08787","arxiv_id":"2607.04834","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"A continuous gauge freedom in complex-pole HEOM controls spectral stability and non-normality, enabling GO-HEOM simulations through previously inaccessible strong-coupling and sub-Ohmic quantum phase transitions.","lead":"Complex-pole hierarchical equations of motion for open quantum systems hide a continuous gauge freedom that reshapes their Liouvillian without changing the physics. Optimizing the gauge yields stable, exact dynamics in strong-coupling and long-memory regimes that previously diverged.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s strongest claim is a clean mathematical observation plus a practical demonstration. The phase-space construction, the recovery of the standard HEOM at the special gauge ϕ=θ, the alternative influence-functional derivation in the SM, and the numerical evidence across two spectral densities all cohere. The reader correctly flags that the one-parameter restriction (ϕ_k=ϕ_r θ_k) is unproven to be globally optimal and that truncation/bond-dimension dependence is not exhaustively mapped; those are real but secondary limitations that do not threaten the existence of the gauge freedom or the concrete stability gains already shown. Because no load-bearing flaw in the central argument was identified, the ACCEPT verdict with high confidence stands unchanged.","tokens_in":19144,"tokens_out":512,"duration_ms":4185,"concrete_test":"Independently re-derive the coefficients A11,A22,A12,A21 of Eq. (9)/SM (S17–S20) from the linear map (7)–(8) and verify that the resulting hierarchy (10) reproduces the conventional complex-pole HEOM exactly when ϕ_k=θ_k; then recompute the BO eigenspectrum of Fig. 2(c,d) at the reported ϕ_r=1.4 and confirm that max(Re λ) remains negative while the physical ⟨σ_z(t)⟩ trajectory is unchanged to machine precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that a continuous gauge freedom exists (Eq. 10), leaves C(t) and the exact reduced dynamics invariant, and can be optimized via a single global scaling ϕ_r to remove spectral divergences (BO) and non-normal error growth (sub-Ohmic)—is supported by the phase-space derivation, the explicit recovery of conventional HEOM at ϕ_k=θ_k, the SM generalized-decomposition route, and the numerical spectra/trajectories in Figs. 2–4 and S1–S3. The reader’s weakest assumption (that the restricted one-parameter family may not be fully optimal) is a genuine practical limitation, but it does not undermine the existence of the gauge or the demonstrated improvements; the paper already notes the larger four-parameter family in SM and deliberately restricts to a usable one-parameter scan. No internal inconsistency or missing condition that would falsify the strongest claim was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript reports a continuous gauge freedom in complex-pole hierarchical equations of motion (HEOM). Starting from a phase-space reformulation of the hierarchy, the authors introduce a one-parameter family of linear transformations of the auxiliary ladder operators (Eqs. 7–8) that leaves the bath correlation function C(t) and the exact reduced dynamics invariant while continuously reshaping the HEOM Liouvillian. The resulting ϕ-dependent hierarchy (Eq. 10) recovers conventional complex-pole HEOM at ϕ_k = θ_k. Optimizing a global scaling ϕ_r of the gauge angles yields GO-HEOM, which simultaneously controls the real parts of the Liouvillian spectrum and the Henrici departure from normality. Numerical demonstrations on the spin-boson model show that GO-HEOM eliminates spectral divergences for strongly coupled Brownian-oscillator baths (Figs. 2–3) and suppresses non-normal error growth for sub-Ohmic baths through the delocalized-to-localized quantum phase transition (Fig. 4), extending the accessible coupling range relative to prior complex-pole HEOM.","tokens_in":19376,"tokens_out":1053,"duration_ms":7969,"significance":"If the gauge equivalence and the reported numerical gains hold, the work supplies a practical, decomposition-independent stabilization strategy for complex-pole HEOM that is immediately compatible with existing AAA/Prony decompositions, MPS/TDVP propagators, and filtering techniques. The explicit separation of spectral divergence from non-normal error amplification clarifies two distinct instability mechanisms that have limited HEOM in strong-coupling and long-memory regimes. The phase-space derivation and the alternative generalized-decomposition route in the SM give a transparent algebraic foundation rather than an ad-hoc fix. The demonstrated access to sub-Ohmic dynamics through the quantum phase transition and to previously divergent Brownian-oscillator reorganization energies is of direct interest for electron-transfer and quantum-critical open-system problems.","major_comments":[{"comment":"The paper deliberately restricts the gauge to the one-parameter family ϕ_k = ϕ_r θ_k (text after Eq. 10 and SM Table I). While the SM correctly notes the larger four-parameter linear transformation (SM Eqs. S35–S43), no numerical evidence is given that the restricted family is near-optimal, nor that the chosen ϕ_r remains stable under changes of hierarchy truncation N_b or MPS bond dimension. A short scan of independent per-mode ϕ_k (or a statement that such a scan yields no further gain) would strengthen the claim that GO-HEOM is a robust, general strategy rather than a convenient but possibly suboptimal slice of the gauge space.","section":null},{"comment":"For the sub-Ohmic case the improvement is attributed to reduced non-normality rather than spectral relocation (Fig. 4b and accompanying text). The only quantitative non-normality diagnostic shown is the Henrici measure for the Brownian-oscillator Liouvillian (Fig. 2b). A corresponding plot (or table) of η_H versus ϕ_r for the sub-Ohmic hierarchy at α = 0.2 would make the mechanistic distinction between the two instability channels fully quantitative and would allow readers to verify that the chosen ϕ_r indeed minimizes non-normality.","section":null}],"minor_comments":[{"comment":"Eq. (10) is dense; a brief parenthetical reminder that the conventional HEOM is recovered at ϕ_k = θ_k (already stated in the text) would help readers navigate the multi-line expression.","section":null},{"comment":"Figure 2(d) caption and panel labels use both φ_r and ϕ_r; consistent notation throughout the figures and SM would avoid minor confusion.","section":null},{"comment":"SM Table I lists optimized ϕ_r for each α; a one-sentence statement in the main text that these values were obtained by the same short-trajectory / spectrum scan described after Eq. 10 would make the protocol fully self-contained.","section":null},{"comment":"The abstract and introduction claim that GO-HEOM is “independent of the bath-correlation decomposition scheme.” A short explicit check (or citation) that the same ϕ_r optimization works for a Prony-fitted decomposition, not only AAA, would make this claim concrete.","section":null},{"comment":"Typographical: “Andr´ es” and similar accented characters appear inconsistently in the author line and SM; a uniform encoding would improve readability.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The central algebraic claim is sound and the numerical gains are clear. The two major comments are incremental and do not threaten the existence of the gauge or the reported improvements; they can be addressed with modest additional diagnostics or a clarifying paragraph. The work is a natural fit for a high-impact chemical-physics or quantum-dynamics venue."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news is that complex-pole HEOM has a continuous, analytically exact gauge freedom (their Eq. 10) that leaves C(t) and the reduced dynamics invariant while letting you reshape both the spectrum and the non-normality of the Liouvillian. Optimizing a single global scaling ϕ_r of the mode phases gives GO-HEOM, which kills the positive-real eigenvalues that blow up strong Brownian-oscillator runs and damps the non-normal error growth that ruins long-time sub-Ohmic trajectories through the delocalized-to-localized transition.\n\nWhat is new is the explicit continuous family and the clean separation of the two instability mechanisms. Earlier stabilization papers (Dunn, the low-T QFPE work, etc.) touched pieces of this, but never wrote down the full gauge or showed that one cheap scan simultaneously fixes both problems. The phase-space derivation is self-contained, recovers ordinary complex-pole HEOM at ϕ_k = θ_k, and the SM even gives the generalized bath-correlation route so you do not have to believe the Fokker–Planck picture. Numerics (Figs. 2–4 plus SM) are consistent: same dynamics when stable, clear improvement when conventional HEOM diverges or drifts, and the method is immediately drop-in with AAA or Prony.\n\nSoft spots are real but secondary. They deliberately restrict to a one-parameter family even though the SM writes the full four-parameter linear map; they never prove that independent per-mode ϕ_k or the larger family cannot do better, nor that the optimal ϕ_r is completely insensitive to truncation or bond dimension. No code is shipped. None of that falsifies the existence of the gauge or the demonstrated gains.\n\nThis is for anyone who actually runs HEOM on structured or strongly coupled baths. The math is solid, the evidence is direct, and the citation pattern is fair. I would send it to referees without hesitation; it is ready for a serious technical review.","headline":"Clean, usable gauge freedom in complex-pole HEOM that actually removes two distinct numerical instabilities and reaches previously inaccessible regimes.","tokens_in":19981,"tokens_out":486,"would_cite":true,"duration_ms":5123,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Complex-pole HEOM hides a continuous gauge that can be tuned to stabilize the Liouvillian without changing the exact dynamics.","keywords":["hierarchical equations of motion","open quantum systems","complex-pole decomposition","gauge freedom","non-normality","spin-boson model","sub-Ohmic bath","Brownian oscillator"],"falsifier":"For a fixed multi-pole decomposition of a strongly coupled Brownian or sub-Ohmic bath, show that independent per-mode optimization of each ϕ_k (or the full four-parameter linear map) yields a generator whose largest real eigenvalue or long-time error growth is substantially smaller than the single-parameter GO-HEOM result reported in the paper.","tokens_in":20038,"feed_emoji":"⚙️","tokens_out":638,"duration_ms":4615,"temperature":0.7,"pith_summary":"Complex-pole hierarchical equations of motion (HEOM) give numerically exact open-system dynamics for arbitrary baths, but their non-Hermitian Liouvillians often produce unstable eigenvalues or amplify numerical errors. This paper shows that those Liouvillians are not unique: a continuous family of analytically equivalent generators all encode the same bath correlation function yet have radically different spectra and degrees of non-normality. The family is parameterized by a gauge angle that can be optimized once, yielding GO-HEOM. The optimized generator removes spectral divergences for strongly coupled Brownian oscillators and suppresses long-time error growth for sub-Ohmic baths, allowing stable simulations through the delocalized-to-localized quantum phase transition at coupling strengths previously inaccessible to HEOM. Because the gauge is independent of how the bath correlation is decomposed, the same fix applies to existing Prony, AAA and Padé implementations.","feed_headline":"Hidden gauge tames unstable open-quantum equations","feed_subtitle":"One tunable angle stabilizes complex-pole HEOM through strong coupling and quantum phase transitions","key_machinery":"The ϕ-dependent gauge transformation of the auxiliary ladder operators (Eq. 7–8 of the main text), which rewrites the Fokker–Planck generator while leaving the bath correlation function invariant and produces the family of HEOM equations (Eq. 10).","core_discovery":"Complex-pole HEOM possesses a previously unknown continuous gauge freedom: a one-parameter family of Liouvillians, all generating identical exact reduced dynamics for a given bath correlation function, whose eigenspectra and non-normality can be continuously reshaped. Optimizing a global scaling of that gauge produces GO-HEOM, which eliminates positive-real eigenvalues for strong Brownian-oscillator coupling and damps non-normal transient error growth for sub-Ohmic environments.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Hidden gauge freedom stabilizes complex-pole HEOM","One-parameter gauge tames HEOM spectral instabilities","GO-HEOM extends exact open-system dynamics via gauge","Gauge optimization eliminates HEOM positive eigenvalues","Continuous HEOM gauge reshapes non-normal error growth"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"That a single global scaling of the gauge angle applied uniformly to every complex pole is already near-optimal for multi-mode baths.","fun_headline_variants_meta":{"raw":{"variants":["Hidden gauge freedom stabilizes complex-pole HEOM","One-parameter gauge tames HEOM spectral instabilities","GO-HEOM extends exact open-system dynamics via gauge","Gauge optimization eliminates HEOM positive eigenvalues","Continuous HEOM gauge reshapes non-normal error growth"]},"model":"grok-4.5","effort":"low","cost_usd":0.00352,"raw_usage":{"total_tokens":1132,"prompt_tokens":770,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":35200000,"prompt_tokens_details":{"text_tokens":770,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":283,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":770,"tokens_out":79,"duration_ms":2718,"temperature":1.0,"reasoning_tokens":283,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T12:47:02.356264+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For a fixed multi-pole decomposition of a strongly coupled Brownian or sub-Ohmic bath, show that independent per-mode optimization of each ϕ_k (or the full four-parameter linear map) yields a generator whose largest real eigenvalue or long-time error growth is substantially smaller than the single-parameter GO-HEOM result reported in the paper.","supporting_citations":[],"review_version":1}