{"id":"b851cf1c-78d7-4dd0-8322-e422009fe6cd","arxiv_id":"2607.04659","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Mode-resolved Krylov complexity and entanglement entropy signal exceptional-point and topological transitions and dynamical phases in the non-Hermitian extended SSH model.","lead":"The paper shows that Krylov spread complexity and entanglement entropy detect topological and exceptional-point transitions in a non-Hermitian extended SSH chain, and that mode-resolved complexity tracks the responsible momenta. It also maps dynamical phases in the purely imaginary spectrum via saturation times of both quantities.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified; the free-fermion solvability assumption is correctly scoped and does not undermine the stated claims.","rationale":"The central claim is that spread complexity (protocol i) and long-time complexity plus entanglement entropy (protocol ii) detect the Hermitian topological transitions and the non-Hermitian exceptional-point-count transitions, that mode-resolved complexity tracks the responsible momenta, and that saturation times distinguish dynamical phases controlled by the slowest decay mode. All of these statements are demonstrated inside the free-fermion setting the authors explicitly adopt. The reader's weakest assumption is therefore accurate as a scope limitation, but it does not constitute a correctness risk for the claims as written. The mathematics is standard, the figures are consistent with the analytic formulae, and the entanglement-entropy numerics supply an independent check of the dynamical-phase structure. Consequently the CONDITIONAL verdict (driven by incremental novelty and lack of public code) needs no adjustment.","tokens_in":23095,"tokens_out":512,"duration_ms":4334,"concrete_test":"Re-derive the saddle-point saturation formula (Eqs. 41–49) for one concrete point inside the purely-imaginary regime (e.g., ta=tb=-1, tc=0, td=0, γ=4.86, h=0) by direct numerical integration of the exact mode-resolved |ΔC(t)| over a dense k-grid; confirm that the extracted t* matches 1/(2Γ(k*)) to within 1 %.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly notes that closed-form C_Ω, C(k;t) and t*=1/(2Γ(k*)) rest on the quadratic free-fermion structure (su(2)×su(2) per k-pair), anti-periodic BC and lowest-weight reference state (Secs. II–III, V). That assumption is load-bearing for the analytic expressions, yet it is not a hidden flaw: the paper never claims results beyond free fermions, the derivations follow standard normal-ordering of su(2) coherent states, and the entanglement-entropy numerics (Gaussian-state QR evolution, L=100) independently corroborate the same dynamical phases and saturation scaling. Intra-phase wiggles in long-time averages are already flagged by the authors as a reason to prefer protocol (i). No internal inconsistency or unsupported extrapolation is present.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies the non-Hermitian extended SSH chain with next-nearest-neighbor hoppings and an imaginary staggered potential. It maps how exceptional points emerge in pairs from Hermitian gap-closing momenta near topological phase boundaries, then analyzes two analytically tractable dynamical protocols via Krylov spread complexity and entanglement entropy: (i) unitary preparation of the non-Hermitian ground state from a product of su(2) lowest-weight states, and (ii) non-unitary evolution under the Hamiltonian. Spread complexity (and its derivative) signals the transitions under protocol (i); long-time averages of complexity and half-chain entanglement entropy do so under protocol (ii). Mode-resolved complexity and associated fidelities identify the critical momenta and track their evolution. In the purely imaginary spectrum regime the authors derive a saddle-point saturation time t* = 1/(2Γ(k*)) controlled by the slowest decay mode and show that both complexity and entanglement entropy exhibit the same dynamical phases, with the critical mode now continuously tunable by the extra hoppings.","tokens_in":23491,"tokens_out":1144,"duration_ms":8849,"significance":"The work cleanly extends the Krylov-complexity diagnostics of Medina-Guerra et al. from the ordinary non-Hermitian SSH model to a richer free-fermion setting that hosts multiple winding numbers and continuously tunable critical modes. The closed-form su(2) coherent-state expressions for C_Ω and C(k;t), the Lambert-W and ε-independent saturation-time formulas, and the independent Gaussian-state entanglement-entropy numerics constitute concrete, reproducible advances. Mode-resolved complexity and the fidelity map supply a transparent momentum-space fingerprint of the transitions that is new relative to the existing literature. Because the model is experimentally relevant (photonic, acoustic, ultracold-atom platforms) and entanglement entropy is more accessible than Krylov complexity, the dynamical-phase characterization has clear observational value within free-fermion non-Hermitian physics.","major_comments":[{"comment":"Section V.A, Eqs. (49)–(53) and Fig. 8: the claim that first and second derivatives of t* signal a dynamical phase transition (t*_1 \to t*_2) is well supported for the ordinary SSH limit, but for the extended model the continuous migration of k* produces only a crossover inside the t*_2 regime (green lines in Fig. 8). The manuscript should state more sharply which derivative discontinuities are true transitions versus smooth crossovers, and whether an order parameter (beyond the location of k*) can be defined that jumps across the t*_1/t*_2 boundary.","section":null},{"comment":"Section III.B and Fig. 4(c,d): the authors correctly note that long-time averages ar C and ar S display intra-phase peaks/valleys that do not coincide with changes in n_EP, and therefore prefer protocol (i). This observation is load-bearing for the claim that C_Ω is the superior indicator; a short quantitative comparison (e.g., the magnitude of the spurious features relative to the true jumps) would strengthen the argument that the preference is not merely qualitative.","section":null}],"minor_comments":[{"comment":"Figure 5 caption and surrounding text: the notation C_Ω ≈ ar C versus C_Ω ≈ 1-ar C is introduced without an explicit definition of the numerical threshold used to draw the blue/red dashed lines; a sentence clarifying the criterion would improve reproducibility.","section":null},{"comment":"Equation (2) and the anti-periodic boundary-condition choice: a brief remark on why anti-periodic rather than periodic conditions are preferred (avoidance of exact k=0,π for even L) would help readers who wish to reproduce the discrete spectra.","section":null},{"comment":"Section IV, fidelity definitions (33)–(35): the small increment δ in t_c and the momentum discretization are not stated numerically; listing the values used for the color plots in Fig. 6 would remove ambiguity.","section":null},{"comment":"Typographical: “UNIT AR Y” and “NON-UNIT AR Y” appear with spaces in section headings; “fidelity” is occasionally written “fidelities” inconsistently when referring to a single map.","section":null},{"comment":"References: the recent experimental proposal for measuring Krylov complexity (Ref. [39]) is cited but not connected to the concrete free-fermion setting of the present work; a short sentence on possible photonic or cold-atom implementations would be useful.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, technically careful extension of the authors’ and Medina-Guerra et al.’s earlier free-fermion Krylov work. Novelty is incremental rather than transformative, but the analytic control and the mode-resolved diagnostics are genuine additions. Fit for a specialized quantum-information / non-Hermitian condensed-matter journal is good; for a broader high-impact venue the free-fermion restriction and the absence of a true interacting or disordered test would be more limiting. No citation or ethical concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a careful, analytically controlled extension of the Medina-Guerra Krylov-spread program from ordinary non-Hermitian SSH to the extended model with next-nearest-neighbor hoppings. What is actually new is the mode-resolved (momentum-resolved) complexity and its fidelities, which track the characteristic momenta and how they evolve with the driving parameter, plus the demonstration that the critical mode k* now varies continuously and that entanglement entropy saturates with the same 1/(2\\Gamma(k*)) scaling. Exceptional points emerge in pairs from the Hermitian gap-closing points, with the count matching twice the winding-number jump; that is cleanly shown.\n\nThe free-fermion structure is used honestly. They rewrite H as su(2)\\times su(2) per momentum pair, start from the product of lowest-weight states, and obtain exact C_\\Omega and C(k;t) via normal-ordering. The saddle-point analysis of |\\Delta C(t)| yields the Lambert-W formula and the \\epsilon-independent t* that match the numerics in Figs. 7–8. The Gaussian-state QR evolution for half-chain EE (L=100) independently recovers the same dynamical phases and saturation, which is useful because EE is more experimentally accessible. Intra-phase wiggles in the long-time averages are already flagged by the authors as a reason to prefer the unitary-preparation protocol; that is proportionate.\n\nSoft spots are minor and scoped. Everything rests on quadratic free fermions, anti-periodic BC and the chosen reference state; the paper never claims otherwise. No public code, and a few free parameters (\\gamma values, the t_d=1.21 cut) are illustrative rather than exhaustive. Citation pattern is appropriate: they build directly on [21,22] and the Hermitian extended-SSH literature without circularity.\n\nThis is for people working on non-Hermitian free-fermion topology or Krylov complexity diagnostics. The math is standard and reproducible from the text. I would send it to peer review; a referee can ask for tighter discussion of the intra-phase features and perhaps a short code note, but the central claims hold. Worth engaging if that is your area.","headline":"Solid free-fermion extension of Krylov-spread diagnostics to the extended non-Hermitian SSH: mode-resolved complexity and EE saturation are the real additions; analytic control is clean and scoped correctly.","tokens_in":24076,"tokens_out":552,"would_cite":true,"duration_ms":5278,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Spread complexity and entanglement entropy detect phase transitions and dynamical phases in a non-Hermitian extended SSH chain.","keywords":["Krylov complexity","spread complexity","mode-resolved complexity","non-Hermitian SSH","exceptional points","entanglement entropy","dynamical phases","saturation time"],"falsifier":"Compute or measure the long-time half-chain entanglement entropy (or the unitary-preparation spread complexity) while sweeping a hopping parameter across a predicted exceptional-point boundary; if the first derivative remains smooth and no change appears in the saturation time of a purely imaginary spectrum, the claimed detection fails.","tokens_in":23971,"feed_emoji":"⚛️","tokens_out":678,"duration_ms":5081,"temperature":0.7,"pith_summary":"The paper studies an extended Su-Schrieffer-Heeger chain that includes next-nearest-neighbor hoppings and a staggered imaginary potential. In the Hermitian limit the model already has several topological phases distinguished by winding numbers; weak non-Hermiticity splits each gap-closing point into a pair of exceptional points. The authors show that Krylov spread complexity, evaluated under two different dynamical protocols, and the long-time half-chain entanglement entropy both jump or change slope exactly where the number of exceptional points changes or where topological winding numbers jump. A mode-resolved version of the complexity further isolates the individual momenta that are responsible for those signatures and tracks how those critical momenta move when parameters are varied. Finally, when the spectrum is purely imaginary, the time at which complexity (and entanglement) saturates is controlled by the slowest-decaying mode; different locations of that mode define distinct dynamical phases that can be read off from either quantity.","feed_headline":"Complexity and entanglement map phases of a non-Hermitian chain","feed_subtitle":"Mode-resolved Krylov complexity and saturation times locate exceptional points and dynamical phases","key_machinery":"Mode-resolved Krylov spread complexity C^s(k;t) obtained from the su(2)×su(2) coherent-state structure of each momentum pair, together with the saddle-point saturation time t* = 1/(2Γ(k*)) set by the slowest-decaying mode of a purely imaginary spectrum.","core_discovery":"Krylov spread complexity of a unitary preparation of the non-Hermitian ground state, as well as the long-time average of spread complexity and of entanglement entropy under non-unitary evolution, locate both the topological phase boundaries of the Hermitian extended SSH model and the boundaries at which the number of exceptional points changes once non-Hermiticity is introduced; mode-resolved complexity identifies the responsible momenta, and the saturation time of both complexity and entanglement is fixed by the inverse of the slowest decay rate Γ(k*).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Krylov complexity signals phases in non-Hermitian SSH chain","Mode-resolved complexity pinpoints exceptional points","Spread complexity and entanglement locate dynamical phases","Long-time complexity tracks non-Hermitian phase boundaries","Mode-resolved Krylov complexity maps SSH transitions"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Everything is derived for a free-fermion quadratic Hamiltonian that factors into independent su(2) algebras per momentum pair, starting from a product of lowest-weight reference states; interactions, disorder or a different initial state would remove the closed-form expressions.","fun_headline_variants_meta":{"raw":{"variants":["Krylov complexity signals phases in non-Hermitian SSH chain","Mode-resolved complexity pinpoints exceptional points","Spread complexity and entanglement locate dynamical phases","Long-time complexity tracks non-Hermitian phase boundaries","Mode-resolved Krylov complexity maps SSH transitions"]},"model":"grok-4.5","effort":"low","cost_usd":0.00585,"raw_usage":{"total_tokens":1567,"prompt_tokens":792,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":58500000,"prompt_tokens_details":{"text_tokens":792,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":718,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":792,"tokens_out":57,"duration_ms":5576,"temperature":1.0,"reasoning_tokens":718,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T15:38:03.810139+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute or measure the long-time half-chain entanglement entropy (or the unitary-preparation spread complexity) while sweeping a hopping parameter across a predicted exceptional-point boundary; if the first derivative remains smooth and no change appears in the saturation time of a purely imaginary spectrum, the claimed detection fails.","supporting_citations":[],"review_version":1}