{"id":"18a153fd-3c7a-44ab-bce6-92deeaf12b1d","arxiv_id":"2607.26586","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The giant reversible piezoelectric response of KNN-BNZ at its polymorphic phase boundary is attributed to a percolating multifractal polar domain whose volume fraction is near the 3D percolation threshold.","lead":"Simulations of a lead-free piezoelectric material show that its best performance appears when a polar network fills the material to just the right fraction — near the level where random networks become fully connected. The result suggests a new design rule: tune the connectivity of polar domains rather than just the composition.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No direct percolation analysis: the near-threshold claim rests on a single hand-defined volume fraction, and D0=2.925 is inconsistent with critical percolation clusters.","rationale":"The reader correctly identified the domain-growing protocol as a weak point and returned CONDITIONAL. My concern is adjacent but more fundamental: even taking the domain-growing protocol at face value, the paper never performs a percolation analysis. It only compares a single measured volume fraction to a textbook threshold, with no finite-size scaling, no error bars, and no percolation observables. The reported D0 = 2.925 is a separate red flag, since it is far from the critical percolation fractal dimension (~2.53) and suggests the domain is not near criticality. However, the qualitative mechanism — a rigid fractal backbone plus compliant nonfractal regions producing reversible strain — is supported by the rotation-angle contrasts and the closed vs open hysteresis loops, so the paper should not be rejected outright. The quantitative percolation-criticality claim needs substantial additional support before it can be accepted. The reader's CONDITIONAL verdict remains appropriate; my stress-test does not move it.","tokens_in":9544,"tokens_out":5308,"duration_ms":54533,"concrete_test":"Reanalyze the saved MD trajectories at T* using a standard cluster-labeling algorithm (e.g., Hoshen-Kopelman) on the polar unit cells and compute percolation observables: spanning probability P(omega_f), mean cluster size chi(omega_f), and the largest-cluster volume fraction as functions of omega_f for system sizes 16^3, 32^3, and 64^3. Perform finite-size scaling to locate the percolation threshold and compare it with the d33 volcano peak at omega_f ≈ 0.371; also measure the global fractal dimension D0 of the largest cluster at the extrapolated p_c. If the scaling collapse places p_c outside the error bars around 0.371, or if D0 is ≈ 3 rather than ≈ 2.53, the near-percolation-criticality interpretation fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is that maximum reversible d33 occurs near the 3D percolation threshold. The evidence is the volcano plot in Fig. 4, which peaks at omega_f ≈ 0.371, compared with p_c ≈ 0.312 from random isotropic percolation. But omega_f is not an occupation probability in a percolation model; it is the volume fraction of the single connected component identified by the custom domain-growing protocol (30-degree angular tolerance, 200-ps correlation). That protocol only outputs a connected, system-spanning 'fractal domain,' so its volume fraction is necessarily above the percolation threshold by construction. A 19% offset from p_c has no meaning without a variance, a spanning-probability curve, or a finite-size scaling analysis. None is reported. Furthermore, at T* = 140 K the paper reports D0 = 2.925, whereas the incipient infinite cluster of 3D percolation has fractal dimension ≈ 2.53. D0 close to 3 indicates a compact structure well above criticality, not a critical percolation cluster. If the identified multifractal domain is not the critical cluster, comparing its volume fraction to p_c is unjustified. Thus the headline 'near-critical polar connectivity' is unsupported by the presented data, even though the qualitative backbone/compliance mechanism may remain plausible.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper uses large-scale molecular dynamics with a machine-learned force field (UniPero) to study the polymorphic phase boundary (PPB) in lead-free KNN–BNZ. It identifies a connected, system-spanning polar domain at T* = 140 K, characterized by a global fractal dimension D0 = 2.925 and a multifractal spectrum width Δα = 1.196. The authors propose that this 'fractal domain' acts as a rigid backbone preserving polar memory, while surrounding nonfractal regions provide compliance, enabling large reversible piezoelectric response. They report d33 = 809.8 pm/V for the [111] initial state, with closed-loop reversibility, whereas the [221] state gives an open loop and d33 = 418.7 pm/V. By pooling states and temperatures, they construct a volcano-shaped plot of d33 versus fractal-domain volume fraction ωf, peaking at ωf ≈ 0.371, which they claim is close to the 3D percolation threshold of 0.312. The central claim is that near-critical percolating connectivity of a multifractal polar domain is the microscopic mechanism for PPB-enhanced piezoelectricity.","tokens_in":9780,"tokens_out":4491,"duration_ms":43799,"significance":"If the percolation-criticality claim were quantitatively established, this would be a conceptually novel mechanism for PPB-enhanced piezoelectricity, moving beyond the standard phase-coexistence picture and offering a design parameter (fractal connectivity) for lead-free piezoelectrics. The qualitative backbone/compliance mechanism is well supported by several internal controls: the closed-loop versus open-loop contrast between [111] and [221] states, the consistent inside/outside rotation-angle asymmetry across more than 20 polar states (Fig. 3f), and the temperature evolution of D0 and Δα. The paper also provides reproducible computational methodology (LAMMPS, a machine-learned potential, explicit finite-size supercell) and validates against experimental dielectric spectra and strain loops. However, the load-bearing quantitative claim—that maximum d33 occurs near the percolation threshold—is not supported by the presented analysis. The volume fraction ωf is not a percolation order parameter, no finite-size scaling or spanning-probability analysis is reported, and the reported D0 = 2.925 is inconsistent with a 3D critical percolation cluster (D ≈ 2.53). These issues are fixable bu","major_comments":[{"comment":"The central claim that maximum reversible d33 occurs near the 3D percolation threshold is not supported by the data. ωf is the volume fraction of the single connected component identified by the custom domain-growing protocol (seed threshold, 30° angular tolerance, 200-ps correlation window), not an occupation probability in a percolation model. The protocol by construction outputs a connected, system-spanning object, so its volume fraction is a cluster property, not a percolation probability. The comparison 0.371 versus 0.312 therefore has no quantitative meaning without a spanning-probability curve, cluster-size distribution, or finite-size scaling analysis, none of which is reported. Moreover, D0 = 2.925 at T* = 140 K (Fig. 2b) is close to the embedding dimension 3, whereas the incipient infinite cluster of 3D percolation has fractal dimension ≈2.53. This indicates a compact, supercri","section":"Abstract; Fig. 4; Supplementary Sect. II.C"},{"comment":"All structural descriptors—D0, Δα, ωf, and the inside/outside partition used in Fig. 3(e)–(f)—depend on hand-chosen thresholds in the domain-growing protocol: the local-polarization seed threshold, the 30° angular tolerance, and the 200-ps correlation window. No sensitivity analysis is reported. Since the volcano peak position in Fig. 4 and the 'close to 0.312' comparison are derived from a single operational definition, the quantitative conclusions are conditional on this protocol. The authors should demonstrate that the main findings are robust over a range of thresholds, or provide a principled, data-driven criterion for selecting them. Without this, the percolation-threshold comparison and the exact value ωf ≈ 0.371 are not meaningful.","section":"Supplementary Sect. II.C; Figs. 2–4"},{"comment":"The volcano plot pools data from different temperatures and polar states, but only reversible polarization-rotation paths are included, with the field direction chosen so that the path avoids trapping by other global states. Because reversibility is established using the same simulations that define the fractal domain, and because the field-direction selection is not specified quantitatively, the dataset may be biased toward favorable paths. Error bars, the number of states per temperature, and the criterion for choosing the field direction are not given. This information is necessary to assess whether the peak at ωf ≈ 0.371 is robust or an artifact of the reversible-path selection.","section":"Fig. 4; Fig. 3(b)–(c)"}],"minor_comments":[{"comment":"The effective d33 is said to be extracted 'from the slope' of the strain–electric-field loop, but a loop has field-dependent slope; please specify the field range or the convention (e.g., unipolar slope near zero field or at Emax) used for the reported values.","section":"Fig. 1(c); page 4"},{"comment":"The manuscript correctly notes that MD underestimates transition temperatures and invokes a temperature rescaling to compare with experiments. This is an acknowledged limitation, but the mapping is not quantified. Please state the rescaling factor or show that the qualitative conclusions are insensitive to the shift.","section":"Page 4; Discussion of temperature rescaling"},{"comment":"D0 = 2.925 is reported to four significant figures without an uncertainty estimate from the box-counting fit. A standard error or confidence interval would help the reader judge whether D0 is statistically distinct from 3.","section":"Fig. 2(b); caption"},{"comment":"The text mentions 'more than 50 distinct directions' while Fig. 3(f) reports 'more than 20' states. Reconcile these numbers and report the exact sample size in the figure or text.","section":"Page 6; Fig. 3(f)"},{"comment":"Minor typos: Ref. [7] 'Naure' should be 'Nature'; Ref. [30] 'molecular synamics' should be 'molecular dynamics'. Also, 'KNN–NBZ' appears once in the text on page 3 and should be 'KNN–BNZ' for consistency.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents a thought-provoking mechanism and has strong computational controls, but the headline percolation-criticality claim is not yet quantitatively supported. I would like the authors to either provide a direct percolation analysis (spanning probability, cluster statistics, finite-size scaling) or tone down the percolation-threshold comparison. The paper's core qualitative message—that a hierarchical polar network can provide both restoring force and compliance—is likely salvageable and potentially important for the field. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper identifies a plausible microscopic mechanism for PPB-enhanced piezoelectricity — a percolating multifractal polar domain that acts as a rigid backbone while surrounding soft regions rotate — but the headline claim that maximum d33 occurs near the 3D percolation threshold is not established. The qualitative story is well supported by the MD evidence; the quantitative percolation-criticality framing is the weak load-bearing beam.\n\nWhat's new: the idea that PPB enhancement is not primarily phase coexistence but a connectivity/compliance tradeoff is new. The authors back it with credible controls: inside-vs-outside rotation angle distributions across more than 20 polar states, the closed-loop [111] vs open-loop [221] contrast, and reproduction of the experimental dielectric plateau, poled/unpoled differences, and order-of-magnitude d33 enhancement. The temperature rescaling is acknowledged up front and is standard for this type of force field. Citation practice looks fine; the MLFF references are to their own prior work but appropriate.\n\nWhere it gets soft: the percolation-criticality claim rests entirely on omega_f, a volume fraction defined by their own domain-growing protocol (30-degree angular tolerance, 200-ps correlation window). No sensitivity analysis is given, and those thresholds are hand-chosen. More importantly, the identified domain is a single connected system-spanning component. Its volume fraction is above the percolation threshold by construction, and D0 = 2.925 is close to 3, not ~2.53 as expected for a critical cluster. That means the structure is compact, not critical. Comparing omega_f = 0.371 to p_c = 0.312 without error bars or a spanning-probability/finite-size analysis is not convincing. The volcano plot itself may just reflect the empirical tradeoff between restoring force and compliance, which is a nice story that doesn't need percolation criticality. The selection of reversible field paths also introduces a conditional bias: only reversible states are included, which is fine for studying reversibility but limits the strength of the d33-vs-omega_f relation.\n\nBottom line: the mechanistic backbone/compliance picture and the dielectric plateau rationalization are worth taking seriously. The percolation-criticality interpretation needs more support. A serious editor should send this to peer review; a referee should ask for sensitivity analysis, a direct percolation analysis (spanning probability, finite-size scaling), and error bars on omega_f and d33, or a tone-down of the criticality claim.","headline":"Convincing qualitative backbone/compliance mechanism for PPB, but the percolation-criticality claim is not yet supported.","tokens_in":10389,"tokens_out":2326,"would_cite":true,"duration_ms":22710,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"At a polymorphic phase boundary in a lead-free piezoelectric, the polar structure is a percolating multifractal network, and the largest reversible piezoelectric response occurs near the three-dimensional percolation threshold.","keywords":["percolation","multifractal","polymorphic phase boundary","piezoelectricity","ferroelectrics","molecular dynamics","polar domains","lead-free piezoelectrics"],"falsifier":"If an independent measure of fractal connectivity (e.g., from X-ray nanodiffraction or PFM imaging across the PPB in KNN-BNZ) showed no percolating fractal polar network with a volume fraction near 0.31–0.37 at the temperature of maximum d33, the central claim would be contradicted. Alternatively, a sensitivity analysis of the domain-growing thresholds that shifts ωf far from 0.312 while retaining the volcano peak would undermine the percolation interpretation.","tokens_in":9289,"feed_emoji":"⚡","tokens_out":5163,"duration_ms":42316,"temperature":0.7,"pith_summary":"This paper attempts to establish that the polymorphic phase boundary (PPB) in the lead-free piezoelectric (K,Na)NbO3–(Bi,Na)ZrO3 is not simply a coexistence region of nearly degenerate ferroelectric phases, but hosts a percolating multifractal polar domain. Using large-scale molecular dynamics, the authors show that this fractal network is characterized by a global fractal dimension D0 and a multifractal spectrum width Δα, and that the reversible piezoelectric coefficient d33 reaches a volcano-shaped peak when the fractal-domain volume fraction ωf is approximately 0.371, close to the three-dimensional percolation threshold of 0.312. The proposed mechanism is a division of labor: the fractal backbone preserves the initial polar state and provides restoring force, while the surrounding nonfractal regions rotate easily under field and produce large strain. If correct, the work would replace the phase-coexistence explanation with a connectivity explanation and offer fractal connectivity as a design parameter for high-performance lead-free piezoelectrics.","feed_headline":"Fractal polar networks explain giant lead-free piezoelectricity","feed_subtitle":"At a polymorphic boundary, the best response emerges when the polar network nears percolation—a new design target.","key_machinery":"The load-bearing object is the polar fractal domain, identified by a domain-growing algorithm that starts from randomly seeded cells with sizable local polarization and adds neighboring cells whose polarization stays within 30° of the domain direction over a 200-ps correlation window. Its structural fingerprints are D0 (global box-counting fractal dimension), Δα (width of the multifractal singularity spectrum, measuring spatial heterogeneity), and ωf (the volume fraction occupied by the fractal domain). The comparison of ωf with the 3D percolation threshold, and the dual-role picture of the fractal backbone (rigid, memory-preserving) versus nonfractal surroundings (compliant, strain-producin","core_discovery":"The central discovery is that at T*=140 K, inside the simulated PPB of KNN-BNZ, the polar microstructure forms a single connected, system-spanning network with fractal dimension D0=2.925 and multifractal spectrum width Δα=1.196—a dense but noncompact backbone with tenuous branches. The authors find that this network acts as a relatively rigid polar memory: dipoles inside the fractal domain rotate less than those outside, and the reversible strain–field loop closes because the backbone restores the initial orientation after field removal. Across more than twenty zero-field polar states and a range of temperatures, the effective reversible d33 follows a single 'volcano' when plotted against th","pith_inferences":["Since the paper's ωf is defined via a specific domain-growing protocol with hand-chosen thresholds (30°, 200 ps), a natural next step would be to test how sensitive the volcano peak position and height are to these thresholds; if the peak robustly sits near percolation across reasonable variations, the claim is strengthened.","Experimental verification could come from high-resolution reciprocal-space mapping or piezoresponse force microscopy across the PPB: if a percolating fractal polar network with volume fraction ≈0.35 is not observed at the temperature of maximum d33, the connectivity explanation would be in doubt.","The temperature rescaling required by the force field implies the simulated T*=140 K corresponds to a higher experimental temperature; if the fractal connectivity criterion is universal, the same volcano relationship may hold when d33 is measured as a function of temperature in a single sample, making ωf a hidden variable that collapses data onto one curve."],"forward_implications":["If the percolating multifractal domain is the microscopic mechanism, then PPB-enhanced giant reversible piezoelectricity is a connectivity effect: near-threshold fractal connectivity enables both reversibility (backbone) and large strain (compliant regions), not simply a flat energy landscape.","The global fractal dimension D0 and multifractal width Δα act as microstructural order parameters that can rationalize the temperature-dependent dielectric plateau and the decoupling between dielectric and piezoelectric peaks.","The optimal ωf≈0.371 provides a quantitative design target: to maximize reversible d33, materials should be tuned so that the fractal polar network occupies close to the percolation-critical volume fraction.","The same composition can show either closed reversible loops or open, trapped loops depending on the initial poling state; this explains the experimentally known poling sensitivity of KNN-based ceramics.","The mechanism is generic: fractal connectivity as a design parameter could apply to other structurally disordered systems where reversible large deformation is sought."],"fun_headline_variants":["Percolating fractal domains boost lead-free piezoelectric response","Fractal percolation threshold dictates lead-free piezo response","Near-percolation fractal networks maximize reversible piezo response","Percolating multifractal backbone drives lead-free piezoelectricity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The comparison with the percolation threshold rests on the particular operational definition of what counts as a 'fractal domain' (a growth rule requiring polarization alignment within 30° over 200 ps), and a different cutoff would shift ωf; in addition, the simulated PPB temperature (≈140 K) is mapped to the experimental one by a rescaling that the force field requires.","fun_headline_variants_meta":{"raw":{"variants":["Percolating fractal domains boost lead-free piezoelectric response","Fractal percolation threshold dictates lead-free piezo response","Near-percolation fractal networks maximize reversible piezo response","Percolating multifractal backbone drives lead-free piezoelectricity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000992,"raw_usage":{"total_tokens":4043,"prompt_tokens":752,"completion_tokens":3291,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":3220}},"tokens_in":496,"tokens_out":3291,"duration_ms":20357,"temperature":1.0,"reasoning_tokens":3220,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T12:49:43.395005+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If an independent measure of fractal connectivity (e.g., from X-ray nanodiffraction or PFM imaging across the PPB in KNN-BNZ) showed no percolating fractal polar network with a volume fraction near 0.31–0.37 at the temperature of maximum d33, the central claim would be contradicted. Alternatively, a sensitivity analysis of the domain-growing thresholds that shifts ωf far from 0.312 while retaining the volcano peak would undermine the percolation interpretation.","supporting_citations":[],"review_version":1}