{"id":"4184ee89-f661-4808-a078-f21ff769ecd9","arxiv_id":"2506.00919","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In the overlap of classical and quantum Yang-Lee critical regions, both scaling functions apply together and yield constraint equations that the order parameter obeys.","lead":"This paper proposes a hybrid scaling mechanism for the overlapping critical regions of classical and quantum Yang-Lee edge singularities, and verifies it numerically on an Ising chain in an imaginary field. The mechanism could help extract quantum phase transition information from finite-temperature experiments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The hybrid 'constraint' Eqs. (17)-(18) is not derived from the scaling forms; it assumes the quantum scaling function has a power-law crossover/pole at the classical critical point, and all support is one-model data collapse with fitted critical points.","rationale":"The reader's weakest assumption identified the same load-bearing point: the constraint equations are obtained by assuming a power-law crossover of one scaling function into the other's exponent, and this is verified only by data collapse in a single model. My analysis confirms that Eqs. (17) and (18) are not logically implied by the scaling forms Eqs. (12), (13), (15), and (16) alone; the T-prefactor in Eq. (17) must be imported by postulating the singular behavior of f3 at the classical critical point. This is not a fatal internal inconsistency, and the exact L=1 solution in Eq. (8) suggests the hybrid form is at least plausible for this model. However, the paper's broader claim of a general mechanism bridging classical and quantum YLES rests on an unproven crossover assumption plus one model's fitted collapse. The reader's conditional verdict remains the right level of confidence: the mechanism is plausible and worth publishing, but the universality claim needs independent support. I therefore do not change the verdict.","tokens_in":16909,"tokens_out":17435,"duration_ms":175578,"concrete_test":"Compute h_L^C(T) and h_L^Q independently from transfer-matrix Lee-Yang zeros (or from the exact L=1 solution Eq. (8)) and test Eq. (17) with no adjustable parameters: plot M/T^{ξ−η} versus h−h_L^C for L=10 and T=0.06–0.2. If the curves fail to collapse once the critical points are not fitted from the same M data, the hybrid constraint is an artifact of parameter fitting. If they collapse, repeat the same zero-parameter test on a second model in the same universality classes to decide universality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Sec. V B the paper states that Eq. (17) is obtained by 'substituting Eq. (15) into Eq. (12).' That substitution is not a derivation. Eq. (12) is M ~ (g_L^C)^{1/δ00} with no T-dependent amplitude, whereas Eq. (17) contains the prefactor T^{ξ−η}; the prefactor can arise only if, in the overlap, the quantum scaling function f3[x] of Eq. (15) has a power-law divergence at the finite argument x_C = (h_L^C − h_L^Q)T^{−β01δ01/(ν01 z01)} with exponent 1/δ00. This is exactly the content of the hybrid mechanism, not a consequence of the two scaling forms. The collapse in Fig. 9(a) therefore mostly verifies that M/T^{ξ−η} versus g_L^C collapses, which is equivalent to assuming the pole of f3 at x_C. Moreover, h_L^C and h_L^Q are located from the same M(T,h) data used to test the collapse, and no error bars are reported, so the data collapse is a consistency check of the ansatz rather than an independent validation. Eq. (18) has the same structure for two-variable f4. If the required power-law crossover does not hold in other models, the central claim that Eqs. (17) and (18) connect classical and quantum YLES in general is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the scaling behavior of the order parameter M in the transverse Ising chain in an imaginary longitudinal field, focusing on overlapping critical regions of classical and quantum Yang-Lee edge singularities (YLES). The authors introduce a 'hybrid scaling mechanism' in which, inside the overlap, the scaling functions of both critical regions apply simultaneously and are mutually constrained. They verify the individual scaling forms for (0+1)D and (1+1)D quantum YLES and for (0+0)D and (1+0)D classical YLES, then propose constraint equations, Eqs. (17) and (18), that connect classical and quantum YLES. The paper claims these constraints allow quantum critical information to be extracted from finite-temperature classical measurements.","tokens_in":17329,"tokens_out":3745,"duration_ms":34878,"significance":"If the hybrid scaling mechanism is correct, the paper would provide a rare link between classical and quantum Yang-Lee criticality and a practical route to extract quantum critical information from finite-temperature data. The paper's strengths include the systematic organization of four different YLES universality classes in one model and the use of literature exponents rather than fitted ones, which makes the reported data collapses genuine tests of the individual scaling forms. The verification of the independent scaling forms, Eqs. (9), (10), (12), (13), (15), and (16), is a useful contribution. However, the central constraint equations are not derived from the scaling forms but assume a power-law crossover of the scaling functions; the numerical support for them is based on collapses using critical points extracted from the same data, without error bars. The generality of the mechanism across models is therefore not established by the manuscript as it stands.","major_comments":[{"comment":"The statement that Eq. (17) is obtained by substituting Eq. (15) into Eq. (12) is not a derivation. Eq. (12) has no temperature prefactor, so Eq. (17) can only result if the quantum scaling function f3[x] in Eq. (15) crosses over to a pure power law x^{1/δ00} at the temperature-dependent argument x_C = (h_C^L − h_L^Q) T^{−β01δ01/(ν01 z01)}. That power-law crossover is precisely the content of the hybrid mechanism, not a consequence of the two scaling forms. The collapse in Fig. 9(a2) therefore verifies the assumed crossover rather than independently confirming a prediction. To support the general claim, the crossover would need to be derived from the scaling forms, or verified in a different model or at a different value of λ.","section":"Sec. V B"},{"comment":"The critical fields h_C^L and h_Q^L in Fig. 9 are located from the same M(T,h) data used to test the collapse, and no error bars or alternative determinations are reported. This makes the data collapse a consistency check rather than an independent validation of Eq. (17). Similarly, the verification of Eq. (7) in Fig. 1(b4) reports a fitted exponent of 2.2311, about 7% off the theoretical value 2.4, with no uncertainty or goodness-of-fit measure. The relation between h_C^∞ and h_Q^∞ is therefore not quantitatively established.","section":"Sec. V B / Fig. 9"},{"comment":"Eq. (18) is a two-variable scaling constraint, but it is tested only along the single path LT^{1/z11} = 2.4. The collapse in Fig. 9(b2) therefore only demonstrates consistency along this chosen curve. To support the claim that Eq. (18) holds generally, the authors should show collapses for multiple values of LT^{1/z11} (for example 1.2, 1.8, and 3.0) and compare the resulting exponents Ω and κ with the theoretical values.","section":"Sec. V B"},{"comment":"The paper claims a general hybrid scaling mechanism for overlapping critical regions, but all quantitative evidence is from a single model at λ = 5 with selected values of L and T. The size and location of the overlapping regions depend on L and T, yet no systematic scan over λ, L, or T is reported. A statement about universality would require at least one additional model or a different λ value to demonstrate that the power-law crossover of the scaling functions is not a special feature of this particular parameter choice.","section":"Secs. III–V"}],"minor_comments":[{"comment":"The title of Sec. IV, 'HYBRIDIZED SCALING MECHANSIM FOR THE CLASSICAL YLES AT FINITE TEMPERATURE', contains a typo: 'MECHANSIM' should be 'MECHANISM'.","section":"Sec. IV"},{"comment":"In the text near Eq. (1), 'YELS' appears where 'YLES' is intended. This typo occurs once in Sec. II A.","section":"Sec. II A"},{"comment":"The caption of Fig. 4 reads 'The rescaled cures according to Eq. (11)'; 'cures' should be 'curves'. The same typo appears in the caption of Fig. 7.","section":"Fig. 4 caption"},{"comment":"The axis label in Fig. 7(a), 'M versus gLCL', appears to be a typo; it should read 'M versus g_C^L'.","section":"Fig. 7(a)"},{"comment":"Eq. (18) is written as a relation between the scaling functions f4 and f2, but the numerical test in Fig. 9(b2) uses the order parameter M. Please state explicitly how Eq. (18) is converted into a plot of rescaled M and how the collapse is performed.","section":"Sec. V B, Eq. (18)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim depends on an assumption (power-law crossover of the quantum scaling function at the classical critical point) that is presented as a derivation. This should be reframed or derived. The numerical evidence would benefit from error bars and from tests at additional parameter values. The manuscript otherwise fits the journal's scope and addresses a timely question."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper so you know the state of play in overlapping Yang-Lee critical regions. The genuinely new thing is Eqs. (17) and (18): constraints linking classical and quantum YLES scaling functions, with the promise that quantum critical information can be extracted from finite-temperature data. That is new relative to Refs. [75,76], which introduced hybrid scaling for two quantum regions and for driven dynamics. The paper does the field a service by systematically checking the individual scaling functions for (0+0)D, (0+1)D, (1+0)D and (1+1)D YLES in the imaginary-field transverse Ising chain, and the collapses in Figs. 3, 6, and 8 look credible. The relations between finite-size and infinite-size critical points, Eqs. (4)–(7), also cleanly verify.\n\nThe soft spot is in the derivation of the constraint. Eq. (17) is obtained by \"substituting Eq. (15) into Eq. (12),\" but that substitution only produces the T^{ξ−η} prefactor if the quantum scaling function f3 has a power-law divergence at the classical critical point with exponent 1/δ00. That is not a formal consequence of the two scaling forms; it is the content of the hybrid mechanism. The collapse in Fig. 9(a2) therefore tests the ansatz, not an independently derived prediction. The same holds for Eq. (18) with f4. Also, h∞C and h∞Q are located from the same M(T,h) data used for the collapse, and no error bars are reported, so the agreement is a consistency check rather than independent validation. The fitted exponent 2.2311 for Eq. (7), 7% below the expected 2.4, is a minor blemish, not a fatal one.\n\nNone of this makes the paper empty. The mechanism is explicit, checkable, and consistent with the data shown. The open question is how general it is: one model, one parameterization, no uncertainty estimate, and no independent test of the crossover assumption. A referee should ask for a second model or at least an analytic demonstration using the L=1 solution, plus error bars or sensitivity analysis. If those come through, this becomes a solid reference for the quantum-to-classical YLES overlap.\n\nVerdict: deserves a serious referee; it should not be desk-rejected. My own reading-group vote is maybe, and I would not cite it in my own next-year work unless I were directly working on Yang-Lee singularities.","headline":"A plausible, well-verified-in-one-model extension of hybrid scaling to classical–quantum Yang-Lee overlaps, but the central constraint equations are an ansatz rather than a derivation.","tokens_in":17771,"tokens_out":2994,"would_cite":false,"duration_ms":28591,"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":"The paper claims that in overlapping critical regions of classical and quantum Yang-Lee edge singularities, both scaling functions apply simultaneously and are linked by constraint equations that let finite-temperature data expose quantum…","keywords":["Yang-Lee edge singularity","hybrid scaling mechanism","overlapping critical regions","quantum phase transition","classical phase transition","non-Hermitian Ising chain","finite-temperature scaling","data collapse"],"falsifier":"Compute the order parameter on a second model with overlapping classical and quantum Yang-Lee regions in the same temperature range and test whether the rescaled curves collapse according to Eqs. (17) and (18). Failure to collapse, or fitted exponents that drift with $\\lambda$ or $T$, would show that the assumed power-law crossover is not universal.","tokens_in":16732,"feed_emoji":"⚛️","tokens_out":6870,"duration_ms":63242,"temperature":0.7,"pith_summary":"The paper sets out to explain what happens when the critical region of a classical Yang-Lee edge singularity and that of a quantum Yang-Lee edge singularity overlap at low but nonzero temperature. It proposes a hybrid scaling mechanism: in the overlap, both universality classes' scaling functions remain valid, and they are linked by constraint equations that mix the critical exponents of the two classes. The paper tests this mechanism on the transverse Ising chain in an imaginary longitudinal field, which realizes (0+0)D and (1+0)D classical YLES at finite temperature and (0+1)D and (1+1)D quantum YLES at zero temperature. Numerical data collapse for all four pairwise overlaps supports the mechanism, most importantly for the classical-quantum overlaps described by Eqs. (17) and (18). If correct, the result gives a practical route to extract quantum critical information from finite-temperature measurements.","feed_headline":"Hybrid scaling unites classical and quantum Yang-Lee critical behavior","feed_subtitle":"Constraint equations let finite-temperature measurements reveal quantum critical exponents.","key_machinery":"The hybrid scaling mechanism is the central device. It has two assertions: (i) in the overlap of two critical regions, the scaling functions of both regions apply to the same data; (ii) those scaling functions obey a constraint obtained by substituting one scaling form into the other, giving hybrid functions whose prefactors and arguments mix exponents from both universality classes. The specific machinery on display is the substitution chain: Eq. (15) into Eq. (12) yields Eq. (17), and Eqs. (13) and (16) yield Eq. (18), with exponents such as $\\xi = \\beta_{01}/(\\nu_{01}z_{01})$ and $\\eta = \\beta_{01}\\delta_{01}/(\\nu_{01}z_{01}\\delta_{00})$ carrying quantum information into classical scaling functions. The numerical verification is data collapse after rescaling by these hybrid exponents.","core_discovery":"The central claim is that the scaling hypothesis can be extended across overlapping critical regions of different universality classes. In the overlap, the order parameter obeys the scaling function of each region simultaneously; substituting one scaling form into the other produces hybrid constraint equations. For the transverse Ising chain in an imaginary longitudinal field, the paper derives and numerically verifies four constraints: Eq. (11) for (0+1)D and (1+1)D quantum YLES, Eq. (14) for (0+0)D and (1+0)D classical YLES, Eq. (17) for (0+0)D classical and (0+1)D quantum YLES, and Eq. (18) for (1+0)D classical and (1+1)D quantum YLES. Because the classical-quantum constraints contain quantum critical exponents, they enable extraction of quantum phase transition properties from finite-temperature data.","pith_inferences":["Because the mechanism is formulated as a generic substitution of scaling forms, the same construction should produce hybrid constraints for any pair of overlapping critical regions, not only Yang-Lee singularities; this is the paper's implicit program.","Equation (17) can be read as an inverse protocol: fitting finite-temperature data to $M = T^{\\xi-\\eta}(g_C^L)^{1/\\delta_{00}}$ yields quantum exponents without ever cooling to $T=0$, an application the authors mention but do not develop.","The paper notes regions where three or four critical regions coexist; a natural extension is to demand joint consistency of several hybrid constraints, which would impose nontrivial relations among exponent ratios.","A direct experimental check could be made in engineered non-Hermitian quantum simulators where the finite-temperature classical-quantum overlap of YLES can be realized, testing whether the data collapse holds beyond this specific chain."],"forward_implications":["The hybrid mechanism, verified here for the static order parameter, extends the earlier hybrid Kibble-Zurek scaling for non-equilibrium dynamics to static critical behavior in the same system.","In the overlapping classical-quantum region, the constraint equations mix exponents of both classes, so finite-temperature measurements of a classical YLES carry quantum YLES exponents.","The same mechanism works for the overlap of two quantum YLES regions and for the overlap of two classical YLES regions, indicating a common pattern across classical and quantum criticality.","The transverse Ising chain in an imaginary longitudinal field thereby becomes a model system that bridges classical and quantum Yang-Lee critical phenomena, and the paper expects the mechanism to be testable in current experimental realizations of YLES.","The approach also suggests a route toward handling regions where three or four critical regions coexist, though the paper notes that such cases need more careful parameter control."],"supporting_citations":[{"why":"Establishes the Yang-Lee edge singularity universality classes and the critical exponents used throughout the scaling functions.","marker":"[3]"},{"why":"Proposes hybrid Kibble-Zurek scaling for overlapping (0+1)D and (1+1)D YLES, the direct antecedent of the static hybrid mechanism.","marker":"[75]"},{"why":"Introduces the hybridized Kibble-Zurek scaling across an overlapping critical region, which the present paper adapts to static overlap.","marker":"[76]"},{"why":"Provides the conformal-analysis scaling functions for the Ising quantum chain in an imaginary field used to write the quantum scaling forms.","marker":"[84]"},{"why":"Gives the real-space renormalization treatment of the transverse-field Ising chain in a complex field that locates finite-size YLES critical points.","marker":"[83]"},{"why":"Supplies the quantum-critical finite-temperature framework that justifies treating temperature as a relevant scaling variable in the overlap.","marker":"[51]"}],"fun_headline_variants":["Hybrid scaling links classical and quantum Yang-Lee singularities","Overlapping Yang-Lee critical regions obey hybrid scaling","Quantum exponents read from classical Yang-Lee scaling","One scaling law for overlapping classical-quantum criticality","Hybrid constraints bridge Yang-Lee classical and quantum edges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the assumption that, inside the overlap, one class's scaling function simply becomes the other class's power law, for example $f_3[x] \\sim x^{1/\\delta_{00}}$ in Eq. (17); if that crossover fails in another model, the mechanism is model-specific.","fun_headline_variants_meta":{"raw":{"variants":["Hybrid scaling links classical and quantum Yang-Lee singularities","Overlapping Yang-Lee critical regions obey hybrid scaling","Quantum exponents read from classical Yang-Lee scaling","One scaling law for overlapping classical-quantum criticality","Hybrid constraints bridge Yang-Lee classical and quantum edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1283,"prompt_tokens":987,"completion_tokens":296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":217}},"tokens_in":603,"tokens_out":296,"duration_ms":3173,"temperature":1.0,"reasoning_tokens":217,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:54:45.074064+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the order parameter on a second model with overlapping classical and quantum Yang-Lee regions in the same temperature range and test whether the rescaled curves collapse according to Eqs. (17) and (18). Failure to collapse, or fitted exponents that drift with $\\lambda$ or $T$, would show that the assumed power-law crossover is not universal.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proposes hybrid Kibble-Zurek scaling for overlapping (0+1)D and (1+1)D YLES, the direct antecedent of the static hybrid mechanism."},{"cited_title":"Zhai, H.-Y","cited_arxiv_id":null,"evidence_quote":"Introduces the hybridized Kibble-Zurek scaling across an overlapping critical region, which the present paper adapts to static overlap."},{"cited_title":"von Gehlen, Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field, J","cited_arxiv_id":null,"evidence_quote":"Provides the conformal-analysis scaling functions for the Ising quantum chain in an imaginary field used to write the quantum scaling forms."},{"cited_title":"Uzelac, R","cited_arxiv_id":null,"evidence_quote":"Gives the real-space renormalization treatment of the transverse-field Ising chain in a complex field that locates finite-size YLES critical points."},{"cited_title":"Sachdev, Quantum Phase Transitions(Cambridge Uni- versity Press, 2011)","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum-critical finite-temperature framework that justifies treating temperature as a relevant scaling variable in the overlap."}],"review_version":1}