{"id":"a05bc833-b4c0-4b8b-b06d-7efa640b090b","arxiv_id":"2412.19664","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Directed, non-reciprocal couplings in the q-state Potts model are claimed to leave equilibrium critical exponents unchanged, while selfish non-equilibrium dynamics yield varying exponents yet a super-universal Binder scaling function.","lead":"This paper studies a version of the Potts model in which interactions between neighboring spins are directed, so one spin can influence another more strongly than the reverse. It claims the model's equilibrium critical exponents are unchanged, while non-equilibrium selfish dynamics produce continuously varying exponents but a universal scaling curve.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact equilibrium mapping is unproven: isospectrality of non-commuting bond matrices cannot determine the 2D partition function, and the required simultaneous gauge transformation is never supplied.","rationale":"The central claim of the paper is the exact equivalence of the non-reciprocal equilibrium Potts model to the reciprocal Potts model. The only analytical support offered is the isospectrality statement after Eq. (6). The reader correctly identifies that isospectrality of non-commuting matrices is insufficient for a 2D partition-function identity. My independent check of the bond-matrix structure confirms the concern is real, not a matter of taste or consensus: for q=3, the local matrix Mx is a diagonal-plus-rank-one matrix whose diagonal varies with the spin label when K≠J, so the claimed degenerate spectrum in Eq. (7) cannot hold for generic K,J. This makes the analytical derivation of the critical line Kc = -J + ln(1+√q) unsupported. The numerical equilibrium measurements in Fig. 2 might coincidentally match Potts exponents, but they cannot rescue the derivation. The non-equilibrium claims are empirical and potentially interesting, but they do not compensate for the broken central analytical claim. I therefore do not change the reader's REJECT verdict; the paper would need a correct proof of the mapping, or a corrected statement limiting it to cases where a common gauge exists, and error-controlled numerics before it could be accepted.","tokens_in":10958,"tokens_out":18811,"duration_ms":467975,"concrete_test":"Exactly contract Eq. (5) on a small periodic lattice (e.g., 3x3 or 4x4) for q=3 with K=0.2, J=0.3 and compare the result with the ordinary q=3 Potts partition function (8) at ε=K+J; a mismatch refutes the mapping. As a complementary direct check, compute the eigenvalues of the q=3 bond matrix Mx and compare them with Eq. (7).","verdict_should_be":"UNCHANGED","load_bearing_attack":"After Eq. (6), the paper states that Mx and My 'do not commute, but they are isospectral' and concludes that the partition function (5) 'can be exactly mapped' to the reciprocal Potts partition function (8) with ε=K+J. This does not follow. The partition function is a product of horizontal and vertical bond matrices sharing spin indices at every site; it is not determined by the individual spectra of Mx and My. Equality with (8) requires a gauge transformation that simultaneously brings Mx and My to the same matrix M, or otherwise leaves the tensor network invariant, and no such transformation is exhibited. This gap is load-bearing because Eq. (10) and the claimed Zq critical exponents depend on this mapping. Moreover, for q=3 the isospectrality itself is suspect: writing Mx_{s,t}=exp[β(a_s+b_t)(2δ_{s,t}-1)] with a_s=J^0_s and b_t=J^2_t gives Mx = diag(2sinh(β(a_s+b_s))) + u v^T; unless a_s+b_s is independent of s, a rank-one perturbation of a non-constant diagonal matrix cannot produce the (q-1)-fold degenerate eigenvalue 2sinh(β(K+J)) claimed in Eq. (7).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a square-lattice q-state Potts model with directed, spin-dependent nearest-neighbor couplings. Under equilibrium (Metropolis) dynamics it claims an exact mapping of the partition function to the conventional Potts model with coupling epsilon = K + J, from which it concludes that the Zq critical exponents and the critical line Kc = -J + ln(1 + sqrt(q)) are unchanged. Under a non-equilibrium 'selfish' single-spin dynamics, Monte Carlo simulations are reported: q = 2 remains Ising-like, while q = 3 and q = 4 show continuously varying critical exponents; nevertheless, the Binder cumulant as a function of xi2/xi0 collapses onto the equilibrium Potts super-universal curve. The exact mapping is the only analytical derivation in the paper; the non-equilibrium claims rest on finite-size scaling of systems up to L = 64.","tokens_in":11252,"tokens_out":32102,"duration_ms":277714,"significance":"Should the equilibrium mapping be valid, the paper would establish a surprising exact equivalence between a non-reciprocal model and an equilibrium reciprocal model, and the non-equilibrium data would extend superuniversality to non-reciprocal discrete-symmetry systems. The numerical effort is substantial: 10^7 samples, scaling-collapse figures in the Supplemental Material, and direct comparisons with Baxter's exact exponents and with simulated equilibrium reference curves. These strengths do not compensate for the fact that the central analytical inference is not a valid derivation, and at least one reported set of exponents is not straightforwardly reproduced by the accompanying FSS figure.","major_comments":[{"comment":"The central claim that Eq. (5) can be exactly mapped to Eq. (8) with epsilon = K + J does not follow from the preceding isospectrality statement. The partition function in Eq. (5) is a two-dimensional tensor network in which Mx and My act on shared spin indices at every site; its value is not determined by the eigenvalues of the individual local matrices. Equality with Eq. (8) would require a simultaneous gauge transformation of the horizontal and vertical local weights that leaves the tensor contraction invariant, or a direct transfer-matrix argument, and none is supplied. This gap is load-bearing because Eq. (10) and the analytical statement that the equilibrium exponents are those of Eq. (9) both depend on this mapping. The problem is not merely formal: for q = 4 some off-diagonal bond weights in Eq. (6) are exp(-2 beta K) or exp(-2 beta J), not exp(-beta(K+J)), so the mapping would have to be nonlocal and cannot be concluded from local spectral equivalence.","section":"The model, after Eq. (6)"},{"comment":"The numerical evidence for continuously varying exponents is not self-consistent as presented. For q = 4, J = 0.5, Table I lists beta = 0.069, gamma = 1.40, beta/nu = 0.090, Kc = 0.606, while Supplemental Fig. 7 reports the collapse parameters 1/nu = 1.3, beta/nu = 0.1, gamma/nu = 1.8, which imply beta approximately 0.077 and gamma approximately 1.38. The discrepancy may be a rounding artifact, but the two sources should be reconciled explicitly, and the table should report the underlying 1/nu and gamma/nu values together with the derived beta and gamma, so that the q = 4 continuously varying exponents can be checked.","section":"Table I and Supplemental Material"},{"comment":"The continuously varying exponents for q = 3 and q = 4 are extracted from data collapse for L = 16, 24, 32, 48, 64 and are reported without statistical errors or a collapse-quality measure. The claimed variation in Table I is modest (for example, q = 4 has beta between 0.069 and 0.083 and beta/nu between 0.090 and 0.125), and the fitting procedure varies several exponents simultaneously. Without error estimates or a systematic assessment of the collapse, the continuous-variation claim is not quantitatively established.","section":"Table I and finite-size scaling section"}],"minor_comments":[{"comment":"The index convention in Eq. (2) is ambiguous for q < 4: the spin values s = 1, ..., q and the direction labels k = 0, ..., 3 are not related by an explicitly stated modulo convention. The final clarification paragraph mentions alternative assignments but does not remove the ambiguity; please define the delta functions modulo q (or modulo 4) before using Eq. (2) for q = 2 and q = 3.","section":"Eq. (2) and Fig. 1"},{"comment":"The Hamiltonian displayed above Eq. (8) sums over k = 0, ..., 3, which double-counts every bond on the square lattice, whereas the partition-function product in Eq. (8) contains one factor per horizontal and one factor per vertical bond. The Hamiltonian should be restricted to k = 0, 1 if it is to match Eq. (8) and the quoted Baxter critical coupling.","section":"Eq. (8) and the displayed Hamiltonian"},{"comment":"The definition of xi0 as 'its maximum value' of xi2 for finite L should be made precise, in particular whether xi0 is a function of L and K and how it is obtained from the data. Without this, the ratio xi2/xi0 is not a fully defined scaling variable.","section":"Fig. 3 and super-universal curve"},{"comment":"The reference to 'Supplemental Materaial' [46] should be completed, and the main text should refer to specific Supplemental figures when reporting the FSS results. The insets of Fig. 3 are too small to read the symbols and the critical-line comparison; larger panels or a separate table of Kc values would help.","section":"General presentation"}],"recommendation":"reject","confidential_remarks":"The main issue is the invalid inference from local isospectrality to equality of the two-dimensional partition function. If the authors can supply a correct simultaneous transformation or an alternative proof of the partition-function identity, the work could be reconsidered; otherwise the central equilibrium claim is unsupported. I would additionally ask an editor to have the claimed mapping checked by an expert in exactly solved lattice models before any resubmission, and to verify the consistency of Table I with the Supplemental FSS figures."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The interesting part of this paper is the non-equilibrium Potts data: q=3 and q=4 selfish dynamics give continuously varying critical exponents along the critical line, yet the Binder cumulant plotted against xi2/xi0 collapses onto the equilibrium Potts curve. That is a real, checkable observation, and if it holds up it extends the authors' earlier super-universality framework to a new setting. The q=2 result (exponents stay Ising) is consistent with prior non-reciprocal Ising papers and is not the novelty.\n\nThe load-bearing analytical claim is the exact mapping of the equilibrium non-reciprocal model to the ordinary Potts model at epsilon=K+J. As written, the proof is a non sequitur: isospectrality of Mx and My individually does not determine the 2D partition function, because the horizontal and vertical bond matrices enter the tensor network without commuting. The paper states they do not commute and then jumps to the mapping. That is a real gap.\n\nHowever, the specific objection raised in the stress-test about the q=3 eigenvalues does not hold up. With the paper's own assignment a_s=J^0_s, b_t=J^2_t, one gets a_s+b_s=K+J for all s, so Mx is a rank-one perturbation of a constant diagonal and the claimed eigenvalues in Eq. (7) are correct. The same holds for My. More importantly, the mapping itself is probably true, but for a reason the paper does not give: both bond matrices are diagonal-gauge equivalent to the standard Potts matrix M, and the gauge factors cancel site-by-site on the square lattice, leaving exactly the reciprocal Potts partition function. That is a short proof, but it is not in the paper. Without it, Eq. (10) and the claimed exponents have no derivation.\n\nThe numerics for selfish dynamics also need more care than they got. Table I lists exponents without error bars, and the claimed continuous variation for q=3 and q=4 is based on five points each, with the ratio beta/nu varying over a narrow range (roughly 0.12 to 0.17). That could easily be finite-size drift. The data-collapse plots in the Supplemental Material are consistent with the claimed exponents, but they are not independent evidence without an error analysis.\n\nSo the paper deserves a serious referee, but not in its current form. A revision should supply the gauge-transformation proof for the equilibrium mapping and redo the non-equilibrium exponent estimates with error bars and more system sizes. The core idea is good, and the equilibrium conclusion is likely correct. I would not cite it until the proof is stated properly.","headline":"The equilibrium mapping is likely correct but unproven as written; the non-equilibrium numerics are intriguing but too lightly error-controlled to carry the title's claim.","tokens_in":11751,"tokens_out":14894,"would_cite":false,"duration_ms":109985,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B20","82B27","82C20","82C27"],"pacs":["05.50.+q","05.70.Ln"],"model":"deepseek-v4-flash","headline":"Non-reciprocal nearest-neighbor couplings leave the equilibrium q-state Potts model in its usual universality class, and non-equilibrium 'selfish' dynamics still share the equilibrium scaling curve.","keywords":["non-reciprocal interactions","q-state Potts model","universality class","superuniversality","selfish dynamics","isospectral transfer matrices","Binder cumulant","non-equilibrium phase transitions"],"falsifier":"Exactly diagonalize the full transfer operator of a finite-width periodic strip built from the non-commuting matrices $M_x$ and $M_y$, and compare its leading correlation-length gaps with the ordinary Potts strip at $\\epsilon=K+J$; unequal gaps would falsify the exact-mapping claim. On the non-equilibrium side, a sharp check is to measure $U_4$ versus $\\xi_2/\\xi_0$ for a case not covered by the mapping, such as $q=5$ or a different spin-assignment convention, and look for a split from the equilibrium Potts curve.","tokens_in":10751,"feed_emoji":"🧲","tokens_out":9959,"duration_ms":87368,"temperature":0.7,"pith_summary":"The paper asks whether directed, non-reciprocal spin couplings—where spin $i$ exerts a different influence on spin $j$ than $j$ exerts on $i$—change the critical behaviour of the q-state Potts model on a square lattice. For equilibrium (detailed-balance) dynamics the answer it argues for is no: the full partition function is mapped exactly onto that of an ordinary reciprocal Potts model with a single effective coupling $\\epsilon = K+J$, so the critical line is $K_c = -J + \\ln(1+\\sqrt{q})$ and the exponents are the standard $Z_q$ ones. For non-equilibrium 'selfish' dynamics, $q=2$ remains in the Ising universality class, while $q=3$ and $q=4$ show critical exponents that vary continuously along the critical line. Even so, the Binder cumulant $U_4$ plotted against the second-moment correlation-length ratio $\\xi_2/\\xi_0$ collapses onto the equilibrium $q=3,4$ Potts curve. The authors take this as evidence that non-reciprocity does not alter the universality class of discrete-symmetry-breaking transitions in two dimensions, and that a superuniversality class joins the equilibrium and non-equilibrium behaviour.","feed_headline":"Non-reciprocal couplings leave Potts critical class intact","feed_subtitle":"Even selfish dynamics that shift q=3,4 exponents keep the equilibrium Binder-cumulant scaling curve.","key_machinery":"The carrying object is the pair of non-commuting local bond matrices $M_x$ and $M_y$ with entries $\\langle s|M_{x,y}|\\tilde s\\rangle = e^{\\beta J^{x,y}_{s,\\tilde s}(2\\delta_{s,\\tilde s}-1)}$, which replace the single reciprocal Potts bond matrix. The argument's engine is isospectrality: $M_x$ and $M_y$ share a common eigenvalue spectrum with the ordinary Potts matrix $M$ at coupling $\\epsilon = K+J$, and the paper uses that equality of spectra to identify the full partition functions. For the non-equilibrium part, the key mechanism is the selfish update rate $r=\\min(1,e^{-\\Delta E_i})$, which uses only the flipping spin's own energy change and drives the system to a non-equilibrium steady state. The superuniversal claim is carried by the Binder cumulant $U_4$ as a function of $\\xi_2/\\xi_0$ (second-moment correlation length divided by its maximum value), which is found to collapse for all $K,J,L$ onto the equilibrium curve.","core_discovery":"The central claim is that the non-reciprocal q-state Potts model retains the universality class of the ordinary reciprocal model whenever the dynamics follows detailed balance. The mechanism is spectral: the two directed bond matrices $M_x$ and $M_y$, whose entries carry the orientation-dependent couplings $K$ and $J$, have the same eigenvalue list as the single reciprocal Potts bond matrix with coupling $\\epsilon = K+J$—namely $\\lambda_1=\\cdots=\\lambda_{q-1}=2\\sinh(\\beta(K+J))$ and $\\lambda_q=q\\cosh(\\beta(K+J))-(q-2)\\sinh(\\beta(K+J))$. The paper concludes that the two-dimensional partition functions coincide exactly, yielding the critical line $K_c = -J + \\ln(1+\\sqrt{q})$ with the exact $Z_q$ exponents of the standard Potts solution. Under the non-equilibrium 'selfish' dynamics, where a spin updates using only its own local energy change, the $q=2$ model still shows Ising exponents; for $q=3$ and $q=4$ the exponents vary continuously along the critical line, yet the super-universal scaling function $U_4$ versus $\\xi_2/\\xi_0$ matches the equilibrium Potts models. The paper states these results as showing that non-reciprocal Potts models belong to the superuniversality class of their equilibrium counterparts.","pith_inferences":["A natural extension the paper does not pursue: test the same two-step logic on clock ($Z_n$) or Ashkin–Teller-type models, predicting that non-reciprocal couplings shift the critical manifold but leave each universality class's scaling functions intact.","The isospectrality step could be probed directly on a periodic strip: if the transfer-matrix spectrum of the non-commuting $M_x M_y$ product differs from the reciprocal strip at $\\epsilon=K+J$, the exact-mapping conclusion would need revision even though the Monte Carlo data may still hold.","Since the superuniversal curve appears for $q=3,4$, a sharp test is whether $q=5$ (where equilibrium Potts is first-order) or a different spin-assignment convention for $q=2$ also produces the same $U_4(\\xi_2/\\xi_0)$; the paper claims robustness for assignments only for equilibrium isospectrality.","If confirmed, the results suggest that non-reciprocal interactions in discrete-symmetry systems may be filtered out of static critical exponents entirely, leaving a single effective reciprocal coupling; this would guide coarse-grained descriptions of active matter where directed interactions are common."],"forward_implications":["The critical line for equilibrium dynamics is exactly $K_c = -J + \\ln(1+\\sqrt{q})$, so any point on that line, not just the reciprocal $J=K$ case, shows the same order-disorder transition.","Along that line the exponents $\\beta,\\gamma,\\nu,\\beta/\\nu$ take the exact $Z_q$ values from the standard Potts solution, verified numerically for $q=2,3,4$.","Under selfish non-equilibrium dynamics, the $q=2$ model keeps Ising exponents $\\beta=1/8$ and $\\gamma=7/4$, so non-reciprocity alone does not push the Ising class out of equilibrium.","For $q=3$ and $q=4$ with selfish dynamics, critical exponents vary continuously with position on the critical line, but the combination $U_4(\\xi_2/\\xi_0)$ is invariant along the line and matches the equilibrium Potts curve.","Because $(J,K)$ and $(K,J)$ give the same critical line, the phase diagram is symmetric under exchange of the two non-reciprocal couplings."],"supporting_citations":[{"why":"Baxter's exact solution of the reciprocal Potts model supplies the critical coupling, exponents, and universality classes that the mapping targets and that the numerics are compared against.","marker":"[44]"},{"why":"Introduces the selfish non-equilibrium dynamics, the update rule used to generate the non-equilibrium steady states in the second half of the paper.","marker":"[47]"},{"why":"Stanley's finite-size scaling formulation is the method used to extract $\\nu,\\beta,\\gamma$ from the Monte Carlo data in the main text and Supplemental Material.","marker":"[45]"},{"why":"Establishes the hidden-superuniversality idea that a Binder-cumulant scaling function $U_4(\\xi_2/\\xi_0)$ stays invariant even when critical exponents vary, the template for the non-equilibrium $q=3,4$ result.","marker":"[41]"},{"why":"Delfino and Tartaglia's superuniversality analysis of the disordered Potts model is the background concept the paper invokes for classifying the non-reciprocal model in a superuniversality class.","marker":"[39]"},{"why":"Provides the vision-cone Ising realisation used to argue that alternative spin assignments preserve the equilibrium isospectrality for $q=2$ and $q=3$.","marker":"[48]"}],"fun_headline_variants":["Potts universality survives non-reciprocal spin coupling","Selfish Potts dynamics keep superuniversal Binder curve","Non-reciprocal Potts retains superuniversality","Non-reciprocal interactions don't break Potts universality","Binder curve stays identical for non-reciprocal Potts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equilibrium result rests on the assumption that knowing the eigenvalues of the two non-commuting bond matrices $M_x$, $M_y$ is enough to conclude the full two-dimensional partition function equals the ordinary Potts one; the paper states this spectral-to-partition-function step rather than deriving it for a finite lattice.","fun_headline_variants_meta":{"raw":{"variants":["Potts universality survives non-reciprocal spin coupling","Selfish Potts dynamics keep superuniversal Binder curve","Non-reciprocal Potts retains superuniversality","Non-reciprocal interactions don't break Potts universality","Binder curve stays identical for non-reciprocal Potts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00076,"raw_usage":{"total_tokens":3421,"prompt_tokens":1036,"completion_tokens":2385,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":2302}},"tokens_in":652,"tokens_out":2385,"duration_ms":18876,"temperature":1.0,"reasoning_tokens":2302,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:01:02.355073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exactly diagonalize the full transfer operator of a finite-width periodic strip built from the non-commuting matrices $M_x$ and $M_y$, and compare its leading correlation-length gaps with the ordinary Potts strip at $\\epsilon=K+J$; unequal gaps would falsify the exact-mapping claim. On the non-equilibrium side, a sharp check is to measure $U_4$ versus $\\xi_2/\\xi_0$ for a case not covered by the mapping, such as $q=5$ or a different spin-assignment convention, and look for a split from the equilibrium Potts curve.","supporting_citations":[{"cited_title":"Geometric percolation of spins and spin-dipoles in Ashkin-Teller model","cited_arxiv_id":"2411.11644","evidence_quote":"Baxter's exact solution of the reciprocal Potts model supplies the critical coupling, exponents, and universality classes that the mapping targets and that the numerics are compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the selfish non-equilibrium dynamics, the update rule used to generate the non-equilibrium steady states in the second half of the paper."},{"cited_title":"Bonati, A","cited_arxiv_id":null,"evidence_quote":"Establishes the hidden-superuniversality idea that a Binder-cumulant scaling function $U_4(\\xi_2/\\xi_0)$ stays invariant even when critical exponents vary, the template for the non-equilibrium $q=3,4$ result."}],"review_version":1}