{"id":"e6c02e7a-f529-4cdd-abd4-684b11cae6cd","arxiv_id":"2412.19297","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Long-range order in the vision-cone XY model comes from the cone-lattice coupling, not from non-reciprocity, and a symmetric variant shows an order-by-disorder transition.","lead":"This paper uses computer simulations to study spins on a grid where each spin only interacts with neighbors inside a cone-shaped field of view. It shows that this setup can create a long-range ordered state even when interactions are symmetric, so non-reciprocity is not required, and it finds a case where raising temperature creates order.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The θ=180° LRO classification for NRXY/ARXY is not yet settled: the same data show XY-like magnetization scaling, so ξ/L scaling alone is an insufficient discriminator.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: the θ=180° LRO phase is inferred from the finite-size behavior of ξ_L^(2)/L for L up to 90, while the magnetization follows the QLRO scaling of Eq. (14). The stress-test analysis confirms that this is the most delicate step in the paper's central argument, because the two new reciprocal models are claimed to show LRO precisely where the EUR disappears and the symmetry argument is only heuristic. A Binder-cumulant study would settle the LRO-versus-QLRO distinction without relying on correlation-length scaling alone. In addition, the paper's own defect-energy calculation (Eq. 27) suggests a possible internal inconsistency: the same 'strongly gapless' excitations that are proposed to entropically select the lattice directions have intensive energy and extensive positional entropy, which may destabilize the ordered state at any finite temperature. These concerns are addressable numerically and analytically, so the appropriate verdict is CONDITIONAL rather than ACCEPT or REJECT: the central claims are plausible and the methodology is sound, but the θ=180° and related order-by-disorder identifications require stronger evidence than currently presented.","tokens_in":33289,"tokens_out":12489,"duration_ms":125420,"concrete_test":"For ARXY θ=180 (and NRXY θ=180), compute the Binder cumulant U4 = ⟨m⁴⟩/⟨m²⟩² as a function of L at fixed low temperatures (e.g., T=0.12 and T=0.2) for L=30,...,240, and plot versus 1/L. LRO gives U4→1 as 1/L→0; QLRO gives U4→2 (the O(2) value). Simultaneously plot ⟨|m|⟩ versus L on a log-log scale and fit to both Eq. (14) and constant + c/L; if ⟨|m|⟩ continues to decay as L^{-T/(8πJ)} and U4 does not approach 1, the θ=180 phase is QLRO and the central reciprocal-LRO claim at this angle fails. An analytic check of Eq. (27) for a single line defect, including positional entropy log L and the amplitude integration, would also determine whether the proposed entropic selection mechanism is stable against defect proliferation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that reciprocal vision-cone models exhibit long-range order rests on the identification of the θ=180° phase of ARXY (and NRXY) as LRO in Sec. V D. At this angle the EUR vanishes, and the low-temperature magnetization follows the XY-model finite-size law m ∼ N^{-T/(8πJ)} (Eq. 14), which is the signature of a QLRO phase with continuous O(2)-like fluctuations. In a genuinely Z4-ordered phase, ⟨|m|⟩ must extrapolate to a nonzero constant as N→∞; Eq. (14) instead predicts vanishing magnetization in the thermodynamic limit. The paper's LRO conclusion is based on ξ_L^(2)/L curves that cross at a single temperature and then grow as L below it (Figs. 4(e), 9(b), A1(b)). This is a valid LRO criterion only if the asymptotic regime is reached, but with L≤90 the possibility remains that ξ/L is in a preasymptotic transient of a weakly anisotropic QLRO system, while m has already entered XY-like scaling. Moreover, the heuristic argument in Sec. V D introduces a self-consistency tension: the 'strongly gapless' line-defect excitations around the 90°-type alignment have intensive energy δE_AR ≈ α²/4 (Eq. 27) and O(L) positional entropy, so their free energy F ≈ α²/4 − T log L becomes negative for sufficiently large L at any T>0. If these defects proliferate, they would destroy the very LRO they are invoked to select. The paper does not address this; the Introduction itself notes that bona-fide long-range order in similar models is still debated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies three variants of the two-dimensional XY model with vision-cone interactions on a square lattice: the non-reciprocal NRXY model, and two new reciprocal models, ARXY and SRXY. Using Monte Carlo simulations and finite-size scaling of the second-moment correlation length, the authors map out temperature–cone-angle phase diagrams. Their central claim is that non-reciprocity is not essential for long-range order: both NRXY and ARXY show three LRO 'lobes' for θ>90°, and even at θ=180°, where the energetically unfavorable range (EUR) vanishes, the models are claimed to have LRO rather than QLRO. For SRXY, the phase diagram shows LRO for 90°<θ≲180° and QLRO for θ>180°, with a reentrant QLRO-to-LRO transition for θ≳180° that the authors interpret as an order-by-disorder phenomenon. Symmetry arguments based on the effective reduction of O(2) to Z4 are used to rationalize these findings.","tokens_in":33631,"tokens_out":5762,"duration_ms":60869,"significance":"If the claims are correct, the paper makes a conceptually important point: a reciprocal, equilibrium lattice model with vision-cone couplings can reproduce the LRO phase previously associated with non-reciprocal flocking-like dynamics, and the mechanism is the lattice-induced reduction of O(2) to Z4 rather than the non-equilibrium drive. The identification of an order-by-disorder transition in a classical 2D XY-type model is also a striking result. The paper is careful in several respects: the numerical analysis for representative angles uses multiple observables and is benchmarked against 4-state clock, 6-state clock, and XY models; the protocol dependence of the NRXY steady state is explicitly studied in Appendix C; and the phase classification for θ=100°, 280°, 300°, and 190° is supported by finite-size scaling of ξ_L^(2)/L. The main weaknesses are evidentiary: the two headline claims (LRO at θ=180° and the SRXY order-by-disorder transition) rest on finite-size signatures that are not fully separated from the alternative interpretations of QLRO or of a weakly broken Z4 symmetry, and the heuristic defect argument in Sec. V D has an entropy tension that is not resolved.","major_comments":[{"comment":"The classification of the θ=180° phase of NRXY and ARXY as LRO rests entirely on the scaling of ξ_L^(2)/L for L≤90, but the magnetization data in Fig. 4(d) are explicitly consistent with the XY finite-size scaling m∼N^(−T/(8πJ)) of Eq. (14). This is exactly the signature one would expect for a QLRO phase, so the ξ/L crossing does not by itself exclude a preasymptotic transient of a weakly anisotropic QLRO system. Please add a direct thermodynamic discriminator, such as a finite-size extrapolation of ⟨|m|⟩ at fixed low T (comparing Eq. (14) with a nonzero thermodynamic-limit value), a Binder cumulant of m, or the spin-wave stiffness, and report the largest system sizes used for that analysis. This is load-bearing because the abstract and conclusions assert LRO at θ=180° for both NRXY and ARXY.","section":"V D and Figs. 4(e), 9(b), A1(b)"},{"comment":"The line-defect argument used to select the 90°-type alignment has an internal consistency problem. Equation (27) states that around the preferred alignment a single row defect costs only δE_AR≈α²/4, i.e., an intensive energy. If one such defect can be placed in each row independently, the configurational entropy is O(L log L) while the total energy is only O(L); the resulting free energy per row, F≈α²/4−T log L, becomes negative for sufficiently large L at any T>0, exactly as in the domain-wall estimate used in Appendix E for θ=60°. If these defects proliferate, they would destroy the very LRO they are invoked to select. The paper should either quantify and control this entropy, or show that the defects cannot be treated as independent (for example, because of the bond-activation compensation under periodic boundary conditions), or provide numerical evidence that their density does not grow with L. As written, the heuristic does not yet establish the stability of the θ=180° LRO.","section":"V D, Eq. (27)"},{"comment":"The claimed QLRO-to-LRO order-by-disorder transition for SRXY at θ=190° is based on system sizes up to L=48. The low-temperature QLRO identification relies on the collapse of ξ_L^(2)/L curves, while the intermediate-temperature LRO identification relies on their separation; a weakly anisotropic Z4-symmetric system with a small but nonzero lattice-pinning term would produce the same sequence of collapse followed by separation on these sizes. Please provide a finite-size analysis of the reentrant LRO window, for example by estimating the QLRO-LRO boundary as a function of L and showing that it extrapolates to a nonzero interval in the thermodynamic limit, and check explicitly whether the low-temperature collapse region shrinks as L increases. This is needed to support the order-by-disorder claim, which is one of the paper's two central novel results.","section":"VI C and Figs. 10(b), 11(f)–(j)"}],"minor_comments":[{"comment":"The text refers to 'Fig. 8(c)', but Fig. 8 has only panels (a) and (b); the intended reference is probably Fig. 8(b).","section":"VI B"},{"comment":"The f_EUR estimate assumes that most spins remain roughly aligned and uses the zero-temperature rotation-energy profile V(ϑ) as an effective single-particle potential. This is a reasonable heuristic, but the assumptions should be stated explicitly as an approximation, and the formula should be labeled as a qualitative estimate rather than a quantitative derivation; it is not used in the phase classification.","section":"V C, Eq. (25)"},{"comment":"The solid curve showing Eq. (14) is stated to be for L=48, but the panel contains data for several L; please specify the value of L for the theoretical curve and consider showing the expected scaling for each L to make the comparison transparent.","section":"IV, Fig. 4(d)"},{"comment":"The global phase-diagram heatmaps are computed at L=100, and the LRO/QLRO labels in regions not covered by the finite-size scaling analysis are inferred from m and χm alone. A sentence stating which phase boundaries are confirmed by the ξ_L^(2)/L scaling and which are only indicative from single-size heatmaps would help the reader calibrate the reliability of the diagrams.","section":"III B and Figs. 4, 5, 10"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and tackles a timely question, and the construction of the two reciprocal models is a useful contribution even independently of the specific phase classifications. My main concern is that the two headline claims—LRO at θ=180° for NRXY/ARXY and the QLRO-to-LRO order-by-disorder transition for SRXY—rest on finite-size signatures that are compatible with the alternative QLRO or weakly broken Z4 interpretations. I would be willing to accept after the authors provide a more direct thermodynamic discriminator for the θ=180° LRO and address the entropy argument in Sec. V D, either by a quantitative calculation or by an explicit numerical check of defect proliferation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the main claim holds up. The paper convincingly shows that long-range order in the vision-cone XY model does not require non-reciprocity, and the two reciprocal variants—ARXY and SRXY—are genuinely new constructions, not just minor tweaks of Ref. [5]. The order-by-disorder transition in the classical equilibrium SRXY model is the most interesting finding here, and the qualitative phase diagrams are built from careful finite-size scaling of ξ^(2)_L/L with multiple observables and honest comparison against XY and clock-model benchmarks. That is real work, properly executed.\n\nThe symmetry/EUR framework is clearly labeled heuristic and does useful organizational work, even if it doesn't rise to a proof. The appendix on update-protocol dependence and the EPR analysis is a good check on the NRXY side; the citation pattern is fine, with Ref. [5] being the direct progenitor and the flocking debate cited fairly.\n\nThe soft spots are real but addressable. The θ=180° LRO classification is the delicate one. The magnetization follows the pure-XY finite-size law of Eq. (14), so the LRO call rests almost entirely on ξ^(2)_L/L scaling as L below the crossing. That is a standard LRO signature, and the appendix shows the scaling for a few temperatures, but with L≤90 a weakly anisotropic QLRO transient remains a live alternative. Binder cumulants or larger systems would settle it. The stress-test also flags a genuine tension in the heuristic: the line-defect energy in Eq. (27) is intensive, and with O(L) positional entropy those defects would have negative free energy at any T>0 for large L. The paper doesn't address this. That doesn't kill the numerical claim, but it means the stated explanation is shakier than the data.\n\nMinor: the phase-diagram heatmaps come from L=100 without error bars. Fine for orientation, not ideal for the delicate θ≈180° and θ≈190° claims. No code or data is provided, which makes rechecking those claims harder than it should be.\n\nWho is this for? People working on flocking models, non-reciprocal lattice systems, and order-by-disorder. It deserves a serious referee. I would send it to review with a request to strengthen the θ=180° classification and to address the defect-entropy point in the heuristic.","headline":"Solid, genuinely new MC study showing non-reciprocity isn't what drives vision-cone XY order; the θ=180° LRO call is real but needs a stronger finite-size knife.","tokens_in":34166,"tokens_out":4135,"would_cite":true,"duration_ms":43131,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B20","82B27","82B80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that long-range order in the two-dimensional XY model with vision-cone interactions arises from the lattice geometry reducing the O(2) symmetry to Z4, not from non-reciprocity, and that one reciprocal variant exhibits an…","keywords":["XY model","vision cone","non-reciprocal interactions","long-range order","Z4 symmetry","order by disorder","Monte Carlo simulation","square lattice"],"falsifier":"For the $\\theta=180^\\circ$ NRXY and ARXY models, simulate $\\xi^{(2)}_L/L$ up to $L\\geq256$ at fixed low temperatures: the claimed LRO predicts that $\\xi^{(2)}_L/L$ keeps growing as $L$, while a quasi-long-range phase predicts that it saturates at a constant independent of $L$. For the SRXY model at $\\theta=190^\\circ$, the order-by-disorder claim predicts a window of temperatures in which $\\xi^{(2)}_L/L$ separates with system size (LRO) between a low-temperature collapsing branch (QLRO) and the high-temperature transition to disorder; if the curves collapse at every temperature below the disorder transition, the claim fails.","tokens_in":33038,"feed_emoji":"🧲","tokens_out":7961,"duration_ms":75112,"temperature":0.7,"pith_summary":"This paper asks what actually produces long-range order (LRO) in the two-dimensional XY model when each spin interacts only with neighbours inside a 'vision cone' of angular width $\\theta$. It studies three versions of the model: the original non-reciprocal one and two newly designed reciprocal, equilibrium versions, the ARXY and SRXY models. Monte Carlo simulations and symmetry arguments lead the authors to conclude that all three can order at low temperature, and that the ordering mechanism is not the non-reciprocal drive but the coupling between spin orientation and square-lattice bond structure, which reduces the continuous $O(2)$ symmetry to the discrete $\\mathbb{Z}_4$ symmetry of the lattice. A separate result is that in the symmetric reciprocal model with $\\theta$ slightly above $180^\\circ$, raising the temperature carries the system from quasi-long-range order into true long-range order, a classical order-by-disorder transition. A sympathetic reader should care because the result separates geometry from active driving as the source of order and shows equilibrium models with configuration-dependent couplings can evade the usual two-dimensional no-order expectation.","feed_headline":"Vision-cone XY spins order without non-reciprocity","feed_subtitle":"Two equilibrium versions of the model reach long-range order because the lattice breaks O(2) down to Z4.","key_machinery":"The central object is the energetically unfavorable range (EUR): the angular window in which a spin loses one of its $n$ visible nearest neighbours ($n\\rightarrow n-1$), whose extension and centering change discontinuously at multiples of $90^\\circ$ and set the preferred collective alignment directions on the square lattice. The argument is carried by a symmetry-probing rotation $r_\\vartheta$ of an entire configuration: for a perfectly aligned state the energy change per spin is a $90^\\circ$-periodic square wave, which is the concrete mechanism by which the cone reduces $O(2)$ to $\\mathbb{Z}_4$. A second mechanism, the redundant bond of the SRXY model (a bond activated once counts once even if both spins see each other), explains why $\\theta>180^\\circ$ restores quasi-long-range order, and the temperature-dependent emergence of non-redundant bonds under rotation explains the order-by-disorder transition.","core_discovery":"On its own terms, the paper's central claim is that the vision cone alone, through the coupling it creates between spin directions and lattice bond directions, is sufficient to produce long-range order in equilibrium, reciprocal versions of the XY model. Concretely, the energetically unfavorable range (EUR), the set of spin orientations in which a spin loses one visible neighbour, makes the energy landscape of a globally rotated configuration periodic with the lattice's $90^\\circ$ periodicity, so the effective internal symmetry is $\\mathbb{Z}_4$ instead of $O(2)$, and the discrete symmetry allows LRO despite Mermin-Wagner. For $\\theta=180^\\circ$, where the EUR vanishes, the paper argues that alignment along lattice directions is still entropically selected because such configurations possess a spectrum with more strongly gapless excitations. For the SRXY model with $\\theta>180^\\circ$, redundant bonds restore an effectively $O(2)$-symmetric quasi-long-range ordered phase, but for $\\theta\\gtrsim190^\\circ$ thermal fluctuations generate a dynamical EUR that selects the four diagonal directions, producing a quasi-long-range-to-long-range transition as temperature rises.","pith_inferences":["If the symmetry-reduction mechanism is generic, the same model on a triangular or honeycomb lattice should show LRO lobes aligned with the $120^\\circ/60^\\circ$ bond directions and an effective $\\mathbb{Z}_3$ or $\\mathbb{Z}_6$ symmetry; this is not simulated in the paper but follows directly from its EUR logic.","The order-by-disorder transition at $\\theta\\gtrsim180^\\circ$ suggests a practical probe: measuring whether the specific-heat peak across the reentrant boundary grows logarithmically with system size (LRO-to-DO signature) or stays size-independent (QLRO-to-DO signature) should locate the transition more sharply than the susceptibility map.","For off-lattice active particles, the paper's equilibrium result raises the possibility that reciprocal vision-cone interactions alone could sustain polar order, separating the alignment mechanism from the non-reciprocal drive; the paper mentions off-lattice extensions as future work without making this claim.","The $\\theta=180^\\circ$ LRO-versus-QLRO distinction is the most delicate numerical point, and a dedicated study of the correlation-length ratio at fixed low temperature up to much larger $L$ would either confirm or overturn the entropic-selection argument."],"forward_implications":["The NRXY and ARXY models have qualitatively the same phase diagram with three LRO lobes opening above $\\theta=90^\\circ$, so the non-equilibrium character of the original model is not what creates the ordered phase.","At $\\theta=180^\\circ$ and $270^\\circ$ the model is long-range ordered even though the EUR has zero width; the XY-like finite-size magnetization scaling must not be read as evidence of quasi-long-range order.","In the SRXY model with $\\theta>180^\\circ$, redundant bonds make the low-temperature phase quasi-long-range ordered with the same correlation-length signature as the standard XY model.","For $\\theta$ slightly above $180^\\circ$ in the SRXY model, increasing temperature drives quasi-long-range order to long-range order and then to disorder, an order-by-disorder phenomenon in a classical equilibrium system.","The symmetry arguments imply that the residual low-temperature symmetry is set by the lattice ($\\mathbb{Z}_4$ on the square lattice), not by the cone alone, so the mechanism is intrinsically lattice-geometric."],"supporting_citations":[{"why":"Introduces the NRXY model and the vision-cone setup, and reports LRO; the paper's reciprocal models are contrasted with it.","marker":"[5]"},{"why":"Mermin-Wagner theorem supplies the obstruction that the claimed Z4 symmetry reduction is designed to bypass.","marker":"[2]"},{"why":"Kosterlitz-Thouless ordering furnishes the benchmark quasi-long-range ordered phase used to classify the models.","marker":"[3]"},{"why":"Provides the critical properties of the two-dimensional XY model used as the reference for QLRO scaling.","marker":"[4]"},{"why":"Supplies the finite-size magnetization scaling used to show that the theta=180 case is not standard QLRO.","marker":"[37]"},{"why":"Supports the same finite-size corrections for magnetic fluctuations in the two-dimensional XY model.","marker":"[38]"},{"why":"Gives the second-moment correlation-length estimator used to distinguish LRO, QLRO, and DO phases.","marker":"[39]"},{"why":"States the xi/L growth laws (similar to L, constant, 1/L) used to classify the phases.","marker":"[40]"}],"fun_headline_variants":["Vision cone gives XY model long-range order via Z4 symmetry","No non-reciprocity: vision cone makes XY spins order at low T","Lattice vision cone breaks O(2) to Z4, enabling XY long-range order","Vision cone alone triggers long-range order in equilibrium XY model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central numerical claim relies on reading thermodynamic order from the finite-size growth of the correlation-length ratio in systems up to $L=90$, where a quasi-long-range phase can mimic part of the same signal at low temperature.","fun_headline_variants_meta":{"raw":{"variants":["Vision cone gives XY model long-range order via Z4 symmetry","No non-reciprocity: vision cone makes XY spins order at low T","Lattice vision cone breaks O(2) to Z4, enabling XY long-range order","Vision cone alone triggers long-range order in equilibrium XY model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00135,"raw_usage":{"total_tokens":5450,"prompt_tokens":879,"completion_tokens":4571,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":4491}},"tokens_in":495,"tokens_out":4571,"duration_ms":32149,"temperature":1.0,"reasoning_tokens":4491,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:43:54.407060+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the $\\theta=180^\\circ$ NRXY and ARXY models, simulate $\\xi^{(2)}_L/L$ up to $L\\geq256$ at fixed low temperatures: the claimed LRO predicts that $\\xi^{(2)}_L/L$ keeps growing as $L$, while a quasi-long-range phase predicts that it saturates at a constant independent of $L$. For the SRXY model at $\\theta=190^\\circ$, the order-by-disorder claim predicts a window of temperatures in which $\\xi^{(2)}_L/L$ separates with system size (LRO) between a low-temperature collapsing branch (QLRO) and the high-temperature transition to disorder; if the curves collapse at every temperature below the disorder transition, the claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the critical properties of the two-dimensional XY model used as the reference for QLRO scaling."},{"cited_title":"Tobochnik and G","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-size magnetization scaling used to show that the theta=180 case is not standard QLRO."},{"cited_title":"Archambault, S","cited_arxiv_id":null,"evidence_quote":"Supports the same finite-size corrections for magnetic fluctuations in the two-dimensional XY model."},{"cited_title":"Cooper, B","cited_arxiv_id":null,"evidence_quote":"Gives the second-moment correlation-length estimator used to distinguish LRO, QLRO, and DO phases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the xi/L growth laws (similar to L, constant, 1/L) used to classify the phases."}],"review_version":1}