{"id":"896610b7-ea81-4f09-94d1-947d1350b8e4","arxiv_id":"2510.23162","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In a measurement-only triangular-lattice toric code, topological-entanglement, 1-form-symmetry string, and logical-loop order parameters transition at different measurement probabilities and with different critical exponents.","lead":"A numerical study of a measurement-only quantum circuit built from the triangular-lattice toric code finds that different topological diagnostics change at different parameter values. The result suggests that entanglement-based and symmetry-breaking-based measures can disagree about where a topological phase transition takes place.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed TEE/1-form-SSB separation may be an FSS artifact: Table 1A has non-physical p_c=1.009 for Z-loop, Fig. 7 variance peak doesn't grow with L, and string vs TEE use different estimators (order parameter vs variance peak).","rationale":"The reader's CONDITIONAL verdict is appropriate and should remain. The paper's central numerical observation is potentially interesting, but the stress-test identifies a more immediate technical problem than footnote [52]: the critical-point comparison mixes estimators (order-parameter mean for strings, variance peak for TEE) and the same FSS pipeline produces at least one unphysical critical point (Z-loop p_c=1.009) and one non-divergent susceptibility peak (X-loop variance). These internal inconsistencies mean the ~0.1 separation in p_g cannot yet be taken as evidence for distinct transitions. The proposed concrete test would settle whether the separation survives a like-with-like comparison. The footnote assumption about the disorder parameter diagnosing 1-form SSB is a genuine interpretive caveat, which the authors explicitly flag; if the FSS artifact is resolved and the separation remains, that caveat becomes the next point to scrutinize. Therefore the verdict stays CONDITIONAL: accept the qualitative phase diagram as tentative, but require the estimator-consistent FSS reanalysis before the non-coincidence claim is accepted.","tokens_in":18407,"tokens_out":23595,"duration_ms":275616,"concrete_test":"Re-analyze the same trajectory data using a single, consistent protocol: (1) compute the variance of χ_X over trajectories (the string susceptibility) and fit it with the same Eq.(5) ansatz and pyfssa settings used for the TEE variance; (2) compute the Binder cumulant of χ_X and locate the crossing point as a second, scale-invariant estimator. If the string susceptibility peak/Binder crossing extrapolate to p_g≈0.846 rather than 0.723, then the claimed ordering is an artifact of using mean-vs-variance estimators. Also refit the Z-loop with p_c constrained to [0,1] and report χ²; an unphysical best fit at 1.009 indicates the unconstrained FSS is overfitting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion that the TEE transition (p_g=0.846/0.876) is distinct from the 1-form-symmetry-breaking transition (p_g≈0.72–0.79) depends on comparing critical points extracted from different finite-size scaling estimators. This comparison is not yet controlled. In Table 1A, the Z-loop FSS returns p_g=1.009±0.017, which lies outside the physical domain p_g∈[0,1] on the line p_x+p_g=1. A fit that locates a transition outside the parameter range is not a valid estimate of a critical point; it is a symptom that the scaling ansatz is being applied to data without a transition. Separately, the text reports that the X-loop variance peaks 'do not develop for increasing system size' (Fig. 7). For a standard continuous transition, the susceptibility peak height grows as L^{γ/ν}; a non-growing peak invalidates the ordinary FSS collapse and makes the quoted p_x_c,loop=0.719±0.003 unreliable. The string transitions are estimated from the mean of the disorder parameter, whereas the TEE transitions are estimated from the variance peak. If finite-size corrections to these two estimators are not identical (and the unphysical fits show they are not), the apparent ~0.1 separation may be a methodological artifact rather than physics. Footnote [52]'s interpretive assumption is a further caveat, but it only matters if the numerical separation survives this estimator test.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a measurement-only circuit (MoC) for the toric code on the triangular lattice, where projective measurements of stabilizers (A_s, B_p) compete with local Pauli measurements, parameterized by probabilities p_g, p_x, p_z. Using stabilizer simulations, the authors compute three independent non-local diagnostics: the topological entanglement entropy (TEE), the Rényi-2 expectation values of open string operators (disorder parameters of the 1-form symmetries), and non-contractible loop operators (logical operators). The main quantitative claim is that on the lines p_x+p_g=1 and p_z+p_g=1, the TEE transitions at p_g ≈ 0.85 and 0.88 with correlation-length exponent ν ≈ 1.7, while the string and loop observables transition at p_g ≈ 0.72–0.79 with ν closer to the 2D percolation value 4/3. From this, the authors conclude in Sec. IV that the TEE phase transition does not coincide with the emergence of spontaneous symmetry breaking (SSB) of the 1-form symmetries; rather, SSB occurs first and the TEE becomes finite later. The triangular lattice is chosen because it is neither self-dual nor bipartite, so the degeneracies of the square-lattice model are absent.","tokens_in":18829,"tokens_out":5109,"duration_ms":60215,"significance":"If the central claim is correct, the paper provides a concrete counterexample to the common assumption that the topological-order transition and the 1-form-SSB transition necessarily occur at the same point. This would be of substantial interest for monitored quantum circuits and for the broader gauge-theory understanding of topological order. The numerical method is well-suited to the problem: stabilizer circuits allow exact trajectory simulations, the TEE is extracted from boundary stabilizer counts on hexagon complexes, and the paper reports bootstrap error estimates and multiple observables. These are genuine strengths. However, the central conclusion rests on finite-size scaling (FSS) analyses whose reliability is not fully established, and on an interpretive premise about 1-form SSB that is stated but not justified. The paper is therefore a promising contribution whose main claim currently outruns the numerical evidence.","major_comments":[{"comment":"The Z-loop FSS row reports p_g = 1.009 ± 0.017, outside the physical range p_g ∈ [0,1] on the line p_x+p_g=1. A critical point cannot lie outside the domain of the tuning parameter; this result indicates that the FSS ansatz is being applied to data that do not contain a genuine transition in the scanned interval. This entry should be removed or refit with a model that respects the domain. More importantly, this unphysical fit undermines confidence in the other FSS results in the same table, since the same procedure is used for all observables.","section":"Table 1A / Sec. III.D"},{"comment":"The text states that the X-loop variance peaks \"do not develop for increasing system size\" while the peak becomes sharper. For a conventional continuous transition, the variance (susceptibility) peak height grows as L^{γ/ν}; a peak that saturates in height cannot be collapsed by the standard FSS ansatz. Therefore the quoted values p_x^c,loop = 0.719 ± 0.003 and ν = 1.27 ± 0.06 are not reliable. The authors should show the peak-height scaling, or replace this estimator with a crossing-based quantity such as a Binder cumulant or a ratio of string correlators.","section":"Sec. III.D, Fig. 7 (center and right panels)"},{"comment":"The central separation between the TEE transition (from variance peaks at p_g = 0.846/0.876) and the string/loop transitions (from means of order parameters at p_g ≈ 0.72–0.79) is extracted with different FSS estimators. Finite-size corrections to variance peaks and order-parameter means are generally different, and the unphysical Z-loop fit shows that not all fits in the family are well-controlled. Without a common estimator or a demonstration that both observables are controlled by the same divergent length scale, the apparent ~0.1 separation does not establish distinct transitions. Please provide, e.g., Binder-cumulant crossings for both TEE and string observables, or a simultaneous collapse using the same scaling variable.","section":"Secs. III.B–III.D and Table I"},{"comment":"The ordering claim \"SSB takes place first, and then the TEE gets a finite value\" relies on the assumption stated in footnote [52] that a vanishing expectation value of the disorder parameter is an exact diagnosis of SSB of the 1-form symmetry, in the same way as for ordinary 0-form symmetry. This assumption is neither proved nor standard in general: a string disorder parameter can vanish for other reasons (e.g., boundary or finite-size effects specific to the monitored trajectory ensemble) without implying true 1-form SSB. The conclusion should be reformulated as a statement about the behavior of the string correlators, or the diagnostic assumption must be justified explicitly.","section":"Footnote [52] and Sec. IV"}],"minor_comments":[{"comment":"There are grammatical slips in the abstract and introduction (e.g., \"gives possibility\", \"takes place\"). These should be corrected for a journal submission.","section":"Abstract and Introduction"},{"comment":"The notation p_x^gc and p_z^gc is confusing: the text says \"for the MoC with p_x+p_g=1 ... p_x^gc = 0.846\". Since the line is p_x+p_g=1, p_x^gc is actually the critical value of p_g on that line, not the probability p_x. Please rename to avoid ambiguity.","section":"Sec. III.B"},{"comment":"The text says \"we fixed the value of p_g with keeping p_x+p_z = p_g\", but the figure captions and the phase diagram use p_x+p_z = 1-p_g. Please reconcile this inconsistency.","section":"Appendix D.1"},{"comment":"The captions contain \"collapse of the date\" which should be \"collapse of the data\". Similar typos appear in the main text (e.g., \"orders\" for \"operators\").","section":"Figure captions (Fig. A5, A6)"},{"comment":"In the FSS ansatz χ_TE(p_g) = L^ζ F((p_g-p_gc)L^{1/ν}), the exponent ζ is not defined or reported. Please clarify what ζ is and whether it is consistent with a dimensionless TEE variance. Also, Table II should explain explicitly what \"TEE(1~6)\" means relative to \"2~6 hexagons\".","section":"Eq. (5) and Table II"},{"comment":"The definition of the Fredenhagen-Marcu operator uses L_{1/2} and L, but the relationship between these lengths is only stated in words. A more precise definition would make the connection to the string order parameters easier to follow.","section":"Appendix C, Eq. (A2)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a topical and potentially important question, and the numerical framework is appropriate. However, the central conceptual claim—that the TEE transition is distinct from the 1-form-SSB transition—is currently supported by FSS results that include at least one unphysical critical point and by an interpretative assumption stated only in a footnote. The paper would merit publication in a strong journal if the authors can provide a controlled common-estimator scaling analysis and either justify or soften the SSB diagnosis. I do not see grounds for rejection, but the present form is not yet conclusive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nWhat you should know about this paper: it studies a measurement-only circuit for the toric code on the triangular lattice, where the lack of self-duality lets the electric and magnetic transitions decouple. The authors report that TEE goes critical at p_g around 0.85 while the string/loop disorder parameters transition around 0.72-0.79, with TEE exponent nu ~ 1.7 versus string/loop nu closer to 4/3. They conclude that the TEE phase transition does not coincide with spontaneous symmetry breaking of the 1-form symmetries. That is an interesting claim, and the triangular-lattice setup is genuinely new as far as I know.\n\nCredit where due: the stabilizer simulation is careful; the TEE is computed with a hexagon-complex partition that avoids some boundary ambiguities; the FSS uses pyfssa with bootstrap errors; and the appendices show time-step checks, alternative FSS schemes, and additional data for both p_x+p_g=1 and p_z+p_g=1. They also explicitly flag the interpretive assumption that a vanishing disorder parameter exactly diagnoses 1-form SSB (footnote 52). That is honest.\n\nThe soft spots are real, though. The central numerical separation is obtained by comparing a variance-peak critical point for the TEE with an order-parameter-mean critical point for the strings. Those are different FSS estimators, and if they have different finite-size corrections, a ~0.1 shift in p_g can appear without physics. The strongest evidence of trouble is Table 1A: the Z-loop FSS gives p_g=1.009 +/- 0.017 on the line p_x+p_g=1. A transition located outside the physical parameter range is not a critical point; it is the scaling ansatz applied to data with no transition. The X-loop variance in Fig. 7 also does not grow with system size, which undermines the quoted p=0.719 +/- 0.003; a standard continuous transition would produce a growing susceptibility peak. So the quantitative split between TEE and 1-form transitions is not yet nailed.\n\nFootnote 52's assumption is a second, smaller caveat: if a disorder parameter can vanish without true 1-form SSB, then the ordering \"SSB first, TEE later\" would not follow. But that only matters if the numerical separation survives the estimator issue.\n\nWho is this for: people working on measurement-only circuits, generalized 1-form symmetries, and monitored quantum memories. It deserves a serious referee. My recommendation: send to review, and ask for a revision that (i) explains or removes the unphysical Z-loop fit, (ii) shows whether the X-loop variance peak grows on larger L, and (iii) re-estimates all critical points with the same estimator (e.g., correlation ratio or crossing of Binder cumulants) before accepting the distinct-transition claim. The paper should also deposit data and code, since the current data-availability statement is \"on request.\"\n\nOverall, a solid and honest numerical study with an interesting but currently under-supported central claim.","headline":"A genuinely new triangular-lattice measurement-only-circuit study with a plausible but under-controlled claim that TEE and 1-form-symmetry critical points split; fitting problems (one critical point outside the physical range, a non-growing variance peak) must be fixed before the quantitative phase diagram is trusted.","tokens_in":19303,"tokens_out":3449,"would_cite":true,"duration_ms":34712,"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 a measurement-only toric code on a triangular lattice, the topological-order transition (as measured by topological entanglement entropy) and the spontaneous breaking of the 1-form symmetries occur at different meas","keywords":["measurement-only circuit","toric code","triangular lattice","topological entanglement entropy","1-form symmetry","stabilizer formalism","logical qubits","percolation transition"],"falsifier":"A concrete check: run the same measurement-only circuit at p_g = 0.80 for system size L = 24 and measure both the open-string disorder parameter and the six-region TEE. The paper predicts the string has essentially vanished while the TEE is still zero; if the string remains substantially nonzero there, or if the two crossing points shift together with increasing L, the claimed separation is an artifact.","tokens_in":18293,"feed_emoji":"⚛️","tokens_out":8272,"duration_ms":78526,"temperature":0.7,"pith_summary":"This paper studies a measurement-only quantum circuit that repeatedly measures the stabilizers of a toric code on a triangular lattice together with single-qubit X and Z measurements. It claims that the steady-state phase transition seen in the topological entanglement entropy (TEE) does not happen at the same parameter value as the transition in the disorder parameters that diagnose spontaneous breaking of the 1-form symmetries. The authors find the symmetry-breaking signal appears first, at measurement strength p_g around 0.72–0.79, while the TEE only switches on around p_g = 0.85–0.88, with a correlation-length exponent near 1.7 that differs from the percolation-like exponent (about 4/3) of the string and loop observables. Because the triangular lattice lacks the self-duality and bipartiteness of the square lattice, the two transitions are not forced to coincide by symmetry, so the setup provides a clean test of whether 1-form symmetry breaking is the origin of topological order. The result challenges a widely held belief and gives a concrete playground for separating the signatures of topological order.","feed_headline":"1-form symmetry breaks before topological order in measurement circuit","feed_subtitle":"Simulation separates the two transitions: symmetry breaking first, topological entanglement later, with distinct exponents.","key_machinery":"The measurement-only circuit itself — projective measurements of toric-code stabilizers and competing local Pauli measurements — is the central object. The key identity is the six-region formula for the topological entanglement entropy, computed from boundary stabilizer counts; the key diagnostic operators are the zigzag string operators, which serve as disorder parameters of the electric and magnetic 1-form symmetries, and the non-contractible loops, which act as logical operators. The triangular lattice is the load-bearing geometric choice: without self-duality or bipartiteness, the electric and magnetic transitions are not forced to coincide, so the different observables can be compared a","core_discovery":"The paper's central claim is that in the steady state of a measurement-only circuit built from triangular-lattice toric-code stabilizers (vertex X-checks, plaquette Z-checks, plus local X and Z measurements), the phase transition in the topological entanglement entropy does not coincide with the spontaneous breaking of the electric and magnetic 1-form symmetries. Using exact stabilizer simulations and finite-size scaling, the authors find TEE critical points p_g^c = 0.846±0.016 (confinement side) and 0.876±0.048 (Higgs side) with correlation-length exponent ν ≈ 1.7, whereas the string disorder parameters (diagnosing 1-form symmetry breaking) and the non-contractible loop order parameters (lo","pith_inferences":["A testable extension: the same protocol on a square lattice, where self-duality forces coincident transitions, should show TEE and symmetry-breaking signals crossing at the same point; observing the triangular-lattice discrepancy there would confirm that the separation is due to lattice asymmetry.","The paper's footnote assumption — that a vanishing disorder parameter is an exact diagnosis of 1-form symmetry breaking — could be checked by comparing the string correlator with a differently normalized disorder parameter in the same simulations; a disagreement in the region p_g ≈ 0.72–0.85 would indicate the ordering is a property of the specific diagnostic rather than of the phase.","If the discrepancy persists in larger systems and in Hamiltonian (non-measurement) versions of the triangular toric code, it would suggest that TEE and 1-form symmetry breaking are generically independent signatures in non-self-dual topological models.","The reported ν ≈ 1.7 for the TEE transition invites comparison with other monitored-circuit critical points; one could look for the same exponent in measurement-only circuits with different stabilizer codes to test whether it is universal across this class."],"forward_implications":["If the discrepancy is real, topological order (as diagnosed by TEE) and 1-form symmetry breaking are not interchangeable diagnostics in monitored circuits; the conventional view that symmetry breaking is the origin of topological order needs qualification.","The TEE critical exponent ν ≈ 1.7 differs from the 2D percolation value 4/3, suggesting the entanglement transition belongs to a different universality class from the string and loop transitions.","The string and loop observables, including the logical operators, transition near the percolation threshold, indicating that emergence of a quantum memory in this circuit is tied to percolation of stabilizer voids rather than to the TEE transition.","The two 1-form symmetry transitions (electric and magnetic) occur at slightly different p_g values on the triangular lattice, a direct manifestation of broken self-duality that could be used to tune the two symmetries independently."],"fun_headline_variants":["Symmetry breaking outruns topological order in measurement circuit","Triangular toric code: TEE and 1-form symmetry part ways","Distinct transitions for topological entanglement and 1-form symmetry","Measurement-only circuit: symmetry break first, topology later","No joint critical point for TEE and 1-form symmetry"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper assumes that a vanishing expectation value of the string disorder parameter — the quantity that diagnoses whether the 1-form symmetry is spontaneously broken — is an exact diagnostic, just as an ordinary order parameter diagnoses 0-form symmetry breaking; if the string can vanish for other reasons, the claimed ordering does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry breaking outruns topological order in measurement circuit","Triangular toric code: TEE and 1-form symmetry part ways","Distinct transitions for topological entanglement and 1-form symmetry","Measurement-only circuit: symmetry break first, topology later","No joint critical point for TEE and 1-form symmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000364,"raw_usage":{"total_tokens":1852,"prompt_tokens":851,"completion_tokens":1001,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":916}},"tokens_in":595,"tokens_out":1001,"duration_ms":10928,"temperature":1.0,"reasoning_tokens":916,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:58:16.882233+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: run the same measurement-only circuit at p_g = 0.80 for system size L = 24 and measure both the open-string disorder parameter and the six-region TEE. The paper predicts the string has essentially vanished while the TEE is still zero; if the string remains substantially nonzero there, or if the two crossing points shift together with increasing L, the claimed separation is an artifact.","supporting_citations":[],"review_version":1}