{"id":"9e3a4bae-5b5b-4591-ac14-e32ae737a56d","arxiv_id":"2412.12738","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Under combined ZZ and X decoherence, the ground state of the transverse-field Ising chain becomes a mixed state whose phase diagram, revealed by Rényi-2 correlators and entanglement entropy, matches the quantum Ashkin-Teller model and includes a strong-to-weak Z2 symmetry-breaking phase.","lead":"This paper studies what happens to a one-dimensional quantum Ising magnet when two types of noise act on it at once. It finds that the resulting mixed state has three phases, including a recently proposed 'strong-to-weak' symmetry breaking phase, and maps the phase diagram onto a known model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The qAT-parent identification is asserted rather than demonstrated; direct overlap with qAT ground states is required before the phase labels are secure.","rationale":"The reader's CONDITIONAL verdict and weakest_assumption both target exactly the qAT-ground-state correspondence, and my independent reading agrees. The filtered state of Eq. (1) is never compared with any qAT ground state in the manuscript. The only cited support for the filtering construction comes from frustration-free models, and Sec. IV explicitly acknowledges that the TFIM is not frustration-free. The qAT coupling constants enter through an unspecified proportionality constant c, so the claimed closeness cannot even be made quantitative without fixing c. A direct overlap or phase-boundary consistency check would settle whether the observed transitions are qAT transitions or a different decoherence-driven mechanism. This concern does not invalidate the numerical observation of region-II order (χII_ZZ large with χI_Z,st small), but it does invalidate the parent-model phase labeling if the check fails. Therefore I recommend no change to the reader's CONDITIONAL verdict.","tokens_in":16822,"tokens_out":7427,"duration_ms":77634,"concrete_test":"Pick the J/h=0.8 sweep and a post-transition point, e.g. pzz=0.4. Construct the normalized filtered MPS |ρD⟩⟩ and, by DMRG, the ground state |ψqAT⟩ of HqAT on the same L=28 ladder with J/h=0.8 and λ given by the proposed relation. Since c is unspecified, first fix c by requiring the qAT transition at J/h=0.8 to coincide with the numerically estimated pc_zz≈0.372; then use the same c to predict the transition points for J/h=1 and 1.2. Also compute the fidelity F=|⟨ψqAT|ρD⟩⟩|² and compare χII_ZZ, χI_Z,st, χI_u, and the entanglement spectrum at representative points in all three regions. If a single c cannot reproduce all three transition points, or if F is not O(1) at the representative point, the qAT identification is unsupported and the phase labels require a different justification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Sec. VIII) is that the filtered state |ρD⟩⟩ is close to the ground state of the quantum Ashkin-Teller model. Section IV introduces the correspondence Jλzz = cτzz(pzz), hλx = cτx(px) as an 'expectation', explicitly conceding that the TFIM is not frustration-free, unlike the toric-code examples where the filtering construction was previously validated. The entire three-phase identification, including SWSSB in region II, is inherited from the qAT phase diagram through this unverified mapping. No numerical evidence is presented that the filtered MPS actually resembles any qAT ground state: all correlators and entanglement entropies are computed on the filtered state alone, and the phase boundaries are matched by eye and by polynomial fits. Moreover, the SWSSB phase is found at large pzz, i.e. large τ, where a small-τ perturbative justification of local filtering would be least reliable. Because the proportionality constant c is unspecified, the proposed relation cannot even be made quantitative without an additional fitting step. The load-bearing assumption is therefore not merely the existence of transitions, which the EE and correlator data do support, but the parent-model interpretation of those transitions as qAT transitions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the one-dimensional transverse-field Ising model (TFIM) under two types of decoherence, ZZ and X channels, using the doubled Hilbert space formalism. The decohered density matrix is vectorized and the decoherence channels become local filtering operators acting on the doubled-space MPS. The authors argue that the resulting filtered state is close to the ground state of the quantum Ashkin-Teller (qAT) model on a ladder, with parameter correspondence J λ_zz = c τ_zz and h λ_x = c τ_x. They numerically compute Rényi-2 correlator susceptibilities, a strange correlator susceptibility, single-chain ZZ correlator susceptibilities, and entanglement entropies for three parameter sweeps (J/h = 0.8, 1.0, 1.2), identifying three phases: a trivial paramagnetic regime, a strong-to-weak symmetry breaking (SWSSB) regime, and a strong-to-trivial SSB regime. The paper concludes that multiple decoherence applied to the ordinary Ising chain produces a SWSSB mixed state whose phase boundaries resemble those of the qAT model.","tokens_in":17106,"tokens_out":2604,"duration_ms":27255,"significance":"If the central identification with the qAT ground state is correct, the paper provides a concrete and tunable decoherence protocol that produces SWSSB from a conventional pure-state model, which is a valuable addition to the growing literature on mixed-state quantum orders. The numerical work is careful in several respects: the MPS implementation uses bond dimension D=200 and truncation 1e-6, the DMRG convergence criterion is stated, and Appendix C compares MPS results with exact diagonalization for L=8 and finds exact agreement for the tested observables. The data and code are deposited on Zenodo, which supports reproducibility. However, the phase identification—and in particular the SWSSB label for region II—is inherited from the qAT phase diagram through an unverified correspondence, so the significance of the numerical transitions depends on strengthening that correspondence with direct evidence.","major_comments":[{"comment":"The load-bearing claim, stated in Sec. VIII, is that the filtered state is 'close to the ground state of the qAT model.' This is supported only by the expectation that the filtering construction, previously validated for frustration-free models, also holds for the non-frustration-free TFIM. No direct numerical evidence is provided that the filtered MPS resembles a qAT ground state: the paper computes correlators and entanglement entropies of the filtered state alone, and matches phase boundaries by eye. I ask the authors to provide a quantitative check, for example the overlap or fidelity between the filtered MPS and a DMRG ground state of H_qAT at the corresponding parameters, or a comparison of local observables and correlation lengths. Without such a check, the assignment of the three phases to qAT regimes I, II, and III—and hence the identification of region II as SWSSB—remains a conjecture.","section":"Sec. IV and Sec. VIII"},{"comment":"The parameter correspondence J λ_zz = c τ_zz and h λ_x = c τ_x contains an unspecified positive constant c. The protocol in Sec. V chooses p_x = 1/2 - (1/2)(1 - 2p_zz)^{1/J} to enforce λ_zz = λ_x, but the actual path in the qAT phase diagram depends on the ratio λ/J, which is fixed only up to the unknown c. The paper does not explain how c is determined or why the selected path in the qAT phase diagram is the relevant one. The authors should either fix c independently or demonstrate that the observed transition points are compatible with qAT phase boundaries for a range of c values.","section":"Sec. IV and Sec. V"},{"comment":"The transition points p_c^zz are estimated from peaks of the entanglement entropy. The peak locations are obtained by fitting a sixth-order polynomial and then extrapolated linearly in 1/L. This procedure is sensitive to the choice of fitting function, and no scaling collapse or finite-size scaling of the order parameters is shown to confirm that the EE peaks correspond to genuine phase transitions rather than crossover behavior. A finite-size scaling analysis for χ_II_ZZ, χ_I_Z,st, and χ_I_u, or a collapse of the EE data, would strengthen the claim that the observed changes are true mixed-state phase transitions.","section":"Sec. VI, Figs. 3 and 4"},{"comment":"For J/h=1, the initial state is the critical Ising state, and the filtered state is claimed to remain critical up to the transition. Appendix D reports c_eff = 1 from a fit of the entanglement entropy, but the paper does not discuss how robust this value is to the fitting range or to finite-size effects. Since the qAT model has a continuously varying central charge, c_eff alone does not identify a specific critical theory; additional evidence, such as a scaling collapse or comparison with known qAT critical correlators, is needed before assigning the critical line to the qAT criticality.","section":"Sec. VI and Appendix D"}],"minor_comments":[{"comment":"There is a typo in the paragraph defining χ_I_u: 'sightly different' should read 'slightly different.'","section":"Sec. V"},{"comment":"The terms 'regime' and 'region' are used inconsistently; for example, the text refers to 'regime I' in Sec. VI but 'Region I' in Table I and Fig. 2. This should be harmonized.","section":"Secs. IV and VII and Table I"},{"comment":"After defining χ_I_Z,st, the text then writes 'χ_I_ZZ,st ~ 0' in one sentence; the subscript is inconsistent with the definition. Please check all subscripts for the strange correlator.","section":"Sec. V and Fig. 5"},{"comment":"The exact-diagonalization comparison is performed only for L=8 and for J=0.1 or J=0.01, i.e., in the paramagnetic phase. It would be useful to state whether the reported exact agreement also holds for the regimes relevant to the SWSSB transition, where the initial state is critical or ferromagnetic.","section":"Appendix C"},{"comment":"The figure caption states L=28, but the text in Sec. VI does not explicitly list the system sizes for the correlator data in Fig. 3. Please state the system sizes used for each panel in the main text or caption.","section":"Fig. 3"}],"recommendation":"major_revision","confidential_remarks":"The central numerical observations—transitions in the filtered MPS with clean EE peaks—are credible and potentially interesting. However, the paper's interpretation rests on the qAT parent Hamiltonian identification, which is asserted rather than demonstrated. The authors should be asked for a direct overlap or a quantitative comparison with qAT ground states. If such a check is not possible, the claims should be substantially softened and the phase diagram presented as a conjecture supported by numerical evidence. This is a load-bearing issue, not a presentation issue, so I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a serious referee. It is a careful numerical study of the 1D TFIM under combined ZZ and X decoherence, and it genuinely does something new—it maps the decohered state through Choi filtering to the quantum Ashkin-Teller model and uses that guide to identify three phases, including a strong-to-weak Z2 SSB region.\n\nWhat the paper does well: the observable set is well chosen. χII_ZZ (Rényi-2) and χI_Z,st (strange correlator) cleanly separate strong from weak SSB, and the three regimes appear consistently across all three sweeps (J/h = 0.8, 1, 1.2). The numerics are solid: D=200, truncation 1e-6, ED cross-checks in Appendix C, and a sensible central-charge extraction in Appendix D that recovers ceff=1 for the doubled critical Ising state. The authors also ship code and data on Zenodo. Credit where due: the paper is transparent that the qAT correspondence is an expectation, not a proved equivalence.\n\nSoft spots, in proportion. The stress-test note lands, but only partially. The mapping Jλzz = c τzz, hλx = c τx involves an unspecified constant c, and the paper never computes a direct overlap between the filtered MPS and an actual qAT ground state. If the filtered state is not near any qAT ground state, the parent-model interpretation loses its footing. That said, the SWSSB labeling does not depend on qAT—it is read directly from the correlators. The qAT analogy is an interpretive wrapper, not the load-bearing wall. The more genuine weakness is the location of the SWSSB phase at large pzz, where the small-τ filtering intuition is least reliable, and the phase-boundary estimates rely on polynomial fits without error bars. These are fixable with a small-L fidelity check against qAT ground states and a proper finite-size analysis.\n\nWho this is for: people working on mixed-state quantum orders, decoherence-induced phases, or SWSSB more generally. It gives a concrete one-dimensional example beyond the toric-code constructions. I would bring it to a reading group. My verdict: send to peer review. The main claim needs either a direct qAT overlap check or a softening of the 'close to the ground state' language, but the phenomenology is real and the numerics are careful.","headline":"Careful numerics on a concrete decohered Ising chain; the SWSSB claim is carried by the correlators, while the qAT parent-model identification is a plausible but unverified interpretive layer.","tokens_in":17585,"tokens_out":2584,"would_cite":true,"duration_ms":24043,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Applying ZZ and X decoherence to a transverse-field Ising chain produces a mixed state close to the quantum Ashkin-Teller ground state, including a strong-to-weak Z2 symmetry-breaking phase.","keywords":["mixed-state quantum order","strong-to-weak spontaneous symmetry breaking","doubled Hilbert space formalism","quantum Ashkin-Teller model","transverse-field Ising model","decoherence channels","matrix product states","Renyi-2 correlators"],"falsifier":"Compute the overlap (or energy difference) between the normalized filtered matrix product state and the density-matrix renormalization group ground state of the quantum Ashkin-Teller model at the corresponding $\\lambda$; if the overlap decays with system size, or if the correlator-extracted phase boundaries do not converge to the Ashkin-Teller boundaries as $L$ grows, the central correspondence is wrong.","tokens_in":16652,"feed_emoji":"⚛️","tokens_out":17535,"duration_ms":123093,"temperature":0.7,"pith_summary":"This paper tries to show that ordinary decoherence applied to the one-dimensional transverse-field Ising chain can create mixed-state quantum order with no pure-state counterpart. In the doubled-Hilbert-space picture, the two decoherence channels (nearest-neighbor $ZZ$ and single-site $X$) become local filtering operators on a two-copy ladder state, and the filtered state is argued to sit close to the ground state of the quantum Ashkin-Teller model. Matrix-product-state numerics then identify three mixed-state phases whose order parameters match the three Ashkin-Teller regimes. The central finding is a strong-to-weak $\\mathbb{Z}_2$ symmetry-breaking (SWSSB) phase, where the strong-symmetry order parameter is finite while the weak-symmetry and single-chain correlators vanish. If correct, this gives a concrete, tunable protocol by which decoherence alone turns the simplest symmetry-breaking quantum chain into a genuinely mixed-state phase.","feed_headline":"Decohered Ising chain yields a strong-to-weak symmetry-breaking phase","feed_subtitle":"Two noise channels turn the Ising ground state into an Ashkin-Teller ladder with a strong-to-weak SSB phase.","key_machinery":"The load-bearing machinery is the doubled Hilbert space vectorization: a density matrix $\\rho$ becomes a vector $|\\rho\\rangle\\rangle$ in a two-leg ladder $\\mathcal{H}_u \\otimes \\mathcal{H}_\\ell$, and each decoherence channel becomes a local operator. For the two channels used here, the combined action is the local filtering operator $\\prod_j e^{\\tau_{zz}\\hat{h}^{zz}_{j,j+1}} e^{\\tau_x \\hat{h}^x_j}$ with $\\tau_{zz,x} = \\tanh^{-1}[p/(1-p)]$, applied to the decoupled two-copy TFIM ground state. The paper's key assumption is that this filtering deforms the MPS into a state close to the ground state of the quantum Ashkin-Teller model, whose phase diagram (paramagnetic, partially ordered/diagonal-$\\mathbb{Z}_2$, and $\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ broken regimes) then labels the mixed-state phases. The order parameters are the Renyi-2 susceptibility $\\chi^{II}_{ZZ}$ (strong-symmetry SSB), the strange-correlator susceptibility $\\chi^{I}_{Z,\\mathrm{st}}$ (weak-symmetry SSB), and the single-chain susceptibility $\\chi^I_u$ (ordinary long-range order), with the doubled-space entanglement entropy marking the transitions.","core_discovery":"The central claim is that the decohered density matrix $\\rho_D = \\mathcal{E}_{ZZ} \\circ \\mathcal{E}_X[\\rho_0]$ is best understood through its doubled-Hilbert-space vector $|\\rho_D\\rangle\\rangle$, which equals the two-copy TFIM ground state $|\\psi_0^*\\rangle|\\psi_0\\rangle$ acted on by local filters $\\prod_j e^{\\tau_{zz}\\hat{h}^{zz}_{j,j+1}} e^{\\tau_x \\hat{h}^x_j}$. The authors claim that this filtered matrix product state approximates the ground state of the quantum Ashkin-Teller model, with couplings related by $J\\lambda_{zz} = c\\,\\tau_{zz}(p_{zz})$ and $h\\lambda_x = c\\,\\tau_x(p_x)$. On this basis they identify three regimes: a trivial paramagnetic mixed state (region I), a strong-to-weak $\\mathbb{Z}_2$ SSB phase (region II) in which the Renyi-2 susceptibility $\\chi^{II}_{ZZ}$ is $O(1)$ while the strange-correlator susceptibility $\\chi^{I}_{Z,\\mathrm{st}}$ and single-chain susceptibility $\\chi^I_u$ vanish, and a strong-to-trivial $\\mathbb{Z}_2$ SSB phase (region III) in which all three are $O(1)$. The entanglement entropy of the renormalized doubled vector peaks at $p_{zz}^c \\approx 0.37$, $0.31$, and $0.39$ for $J/h = 0.8$, $1$, and $1.2$, marking phase transitions the authors take to match the Ashkin-Teller diagram. The conclusion is that decoherence, viewed as local filtering, is a concrete route to SWSSB from an ordinary Ising chain.","pith_inferences":["Because regime II of the Ashkin-Teller model is a spin-glass-type phase, the SWSSB region likely reflects a glassy, non-commuting order in the original density matrix; a direct test would be to check whether $\\rho_D$ shows Edwards-Anderson-type freezing of local Renyi-2 correlators, not just the doubled-space $\\chi^{II}_{ZZ}$.","The strong symmetry of the decoherence channel suggests the SWSSB phase should be robust to weak-symmetric perturbations of the channel; adding a small weak-symmetric noise channel and checking whether $\\chi^{II}_{ZZ}$ stays finite would test that robustness.","The numerics at $J/h=1$ show $c_{\\mathrm{eff}}=1$ before the transition; one testable extension is to measure the central charge at the extrapolated $p_{zz}^c$ and check whether it flows to the $\\mathbb{Z}_2$-orbifold boson CFT expected at the Ashkin-Teller transition.","The filtering/parent-Hamiltonian logic could be inverted: choose any ladder Hamiltonian whose ground state is a doubled pure state, identify the local filters that deform it, and the corresponding single-chain decoherence protocol should produce the same mixed-state phase diagram."],"forward_implications":["Decoherence strengths $p_{zz}$ and $p_x$ act as a tunable coupling $\\lambda$ in the Ashkin-Teller parent model, so the mixed-state phase diagram can be navigated by changing noise rates rather than Hamiltonian parameters.","The middle regime is a genuine strong-to-weak $\\mathbb{Z}_2$ SSB phase: finite $\\chi^{II}_{ZZ}$ with vanishing $\\chi^{I}_{Z,\\mathrm{st}}$ and $\\chi^I_u$ means the strong symmetry is broken while the weak (diagonal) symmetry is restored, an order with no pure-state analogue.","The phase boundaries obtained from entanglement-entropy peaks ($p_{zz}^c \\approx 0.37$ for $J/h=0.8$, $0.31$ for $J/h=1$, and $0.39$ for $J/h=1.2$ in the thermodynamic limit) locate the decoherence-induced transitions.","At the critical point $J/h=1$, the state remains critical up to $p_{zz} \\approx 0.3$ with effective central charge $c_{\\mathrm{eff}} = 1$ and is then driven into the gapped SWSSB region, showing a noise-induced transition out of criticality.","The same filtering formalism is expected to apply to other spin models such as the $XXZ$ chain, where the doubled-space ladder picture can reveal mixed-state phases not yet known."],"supporting_citations":[{"why":"Supplies the quantum Ashkin-Teller phase diagram that labels the three mixed-state regimes.","marker":"[39]"},{"why":"Supplies the local filtering technique for deforming matrix product states toward a perturbed parent Hamiltonian.","marker":"[36]"},{"why":"Provides the earlier filtering-to-perturbed-ground-state example (toric code in a field) that motivates the Ashkin-Teller correspondence.","marker":"[38]"},{"why":"Supplies the doubled-space correlator framework and the strong-symmetry SSB order parameter used in the numerics.","marker":"[23]"},{"why":"Defines strong-to-weak spontaneous symmetry breaking and the weak-symmetry order parameters that identify region II.","marker":"[24]"},{"why":"Provides the purification-perspective characterization of strong symmetry breaking behind the SWSSB diagnostics.","marker":"[25]"},{"why":"Supplies the strange-correlator construction used as the weak-symmetry order parameter.","marker":"[17]"},{"why":"Establishes the isomorphism that maps density matrices to vectors in the doubled Hilbert space, the backbone of the formalism.","marker":"[34]"},{"why":"Gives the conjugate form of that isomorphism used to vectorize states and decoherence channels.","marker":"[35]"},{"why":"Introduces the strong versus weak symmetry classification for mixed states on which the phase definitions rely.","marker":"[22]"}],"fun_headline_variants":["Two decoherence channels turn Ising into strong-to-weak phases","Strong-to-weak symmetry breaking emerges from decohered Ising chain","Decoherence filters Ising state into Ashkin-Teller order","Ising under two noises reveals strong-to-weak quantum order","Strong-to-weak SSB phase from decohered Ising chain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the assumption that the local filtering deforms the doubled-space matrix product state into a state close to the quantum Ashkin-Teller ground state, with the coupling correspondence $J\\lambda_{zz} = c\\,\\tau_{zz}(p_{zz})$ and $h\\lambda_x = c\\,\\tau_x(p_x)$ holding at least qualitatively; if that proximity fails, the SWSSB identification loses its parent-model justification.","fun_headline_variants_meta":{"raw":{"variants":["Two decoherence channels turn Ising into strong-to-weak phases","Strong-to-weak symmetry breaking emerges from decohered Ising chain","Decoherence filters Ising state into Ashkin-Teller order","Ising under two noises reveals strong-to-weak quantum order","Strong-to-weak SSB phase from decohered Ising chain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001177,"raw_usage":{"total_tokens":4978,"prompt_tokens":1174,"completion_tokens":3804,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":790,"completion_tokens_details":{"reasoning_tokens":3711}},"tokens_in":790,"tokens_out":3804,"duration_ms":23463,"temperature":1.0,"reasoning_tokens":3711,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:46:52.474442+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the overlap (or energy difference) between the normalized filtered matrix product state and the density-matrix renormalization group ground state of the quantum Ashkin-Teller model at the corresponding $\\lambda$; if the overlap decays with system size, or if the correlator-extracted phase boundaries do not converge to the Ashkin-Teller boundaries as $L$ grows, the central correspondence is wrong.","supporting_citations":[{"cited_title":"Kohmoto, M","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum Ashkin-Teller phase diagram that labels the three mixed-state regimes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the doubled-space correlator framework and the strong-symmetry SSB order parameter used in the numerics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the purification-perspective characterization of strong symmetry breaking behind the SWSSB diagnostics."},{"cited_title":"Choi, Completely positive linear maps on complex ma- trices, Lin","cited_arxiv_id":null,"evidence_quote":"Establishes the isomorphism that maps density matrices to vectors in the doubled Hilbert space, the backbone of the formalism."},{"cited_title":"de Groot, A","cited_arxiv_id":null,"evidence_quote":"Introduces the strong versus weak symmetry classification for mixed states on which the phase definitions rely."}],"review_version":1}