{"id":"011547e5-3dcb-45f3-ad7e-9e4ded0f1668","arxiv_id":"2502.02645","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Weak measurements drive interacting one-dimensional Dirac fermions through a BKT-type localization transition, and in the non-interacting limit any nonzero measurement rate yields a finite correlation length with no entanglement transition.","lead":"This paper analyzes one-dimensional interacting Dirac fermions that are continuously monitored by weak particle-number measurements, and derives the phase diagram for their density correlations. It shows that attractive interactions protect an algebraic-correlation phase up to a critical measurement rate, while free fermions localize for any nonzero rate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The γ_c = 0 and log ξ ~ 1/γ results hinge on a scale-dependent complex spacetime rotation (Sec. III D) whose effect on the beta functions is not derived; this is the load-bearing step.","rationale":"The paper's central derivation is carefully built: exact bosonization of the monitored Thirring model, a replica Keldysh action, exact integration of the k = 0 heating mode (App. F), and a controlled Gaussian fixed point. The strongest claim, γ_c = 0 in the free limit with log ξ ~ 1/γ, is a refinement of Ref. [13] and is qualitatively consistent with the weak-localization scenario in Refs. [14,15]. I read the complex-rescaling step in Sec. III D as the single point where the argument is heuristic rather than derived. The rotation angle βσ = s Im(1/Kσ) is applied at every RG step to restore the Hermiticity symmetry that the truncated complex flow violates. Since beta functions are scheme-dependent before critical exponents are extracted, and since the location γ_c = 0 is a statement about the bare parameter mapping, one needs to know that the rotation does not change the initial conditions at O(γ^2). The paper states this but does not demonstrate it. The reader's secondary worry about O(γ^2) terms in X is less severe: at Δg = 0, X0 ~ -cγ^2 and δ ~ dγ^2, so X0/√δ is an O(1) constant; it changes the prefactor of log ξ ~ const/γ but cannot restore a finite γ_c unless the sign of δ changes. That sign change is exactly what the unrotated flow could produce, so the rotation remains the load-bearing issue. A direct two-loop or unrotated Wilsonian check, or an independent free-fermion numerical test of the γ_c = 0 prediction, would settle it. I do not see a reason to change the CONDITIONAL verdict.","tokens_in":36292,"tokens_out":14388,"duration_ms":150215,"concrete_test":"Re-derive the beta functions for the full complex couplings (Re Kσ, Im Kσ, Re λσ, Im λσ) directly from Eq. (39) using the same shell integration as App. G, but without imposing the βσ rotation; then check two things: (i) whether the Hermiticity manifold K+ = K_-*, λ+ = -λ_-* is invariant under the unrotated flow, and (ii) whether for Δg = 0 the RG scale s* at which Re K diverges scales as 1/γ, as 1/γ^2, or not at all. If the unrotated flow gives a finite γ_c or a different exponent, the rotation is not a harmless scheme choice and the central claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result that γ_c = 0 and log ξ ~ 1/γ for Δg = 0 rests on converting the complex flow (44)-(45) into the real BKT flow (46)-(48) by the scale-dependent complex rescaling (x, t) → (x, t)e^{iβσ}/b with βσ = s Im(1/Kσ) (Sec. III D). This is the least secure step of the argument. The paper does not show that this rotation is a symmetry of the Wilsonian effective action, that it commutes with the momentum-shell integration used to derive (44)-(45), or that the phase of the rescaling has no effect on observables. If the rotation is merely a scheme choice, the bare values X ~ Δg + O(γ^2) and Y^2 ~ m^4γ^2 could acquire O(γ^2) corrections from the imaginary parts of K and λ, which would change the sign of δ in Eq. (50) near Δg = 0 and could move γ_c away from zero. The O(γ^3) terms dropped in (44)-(45) are also uncontrolled at the scale s* ~ 1/γ, where λ may no longer be small. The O(γ^2) correction to X itself is not the main danger: for Δg = 0, X0/√δ is O(1), so it only changes the prefactor in log ξ ~ const/γ, not the scaling. The rotation step is where the physically contentful prediction enters.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes a continuum model of monitored interacting Dirac fermions (Thirring model) in one dimension using a replica Keldysh path integral. The central claims are: (i) for attractive interactions, a BKT-type transition occurs at a critical measurement rate γ_c, separating an algebraic critical phase from a localized phase with exponentially decaying density-density correlations; (ii) in the non-interacting limit γ_c = 0, so any nonzero measurement strength gives a finite correlation length with log ξ ~ 1/γ; (iii) along the non-interacting line, the entanglement entropy obeys an area law for any γ > 0 in the thermodynamic limit. The derivation combines exact bosonization, a replica Keldysh action, integration of a divergent center-of-mass mode, and a second-order perturbative RG. The critical step is a complex space-time rotation used at each RG step to convert complex flow equations into the real BKT flow.","tokens_in":36549,"tokens_out":7468,"duration_ms":72020,"significance":"If the central results hold, the paper provides a rare analytical treatment of measurement-induced phase transitions in a continuum interacting fermion model, going beyond non-interacting lattice calculations. The prediction that weak measurements in the free case are a doubly fine-tuned critical endpoint with log ξ ~ 1/γ is conceptually striking and connects to weak-localization phenomenology. The authors are careful to derive the replica master equation, the bosonized action, and the Gaussian observables in appendices, and they compute the key integral I=0.07805 explicitly. The main physical claim depends on a nontrivial complex Wick rotation whose validity is not established in the manuscript, which is the reason for my conditional assessment.","major_comments":[{"comment":"The conversion of the complex flow equations (44)-(45) into the real BKT equations (46)-(48) is the load-bearing step for the entire phase diagram, including γ_c=0 and log ξ ~ 1/γ. This conversion is achieved by the scale-dependent complex space-time rescaling (x,t) → (x,t)e^{iβσ}/b with βσ = s Im(1/Kσ). The paper does not show that this rotation is a symmetry of the Wilsonian effective action, that it commutes with the momentum-shell integration used to derive (44)-(45), or that the rotated coordinates still describe the same physical correlation length. The assertion that this rotation is 'uniquely defined' is not substantiated, and no alternative scheme-independence check is provided. This is a central technical gap, not a presentation issue.","section":"Sec. III D, Eqs. (44)-(48)"},{"comment":"The claim that δ = Y^2 - 2X^2 is positive for Δg=0 and hence γ_c=0 relies on the bare values X=O(γ^2) and Y^2 ~ m^4 γ^2. While the O(γ^4) terms in X^2 are indeed subleading to Y^2 near γ=0, the O(γ^3) terms neglected in the flow equations (44)-(45) are not controlled at the RG scale s* ~ 1/γ, where X(s) diverges and the perturbative expansion in the nonlinearity is no longer small. The authors should quantify the size of the neglected terms at s* or argue why the standard BKT extraction of the correlation length from the one-loop flow remains valid in this complex-coupling non-equilibrium setting.","section":"Sec. III E, Eq. (50) and following discussion"},{"comment":"The identification of the RG scale s* with the physical correlation length ξ requires that the complex rescaling factor e^{iβσ} in the step (x,t) → (x,t)e^{iβσ}/b does not alter the mapping between the RG time s and the physical distance. The paper states that this phase 'does not need to be tracked' and 'does not enter' the computation of observables, but this is stated without proof. Since βσ is scale-dependent, the relation between s and the original coordinates could acquire an additional factor, potentially affecting the numerical coefficient in log ξ ~ 1/γ and the identification of the critical point. A derivation or an explicit argument that the rotation is a pure gauge is needed.","section":"Sec. III D, paragraph after Eq. (43), and Sec. III E, Eq. (52)"}],"minor_comments":[{"comment":"The transition from the action (35) to the Euclidean form (39) involves a Wick rotation t → iσt and a simultaneous rescaling of space and time that depends on the complex phase of ησ. The text would benefit from stating the precise transformation of the fields and the measure under this combined rotation, rather than referring only to Fig. 5.","section":"Sec. III D, Eq. (39)"},{"comment":"The constant I is defined by a double integral that is evaluated numerically. The paper reports the numerical value 0.07805, but it would be helpful to state the integration limits and the numerical method used, and to note the claimed fifth-digit agreement between the x^2 and t^2 integrals.","section":"App. G, Eq. (G20)"},{"comment":"The statement that the measurement operator is 'unique up to a global prefactor' is slightly imprecise, since the relative coefficient m between O1 and O2 is a free parameter that is later assumed O(1). The text should clarify that the global prefactor is absorbed into γ while m remains a physical parameter.","section":"Sec. II B, Eq. (10)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central results are analytically derived with significant technical detail, but the complex Wick rotation in Sec. III D is the least secure step. The authors should either provide a rigorous justification for this rotation or soften the claims that depend on it. I would also suggest that the authors compare their γ_c=0 prediction with the lattice results of Refs. [14,63,99] more explicitly, and comment on the extent to which those results also rely on a similar analytic continuation. The paper is likely to be of interest to the readership of the journal, but the technical gap should be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Müller–Buchhold–Diehl paper. My take: the central phase diagram is probably right, and the paper is a real advance over their PRX 2021 work, but the most interesting claims rest on one RG step that is justified heuristically rather than derived. I would send it to a serious referee, with Sec. III D earmarked as the thing to examine closely.\n\nWhat is actually new: the replica-limit Keldysh formulation, the exact decoupling of the k=0 mode, the Hermiticity-preserving complex space-time rotation that converts the complex BKT flow into the real BKT flow, the critical-endpoint interpretation for free fermions with gamma_c=0, the modified correlation-length scaling log xi ~ 1/gamma, and the no-entanglement-transition statement along the noninteracting line. These are concrete additions, not repackaging. The appendices are careful: the Gaussian correlation functions, the integration over the infinite-temperature mode, and the second-order sine-Gordon RG are all written out, and the constant I=0.07805 is a computed integral, not a fit. The paper also gives explicit power-counting reasons why O(gamma^2) corrections to X should not change the leading behavior near Delta g=0, and it is honest that the interacting entanglement entropy remains open.\n\nThe soft spot is exactly the one flagged in the stress test: Sec. III D. The scale-dependent rotation (x,t)->(x,t)e^{i beta_sigma}/b with beta_sigma = s Im(1/K_sigma) is load-bearing. It is introduced to preserve the Hermiticity symmetry between the two Keldysh contours, and it does remove the oscillatory terms, but the paper does not show that this rotation commutes with the momentum-shell integration or that it is a legitimate scheme choice for computing observables. The worry about O(gamma^2) shifts changing the sign of delta near Delta g=0 is real, though I think it is less dangerous than it looks: the leading gamma^2 contribution to delta is positive, and X^2 enters only at gamma^4 unless the rotation generates an O(gamma^2) correction to X. That correction is precisely what is not controlled. So the right verdict is conditional, not reject.\n\nThe citation pattern is fine. The earlier PRX 2021 is prior work by the same group, and the paper states plainly what it adds. There is no fitting to external data and no circular derivation beyond the usual self-reference.\n\nWho this is for: anyone working on monitored fermions, measurement-induced transitions, or weak localization in one dimension. They will read it. I would recommend acceptance after major revision, with the referee asking either for a derivation of the rotation or for a numerical check of gamma_c=0 in a discretized version of the model.","headline":"A genuine refinement of the group's earlier Dirac-fermion result, with a plausible critical-endpoint scenario for free fermions, but the load-bearing complex RG rotation in Sec. III D is asserted rather than derived.","tokens_in":37136,"tokens_out":3296,"would_cite":true,"duration_ms":35198,"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":"Monitored one-dimensional Dirac fermions undergo a Berezinskii-Kosterlitz-Thouless transition: attractions keep correlations algebraic below a critical measurement rate, while any nonzero rate localizes free fermions.","keywords":["measurement-induced phase transition","monitored Dirac fermions","Thirring model","BKT transition","Keldysh path integral","bosonization","Luttinger liquid","entanglement entropy"],"falsifier":"Simulate the bosonized replica action, or a discretized Dirac chain with weak monitoring, and compute the stationary density-density correlation function $C(x)$: along the free line $\\Delta g = 0$ the claim requires exponential decay with $\\log \\xi \\sim 1/\\gamma$ for every nonzero $\\gamma$, so a single value of $\\gamma$ with algebraic decay would falsify $\\gamma_c = 0$; for $\\Delta g < 0$ the claim requires algebraic decay for some $\\gamma > 0$ below a critical rate, so observing exponential decay at arbitrarily small $\\gamma$ for fixed $\\Delta g < 0$ would falsify the BKT phase. A complementary analytic check computes the $O(\\gamma^2)$ corrections to $\\delta$ from the second-order flow of $\\mathrm{Re}\\,K_\\sigma$ and asks whether $\\delta$ stays positive at $\\Delta g = 0$.","tokens_in":36023,"feed_emoji":"📐","tokens_out":12812,"duration_ms":104400,"temperature":0.7,"pith_summary":"The paper studies one-dimensional interacting Dirac fermions (the massless Thirring model) subject to continuous, weak local measurements of particle number, and it works out the stationary phase diagram of the monitored ensemble. The central claim is a Berezinskii-Kosterlitz-Thouless (BKT) transition, whose hallmark is a correlation length diverging faster than any power law: attractive interactions stabilize a critical phase with algebraically decaying density-density correlations for measurement rates below a critical value $\\gamma_c$, while stronger measurements produce a localized phase with a finite correlation length and exponentially decaying correlations. In the non-interacting limit the critical rate collapses to $\\gamma_c = 0$, so any nonzero measurement strength localizes the system, with the correlation length diverging only logarithmically, $\\log \\xi \\sim 1/\\gamma$. The paper thereby identifies weak measurements of free fermions as an implicit double fine-tuning to the critical endpoint of the BKT transition, and it shows that along the non-interacting line the entanglement entropy obeys an area law, with no measurement-induced transition. A sympathetic reader should care because this is one of the few monitored many-body systems where the full phase diagram, the correlation functions, and (for free fermions) the entanglement can be obtained analytically, exposing a sharp difference between free and interacting monitored fermions.","feed_headline":"Free Dirac fermions localize under any nonzero measurement","feed_subtitle":"Interactions protect algebraic correlations up to a critical rate; without them, any measurement localizes.","key_machinery":"The load-bearing object is the replicated Keldysh path-integral action of the bosonized model. After the $k=0$ center-of-mass (heating) mode is integrated out exactly, the action reduces to a complex sine-Gordon theory whose forward and backward contours decouple: on each contour $S_\\sigma[\\varphi_\\sigma] = \\frac{K_\\sigma}{2\\pi}\\sum_{k>0}\\int_{p,\\omega} \\varphi_\\sigma^{(k)*}(p^2+\\omega^2)\\varphi_\\sigma^{(k)} + i\\lambda_\\sigma \\sum_{r\\neq r'}\\int_{x,t} \\cos 2(\\varphi_\\sigma^{(r)}-\\varphi_\\sigma^{(r')})$, with $K_\\sigma = g(1 - 2i\\sigma\\gamma/\\pi v g)$ and $\\lambda_\\sigma = im^2\\gamma$. The argument is carried by a scale-dependent complex Wick rotation that rescales space and time by an angle $\\beta_\\sigma = s\\,\\mathrm{Im}(1/K_\\sigma)$ at every RG step; this rotation preserves the exact Hermiticity symmetry $S[\\varphi_+,\\varphi_-] = -S^*[\\varphi_-,\\varphi_+]$ and converts the complex flow into the real BKT equations $\\partial_s Y = 2XY$, $\\partial_s X = Y^2$, with conserved $\\delta = Y^2 - 2X^2$. The sign of the bare value $\\delta \\simeq (8m^4 I/\\pi^3)\\gamma^2/v^2 - 2\\Delta g^2$ decides whether the cosine term is relevant: negative $\\delta$ yields a line of fixed points (algebraic correlations), positive $\\delta$ yields a runaway flow and a finite correlation length, and the leading $\\gamma^2$ term in $\\delta$ is exactly what forces $\\gamma_c = 0$ at $\\Delta g = 0$.","core_discovery":"At the paper's core is the claim that the competition between unitary evolution and local particle-number measurements in one-dimensional Dirac fermions is governed by a sine-Gordon-type nonlinearity in replica space, whose relevance is decided by a BKT flow. For attractive interactions (Luttinger parameter $g < 1$) and measurement strength $\\gamma$ below a critical $\\gamma_c \\sim -\\Delta g$, the nonlinearity is irrelevant: density-density correlations decay algebraically, $C(x) \\simeq -c/2\\pi^2 x^2$, and the correlation length is infinite. For $\\gamma > \\gamma_c$ the nonlinearity flows to strong coupling, producing a finite correlation length $\\xi$ and exponentially decaying correlations, with the essential BKT scaling $\\log \\xi \\sim 1/\\sqrt{\\gamma - \\gamma_c}$. The paper's central new prediction is the non-interacting limit $g = 1$: because the control parameter is $\\delta \\sim \\gamma^2$ there rather than $\\delta \\sim \\gamma - \\gamma_c$, the critical point shifts to $\\gamma_c = 0$ and the correlation length diverges only as $\\log \\xi \\sim 1/\\gamma$, matching weak-localization phenomenology. Along the free-fermion line the system is therefore localized for every nonzero $\\gamma$ in the thermodynamic limit, and the entanglement entropy, computed from density correlations via the Klich-Levitov relation, obeys an area law with no entanglement transition.","pith_inferences":["A concrete experimental signature follows from the paper's logic even though the authors do not spell it out: prepare a nearly free one-dimensional gas, monitor its density weakly, and measure the density-density correlations; they should turn from algebraic to exponential at a distance that grows like $\\exp(1/\\gamma)$, so the algebraic phase is visible only as $\\gamma \\to 0$.","The same formalism implies an operator-asymmetry test: monitoring the current density instead of the particle density should flip the sign of the interaction axis (the paper's duality), so on a single experimental platform the two monitoring channels should yield phase diagrams that are mirror images in $\\Delta g$.","An implicit consequence of the critical-endpoint picture is that the free monitored ensemble is conformal (central charge $c = 1$) only at exactly $\\gamma = 0$; any nonzero $\\gamma$ flows to a gapped, area-law state, so the usual free-fermion criticality and the monitored ensemble belong to the same universality class only in the zero-measurement limit.","A numerical check that would go beyond the paper: compute the stationary covariance matrix of a weakly monitored tight-binding chain with approximately linear dispersion and extract $\\xi$; if $\\log \\xi$ is linear in $1/\\gamma$ over a wide range, the continuum prediction is robust to the lattice regularization, and if not, the $\\gamma_c = 0$ endpoint is a continuum artifact."],"forward_implications":["For any attractive interaction $\\Delta g < 0$ there is a genuine BKT transition at a nonzero critical rate $\\gamma_c \\simeq -\\Delta g\\,\\pi^{3/2}/(2m^2 I)$: correlations are algebraic below it and exponentially decaying above it, with $\\log \\xi \\sim 1/\\sqrt{\\gamma - \\gamma_c}$.","For free Dirac fermions the critical rate collapses to $\\gamma_c = 0$: every nonzero measurement strength produces a finite correlation length, with the modified scaling $\\log \\xi \\sim 1/\\gamma$ rather than the BKT form.","Free monitored fermions are localized in the thermodynamic limit for any $\\gamma > 0$: entanglement entropy saturates to an area law, so no entanglement transition occurs as a function of $\\gamma$.","Below the correlation length the entanglement entropy retains the free-fermion logarithmic form $S_{\\mathrm{vN}} \\simeq (c/3)\\log L$ with $c = \\mathrm{Re}(1/K_+) \\to 1$ as $\\gamma \\to 0$, so the free point is a doubly fine-tuned critical endpoint of the BKT line.","By the duality $g \\leftrightarrow 1/g$, $\\hat\\theta \\leftrightarrow \\hat\\varphi$, the same BKT physics describes monitoring the current density with repulsive interactions, so a symmetric phase diagram is expected when both density and current measurements are applied at equal rates."],"supporting_citations":[{"why":"The prior effective theory for monitored Dirac fermions whose BKT flow this work refines, and the reference against which the replica-limit construction is compared.","marker":"[13]"},{"why":"The theory of monitored free fermions whose weak-localization scaling $\\log \\xi \\sim 1/\\gamma$ the $\\gamma_c = 0$ result is explicitly matched to.","marker":"[14]"},{"why":"The field theory of monitored interacting lattice fermions whose symmetric phase diagram and volume-law phase are compared with the present continuum results.","marker":"[16]"},{"why":"One of the canonical BKT papers supplying the flow equations and the essential correlation-length scaling used throughout.","marker":"[72]"},{"why":"The exactly solvable Luttinger model whose bosonized Hamiltonian defines the unitary part of the dynamics and the Luttinger parameter $g$.","marker":"[85]"},{"why":"Identifies the model as the massless Thirring model, the relativistic field theory whose monitored version is studied here.","marker":"[86]"},{"why":"Supplies the normal-ordering strategy used in the second-order renormalization of the sine-Gordon action in Appendix G.","marker":"[92]"},{"why":"The Klich-Levitov relation connecting particle-number cumulants to the von Neumann entropy, used to obtain entanglement in the non-interacting case.","marker":"[93]"}],"fun_headline_variants":["Free Dirac fermions localize under any measurement rate","Interactions delay localization in monitored Dirac fermions","BKT physics sets critical measurement for Dirac fermions","Weak measurements freeze free fermions at any strength","Monitor Dirac fermions: free case localizes instantly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on one renormalization-group prescription: at every step, space and time are rotated by a complex angle chosen to keep the density operator Hermitian, which converts the complex flow into the standard real Berezinskii-Kosterlitz-Thouless flow equations. If that rotation is not a legitimate scheme, or if dropped second-order terms in the measurement rate change which of the two phases is favored near zero interaction, the predicted phase diagram does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Free Dirac fermions localize under any measurement rate","Interactions delay localization in monitored Dirac fermions","BKT physics sets critical measurement for Dirac fermions","Weak measurements freeze free fermions at any strength","Monitor Dirac fermions: free case localizes instantly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1519,"prompt_tokens":1012,"completion_tokens":507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":433}},"tokens_in":628,"tokens_out":507,"duration_ms":4913,"temperature":1.0,"reasoning_tokens":433,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T11:35:08.357500+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the bosonized replica action, or a discretized Dirac chain with weak monitoring, and compute the stationary density-density correlation function $C(x)$: along the free line $\\Delta g = 0$ the claim requires exponential decay with $\\log \\xi \\sim 1/\\gamma$ for every nonzero $\\gamma$, so a single value of $\\gamma$ with algebraic decay would falsify $\\gamma_c = 0$; for $\\Delta g < 0$ the claim requires algebraic decay for some $\\gamma > 0$ below a critical rate, so observing exponential decay at arbitrarily small $\\gamma$ for fixed $\\Delta g < 0$ would falsify the BKT phase. A complementary analytic check computes the $O(\\gamma^2)$ corrections to $\\delta$ from the second-order flow of $\\mathrm{Re}\\,K_\\sigma$ and asks whether $\\delta$ stays positive at $\\Delta g = 0$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"One of the canonical BKT papers supplying the flow equations and the essential correlation-length scaling used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The exactly solvable Luttinger model whose bosonized Hamiltonian defines the unitary part of the dynamics and the Luttinger parameter $g$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the model as the massless Thirring model, the relativistic field theory whose monitored version is studied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Klich-Levitov relation connecting particle-number cumulants to the von Neumann entropy, used to obtain entanglement in the non-interacting case."}],"review_version":1}