{"id":"9908c54c-ab45-4b2b-8a73-2fdd0555c3cc","arxiv_id":"2412.14165","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new framework computes generalized charged moments and symmetry-resolved Rényi entropies for arbitrary excited states of the free compact boson CFT, benchmarked against the XX chain.","lead":"The paper defines symmetry-resolved generalized entropies, which split entanglement measures by charge sector for excited states and out-of-equilibrium settings. It derives explicit formulas for these quantities in a free boson conformal field theory and checks them against exact lattice calculations in the XX spin chain.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The C′→plane replacement below Eq. (47) is asserted, not derived; free-boson zero-mode corrections on the annulus are ∼1/log(ℓ/ε), so the 1/log expansions in §§6–7 can be missing boundary terms.","rationale":"The paper's central deliverable is the explicit formulas (58) and (81) for the n=1 and n=2 normalized generalized charged moments at leading order in the chord length. The lattice benchmarks and recovery of Refs. [18,20] give independent support for the leading-order polynomial form in the cases tested. The remaining soft spot is the unproven replacement of C′ correlators by plane correlators below Eq. (47). This is exactly the step that turns Fn into a polynomial in θ, and it is also what allows the 1/log expansions in Sections 6 and 7. For the free compact boson, the annulus zero-mode sector generates 1/log(ℓ/ε) corrections to vertex correlators; these are not computed. If such a correction contains θ^2, the quoted variance shift and 1/log terms shift by an undetermined constant. This does not invalidate the leading-order formulas for the benchmarked states, so the verdict remains CONDITIONAL rather than REJECT. The proposed annulus computation is a single, self-contained check that isolates the C′→plane approximation from lattice finite-size effects. If the 1/Λ correction vanishes at O(θ^2) or is O(1/Λ^2), the concern is resolved; otherwise the subleading physical predictions in §§6–7 need revision.","tokens_in":38224,"tokens_out":10879,"duration_ms":108406,"concrete_test":"Compute exactly, for the n=1 primary case with |ψ1⟩=|ψ2⟩=V_α|0⟩ and α≠0 at small θ, the normalized chiral charged moment on the annulus of modulus Λ=log(ℓ/ε), using the free-boson mode expansion of Sec. 3 and keeping the compact zero-mode/momentum sum. Expand the result in 1/Λ and compare the coefficients of θ^2 and θ^4 with those obtained from the plane-replacement formulas (52) and (58). If the difference is O(1/Λ^2) or higher, the polynomials (58) and (81) are safe at the orders used; if it is O(1/Λ) with an O(θ^2) piece, the boundary suppression claim fails and the subleading expansions in §§6–7 require correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is in Sec. 4.1, just below Eq. (47), and its n=2 analogue in Sec. 5.1: correlators on C′ (the plane with disks of radius δ∼ε/ℓ cut around 0 and ∞) are replaced by ordinary plane correlators, with the statement that entanglement-cut boundary effects are suppressed by powers of log(ℓ/ε). Eq. (48) then fixes the two-point functions used in every Wick contraction entering Eqs. (50), (58), (74), and (81). The suppression is not demonstrated. In the free compact boson this matters because the annulus has a compact zero/momentum mode whose contributions to correlators of vertex operators with charges βθ/(2π) and ±α_i are of order 1/log(ℓ/ε), not merely ε/ℓ. The symmetry twist operators V_{±βθ/2π} sit at y0≈δ and y0′≈1/δ, i.e. on the cut boundaries, so their OPEs with the bulk insertions are exactly where boundary-condition dependence can enter. If a 1/log correction carries an O(θ^2) piece, then h2 in Eq. (92), the variance shift b1 in Eq. (96), and the 1/log expansions in Eqs. (93), (99), and (102) are incomplete. The lattice benchmarks (Figs. 4–5, L=64) check only selected level-2 descendants at two θ values and cannot isolate a 1/log boundary term from lattice finite-size and parity effects.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces the notion of symmetry-resolved generalized entropies, which are intended as building blocks for studying symmetry-resolved entanglement of excited states and its out-of-equilibrium dynamics. The central technical object is the normalized generalized charged moment F_n(θ; ψ_1, ..., ψ_{2n}) of Eq. (4). For the free compact boson CFT, the authors derive explicit sum formulas for the n=1 and n=2 chiral moments, Eq. (58) and Eq. (81), which at leading order in the chord length ℓ take the form of finite polynomials in the flux θ. These formulas are benchmarked against exact XX-chain lattice computations (Figs. 4 and 5) and against previously known primary-field results (Eqs. (64), (85), (86), (88)). The paper then applies the moments to compute the generalized subsystem charge distribution and the symmetry-resolved generalized second Rényi entropy, obtaining expansions in 1/log(ℓ/ε).","tokens_in":38436,"tokens_out":6332,"duration_ms":55824,"significance":"If the results hold, the paper provides a substantial extension of symmetry-resolved entanglement techniques from primary states to arbitrary descendant states in a CFT, which is directly relevant to Luttinger-liquid physics and to the program of computing entanglement dynamics via generalized entropies. The explicit Wick-contraction sums in Eqs. (58) and (81) are new, and their validation against exact lattice data and against independent published special cases is a genuine strength. The definition of symmetry-resolved generalized entropies is natural and likely to be reused. However, the practical value for out-of-equilibrium settings rests on the 1/log(ℓ/ε) expansions in Sections 6 and 7, and those expansions are the part of the paper that is least supported by derivation or numerics.","major_comments":[{"comment":"The replacement of correlation functions on C′ (the plane with two disks cut out around 0 and ∞, representing the regularized entanglement cuts) by ordinary plane correlators is asserted with the statement that the effects of the entanglement-cut boundary conditions are 'suppressed by powers of log(ℓ/ε)', but no derivation or quantitative estimate is provided. In the free compact boson, the annulus has a compact zero mode whose contributions to correlators of vertex operators with zero total charge are of order 1/log(ℓ/ε), not exponentially small. Since the twist operators V_{±βθ/(2π)} are inserted at y0 ≈ δ and y0′ ≈ 1/δ, i.e., on the cut boundaries, their OPEs with the bulk insertions are exactly where boundary-condition dependence can enter. A correction of order 1/log(ℓ/ε) carrying an O(θ^2) piece would change h2 in Eq. (92), the variance shift b1 in Eq. (96), and the 1/log expansions in Eqs. (93), (99), (102), and (104), all of which are presented as physical results of the framework. The lattice benchmarks in Figs. 4 and 5 (L=64) test only selected level-2 descendants at two values of θ and cannot isolate a 1/log boundary term from lattice finite-size and parity effects. The authors should either provide an explicit boundary CFT computation showing that the boundary corrections are subleading at the orders kept, or include the 1/log corrections and re-derive the expansions in Sections 6 and 7.","section":"Sec. 4.1 (below Eq. (47)); Sec. 5.1 (below Eq. (72))"}],"minor_comments":[{"comment":"The abstract contains formatting artifacts from the LaTeX source ('W e', 'T he', and stray spaces), which should be cleaned before final submission.","section":"Abstract"},{"comment":"The phrase 'the regularization-dependent corrections are power-law suppressed by log ℓ/ε' is ambiguous; it should read 'suppressed by powers of 1/log(ℓ/ε)'.","section":"Sec. 3.1, bullet 3"},{"comment":"The deletion notation R_{k1,...,\\emptyset ki,...,kM} is not defined; please describe the deletion operation explicitly, for example by placing a hat over the deleted entry.","section":"Eq. (58) and similar formulas"},{"comment":"The insets showing imaginary parts are small and hard to read; consider enlarging them or presenting the imaginary parts in separate panels.","section":"Figs. 4 and 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript builds heavily on the authors' own prior framework of generalized entropies [63] and on symmetry-resolved entropies [25], but the new explicit sum formulas and benchmarks go beyond those works. The main concern is the unproven suppression of boundary-condition corrections, which affects the 1/log expansions that are essential for the advertised application to out-of-equilibrium dynamics. This is a fixable issue within the scope of the manuscript, but it needs to be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper introduces symmetry-resolved generalized entropies and computes real new objects: explicit sum formulas for the n=1 and n=2 generalized charged moments of the free compact boson for arbitrary descendant states, with lattice benchmarks. That is a legitimate contribution and I think it warrants peer review. The second thing is that the construction depends on an approximation that is asserted, not proved, and the paper's advertised dynamical results are not actually in it.\n\nWhat's new and good: the definitions of Sn(q; ψ1,...,ψ2n) and the normalized charged moments Fn; the Wick-contraction sums in Eqs. (58) and (81); the reduction to the known primary-field results of Refs. [18,20]; and the XX-chain checks in Figs. 4–5. The factorization into left/right movers is clean, and the generating-function reading of F1 for the subsystem charge distribution is useful. The 1/log expansions for the charge distribution and second Rényi entropy are a natural application, with a variance shift that people will care about.\n\nSoft spots, in proportion. The main one is below Eq. (47): correlators on C′ are replaced by plane correlators on the strength of the statement that entanglement-cut boundary effects are suppressed by powers of 1/log(ℓ/ε). That is load-bearing. In this theory the compact zero mode gives 1/log corrections to vertex correlators on the annulus, so the suppression is not automatic. If those corrections contribute to the ratio at the orders kept, the 1/log expansions in §§6–7 are incomplete. The lattice benchmarks are reassuring but only cover a couple of level-2 descendant states at two θ values; they cannot isolate such a boundary term. Second, the abstract promises a framework for dynamical evolution, but the paper stops at static building blocks; that is a scope gap, and the abstract oversells it. Third, no data or code ship, and the n=2 algebra is long enough that independent reproduction is a real task.\n\nMy read: the sum formulas are likely correct as leading-order results, and the paper deserves a serious referee. But the referee should be asked to press on Section 4.1 and the annulus zero-mode issue before the 1/log predictions are taken as gospel. I'd send it with that caveat, and I'd cite the sum formulas if I worked in this area.","headline":"A real technical advance on symmetry-resolved entanglement for descendant states, with a load-bearing boundary-condition approximation that needs scrutiny before the 1/log expansions are trusted.","tokens_in":39077,"tokens_out":4117,"would_cite":true,"duration_ms":38305,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives explicit sum formulas for the normalized generalized charged moments of the free compact boson CFT, making symmetry-resolved entanglement computable for arbitrary descendant states and matching XX-chain lattice data.","keywords":["symmetry-resolved entanglement","generalized Rényi entropies","charged moments","free compact boson CFT","XX spin chain","full counting statistics","conformal field theory","entanglement equipartition"],"falsifier":"Evaluate the full cut-plane correlator in Eq. (47), including the boundary conditions on the two disks, and compare it with the plane-correlator approximation used in the paper: if the difference is not suppressed by at least one power of $1/\\log(\\ell/\\epsilon)$ at the orders retained, the sum formulas for $F_n$ and the derived variance shift would be incomplete. On the lattice side, exact diagonalization of the XX chain at larger flux $\\theta$ or for $n=2$ with additional descendant states would expose any missing $\\theta$-dependence in the polynomial.","tokens_in":37927,"feed_emoji":"⚛️","tokens_out":9474,"duration_ms":70932,"temperature":0.7,"pith_summary":"This paper introduces symmetry-resolved generalized Rényi entropies, which split the entanglement between a subsystem and its complement according to the sector of a global $U(1)$ charge while allowing the replica in- and out-states to be arbitrary excited states. These quantities are the building blocks for studying symmetry-resolved entanglement in excited states and in symmetry-preserving out-of-equilibrium dynamics. The paper claims that in the free massless compact boson CFT these entropies are controlled by normalized generalized charged moments, and it derives explicit sum formulas for the $n=1$ and $n=2$ moments, Eqs. (58) and (81), valid for arbitrary descendant states. At leading order in the chord length the moments are finite polynomials in the Aharonov–Bohm flux $\\theta$, and the formulas reproduce known primary-state results and match exact XX-chain lattice computations. This gives a route to the full counting statistics of the subsystem charge and to the time evolution of symmetry-resolved entanglement after symmetric quenches.","feed_headline":"Explicit sums resolve entanglement by charge for excited states","feed_subtitle":"Compact boson CFT yields Eq. (58) and Eq. (81), benchmarked on the XX chain.","key_machinery":"The central object is the normalized generalized charged moment $F_n(\\theta;\\psi_1,\\ldots,\\psi_{2n})$, defined in Eq. (19) as the ratio of the trace of $n$ glued generalized density matrices with an insertion of the subsystem charge operator $e^{i\\theta Q_A}$ to the ground-state $n$-th charged moment. The computation uses the replica trick: the moment is a partition function on an $n$-fold branched cover of the cylinder, with vertex operators and $\\partial\\varphi$ insertions at the infinities and the $U(1)$ twist operators at the entanglement cuts. A conformal transformation, Eq. (40) for $n=1$ and Eq. (71) for $n=2$, maps this geometry to a branched cover of the plane, and the correlation-function ratio is expanded into Wick contractions. The sum formulas (58) and (81) organize these contractions into a polynomial in $\\theta$: the $n=1$ case simplifies because contractions of $\\partial\\varphi$ with vertex operators combine into a phase $e^{i\\beta\\theta r\\alpha}$ times mode-dependent factors $L(k_i)$, while the $n=2$ case requires the full functions $L_{k_j}(\\theta)$ and $\\tilde{L}_{k_j}(\\bar{\\alpha})$.","core_discovery":"For a $(1+1)$-dimensional free massless compact boson with a $U(1)$ winding symmetry, the normalized generalized charged moment $F_n(\\theta;\\psi_1,\\ldots,\\psi_{2n})$ — the ratio of a charged replica trace built from $2n$ states to the ground-state charged moment — can be evaluated as an explicit sum over Wick contractions. The central formulas are Eq. (58) for $n=1$ and Eq. (81) for $n=2$: they give $F_n$ for arbitrary descendant states as a finite polynomial in $\\theta$ at leading order in the chord length $\\ell$, with coefficients that are trigonometric functions of the subsystem size ratio $r$. The $n=1$ formula is organized recursively by a pairing function $R$ and single-mode functions $L(k_i)$, while the $n=2$ formula additionally tracks contractions with the vertex-operator insertions through functions $L_{k_j}(\\theta)$ and $\\tilde{L}_{k_j}(\\bar{\\alpha})$. These sums reproduce the known primary-field charged moments from Refs. [18,20] and match exact lattice data in the XX chain for level-2 chiral states in Figures 4 and 5.","pith_inferences":["If the same Wick-contraction framework carries over to other CFTs, the natural next test is the Ising model's $\\mathbb{Z}_2$ symmetry resolution for descendant states, whose analogous sums should produce polynomial charged moments with different trigonometric coefficients.","The predicted variance shift in the subsystem charge distribution is a sharp experimental signature: a quantum-gas microscope measuring charge fluctuations in a one-dimensional Bose gas after exciting a Luttinger-liquid state should see the distribution width deviate from the ground-state Gaussian.","The $r=1/2$ simplification in Appendix A.2 suggests that half-system bipartitions may admit closed forms for general $n$, which would provide a cheap diagnostic of the entire construction before more general geometries are attempted."],"forward_implications":["For any descendant state of the compact boson CFT, the $n=1$ and $n=2$ symmetry-resolved generalized Rényi entropies can be written down directly from the sum formulas, extending previous results that were limited to primary states.","The $n=1$ moment is a generating function, so the full counting statistics of the subsystem $U(1)$ charge in an excited state follows immediately, with a mean shifted to $r m$ and a variance shifted by $-2\\pi^2 h_2$ relative to the ground state.","Entanglement equipartition across charge sectors is broken at order $1/(\\log \\ell')^2$ by universal terms, and the excited-state symmetry-resolved second Rényi entropy acquires a double-logarithmic correction.","The same building blocks combine with a numerical time-evolution scheme to track symmetry-resolved entanglement and charge statistics after quantum quenches that preserve the symmetry."],"supporting_citations":[{"why":"introduces the charged moments that the generalized moments defined here extend","marker":"[6]"},{"why":"computes symmetry-resolved entropies for primary excited states, whose charged moments this paper must reproduce","marker":"[18]"},{"why":"derives symmetry-resolved relative entropies, recovered here as special cases of the n=1 and n=2 formulas","marker":"[20]"},{"why":"defines generalized Rényi entropies and supplies the replica and lattice techniques adapted in this paper","marker":"[63]"},{"why":"provides the boundary CFT framework and the ground-state charged moment formula used for the normalization","marker":"[25]"},{"why":"gives the free-fermion lattice results used to interpret the parity oscillations and variance shift","marker":"[10]"},{"why":"maps XX-chain eigenstates to bosonic CFT states, underpinning the numerical benchmarks","marker":"[67]"},{"why":"applies generalized entropies to post-quench Rényi growth, the target application of this framework","marker":"[84]"}],"fun_headline_variants":["Explicit sums give symmetry-resolved entropies for any descendant state","Charge-resolved entropies from explicit sums: CFT benchmarked on XX chain","Exact formulas for charge-resolved entropies in compact boson CFT","Explicit sums resolve entanglement by charge, verified on XX chain","Symmetry-resolved entropies from explicit sums: CFT meets XX spin chain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that correlation functions on the cut plane can be replaced by ordinary plane correlators, with the entanglement-cut boundary conditions contributing only corrections suppressed by powers of $\\log(\\ell/\\epsilon)$; if that suppression fails at the orders the paper keeps, the polynomial-in-$\\theta$ formulas, the $1/\\log$ expansions, and the variance shift would be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Explicit sums give symmetry-resolved entropies for any descendant state","Charge-resolved entropies from explicit sums: CFT benchmarked on XX chain","Exact formulas for charge-resolved entropies in compact boson CFT","Explicit sums resolve entanglement by charge, verified on XX chain","Symmetry-resolved entropies from explicit sums: CFT meets XX spin chain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001172,"raw_usage":{"total_tokens":4829,"prompt_tokens":911,"completion_tokens":3918,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":3816}},"tokens_in":527,"tokens_out":3918,"duration_ms":26420,"temperature":1.0,"reasoning_tokens":3816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:25:16.105181+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the full cut-plane correlator in Eq. (47), including the boundary conditions on the two disks, and compare it with the plane-correlator approximation used in the paper: if the difference is not suppressed by at least one power of $1/\\log(\\ell/\\epsilon)$ at the orders retained, the sum formulas for $F_n$ and the derived variance shift would be incomplete. On the lattice side, exact diagonalization of the XX chain at larger flux $\\theta$ or for $n=2$ with additional descendant states would expose any missing $\\theta$-dependence in the polynomial.","supporting_citations":[{"cited_title":"Generalized entanglement entropies in two-dimensional conformal field theory","cited_arxiv_id":"2112.09000","evidence_quote":"defines generalized Rényi entropies and supplies the replica and lattice techniques adapted in this paper"},{"cited_title":"Post-Quantum Quench Growth of Renyi Entropies in Low Dimensional Continuum Bosonic Systems","cited_arxiv_id":"2112.04412","evidence_quote":"applies generalized entropies to post-quench Rényi growth, the target application of this framework"}],"review_version":1}