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This paper derives the first analytic formula for Rényi entanglement asymmetry in an interacting equilibrium spin chain with explicitly broken U(1), expressing the asymmetry constant directly through the static susceptibility of the broken

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2026-08-01 01:50 UTC pith:VZ5ZLMHS

load-bearing objection First analytic equilibrium entanglement asymmetry for an interacting gapped spin chain with broken U(1), built from a sine-Gordon sum rule and validated three ways; the load-bearing replica-uniformity proof sits in the SM and should be checked. the 3 major comments →

arxiv 2607.25625 v1 pith:VZ5ZLMHS submitted 2026-07-28 cond-mat.stat-mech hep-thquant-ph

Entanglement asymmetry in the gapped XYZ spin-frac12 chain

Felipe Taha Sant'Ana This is my paper
classification cond-mat.stat-mech hep-thquant-ph PACS 75.10.Pq03.65.Ud
keywords entanglement asymmetryXYZ spin chainRényi entanglement entropysine-Gordon form factorsstatic susceptibilityquantum Mpemba effectmatrix product statesU(1) symmetry breaking
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The central claim is that in the gapped, U(1)-breaking phase of the interacting XYZ chain, the Rényi entanglement asymmetry of a large interval has a universal form: it grows as half the logarithm of the interval length, with the additive constant set by the static susceptibility of the broken magnetization divided by the kink mass. This is the first analytic equilibrium asymmetry for an interacting model with explicitly broken U(1), going beyond free-fermion and perturbative critical results. The derivation ties the asymmetry to a susceptibility via a charged-moment identity, evaluates that susceptibility from sine-Gordon form factors, and verifies the result against infinite-system tensor-network simulations. If correct, the formula makes the asymmetry a direct, parameter-free probe of symmetry-breaking strength and provides the equilibrium baseline for Mpemba-type relaxation studies.

Core claim

For a large interval of length ℓ in the gapped XYZ chain, the paper establishes the master formula ΔS_A^(n)(ℓ) = ½ log(πℓ χzz) + log n/[2(n−1)] + O(1/(Mℓ)), where χzz is the static susceptibility of the broken σz charge and M is the kink mass. It then evaluates the universal ratio c(β²) = χzz/M in the sine-Gordon scaling limit, obtaining a lower bound from the two-kink plus first-breather form-factor channels, a two-sided corridor for the multi-particle tail on the attractive side, and a tangent law at the free-fermion point: c(β²) = 1 + (π/2 − 1)(ξ_sG − 1) + O((ξ_sG − 1)²). The multi-particle tail is identified with a four-kink term that vanishes quadratically away from the free-fermion poi

What carries the argument

The central mechanism is a charged-moment identity, proved for fermionic Gaussian states and injective matrix-product ground states, which identifies the curvature of the charged moment with the static susceptibility χzz of the broken charge. A replica-separation uniformity statement for the phase-dressed transfer operator forces the n-dependence log n/[2(n−1)] through a Gaussian integral over the cycle-graph Laplacian. The susceptibility is then evaluated by a sine-Gordon non-conservation sum rule: the two-kink and one-breather form factors provide a lower bound on c(β²), while two moment sum rules confine the leftover multi-particle contribution to a corridor. Exact lattice masses from the

Load-bearing premise

The formula stands or falls on the assumption that the phase-dressed transfer operator of the replicated chain has a single leading eigenvalue entering one replica at a time; if that replica-separation uniformity fails, neither the log-length coefficient nor the n-dependent offset is forced.

What would settle it

In the same tensor-network setup, compute the difference ΔS_A^(3)(ℓ) − ΔS_A^(2)(ℓ) for a long interval at a fixed interacting point; the master formula predicts that this difference approaches log 3/4 − log 2/2 plus a constant independent of ℓ up to O(1/(Mℓ)) corrections. A residual ℓ-dependence, or an offset that moves with the anisotropy γ, would falsify the replica-separation uniformity assumption.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The additive constants of ΔS_A^(n)(ℓ) are fixed for every Rényi index, so measuring the asymmetry at one interval length determines the entire large-ℓ curve.
  • The offset log n/[2(n−1)] is independent of the model parameters, making the n-dependence a universal signature of the single-eigenvalue replica mechanism.
  • Because χzz controls the asymmetry amplitude, the static susceptibility of the broken charge can be extracted from entanglement data without applying an external field or charge perturbation.
  • The tangent law at the free-fermion point and the quadratic onset of the four-kink tail single out that point as a special marginal locus where the asymmetry constant is exactly 1.
  • The master formula provides the equilibrium baseline against which post-quench asymmetries and quantum-Mpemba-type effects in this chain should be compared.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the charged-moment identity and replica-separation uniformity hold for other gapped symmetry-broken ground states with a single dominant transfer eigenvalue, the same structure—half log ℓ plus a susceptibility constant plus an n-only offset—would extend to other integrable and non-integrable chains, making χzz/M a general asymmetry amplitude.
  • Beyond the paper: the quadratic vanishing of the multi-particle tail at the free-fermion point suggests a measurable cusp or softening in the asymmetry constant as the marginal point is approached; high-resolution probes of the charge structure factor could look for it.
  • Beyond the paper: measuring ΔS_A^(n)(ℓ) at two Rényi indices in a cold-atom quantum gas microscope could isolate χzz directly from entanglement data, offering a route to detect symmetry breaking without coupling to the broken charge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the Rényi entanglement asymmetry ΔS_A^(n) of a large interval in the gapped U(1)-breaking phase of the XYZ spin-1/2 chain. The central result is Eq. (4), which states that ΔS_A^(n)(ℓ) = ½ log(πℓ χ_zz) + log n/[2(n−1)] + O(1/(Mℓ)), where χ_zz is the static susceptibility of the broken charge and M is the kink mass. The coefficient c(β²)=χ_zz/M is evaluated in the sine-Gordon limit through a non-conservation sum rule using Lukyanov form factors: the two-kink and one-breather channels provide a lower bound (Eqs. (5)–(6)), two moment sum rules confine the multi-particle remainder to a two-sided corridor on the attractive side (Eq. (7)), and a four-kink estimate is used near the free-fermion point. The kink mass is taken from the Baxter/JKM exact lattice solution, and iDMRG simulations validate the master formula by direct charge-projector and charged-moment constructions.

Significance. If correct, the paper provides the first analytic equilibrium entanglement asymmetry for an interacting model with explicitly broken U(1), going beyond free-fermion results and the perturbative critical XXZ calculation. The manuscript has clear strengths: the master formula is cross-checked by three independent routes (free-fermion transfer matrix, direct variance/inversion of c1, and the n=2/3 combination c1^(2)−c1^(3)=½log2−¼log3), the lattice masses are exact from Baxter/JKM, and the iMPS validation uses directly measured correlator sums and an exact mass rather than a fit to the formula being tested. The code and data are deposited. The main reservations concern the depth of the proof of the central MPS identities, the treatment of two repulsive-side data points that fall below the claimed lower bound, and the unquantified four-kink amplitude.

major comments (3)
  1. [Section 2, after Eq. (3)] Eq. (4) rests on two MPS identities: V=Var(Q_A)[1+O(ℓ^-1)] and the replica-separation uniformity V_d=V for every integer n≥2. The main text states these with proofs deferred to SM Secs. 1.1–1.2. The one-sentence justification that the leading phase-dressed eigenvalue enters the replicated moment 'one replica at a time' is a structural spectral claim. If wrong, the coefficient of log ℓ could acquire n-dependence and the offset log n/[2(n-1)] would not be forced. The n=2 and n=3 difference tests only the offset difference; a common n-dependent rescaling would preserve that difference, so current numerics do not isolate this assumption. The published version should give a self-contained derivation of both identities, since this is load-bearing for Eq. (4).
  2. [Section 5, Numerical Validation] The iDMRG determinations at ξ_sG=1.212 and 1.4813 fall below the two-kink+B1 lower bound. The text ascribes this to γ→0 extrapolation systematics without a quantitative model. Because the extrapolation on the repulsive side is long and the deviations are in the direction that weakens the bound, the claim that the bound is satisfied to within combined uncertainty cannot be checked from the presented data. Please provide the extrapolation-error model, show the raw γ-dependence, or explicitly lower the status of the repulsive-side bound/corridor claims.
  3. [Section 3, four-kink estimate after Eq. (7)] The amplitude A≈0.08 is described as a Monte-Carlo estimate of the leading four-kink form factor, but no error bar, integration parameters, or independent cross-check is given. The statement that this estimate captures the small excess seen near the free-fermion point is used quantitatively to support the tail, and the bound A≥4A^-1≈0.012 also relies on the same quantity. An uncertainty and a description of the evaluation are necessary before this can count as evidence; otherwise it should be flagged as an illustrative estimate.
minor comments (5)
  1. [Section 3, around Eq. (7)] The text says 'Two moment sum rules make this deficit two-sided', but Eq. (7) gives a two-sided bound only for ξ_sG<1; on the repulsive side it is one-sided (c_tail ≥ 4r_-1). Please rephrase to avoid overstating the constraint on the repulsive side.
  2. [Section 4, Eq. (10)] The symbols C_σ+σ+(Δz) and κ(β²) are used without definition in the main text. Please define the amplitude of the breaking coupling and the sine-Gordon mass–coupling constant explicitly, and clarify the relation between C and C_σ+σ+.
  3. [Section 5, charged-moment construction] The phrase 'the α=±π(σz-string) region of the circle average' is unexplained. Define the operator and why its contribution is exponentially suppressed.
  4. [Eqs. (5) and (9)] Check notation: in Eq. (5), the denominator of F(θ) should be parenthesized as (cosh θ + iπ/(2ξ_sG)); in Eq. (9), 'atanh k′_1' is unclear. Use a standard notation with an explicitly indicated argument.
  5. [Introduction, first paragraph] 'One Néel-x vacuum' should be clarified: the ground state is twofold degenerate in the U(1)-broken phase. State which member is chosen and whether the asymmetry is independent of this choice.

Circularity Check

0 steps flagged

No circularity: the master formula is derived from independent analytic inputs and validated against distinct iDMRG-observed quantities.

full rationale

The derivation chain is not circular. The susceptibility chi_zz is evaluated from sine-Gordon form factors (Lukyanov, Fehér–Takács) and the kink mass M from the Baxter/JKM solution; neither input is fitted to the target asymmetry. The iDMRG validation uses directly measured correlator sums, the exact mass from Eq. (9), and independently constructed charged moments, not the master formula being tested. The only deferred items (V=Var(Q_A) and V_d=V) are structural identities asserted for Gaussian/injective-MPS ground states with proofs in the SM; they may be assumptions that could fail, but the text does not define the target result through them, and no main-text equation reduces Eq. (4) to a fit or to a self-citation. The small four-kink amplitude A is estimated perturbatively/Monte-Carlo and separately bounded, and it does not enter the central two-kink+B1 lower bound. There are no self-citations by the author; all external form-factor, mass, and vacuum-expectation-value inputs are independent. The two repulsive iDMRG points lying below the bound are explicitly attributed to gamma→0 extrapolation systematics rather than used as theory input. No circular step can be exhibited with the required quote-and-reduction standard, so the appropriate score is 0.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No genuinely new entities are postulated. The two-kink and B1 channels are standard sine-Gordon excitations. The master formula carries no fitted parameters; the only estimated coefficient is the small four-kink amplitude A. The principal unproved-in-text assumptions are the sine-Gordon description, the external form-factor bootstrap, and the MPS replica-uniformity theorem deferred to the Supplemental Material.

free parameters (2)
  • four-kink amplitude A = ≈0.08 (Monte-Carlo estimate)
    Leading O((ξsG−1)²) coefficient of the multi-particle tail; used to describe the small iDMRG excess near the free-fermion point. Only the lower bound 4r_{-1}≈0.012 is proven; the quoted 0.08 is an estimate, not a closed-form result.
  • γ→0 extrapolation fit coefficients = not quoted (spread sets quoted errors)
    Polynomial and umklapp-corrected fits to iDMRG data are used to extrapolate c(γ)=χzz/M to γ→0. These coefficients are empirical but do not enter the analytic master formula; they affect only the numerical comparison.
axioms (5)
  • domain assumption Scaling limit of the γ-perturbed XXZ/XYZ chain is sine-Gordon with β²=1−arccos(∆z)/π and ξsG=β²/(1−β²).
    Used throughout Eqs. (5)–(8) and in the tangent law; nonperturbative field-theory description of the lattice model.
  • domain assumption The breaking vertex e^{iβϕ} carries charge q=2 and coupling µ=γC; C and kink mass M come from the Baxter/JKM exact solution (Eqs. (9)–(10)).
    External exact integrability results supply the lattice-to-continuum dictionary; central to converting χzz into c(β²).
  • domain assumption Lukyanov's two-soliton and one-breather form factors, with F as the minimal CDD solution, are the correct matrix elements of sinβϕ.
    Eqs. (5)–(6) are assembled from [12,13]; the lower-bound property relies on excluding CDD dressing via the UV growth bound, whose proof is in SM Sec. 2.1.
  • domain assumption For injective MPS with a gapped transfer matrix, V=Var(Q_A) and the replica-separation uniformity V_d=V hold for every integer n≥2.
    Fixes the subleading constant and n-dependence of Eq. (4); proof deferred to SM Secs. 1.1–1.2 and not independently verified here.
  • domain assumption Equation-of-motion/Hellmann–Feynman identity eliminating Gβ in favour of M and µ, equivalent to known sine-Gordon bulk energy.
    Used to evaluate the non-conservation sum rule; cross-checked against the LZ vacuum expectation value [16].

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read the original abstract

The entanglement asymmetry measures how strongly a symmetry is broken inside a subsystem. Analytic results at equilibrium have so far covered free theories and, perturbatively, the critical XXZ chain. We compute the R\'enyi entanglement asymmetries of a large interval in the gapped, $U(1)$-breaking phase of the interacting XYZ chain. The calculation combines three ingredients. A charged-moment identity, which we prove for fermionic Gaussian and for injective matrix-product ground states, ties the asymmetry to the static susceptibility of the broken charge. A non-conservation sum rule then evaluates the susceptibility from sine-Gordon form factors, its two-kink and one-breather channels providing a lower bound on the universal amplitude. The Baxter--Johnson--Krinsky--McCoy solution supplies the kink mass for different couplings. Infinite-system density-matrix renormalization group simulations built on these masses reproduce the master formula.

Figures

Figures reproduced from arXiv: 2607.25625 by Felipe Taha Sant'Ana.

Figure 1
Figure 1. Figure 1: FIG. 1. The universal coefficient [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The R´enyi entanglement asymmetry ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

discussion (0)

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Reference graph

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