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arxiv: 2604.05356 · v1 · submitted 2026-04-07 · ❄️ cond-mat.stat-mech · math-ph· math.MP· quant-ph

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Entanglement in the open XX chain: R\'enyi oscillations, hard-edge crossover, and symmetry resolution

Miguel Tierz
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Pith reviewed 2026-05-10 19:36 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MPquant-ph
keywords Rényi entanglement entropyopen XX chainHankel determinanthard-edge asymptoticssymmetry resolutionToeplitz-plus-Hankelspin chain entanglement
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The pith

Closed-form asymptotics for Rényi entanglement entropies of the open XX chain follow from mapping boundary correlations to a Hankel determinant.

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

The paper establishes explicit asymptotic formulas for the Rényi entanglement entropies in the open XX spin-1/2 chain. It achieves this by converting the determinant of the boundary correlation matrix into a Hankel determinant with a positive weight function. Large-size behavior then follows from known Riemann-Hilbert results, with the Szegő function supplying the leading oscillatory amplitude and phase. A single scaling variable organizes the crossover as the Fermi momentum nears the band edge, and the framework recovers the equipartition offset and halved width for symmetry-resolved entropies under open boundaries.

Core claim

We derive closed-form asymptotic formulas for the Rényi entanglement entropies of the open XX spin-1/2 chain by mapping the underlying determinant of the boundary correlation matrix (which has Toeplitz-plus-Hankel structure) to a Hankel determinant with a positive weight whose large-size asymptotics follow from known Riemann-Hilbert results. An explicit evaluation of the Szegő function yields the leading 2k_F oscillatory amplitude and phase. A single variable s = 2ℓ sin(k_F/2) organizes the hard-edge crossover as the Fermi momentum approaches the band edge.

What carries the argument

The mapping of the Toeplitz-plus-Hankel determinant of the boundary correlation matrix to a Hankel determinant with positive weight, whose asymptotics are given by Riemann-Hilbert analysis.

Load-bearing premise

The weight function obtained from the open XX chain boundary correlations must meet every technical condition of the Riemann-Hilbert asymptotic theorems without extra corrections that would change the leading oscillatory amplitude or hard-edge crossover.

What would settle it

Numerical evaluation of the Rényi entropy determinant for large but finite ℓ near the band edge, testing whether the oscillation amplitude follows the predicted s to the power of plus or minus one over alpha.

Figures

Figures reproduced from arXiv: 2604.05356 by Miguel Tierz.

Figure 1
Figure 1. Figure 1: Interval geometries used throughout. (a) Boundary block A = [1, ℓ]. (b) Detached block A = [ℓ0 + 1, ℓ0 + ℓ] at distance ℓ0 from the open end. The hard-edge scaling variable is s = 2ℓ sin(kF /2); the detached-block geometry enters through x = ℓ 2/(2ℓ0 + ℓ) 2 . The paper is organized as follows. Section 2 sets up the determinant representation and the even-symbol map to a Hankel determinant with a positive w… view at source ↗
Figure 2
Figure 2. Figure 2: Deift–Its–Krasovsky even-symbol map from the unit circle to the real interval. The occupied arc θ ∈ [−kF , kF ] on z = e iθ maps to x = cos θ ∈ [−1, 1] with a single internal jump at x0 = cos kF . Under λ = − coth σ with σ > 0, the jump ratio becomes e 2σ > 0 and the weight acquires a definite sign, yielding (after an overall flip) a positive Hankel weight on [−1, 1]. where the prefactor 2 ℓ 2 /πℓ is a nor… view at source ↗
Figure 3
Figure 3. Figure 3: Detached block: ratio of the fixed-kF oscillation amplitude Aα(kF ; x)/A(OBC) α (kF ) versus x = ℓ 2/(2ℓ0 + ℓ) 2 for α = 1, extracted as a median over a midrange of fillings. At fixed kF in the interior of the band, the oscillatory coefficient is numerically consistent with the factorized form Aα(kF ; x) ≈ A (OBC) α (kF )|B(x)|, φα(kF ; x) ≈ φα(kF ) + arg B(x), (18) where B(1) = 1 and B(x) → 0 as x → 0 [P… view at source ↗
Figure 4
Figure 4. Figure 4: Scaling collapse of the smooth backbone Fα(s) for α = 1, boundary block A = [1, ℓ]. We plot ∆Sα ≡ Sα − 1 12 (1 + 1/α)ln s against s = 2ℓ sin(kF /2) on a linear scale. Thin transparent lines: raw numerical data (2kF oscillations visible, largest at small s). Thick lines: demodulated backbone obtained by Savitzky–Golay filtering. The ℓ = 512 and 1024 backbones collapse onto a common curve, supporting the mat… view at source ↗
Figure 5
Figure 5. Figure 5: Oscillation envelope Aα(s) for α = 1, extracted by local cosine/sine demodulation. Dots: raw local-fit amplitude at each evaluation point; solid curves: lightly smoothed result. Dashed guide lines show the predicted s ±1/α power laws. Three system sizes ℓ = 512, 1024, 2048 are indistinguishable, confirming the envelope power laws (20) (see also [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left–right edge universality for α = 1 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Charged-moment numerics for the boundary block. (a) Gaussian curvature −∂ 2 ϕ log Zα(ϕ)|ϕ=0 versus ln ℓ for kF = π/3 and α = 1, 2, 3. Solid lines: asymptotic slope K/(2π 2α); dashed: the previously used K/(4π 2α). (b) Oscillation amplitude ratio |osc.(ϕ)|/|osc.(0)| extracted by local demodulation of log Zα(ϕ) at α = 2, kF = π/3, ℓ ≈ 400. The 2kF oscillation amplitude grows by a factor ≈ 5.5 from ϕ = 0 to ϕ… view at source ↗
Figure 8
Figure 8. Figure 8: Hard-edge log fingerprint: (Fα(s) − F(0) α )/s2 versus log s in the overlap regime shows the expected s 2 log s behavior. B Proof sketch for Proposition 1 The charged moments [23, 24] are handled by the same even-symbol map to a Hankel determinant as the neutral ones, with the only new ingredient being the U(1) phase at the internal jump. The steps are as follows. (i) Positive Hankel weight. After the chan… view at source ↗
Figure 9
Figure 9. Figure 9: Normalized oscillation envelopes for α = 1. 10 0 s 5 × 10 2 6 × 10 2 7 × 10 2 (s)/s 1/ Edge plateau, = 2, = 512 (a) Aα(s)/s1/α vs. s for α = 2. 0.12 0.10 0.08 0.06 0.04 0.02 1/s 2 0.170 0.175 0.180 0.185 0.190 0.195 0.200 (2 s)1/ (s) Interior plateau, = 2, = 512 (b) (2s) 1/αAα(s) vs. 1/s2 for α = 2 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalized oscillation envelopes for α = 2. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Normalized oscillation envelopes for α = 1 2 . 10 0 s 10 1 7 × 10 2 8 × 10 2 9 × 10 2 (s)/s 1/ Edge plateau, = 3, = 512 (a) Aα(s)/s1/α vs. s for α = 3. 0.12 0.10 0.08 0.06 0.04 0.02 1/s 2 0.150 0.155 0.160 0.165 0.170 (2 s)1/ (s) Interior plateau, = 3, = 512 (b) (2s) 1/αAα(s) vs. 1/s2 for α = 3 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Normalized oscillation envelopes for α = 3. 0 2 4 6 8 10 12 14 16 18 s = 2 sin(kF/2) 0.1 0.2 0.3 0.4 0.5 0.6 S = S 1 12 (1 + 1/ )ln s Smooth backbone (s), = 2 = 256 = 512 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Edge double-scaling collapse of the smooth part for α = 2. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Oscillation envelope for α = 2. 0 2 4 6 8 10 12 14 16 18 s = 2 sin(kF/2) 0.35 0.40 0.45 0.50 0.55 0.60 0.65 S = S 1 12 (1 + 1/ )ln s Smooth backbone (s), = 0.5 = 256 = 512 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Edge double-scaling collapse of the smooth part for α = 1 2 . 10 0 10 1 s = 2 sin(kF/2) 10 5 10 3 10 1 10 1 10 3 (s) Oscillation envelope, = 0.5 = 256 = 512 s+1/ s 1/ [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Oscillation envelope for α = 1 2 . 25 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Edge double-scaling collapse of the smooth part for α = 3. 10 0 10 1 s = 2 sin(kF/2) 10 2 10 1 (s) Oscillation envelope, = 3 = 256 = 512 s+1/ s 1/ [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Oscillation envelope for α = 3. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
read the original abstract

We derive closed-form asymptotic formulas for the R\'enyi entanglement entropies of the open XX spin-$1/2$ chain by mapping the underlying determinant of the boundary correlation matrix (which has Toeplitz-plus-Hankel structure) to a Hankel determinant with a positive weight whose large-size asymptotics follow from known Riemann--Hilbert results. An explicit evaluation of the Szeg\H{o} function yields the leading $2k_F$ oscillatory amplitude and phase. A single variable $s = 2\ell \sin(k_F/2)$ organizes the hard-edge crossover as the Fermi momentum approaches the band edge: the oscillation envelope obeys $s^{\pm1/\alpha}$ power laws and $\ln s$ is the natural leading logarithm for a clean data collapse. For detached blocks the oscillatory amplitude is numerically consistent with a factorization through the conformal cross-ratio. The same framework recovers the open-boundary-condition (OBC) equipartition offset $-\tfrac{1}{2}\log\log\ell$ for symmetry-resolved entropies, together with the known halving of the Gaussian width relative to the periodic chain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper derives closed-form asymptotic formulas for the Rényi entanglement entropies of the open XX spin-1/2 chain. It maps the determinant of the boundary correlation matrix (with Toeplitz-plus-Hankel structure) to a Hankel determinant with a positive weight, then invokes known Riemann-Hilbert asymptotics to obtain the leading 2k_F oscillatory amplitude and phase via the Szegő function. A single scaling variable s = 2ℓ sin(k_F/2) organizes the hard-edge crossover with s^{±1/α} envelopes, and the framework recovers the OBC symmetry-resolved offset -½ log log ℓ together with the halved Gaussian width.

Significance. If the mapped weight satisfies the full set of technical hypotheses required by the cited Riemann-Hilbert theorems, the results supply explicit, parameter-free expressions for the oscillatory and crossover contributions to Rényi entropies in open chains. The compact organization via s and the recovery of known OBC symmetry-resolution features would constitute a useful advance in the exact-asymptotics literature for integrable spin chains.

major comments (1)
  1. [Mapping to Hankel determinant (near Eq. (weight definition))] The central mapping from the Toeplitz-plus-Hankel boundary-correlation determinant to a Hankel determinant with positive weight is asserted in the abstract and used to invoke the Riemann-Hilbert asymptotics, but the manuscript provides no explicit verification that the extracted weight satisfies positivity, analyticity, absence of zeros in the support, and the precise endpoint regularity conditions required by the referenced RH theorems. Without this check, it remains possible that model-specific subleading corrections modify the claimed leading oscillatory amplitude or the s^{±1/α} crossover envelopes.
minor comments (1)
  1. [Abstract and introduction] The abstract states that the weight is positive but does not display the explicit functional form; including the weight function (or its defining integral representation) in the main text would allow readers to confirm the technical conditions directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of our results, and recommendation for major revision. We address the single major comment below and will implement the necessary changes in the revised manuscript.

read point-by-point responses

  1. Referee: [Mapping to Hankel determinant (near Eq. (weight definition))] The central mapping from the Toeplitz-plus-Hankel boundary-correlation determinant to a Hankel determinant with positive weight is asserted in the abstract and used to invoke the Riemann-Hilbert asymptotics, but the manuscript provides no explicit verification that the extracted weight satisfies positivity, analyticity, absence of zeros in the support, and the precise endpoint regularity conditions required by the referenced RH theorems. Without this check, it remains possible that model-specific subleading corrections modify the claimed leading oscillatory amplitude or the s^{±1/α} crossover envelopes.

    Authors: We agree that the manuscript asserts the mapping without a self-contained verification of the technical hypotheses required by the cited Riemann-Hilbert theorems. In the revised version we will add an appendix that (i) explicitly constructs the weight w(x) from the boundary correlation matrix of the open XX chain, (ii) proves positivity by direct inspection of the resulting expression (which is a positive multiple of a squared modulus arising from the Jordan-Wigner fermions), (iii) establishes analyticity in a neighborhood of the support together with the absence of zeros on the support, and (iv) confirms the endpoint regularity (square-root vanishing or milder) matches the hypotheses of the referenced theorems. With these conditions verified, the Szegő-function evaluation of the oscillatory amplitude and the s^{±1/α} hard-edge envelopes remain the leading asymptotics; no model-specific subleading corrections alter them. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external RH asymptotics to mapped determinant

full rationale

The paper obtains its closed-form Rényi entropy asymptotics by first mapping the boundary correlation matrix determinant (Toeplitz-plus-Hankel) to a Hankel determinant with positive weight, then invoking established Riemann-Hilbert results for the large-N behavior of such determinants, including the Szegő function that supplies the explicit 2k_F amplitude. The organizing variable s = 2ℓ sin(k_F/2), the s^{±1/α} envelopes, the ln s collapse, the OBC symmetry-resolved offset, and the factorization for detached blocks all follow from applying those external theorems to the mapped weight. No step reduces the claimed formulas to parameters fitted from the same data, to a self-citation chain, or to an ansatz defined in terms of the target result. The derivation therefore remains independent of its own outputs and rests on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the determinant mapping and the applicability of known Riemann-Hilbert asymptotics for Hankel determinants; no free parameters or new entities are introduced.

axioms (1)
  • standard math Large-N asymptotics of Hankel determinants with positive weight are given by Riemann-Hilbert theory
    Invoked directly for the leading behavior after the mapping from the Toeplitz-plus-Hankel correlation matrix.

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