Symmetry resolved entanglement entropy after an inhomogeneous quench

Hui-Huang Chen; Jiayu Yin; Ming Zhang; Xin-Liang Zhou

REVIEW 2 major objections 4 minor 54 references

Symmetry resolved entanglement entropy after an inhomogeneous quench

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper derives exact symmetry-resolved entanglement entropy after a uniform-to-SSD quench in a free-fermion CFT: log t growth with a subleading q^2 term that breaks equipartition.

desk verdict First SREE result for the SSD quench, but the advertised O(1/log) equipartition breaking is actually O(1/log^2), and the numerics never test the q-dependence. read the letter →

arxiv 2504.14661 v2 pith:6IRN6SL5 submitted 2025-04-20 hep-th cond-mat.stat-mech

Hui-Huang Chen , Xin-Liang Zhou , Jiayu Yin , Ming Zhang This is my paper · ORCID

classification hep-thcond-mat.stat-mech

keywords symmetry-resolvedentanglemententropysine-squaredeformationinhomogeneousquantumquenchfree-fermionCFTchargedmomentsequipartitionofMöbiusconformalfieldtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies what happens to symmetry-resolved entanglement entropy when a one-dimensional critical free-fermion system, initially in the ground state of the uniform Hamiltonian, is suddenly evolved with the Möbius (or sine-square-deformed, SSD) Hamiltonian. The authors derive a closed-form expression for the charged moments of the reduced density matrix and, from them, the entropy $S_A(q,t)$ in each charge sector. They find that at late times the sector-resolved entropy grows as $\log t$, matching the total entropy, so equipartition holds at leading order; the first sector-dependent correction is proportional to $q^2$ and breaks equipartition at subleading order. Exact free-fermion numerics agree with the formula.

What carries the argument

The load-bearing object is the fluxed twist field $T_{n,\alpha}$, which inserts both the replica permutation and the U(1) phase $e^{i\alpha Q_A}$; diagonalizing the replica twist matrix and bosonizing turns its correlation function into products of vertex-operator two-point functions. The second ingredient is the Möbius conformal map that rewrites evolution under $H_1(\theta)$ as a dilatation in the $\zeta$-plane with scale $\lambda = e^{2\pi\tau/(L\cosh 2\theta)}$. In the SSD limit $\theta\to\infty$ these steps produce the Gaussian charged moments of Eq. (4.23), and the $n$-derivative of $b_n(t)$ at $n=1$, denoted $b(t)$, fixes both the $q$-independent subleading terms and the $q^2$ coefficient in Eq. (4.33).

What would settle it

Compute the coefficient of $q^2$ in $S_A(q,t)-S_A(t)$ numerically at fixed large $t$ and increasing $L$ (for example $L$ from $10^3$ to $10^5$ with $q=1,2$). Equations (4.24)-(4.25) give $b_1(t)+b(t)=$ constant, so within the paper's Gaussian approximation the coefficient should scale as $1/\log^2 L$, not $1/\log L$; measuring the exponent directly separates the two. Independently, evaluating the $\alpha^4$ term in $\Upsilon_n(\alpha)$ and asking whether it shifts the $q^2$ coefficient at that same order would settle whether Eq. (4.33) is the exact asymptotic form or only the Gaussian leading approximation.

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Extended reading notes

Core claim

The central claim is that after the uniform-to-SSD quench, the charged moments $Z_n(\alpha,t)$ are Gaussian in the flux $\alpha$, $$Z_n(\$\alpha$,t) \simeq Z_n(0,t) $e^{{-\alpha^2 b_n(t)/2}}$,$$ with $b_n(t)$ given by a logarithm of the time-dependent geometric factors $r(t),s(t)$ minus the non-universal constant $\gamma(n)$. Fourier transforming this Gaussian and taking the replica limit $n\to 1$ yields $$S_A(q,t) = S_A(t) - \frac{1}{2}\log(2\pi b_1(t)) + \frac{b(t)}{2 b_1(t)} - \frac{b(t)+b_1(t)}{2 b_1(t)^2} $q^{2}$.$$ The leading term grows as $\log t$ and is independent of $q$; the $q^2$ term is the first sector-dependent correction, which the paper reports at order $1/\log L$. The same calculation also explains why revivals are absent: the SSD boundary suppresses quasiparticle reflection, so the entropy never saturates.

Load-bearing premise

The derivation assumes the flux dependence of the non-universal constant $\Upsilon_n(\alpha)$ is fully captured by its $\alpha^2$ term, so the charged moments are exactly Gaussian; if the neglected $\alpha^4$ corrections contribute to the $q^2$ coefficient at the same order in $1/\log L$, the predicted equipartition breaking is not a controlled asymptotic result.

Editorial extensions

If this is right

At late times and for large $L$, every charge sector's entanglement entropy grows as $\log t$ with the same leading coefficient as the total entropy, so there is no saturation and no revival peak.
Equipartition holds at leading order in $1/\log L$; sector dependence first enters through the $q^2$ term, so observables that mix sectors will see the breaking only at subleading order.
The coefficient of the $q^2$ correction is explicitly computable from $r(t)$, $s(t)$, and the non-universal constants $\gamma(1)$ and $\gamma'(1)$, giving a parameter-free prediction for any subsystem length $l$ and time $t$.
The same fluxed twist-field computation works for the whole Möbius family of quenches (finite $\theta$), not only the SSD limit.
The free-fermion numerical method, evolving the correlation matrix as $C(t)=e^{iht}C(0)e^{-iht}$, applies to arbitrary inhomogeneous post-quench Hamiltonians and can benchmark other protocols.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $b_1(t)+b(t)$ is a constant in the paper's formulas, the $q^2$ term in Eq. (4.33) decays as $1/\log^2 L$ rather than $1/\log L$; one can check numerically which scaling the exact free-fermion result follows.
The Gaussian form of the charged moments implies the full counting statistics of the charge $Q_A$ in the subsystem is approximately Gaussian with variance $b_1(t)\sim (4\pi^2)^{-1}\log L$, linking symmetry-resolved entropy directly to charge fluctuations.
The same Gaussian machinery should give the symmetry-resolved Rényi entropies for $n\neq 1$ and charged entanglement negativity after this quench; their $q$-dependence will be controlled by the same $b_n(t)$ and can be derived without new ingredients.
A natural extension is a quench between two different Möbius parameters or a local joining quench governed by the SSD Hamiltonian; the conformal map changes but the calculation structure should survive, with a different non-universal constant.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this is the first symmetry-resolved entanglement entropy (SREE) computation for the uniform-to-SSD (Möbius) quench in a free fermion CFT. The authors use fluxed twist fields plus the known Möbius conformal map, get a closed-form charged moment, and derive an explicit SREE formula with log t growth and a q^2 correction to equipartition. That is a genuine new result, and the t=0 limit correctly reduces to the known equilibrium charged moments, which is a good sanity check.

The soft spots are real but contained. The stress-test note is right: the paper claims the equipartition-breaking term is O(1/log L), but from their own Eqs. (4.24)-(4.25), b_1(t)+b(t) is a time- and L-independent constant. So the q^2 term in (4.33) is suppressed as 1/log^2 L (or 1/log^2 t at late times), not 1/log. This appears in the abstract, Sec. 4.4, and the conclusion. The O(alpha^4) terms dropped in (4.20) are not the culprit; for fixed q they contribute at 1/log^3 after the n->1 derivative. So it is an internal inconsistency in the retained expression, not a neglected correction. The formula itself may still be correct; the claimed order of the symmetry-breaking effect is off by one log.

Two other issues. First, the finite-theta Möbius result is asserted but never shown; the paper jumps straight to the SSD limit after saying the general expression is too complicated to report. That is a gap in derivability, though not fatal. Second, the numerical benchmark only shows q=1. The q^2 term is the whole point, and footnote 6 admits q>=2 needs larger L than they could handle. They also never state which theta their lattice Hamiltonian uses; if they did not actually take the SSD limit in the numerics, the comparison is weaker than claimed.

Who is this for? People working on symmetry-resolved entanglement dynamics and inhomogeneous quenches. The method is a straightforward but useful combination of existing machinery, and the log t growth plus suppressed revivals is consistent with earlier SSD quench results. The central formula (4.33) is probably correct, and the flaws are addressable in revision.

Recommendation: send to peer review. A referee should ask the authors to correct the scaling statement, include the finite-theta expression or a clear derivation, and provide at least some q=0 and q=2 data with a specified lattice Hamiltonian. With those changes it would be a solid paper; as it stands, the advertised main effect is overstated.

Referee Report

2 major / 4 minor

Summary. The manuscript studies the time evolution of symmetry-resolved entanglement entropy (SREE) after a quench from the uniform Hamiltonian to a Möbius/SSD deformed Hamiltonian in a critical free-fermion chain with open boundaries. The authors use the fluxed twist-field method, bosonization, and a sequence of conformal maps to obtain the time-dependent charged moments (Eq. (4.19)), approximate their α-dependence by a Gaussian (Eqs. (4.22)–(4.23)), and derive a closed-form SREE, Eq. (4.33). They claim logarithmic growth S_A(q,t) ∼ log t at late times, leading-order equipartition, and a subleading q² correction that breaks equipartition at O(1/log L). The predictions are benchmarked against exact lattice numerics in Fig. 1.

Significance. If the derivation holds, the paper would provide one of the first exact CFT treatments of symmetry-resolved entanglement in an inhomogeneous quench, with an analytic formula containing no fitted parameters and an independent exact numerical benchmark. The broad physical picture — leading-order equipartition with subleading symmetry dependence — is plausible and consistent with the free-fermion structure. However, the specific order of the equipartition breaking is misidentified in the text, and once that order is corrected the Gaussian truncation leaves uncontrolled terms of the same order in 1/log L. These issues affect the central quantitative claim and require revision before the paper can be accepted.

major comments (2)

[Sec. 4.4 and discussion after Eq. (4.33)] The asserted O(1/log L) equipartition-breaking scaling is inconsistent with the displayed formula. From Eqs. (4.24) and (4.25), b_1(t)+b(t) = −γ(1)−γ′(1) is independent of t and L. The q² term in Eq. (4.33) is therefore C/[2 b_1(t)²] with C = −γ(1)−γ′(1), which is O(1/log² L) at fixed t and O(1/log² t) at late times, not O(1/log L) as stated in Sec. 4.4, in the sentence following Eq. (4.33), and in the Conclusion. The numerical comparison in Fig. 1 plots S_A(q,t) versus t and is unlikely to distinguish 1/log t from 1/log² t; the authors should correct the scaling statements and present data or a scaling analysis that actually tests the corrected order.
[Secs. 4.2–4.3, Eqs. (4.20)–(4.23) and (4.33)] Once the q² term is recognized as O(1/log² L), the Gaussian truncation is not sufficient to determine the SREE at that order. The full Υ_n(α) in Eq. (4.18) contains O(α⁴) terms; writing log Z_n(α) = log Z_n(0) − b_n α²/2 + c_n α⁴ + ⋯ with c_n = O(1), the Fourier transform in Eq. (4.28) generates a q⁴ contribution with coefficient c_n/b_n² = O(1/log² L). After the n → 1 derivative this contributes to S_A(q,t) at the same order in 1/log L as the retained q² term of Eq. (4.33), for fixed q = O(1). Thus Eq. (4.33), truncated at q², is not the complete SREE to the order at which equipartition is broken unless the α⁴ and higher terms are shown to be subleading in the appropriate regime, which the manuscript does not do.

minor comments (4)

[Abstract] The phrase "we derive an exact expression for the SREE" overstates the result: Eq. (4.22) is obtained by expanding Υ_n(α) through order α², and the subsequent SREE formula inherits this approximation.
[Eq. (4.29)] The error-function arguments appear to be mistyped: the Fourier integral over [−π, π] gives arguments (b_n π ± i q)/√(2 b_n), not b_n π ± i q√(2 b_n). The large-b_n asymptotics are unaffected, but the finite-b_n expression should be corrected.
[Throughout] There are several typographical errors, including "Specificly" (Sec. 4), "aprears" (after Eq. (4.33)), "Guassian" (after Eq. (4.23)), "coulumn" (Sec. 5), and "finial" (after Eq. (4.13)).
[Footnote 6] The statement that q ≥ 2 requires larger L would benefit from a quantitative convergence test or an estimate of the required system size, since the q-dependent corrections are the central object of the paper.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the time-dependent SREE follows from a conformal-mapping calculation with equilibrium Fisher-Hartwig constants fixed at t=0, and the final formula is benchmarked against independent exact numerics.

full rationale

The derivation is essentially self-contained in the sense relevant to circularity. The charged moments are obtained by evolving the fluxed twist field under the SSD/Mobius Hamiltonian via a chain of conformal mappings (Sec. 4.1), giving Eq. (4.19). The non-universal constant Υ(n,α) is imported from the equilibrium Fisher-Hartwig result [50] and fixed by matching Eq. (4.16) to Eq. (4.17) at t=0. This is an external calibration, not a fit to the quench data; the time dependence of the final SREE formula Eq. (4.33) enters through r(t) and s(t) derived from the conformal map, and the prediction is then compared with exact free-fermion numerics in Sec. 5. No parameter of the final formula is fitted to the post-quench data. The self-citations [27, 29, 38, 43] appear in the introduction as context and are not load-bearing for the central derivation. We note, without counting it as circularity, that the claimed O(1/log L) scaling of the equipartition-breaking term is not supported by the paper's own definitions: from Eqs. (4.24) and (4.25), b1(t)+b(t) = −γ(1)−γ'(1), which is time-independent, so the q^2 term in Eq. (4.33) is actually O(1/log^2 L). That is a scaling/correctness issue, not a reduction of the prediction to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard CFT solvability of the SSD quench, the Fisher-Hartwig non-universal constant imported from prior literature, and the truncation of the flux expansion to second order. No parameters are fitted to the time-dependent data; the constants come from equilibrium matching. The main structural risk is the uncontrolled O(α⁴) contribution at the order of the claimed subleading correction.

assumptions (4)

domain assumption Time evolution under the Möbius/SSD Hamiltonian acts as a pure dilatation in the ζ-plane (Eq. 4.9).
This solvability property is the basis for evaluating the time-dependent one-point function of the fluxed twist field; it follows from the Virasoro decomposition of H1(θ).
ad hoc to paper The non-universal constant Υ(n,α), fixed at t=0 by matching to the equilibrium charged moments, remains unchanged during the SSD time evolution.
The UV cutoff ε is eliminated using Eq. (4.16)-(4.19); the same Υ(n,α) is used for all t, without independent justification that the inhomogeneous deformation does not alter short-distance regularization.
standard math Fisher-Hartwig conjecture determines the equilibrium charged moments (Eqs. 4.17-4.18).
Imported from [50,52]; used to fix the non-universal constant and the UV cutoff.
ad hoc to paper Dropping O(α^4) terms in Υ_n(α) is controlled at the order to which the SREE is computed.
The Gaussian approximation in Eq. (4.22)-(4.23) requires this. The paper's claimed O(1/log L) order for the q² correction would justify it, but the derived coefficient is O(1/log² L), where O(α^4) terms contribute at the same order.

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Pith. "Pith review of Symmetry resolved entanglement entropy after an inhomogeneous quench." pith.science (2026). https://pith.science/paper/6IRN6SL5

@misc{pith2026250414661,
  author       = {Pith},
  title        = {Pith review of: Symmetry resolved entanglement entropy after an inhomogeneous quench},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6IRN6SL5}},
  note         = {Machine review of arXiv:2504.14661}
}

abstract

We investigate the non-equilibrium time evolution of symmetry-resolved entanglement entropy (SREE) following an inhomogeneous quench in a critical one-dimensional free fermion system. Using conformal field theory, we derive an exact expression for the SREE and analyze its behavior. We find that, at leading order in the long-time limit, the SREE grows logarithmically as $\log t$. While the equipartition of entanglement holds at leading order, we identify subleading corrections that break it. Our numerical simulations corroborate the analytical predictions with excellent agreement.

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