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Time Evolution of the Symmetry Resolved Entanglement Entropy after a Mass Quench

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper shows that after a sudden mass change in the Ising field theory, each Z2 symmetry sector's Rényi entropy grows linearly in time at the same rate as the total entropy, with subleading oscillations.

desk verdict Solid form-factor computation of T_mu after a mass quench; the Renyi-level result is robust, but the abstract overclaims the von Neumann limit that the paper itself cannot take. read the letter →

arxiv 2502.06612 v2 pith:OLPC7APM submitted 2025-02-10 hep-th cond-mat.stat-mechquant-ph

Federico Rottoli , Michele Mazzoni , Fabio Sailis , Olalla A. Castro-Alvaredo This is my paper · ORCID
classification hep-thcond-mat.stat-mechquant-ph
keywords symmetry-resolvedentanglemententropymassquenchIsingfieldtheorybranchpointtwistfieldsformfactorsquasiparticlepictureRényiZ2symmetry
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 computes the post-quench time evolution of the symmetry-resolved entanglement entropy in the Ising field theory after the mass is suddenly changed. Using form factors of the composite branch point twist field, the authors derive an explicit closed-form expression for the normalized one-point function at large times, up to $O((mt)^{-3})$ and second order in the quench strength. The central result is that each Z2 symmetry sector's Rényi entropy grows linearly in time at exactly the same rate as the total entropy, with subleading oscillatory corrections of frequency $2m$. A sympathetic reader cares because this is the first twist-field computation of out-of-equilibrium symmetry-resolved entropy, and it confirms what the quasiparticle picture predicts for the linear growth while exposing corrections that the quasiparticle ansatz cannot see.

What carries the argument

The central object is the composite branch point twist field $T_\mu$, the Z2-charged version of the branch point twist field $T$; its replica exchange relations differ from those of $T$ by a sign, and its two-particle form factors have Pfaffian form with $n$ replaced by $n/2$ in the minimal factor. The calculation uses the boundary-state form of the post-quench state, expands the one-point function in the squeezing kernel $K(\theta)$ to order $K^2$, regularises kinematic singularities with $\kappa$-shifts, and then resums the leading terms into an exponential. The piece that carries the new physics is the ratio $R_n=Z_n(1)/Z_n(0)$, which isolates the oscillatory correction and is the quantity that feeds into the symmetry-resolved entropies. The resummation step borrows the argument proven for the ordinary field $T$ in [42].

What would settle it

Compute the charged moment $Z_n(1)$ directly on the lattice (transverse-field Ising chain) for a sudden mass quench, evaluate the normalized one-point function at large $mt$ and small $\alpha$, and compare the coefficient of $\cos(2mt-\pi/4)/(mt)^{3/2}$ with $-\alpha/(8\sqrt{\pi}\,n\sin^2(\pi/(2n)))$; a discrepancy in that coefficient, or a power-law decay different from $(mt)^{-3/2}$, would falsify the exponentiated formula, while the linear growth rate $n\Gamma/2$ would survive.

Watch

Extended reading notes

Core claim

The paper establishes Eq. (55): the normalized one-point function of the composite branch point twist field $T_\mu$ after a mass quench behaves as $\exp[-n\Gamma mt/2 - n\alpha^2/(64\pi mt) - \alpha\cos(\pi/(2n))/(8\sqrt{\pi}\,n\sin^2(\pi/(2n)))\,\cos(2mt-\pi/4)/(mt)^{3/2}]$, with $\Gamma$ and $\alpha$ fixed by the pre- and post-quench masses. Since the symmetry-resolved partition functions are discrete Fourier transforms of the one-point functions of $T$ and $T_\mu$, this exponential governs both total and charged moments. Consequently the symmetry-resolved Rényi entropies grow linearly in time with the same rate $n\Gamma/2$ as the total entropy, and the symmetry sectors differ only through oscillations of order $(mt)^{-3/2}$ encoded in the ratio $R_n=Z_n(1)/Z_n(0)$. The result provides the first explicit branch point twist field computation of the symmetry-resolved entanglement entropy dynamics and matches the leading quasiparticle prediction, while the oscillatory correction is non-analytic in the post-quench particle density and therefore invisible to the quasiparticle ansatz.

Load-bearing premise

The decisive assumption is that the way infinitely many correction terms collapse into a simple exponential for the uncharged twist field also works for the charged composite field; the paper borrows that collapse from the earlier calculation instead of proving it, and the charged field's form factors behave differently at large rapidities, so if the collapse fails the closed formula is only a low-order approximation.

Editorial extensions

If this is right

  • Each Z2-symmetry sector's Rényi entropy grows linearly in time at the same rate as the total Rényi entropy, so the quench does not distinguish the sectors in its leading growth.
  • The leading nontrivial difference between sectors is an oscillation $\cos(2mt-\pi/4)$ with amplitude decaying as $(mt)^{-3/2}$, whose coefficient depends on $n$ through $\sin^2(\pi/(2n))$.
  • Equipartition between the two symmetry sectors holds up to subleading corrections for small UV cutoff $\varepsilon$ and intermediate $n$, consistent with earlier free-fermion studies.
  • The one-point functions of $T$, $T_\mu$ and the order field $\mu$ all decay at the same exponential rate $\Gamma$, suggesting a common relaxation rate for symmetry fields in the theory.
  • The oscillatory correction is non-analytic in the post-quench occupation density, so the quasiparticle ansatz, which is analytic in that density, cannot reproduce it.

Reading between the lines

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

  • If the exponentiation in Eq. (55) holds to all orders in the low-density expansion, then the whole large-time charged moment is controlled by just two numbers, $\Gamma$ and $\alpha$; a numerical check of the $(mt)^{-3/2}$ amplitude for two different masses would test this stronger statement.
  • The non-commutativity of the limits $n\to 1$ and $\kappa\to -i\eta/2$ noted in the paper suggests that the von Neumann limit should be taken through a finite-size regularisation rather than by analytic continuation in $n$.
  • The half-space geometry probes only the light-cone growth $2mt$; adapting the same form-factor computation to a finite interval of length $\ell$ should produce a linear regime of duration $\ell/(2m)$, which the current one-point function setup cannot see.
  • The matching of the linear term with a small-density expansion of the quasiparticle prediction may fail at higher orders in density, since the first oscillatory correction is non-analytic; this suggests an order-by-order comparison in the density could reveal where the two pictures diverge.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper computes the time-dependent one-point function of the composite branch point twist field T_mu in the Ising field theory after a mass quench, using the form-factor expansion of the post-quench squeezed state. The authors evaluate the O(K) and O(K^2) contributions explicitly with kappa-regularization, obtaining the leading exponential decay, the power-law correction proportional to alpha^2/(mt), and the oscillatory (mt)^{-3/2} term, assembled in the closed form (55). From this they derive the charged moments and the symmetry-resolved Rényi entropies for the Z_2 symmetry sectors, concluding that these entropies grow linearly in time with the same rate Gamma as the total Rényi entropy, with subleading oscillatory corrections, and that the result is consistent with the quasiparticle picture.

Significance. If the central formula (55) and its consequences are valid at the stated orders, the paper provides a valuable explicit QFT computation of symmetry-resolved entanglement dynamics after a global quench. The technical computation is largely self-contained and parameter-free: the linear growth rate Gamma is a computed integral of the quench kernel, not a fitted parameter, and the O(K^2) residue computation is shown in detail. The n=1 reduction to the magnetization result of [43] is a useful cross-check. The quasiparticle comparison in Section 5 correctly identifies which part of the result is and is not captured by the quasiparticle ansatz. However, as discussed in the major comments, the advertised claim about the von Neumann symmetry-resolved entropy is stronger than what the computation actually establishes, and the all-orders exponentiation in (55) is imported from [42] rather than proved for T_mu. These issues are localizable and fixable, but they affect the paper's central message as it currently stands.

major comments (3)
  1. [Abstract and Section 6, with Eqs. (2), (55), (56), (58), (61)] The abstract and title claim that the 'symmetry resolved entropy' grows linearly in time, but the computation only establishes this for the Rényi symmetry-resolved entropy S_n(q) at integer n>1. Equation (2) defines the von Neumann SREE as S(q)=lim_{n->1} S_n(q), and Section 6 explicitly states that 'we could not find a well-defined n->1 limit of our entropies'. The obstruction is visible in Eq. (56): the corrected vacuum expectation value contains A=1/(2 sin(pi/n)) integral K^2, which diverges as n->1, and the footnote after Eq. (29) shows that the limits n->1 and kappa->-i eta/2 do not commute for T_mu. Consequently Eq. (58) and the entropy formulas (61)-(62) rely on an unproven regularisation of the n->1 limit. The authors should either provide a controlled limiting procedure or explicitly restrict the paper's claims to Rényi entropies with integer n>1, revising the abstract and title accordingly.
  2. [Section 4, Eq. (55)] The passage from Eq. (54) to the exponentiated formula (55) is justified only by the sentence 'by employing identical resummation arguments as presented in [42]'. This transfer is not automatic for T_mu: the two-particle form factor has different large-rapidity asymptotics, w(theta)-> +/- i/n (Eq. (19)), and the limits n->1 and kappa->-i eta/2 do not commute for T_mu, as noted after Eq. (29). At O(K^2), where all terms in Eq. (54) have been computed directly, the exponentiated form is credible and the linear growth and oscillatory corrections follow. But the all-orders exponential resummation is an imported assumption. Please either supply the proof, state that Eq. (55) is a conjecture beyond O(K^2), or formulate the main result as an O(K^2) statement only.
  3. [Section 4.1, Eqs. (61)-(62)] The expansion leading to Eq. (62) and the subsequent statement that the symmetry-resolved Rényi entropies grow at the same rate as the total entropy assume that R_n and R_1 are small and that the term R_n/(1-n) is subleading. The paper itself notes that this requires delicate conditions on n and the UV cut-off epsilon. More importantly, Eq. (62) uses Z_1(0)=1 and the corresponding normalisation for Z_1(1), which the authors state they cannot obtain analytically from their formulas ('on physical grounds ... even if we can not show this analytically'). Since this normalisation enters the definition of the SREE in Eq. (61), the quantitative claim for the symmetry-resolved entropies depends on an additional physical input that should be stated as an assumption rather than a derived result.
minor comments (4)
  1. [Section 1, Eq. (4)] The notation 'nxOmega|T(0,t)|Omega_yn' and similar expressions is confusing because the subscript n appears on different objects; please define the n-replica state |Omega>_n once and use a consistent notation for the n-fold copy throughout.
  2. [Footnote after Eq. (29) and Section 4] The footnote states that if one sets n=1 from the start in Eq. (28) one finds A=0, while Section 4 says 'we know from direct computation for mu that A should go to zero'. These two statements should be reconciled, since the former is a statement about a limit of the T_mu computation and the latter is an independent property of the magnetization.
  3. [Section 5, Eq. (65)] The notation Z_n(a) with a=0,1 is used in Eq. (65) before the meaning of the parameter a is explained; please introduce the notation explicitly before the formula and state that Z_n(1) denotes the charged moment with the Z_2 charge insertion.
  4. [References] References [49] and [67] are the same work, and likewise [50] and [68] are the same work; please consolidate duplicate entries in the bibliography.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the T_mu computation is direct and parameter-free; the only self-citation (the [42] resummation) is not load-bearing at the claimed O(K^2) order, and the acknowledged n->1 failure is a limitation rather than a circular reduction.

full rationale

The central result, Eq. (55), is obtained by direct form-factor computation for the composite twist field T_mu. The O(K) contribution is computed in Eq. (23), the O(K^2) connected term in Eq. (47), the residue term producing -n Gamma m t / 2 in Eq. (41), and the disconnected terms in Eqs. (33) and (50). The growth rate Gamma in Eq. (42) is an integral of the quench kernel K, not a fitted parameter, and no quantity appearing in the final formula is calibrated to the data being predicted. The paper does invoke prior work by overlapping authors: the resummation step is imported by the sentence 'by employing identical resummation arguments as presented in [42] we can show that' (Section 4), and the T_mu form factors are cited from [26] and [61], with [26] sharing authors. However, Eq. (55) is explicitly stated only at orders O((mt)^{-3}) and O(K^2). At that order the exponential is just a rearrangement of the O(K^2) terms already computed in Eq. (54): exp(A cos(pi/n)) = 1 + A cos(pi/n) + O(K^4) and exp(C) reproduces the computed 1 + C + C^2/2 + ... combination. The [42] resummation therefore does not force the quantitative O(K^2) result, so the self-citation is minor and not load-bearing. The paper also honestly acknowledges that 'we could not find a well-defined n to 1 limit of our entropies' (Section 6), which is a limitation of the abstract's unqualified 'symmetry resolved entropy' wording but not a circular step. No fitted input is renamed as a prediction, and the quasiparticle comparison in Section 5 provides an independent consistency check of the linear growth rate. Overall the derivation is self-contained at the order claimed; score 2 reflects only the minor self-citation in the resummation transfer.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The ledger is clean: no parameter is fitted to data. The quench strength alpha and the mass ratio are physical inputs; Gamma and A are computed integrals of the kernel K, not fitted constants. The main load-bearing inputs are prior form-factor results and the imported resummation from [42]. No new particles, forces, or fields are introduced; T_mu is an existing composite twist field from the prior literature.

assumptions (6)
  • domain assumption The post-quench initial state has the boundary-state form (5) with the kernel K(theta) of Eq. (6), taken from [54,55].
    All subsequent form-factor integrals use this representation of the quench state; if K(theta) were incorrect, the growth rate and oscillatory corrections would change. The derivation is cited, not repeated.
  • domain assumption The form factors of the composite twist field T_mu are given by the Pfaffian formula (17) with the two-particle kernel w(theta) of Eq. (18), taken from [61] and [26].
    The paper relies on the prior form-factor bootstrap for T_mu, including its different large-rapidity asymptotics, without re-deriving it.
  • standard math kappa-regularization, shifting coincident rapidities by kappa and then taking kappa to 0, gives the correct finite result after cancellation of kinematic-pole divergences.
    This is a standard regularization in this literature, and the paper explicitly shows the pole cancellations between Eqs. (29) and (41), but it is still a regularization assumption.
  • domain assumption The all-orders exponential resummation shown for T in [42] also holds for T_mu, so the low-order series (54) exponentiates into Eq. (55).
    The paper imports 'identical resummation arguments' from [42] without proving the transfer to the different form factors of T_mu. This is a load-bearing premise for the closed-form result.
  • ad hoc to paper The physical normalization at n = 1 can be imposed by setting tau'_1 = 1 and Z_1(0) = 1, even though the analytic continuation is not obtained from the formulas.
    The authors state that they cannot show these values analytically from their formulas, but they use such normalizations to define R_1 and the entropies. This is an admitted ad hoc step.
  • standard math Stationary-phase evaluation of the large-mt oscillatory integrals is valid, including the double time-derivative step for R(t) in Section 3.2.
    The asymptotic (mt)^-3/2 and (mt)^-1 corrections are obtained by expanding around theta = 0 and dropping higher orders; this is standard asymptotic analysis but controls the claimed subleading terms.

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Pith. "Pith review of Time Evolution of the Symmetry Resolved Entanglement Entropy after a Mass Quench." pith.science (2026). https://pith.science/paper/OLPC7APM

@misc{pith2026250206612,
  author       = {Pith},
  title        = {Pith review of: Time Evolution of the Symmetry Resolved Entanglement Entropy after a Mass Quench},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLPC7APM}},
  note         = {Machine review of arXiv:2502.06612}
}
read the original abstract

In this paper we investigate the properties of the symmetry resolved entanglement entropy after a mass quench in the Ising field theory. Since the theory is free and the post-quench state known explicitly, the one-point function of the relevant (composite) branch point twist field can be computed using form factor techniques, similar to previous work on the branch point twist field and the magnetisation, respectively. We find that the symmetry resolved entropy grows linearly in time at the same rate as the total entropy, and that there are sub-leading oscillatory corrections. This result provides the first explicit computation of the out-of-equilibrium dynamics of the symmetry resolved entropy employing twist fields in quantum field theory and is consistent with existing results based on the quasiparticle picture.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Toy Model for Topological Entanglement Features in 1+1D Integrable Quantum Field Theory

    hep-th 2026-07 conditional novelty 6.0 of 10

    In the Federbush model, branch-point twist field form factors and first-order quench corrections are independent of the topological coupling λ, so Rényi entropies of the infinite-volume vacuum match two free Dirac fermions.

Reference graph

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