Entanglement asymmetry in periodically driven quantum systems

K. Sengupta; Suchetan Das; Tista Banerjee

REVIEW 2 major objections 5 minor 9 cited by

Entanglement asymmetry in periodically driven quantum systems

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Pith's one-line read Periodically driven quantum systems can restore broken symmetry at special drive frequencies and show a quantum Mpemba effect; in driven conformal field theories the entanglement asymmetry grows logarithmically in the heating phase.

desk verdict First real study of entanglement asymmetry in Floquet systems; the spin-chain results are solid, but the CFT scaling laws rest on an explicitly assumed and unproven branch-independence that needs to be closed or tested. read the letter →

arxiv 2412.03654 v3 pith:XQXHLK4K submitted 2024-12-04 quant-ph cond-mat.str-elhep-th

Tista Banerjee , Suchetan Das , K. Sengupta This is my paper · ORCID

classification quant-phcond-mat.str-elhep-th PACS 03.65.Ud11.25.Hf

keywords entanglementasymmetryperiodicallydrivenquantumsystemsFloquetHamiltonianemergentsymmetryMpembaeffectconformalfieldtheoryXYspinchainRydbergatom

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 asks how the entanglement asymmetry $\Delta S_n$ — a measure of how much a state breaks a symmetry, computed from the difference between projected and unprojected R\'enyi entropies — evolves when a quantum system is driven periodically. The authors show that in driven spin chains, both integrable (XY) and constrained non-integrable (Rydberg/PXP), special drive frequencies make the first-order Floquet Hamiltonian conserve the symmetry charge, so the asymmetry decays to zero and does so faster for initially more asymmetric states: the quantum Mpemba effect. For a driven conformal field theory on a strip, they derive an exact expression and find that long-time behavior is set by the Floquet phase: $\Delta S_n$ grows as $\tfrac12\ln(mT)$ in the heating phase, as $\tfrac12\ln(\ln mT)$ on the critical line, and merely oscillates in the non-heating phase. The paper thus extends symmetry-restoration and Mpemba physics from quenches to periodic drives and gives analytically controlled predictions for a field-theory setting.

What carries the argument

The argument runs on two mechanisms. For the spin chains the load-bearing piece is first-order Floquet perturbation theory in the large-drive-amplitude regime: because the zeroth-order evolution over one cycle is the identity, the leading effective Hamiltonian is $H_F^{(1)}=\frac1T\int_0^T dt\, U_0^\dagger(t)(H(t)-H_0)U_0(t)$, and at $g_1/(\hbar\omega_D)=p$ (XY) or $\Delta_1T/(2\hbar)=p\pi$ (Rydberg) this object conserves the relevant charge. The asymmetry is then extracted from the fermionic correlation matrix (XY) or from exact diagonalization (Rydberg), with the quasiparticle formula $\Delta S_2(mT)=-\int\frac{d\alpha}{2\pi}\exp[(A_2(\alpha)+B_2(\alpha,mT))\ell]$ reproducing the decay at the resonance. For the CFT, the central object is the composite twist operator with vertex operators at the endpoints of the interval; the drive is encoded as an $SU(1,1)$ M\"obius map $\zeta\mapsto (a_m\zeta+b_m)/(c_m\zeta+d_m)$, and the key identity fixes $\beta_m=(2L/\pi)^2|X_m||Y_m|/(\mathrm{Im}\sqrt{X_mY_m^*})^2$, whose growth rate determines $\Delta S_n\simeq\tfrac12\ln(c_0\ln\beta_m/n^2)$ through Eq. (45).

What would settle it

A direct test is to compute $\Delta S_n(mT)$ for a square-pulse-driven CFT on a strip using each branch of $\sqrt{\zeta_m}-\sqrt{\zeta_m^*}$ and check that their large-$m$ limits coincide up to subleading terms; a branch-dependent answer would invalidate Eq. (45). On the spin-chain side, at $g_1/(\hbar\omega_D)=p$ with large $g_1$, one can measure $\Delta S_2(mT)$ on the prethermal plateau: if it does not decay to zero, or if the initially more asymmetric state does not relax faster than the less asymmetric one, the emergent-symmetry mechanism is refuted.

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Extended reading notes

Core claim

The central claim is that periodic driving can generate an approximate emergent symmetry in the Floquet Hamiltonian — a symmetry possessed by neither the undriven Hamiltonian nor the initial state — and that this symmetry dictates how entanglement asymmetry $\Delta S_n$ evolves. For the square-pulse-driven XY chain, when $g_1/(\hbar\omega_D)=p$ with integer $p$, the leading Floquet term $H_F^{(1)}$ commutes with $\tau_z$, so $\Delta S_2(mT)\to 0$ at late stroboscopic times; higher-order terms break this symmetry only at order $1/(\hbar\omega_D)$, so the restoration survives over long prethermal timescales, and initial states with more symmetry breaking relax faster, the quantum Mpemba effect. Exact diagonalization of the driven Rydberg (PXP) chain shows the same behavior at $\Delta_1 T/(2\hbar)=p\pi$, again traced to $[H_F^{(1)},\sigma^z_j]=0$. For a driven CFT on a strip, the paper derives an exact expression for $\Delta S_n(mT)$ from a composite twist operator with vertex insertions at the interval endpoints; the stroboscopic drive acts as an $SU(1,1)$ M\"obius transformation, and the asymmetry reads $\Delta S_n\simeq \tfrac12\ln\gamma_m$ with $\gamma_m=c_0\ln\beta_m/n^2$. In the heating phase $\beta_m$ grows exponentially in $mT$, giving $\Delta S_n\sim \ln(mT)$; on the critical line $\beta_m\sim m^4$, giving $\Delta S_n\sim \ln(\ln mT)$; in the non-heating phase the coefficients remain oscillatory and $\Delta S_n$ performs small-amplitude oscillations about its initial value.

Load-bearing premise

The CFT derivation assumes that, for the square-pulse protocol, both branches of the multivalued factor $(\sqrt{\zeta_m}-\sqrt{\zeta_m^*})^{-2h_j}$ give the same answer in the large-$m$ limit; if the branch choice mattered, the single-valued expression for $\beta_m$ and the resulting $\ln(mT)$ and $\ln(\ln mT)$ laws would not be justified.

Editorial extensions

If this is right

At drive frequencies where the first-order Floquet Hamiltonian conserves the charge, driven XY and Rydberg chains show dynamical symmetry restoration, with $\Delta S_2\to 0$ over prethermal timescales.
At those same frequencies the relaxation is Mpemba-like: a state with a larger initial symmetry breaking decays to the symmetric state faster than one with smaller breaking.
Away from the special frequencies the spin-chain asymmetry does not vanish; for the XY chain it saturates to a finite value, and for the Rydberg chain it shows Rabi-frequency oscillations about a finite value.
For driven CFTs, the behavior of $\Delta S_n$ distinguishes the Floquet phases: logarithmic growth in the heating phase, $\ln\ln mT$ growth on the critical line, and persistent small oscillations in the non-heating phase, so the asymmetry is a phase diagnostic.
The equilibrium strip result $\Delta S_n\simeq\tfrac12\ln\ln(\ell/a)$ is recovered in the undriven limit and agrees with the boundary-CFT scaling found for intervals on a semi-infinite line.

Reading between the lines

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

Editorial inference: the logarithmic-versus-log-log crossover of $\Delta S_n$ in the driven CFT gives an experimentally accessible marker of the critical line, since symmetry-resolved R\'enyi entropies can be measured in cold-atom and trapped-ion simulators.
Editorial inference: the resonance condition $g_1/(\hbar\omega_D)=p$ is not special to the XY model; any integrable chain whose Floquet evolution has an $SU(2)$ structure should show the same emergent symmetry and Mpemba effect at integer drive ratios.
Editorial inference: because the emergent symmetry is broken by higher-order terms, the spin-chain predictions describe a prethermal transient; at sufficiently late times $\Delta S_2$ should eventually rise again, and the rise time as a function of $1/\omega_D$ is a testable prediction the paper leaves implicit.
Editorial inference: the non-heating phase's oscillating asymmetry suggests periodic driving can also suppress symmetry restoration, offering a Floquet-engineering knob to freeze or release symmetry breaking on demand.

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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

Two things you should know. First, this is the first paper to push entanglement asymmetry into the Floquet setting, and the spin-chain half of it is genuinely good. The driven XY chain shows dynamical symmetry restoration at special frequencies, and the paper nails the mechanism: at g1/(ℏω_D)=p the leading Floquet Hamiltonian develops an emergent approximate τ_z conservation, and they demonstrate this three independent ways — exact free-fermion numerics, first-order Floquet perturbation theory, and a quasiparticle counting formula. The agreement in Fig. 1 is convincing. The Rydberg chain ED is smaller (L=24, ℓ=4) but shows the same physics in a prethermal window, and that is plausible.

Second, the CFT part is elegant but more fragile than the abstract implies. The central results are ΔS ~ ln(mT) in the heating phase and ~ ln(ln mT) on the critical line. These follow from Eq. 45, which comes from Eq. 44, which comes from Eq. 43. And Eq. 43 explicitly depends on a branch choice for (√ζ_m − √ζ*_m)^{-2h_j}. The paper says: 'We shall, in the rest of this work, assume that the latter property holds' — i.e., that both branches give the same result at large m for the square-pulse protocol. That is not shown, and it is load-bearing. As the stress-test note observes, a sign flip in the square root introduces a phase e^{-2πi h_j}; since h_j carries the α-dependent term Δ_j(α)/n^2, that phase lands inside the α-integrals that produce the Gaussian evaluation. Without branch-independence, the functional forms are unsupported. The paper does not provide a numerical check of the CFT prediction either, so the log and log-log growths are unverified predictions rather than established results.

Also minor: 'generic CFT' overstates; the derivation assumes Δ(α) quadratic in α, which covers c=1 free bosons and SU(2)_k WZW, not all CFTs. No code or data is shipped; the free-fermion results are at least reproducible from the formulas, but the CFT appendix would benefit from the missing step.

Overall: I would send this to referees. The XY chain results and the emergent-symmetry/Mpemba mechanism are worth publishing. But the CFT section needs either a proof of the branch claim or an explicit numerical test (e.g., a free-fermion lattice approximation of the driven strip), and the abstract should be narrowed accordingly. This is a conditional accept after revision, not a full accept as is.

Referee Report

2 major / 5 minor

Summary. The paper studies the dynamics of the entanglement asymmetry ΔS_n in three periodically driven systems: an integrable XY spin chain, a non-integrable Rydberg (PXP) chain, and a conformal field theory (CFT) on a strip. For the spin chains, the authors identify special drive frequencies at which the leading-order Floquet Hamiltonian has an emergent approximate symmetry, leading to dynamical symmetry restoration (ΔS_2 → 0) and a quantum Mpemba effect. For the driven CFT, the paper derives the scaling laws ΔS_n ~ ln(mT) in the heating phase and ΔS_n ~ ln(ln mT) on the critical line, with small oscillations in the non-heating phase, and provides a phase diagram. The XY chain results are supported by exact numerics, first-order Floquet perturbation theory, and a quasiparticle picture; the Rydberg chain is treated by exact diagonalization for L=24; the CFT results are analytic up to an explicitly stated branch-independence assumption in Eq. (43).

Significance. If correct, the results give a systematic picture of how periodic driving controls entanglement asymmetry, and they identify a concrete mechanism (an emergent approximate symmetry of the Floquet Hamiltonian) that generates both symmetry restoration and the quantum Mpemba effect in driven integrable and non-integrable models. The CFT analysis is the first to provide an analytic prediction for entanglement asymmetry in driven CFTs, with distinct logarithmic and log-log growth laws in the heating and critical phases. The paper also contains a useful cross-check of three independent methods for the XY chain, which is a strength. However, the CFT scaling laws rest on an unproven branch-independence assumption, and the XY-chain Mpemba demonstration is not presented in the standard fixed-Hamiltonian form, so the significance is conditional on these points being resolved.

major comments (2)

[§4.2.1, after Eq. (43)] The branch-independence assumption is load-bearing but unproven. The text states, after Eq. (43), that for the square-pulse protocol both branch choices of (√ζ_m − √ζ*_m)^{−2h_j} give identical results in the large m limit, and then says: 'We shall, in the rest of this work, assume that the latter property holds.' No derivation or numerical check is provided. For the square-pulse protocol the coefficients a_m, b_m, c_m, d_m are complex (Eqs. 54–56), so ζ_m is complex and flipping the sign of √ζ_m multiplies the correlator by e^{−2πi h_j}. Since h_j contains the α-dependent term Δ_j(α)/n^2, this phase is not a constant factor across the α-integrals in Eq. (33) and could affect the Gaussian evaluation that produces Eq. (45). The scaling laws ΔS_n ~ ln m and ΔS_n ~ ln ln m follow from the m-dependence of β_m in Eq. (44), which in turn relies on this branch choice. The authors should either prove the branch independence for the square-pulse protocol or verify numerically that the two branches yield the same result for the integral in Eq. (33) in the large-m limit; without this, the central CFT results are not rigorously established.
[§2, Fig. 1(a)] The demonstration of the quantum Mpemba effect in the driven XY chain compares curves with different g0 and κ0 values (green vs. blue symbols). These differences change not only the initial state but also the Hamiltonian, since g0 appears in the drive protocol and in the Floquet Hamiltonian. The quantum Mpemba effect is normally defined for a fixed Hamiltonian with initial states of different symmetry breaking. The crossing in Fig. 1(a) could therefore reflect the different Floquet dynamics rather than the initial-state effect. The authors should either show the crossing for different initial states at fixed drive parameters (for example, by varying κ0 or using the initial-state parameters of Eq. 15 while keeping g0, g1, and ω_D fixed), or explicitly justify why the comparison in Fig. 1(a) isolates the Mpemba effect.

minor comments (5)

[§1, Abstract and body] There are several typos: 'it's initial value' should be 'its initial value' (appears in the abstract and in §5); 'CFTS' should be 'CFTs'; 'Renyi' should be 'Rényi' in most places.
[§2, Eq. (12)] The definition of θ_k(T) uses arccos, which is multi-valued; the authors should specify the branch (e.g., principal value) to avoid ambiguity in the quasienergy dispersion.
[§3, Fig. 3 and text] The Rydberg-chain results are limited to L=24 and subsystem size ℓ=4. The claim of symmetry restoration over 'exponentially large' prethermal timescales would be stronger if the authors showed data for a second system size or provided an estimate of the prethermal time from higher-order perturbation theory, even though the present evidence is plausible.
[§4.2.3, Eqs. (57)–(65)] The notation in Eq. (57) '2(L/(aπ)^2)' is ambiguous: it should be written as 2(L/(aπ))^2 or 2L^2/(a^2 π^2) to avoid confusion about the powers of a.
[§4.2.1] After Wick rotation to real time, the authors state that 'am = d*_m and bm = c*_m' but do not explicitly show that these relations hold for the square-pulse protocol; it would be helpful to state that this follows from unitarity of U(mT,0).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the CFT branch-independence caveat is an unproven technical assumption, not a self-referential step.

full rationale

The paper's derivation chain is self-contained and non-circular. The XY-chain results are obtained from exact Floquet evolution of the free-fermion correlation matrix (Eqs. 12-20), with no parameter fitted to the target quantity ΔS; the special-frequency symmetry [τz, H_F^(1)] = 0 is derived from first-order Floquet perturbation theory (Eq. 23), and the resulting ΔS→0 and Mpemba crossing are verified by exact numerics (Figs. 1-2), so the prediction is not the input. The Rydberg-chain analysis uses exact diagonalization of the time-dependent Hamiltonian; H_F^(1) from Refs. [69,70] is an explanatory mechanism whose stated assumptions (large Δ1) do not include the target ΔS behavior, and Ref. [38]'s prethermal timescale is auxiliary rather than load-bearing. The CFT result is derived from the composite-twist and SU(1,1) machinery of Refs. [21,71]; the equilibrium strip result (Eq. 40) independently matches Ref. [24], and the driven scaling laws (Eqs. 63 and 65) follow algebraically from the exact Floquet coefficients. The only flagged weakness is the explicit assumption after Eq. 43 that the two branches of (√ζ_m − √ζ*_m)^{-2h_j} agree in the large-m limit for the square-pulse protocol; that is an unproven technical assumption, not a circular reduction, since the claimed scalings are consequences of Eq. 44 rather than inputs to it. Self-citations supply standard Floquet Hamiltonians and prethermal results, but they do not assume the ΔS behavior the paper derives.

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

The central claims rest on standard many-body methods (free-fermion correlators, ED) and standard CFT technology (twist operators, boundary states, SU(1,1) Floquet maps). No new free parameters are fitted; drive amplitudes, frequencies, and initial-state angles are physical choices. The main non-standard assumptions are the prethermal dominance of H^(1)_F, the quadratic form of the flux dimension, and the branch-choice assumption in the driven CFT derivation.

assumptions (4)

domain assumption The composite twist operator with flux insertions transforms as a primary operator under the driven SU(1,1) conformal transformation.
Used in Eq. 43 to obtain the stress tensor expectation value for the driven CFT; standard in Floquet CFT literature (Ref [71]).
domain assumption The conformal dimension of the flux-insertion operator is quadratic in the flux parameter: Delta(alpha) = c0 alpha^2 for U(1)-symmetric CFTs.
Used to evaluate the alpha-integrals in Eqs. 39 and 45; holds for free boson and WZW models, not for a generic CFT.
ad hoc to paper For the square-pulse protocol, the two branch choices of (sqrt(zeta_m) - sqrt(zeta*_m))^{-2 h_j} give identical results in the large m limit.
Stated in Sec. 4.2.1 after Eq. 43; the authors assume this property holds for their protocol.
domain assumption Higher-order terms in the Floquet Hamiltonian are negligible over the observation timescales, so the emergent symmetry of H^(1)_F controls the dynamics (prethermal assumption).
Underlies the symmetry restoration and Mpemba effect claims in Secs. 2 and 3; supported by exponential prethermal timescale estimates cited from Refs [38,68,69,70].

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Pith. "Pith review of Entanglement asymmetry in periodically driven quantum systems." pith.science (2026). https://pith.science/paper/XQXHLK4K

@misc{pith2026241203654,
  author       = {Pith},
  title        = {Pith review of: Entanglement asymmetry in periodically driven quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQXHLK4K}},
  note         = {Machine review of arXiv:2412.03654}
}

abstract

We study the dynamics of entanglement asymmetry in periodically driven quantum systems. Using a periodically driven XY chain as a model for a driven integrable quantum system, we provide semi-analytic results for the behavior of the dynamics of the entanglement asymmetry, $\Delta S$, as a function of the drive frequency. Our analysis identifies special drive frequencies at which the driven XY chain exhibits dynamic symmetry restoration and displays quantum Mpemba effect over a long timescale; we identify an emergent approximate symmetry in its Floquet Hamiltonian which plays a crucial role for realization of both these phenomena. We follow these results by numerical computation of $\Delta S$ for the non-integrable driven Rydberg atom chain and obtain similar emergent-symmetry-induced symmetry restoration and quantum Mpemba effect in the prethermal regime for such a system. Finally, we provide an exact analytic computation of the entanglement asymmetry for a periodically driven conformal field theory (CFT) on a strip. Such a driven CFT, depending on the drive amplitude and frequency, exhibits two distinct phases, heating and non-heating, that are separated by a critical line. Our results show that for $m$ cycles of a periodic drive with time period $T$, $\Delta S \sim \ln mT$ [$\ln (\ln mT)$] in the heating phase [on the critical line] for a generic CFT; in contrast, in the non-heating phase, $\Delta S$ displays small amplitude oscillations around it's initial value as a function of $mT$. We provide a phase diagram for the behavior of $\Delta S$ for such driven CFTs as a function of the drive frequency and amplitude.

Figures

Figures reproduced from arXiv: 2412.03654 by the authors.

**Figure 1.** (a) Plot of ∆S2 (mT) for ħhωD/J = g1 = 20 as a function of the number of drive cycles m showing symmetry restoration at late times. The green triangles (squares) correspond to results from first-order perturbation theory (exact numerics) for g0 = 0.5 and κ0 = 0.6; the corresponding blue symbols show similar plots for g0 = 1.2 and κ0 = 0.6. The dotted line corresponds to results from the quasiparticle picture using E… view at source ↗

**Figure 2.** (a) Plot of ∆S avg 2 for m2 = 800 and m1 = 700 as a function of ħhωD/J for g0 = 0.5, κ0 = 0.6 and g1 = 20 showing dynamical symmetry restoration at special drive frequencies for which g1 = p(ħhωD/J) with p ∈ Z. The dips in the figure correspond to p = 1 and 2. (b) Plot of ∆S avg 2 as a function of drive frequency g1 with g1 = ħhωD/J. The plot shows the absence of dynamical symmetry restoration at smaller g1 indicati… view at source ↗

**Figure 3.** (a) Plot of ∆S2 (mT) as a function of m for ℓ = 4 and θ = π/5,π/10 for ħhωD/w = 15 and ∆1/w = 20. The plots shows a finite ∆S2 for all m and rapid oscillations in short time scales. The inset shows the frequency of these oscillations as function of ∆; the dotted line corresponds to the Rabi oscillation frequency ω(1) r computed from H (1) F . (b) Similar plots for ħhωD = ∆1 = 20w showing approximate dynamical symmet… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (a) Phase diagram showing the heating (red) and the non-heating (yellow) [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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