Information scrambling and entanglement dynamics in Floquet Time Crystals

Fernando Iemini; Himanshu Sahu

REVIEW 6 major objections 6 minor 105 references

Information scrambling and entanglement dynamics in Floquet Time Crystals

T0 review · 6 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In Floquet time crystals, information scrambling is frozen first, then grows logarithmically along a protected direction.

desk verdict Plausible numerics for a new direction-dependent scrambling signature in Floquet time crystals, but the headline envelope rests on L=8 alone and the appendix has fixable errors. read the letter →

arxiv 2411.13469 v2 pith:X6NBMLH6 submitted 2024-11-20 quant-ph cond-mat.dis-nncond-mat.stat-mech

Himanshu Sahu , Fernando Iemini This is my paper · ORCID

classification quant-phcond-mat.dis-nncond-mat.stat-mech

keywords out-of-time-orderedcorrelatorsFloquettimecrystalmany-bodylocalizationℓ-bitentanglemententropyinformationscramblingdisorderedspinchainperioddoubling

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 quantum information spreads in a disordered spin chain that forms a Floquet time crystal, a phase that oscillates with twice the period of its driving. The authors compute out-of-time-ordered correlators and entanglement entropy and find that the time crystal has a preferred 'quasi-protected' direction along which scrambling is unusually slow. Along that direction, the correlator stays frozen while the period-doubling order is stable, then grows logarithmically as the order melts, and finally all spin distances collapse onto a single distance-independent growth curve. Along the orthogonal direction, scrambling looks like that of a usual many-body localized system. If true, the results show that time-crystalline order leaves a fingerprint on how quantum information gets scrambled, and that the melting of the time crystal can be read off from the merging of OTOC curves.

What carries the argument

The argument is carried by the $\ell$-bit phenomenological picture of many-body localization, applied to a Floquet system. The local spin operators are expanded in a basis of quasi-local integrals of motion $\hat\tau_j^\alpha$ with exponentially decaying coefficients; the preferred z-direction is the one along which the period-doubling magnetization is protected. The OTOC, defined as the squared commutator $C_{\alpha\beta}(\ell,t)=\frac12\langle[\hat\sigma^\alpha_\ell(t),\hat\sigma^\beta_1]^2\rangle$, measures how much of the operator leaks into orthogonal directions. The timescales $T_1$, $T_2$, $T_3$ — set by the exponential stability of the period-doubling order — organize the different behaviors. In the exactly solvable limit of perfect spin flips and no transverse field, the OTOCs vanish identically, which serves as the reference point from which the slow scrambling develops.

What would settle it

Measure the OTOC curves $C_{zz}(\ell,t)$ at system sizes beyond $L=10$: if the late-time growth rates for different distances do not collapse onto one envelope, or if the period-doubling window $T_2$ does not grow exponentially with $L$, the central claim collapses.

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Extended reading notes

Core claim

The paper's central claim is that in a Floquet time crystal stabilized by many-body localization, the out-of-time-ordered correlator $C_{zz}(\ell,t)$ along the quasi-protected $\ell$-bit direction evolves through three distinct regimes. For times $t<T_1$ it undergoes the same initial relaxation seen in all phases. During the stable period-doubling window $T_1<t<T_2$, the correlator is essentially frozen at small, distance-dependent plateaus, because a conserved piece of the $\hat\sigma^z$ operator overlaps with the local integrals of motion. In the decoherence regime $T_2<t<T_3$, the correlator grows logarithmically, $C_{zz}(\ell,t)\sim e^{-b_\ell \ell} + c \log(t)^d$, with the logarithmic term independent of distance once the correlation wavefront has traversed the chain. In the late thermal regime, all distances merge into a single envelope and saturate. The entanglement entropy grows logarithmically throughout and saturates to a thermal volume law at exponentially long times.

Load-bearing premise

The central claim rests on the assumption that the Floquet time crystal at the simulated parameters is properly described by a complete set of local integrals of motion — the $\ell$-bits — with a preferred z-direction, and that the period-doubling time grows exponentially with system size; the numerics only reach $L=8$ and $L=10$, so this exponential scaling and the $\ell$-bit structure are inherited from prior MBL phenomenology rather than demonstrated here.

Editorial extensions

If this is right

The protected $\ell$-bit direction makes scrambling strongly anisotropic: $C_{xx}$ can be orders of magnitude larger than $C_{zz}$ while the magnetization still exhibits period doubling.
The logarithmic growth of entanglement entropy persists over the entire prethermal window, so the time crystal is 'slow' not only in its magnetization but also in how it builds up entanglement.
The late-time distance independence of $C_{zz}$ means that once the wavefront has crossed the chain, the scrambling rate is set by the decoherence process itself, not by the distance between probes.
OTOC measurements on existing superconducting-qubit platforms that realize Floquet time crystals could observe the frozen plateau and the later envelope merge directly.

Reading between the lines

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

The collapse of all $C_{zz}(\ell,t)$ curves onto one envelope could serve as a practical diagnostic for the onset of time-crystal melting, since the merging time should correlate with $T_2$.
The anisotropy in scrambling suggests that other quantities, such as operator spreading widths or Krylov complexity, will also be direction-dependent inside the FTC; the paper does not compute those.
A natural testable extension is to drive the system closer to the ergodic side of the transition, where the prediction would be that the envelope merging disappears and is replaced by the usual ballistic light cone.
The initial-state-independence claim could be checked against tilted or N\'eel initial states; if logarithmic growth rates vary with the tilt angle, the $\ell$-bit description may need refinement.

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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

Colleague,

The new content here is the direction-dependent OTOC phenomenology in an MBL-stabilized Floquet time crystal: along the z/ℓ-bit direction the OTOC freezes during the period-doubling window, then grows logarithmically as the magnetization decoheres, and in the late regime all distances merge into a single distance-independent envelope. That envelope is the paper's most interesting claim and also its most fragile one.

The paper does several things well. The model is a standard kicked disordered Ising chain, the ED setup is clear, and the qualitative distinction between Cxx and Czz is convincing in the figures. The single-realization panels showing plateaus with quadratic growth are a nice touch. The ℓ-bit interpretation is reasonable and consistent with the prior MBL literature; the analytic appendix for the fine-tuned case is a useful sanity check.

The soft spots are real but mostly fixable. First, the envelope claim is demonstrated only for L=8; there are no L=10 OTOC curves for Czz, even though the time-scale hierarchy is fitted using L=8 and L=10. At L=8 all distances are comparable to the system size, so the collapse of the increments could partly reflect trivial convergence to the common thermal maximum. This needs a finite-size check before the abstract's headline survives. Second, the appendix has internal inconsistencies: it sets ϕ=π/2 but uses U_K^† σ^z U_K = −σ^z, which holds for ϕ=π, and the disorder average in Eq. (A7) does not follow from averaging cos(4J0 t) over J0 ∈ [J/2, 3J/2] — I get 1 − [sin(6Jt) − sin(2Jt)]/(4Jt). These are errors, not just typos in notation. Third, the claim that results are independent of the initial state is asserted without systematic checks. Fourth, the fitted growth exponents have no error bars; for a numerical phenomenology paper that is acceptable, but the precision should be stated.

The circularity concern is milder than the reader's note suggests. The ℓ-bit picture is a standard framework, and the OTOC is a different observable from the magnetization; using the magnetization to identify the protected direction and then interpreting the OTOC in that basis is a reasonable physics step. Still, the paper could acknowledge the dependence more explicitly.

Who is this for: people working on scrambling, MBL, and driven systems. It does not open a new branch, but it gives a concrete characterization of information flow in FTCs that is absent from the cited literature. I would send this to a serious referee. The main revisions: add L=10 OTOC data for the envelope, correct the appendix, and report statistical precision. A desk reject would be wrong; the qualitative claims are plausible and the topic is active.

Referee Report

6 major / 6 minor

Summary. This manuscript studies information scrambling and entanglement growth in a disordered Floquet spin-1/2 chain in an MBL-stabilized discrete time crystal (FTC) phase. Using numerical simulations for L=8 and L=10 spin chains with 1000 disorder realizations, the authors report a hierarchy of stroboscopic time scales in the magnetization (initial decay, period-doubling plateau, decoherence, thermalization) and direction-dependent OTOC dynamics: Cxx shows MBL-like logarithmic growth while Czz remains quasi-frozen and later grows logarithmically, with all Czz(l,t) curves merging into a distance-independent envelope in the decoherence regime. The entanglement entropy is reported to grow logarithmically until thermal volume-law saturation. The paper also contains an analytic treatment of a fine-tuned no-scrambling limit (Appendix A) and a comparison with the ergodic phase (Appendix B).

Significance. If the reported phenomenology is robust, it would extend the MBL scrambling picture to Floquet time crystals and identify a genuinely new signature: a preferred quasi-protected direction with frozen-then-logarithmic scrambling and distance-independent late-time growth, distinct from both conventional MBL and thermal behavior. The paper's strengths are that it studies a concrete, experimentally relevant model, averages over 1000 disorder realizations, and connects the results to the established l-bit/MBL phenomenology. The main limitations are the small system sizes (L=8 and L=10), the fitted rather than derived growth laws, and the absence of finite-size scaling for the central OTOC envelope claim, so the conclusions currently rest on evidence that is suggestive but not yet demonstrative.

major comments (6)

[III.B, Fig. 1] The central claim that in the decoherence window all Czz(l,t) curves merge into a distance-independent envelope is demonstrated only for L=8; no L=10 OTOC curves or finite-size collapse are shown. At L=8, all distances are comparable to the system size, and with the quoted exponents the decoherence window T2<t<T3 spans only about 1.8 decades (T2~31, T3~2000 for L=8; T2~74, T3~13000 for L=10). The apparent merging may therefore reflect trivial saturation to a common thermal maximum after the wavefront has crossed the chain, rather than a genuine distance-independent scrambling rate. Please provide Czz and Cxx data for L=10 and, if possible, L=12, with error bars, and show that the envelope structure persists away from the trivial saturation regime.
[III.A] The exponential time scales T2~exp(beta L) with beta~0.43 and T3~exp(beta' L) with beta'~0.95 are obtained from only L=8 and L=10. Two points cannot validate an exponential law, and the manuscript does not report the fitting procedure, residuals, or statistical uncertainties. This exponential hierarchy is load-bearing for calling the z-direction quasi-protected. Please compute T2(L) and T3(L) for at least three system sizes, or provide a scaling collapse, and report uncertainties on beta and beta'.
[Appendix A] The exact OTOC calculation is internally inconsistent. It begins with hx=0 and phi=pi/2, but then uses U_K^dagger sigma^z U_K = -sigma^z, which holds only for phi=pi (for phi=pi/2, U_K^dagger sigma^z U_K = -sigma^y). It also defines the one-period evolution as (U_F U_K)^dagger sigma^z (U_F U_K), which does not match the Heisenberg evolution generated by U_F in Eq. (2). In addition, the disorder average in Eq. (A7) does not follow from Cxx(1,t)=1-cos(4 J0 t): the correct average is 1 - cos(4 J t) sin(2 J t)/(2 J t), not 1 - cos(6 J t) sin(2 J t)/(2 t). This appendix should be corrected or removed; as written, it does not support the statement that the perfect-FTC limit has no scrambling.
[III.B, Eq. (7)] The l-bit interpretation is partly circular. The protected z-direction and the small overlap of sigma^z_i onto orthogonal tau^x_j and tau^y_j operators are inferred from the smallness of Czz itself, which is exactly the quantity the l-bit picture is invoked to explain. The coefficients c^{[i,alpha]}_{j,beta} in Eq. (7) are not computed independently. To make this explanation load-bearing, extract the l-bit overlaps from the model (for example, by constructing approximate l-bits or analyzing the diagonal ensemble) and show that they quantitatively reproduce the relative magnitudes of Cxx and Czz and the distance dependence of A_zz(l).
[III.C, Figs. 3-4] The claim that the entanglement entropy grows logarithmically over all time until thermal volume-law saturation is based on small system sizes and fitted scalings. In Fig. 4, the saturation time is reported to grow exponentially with L, but only a few system sizes are shown, no error bars are given, and the logarithmic-growth claim is not supported by a collapse or by an independent check of the time-window where the growth is purely logarithmic. Please provide error bars and a finite-size scaling analysis for S(t) and tsat(L), analogous to the analysis presented for the ergodic phase in Appendix B.
[III, initial-state independence] The statement 'our results are independent of the choice of initial state' is asserted without systematic evidence. All reported scalings are obtained for a single initial product state with theta=pi/8. Since OTOC growth rates and entanglement growth in disordered systems can depend on the initial state's overlap with the l-bit structure, please demonstrate the key claims (the Czz envelope, the log-growth exponents, and the T2/T3 scaling) for at least one additional initial state, or explicitly restrict the claims to the class of initial states simulated.

minor comments (6)

[Abstract] The phrase 'entanglement of entropy' should be 'entanglement entropy'.
[II and III] The exact value of the kicking phase phi used in Figs. 1 and 4 is not stated (the text only says phi~pi); please give it explicitly, especially because Appendix A uses phi=pi/2.
[III.A] The time scales T1, T2, and T3 are defined only qualitatively by visual inspection of Fig. 1; please provide operational definitions (for example, threshold crossings of Z(t)) so that the fits are reproducible.
[III.B] Please reconcile the notation A_alpha beta(l) ~ exp(-v_alpha beta l) in Eq. (6) with Czz(l,t) ~ exp(-b_l l) + c log(t)^d; the subscript l in b_l is easily confused with the distance variable l.
[III.C, Eq. (8)] Because S is defined with an explicit division by L_A, it is an entanglement entropy density rather than the total entanglement entropy; please state this explicitly, since the 'volume-law saturation' and comparisons with thermal values depend on the normalization.
[Fig. 4] The vertical dotted lines labeled T1, T2, and T3 should be defined in the caption; currently they are mentioned only in the text and their numerical values are not given.

Circularity Check

0 steps flagged · score 1.0 of 10

No constructional circularity: OTOC results are independently simulated and the ℓ-bit picture is an interpretive overlay; peripheral self-citations are not load-bearing.

full rationale

The paper's central observable results—the frozen-then-logarithmic Czz(ℓ,t) growth, the late-time distance-independent envelope, and the logarithmic entanglement growth—are obtained by direct numerical simulation of the Floquet unitary in Sec. II, not by construction from the ℓ-bit expansion or from the fitted time scales. The ℓ-bit decomposition in Eq. (7) is introduced after the OTOC data as an interpretive 'picture' ('Alternatively, one could also understand this regime by recalling an ℓ-bit phenomenological model for MBL-like dynamics'), with the coefficients not fitted to Czz. The smallness of Czz is attributed to the independently observed period-doubled z-magnetization conservation ('These small values arise due to the (period doubled) conservation of the z-magnetization'), a magnetization observable that is measured separately from the OTOC; this is explanatory consistency, not a definitional reduction. The late-time envelope is likewise an observed merge of the numerically computed correlators ('the correlator Czz(ℓ,t) have all the same growth and merge into a single one forming an envelope-like structure'), supported by the wavefront argument, and is not defined as equal to a fitting function. The fitted quantities T2∼e^{βL}, T3∼e^{β′L}, and the inset log-power fits are fits to data, but they are not presented as predictions derived from the theory. The only self-citations (e.g., Refs. [103] and [105]) appear in the discussion of future Krylov-complexity directions and carry no load in the derivation. Finite-size limitations (L=8 and L=10) are a correctness risk, not circularity. The paper is therefore self-contained against its own central claims; score 1 reflects only the presence of peripheral, non-load-bearing self-citations.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central results depend on free parameters fitted to the numerical data (growth exponents, spatial decay rates, envelope parameters, time-scale exponents) and on the standard MBL ℓ-bit picture, which is imported from prior literature rather than established for this Floquet model. No new physical entities are introduced, so the invented-entities ledger is empty.

free parameters (5)

OTOC growth exponent a_{αβ}(ℓ) = fitted, non-universal
In Eq (6), C_{αβ}(ℓ,t) ∼ A_{αβ}(ℓ) log(t)^{a_{αβ}(ℓ)}; exponents fitted from inset of Fig 1.
OTOC spatial decay rate v_{αβ}(ℓ) = fitted
A_{αβ}(ℓ) ∼ e^{-v_{αβ} ℓ}.
Late-time envelope amplitude c and growth exponent d = c,d fitted, claimed independent of ℓ
Czz(ℓ,t) ∼ e^{-bℓ} + c log(t)^d in the decoherence regime.
Magnetization time-scale exponents β, β' = β ≈ 0.43, β' ≈ 0.95
T2 ∼ e^{βL}, T3 ∼ e^{β'L}, extracted from finite-size data (Section III.A).
Entanglement growth-time exponent p = 0 < p ≤ 1
log t* ∝ (hx)^{-p} (Section III.C); p is not fixed independently.

assumptions (4)

domain assumption Existence of a full set of local integrals of motion (ℓ-bits) in the FTC phase, with a preferred z-direction and exponentially localized overlaps.
Used in Eq (7) and the discussion of Section III.B to explain the direction-dependent OTOC growth; imported from MBL phenomenology (refs 94-98) and not derived here.
domain assumption Stability of the FTC phase with period-doubling time T2 ∼ e^{βL} for L → ∞.
Assumed from prior FTC/MBL results (refs 2,90) to interpret finite-size data for L=8,10.
domain assumption Disorder average over 10^3 realizations and the chosen initial product states are representative of the ensemble.
Section III states averaging over 10^3 disorder realizations and θ=π/8 product states; no convergence or error-bar checks are shown.
ad hoc to paper The fitted functional forms (log^a t, e^{-vℓ}, envelope) are valid over the asymptotic regimes they describe.
The claims of logarithmic growth and distance-independent merging rest on fits over finite time windows, not on a derivation.

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Cite this review

Pith. "Pith review of Information scrambling and entanglement dynamics in Floquet Time Crystals." pith.science (2026). https://pith.science/paper/X6NBMLH6

@misc{pith2026241113469,
  author       = {Pith},
  title        = {Pith review of: Information scrambling and entanglement dynamics in Floquet Time Crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6NBMLH6}},
  note         = {Machine review of arXiv:2411.13469}
}

abstract

We study the dynamics of out-of-time-ordered correlators (OTOCs) and entanglement of entropy as quantitative measures of information propagation in disordered many-body systems exhibiting Floquet time-crystal (FTC) phases. We find that OTOC spreads in the FTC with different characteristic timescales due to the existence of a preferred ``quasi-protected'' direction - denoted as $\ell$-bit direction - along which the spins stabilize their period-doubling magnetization for exponentially long times. While orthogonal to this direction the OTOC thermalizes as an usual MBL time-independent system (at stroboscopic times), along the $\ell$-bit direction the system features a more complex structure. The scrambling appears as a combination of an initially frozen dynamics (while in the stable period doubling magnetization time window) and a later logarithmic slow growth (over its decoherence regime) till full thermalization. Interestingly, in the late time regime, since the wavefront propagation of correlations has already settled through the whole chain, scrambling occurs at the same rate regardless of the distance between the spins, thus resulting in an overall envelope-like structure of all OTOCs, independent of their distance, merging into a single growth. Alongside, the entanglement entropy shows a logarithmic growth over all time, reflecting the slow dynamics up to a thermal volume-law saturation.

Figures

Figures reproduced from arXiv: 2411.13469 by the authors.

**Figure 1.** (Left) The dynamics of magnetization Z(t) in the FTC phase, and its different characteristic time scales Ti (vertical dotted lines). The magnetization displays a period doubling plateau in the fully stable regime of the time crystal, which later slowly decoheres and thermalises at late times. We show the corresponding dynamics of OTOCs along the (center) z-direction (Czz(l, t)) and (right) x-direction (Cxx(l, t)), f… view at source ↗

**Figure 2.** The dynamics of OTOCs Cαβ(l, t) for single realizations. We set the system deep in the FTC phase (with parameters J = h z = 1, h x = 0.025 and L = 8 spins) highlighting the OTOC in the (long) period doubling regime, t < T2. We show in panels (a) and (b) two typical dynamics for different single disorder realizations. While different OTOC directions show a qualitative similar behavior (apart from their orders of ma… view at source ↗

**Figure 3.** Dynamics of entanglement entropy S(t) deep in the FTC phase (h x ≈ 0), for varying h x fields with a fixed system size L = 8, and couplings J = h z = 1. The inset shows the subtracted entanglement entropy δS = Shx (t) − Shx=0(t) with a rescaled time t/t∗ . in a MBL system [87]. The entanglement entropy remains constant and equal to the no transverse field case, S(h x = 0), up to a growth time t ∗ . The growth time … view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Late-time dynamics of entanglement entropy in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: The dynamics associated with melted FTC (ergodic phase) for system-size [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: (a) The entanglement entropy S(t) in melted FTC phase for the varying system-sizes L, and parameter values J = 1 = h z , h x = 0.5 and the kicking phase ϕ = π/2 + 0.4. (b) The saturation time plotted as function of system-size L, where the dash-dotted lines shows the f…

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