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Periodically and aperiodically Thue-Morse driven long-range systems: from dynamical localization to slow dynamics

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Pith's one-line read Time-periodic and Thue-Morse electric-field drives renormalize every hopping amplitude in a power-law chain by a sinc factor, so at special amplitude-to-frequency ratios the clean long-range system is exactly dynamically localized, while…

desk verdict Clear new EDL condition for Thue-Morse driven long-range chains; the drive-induced fractal phase in the PLRBM needs stronger finite-size evidence before it carries the abstract. read the letter →

arxiv 2412.19736 v2 pith:WMCIYUA7 submitted 2024-12-27 cond-mat.dis-nn cond-mat.str-el

Vatsana Tiwari , Devendra Singh Bhakuni , Auditya Sharma This is my paper · ORCID
classification cond-mat.dis-nncond-mat.str-el
keywords power-lawrandombandedmatrixdynamicallocalizationThue-MorsedrivingFloquetengineeringfractaldimensionprethermalizationlevelspacingratiolong-rangehopping
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 when a one-dimensional chain with power-law hopping, whose strength decays as $1/r^\alpha$, is shaken by a time-periodic or aperiodic electric field. The authors show that the field renormalizes every hopping amplitude by a sinc factor, so at specially chosen amplitude-to-frequency ratios all hopping vanishes at once and the chain is exactly dynamically localized: a wave packet never spreads, no matter how long the hopping range. On the delocalized side of the static phase diagram, periodic driving turns the disordered model into a weakly multifractal or fractal phase, with transport slowing from ballistic to diffusive or subdiffusive. Under a Thue-Morse aperiodic drive, the clean long-range model still shows exact dynamical localization at the same kind of special points, while the disordered model shows prethermal plateaus followed by subdiffusive relaxation. Overall, the paper maps how drive parameters control transport in long-range and quasiperiodic systems, from frozen dynamics to slow relaxation.

What carries the argument

The load-bearing object is the effective Hamiltonian obtained from the Baker-Campbell-Hausdorff expansion of the one-cycle (or one Thue-Morse block) evolution operator. For the clean chain, each hopping $J_p$ is multiplied by $\sin(\phi_p)/\phi_p$, with $\phi_p$ proportional to $pFT$, and exact dynamical localization occurs exactly when all these renormalized hoppings vanish at once at $F=n\omega$ (Thue-Morse) or $F=2m\omega$ (square wave). For the disordered model, the same expansion yields a zeroth-order term $H_0$ with renormalized hoppings plus higher-order corrections $H_1$; at the vanishing points only $H_1$ survives, and its frequency-dependent corrections govern the slow dynamics. Static phase identification uses the level-spacing ratio of the Floquet quasienergy spectrum and the system-size scaling of the generalized inverse participation ratio $I_q \sim L^{-\tau_q}$, with fractal dimension $D_q = \tau_q/(q-1)$ distinguishing localized ($D_q=0$), delocalized ($D_q=1$), and fractal ($0<D_q<1$) eigenstates.

What would settle it

Compute the generalized participation ratios $I_q$ for $q=2,3,4$ at $L=400,700,1000,1500,2000$ with several hundred disorder realizations and perform a finite-size scaling collapse; the fractal phase is real only if $D_q$ extrapolates to a constant in $(0,1)$ as $L\to\infty$, whereas a crossover would push $D_q$ toward $0$ or $1$.

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Extended reading notes

Core claim

The paper's central claim is that an electric-field drive acts on a power-law hopping model through a multiplicative renormalization of each hopping amplitude: for square-wave driving the effective amplitude is $J_p^{\rm eff} = J_p \sin(pFT/4)/(pFT/4)$ and for Thue-Morse driving it is $J_p^{\rm eff} = J_p \sin(pFT)/(pFT)$, with $J_p = J/p^\alpha$. Because the sine factor vanishes for every $p$ simultaneously at $F=2m\omega$ (square wave) or $F=n\omega$ (Thue-Morse), the clean long-range chain exhibits exact dynamical localization for arbitrary $\alpha$; away from these points it is ballistic. For the disordered power-law random banded matrix model, the authors find a drive-induced intermediate phase on the delocalized side ($\alpha<1$): Floquet eigenstates have fractal dimension $0<D_q<1$ that is nearly independent of $q$ for larger $\alpha$, indicating weak multifractality crossing over to a fractal phase, with transport that is diffusive to subdiffusive. On the localized side ($\alpha>1$) the driven model stays localized, with logarithmic spreading of $X(t)$ and $S(t)$. The disordered Thue-Morse-driven model shows diffusive relaxation to the infinite-temperature state on the delocalized side and a prethermal plateau followed by subdiffusion on the localized side, in contrast to the Thue-Morse-driven Aubry-André-Harper model, where even the delocalized side develops a prolonged prethermal plateau.

Load-bearing premise

The existence of the drive-induced fractal phase rests on power-law fits of participation ratios at only two moment orders and three system sizes with no finite-size scaling collapse, so the intermediate fractal dimension could be a finite-size crossover between the static delocalized and localized phases.

Editorial extensions

If this is right

  • At the zeros of the renormalized hopping, a clean long-range chain of any $\alpha$ is completely frozen: $X(t)$ remains at its initial width while the drive runs.
  • Tuning slightly away from the zero point restores full ballistic transport, so the drive acts as a sharp on/off switch for transport in the clean model.
  • Periodic driving replaces ballistic spreading with diffusive ($\beta=1/2$) or subdiffusive ($\beta<1/2$) transport on the delocalized side of the disordered model, with the exponent decreasing as $\alpha$ approaches the static transition.
  • Thue-Morse driving produces exponentially long prethermal plateaus ($\tau_h \propto e^{\omega}$) on the localized side, followed by subdiffusive relaxation to the infinite-temperature state.
  • The same aperiodic drive suppresses transport even in the delocalized phase of the Aubry-André-Harper model, creating a prolonged prethermal plateau before subdiffusive growth.

Reading between the lines

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

  • If the fractal phase survives thermodynamic-limit scaling, the periodic drive offers a parameter-free route to engineering subdiffusive transport exponents by tuning $F/\omega$ and $\alpha$ in a non-interacting chain.
  • The sinc-factor structure suggests that any driving protocol whose Fourier spectrum contains a common zero for all hopping distances, not only square-wave and Thue-Morse, should produce exact dynamical localization; testing a second aperiodic sequence such as Fibonacci driving would map the boundary of the phenomenon.
  • The exponentially long heating time in the Thue-Morse-driven disordered model implies the prethermal plateau may be observable in current cold-atom or trapped-ion simulators with power-law interactions, where the drive frequency can be large compared with local bandwidths.
  • The contrast between the PLRBM and Aubry-André-Harper results hints that the nature of the static eigenstates controls whether the delocalized side develops a prethermal plateau, a distinction that could be probed by measuring the level-spacing statistics of the driven system.
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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 / 6 minor

Summary. This paper studies the interplay of time-periodic and aperiodic Thue-Morse electric-field driving with power-law hopping disorder in a one-dimensional fermionic chain (the PLRBM model). For periodic square-wave driving, the authors analyze the Floquet operator and report a drive-induced weak-multifractal/fractal phase on the delocalized side (α<1) of the static transition, with diffusive-to-subdiffusive transport, while the localized side (α>1) remains localized with logarithmic transport. For the clean long-range chain under Thue-Morse driving, they derive an effective Hamiltonian with renormalized hopping J_eff^p = J_p sin(pFT)/(pFT) and claim exact dynamical localization at F=nω. For the disordered Thue-Morse-driven chain, they report diffusive relaxation on the delocalized side and a prethermal plateau followed by subdiffusion on the localized side, and they compare with a Thue-Morse-driven Aubry-André-Harper model. The central claims are the drive-induced fractal phase and the exact dynamical localization condition.

Significance. The paper addresses a timely question: how long-range hopping and temporal driving combine to produce non-equilibrium phases. The strongest feature is that the EDL condition F=nω is a parameter-free prediction from the renormalized-hopping formula, and the numerical saturation data in Fig. 9 are consistent with it. The manuscript also provides transport and entanglement data across a broad parameter range. However, the evidence for the drive-induced fractal phase, highlighted in the abstract and conclusions, is not quantitatively sufficient as presented; the Dq analysis uses only two q values and three system sizes without error bars or finite-size scaling. If the fractal phase is established by additional scaling analysis, the paper would constitute a useful contribution to the Floquet engineering of long-range disordered systems.

major comments (3)
  1. [Section III.A, Eq. (19), Fig. 4] The claim that the periodically driven PLRBM model exhibits a drive-induced fractal phase on the delocalized side (0<Dq<1 independent of q for α=0.5–0.85) is based on power-law fits of I2 and I3 over only L=400, 700, 1000, with 100 disorder realizations and no error bars. With only q=2 and q=3, the distinction drawn in Fig. 4(c,f) between 'weak multifractality' (Dq varying with q) and 'fractal behavior' (Dq constant in q) is not established even in principle, and the intermediate ⟨r⟩ values in Fig. 3(a) are equally consistent with a finite-size crossover between the delocalized and localized PLRBM phases. Please provide error bars, additional q values, and a finite-size scaling analysis that demonstrates a stable thermodynamic-limit Dq.
  2. [Section IV.A, Eqs. (21)-(26)] The derivation of the exact dynamical localization condition for the Thue-Morse driven clean chain uses a Baker-Campbell-Hausdorff expansion truncated at leading order. The paper then states that 'at the zeros of J_eff^p, F=nω, one can observe the phenomenon of exact dynamical localization' and 'the transport of the system ceases.' Since higher-order terms in the BCH expansion are not shown to vanish at those points, the word 'exact' is not justified by the presented derivation. Please either supply an exact argument (for example, a momentum-space integration) or qualify the claim as holding within the high-frequency expansion.
  3. [Section III.A, Fig. 3(a)] The identification of an intermediate phase for α<1 from ⟨r⟩ values between 0.386 and 0.529 is made without a finite-size analysis. Since the system sizes are L=1024 and the disorder average is only 100 realizations, the intermediate values could be a crossover effect; please show ⟨r⟩ vs L for representative α values and DL/ADL tunings.
minor comments (6)
  1. [Section II, Eq. (3)] The definitions of U_+ and U_- in Eq. (3) are not carried consistently into Section IV.A, where UA and UB appear; please make the notation uniform.
  2. [Fig. 3(a)] Please add a horizontal line at the COE value ⟨r⟩=0.529 to aid comparison with the Poisson line at 0.386.
  3. [Fig. 4] The power-law fits in panels (a,b,d,e) are quoted without their fitted exponents or goodness-of-fit; please report τ_q and R² or similar.
  4. [Section IV.B, Fig. 10 insets] The conclusion τ_h ∝ exp(ω) is drawn from a log-linear plot with four points and no error bars; please include the fit and uncertainties.
  5. [Section III.B] There are several typos, e.g., 'inifinite' for 'infinite' and 'aymptotic' for 'asymptotic'; a careful proofread is needed.
  6. [Fig. 1 table] The table entries such as 'Subdiffusive Prethermalization Delocalization' are confusing; consider rewording to distinguish the prethermal plateau from the eventual delocalization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EDL conditions and effective hoppings are derived analytically, and the fractal-phase claim, though numerically underpowered, is an interpretation of direct Floquet data.

full rationale

The derivation chain rests on BCH/Magnus expansions rather than on fitting or self-citation. The periodically driven renormalized hopping, J_eff^p = J_p sin(pFT/4)/(p^alpha (pFT/4)) (Eqs. 14-15), and the Thue-Morse J_eff^p = J_p sin(pFT)/(pFT) (Eq. 26) are obtained algebraically in the text; the EDL zeros F = 2mω and F = nω are analytic consequences and are then checked against independent time evolution (Figs. 8-9). No parameter is fitted to the quantity it is said to predict. The fractal/multifractal phase classification in Sec. III.A is inferred from power-law fits of I2 and I3 for q = 2, 3 at L = 400, 700, 1000; this is numerically underpowered and a real correctness risk, but it is not circular: the classification is an interpretation of numerically computed Floquet eigenstates, not an input to the calculation. The self-citations [35, 36, 86] support related prior results, but the expressions used here are rederived in the main text and appendices, so those self-citations are not load-bearing. No uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed. I therefore find no circular step.

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

The central derivations introduce no fitted constants: the renormalized hopping formulas are fixed functions of J, F, T, and α. The load-bearing assumptions are standard numerical and analytical ones, chiefly the truncation of the Magnus expansion and the reliability of finite-size spectral diagnostics.

assumptions (5)
  • domain assumption The BCH/Magnus expansion truncated at leading order yields the effective Hamiltonians in Eqs. (10), (12), and (26), and higher-order terms H1 can be neglected at high frequency.
    Used to derive the renormalized hopping amplitudes and the EDL condition. For the clean Thue-Morse case the zero-hopping point can be confirmed by exact momentum-space integration, but the paper does not show that proof and applies the truncated expansion to the disordered case.
  • domain assumption The undriven PLRBM model has a delocalization-to-localization transition at α=1, with delocalized behavior for α<1 and localized behavior for α>1.
    Taken from Refs. [71-73] and used throughout to benchmark the driven phases; the driven results are interpreted against this static phase diagram.
  • domain assumption Spectral diagnostics (level-spacing ratio and generalized IPR scaling) at system sizes up to L=1024 and 100 disorder samples distinguish localized, fractal/multifractal, and delocalized phases without significant finite-size corrections.
    Underpins the central fractal-phase claim; no finite-size scaling or error analysis is provided.
  • domain assumption The Fourier spectrum of the Thue-Morse sequence, with both low- and high-frequency components, controls the plateau and subdiffusion, and the heating time grows exponentially with driving frequency.
    Used in Sections IV.B and IV.C to explain the prethermal plateau and τh∝exp(ω); asserted qualitatively rather than derived.
  • standard math Entanglement entropy computed from the single-particle correlation matrix via Eq. (6) correctly captures the dynamics and the infinite-temperature (Page) value.
    Standard for non-interacting fermions and supported by Refs. [78,80,81].

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Pith. "Pith review of Periodically and aperiodically Thue-Morse driven long-range systems: from dynamical localization to slow dynamics." pith.science (2026). https://pith.science/paper/WMCIYUA7

@misc{pith2026241219736,
  author       = {Pith},
  title        = {Pith review of: Periodically and aperiodically Thue-Morse driven long-range systems: from dynamical localization to slow dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMCIYUA7}},
  note         = {Machine review of arXiv:2412.19736}
}
abstract

We investigate the electric-field driven power-law random banded matrix(PLRBM) model where a variation in the power-law exponent $\alpha$ yields a delocalization-to-localization phase transition. We examine the periodically driven PLRBM model with the help of the Floquet operator. The level spacing ratio and the generalized participation ratio of the Floquet Hamiltonian reveal a drive-induced fractal phase accompanied by diffusive transport on the delocalized side of the undriven PLRBM model. On the localized side, the time-periodic model remains localized - the average spacing ratio corresponds to Poisson statistics and logarithmic transport is observed in the dynamics. Extending our analysis to the aperiodic Thue-Morse (TM) driven system, we find that the aperiodically driven clean long-range hopping model (clean counterpart of the PLRBM model) exhibits the phenomenon of \textit{exact dynamical localization} (EDL) on tuning the drive-parameters at special points. The disordered time-aperiodic system shows diffusive transport followed by relaxation to the infinite-temperature state on the delocalized side, and a prethermal plateau with subdiffusion on the localized side. Additionally, we compare this with a quasi-periodically driven AAH model that also undergoes a localization-delocalization transition. Unlike the disordered long-range model, it features a prolonged prethermal plateau followed by subdiffusion to the infinite temperature state, even on the delocalized side.

Figures

Figures reproduced from arXiv: 2412.19736 by the authors.

Figure 1
Figure 1. Schematic representation of our main findings and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Quasienergy spectrum of periodically driven [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (a) Average gap ratio ⟨r⟩ with long-range exponent α for the driving-parameters tuned at drive-amplitude F = 1.3ω, F = 2ω. Inset(a). Average gap ratio ⟨r⟩ with drive-parameters F/ω for long-range exponent α = 0.67, 1.4. The other system parameters are driving-frequency ω = 2, and system-size L = 1024. The black dashed line corresponds to the average gap ratio for Poisson statistics, ⟨r⟩ = 0.386. (b) η obtained from … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Generalized IPR and fractal dimension Dq for periodically driven PLRBM model. (a,b) I3 and I2 vs system-size L for drive-parameter F = 1.3ω. Black dashed line shows power-law fit (Iq ∼ AL−τq ) to characterize the features of Floquet￾eigenstates. (d,e) I3 and I2 vs syst…
Figure 5
Figure 5. Figure 5: (a-d). Dynamics of periodically driven system tuned at DL points [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Dynamics of root-mean-squared-width X(t) for pe￾riodically driven system tuned at DL points F = 2ω for dif￾ferent long-range exponents α. The dashed lines correspond to power-law fit X(t) ∝ t β for the curves plotted in the same color. The other parameters are system-s…
Figure 7
Figure 7. Figure 7: (a,b). Saturation values of root-mean squared displacement [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Root mean squared displacement for Thue-Morse [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: (a) Oscillatory behavior of saturation values of root-mean squared displacement [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Dynamics of Thue-Morse driven system tuned at dynamical localization point ( [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Dynamics of Thue-Morse driven system tuned at away from dynamical localization point ( [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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