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Locality and Heating in Periodically Driven, Power-law Interacting Systems

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Periodically driven power-law interacting systems heat up only after a time exponentially long in the drive frequency: for α>D under linear response, and for α>2D for generic drives.

desk verdict Solid advance on heating times for power-law Floquet systems; the main results hold up, but the conjectured tight Lieb-Robinson bound in Sec. V is too strong at short times and should be repaired. read the letter →

arxiv 1908.02773 v2 pith:HMQ4MDLV submitted 2019-08-07 quant-ph

Minh C. Tran , Adam Ehrenberg , Andrew Y. Guo , Paraj Titum , Dmitry A. Abanin , Alexey V. Gorshkov This is my paper · ORCID
classification quant-ph
keywords Floquetsystemsprethermalizationheatingtimepower-lawinteractionsLieb-RobinsonboundslinearresponsetheoryMagnusexpansionlong-rangeinteracting
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 when a periodically driven quantum system whose interactions decay with distance as $1/r^{\alpha}$ avoids rapid heating to infinite temperature. It proves two exponential-in-frequency lower bounds on the heating time: one under linear response theory, valid for $\alpha>D$, and one for generic (possibly strong) drives, valid for $\alpha>2D$. The mechanism is locality: the drive can only absorb energy through correlated clusters of sites, and Lieb-Robinson bounds adapted to power-law interactions control how many clusters contribute. The paper also generalizes several two-body Lieb-Robinson bounds to $k$-body interactions, which yields a stronger heating-time bound than previously known, and conjectures that the remaining gap between $\alpha>D$ and $\alpha>2D$ is an artifact of the available bounds rather than of the physics.

What carries the argument

The argument is carried by two constructions. In Section IV, heating is quantified through the dissipative response function $\sigma(\omega)=\sum_{i,j}\sigma_{ij}(\omega)$: diagonal entries are exponentially small by an eigenstate argument, and off-diagonal entries are bounded using a Lieb-Robinson bound with a logarithmic light cone for $\alpha>D$, yielding Eq. (29). In Section V, a periodic unitary $Q(t)=e^{\Omega(t)}$ is chosen order by order in the period $T$; the transformed Hamiltonian splits into a time-independent effective Hamiltonian $H_*$ (itself power-law with exponent $\alpha$) and a residual drive whose local norm is exponentially small in the frequency $\omega_*\propto 1/T$. Lieb-Robinson bounds—logarithmic cones for $\alpha>D$, algebraic cones for $\alpha>2D$—then control how much the residual drive can affect a local observable, producing the heating-time estimates.

What would settle it

Run a finite-size simulation of a one-dimensional spin chain with $1/r^{\alpha}$ couplings ($1<\alpha<2$), start it in a thermal state of $H_0$, apply a weak cosine drive of frequency $\omega$, and measure the steady-state energy absorption rate; if the rate decays polynomially rather than exponentially with $\omega$, the linear-response bound of Eq. (29) is falsified.

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Extended reading notes

Core claim

The central claim is that a $D$-dimensional spin system with power-law interactions $1/r^{\alpha}$, driven periodically by a local drive, has a heating time exponentially large in the drive frequency $\omega$ whenever $\alpha$ exceeds a critical value. Under linear response theory (a weak harmonic drive and an initial thermal state), the paper proves this for all $\alpha>D$, bounding the heating rate by $C N \exp[-(1-D/\alpha)\kappa\omega]$ (Eq. (29)). For generic drives, the paper constructs an effective time-independent Hamiltonian $H_*$ through a Magnus-like expansion; the residual time-dependent part has local norm bounded by $C\lambda e^{-\kappa'\omega_*}$, and combining this with Lieb-Robinson bounds gives exponentially long heating times for $\alpha>2D$ (Eqs. (48), (49)). The paper further generalizes recent Lieb-Robinson bounds from two-body to $k$-body interactions, and shows that if a conjectured tight Lieb-Robinson bound of the form $\|[A(t),B]\|\le C\|A\|\|B\|(t^{\beta}/r)^{\alpha}$ (Eq. (50)) existed, the $\alpha>D$ result would extend to generic drives, closing the gap.

Load-bearing premise

The $\alpha>D$ result assumes linear response theory—a weak harmonic drive and an initial thermal state—and for strong drives the proof covers only $\alpha>2D$ unless the conjectured tight bound on the spread of quantum signals (Eq. (50)) is true.

Editorial extensions

If this is right

  • Rapidly driven platforms with power-law interactions—trapped ions, Rydberg atoms, polar molecules—can host prethermal Floquet phases for exponentially long times when α>2D (for strong drives) or α>D (for weak drives).
  • The generalized k-body Lieb-Robinson bounds are tools in their own right, applicable to error bounds for digital quantum simulation and to limits on information propagation in long-range systems.
  • The effective Hamiltonian H* inherits the power-law structure of the original Hamiltonian, so the prethermal regime preserves long-range physics rather than becoming effectively short-range.
  • For D<α<2D, the paper predicts exponential heating time but leaves the strong-drive case contingent on a tight Lieb-Robinson bound; the gap is expected to vanish once such a bound is proven.

Reading between the lines

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

  • If the conjectured tight Lieb-Robinson bound is true, the same Magnus construction would likely yield prethermalization for all α>D, making exponential heating time a universal feature of power-law interactions.
  • The exponential suppression of heating under weak drives should be directly measurable in current trapped-ion simulators with tunable α: measuring absorbed power per cycle versus ω for α just above D would test Eq. (29).
  • The emphasis on light-cone shape suggests a general principle: any interaction ensemble whose Lieb-Robinson light cone grows at most algebraically should exhibit exponentially long heating times under local periodic drives, so the result may extend to other long-range models.
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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

1 major / 3 minor

Summary. The paper studies heating times in periodically driven quantum spin systems on a D-dimensional lattice with power-law interactions decaying as 1/r^alpha. The main results are: (i) under linear response theory, for a weak harmonic drive and an initial thermal state, the heating rate is bounded by C N exp[-(1-D/alpha) kappa omega] for alpha>D, giving an exponentially long heating time in the drive frequency; (ii) for generic (possibly strong) drives, a Magnus-like expansion combined with Lieb-Robinson bounds yields a quasi-conserved effective Hamiltonian and an exponentially long heating time for alpha>2D; (iii) several Lieb-Robinson bounds are generalized from two-body to k-body interactions, including the Tran et al. bound; and (iv) the paper conjectures that a tight Lieb-Robinson bound for alpha>D would close the gap between the alpha>D linear-response result and the alpha>2D generic-drive result, and it presents a derivation of this gap-closing statement assuming such a bound. The appendices contain the detailed proofs of the Lieb-Robinson generalizations, the linear-response estimate, and the Magnus expansion estimates.

Significance. If the results hold, they extend the finite-range prethermalization and heating-suppression results to power-law interacting systems with a sharp alpha-dependent threshold, matching numerical evidence and clarifying the role of locality in Floquet heating. The k-body generalization of state-of-the-art Lieb-Robinson bounds is a useful technical contribution in its own right, as such bounds are likely to find applications beyond this paper. The authors are appropriately careful to label the tight-bound statement as a conjecture and to distinguish the proven alpha>2D result from the conjectured alpha>D extension. The main theorems are supported by detailed appendices, and the paper is honest about the assumptions entering the linear-response analysis. However, as discussed below, the conjectured tight bound in Section V is not valid in the form written, which affects the gap-closing demonstration as stated, though it does not invalidate the two main theorems.

major comments (1)
  1. [Section V, Eq. (50) and the derivation of Eq. (55)] The conjectured tight Lieb-Robinson bound in Eq. (50), ||[A(t),B]|| <= C ||A|| ||B|| (t^beta/r)^alpha, cannot hold for all t for a power-law Hamiltonian that contains the direct two-body interaction. For two operators supported a distance r apart, the first-order short-time expansion gives ||[A(t),B]|| >= c t / r^alpha for sufficiently small t. Since the paper's choices beta=1/(alpha-D) for D<alpha<D+1 and beta=1 for alpha>D+1 both satisfy alpha beta > 1, the conjectured bound is smaller than the direct lower bound for all sufficiently small t. Therefore Eq. (50) is too strong as a Lieb-Robinson bound. Because the derivation of the gap-closing estimate Eq. (55) integrates the conjectured bound from s=0, the demonstration that the gap vanishes does not follow as written. The problem is repairable by replacing t^beta with max(t,t^beta) or by treating the s<1 contribution separately with the direct-interaction lower bound; the exponential-in-omega conclusion should survive. This issue does not affect the linear-response result of Section IV or the alpha>2D result in Section V that uses the published Lieb-Robinson bounds.
minor comments (3)
  1. [Appendix B, Eq. (B10)] In the far-distance sum, the exponential term should carry a factor of r_*^{D-1} from the D-dimensional density of sites, so the second term is C N r_*^{D-1} e^{-mu r_*} rather than C N e^{-mu r_*}. As written, Eq. (B10) is not a valid upper bound. Because the exponential is subdominant relative to the algebraic term in the regime of interest, the final bound Eq. (B12) remains valid once the prefactor is restored, but the displayed inequality should be corrected.
  2. [Appendix A] The generalization of the Gong et al. bound to k-body interactions is carried out explicitly only for D=1, with the statement 'The proof for D>1 follows a very similar analysis.' Since this generalization is subsequently used in the main text for general D, please provide the D>1 convolution bound and the corresponding summation factor in at least as much detail as the one-dimensional case.
  3. [Section V, discussion after Eq. (55)] The sentence 'Recall that the best values we can hope for beta are beta=1/(alpha-D) when D+1>alpha>D and beta=1 when alpha>D+1' is used to infer an exponential heating time for all alpha>D. Once Eq. (50) is corrected as suggested in the major comment, it would be helpful to state explicitly that the corrected bound still yields the same beta and the same exponential dependence up to a constant in the time exponent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the heating-time bounds are derived from separately proven Lieb-Robinson bounds and self-contained estimates; the tight-bound conjecture is explicitly conditional.

full rationale

We find no step in which a prediction reduces by construction to an input. In Sec. IV, Eq. (29) combines (i) the exponential smallness of individual response-matrix entries, Eq. (26), proved for power-law H0 in App. B1 from a commutator norm bound, and (ii) a distance suppression from the Gong et al. Lieb-Robinson bound, Eq. (B9). The optimization r_* = exp(kappa omega/alpha) is not a fit of the final decay exponent; it is a saddle-point choice between two independently bounded contributions. In Sec. V, the smallness of the residual drive V' in Eq. (45) follows from Lemmas 1 and 2 proven in App. C; it does not assume an exponentially long heating time. The commutator bound on delta, Eq. (46), is then estimated with published Lieb-Robinson bounds (Refs. 19 and 20), including the two-body bound of Tran et al. [20] generalized to k-body interactions in Sec. III by an explicit truncation proof. Ref. [20] is a self-citation, but it is a published, parameter-free theorem proved independently, and the present generalization is also proved here, so the citation is real evidence rather than a circular premise. The conjectured tight bound Eq. (50) and the resulting gap-closing estimate Eq. (55) are explicitly labeled a conjecture and are used only conditionally; even if Eq. (50) cannot hold at very short times, that is a soundness concern, not a circularity, and the paper does not count the conjectured statement as a theorem. No fitted input is renamed as a prediction.

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

The derivation rests on standard domain assumptions about power-law lattice systems and existing Lieb-Robinson bounds. The only ad hoc addition is the conjectured tight LR bound, which is clearly labeled and used conditionally. No free parameters are fitted to data and no new physical entities are introduced.

assumptions (5)
  • domain assumption Power-law Hamiltonians with alpha > D have finite local norm lambda = 1 + sum_{j != i} 1/r_ij^alpha (Definition 1 and Eq. (A3)).
    The proofs require the summed interaction strength at each site to be finite; this fails for alpha <= D.
  • domain assumption Linear response theory gives the energy absorption rate; the drive is weak and the state is initially thermal (Section IV).
    The alpha > D heating-time claim is derived from the dissipative response function, which is only the lowest-order-in-g description.
  • domain assumption Repeated commutators of power-law Hamiltonians grow at most as k! lambda^k (Lemma 3, Appendix E).
    Used to bound ad^k_{H0} O_i and to show G_q and Omega_q remain power-law; depends on the reproducibility condition sum_l 1/(r_il^alpha r_lj^alpha) <= lambda1 / r_ij^alpha.
  • domain assumption Existing Lieb-Robinson bounds with algebraic light cones hold for the effective Hamiltonians (Refs [19,20] and their k-body generalizations).
    The Magnus-based heating time is obtained by inserting these LR bounds into the commutator in Eq. (46).
  • ad hoc to paper The conjectured tight Lieb-Robinson bound (Eq. (50)) of the form ||[A(t),B]|| <= C ||A|| ||B|| (t^beta/r)^alpha for all alpha > D.
    Used only for the conditional result that the alpha > D range extends to generic drives; explicitly flagged as a conjecture.

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Cite this review

Pith. "Pith review of Locality and Heating in Periodically Driven, Power-law Interacting Systems." pith.science (2026). https://pith.science/paper/HMQ4MDLV

@misc{pith2026190802773,
  author       = {Pith},
  title        = {Pith review of: Locality and Heating in Periodically Driven, Power-law Interacting Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMQ4MDLV}},
  note         = {Machine review of arXiv:1908.02773}
}
abstract

We study the heating time in periodically driven $D$-dimensional systems with interactions that decay with the distance $r$ as a power-law $1/r^\alpha$. Using linear response theory, we show that the heating time is exponentially long as a function of the drive frequency for $\alpha>D$. For systems that may not obey linear response theory, we use a more general Magnus-like expansion to show the existence of quasi-conserved observables, which imply exponentially long heating time, for $\alpha>2D$. We also generalize a number of recent state-of-the-art Lieb-Robinson bounds for power-law systems from two-body interactions to $k$-body interactions and thereby obtain a longer heating time than previously established in the literature. Additionally, we conjecture that the gap between the results from the linear response theory and the Magnus-like expansion does not have physical implications, but is, rather, due to the lack of tight Lieb-Robinson bounds for power-law interactions. We show that the gap vanishes in the presence of a hypothetical, tight bound.

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Forward citations

Cited by 2 Pith papers

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    ( 29) We now provide a rigorous proof of Eq

    Proof of Eq. ( 29) We now provide a rigorous proof of Eq. ( 29) in the main text. Equation ( B2) says that the ( i,j ) entry of σ([ω,ω +δω]) is exponentially suppressed. In principle, summing over all i,j implies that σ([ω,ω +δω]) is also exponentially small as a function of ω...

  40. [49]

    Structure of Gq for q < q max First, we prove the statement of Lemma 1 that the operatorsGq are also power-law Hamiltonians for all q < qmax. Proof. We proceed by induction and assume that Lemma 1 holds for all q up to q =q0 − 1 for some q0 ≥ 1. We now prove that it also holds...

  41. [50]

    ( 44), which is a similar result to Lemma 1, but for q ≥qmax =ω∗

    Structure of Gq for q ≥ qmax We now prove Eq. ( 44), which is a similar result to Lemma 1, but for q ≥qmax =ω∗. Lemma 2. For all q ≥ qmax = ω∗, Gq ∈ Ce −κ′qHα, whereC and κ′ are constants. 14 Proof. Let us first look at the first term in Eq. ( 33): Gq,1 = q∑ k=1 (−1)k k! ∑ 1≤i1,...

  42. [51]

    [ 23]’s bound First, we consider a generalization of the bound in Gong et al

    Using Gong et al. [ 23]’s bound First, we consider a generalization of the bound in Gong et al. [23] [See also Eq. ( A12)]. The bound holds for α>D , has a logarithmic light cone t /greaterorsimilarlogr, and is extended to many-body interactions. To bound the com- mutator norm...

  43. [52]

    [ 19]’s bound Instead of using Gong et al

    Using Else et al. [ 19]’s bound Instead of using Gong et al. ’s bound, we now use the bound in Else et al. [19], which already holds for many- body interactions. The bound states that when |X| = 1, ‖[A(t),B]‖≤C‖A‖‖B‖ { exp ( vt−r1−σ) +(vt)1+D/(1−σ) rσ(α−D) } , (D9) where 1 >σ ...

  44. [53]

    [20]’s bound In addition to Else et al

    Using Tran et al. [20]’s bound In addition to Else et al. [19]’s bound, we can also use the bound in Tran et al. [20] [see also Eq. ( 24) for a generalization to k-body interactions], which also works for α > 2D. Compared to the bound in Else et al. , the bound in Tran et al. ...

  45. [54]

    In Appendix E 2, we present some bounds on discrete sums

  46. [55]

    We recall from the main text that Hα is the set of power-law Hamiltonians with the exponent α

    Properties of the set Hα of power-law Hamiltonians In this section, we explore some properties of Hα that are useful for proving that the effective Hamiltonian is also power-law [See Appendix C]. We recall from the main text that Hα is the set of power-law Hamiltonians with the...

  47. [56]

    i, j ∈ X and i, j / ∈ Y , hold and ξ1 = 0 otherwise

    For example, ξ1 = 1 if all of the conditions in the first row, i.e. i, j ∈ X and i, j / ∈ Y , hold and ξ1 = 0 otherwise. as a sum over the nine cases: ∑ Z∋i,j ‖hZ‖ = ∑ X∪Y ∋i,j ‖[aX,b Y ]‖ ≤ 2 ∑ X ∑ Y ‖aX ‖ ‖bY ‖ξ(X ∩Y ⁄= ∅)ξ(X ∪Y ∋i,j ) = 2 9∑ n=1 ∑ X ∑ Y ‖aX ‖ ‖bY ‖ξ(X ∩Y ⁄= ...

  48. [57]

    Bounds on discrete sums In this section, we provide bounds on some discrete sums used in the main text. Lemma 4. For all 1 ≤ k ≤ q, we have the following inequalities: ∑ 1≤i1,...,ik≤q i1+···+ik=q k∏ j=1 ij! ≤ q! (k − 1)!, (E10) ∑ 0≤i1,...,ik≤q i1+···+ik=q k∏ j=1 ij! ≤ 2kq!. (E...

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Reviewed August 14, 2026 · model on record in the stance chip above.