{"id":"7d7ba228-c752-439b-aa53-62c67d67aab2","arxiv_id":"1908.02773","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Power-law interacting, periodically driven systems exhibit heating times exponential in drive frequency for alpha > D in linear response and alpha > 2D in general, with the gap attributed to the absence of tight Lieb-Robinson bounds.","lead":"Periodically driven quantum systems whose interactions decay as a power law with distance stay cold for exponentially long times when the decay is fast enough. The paper proves this for alpha > D under weak drives and for alpha > 2D in general, and traces the remaining gap to a missing mathematical bound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conjectured tight LR bound Eq. (50) violates the direct-interaction lower bound at short times, so the gap-closing argument in Sec. V does not hold as stated; main theorems are unaffected.","rationale":"The reader identified the conjectured tight LR bound, Eq. (50), as the weakest assumption for the intermediate regime D < alpha < 2D. My reading confirms that this is where the paper is least secure, but sharpens the concern: it is not merely that Eq. (50) is unproven; as written it is demonstrably invalid at short times. This matters because the abstract explicitly claims that the gap vanishes in the presence of a hypothetical tight bound, and Eq. (55) is the only quantitative support for that claim. However, the two main proven statements—linear response for alpha > D (Eq. (29)) and Magnus-type expansion for alpha > 2D (Eqs. (48)-(49))—use the existing LR bounds of Gong et al., Else et al., and Tran et al., and do not invoke Eq. (50). I checked the algebra in Appendices B and D at a high level and found no other showstopper; the paper is honest about what is proven versus conjectured. Because the flaw is confined to a clearly labeled conjectural section and is easily repaired, I would not reject the paper or mark it unverified, but I would ask for the conjecture/demonstration to be revised or explicitly restricted to t >= 1 with a separate small-time estimate. Hence CONDITIONAL rather than UNCHANGED.","tokens_in":27763,"tokens_out":33288,"duration_ms":403229,"concrete_test":"Analytically check Eq. (50) at short time: take D=1, alpha=2, choose A = sigma_i^x and B = sigma_j^y with |i-j|=r, and include in H a term (1/r^2) sigma_i^z sigma_j^z. Compute ||[A(t),B]|| to first order in t; it is 4t/r^2 for small t, while Eq. (50) with beta=1 gives (t/r)^2 = t^2/r^2, which is smaller whenever t<1. More generally, test the necessary condition C(t,r) >= c t/r^alpha as t -> 0 for any proposed LR bound; if the bound behaves as t^(alpha*beta)/r^alpha with alpha*beta > 1, it fails. Re-run the derivation of Eq. (55) with the corrected bound C max{t, (t^beta/r)^alpha}; the conclusion t_* exponential in omega survives with modified constants.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section V's derivation of an exponentially long heating time for D < alpha < 2D rests on the conjectured tight Lieb-Robinson bound Eq. (50): ||[A(t),B]|| <= C ||A|| ||B|| (t^beta / r)^alpha. This specific form cannot be a valid LR bound for all t. A two-body power-law term of strength 1/r^alpha between the supports of A and B produces, at first order in t, ||[A(t),B]|| >= c t / r^alpha. The paper's own choices beta = 1/(alpha - D) for D < alpha < D+1 and beta = 1 for alpha > D+1 both give alpha*beta > 1, so for every sufficiently small t < 1 the conjectured bound is C t^(alpha*beta)/r^alpha < c t/r^alpha. Thus Eq. (50) is too strong at short times and cannot be satisfied by any power-law Hamiltonian containing the direct interaction. The subsequent estimate Eq. (55), ||delta|| <= C exp(-kappa' omega_*) t^(beta D + 1), integrates the commutator from s = 0 using Eq. (50), so the 'gap vanishes' demonstration does not follow as written. The flaw is repairable by splitting off s < 1 with the direct-interaction bound or replacing t^beta by max{t, t^beta}, and it does not affect Section IV (alpha > D under linear response) or the Section V result for alpha > 2D using the Gong/Else/Tran bounds.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28110,"tokens_out":8780,"duration_ms":98505,"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":[{"comment":"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.","section":"Section V, Eq. (50) and the derivation of Eq. (55)"}],"minor_comments":[{"comment":"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.","section":"Appendix B, Eq. (B10)"},{"comment":"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.","section":"Appendix A"},{"comment":"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.","section":"Section V, discussion after Eq. (55)"}],"recommendation":"major_revision","confidential_remarks":"The main theorems (Section IV for alpha>D under linear response, Section V for alpha>2D for generic drives) appear sound and well supported. The flaw in the conjectured tight bound is local and repairable, so this is not a rejection. However, because the paper's explicit claim that the gap vanishes in the presence of the stated tight bound is not valid as written, the manuscript needs a revision before it can be accepted. I would be willing to review a revised version addressing the Eq. (50) issue and the minor corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the two main claims are in good shape — an exponentially long heating time for alpha > D under linear response, and one for alpha > 2D via the Magnus construction — and the paper deserves a serious referee. The genuinely new pieces are the clean linear-response result (no disorder needed, unlike Ho et al.) and the k-body generalizations of the Gong et al. and Tran et al. Lieb-Robinson bounds. The Magnus part also gets a better heating-time exponent than Kuwahara et al. by feeding in those bounds. Credit where due: the proofs are mostly detailed, the scope of each claim is stated, and the conjecture is clearly labeled as a conjecture.\n\nThe soft spot is the conjectured tight LR bound, Eq. (50). The stress-test note is right: for a direct two-body interaction of strength 1/r^alpha between the supports, first-order perturbation gives a lower bound ~ t/r^alpha. With the paper's beta = 1/(alpha D) or beta = 1, alpha beta > 1, so for sufficiently small t the conjectured bound C t^(alpha beta)/r^alpha is smaller than the physical lower bound. Eq. (50) is therefore not a valid LR bound for all times, and the derivation of Eq. (55) — which integrates the commutator from s = 0 — does not go through as written. This is repairable (replace t^beta by max{t, t^beta}, or split the s integral), and it does not touch Section IV or the alpha > 2D result in Section V. Since that part of Section V is explicitly a \"gap vanishes if\" argument, I would call this a moderate flaw in a secondary thread, not a fatal one.\n\nTwo smaller issues: the D > 1 extension of the convolution bound in Appendix A is only sketched, and Eq. (B10) drops an r^(D-1) prefactor that is indeed subdominant. Neither affects the exponential claims.\n\nThe paper is honest about the alpha > D versus alpha > 2D gap, and the citation pattern is fine. Ref. [20] is independently published and is itself generalized here, so the self-citation is not a problem.\n\nWho is this for: people working on prethermalization in long-range interacting systems, trapped-ion and Rydberg experiments, and Lieb-Robinson bounds for power-law interactions. I would send it to a competent referee; with a modest revision fixing the conjecture statement, it should be publishable.","headline":"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.","tokens_in":28643,"tokens_out":2165,"would_cite":true,"duration_ms":23739,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Floquet systems","prethermalization","heating time","power-law interactions","Lieb-Robinson bounds","linear response theory","Magnus expansion","long-range interacting systems"],"falsifier":"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.","tokens_in":27570,"feed_emoji":"⚛️","tokens_out":15009,"duration_ms":132390,"temperature":0.7,"pith_summary":"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.","feed_headline":"Driven power-law systems heat up only after an exponential time","feed_subtitle":"Results cover weak drives for α>D and strong drives for α>2D, extending prethermalization to long-range systems.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the linear-response bound on the response function for finite-range interactions that the paper extends to power-law interactions","marker":"[5]"},{"why":"supplies the Magnus-like prethermalization construction (periodic unitary, effective Hamiltonian) that Section V generalizes to power-law systems","marker":"[6]"},{"why":"established the prior exponential heating time for power-law interactions with α>2D that this paper improves","marker":"[8]"},{"why":"gave the prior exponential heating result for disordered power-law systems with α>D/2 that the paper extends to clean systems for α>D","marker":"[17]"},{"why":"provides the many-body Lieb-Robinson bound with algebraic light cone for α>2D used to derive Eq. (48)","marker":"[19]"},{"why":"provides the tighter algebraic light cone bound that the paper generalizes to k-body interactions and uses for Eq. (49)","marker":"[20]"},{"why":"supplies the first power-law Lieb-Robinson bound with logarithmic light cone for α>D, foundational for the α>D analysis","marker":"[22]"},{"why":"gives the logarithmic-cone Lieb-Robinson bound used to control off-diagonal response terms in Section IV and generalized to k-body interactions","marker":"[23]"}],"fun_headline_variants":["Power-law systems heat up exponentially slowly under periodic drives","Exponential heating times for driven systems with power-law interactions","Periodic driving delays heating in power-law interacting systems","Heating time exponential in frequency for α>D power-law drives","Exponential heating delay in periodically driven power-law systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Power-law systems heat up exponentially slowly under periodic drives","Exponential heating times for driven systems with power-law interactions","Periodic driving delays heating in power-law interacting systems","Heating time exponential in frequency for α>D power-law drives","Exponential heating delay in periodically driven power-law systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000295,"raw_usage":{"total_tokens":1739,"prompt_tokens":998,"completion_tokens":741,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":661}},"tokens_in":614,"tokens_out":741,"duration_ms":8178,"temperature":1.0,"reasoning_tokens":661,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:35:40.562574+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Therefore, [AM,B ] = 0, and C(t,r ) is at most the total error of the approximation, i.e","cited_arxiv_id":null,"evidence_quote":"supplies the linear-response bound on the response function for finite-range interactions that the paper extends to power-law interactions"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gave the prior exponential heating result for disordered power-law systems with α>D/2 that the paper extends to clean systems for α>D"},{"cited_title":"Because f (τ,ℓ ) is a decreasing function of ℓ, the bound Eq","cited_arxiv_id":null,"evidence_quote":"provides the many-body Lieb-Robinson bound with algebraic light cone for α>2D used to derive Eq. (48)"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the tighter algebraic light cone bound that the paper generalizes to k-body interactions and uses for Eq. (49)"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the logarithmic-cone Lieb-Robinson bound used to control off-diagonal response terms in Section IV and generalized to k-body interactions"}],"review_version":1}