{"id":"0cc24728-0db5-4c8b-adeb-53ea301409c1","arxiv_id":"1908.07530","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves, with one explicit assumption in the intermediate regime, that prethermal Floquet phases exist for power-law interacting systems with exponent alpha > d, and predicts a disorder-free one-dimensional prethermal discrete time crystal for 1 < alpha < 2.","lead":"This paper reports a proof that fast periodic driving can stabilize long-lived out-of-equilibrium phases in quantum systems with long-range interactions, and predicts a one-dimensional time crystal that needs no disorder. It matters because trapped-ion and Rydberg atom platforms naturally have long-range interactions, making the predicted prethermal time crystal a concrete target for quantum simulation experiments.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline 'prove ... alpha > d' overreaches: for d < alpha < 2d, including the 1D PDTC, the paper only proves short-time local dynamics (Theorem 3) and assumes, without proof, relaxation to the Gibbs state of D*.","rationale":"The reader's weakest assumption exactly identifies the same load-bearing gap: the Gibbs-relaxation assumption for d < alpha < 2d. My reading of the theorems confirms this is the single most important weakness. Theorem 3 only guarantees short-time accuracy, and the paper's own starred caveat in Sec. II.3 and Fig. 1(c) concedes the extra assumption. The numerical evidence for the 1D PDTC is suggestive and consistent, but it is not a proof and does not remove the need for the assumption in the thermodynamic limit. No other internal inconsistency appeared: the alpha > 2d theorem is carefully constructed, the R-ranged operator norm is a sensible technical tool, and the numerics are broadly consistent with the claimed phase diagram. Therefore the reader's CONDITIONAL verdict is appropriate, with the condition being independent verification of the Gibbs-relaxation step (or an improved Lieb-Robinson bound for d < alpha < 2d). I would not strengthen to REJECT because the overclaim is explicitly flagged in the paper, and the regime alpha > 2d appears sound. The concrete test I propose directly targets the unproven assumption in the parameter regime that matters most for the paper's headline prediction.","tokens_in":44681,"tokens_out":2397,"duration_ms":464230,"concrete_test":"For the model in Eq. (46) with 1 < alpha < 2 (e.g., alpha = 1.13) and moderate sizes (L = 12-16), construct D* numerically to high order in the Floquet-Magnus expansion (or via the iterative frame rotation of Theorem 1) and compare, at times tau_pre < t < tau*, the driven system's local observables (e.g., and M(t)) with the Gibbs ensemble e^{-beta D*}/Z at the beta set by the initial energy density. If the difference does not shrink with increasing drive frequency and system size, the entropy-maximization assumption fails and the PDTC claim in this regime is unsupported. A complementary check: compare against the same Gibbs ensemble for 2 < alpha < 4 (where Theorem 2 applies) to confirm the numerical procedure detects a real distinction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract and Conclusion claim proven prethermal phases for all alpha > d, but the analytical results only establish that local observables follow the prethermal Hamiltonian for alpha > 2d (Theorem 2, Sec. III.4.2). For d < alpha < 2d, Theorem 3 bounds the difference between the true and approximate dynamics only for times m*lambda <= C7, i.e. a few drive periods, not the full prethermal window. To bridge this gap, Sec. II.3 and Sec. III.4.3 explicitly assume that 'local observables still relax to the Gibbs state of D*' because it maximizes entropy under energy conservation. This is a physical conjecture, not a theorem, and the paper marks it with a star in Fig. 1(c). The 1D PDTC is predicted precisely in the regime 1 < alpha < 2, where the assumption is most needed: without it, nothing in the proof guarantees that the system prethermalizes into the symmetry-broken sector of D* or that the magnetization remains period-doubled until tau*. The finite-size rigorous statements in Sec. II.2 require omega >> log(Lambda) and only give tau_O ~ Lambda^{-1} e^{omega/J}, so they do not establish the thermodynamic-limit phase either. Thus the central claim 'prove ... alpha > d' is not supported by the theorems in the regime that hosts the paper's flagship prediction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an analytic construction of a prethermal Floquet Hamiltonian D* for periodically driven, long-range interacting systems, while keeping track of the spatial locality of the interactions. Theorem 1 provides a locality-preserving rotation such that the driven evolution is exponentially well approximated by X exp(-iD*T). Theorem 2 proves that local observables follow D* throughout the prethermal window when the effective range exponent satisfies gamma* > d, which for two-body power-law interactions corresponds to alpha > 2d. Theorem 3 gives a short-time approximation for d < alpha < 2d, and the paper argues that in this regime local observables should relax to the Gibbs state of D* by entropy maximization under approximate energy conservation. The authors also present Krylov-subspace numerics for a one-dimensional long-range spin chain, reporting a disorder-free prethermal discrete time crystal for 1 < alpha < 2, together with quantum Monte Carlo estimates of the energy-density transition. The abstract and conclusion claim a proof of prethermal phases for all alpha > d.","tokens_in":44909,"tokens_out":5421,"duration_ms":57600,"significance":"If the analytic results are correct, this is a substantial step: it extends the emergent-symmetry framework for prethermal Floquet phases from short-range to long-range interactions, and it explains why a disorder-free one-dimensional prethermal discrete time crystal can exist. The paper's strengths include the explicit R-ranged operator formalism, the improved short-range iteration with n* free of logarithmic corrections, the detailed bounds in Appendices A-D, and the large-scale numerical study with independent QMC input. However, the rigorous proof of local-dynamics approximation is confined to alpha > 2d; the flagship 1 < alpha < 2 prediction rests on an explicitly stated but unproved entropy-maximization/Gibbs-relaxation assumption. The manuscript would be fully appropriate for publication once the claims are restated precisely and the conditional nature of the d < alpha < 2d phase is acknowledged in the abstract and conclusion.","major_comments":[{"comment":"The headline claim \"we prove the existence of non-equilibrium phases ... with power-law exponent alpha > d\" is not supported by the theorems as stated. Theorem 2 (Sec. III.4.2, Eq. (40)) requires gamma* > d, which for two-body power-law interactions corresponds to alpha > 2d. In the complementary regime d < alpha < 2d, Theorem 3 (Sec. III.4.3, Eq. (41)) bounds local observables only for m lambda <= C7, i.e. a few periods, and the extension to the full prethermal window explicitly invokes the unproven assumption that local observables relax to the Gibbs state of D* (Sec. II.3; star in Fig. 1(c)). The 1D PDTC prediction lives precisely in 1 < alpha < 2, so the abstract and conclusion overstate the rigorous content. I recommend rewording the claim to \"prove for alpha > 2d and predict, under a stated entropy-maximization assumption, for d < alpha < 2d.\"","section":"Abstract; Sec. II.3; Sec. III.4.3; Conclusion"},{"comment":"Even granting the Gibbs-relaxation assumption, the argument that the system exhibits period-doubled magnetization until tau* requires that the prethermal dynamics select a single symmetry-breaking sector of D*. The paper asserts this in Sec. II.1 (\"rho can instead approach the equilibrium state within a particular symmetry-breaking sector\"), but supplies no argument for why the driven dynamics, which has the symmetric Gibbs state as a fixed point of X exp(-iD*T), breaks this symmetry in the thermodynamic limit within the prethermal window. Since this sector selection is the mechanism underlying the PDTC, it should be explicitly identified as part of the conjecture for d < alpha < 2d rather than presented as an immediate consequence of energy conservation.","section":"Sec. II.1 and Sec. III.4.3"},{"comment":"The numerical inference of an energy-density phase transition in the PDTC is indirect. The quantum Monte Carlo estimate in Appendix H computes the transition for the zeroth-order Hamiltonian D, not for the full frequency-dependent D*, and uses sign-flipped couplings and a modified long-range profile (Eq. (H2)). The comparison in Fig. 4(b) therefore tests whether the onset of exponential tau_TC tracks the transition of D, which is plausible only to the extent that D* is close to D. The paper does not provide a direct numerical construction or estimate of D*'s transition, so the statement that the observed crossover matches the D* transition should be softened or supplemented with a D*-based check.","section":"Sec. IV.3; Appendix H; Fig. 4"}],"minor_comments":[{"comment":"The bounds on ||V*|| and ||E*|| are written as mu (1/2)^{-n*}, which grows exponentially in n*; from the surrounding text and Eq. (A46) the intended exponent is (1/2)^{n*}.","section":"Appendix A, Eqs. (A32)-(A33)"},{"comment":"In the geometric-series evaluation following Eq. (26), a factor e^{sigma(gamma - alpha + d)} appears to be missing in the numerator; the numerical value of the bound should be checked.","section":"Sec. III.2, Eq. (27)"},{"comment":"Theorem 4 states a bound with norm ||.||_{kappa*,gamma*} although the short-range norm introduced in this appendix has only a kappa parameter; this appears to be a typographical carryover from Theorem 1.","section":"Appendix A, Eq. (A14)"},{"comment":"The caption and text report that the extracted J_local is larger in the long-range model, but no explicit values or fitting ranges are given; stating the extracted J_local would make the exponential scaling in Fig. 3 quantitatively reproducible.","section":"Sec. IV.2; Fig. 3"},{"comment":"The statement \"for small enough lambda, m_max > 1, so one can at least accurately describe the dynamics ... during a single driving period\" is correct but should be accompanied by the observation that this window is parametrically shorter than the prethermal window tau* ~ 2^{n*}, since otherwise the reader may overestimate what Theorem 3 establishes.","section":"Sec. III.4.3, Eq. (41)"}],"recommendation":"major_revision","confidential_remarks":"The proof of Theorem 2 relies on Lemma 3 in Appendix C, which is presented as a corollary of Ref. [80], an arXiv preprint by three of the present authors (Else, Machado, Nayak, Yao). If that preprint has not been independently accepted, Theorem 2 is not self-contained by the usual journal standards. I would ask the editor to verify the status of Ref. [80] before final acceptance. The related complementary work by Tran et al. (Ref. [98]) is appropriately acknowledged in the note added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a strong paper with a real technical contribution, but its headline claim is slightly ahead of its theorems. The R-ranged potential construction is genuinely new and does what prior frame-rotation proofs couldn't: it keeps track of spatial range separately from support size, so the prethermal Hamiltonian inherits power-law decay. For alpha > 2d the local-dynamics statement (Theorem 2) is argued carefully, and the numerics are solid. The 1D PDTC is a nice prediction, and the comparison with the MBL time crystal is a useful addition.\n\nThe soft spot is exactly where the abstract goes further than the body. The abstract says 'prove ... alpha > d'. What is proven for d < alpha < 2d, including the 1 < alpha < 2 regime where the time crystal lives, is much weaker: Theorem 3 only controls local observables for m*lambda <= C7, a few periods. To get from there to a prethermal phase, Sec. II.3 and Sec. III.4.3 assume that local observables relax to the Gibbs state of D*, on entropy-maximization grounds. That is a physical conjecture, not a theorem, and it is flagged with a star in Fig. 1(c). It may well be right, but the abstract and Conclusion present it as established. The finite-size rigorous statements require omega >> log Lambda and only give tau_O ~ Lambda^{-1} e^{omega/J}, so they don't close the thermodynamic-limit gap either.\n\nOther soft spots are minor. The analytic proof leans on a Lieb-Robinson bound from the authors' own group; self-citation of that kind is normal, but the bound should be checked independently before the alpha > 2d theorem is relied on at full weight. No code or data are shipped; the numerics are described well enough to reproduce in principle, but the raw data would help.\n\nNet: a serious paper, worth refereeing and citing. I disagree with the reader's soundness score only in degree: I would call the construction solid and the overreach localized to the abstract and conclusion. The paper deserves a serious referee, with the main request being that the authors either prove or clearly label the Gibbs-relaxation assumption in every statement about d < alpha < 2d, and soften the abstract accordingly.","headline":"Real new construction, solid numerics; abstract overstates what is proven for dd$, including a disorder-free one-dimensional prethermal time crystal.","keywords":["prethermalization","discrete time crystals","long-range interactions","Floquet phases of matter","power-law interactions","emergent symmetry","nonequilibrium phases of matter","many-body localization"],"falsifier":"Prepare the one-dimensional long-range Floquet spin chain described in the paper with $1<\\alpha<2$, choose an initial state below the ferromagnetic critical energy density, and measure the period-doubling magnetization and a local observable's evolution for times up to $e^{\\omega/J}$. If the exact Floquet dynamics deviate from the evolution generated by $D^*$ on a time scale much shorter than the heating time, or if the period-doubling order decays at the prethermalization time instead of surviving to $\\tau_*\\sim e^{\\omega/J}$, the central claim is falsified. More directly, compare the local reduced state at intermediate times with the canonical ensemble of $D^*$ at the initial energy density; a mismatch in the $d<\\alpha<2d$ regime disproves the assumed relaxation.","tokens_in":44443,"feed_emoji":"🕐","tokens_out":9900,"duration_ms":93672,"temperature":0.7,"pith_summary":"This paper proves that a periodically driven quantum system whose interactions decay as a power law can host long-lived, non-equilibrium phases of matter before it heats, as long as the decay exponent $\\alpha$ exceeds the spatial dimension $d$. Previous proofs of prethermal phases required short-range interactions because the effective prethermal Hamiltonian could only be shown to generate local dynamics in that setting. The paper removes that restriction for $\\alpha>2d$ and, under one stated assumption about relaxation, for all $\\alpha>d$. It then predicts a disorder-free prethermal discrete time crystal in one dimension for $1<\\alpha<2$, a phase that equilibrium physics, many-body localization, and short-range prethermal Floquet systems all forbid. The predicted crystal is distinguished from an MBL time crystal by an energy-density phase transition and an exponentially long but finite lifetime.","feed_headline":"Prethermal phases persist in long-range driven matter for any α > d","feed_subtitle":"Periodically kicked long-range systems can host a disorder-free 1D time crystal, forbidden with short-range interactions","key_machinery":"The load-bearing object is a class of range-indexed potentials built from $R$-ranged sets: a set of lattice sites is $R$-ranged if any two of its sites can be joined by a chain of in-set hops of length at most $R$. Hamiltonian terms are organized by range $R_l=e^{\\sigma l}$ and weighted by a two-parameter norm that penalizes support size exponentially and spatial range as a power law. Iterating the frame-rotation construction for prethermalization while tracking this norm shows that $D^*$, $E^*$, and $V^*(t)$ inherit the original power-law decay, so the prethermal Hamiltonian stays local in the relevant sense. For $\\gamma>d$, this makes a power-law-light-cone Lieb-Robinson bound applicable, which is what upgrades energy conservation to genuine approximation of local observables. For $\\gammad$, the system possesses a prethermal Hamiltonian $D^*$ whose energy is conserved for an exponentially long time, and which, for $\\alpha>2d$, also correctly generates the dynamics of local observables throughout the prethermal window. For $d<\\alpha<2d$, the paper proves only short-time accuracy of local dynamics and assumes that local observables relax to the canonical ensemble of $D^*$ because that state maximizes entropy under the conserved energy. Under that assumption, prethermal phases exist for all $\\alpha>d$. The paper further predicts that in one dimension with $1<\\alpha<2$, where the prethermal Hamiltonian has a finite-temperature ferromagnetic transition, the driven system realizes a disorder-free prethermal discrete time crystal whose period-doubling order survives until the heating time $\\tau_*\\sim e^{\\omega/J}$, rather than melting at the prethermalization time.","pith_inferences":["If the assumed relaxation in the $d<\\alpha<2d$ window is confirmed, the same range-indexed construction should carry other prethermal phases, including symmetry-protected topological order, to long-range interactions.","The same technique extends to undriven static systems with a near-integer spectral operator, so a prethermal continuous time crystal should also exist in long-range interacting systems.","One clean test of the assumption is to compare the reduced density matrix of a small subsystem under exact Floquet evolution with the canonical ensemble of $D^*$ at the same energy density; agreement only for $\\alpha>2d$ and breakdown in $1<\\alpha<2$ would mark exactly where the proof relies on unproven relaxation.","The two-parameter norm may be useful beyond driven systems, wherever few-body terms act across arbitrarily large distances and one needs separate control of support size and spatial range."],"forward_implications":["For $\\alpha>2d$, long-range Floquet systems will host prethermal phases whose local dynamics are generated by $D^*$ for exponentially long times, with no disorder or many-body localization required.","In finite-size systems, the paper proves prethermal phases for every $\\alpha>d$ without the relaxation assumption, provided the drive frequency is large compared with the logarithm of the system volume.","A one-dimensional spin chain with interactions $\\propto |i-j|^{-\\alpha}$, $1<\\alpha<2$, should show period doubling that survives until the heating time whenever the initial state has energy density below the ferromagnetic critical value.","The prethermal time crystal and the trivial phase are separated by a sharp transition in initial energy density, unlike the MBL time crystal, which exists for generic initial states.","Platforms with power-law interactions, such as trapped-ion-style spin chains, can in principle observe the prethermal time crystal without introducing disorder."],"supporting_citations":[{"why":"Supplies the prethermal emergent-symmetry framework that the long-range construction extends.","marker":"[36]"},{"why":"Provides the iterative frame-rotation prethermalization theorem whose locality tracking is generalized.","marker":"[40]"},{"why":"Establishes exponentially slow heating for long-range systems with $\\alpha>d$, used for the conserved-energy step.","marker":"[38]"},{"why":"Gives the companion rigorous energy-absorption bound supporting the long-range heating-time estimate.","marker":"[39]"},{"why":"Provides the power-law-light-cone Lieb-Robinson bound used to prove local-observable tracking for $\\gamma>d$.","marker":"[80]"},{"why":"Proves the finite-temperature symmetry-breaking transition in one-dimensional long-range Ising models, which the 1D prethermal time crystal relies on.","marker":"[75]"},{"why":"Supplies earlier numerical evidence of exponentially slow heating in long-range Floquet systems that the present numerics build on.","marker":"[84]"},{"why":"Establishes discrete-time-crystal phase structure in driven systems, providing the comparison class for the prethermal time crystal.","marker":"[28]"}],"fun_headline_variants":["Long-range interactions enable prethermal time crystals","Disorder-free time crystal emerges in driven 1D long-range chain","Prethermal phases persist for α>d in driven long-range matter","Exponentially long-lived prethermal order in long-range driven systems","Forbidden phase realized: prethermal time crystal without disorder"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, for $d<\\alpha<2d$, local observables in the prethermal window relax to the thermal equilibrium state of the prethermal Hamiltonian because that state maximizes entropy under the conserved energy; the paper proves only short-time accuracy in this regime, so if that relaxation fails, the existence proof for these phases, including the one-dimensional disorder-free prethermal time crystal, does not go through.","fun_headline_variants_meta":{"raw":{"variants":["Long-range interactions enable prethermal time crystals","Disorder-free time crystal emerges in driven 1D long-range chain","Prethermal phases persist for α>d in driven long-range matter","Exponentially long-lived prethermal order in long-range driven systems","Forbidden phase realized: prethermal time crystal without disorder"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001044,"raw_usage":{"total_tokens":4330,"prompt_tokens":827,"completion_tokens":3503,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":3418}},"tokens_in":443,"tokens_out":3503,"duration_ms":26792,"temperature":1.0,"reasoning_tokens":3418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:05:15.880728+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Prepare the one-dimensional long-range Floquet spin chain described in the paper with $1<\\alpha<2$, choose an initial state below the ferromagnetic critical energy density, and measure the period-doubling magnetization and a local observable's evolution for times up to $e^{\\omega/J}$. If the exact Floquet dynamics deviate from the evolution generated by $D^*$ on a time scale much shorter than the heating time, or if the period-doubling order decays at the prethermalization time instead of surviving to $\\tau_*\\sim e^{\\omega/J}$, the central claim is falsified. More directly, compare the local reduced state at intermediate times with the canonical ensemble of $D^*$ at the initial energy density; a mismatch in the $d<\\alpha<2d$ regime disproves the assumed relaxation.","supporting_citations":[],"review_version":1}