{"id":"8d6114e3-5c94-4ed7-8cb4-6557b785b080","arxiv_id":"2504.19378","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact diagonalization shows a period-doubling discrete time crystal in the driven quantum Sherrington-Kirkpatrick model, whose stability tracks the Shannon entropy of the static spin-glass eigenstates.","lead":"This paper reports that a periodically driven quantum Sherrington-Kirkpatrick spin glass, with random all-to-all Ising couplings, shows a stable period-doubling response, the signature of a discrete time crystal. It matters because stable time crystals were thought to require short-range interactions, so the result points to a new class of long-range random systems for time-crystal physics.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The DTC conclusion rests on finite-time autocorrelations and static H2 diagnostics at L<=14; no Floquet spectral order or lifetime scaling is shown, so a prethermal period-doubling transient is not excluded.","rationale":"I agree with the reader's conditional verdict. The finite-time data and multiple protocols are genuine evidence for a long-lived period-doubling regime, but the paper's thermodynamic claim requires the non-ergodicity of H2 to survive and the Floquet evolution to remain non-heating. Neither is demonstrated. A Floquet spectral analysis or a lifetime scaling test would settle the issue. Because this concern is the same missing support identified by the reader, the verdict should remain conditional; no change is required.","tokens_in":16527,"tokens_out":10524,"duration_ms":120929,"concrete_test":"For protocol I at J=2.0 and random hx=0.2, compute the order parameter O_N and the late-time envelope of |C(nT)| for N=10^2, 10^3, and 10^4 periods at L=10, 12, 14, and 16 (using Krylov time evolution for L=16). If the envelope decays in N or if the extracted decay rate does not decrease systematically with L, the observed period doubling is a prethermal finite-size effect rather than a DTC phase; if instead the envelope is flat and the decay rate shrinks with L, the DTC interpretation is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a robust DTC phase in the thermodynamic limit. The evidence is persistent period doubling in C(t) for up to 600T, an order parameter O>0.5 at L=14, and static H2 diagnostics (Poisson level spacing, low Shannon entropy) for L<=14. The weakest link is the inference from those finite-size static diagnostics to the absence of Floquet heating. In the all-to-all SK model each spin is coupled to O(L) others, so perturbative resonances proliferate with L; the paper itself cites the delocalization tendency of random long-range spin-preserving interactions (refs. [45,51]). Poisson statistics and S/ln(2^L)<0.5 at L=10-14 do not establish MBL in the thermodynamic limit. Moreover, a DTC is a property of the Floquet unitary U_F, not just of H2: one must show that Floquet eigenstates come in symmetry-related pairs with long-range order. The paper never analyzes the Floquet spectrum or the system-size dependence of the period-doubling lifetime. Since prethermal plateaus can persist for exponentially long times, t=600T (or the unshown claim of 10^4 periods) cannot distinguish a transient from an asymptotic phase.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the periodically driven quantum Sherrington-Kirkpatrick (SK) model under three two-step driving protocols: a spin-flip pulse followed by evolution under H2, which contains random all-to-all Ising zz couplings plus either a transverse field (protocol I), nearest-neighbor XY coupling (protocol II), or only Ising zz interactions (protocol III). Starting from random product states in the z basis, the authors compute stroboscopic spin autocorrelations and a DTC order parameter, observing period-2T oscillations with a peak at half the drive frequency. They identify DTC regions in the J-hx and J-g planes and correlate them with the level-spacing ratio and Shannon entropy of the eigenstates of H2. The central claim is that the quantum SK model hosts a robust DTC phase despite all-to-all random long-range interactions, that randomness in the transverse field or XY coupling broadens the DTC region, and that uniform power-law spin-preserving interactions do not stabilize a DTC except in the presence of random XY couplings.","tokens_in":16772,"tokens_out":7941,"duration_ms":87517,"significance":"If correct, the result would substantially extend the DTC paradigm from short-range disordered systems to all-to-all random spin glasses and would identify random spin-flip terms as an important stabilizing ingredient. The paper's strengths are the use of three driving protocols, a direct comparison with uniform power-law models, and the use of independent static diagnostics (r-ratio and Shannon entropy) that are not fitted to the dynamical order parameter. However, the evidence for a thermodynamic DTC phase is currently indirect: the system sizes are limited to L=14 (L=20 for protocol III), the evolution times are finite (150-600 periods, with a mention of 10^4 periods but no data shown), no Floquet spectral analysis is presented, and no error bars are given for the phase boundaries. The paper is therefore a useful numerical study whose central claim needs additional support before it can be accepted at the level of a robust phase.","major_comments":[{"comment":"The inference from finite-time period doubling to a 'robust DTC phase' in the thermodynamic limit is not yet supported. The largest systems are L=14 (L=20 only for protocol III), the displayed evolution times are 150-600 periods, and the statement in the text that oscillations persist to a time scale of 10^4 is not backed by a figure or a quantitative analysis. Since prethermal time-crystalline plateaus can persist for times exponential in a control parameter, the observed C(t) and order parameter O do not by themselves distinguish a genuine DTC from a transient. I request a finite-size scaling of the period-doubling lifetime at fixed parameters (for example, the time at which O decays below a threshold as a function of L) or an analysis of the Floquet spectrum, showing symmetry-related Floquet eigenstate pairs and long-range order in the eigenstates. Without one of these, the thermodynamic 'phase' claim should be softened to a finite-time, finite-size observation.","section":"Section III, protocols I-III (Figs. 2, 3, 6, 8)"},{"comment":"The claimed phase boundaries hc, gc, and the epsilon-window in protocol III are extracted by eye from color plots of the order parameter, with no quantitative criterion and no statistical uncertainty. The order parameter is computed for a single initial state in the phase diagrams, averaged over 25-50 disorder realizations, and the threshold used to call a region 'DTC' (the text mentions O>1/2) is not defined precisely. Since a central conclusion is that the DTC region is broader for random transverse fields and random XY couplings, the authors should define the criterion (for example, time-averaged O>0.5 over a fixed number of periods) and report the disorder and initial-state statistics, including the variance of O across realizations.","section":"Section III, phase diagrams (Figs. 3, 5, 6, 8)"},{"comment":"The non-ergodicity of H2 is established only at L=10-14 via the level-spacing ratio and the normalized Shannon entropy, and the paper then uses this to argue that localized eigenstates of H2 stabilize the DTC. For all-to-all couplings, the number of perturbative resonances grows with system size, and the manuscript itself cites works (refs. [45,51]) showing a delocalization tendency of random long-range spin-preserving interactions. Poisson level statistics at L<=14 and S/ln(2^L)<0.5 are therefore not sufficient to conclude MBL-like non-ergodicity in the thermodynamic limit. The sentence in Section III, 'If H2 represents a non-ergodic system whose eigenstates are MBL', is an assumption rather than a demonstrated fact. Please add finite-size scaling of the r-ratio and Shannon entropy at fixed parameters, or explicitly restrict the claims to finite-size non-ergodicity. This is load-bearing because the stability of the DTC is attributed to this non-ergodicity.","section":"Section III and Appendix A (Figs. 4, 7, 10)"}],"minor_comments":[{"comment":"The statement that the DTC order parameter 'follows Shannon entropy very closely' is qualitative; a scatter plot of O versus S/ln(2^L) across the parameter plane, or a correlation measure, would make this quantitative.","section":"Section III, Fig. 3"},{"comment":"The choice N=300 in the time-averaged order parameter is not discussed; please state the time window in periods and justify that the average has converged.","section":"Section II, Eq. (4)"},{"comment":"The negative results for power-law interacting systems are presented for alpha=1,2 and L=14; a brief comment on finite-size effects for these conclusions would be helpful, since the absence of period doubling is itself a finite-size statement.","section":"Section IV, Fig. 9"},{"comment":"The uniform-XY data in Appendix B are averaged over only 10 disorder realizations; it would be useful to report error bars or at least state that the qualitative conclusions are stable across the realizations.","section":"Appendix B, Fig. 11"},{"comment":"The text uses 'MBL' for an all-to-all model without spatial locality; conventional MBL refers to local Hamiltonians, so 'non-ergodic' or 'localized' would be safer terminology unless the authors define precisely what they mean by MBL in this context.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main physical question is interesting and the paper contains useful numerical evidence, but the central claim of a robust thermodynamic DTC phase is stronger than what the finite-size, finite-time data currently establish. The requested Floquet spectral analysis and lifetime/finite-size scaling are natural extensions rather than impossible demands; if they cannot be provided, the manuscript should be revised to present the results as finite-time period doubling in finite systems, which would be a less striking but still publishable claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nFirst thing to know: this is a careful ED study of a question nobody had asked—whether the all-to-all random SK model supports a DTC. The authors report yes, for three drive protocols, with random transverse field or random XY coupling broadening the DTC window. That is a genuinely new numerical result, and the comparison with uniform power-law models is a useful addition: it shows random Ising couplings matter, and random XY can even stabilize a DTC in an otherwise uniform power-law Ising chain.\n\nWhat's good: the model choice is apt, the protocols are described cleanly, and the diagnostics are standard. The correlation between the DTC order parameter and low Shannon entropy of H2 is a nice, honest mechanistic story. They also explicitly say not all of the non-ergodic region becomes DTC, which is the right kind of nuance.\n\nThe soft spot is the extrapolation. The central claim is a robust DTC phase in the thermodynamic limit. The evidence is persistent period doubling up to ~600T (or 10^4 periods in one unreported check), an order parameter O>0.5 at L=14, and static H2 diagnostics—Poisson level spacing and low Shannon entropy—for L up to 14. But none of that rules out a prethermal DTC: there is no Floquet spectral analysis (no symmetry-paired eigenstates of U_F), no system-size scaling of the DTC lifetime, and the paper itself cites the delocalization tendency of random long-range spin-preserving interactions. Poisson r at L=14 is far from a proof of MBL in an all-to-all model. The O>0.5 threshold is also used without error bars, and no code or data are provided.\n\nNone of these are fatal. The numerical data are consistent with the paper's reading; they just don't exclude the prethermal alternative. A serious referee could ask for Floquet eigenstate pairing, lifetime scaling with L, and error bars before endorsing the thermodynamic claim.\n\nThis paper is for the MBL/DTC and spin-glass dynamics crowd. I'd bring it to a reading group as a case study in how far finite-size ED can push a DTC claim. Send it to peer review: the question is important enough, and the evidence is sharp enough, to warrant referee time even though the conclusion as stated is stronger than the data justify. I'd be inclined to accept after major revision.","headline":"Careful ED study of DTC in the SK model, but the thermodynamic-phase claim outruns L<=14 and 600T; worthwhile refereeing.","tokens_in":17295,"tokens_out":3440,"would_cite":true,"duration_ms":33007,"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":"This paper argues that a periodically driven quantum Sherrington-Kirkpatrick spin glass can host a stable discrete time crystal phase despite its all-to-all random Ising interactions.","keywords":["discrete time crystal","Sherrington-Kirkpatrick model","quantum spin glass","many-body localization","Floquet driving","long-range interactions","time-translation symmetry breaking","Shannon entropy"],"falsifier":"At a claimed DTC point, for example J=2.0 with random $h^x$ of strength 0.2, measure the decay time of the period-doubling autocorrelation for L=10, 12, 14, and 16: if the decay time does not grow with system size, or if the eigenstates of the full Floquet unitary $U_F$ show level repulsion instead of two separated localized sectors whose eigenphases differ by $\\pi$, then the order is prethermal, not a stable DTC.","tokens_in":16319,"feed_emoji":"💎","tokens_out":11117,"duration_ms":102827,"temperature":0.7,"pith_summary":"Discrete time crystals are periodically driven phases whose observables spontaneously oscillate at twice the drive period, breaking discrete time-translation symmetry. This paper argues that the quantum Sherrington-Kirkpatrick (SK) model, an Ising spin glass with all-to-all random couplings, supports a stable discrete time crystal despite the long-range nature of its interactions. Across three driving protocols, the spin autocorrelation shows persistent period-doubling oscillations, and the parameter region where they appear coincides with low Shannon entropy and Poisson level statistics of the undriven Hamiltonian's eigenstates. If correct, this shows that random long-range couplings do not preclude time-crystalline order, overturning the usual assumption that stable time crystals require short-range interactions.","feed_headline":"All-to-all spin glass can host a stable time crystal","feed_subtitle":"All-to-all quantum spin glass keeps period-doubling despite fully random long-range couplings.","key_machinery":"The load-bearing object is the two-step Floquet unitary $U_F = e^{-iH_2} e^{-iH_1}$, with $H_1 = \\frac{\\pi}{2}\\sum_i \\sigma_i^x$ acting as a perfect spin flip and $H_2$ the quantum SK Hamiltonian, namely $\\sum_{i<j} J_{ij}\\sigma_i^z\\sigma_j^z$ plus either a transverse field $\\sum_i h_i^x \\sigma_i^x$ or nearest-neighbour XY coupling $\\sum_i g_i(\\sigma_i^x\\sigma_{i+1}^x+\\sigma_i^y\\sigma_{i+1}^y)$. The diagnostic is the stroboscopic spin autocorrelation $C(t) = \\langle \\psi_0|\\sigma_i^z(t)\\sigma_i^z(0)|\\psi_0\\rangle$, whose persistent period-2 oscillations with a sharp Fourier peak at half the drive frequency define the DTC. The mechanism invoked is eigenstate localization of $H_2$: the normalized Shannon entropy $S(E_n)/\\ln(2^L)$, with $S(E_n)=-\\sum_i |\\psi_n(i)|^2 \\ln|\\psi_n(i)|^2$, and the level-spacing ratio $r_n$ are used to show that non-ergodic eigenstates keep the spin-flip information from heating, and the DTC order parameter $O = \\frac{1}{N}\\sum_n [(-1)^n C(nT)-C(nT)]$ tracks the low-entropy regime.","core_discovery":"The central claim is that the periodically driven quantum SK model exhibits a genuine DTC phase for a finite range of transverse-field or XY-coupling strengths, even though every spin pair is randomly coupled. The stability condition is traced to the eigenstates of H2, the undriven quantum SK Hamiltonian in the second half of the drive: when those eigenstates are many-body localized, with low normalized Shannon entropy and Poisson level-spacing ratio, an initial state overlaps only a small fraction of them, so the pure spin-flip pulse in the first half of the drive produces period-2 autocorrelation oscillations that persist without visible decay; when the eigenstates are ergodic, the oscillations decay. The DTC occupies only part of the non-ergodic region, and the DTC order parameter closely follows the eigenstate Shannon entropy of H2. In a comparison with power-law interacting models, uniform long-range couplings fail to stabilize the DTC under most protocols, while adding random XY couplings can induce it even there.","pith_inferences":["An extension the authors do not pursue: the phase boundary $h_c(J)$ could be estimated from the mobility edge of H2; if the Shannon-entropy contour of 0.5 at fixed J coincides with the DTC boundary at larger L, the correspondence would be a predictive tool.","If the finite-size non-ergodicity does not survive the thermodynamic limit, the observed DTC would instead be a very long prethermal plateau typical of all-to-all models; a trapped-ion or Rydberg simulator with programmable all-to-all couplings could distinguish the two by measuring autocorrelation decay over tens of thousands of cycles.","The random-XY result suggests a broader design rule: adding short-range random hopping to a non-thermalizing long-range spin Hamiltonian may generically stabilize period doubling, connecting DTC physics to random-hopping localization."],"forward_implications":["If the central claim is correct, the class of DTC-supporting systems extends from short-range disordered chains to all-to-all random spin glasses, so short-range interactions are not a necessary ingredient.","The DTC order parameter closely following the Shannon entropy of H2 means the DTC parameter window can be predicted from eigenstate localization of the undriven Hamiltonian alone.","In the third driving protocol the DTC phase width in the spin-flip error $\\epsilon$ is roughly independent of the SK coupling strength J, unlike protocols I and II, so the robustness window is set by the drive itself.","Random XY couplings stabilize period doubling even when the long-range Ising part is uniform, so a short-range random spin-flip term can rescue DTC order in otherwise uniform long-range models."],"supporting_citations":[{"why":"Defines the all-to-all random Ising couplings $J_{ij}$ with zero mean and variance $J/\\sqrt{L}$, the model under study.","marker":"[52]"},{"why":"Establishes the Floquet many-body-localization paradigm for period-doubling time crystals that this paper extends to fully connected random couplings.","marker":"[16]"},{"why":"Supplies the rigidity criterion for DTC order in disordered Floquet systems, which the paper leans on for identifying a stable phase.","marker":"[17]"},{"why":"States the common belief that stable DTCs require short-range interactions, the view this paper directly challenges.","marker":"[21]"},{"why":"Argues that random long-range spin-preserving interactions can avoid delocalization, supporting the premise that the quantum SK model can be non-ergodic.","marker":"[45]"},{"why":"Shows enhanced non-ergodicity in the quantum SK model, cited as evidence that long-range spin-preserving interactions strengthen localization.","marker":"[57]"},{"why":"Establishes non-ergodic eigenstates in the quantum SK model, providing the basis for linking H2 eigenstates to DTC stability.","marker":"[58]"},{"why":"Characterizes localization in random XY chains, used to explain why random XY couplings produce strongly localized eigenstates and thus a broad DTC phase.","marker":"[61]"},{"why":"Demonstrates prethermal DTC in clean long-range interacting systems, providing the comparison baseline that distinguishes the random-SK DTC from prethermal order.","marker":"[41]"}],"fun_headline_variants":["Time crystal emerges in all-to-all quantum spin glass","Quantum SK model supports robust discrete time crystal","Random all-to-all couplings yield stable time-crystal phase","Discrete time crystal found in fully connected spin glass"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the undriven spin-glass Hamiltonian H2 remains non-thermalizing for system sizes larger than the L=14 simulated here, so that it never heats up under the periodic drive and the period-doubling seen in finite-time simulations is a true phase rather than an extremely long transient.","fun_headline_variants_meta":{"raw":{"variants":["Time crystal emerges in all-to-all quantum spin glass","Quantum SK model supports robust discrete time crystal","Random all-to-all couplings yield stable time-crystal phase","Discrete time crystal found in fully connected spin glass"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000262,"raw_usage":{"total_tokens":1615,"prompt_tokens":984,"completion_tokens":631,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":569}},"tokens_in":600,"tokens_out":631,"duration_ms":6922,"temperature":1.0,"reasoning_tokens":569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:55:11.258885+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At a claimed DTC point, for example J=2.0 with random $h^x$ of strength 0.2, measure the decay time of the period-doubling autocorrelation for L=10, 12, 14, and 16: if the decay time does not grow with system size, or if the eigenstates of the full Floquet unitary $U_F$ show level repulsion instead of two separated localized sectors whose eigenphases differ by $\\pi$, then the order is prethermal, not a stable DTC.","supporting_citations":[{"cited_title":"Ippoliti, K","cited_arxiv_id":null,"evidence_quote":"States the common belief that stable DTCs require short-range interactions, the view this paper directly challenges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Argues that random long-range spin-preserving interactions can avoid delocalization, supporting the premise that the quantum SK model can be non-ergodic."},{"cited_title":"Mukherjee, S","cited_arxiv_id":null,"evidence_quote":"Shows enhanced non-ergodicity in the quantum SK model, cited as evidence that long-range spin-preserving interactions strengthen localization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes non-ergodic eigenstates in the quantum SK model, providing the basis for linking H2 eigenstates to DTC stability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterizes localization in random XY chains, used to explain why random XY couplings produce strongly localized eigenstates and thus a broad DTC phase."},{"cited_title":"Machado, D","cited_arxiv_id":null,"evidence_quote":"Demonstrates prethermal DTC in clean long-range interacting systems, providing the comparison baseline that distinguishes the random-SK DTC from prethermal order."}],"review_version":1}