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Discrete Time Crystal in quantum Sherrington-Kirkpatrick model

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Pith's one-line read 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.

desk verdict Careful ED study of DTC in the SK model, but the thermodynamic-phase claim outruns L<=14 and 600T; worthwhile refereeing. read the letter →

arxiv 2504.19378 v2 pith:WTNODECN submitted 2025-04-27 cond-mat.dis-nn

Aarya Bothra , Arti Garg This is my paper · ORCID
classification cond-mat.dis-nn
keywords discretetimecrystalSherrington-Kirkpatrickmodelquantumspinglassmany-bodylocalizationFloquetdrivinglong-rangeinteractionstime-translationsymmetrybreakingShannonentropy
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 5 minor

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.

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 (3)
  1. [Section III, protocols I-III (Figs. 2, 3, 6, 8)] 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.
  2. [Section III, phase diagrams (Figs. 3, 5, 6, 8)] 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.
  3. [Section III and Appendix A (Figs. 4, 7, 10)] 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.
minor comments (5)
  1. [Section III, Fig. 3] 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.
  2. [Section II, Eq. (4)] 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.
  3. [Section IV, Fig. 9] 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.
  4. [Appendix B, Fig. 11] 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.
  5. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DTC order parameter and the eigenstate diagnostics are independently computed, and no fitted quantity is renamed as a prediction.

full rationale

None of the paper's load-bearing steps reduces to its own inputs. The DTC order parameter O (Eq. 4) is computed directly from the stroboscopic autocorrelation C(nT), which is obtained by exact-diagonalization time evolution under the Floquet unitary. The Shannon entropy and level-spacing ratio are independent static diagnostics of the eigenstates of H2, computed from a separate spectral analysis. The observed overlap between the O>0.5 region and the S/ln(2^L)<0.5 region is presented as an empirical correlation, not as a derivation of one from the other; neither quantity is fitted to reproduce the other. The paper's self-citations (refs. 48, 50, 51, 57) supply background on SK-model non-ergodicity and long-range MBL, but the central DTC evidence is this paper's own numerical simulation, so those citations are not load-bearing. Concerns about finite-size scaling, prethermal transients, and the absence of a Floquet spectral analysis are legitimate scientific risks, but they are not circularity.

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

No free parameters are fitted to reproduce the DTC signal; J, hx, g and epsilon are scanned Hamiltonian parameters. The paper postulates no new particles or forces. Its theoretical input consists of standard ED linear algebra plus domain assumptions about non-ergodicity, MBL protection of DTC, and the operational O>0.5 criterion.

assumptions (5)
  • standard math Exact diagonalization gives exact finite-size dynamics for the driven SK model.
    Used in Sec. III to construct U2|Psi> = sum_n exp(-i E_n) c_n |Phi_n>; standard linear algebra.
  • domain assumption H2 has a non-ergodic eigenstate phase in the thermodynamic limit.
    Finite-size level spacing and Shannon entropy (Figs. 4, 7, 10) are used to infer a non-ergodic phase; no rigorous thermodynamic-limit argument is given.
  • domain assumption Non-ergodic eigenstates of H2 prevent heating under the periodic drive and stabilize the DTC.
    Invoked in Sec. III: "we associate the robustness of the observed DTC phase with the properties of eigenstates of H2"; imported from MBL-DTC literature.
  • domain assumption Watanabe-Oshikawa no-go theorem does not forbid discrete time crystals in long-range systems.
    Stated in the Introduction to motivate why a DTC in the SK model can exist; the theorem's scope is assumed.
  • domain assumption The order parameter O with N=300 and the O>0.5 threshold define the DTC phase.
    Phase diagrams (Figs. 3, 6, 8) are generated from this operational definition; its relation to spontaneous symmetry breaking is not established.

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Cite this review

Pith. "Pith review of Discrete Time Crystal in quantum Sherrington-Kirkpatrick model." pith.science (2026). https://pith.science/paper/WTNODECN

@misc{pith2026250419378,
  author       = {Pith},
  title        = {Pith review of: Discrete Time Crystal in quantum Sherrington-Kirkpatrick model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTNODECN}},
  note         = {Machine review of arXiv:2504.19378}
}
abstract

Discrete time crystals (DTC) have emerged as a significant phase of matter for out-of-equilibrium many-body systems. We study how long-range interactions and disorder contribute to the stability of the DTC phase. Generally, a stable DTC phase is believed to be realized in disordered systems with short-range interactions. In this work, we study periodically driven quantum Sherrington-Kirkpatrick (SK) model of Ising spin-glass in which all spins are randomly coupled. We investigate the possibilities of DTC phase in the SK model within three different driving protocols and found that the quantum SK model exhibits a robust DTC phase despite the long-range nature of interactions. The DTC phase in quantum SK model persists for a larger range of parameters if the $XY$ coupling or a transverse field is also random. This suggests that disorder in the $XY$ coupling or transverse field is also crucial for stabilising a broad DTC phase, despite the SK model's random couplings. Our analysis shows that the stability of the DTC phase is determined by the non-ergodic nature of the eigenstates of the quantum SK model, with the DTC order-parameter closely following the Shannon entropy of eigenstates. We compare the periodically driven SK model to alternative models of long-range interactions with uniform coefficients and found that the DTC phase is absent in these models for most of the driving protocols.

Figures

Figures reproduced from arXiv: 2504.19378 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic phase diagram of periodically driven [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Left panel shows autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) Order Parameter, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: First panel shows the level spacing ratio averaged over middle 500 eignstates of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The left panel shows auto-correlation function [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: First plot shows the auto-correlation function [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Level spacing ratio and normalized Shannon [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Auto-correlation function [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Auto-correlation function [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: First two panels show level spacing ratio and [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Auto-correlation function [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

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    2000

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Reviewed August 16, 2026 · model on record in the stance chip above.