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A single spin system can host two different time crystals at once

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2026-08-03 14:19 UTC pith:2SF433OK

load-bearing objection A plausible numerical claim of coexisting DTC orders under nonuniform drive, but the missing regional spectra and the projection-artifact concern make it unverified as written. the 4 major comments →

arxiv 2512.20603 v2 pith:2SF433OK submitted 2025-12-23 quant-ph cond-mat.quant-gascond-mat.stat-mech

Coexistence of inequivalent time-crystalline orders in a Floquet collective spin system

Shashank Mishra , Sayan Choudhury This is my paper
classification quant-ph cond-mat.quant-gascond-mat.stat-mech
keywords discrete time crystalFloquet drivingLipkin-Meshkov-Glick modelphase coexistencechimera statesemiclassical dynamicsout-of-time-order correlatorsubharmonic response
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether a single, globally coupled spin system can break discrete time-translation symmetry in two inequivalent ways in different spatial regions at the same time. It argues yes: applying a periodic drive whose amplitude differs between two halves of an all-to-all interacting spin model—the Lipkin-Meshkov-Glick model—drives one half into a period-6 discrete time crystal and the other into a period-2 time crystal under identical global parameters. The coexistence is established in the thermodynamic limit via semiclassical equations and confirmed at finite size by exact diagonalization, with the fidelity out-of-time-order correlator matching the semiclassical stability map. The authors also find chimera-like states in which a time crystal coexists with a Floquet-synchronized period-T phase. If correct, this is a temporal analogue of equilibrium phase coexistence and a new route to engineering nested time-crystalline order.

Core claim

The central claim is that a spatially nonuniform drive on an all-to-all interacting spin model produces spatially separated, dynamically stable subharmonic responses with different periods. Concretely, at interaction strength J=0.5 and drive amplitude h1=0.17, a window around h2≈0.5 contains two contiguous regimes: one subregion locks to 6T, the other to 2T. The paper argues this is genuine coexistence of distinct discrete time-crystalline orders, not a single phase with an extra harmonic component, because the route from coexistence to a single-DTC phase passes through regimes where subharmonic order is lost. Additional coexistence windows appear (e.g., 2-DTC with 10-DTC and 2-DTC with 6-DT

What carries the argument

The driving protocol splits N spins into two equal halves with drive amplitudes h1 and h2; the Floquet operator alternates a global Ising interaction with the two-half transverse kicks. Dynamics are reduced to six semiclassical equations for the collective magnetizations of the two halves, which become exact as N→∞. Stability is diagnosed with the semiclassical decorrelator D(n), measuring divergence of nearby trajectories, and its quantum counterpart, the fidelity out-of-time-order correlator (FOTOC). Subharmonic periods are read off from the discrete Fourier transform of stroboscopic magnetization in each region. The machinery's role is to identify, for each (h1,h2), whether the trajectori

Load-bearing premise

The phase classification depends on treating 'vanishing' decorrelator values as stable time-crystalline order, but the paper never states a numerical threshold, so the boundaries of the coexistence regions and their robustness are not sharply defined.

What would settle it

Take the fixed line h1=0.17, scan h2 finely across 0.4–0.6, and compute the long-time-averaged decorrelator with a pre-registered threshold (e.g., D<10^-4) for stable order; if the two contiguous stable windows with DFT peaks at 6T and 2T do not both appear, or if a single region shows both peaks simultaneously, the coexistence claim is refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the coexistence is correct, a single all-to-all coupled system can be in two different time-crystalline phases at once, extending the concept of phase coexistence to the temporal domain.
  • The dependence of one half's subharmonic response on the other half's drive amplitude (e.g., h2 changes region 1's period) shows long-range interactions can mediate nonlocal control of time-crystalline order.
  • The matching semiclassical decorrelator and finite-size FOTOC maps provide a practical diagnostic for detecting coexisting time crystals in experiments.
  • The narrow parameter windows for coexistence suggest that spatial drive inhomogeneity is a switchable control knob: small changes in one drive amplitude select between coexisting 2T/6T, 2T/10T, or single-phase behavior.
  • The chimera DTC (time crystal alongside period-T synchronization) extends the known chimera state phenomenology to ordered-ordered coexistence, not just order-disorder.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next step the paper does not take is to compute a spatial order parameter for each subregion and check whether the coexistence domain obeys a Gibbs-phase-rule-like condition; if it does, the temporal coexistence might be describable by an effective free energy.
  • The claim that coexistence is not a trivial harmonic component could be tested further by measuring the phase coherence between the two subharmonic responses; genuine independent symmetry breaking would show a fluctuating relative phase, whereas a harmonic component would lock it.
  • The same two-half drive construction could be applied to other long-range models (e.g., central-spin or cavity-mediated spin ensembles) to see whether the coexistence windows are universal in the large-N limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript studies a Floquet LMG model in which two equal-size regions of a globally coupled spin system are driven by different transverse field strengths h1 and h2. Using semiclassical equations of motion in the thermodynamic limit and exact diagonalization for N=100 spins, the authors construct phase diagrams from time-averaged decorrelators and FOTOCs. They claim that for h1=0.17 and h2≈0.5 one region stabilizes a 6-DTC while the other supports a 2-DTC, and that this coexistence is not a trivial harmonic of a single higher-period phase. Additional chimera-like states, where a DTC coexists with a period-T synchronized region, are also reported. The central claim is that a uniformly coupled collective spin system can spontaneously break discrete time-translation symmetry in two inequivalent, spatially separated ways.

Significance. If fully substantiated, the result would be a significant extension of time-crystalline order to spatially coexisting phases and would constitute a temporal analogue of equilibrium phase coexistence in a single globally coupled system. The paper has notable strengths: the semiclassical approach is validated against the published higher-order DTC hierarchy of Ref. [25], no free parameters are fitted, and the finite-size quantum calculations provide a cross-check. The main claim, however, is not established in the current manuscript because the regional spectral evidence, the definition of the stability threshold, and the direct refutation of the harmonic-projection concern are all missing or deferred to the supplemental material.

major comments (4)
  1. [Co-existing and Chimera-like DTC Order; Fig. 4] The central coexistence claim rests on regional DFT spectra that are not shown. The text states that 'the simultaneous presence ... is unambiguously established by the regional DFT spectra of L_z^(1) and L_z^(2), shown in ref. [43]' and that regional DFT plots are 'provided in the ref. [43]'. Ref. [43] is a placeholder for supplemental material, so the manuscript as submitted does not contain the evidence that distinguishes which region has which period. Since the total magnetization cannot resolve region-specific periodicities, the 6-DTC/2-DTC coexistence for h1=0.17, h2≈0.5 is not verifiable. Please include the regional DFT density plots (or equivalent) for this parameter set, preferably in an appendix.
  2. [Fig. 2; decorrelator threshold] The identification of stable time-crystalline regions relies on 'vanishing values' of the time-averaged decorrelator in Fig. 2, but no numerical threshold is defined. The text says small D(n) implies stable dynamics and finite values imply chaotic dynamics, but without a cutoff the phase boundaries — and therefore the existence and extent of the coexistence regions — are not well-defined. Specify the threshold used and demonstrate that the reported phase boundaries are stable under reasonable variations of that threshold. A post hoc threshold would make the claimed coexistence regimes arbitrary.
  3. [Conclusion: 'trivial harmonic' argument] The argument that the coexistence 'cannot be interpreted as a trivial harmonic component of a single higher-period phase' is not supported by the accompanying statement that transitions between coexistence and single-DTC regions pass through intermediate loss of subharmonic order. That statement concerns parameter-space boundaries, not the internal structure of the coexistence regime. A single period-6 global orbit can produce a period-2 magnetization in one coordinate (e.g., L2_z(t+2T)=L2_z(t)) while another coordinate has period 6. To exclude this projection artifact, the manuscript must show the regional Fourier content: for example, that the region claimed to be 2-DTC has a dominant 2T peak and no 6T subharmonic peak, or introduce independent order parameters for the 2T and 6T sectors and show they are nonzero in different regions. Without this, the central claim reduces to an unveri
  4. [Model: equations of motion] The semiclassical equations of motion, which generate the thermodynamic-limit phase diagram and the DFT spectra, are not presented in the main text; the manuscript says 'A complete derivation of these equations is provided in ref. [43]'. Ref. [43] is not an accessible appendix. Since the central results are numerical outputs of these equations, the equations (or at least their explicit form) should be included in an appendix so that the reader can check the dynamics and reproduce the phase diagram. This is a reproducibility issue that is load-bearing for the thermodynamic-limit claim.
minor comments (5)
  1. [Throughout] There are numerous typos, including 'Intorduction', 'semicalssically', 'stability of the stability', 'withe', and inconsistent spelling of 'decorrelator'/'decorrelator'. The paper should be carefully proofread.
  2. [Eq. (4), FOTOC] The definition of the fidelity OTOC is nonstandard; the text uses W = exp(iϵ(Sz1+Sz2)) but does not clarify how F(t) relates to the conventional OTOC or what value of ϵ is used. Please add a brief explanation and specify parameters.
  3. [Fig. 4] The caption and text do not specify how the DFT amplitudes are normalized or which color scale is used. This makes it difficult to compare the semiclassical and exact-diagonalization panels quantitatively.
  4. [Fig. 1] The caption does not define the horizontal axis or the units; the reader is left to infer that it is h for the uniform drive. Please specify the parameter range and the vertical quantities in the caption.
  5. [References [36,37]] The distinction from previous chimera-DTC works is described only qualitatively. A short quantitative comparison (e.g., model differences, order parameters, or period combinations) would strengthen the novelty statement.

Circularity Check

0 steps flagged

No significant circularity: the coexistence claim is derived from the Hamiltonian dynamics and validated against an independent benchmark, not from fitted parameters or self-citations.

full rationale

The paper's derivation chain is self-contained. The only inputs are the physical parameters (J, h1, h2) in the Floquet Hamiltonian (Eqs. 1-2); no constants are fitted to the data and no observable is used to define the model. The semiclassical equations are derived from the Hamiltonian and the regional magnetizations are computed by evolving those equations; the periodicities (2-DTC, 6-DTC, etc.) are read from DFT peaks of the stroboscopic magnetization, which is an emergent output rather than an input. The decorrelator and FOTOC are used as stability diagnostics, not as definitions of the periods themselves. The method is validated in the uniform-drive limit by reproducing the known higher-order DTC hierarchy of Pizzi et al. [25], which is independent of the present authors. The self-citations ([21], [22], [28], [41], [42]) appear only as background references for DTC mechanisms and LMG-model context; none of these carry the load of the coexistence claim. The conclusion's assertion that the coexistence is not a trivial harmonic component of a single higher-period phase is arguably under-supported, and the regional DFT spectra are deferred to the supplemental material, but this is an evidentiary weakness rather than a circular reduction. The paper does not define X in terms of Y, fit a parameter and call it a prediction, or rely on an author-invoked uniqueness theorem. Thus no circular step is established.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper's claims rest on the standard semiclassical treatment of the LMG model, the decorrelator/FOTOC stability diagnostics, and the DFT-peak classification of DTC periods. None of these are formal theorems; they are domain assumptions of the numerical analysis. No free parameters are fitted to target results; h1, h2, and J are physical inputs scanned to produce the phase diagram.

axioms (5)
  • domain assumption In the thermodynamic limit, the dynamics of the two collective spins L1 and L2 is exactly captured by six coupled semiclassical equations.
    Invoked in the Model section; the derivation is deferred to ref. [43]. Standard for all-to-all LMG models, but not shown in the main text.
  • domain assumption Vanishing decorrelator D(n) between two initially nearby trajectories signals stable time-crystalline dynamics; finite D(n) signals chaos.
    Used in Fig. 2 to define stable phases; no quantitative threshold is specified.
  • domain assumption Small FOTOC values correspond to stable time-crystalline dynamics and mirror the semiclassical decorrelator.
    Used to argue quantum stability; validation is shown only for uniform drive (Fig. 1) and then asserted to mirror for nonuniform drive (Fig. 3).
  • domain assumption Discrete Fourier transform peaks of stroboscopic magnetization at period qT identify a q-DTC and distinguish coexisting orders.
    Used to classify 2-DTC, 6-DTC, and 10-DTC phases from DFT density plots.
  • domain assumption N=100 exact diagonalization is representative of experimentally relevant finite-size behavior and of the thermodynamic limit.
    Finite-size comparison in Fig. 4; no finite-size scaling analysis is shown.

pith-pipeline@v1.3.0-alltime-deepseek · 5907 in / 10664 out tokens · 100079 ms · 2026-08-03T14:19:21.879371+00:00 · methodology

0 comments
read the original abstract

We investigate the dynamical phases that emerge in collective spin models subjected to a spatially non-uniform periodic drive. Taking the paradigmatic Lipkin-Meshkov-Glick (LMG) model as a concrete platform, we establish that a rich landscape of dynamical phases emerges when two regions of the system are driven with different field strengths, $h_1$ and $h_2$. Remarkably, despite the `all-to-all' nature of the interactions, the system can be driven into dynamical phases characterized by distinct kinds of discrete time crystal (DTC) orders in different parts of the system. Apart from these coexisting DTCs, tuning the driving field leads to the emergence of phases where DTCs coexist with Floquet-synchronized or oscillatory phases; the former has been dubbed a chimera DTC. Finally, we demonstrate that a tunable set of global DTC phases emerges when $h_1$ and $h_2$ are proximate. Crucially, these dynamical regimes can be observed both for experimentally relevant finite-size systems and in the thermodynamic limit. Our results establish spatially structured driving as a powerful route to realize non-equilibrium phase coexistence in collective spin systems.

Figures

Figures reproduced from arXiv: 2512.20603 by Sayan Choudhury, Shashank Mishra.

Figure 1
Figure 1. Figure 1: FIG. 1. Average Decorrelator and FOTOC (500-1000 cycles) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Time-averaged Decorrealtor (from [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Density plot of FOTOC average (500-1000 periods) [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. DFT density of the stroboscopic magnetization [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

discussion (0)

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Reference graph

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