Empirical validation of long-range order and hyperuniformity induced by inhomogeneous periodic driving in low-dimensional classical systems

Scaling theory and the exactly solvable spherical model predict that a driving force which is periodic in time but spatially inhomogeneous, $h(\mathbf{x},t)\propto\cos(\omega_0 t+\psi(\mathbf{x}))$ with a quenched random phase $\psi$, lowers the lower critical dimension for continuous symmetry breaking to zero at zero temperature, so true long-range order can exist in one and two dimensions, and that the conserved-order-parameter version is hyperuniform with $S(q)\propto q^2$ (Ikeda and Kuroda, Phys. Rev. E 110, 024140 (2024)). The authors explicitly left these predictions - the phase diagrams, the altered lower critical dimensions, and the hyperuniform scaling - unverified by simulation or experiment. Later work has numerically confirmed the central two-dimensional ordering scenario for athermal chiral active particles, but the predicted phase diagrams of the driven model itself at finite size and finite $n$, the one-dimensional case, any experimental confirmation, and the robustness to relaxing the idealizing assumptions (spatially uncorrelated driving phases, quadratic free energy, perfectly coherent drive) remain open. The problem is to determine, by direct simulation beyond the thermodynamic and large-$n$ limits and by experiment, whether inhomogeneous oscillatory driving genuinely stabilizes low-dimensional order and hyperuniformity as predicted, and how far these predictions survive relaxing the model assumptions.

Background

In equilibrium systems with short-range interactions, a continuous symmetry cannot be spontaneously broken in one or two spatial dimensions: the Nambu-Goldstone modes (the gapless long-wavelength deformation modes of an ordered phase) fluctuate so strongly that the order parameter averages to zero in the thermodynamic limit. This is the Hohenberg-Mermin-Wagner theorem (Mermin and Wagner 1966; Hohenberg 1967). The theorem is an equilibrium statement and does not bind nonequilibrium systems; several driven or active models are known to evade it, for example flocks, models with anisotropic or anticorrelated noise, and sheared systems.

Ikeda and Kuroda (2024) analyzed a previously unstudied class of evasion: forcing that is periodic in time but spatially inhomogeneous, $h(\mathbf{x},t)=\sqrt{2D}\,\cos\big(\omega_0 t+\psi(\mathbf{x})\big),$ where $D$ is the drive strength, $\omega_0$ the drive frequency, and $\psi(\mathbf{x})$ a quenched random phase (quenched: fixed once and for all when the system is prepared, not fluctuating in time), drawn uniformly from $[0,2\pi]$ at each point. Such drives model chiral active matter, in which each self-propelled particle moves along a circle whose phase is fixed by its initial orientation, and actively deforming particles such as tissue-cell models whose diameters oscillate with independent phases (Tjhung and Berthier 2017).

A scaling analysis that assumes only the existence of a Nambu-Goldstone mode gives the lower critical dimension $d_l$ (the dimension below which long-range order is impossible): $d_l=2$ when thermal noise is present ($T>0$, $\omega_0\neq0$), as in equilibrium; $d_l=0$ at zero temperature with nonzero drive frequency, so long-range order can occur in every dimension including $d=1,2$; and $d_l=4$ for a static inhomogeneous drive ($\omega_0=0$), where the drive acts as a static random field and the Imry-Ma argument - the argument that a static random field destabilizes continuously broken order in dimensions below four - applies (Imry and Ma 1975). To obtain detailed phase behavior, the authors solved the mean spherical model - the Berlin-Kac model in which the order parameter $\phi(\mathbf{x},t)$ obeys the global constraint $\int d\mathbf{x}\,\phi^2=N$, exactly solvable and equivalent to the $n\to\infty$ limit of the O($n$) model - driven by this force, with model-A dynamics (relaxational dynamics for a non-conserved order parameter, such as magnetization) and model-B dynamics (conserved dynamics, such as density, evolving through the divergence of a current). They computed the critical drive strength $D_c$ as a function of dimension, temperature and frequency, and found that for the conserved case at $T=0$ the static structure factor $S(q)$ (the Fourier transform of the equal-time density-density correlation function) obeys $S(q)\propto q^2$ at small wave number $q$: hyperuniformity, the anomalously strong suppression of long-wavelength density fluctuations that is characteristic of perfect crystals. At $\omega_0=0$ the conserved model instead shows divergent $S(q)\sim q^{-2}$ and a lower critical dimension of 6.

These predictions rest on four idealizations: the thermodynamic limit, the large-$n$ (spherical) limit, a spatially uncorrelated driving phase $\psi(\mathbf{x})$, and a purely quadratic (Gaussian) free energy. The authors close by asking that the theory be tested through the translational order parameters in numerical simulations and experiments on periodically driven systems, naming crystal phases of chiral active particles and actively deforming particles as candidate platforms.

Problem Statement

Does spatially inhomogeneous but temporally periodic driving genuinely stabilize continuous symmetry breaking in low-dimensional classical systems, as predicted by the scaling and spherical-model analyses of Ikeda and Kuroda? Establish this empirically and determine how robust the predictions are, through three linked sets of results that jointly answer the question.

(1) Direct numerical verification beyond the thermodynamic and large-$n$ limits. Simulate the periodically driven model at finite $n$ and finite system size - the spherical/O($n$) model itself under model-A and model-B dynamics with the drive $h(\mathbf{x},t)=\sqrt{2D}\cos(\omega_0 t+\psi(\mathbf{x}))$ and quenched random phases $\psi$, or faithful microscopic realizations such as chiral active or actively deforming particles - and test the specific predictions: (i) an ordered phase exists at $T=0$ with $\omega_0\neq0$ in $d=1$ and $d=2$ (predicted lower critical dimension $d_l=0$) with a finite critical drive strength $D_c$; (ii) any nonzero thermal noise destroys long-range order in $d\leq2$ ($d_l=2$ at $T>0$); (iii) the crossover to random-field behavior as $\omega_0\to0$, with $d_l=4$ for the non-conserved and $d_l=6$ for the conserved model; (iv) hyperuniform density fluctuations $S(q)\propto q^2$ for the conserved order parameter at $T=0$, $\omega_0\neq0$, lost at $T>0$ and replaced by divergent $S(q)\sim q^{-2}$ at $\omega_0=0$. Where the numerics disagree with the analytical phase diagrams, characterize the discrepancy (finite-$n$ crossover, finite-size effect, or failure of the theory).

(2) Experimental realization. Realize a one- or two-dimensional system driven by spatially inhomogeneous periodic forces under zero or effectively suppressed thermal noise - candidate platforms include chiral active particles and actively deforming particles or tissues - and measure whether true long-range translational order (an order parameter tending to a nonzero constant with increasing system size) and hyperuniform density fluctuations develop, quantifying the order-parameter scaling and $S(q)$ at small wave number.

(3) Robustness of the mechanism. Determine how the altered lower critical dimensions, the phase boundaries and the hyperuniform scaling change when the idealizing assumptions are relaxed: nonzero spatial correlations in the driving phase field $\psi(\mathbf{x})$; nonlinear (non-quadratic) free energies or anharmonic interactions; and driving with a finite coherence time rather than a perfectly periodic force.

The common objective is to establish whether inhomogeneous oscillatory driving is a genuine and robust route to continuous symmetry breaking in low dimensions, or an artifact of the spherical and thermodynamic limits. The numerical confirmation of two-dimensional long-range order in athermal chiral active particles already achieved by Kuroda, Kawasaki and Miyazaki (2025) counts as prior progress toward component (1). All three components are required for a resolution, together with each named item inside them - the four predictions (i)-(iv) of the numerical verification and each of the three relaxations of the robustness study. Component (2) does not require an executed measurement: a quantitatively specified experimental design that fixes the platform, the drive and noise conditions, and the order-parameter and $S(q)$ measurement protocol resolves that component.

References

1. Harukuni Ikeda, Yuta Kuroda. Continuous symmetry breaking of low-dimensional systems driven by inhomogeneous oscillatory driving forces. Physical Review E 110, 024140 (2024). DOI: 10.1103/PhysRevE.110.024140. https://doi.org/10.1103/PhysRevE.110.024140 (arXiv:2304.14235, https://arxiv.org/abs/2304.14235)
2. Yuta Kuroda, Takeshi Kawasaki, Kunimasa Miyazaki. Long-range translational order and hyperuniformity in two-dimensional chiral active crystal. Physical Review Research 7, L012048 (2025). DOI: 10.1103/PhysRevResearch.7.L012048. https://doi.org/10.1103/PhysRevResearch.7.L012048 (arXiv:2402.19192, https://arxiv.org/abs/2402.19192)
3. Yann-Edwin Keta, Silke Henkes. Long-range order in two-dimensional systems with fluctuating active stresses. Soft Matter 21, 5710-5719 (2025). DOI: 10.1039/D5SM00208G. https://doi.org/10.1039/D5SM00208G (arXiv:2410.14840, https://arxiv.org/abs/2410.14840)
4. Zhu-Qin Li, Qun-Li Lei, Yu-qiang Ma. Fluidization and anomalous density fluctuations in 2D Voronoi cell tissues with pulsating activity. Proceedings of the National Academy of Sciences 122, e2421518122 (2025). DOI: 10.1073/pnas.2421518122. https://doi.org/10.1073/pnas.2421518122 (arXiv:2402.02981, https://arxiv.org/abs/2402.02981)
5. Raphaël Maire, Ludivine Chaix. Hyperuniformity and conservation laws in non-equilibrium systems. The Journal of Chemical Physics 163, 214507 (2025). DOI: 10.1063/5.0300881. https://doi.org/10.1063/5.0300881 (arXiv:2509.04242, https://arxiv.org/abs/2509.04242)
6. Raphaël Maire, Andrea Plati. Enhancing (quasi-)long-range order in a two-dimensional driven crystal. The Journal of Chemical Physics 161, 054902 (2024). DOI: 10.1063/5.0217958. https://doi.org/10.1063/5.0217958 (arXiv:2405.05621, https://arxiv.org/abs/2405.05621)
7. Raphaël Maire, Andrea Plati, Frank Smallenburg, Giuseppe Foffi. Self-assembly of quasicrystals under cyclic shear. Soft Matter 22, 3346-3358 (2026). DOI: 10.1039/D6SM00068A. https://doi.org/10.1039/D6SM00068A (arXiv:2601.18403, https://arxiv.org/abs/2601.18403)
8. Yiwei Zhang, Étienne Fodor. Pulsating Active Matter. Physical Review Letters 131, 238302 (2023). DOI: 10.1103/PhysRevLett.131.238302. https://doi.org/10.1103/PhysRevLett.131.238302
9. Elsen Tjhung, Ludovic Berthier. Discontinuous fluidization transition in time-correlated assemblies of actively deforming particles. Physical Review E 96, 050601 (2017). DOI: 10.1103/PhysRevE.96.050601. https://doi.org/10.1103/PhysRevE.96.050601
10. Tzer Han Tan, Alexander Mietke, Junang Li, Yuchao Chen, Hugh Higinbotham, Peter J. Foster, Shreyas Gokhale, Jörn Dunkel, Nikta Fakhri. Odd dynamics of living chiral crystals. Nature 607, 287-293 (2022). DOI: 10.1038/s41586-022-04889-6. https://doi.org/10.1038/s41586-022-04889-6 (arXiv:2105.07507, https://arxiv.org/abs/2105.07507)
11. Mingji Huang, Wensi Hu, Siyuan Yang, Quan-Xing Liu, H. P. Zhang. Circular swimming motility and disordered hyperuniform state in an algae system. Proceedings of the National Academy of Sciences 118, e2100493118 (2021). DOI: 10.1073/pnas.2100493118. https://doi.org/10.1073/pnas.2100493118
12. Jing Wang, Zihao Sun, Huaicheng Chen, Gao Wang, Duyu Chen, Guo Chen, Jianwei Shuai, Mingcheng Yang, Yang Jiao, Liyu Liu. Active Hyperuniform Networks of Chiral Magnetic Micro-Robotic Spinners. arXiv:2505.21820 (2025). https://arxiv.org/abs/2505.21820
13. Kangeun Jeong, Yuta Kuroda, Yuki Asatani, Takeshi Kawasaki, Kunimasa Miyazaki. Crystallization of chiral active Brownian particles at low densities. Physical Review Research 7, 043212 (2025). DOI: 10.1103/yntx-ss9r. https://doi.org/10.1103/yntx-ss9r (arXiv:2506.20230, https://arxiv.org/abs/2506.20230)
14. Yuta Kuroda, Takeshi Kawasaki, Kunimasa Miyazaki. Singular density correlations in chiral active fluids in three dimensions. Physical Review E 112, 045426 (2025). DOI: 10.1103/25s2-6y3m. https://doi.org/10.1103/25s2-6y3m
15. Connor Roberts, Gunnar Pruessner. Exact results and instabilities in the harmonic approximation of active crystals. The Journal of Chemical Physics 164, 134108 (2026). DOI: 10.1063/5.0314725. https://doi.org/10.1063/5.0314725 (arXiv:2512.01539, https://arxiv.org/abs/2512.01539)
16. N. D. Mermin, H. Wagner. Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models. Physical Review Letters 17, 1133 (1966). DOI: 10.1103/PhysRevLett.17.1133. https://doi.org/10.1103/PhysRevLett.17.1133
17. P. C. Hohenberg. Existence of Long-Range Order in One and Two Dimensions. Physical Review 158, 383 (1967). DOI: 10.1103/PhysRev.158.383. https://doi.org/10.1103/PhysRev.158.383
18. T. H. Berlin, M. Kac. The Spherical Model of a Ferromagnet. Physical Review 86, 821 (1952). DOI: 10.1103/PhysRev.86.821. https://doi.org/10.1103/PhysRev.86.821
19. Yoseph Imry, Shang-keng Ma. Random-Field Instability of the Ordered State of Continuous Symmetry. Physical Review Letters 35, 1399 (1975). DOI: 10.1103/PhysRevLett.35.1399. https://doi.org/10.1103/PhysRevLett.35.1399