Abstract
Order and disorder are central concepts in condensed-matter physics. Crystals break translational and rotational symmetries, whereas quasicrystals challenge this paradigm with forbidden rotational symmetries and aperiodicity. Here we report a distinct ordered state—ideal non-crystals—characterized by optimal steric order without symmetry breaking. Steric optimization yields ideal non-crystals as a thermodynamically favoured limiting state, accompanied by maximal steric order that may serve as a true order parameter for the glass transition. Despite their apparent disorder, they exhibit long-range orientational correlations, quantified via a specific path-integral-like approach. Ideal non-crystals possess distinct properties, including Debye-like phononic modes, affine elasticity, thermodynamic ultrastability and long-wavelength density uniformity, reminiscent of hyperuniformity. By uncovering a distinct form of entropy-driven ordering in sterically optimized materials, this work expands the landscape of ordered states and provides a framework for designing amorphous materials with crystal-like mechanical and thermal properties free from the anisotropy inherent in crystals.
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Data availability
The data that support the findings of this study are available in the Article and the Supplementary Information. Source data are provided with this paper.
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The codes used to generate the results in this paper are available from the corresponding authors upon request.
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Acknowledgements
We thank J. Russo and R. Ni for their helpful discussions. X.F., D.X., J.Z., N.X. and H. Tong acknowledge support from the National Natural Science Foundation of China (grant numbers 12274392, 12334009 and 12074355). H. Tanaka acknowledges support from the Grant-in-Aid for Specially Promoted Research (JSPS KAKENHI grant number JP20H05619) from the Japan Society for the Promotion of Science (JSPS). We also thank the Supercomputing Center of the University of Science and Technology of China and the Hefei Advanced Computing Center for the computer time.
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H. Tong conceived the project. H. Tong, H. Tanaka and N.X. supervised the project. X.F. performed the simulations and data analysis. D.X. contributed to coding at the initial stage of the project. J.Z. contributed to the analysis of jamming scaling and hyperuniformity. All authors discussed the results. X.F., H. Tong, H. Tanaka and N.X. wrote the manuscript.
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Extended data
Extended Data Fig. 1 Evolution of steric order Θ during the optimisation procedure.
The steric order Θ as a function of iteration steps Nstep for 10 independent realisations (background) and after the ensemble average (bold red circles). The convergence is rapidly achieved after around five iterations, indicating that the system is only weakly perturbed from its initial state.
Extended Data Fig. 2 System size dependence of structural order parameters.
The system size dependence of the average steric order parameter Θ (a) and the hexatic bond-orientational order parameter Ψ6 (b) for three temperatures at which the system can equilibrate via swap Monte Carlo simulations. The data points are presented as mean values ± s.d. of measurements taken from 1000, 200, and 100 samples for N = 1024, 4096, and 16384 systems, respectively. Both results indicate the absence of apparent finite-size effects. In particular, the value of Ψ6 is quite low at low temperatures and independent of the system size, suggesting that the hexagonal crystalline order is not favoured in our system.
Extended Data Fig. 3 Characteristic temperatures from glassy dynamics.
The dynamics of our ideal-non-crystal system is studied using molecular dynamics simulations via LAMMPS. The structure relaxation is measured by the self-intermediate scattering function \({F}_{s}(k,t)=\langle {\Sigma }_{j}\,\exp ({\rm{i}}\cdot [{\underline{{\boldsymbol{r}}}}_{\rm{j}}({\rm{t}})-{\underline{{\boldsymbol{r}}}}_{\rm{j}}(0)])/{\rm{N}}\rangle\), where k = ∣k∣ corresponds to the first peak of the static structure factor and \(\langle \cot \rangle\) denotes the time average. The relative position \({\underline{{\boldsymbol{r}}}}_{j}(t)={{\boldsymbol{r}}}_{j}(t)-{\Sigma }_{k}\,{{\boldsymbol{r}}}_{k}(t)/{n}_{j}\) is used to remove long-wavelength Mermin-Wagner fluctuations in 2D, with the summation running over all neighbours of particle j. The structure relaxation time τα is defined by Fs(k, τα) = e−1. a, Self-intermediate scattering function Fs(k, t) for different temperatures. The dashed line indicates Fs(k, t) = e−1. b, τα as a function of 1/T. The solid line shows an Arrhenius fit to the high-temperature data \({\tau }_{\alpha } \sim \exp (\Delta E/T)\). The estimated onset temperature of sluggish glassy dynamics Ton = 2.4 × 10−3 is indicated by the dashed line. c, τα as a function of T. The solid line is a fit of data below Ton according to the Vogel-Fulcher-Tammann (VFT) law \({\tau }_{\alpha } \sim \exp [D{T}_{{\rm{VFT}}}/(T-{T}_{{\rm{VFT}}})]\), from which we extract the hypothesised ideal glass transition temperature TVFT = 9.12 × 10−4, with D as a fitting parameter. Ton and TVFT provide two reference temperatures for our system.
Extended Data Fig. 4 Cooling rate dependence of thermodynamic quantities.
The temperature dependence of potential energy per particle E (a), pressure p (b), steric order parameter Θ (c), and hexatic bond-orientational order parameter Ψ6 (d) for cooling rates covering three orders of magnitude. For particle \(j,{\Psi }_{6}^{j}=| {\Sigma }_{k}{e}_{jk}^{6i\theta }/{n}_{j}|\), where nj is the number of nearest neighbors of particle j, and θjk is the angle of the bond rjk = rj − rk with respect to the x-axis. Insets provide enlarged views of the low-temperature regime. The dashed line indicates the lowest temperature, T = 2 × 10−4, down to which results from different cooling rates converge, ensuring equilibration by swap Monte Carlo simulations. Importantly, in panel d, Ψ6 shows a peak around T = 1.4 × 10−3, below which Ψ6 significantly decreases with temperature, indicating that the hexatic crystalline order is thermodynamically unfavourable in our system.
Extended Data Fig. 5 Application of C(r) in two-dimensional melting of monodisperse crystal.
Monodisperse crystals with the same interacting potential and packing fraction as our ideal-non-crystal system are heated and then cooled down using molecular dynamics simulations, with a constant heating (cooling) rate dT/dt = 10−9. a, Temperature dependence of potential energy per particle E during heating (red line) and cooling (black line) processes. The evolution of E during melting resembles that of ideal non-crystals, suggesting a similar physical mechanism of melting. Marginal hysteresis is observed between cooling and heating, differing from the ideal-non-crystal system. For state points indicated in a, the structural correlations are characterized in three different ways: The conventional correlation function of the hexatic order parameter \({G}_{6}(r)=\langle {\Psi }_{6}^{* }(r){\Psi }_{6}(0)\rangle\) (b), the correlation of Ψ6 along the coherent path \({G}_{6}^{p}(r)\) (c), and the path-integral-like correlation function C(r) defined in this work (d). The dashed line is the power-law scaling predicted by the KTHNY theory of two-dimensional melting from hexatic phase to liquids. A close comparison confirms prefect agreement between them, validating the efficacy of our methodology. In essence, our path-integral-like scheme, based on the definition of the coherent path and the corresponding correlation function, captures the structural coherence from the steric constraint of triangle units. For monodisperse systems, the steric order reduces to the hexatic orientational order, making C(r) and G6(r) give essentially the same information.
Extended Data Fig. 6 Heating rate dependence of melting behaviour of ideal non-crystals.
a, Evolution of potential energy per particle E when heated from ideal-non-crystal configurations using normal MD with a heating rate dT/dt = 10−9 and then cooled down using SMC with a cooling rate dT/dt = 10−10. Compared to Fig. 2c, where a heating rate of dT/dt = 10−10 is used, the steep increase of E takes place at a higher temperature, but the overall behaviours are the same. This indicates a nonequilibrium nature of the melting process. b, The path-integral-like correlation function C(r) for state points indicated in a (note that the temperatures are different from Fig. 2d for the same colour). The dashed line indicating C(r) ~ r−0.25 is plotted as a reference. Here, because of the faster heating rate, the system cannot be well equilibrated (even in the metastable sense) during melting. Therefore, the mixed behaviour of C(r) with medium-range power-law-like correlation and long-range exponential decay at intermediate temperatures might be due to the coexistence of fluid-like and solid-like components. While melting behaviour deserves further careful investigations, the ultrastability and long-range structural correlation in ideal-non-crystal states are clear from these analyses.
Extended Data Fig. 7 The structure and hyperuniformity of weakly polydisperse crystals.
We characterise crystals with extremely weak polydispersities to demonstrate their similarity to ideal non-crystals. a, Visualisation of a typical crystalline configuration with a polydispersity Δ = 0.231% and steric order Θ = 1.58 × 10−3 (approximately the same as ideal non-crystals). The packing fraction is set to ϕ = 0.92 and particles are coloured according to their steric order Θ. b, The bare configuration corresponding to a without colour coding, which is visually indistinguishable from a perfect hexagonal crystal. This plot gives some intuitive sense of how ordered the obtained ideal non-crystals are compared to the underlying perfect state (Θ=0). c, Spectral density χV(k) for weakly polydisperse crystals with different degree of steric order Θ. The corresponding polydispersities are Δ = 1.155%, 0.577%, 0.231%, 0.115%, and 0.058% for decreasing Θ. d, The plateau value of spectral density when approaching the low-k limit χV(k → 0) as a function of steric order Θ. The dashed line is a power-law extrapolation fitting of the data χV(k → 0) ~ Θ2.
Supplementary information
Supplementary Information
Supplementary Sections 1–6 and Figs. 1–19.
Supplementary Video 1
Extension of the coherent path in an ideal-non-crystal configuration.
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Fan, X., Xu, D., Zhang, J. et al. Ideal non-crystals as a distinct form of ordered states without symmetry breaking. Nat. Mater. (2026). https://doi.org/10.1038/s41563-026-02496-8
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DOI: https://doi.org/10.1038/s41563-026-02496-8
