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Dual-symmetry-guided assembly of complex lattices

Nature (2026)Cite this article

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Abstract

Complex lattices that combine low- and high-order rotational symmetries underpin functional materials ranging from kagome superconductors1,2,3 to auxetic mechanical networks4 and photonic crystals with topologically protected states5,6,7. However, assembling such structures typically requires anisotropic particle shapes, directional bonding or fully imposed templates8,9,10,11, which often suffer from severe kinetic frustration and defect trapping. Here we introduce a dual-symmetry-guided (DSG) principle that exploits the geometric self-duality of a target tiling. By decomposing the structure into two mutually dual sublattices of lower symmetry and sparsely pinning only one sublattice using optical traps in a colloidal monolayer, the complementary sublattice spontaneously self-organizes through purely isotropic repulsive interactions, thereby reconstructing the full lattice. Using this minimal guidance strategy, we experimentally realize, and corroborate with simulations, a broad class of complex Archimedean lattices as well as two-dimensional quasicrystalline structures. DSG reveals lattice-dependent thermal stability while preserving interconnected free volume for mobile particles, enabling efficient defect relaxation and kinetically accessible assembly even under strong pinning conditions. We show that full pinning corresponds to a special limiting case of DSG, and that reformulating conventional templating protocols within the DSG framework systematically reduces kinetic barriers and suppresses defect formation. By decoupling structural complexity from interaction anisotropy, DSG provides a general and experimentally accessible route to complex-symmetry materials with programmable structural and physical properties.

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Fig. 1: Implementation of the DSG strategy through colloidal experiments.
Fig. 2: Archimedean lattices stabilized using the DSG strategy.
Fig. 3: Effective parameter space for the DSG strategy.

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Data availability

The data that support the findings of this study are available via Figshare at https://doi.org/10.6084/m9.figshare.31120829. A representative minimal dataset (experimental and simulation) is provided. Owing to their large file size, the full raw experimental videos and complete simulation trajectories are available from the corresponding author upon reasonable request.

Code availability

The code used to generate and analyse the data in this study is available via Figshare at https://doi.org/10.6084/m9.figshare.31120829.

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Acknowledgements

P.T. acknowledges the National Natural Science Foundation of China (number 12425503), the Space Application System of China Manned Space Program (KJZ-YY-NLT0501), the Innovation Program of Shanghai Municipal Education Commission (number 2023ZKZD06), and the Shanghai Pilot Program for Basic Research-FuDan University number 21TQ1400100 (22TQ003). H.T. acknowledges a Grant-in-Aid for Specially Promoted Research (JP20H05619) from the Japan Society of the Promotion of Science (JSPS). Y.M. acknowledges National Natural Science Foundation of China (number 12347102), the Quantum Science and Technology-National Science and Technology Major Project (number 2024ZD0300101), and the Natural Science Foundation of Jiangsu Province (number BK20233001). H.F. acknowledges China Postdoctoral Science Foundation funded project (number 2025M773335). J.H. acknowledges the National Natural Science Foundation of China (number 12320101004).

Author information

Author notes

  1. These authors contributed equally: Huang Fang, Xiaotian Li, Wensi Sun

Authors and Affiliations

  1. State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai, People’s Republic of China

    Huang Fang, Xiaotian Li, Wensi Sun, Chengxin Wang, Nuo Chen, Yining Gan, Jiping Huang & Peng Tan

  2. National Laboratory of Solid State Microstructures, Collaborative Innovation Center of Advanced Microstructures, and School of Physics, Nanjing University, Nanjing, People’s Republic of China

    Yuqiang Ma

  3. Jiangsu Physical Science Research Center, Nanjing, People’s Republic of China

    Yuqiang Ma

  4. Hefei National Laboratory, Shanghai, People’s Republic of China

    Yuqiang Ma

  5. Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan

    Hajime Tanaka

  6. Department of Fundamental Engineering, Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

    Hajime Tanaka

  7. Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai, People’s Republic of China

    Peng Tan

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Contributions

P.T., H.T. and Y.M. conceived of and supervised the research. H.F., X.L., W.S., C.W. and N.C. performed the experiments, simulations and data analysis. Y.G. and J.H. contributed to data analysis. H.F., H.T. and P.T. wrote the paper.

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Correspondence to Yuqiang Ma, Hajime Tanaka or Peng Tan.

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Extended data figures and tables

Extended Data Fig. 1 Two-dimensional charged colloidal experiments.

a Side-view fluorescence confocal image of the experimental system (Leica SP8). Colloidal particles are labelled with NBD dye (excitation: 488 nm; emission: 546 nm), while the aqueous phase is labelled with Rhodamine B (excitation: 550 nm; emission: 610 nm), demonstrating confinement of particles at the oil–water interface. b Snapshots of representative configurations during the self-organization experiment. Upper left: initial hexagonal monolayer; lower left: transformation to square lattice upon square-lattice trap activation; right: transformation to honeycomb lattice upon a sparsely pinned hexagonal-lattice trap activation. An optical tweezer-defined “fence” confines particle density to ρ ≈ 0.03 μm−2. The fence geometry is adjusted to match the boundary of the target Archimedean tiling, ensuring a constant particle number during assembly. Scale bar: 10 μm. c Distribution of positional fluctuations of individual particles from panel b over a 5-minute interval. Trapped particles show highly restricted motion, with normalised positional fluctuations std(ri)/a < 0.03, where a is the characteristic interparticle spacing. d Measured interparticle repulsive force decays as F(r) ∝ 1/r4, consistent with dipolar interactions arising from image-charge effects at the oil–water interface, as confirmed via 30 independent catch-and-release experiments. e Measured potential well profile generated by a single optical trap at 10 mW laser power (Tweez 250si, Aresis). The solid line shows a harmonic fit to the experimental data, validating the effective Gaussian trapping model used in simulations.

Extended Data Fig. 2 Universality of the DSG design principle.

Each row corresponds to a distinct non-Archimedean lattice that combines multiple local symmetries. Columns (from left to right) show the target lattice {VT} and the selected sublattice {V0}, the stabilised configurations obtained from stability tests in experiments and simulations, and, where applicable, the configurations self-assembled in simulation tests. Orange dots mark the sublattice {V0}, while green dots indicate vertices shared by both the dual lattice D({V0}) and the target lattice {VT}. a: Cairo tiling; b: 8-fold two-dimensional quasicrystal; c: Penrose tiling (10-fold quasicrystal); d: 12-fold two-dimensional quasicrystal. In all cases, the obtained configurations closely reproduce the intended target structures, demonstrating that DSG is not restricted to Archimedean lattices or periodic order. Simulation parameters for both stability and self-assembly tests: Atrap = 30kBT, Γ = 80. Experimental parameters are similar to those displayed in Fig. 3: Atrap ≈ 30kBT, Γ ≈ 60.

Extended Data Fig. 3 Dynamic behaviour in the colloidal supersolid.

Mean-squared displacement (MSD) of mobile particles under DSG pinning as a function of the elapsed time t/t0 (\({t}_{0}\equiv \sqrt{m{\sigma }^{2}/{k}_{{\rm{B}}}T}\)). a is the bond length of the target perfect Archimedean lattice. The slope α characterises the MSD scaling behaviour across distinct dynamical regimes. At short times, α ≈ 2, indicating ballistic-like motion within local cages, whereas at long times α ≈ 1, corresponding to diffusive transport across symmetry-equivalent dual sites. The shoulder connecting the two regimes reflects confined oscillatory motion prior to inter-site hopping. The inset shows representative collective rearrangement modes characteristic of the DSG-induced supersolid state for different lattice types. Orange dots denote pinned particles, and green dots denote mobile particles. a: (63); b: (33.42); c: (32.4.3.4); d: (3.4.6.4). Atrap = 30kBT, Γ = 80.

Extended Data Fig. 4 Free-energy landscape connecting the ground state and nearby metastable states.

Snapshots show representative configurations associated with each node. Solid lines represent free-energy profiles along transition pathways. a: Honeycomb lattice under full pinning; b: Honeycomb lattice under DSG pinning; c: (3.4.6.4) lattice under full pinning; d: (3.4.6.4) lattice under DSG pinning. For both lattice types, full pinning requires particles to escape deep optical traps in order to anneal distant interstitial–vacancy defect pairs, resulting in free-energy barriers exceeding 20kBT (bold orange lines in a and c). Under DSG pinning, the corresponding rearrangements proceed through interconnected free volumes without trap escape, yielding barriers of only a few kBT or effectively barrier-free pathways (bold orange lines in b and d). These results quantitatively demonstrate the kinetic advantage of DSG over full pinning, providing a microscopic explanation for the enhanced defect relaxation observed in Fig. 3. Details of the free-energy calculations are provided in Supplementary Information (Fig. S12). Simulation parameters: Atrap = 30kBT, Γ = 50.

Extended Data Fig. 5 Photonic and phononic band structures of Archimedean lattices constructed by DSG.

a Photonic band structure (TM mode) of the ideal and DSG-enabled (3.4.6.4) lattices calculated using COMSOL Multiphysics. The particle diameter is σ = 0.9, a and the refractive index is n = 3.0, where a denotes the characteristic bond length. While the ideal lattice exhibits no complete band gap, the lattice constructed via DSG displays a narrow but finite photonic band gap, arising from DSG-induced geometric distortions. b Photonic band structure (TM mode) of (4.82) lattices with and without helper (centre-filling) particles. The centre-filled (4.82) lattice exhibits an additional photonic band gap that is absent in the unfilled structure, demonstrating the role of DSG-compatible repulsers in tailoring optical properties. Here σ = 0.8, a and n = 3.0. c Vibrational density of states (VDOS) and dispersion relations of the (33.42) lattice under partial (DSG) and full pinning. The dispersion relations are obtained from the current–current correlation function c(k, ω), with the wavevector k averaged over 10 distinct directions (Methods). Partial DSG pinning opens a phononic band gap that is absent under full pinning, highlighting the sensitivity of vibrational spectra to symmetry-selective constraints and kinetic accessibility. Inset schematics show the corresponding lattice configurations, with pinned particles indicated by red circles. Simulation parameters: Atrap = 50kBT, Γ = 60.

Extended Data Table 1 Summary of DSG performance across lattice types and interaction models
Full size table

Supplementary information

Supplementary Information (download PDF )

Supplementary Information including Supplementary Tables 1 and 2, and Figs. 1–12.

Supplementary Video 1 (download MP4 )

DSG-stabilized Archimedean lattices. Bright-field microscopy recording showing the formation and stabilization of Archimedean lattices under DSG pinning.

Supplementary Video 2 (download MP4 )

DSG-stabilized Cairo tiling and quasicrystals. Bright-field microscopy recording demonstrating DSG-guided formation of Cairo tiling and two-dimensional quasicrystals.

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Fang, H., Li, X., Sun, W. et al. Dual-symmetry-guided assembly of complex lattices. Nature (2026). https://doi.org/10.1038/s41586-026-10364-3

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