• Landau, L. The theory of phase transitions. Nature 138, 840–841 (1936).

    Article 
    ADS 

    Google Scholar
     

  • Post, K. W. et al. Coexisting first- and second-order electronic phase transitions in a correlated oxide. Nat. Phys. 14, 1056–1061 (2018).

    Article 

    Google Scholar
     

  • Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973).

    Article 
    ADS 

    Google Scholar
     

  • Thouless, D. J. Long-range order in one-dimensional Ising systems. Phys. Rev. 187, 732 (1969).

    Article 
    ADS 

    Google Scholar
     

  • Kosterlitz, J. M. Nobel lecture: topological defects and phase transitions. Rev. Mod. Phys. 89, 040501 (2017).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Sone, K. et al. Nonlinearity-induced topological phase transition characterized by the nonlinear Chern number. Nat. Phys. 20, 1164–1170 (2024).

    Article 

    Google Scholar
     

  • Qi, X. & Zhang, S. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article 
    ADS 

    Google Scholar
     

  • Wachowiak, A. et al. Direct observation of internal spin structure of magnetic vortex cores. Science 298, 577–580 (2002).

    Article 
    ADS 

    Google Scholar
     

  • Naumov, I. I., Bellaiche, L. & Fu, H. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737–740 (2004).

    Article 
    ADS 

    Google Scholar
     

  • Tang, Y. L. et al. Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348, 547–551 (2015).

    Article 
    ADS 

    Google Scholar
     

  • Yadav, A. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).

    Article 
    ADS 

    Google Scholar
     

  • Das, S. et al. Observation of room-temperature polar skyrmions. Nature 568, 368–372 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Han, L. et al. High-density switchable skyrmion-like polar nanodomains integrated on silicon. Nature 603, 63–67 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Wang, J. et al. Polar Solomon rings in ferroelectric nanocrystals. Nat. Commun. 14, 3941 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Sánchez-Santolino, G. et al. A 2D ferroelectric vortex pattern in twisted BaTiO3 freestanding layers. Nature 626, 529–534 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Yadav, A. K. et al. Spatially resolved steady-state negative capacitance. Nature 565, 468–471 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Das, S. et al. Local negative permittivity and topological phase transition in polar skyrmions. Nat. Mater. 20, 194–201 (2021).

    Article 

    Google Scholar
     

  • Shao, Y. T. et al. Emergent chirality in a polar meron to skyrmion phase transition. Nat. Commun. 14, 1355 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Wang, S. et al. Giant electric field-induced second harmonic generation in polar skyrmions. Nat. Commun. 15, 1374 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Li, Q. et al. Subterahertz collective dynamics of polar vortices. Nature 592, 376–380 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Geng, W. R. et al. Dipolar wavevector interference induces a polar skyrmion lattice in strained BiFeO3 films. Nat. Nanotechnol. 20, 366–373 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Tang, Y. Y. et al. Organic ferroelectric vortex-antivortex domain structure. J. Am. Chem. Soc. 142, 21932–21937 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Guo, M. et al. Toroidal polar topology in strained ferroelectric polymer. Science 371, 1050–1056 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Avron, J. E., Seiler, R. & Simon, B. Homotopy and quantization in condensed matter physics. Phys. Rev. Lett. 51, 51–53 (1983).

    Article 
    ADS 

    Google Scholar
     

  • Li, L. et al. Defect-induced hedgehog polarization states in multiferroics. Phys. Rev. Lett. 120, 137602 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Luk’yanchuk, I. et al. Hopfions emerge in ferroelectrics. Nat. Commun. 11, 2433 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Wang, Y. J. et al. Polar meron lattice in strained oxide ferroelectrics. Nat. Mater. 19, 881–886 (2020).

    Article 

    Google Scholar
     

  • Zhou, L. et al. Harness of room-temperature polar skyrmion bag in oxide superlattice. Nat. Commun. 16, 9911 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Zheng, W. et al. Skyrmion nanodomains in ferroelectric–antiferroelectric solid solutions. Nat. Mater. 24, 1424–1432 (2025).

    Article 

    Google Scholar
     

  • Catalan, G. et al. Domains in three-dimensional ferroelectric nanostructures: theory and experiment. J. Phys. Condens. Matter 19, 132201 (2007).

    Article 
    ADS 

    Google Scholar
     

  • Hong, Z. et al. Stability of polar vortex lattice in ferroelectric superlattices. Nano Lett. 17, 2246–2252 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Hong, Z. & Chen, L. Blowing polar skyrmion bubbles in oxide superlattices. Acta Mater. 152, 155–161 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Stoica, V. A. et al. Optical creation of a supercrystal with three-dimensional nanoscale periodicity. Nat. Mater. 18, 377–383 (2019).

    Article 

    Google Scholar
     

  • Wang, B., Gu, Y., Zhang, S. & Chen, L. Flexoelectricity in solids: progress, challenges and perspectives. Prog. Mater. Sci. 106, 100570 (2019).

    Article 

    Google Scholar
     

  • Li, Q. et al. Quantification of flexoelectricity in PbTiO3/SrTiO3 superlattice polar vortices using machine learning and phase-field modeling. Nat. Commun. 8, 1468 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Morozovska, A. et al. Chiral polarization textures induced by the flexoelectric effect in ferroelectric nanocylinders. Phys. Rev. B 104, 054118 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Janquera, J. et al. Topologial phases in polar oxide nanostructures. Rev. Mod. Phys. 95, 025001 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Chen, P. et al. Atomic imaging of mechanically induced topological transition of ferroelectric vortices. Nat. Commun. 11, 1840 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Zhao, H. J. et al. Dzyaloshinskii–Moriya-like interaction in ferroelectrics and antiferroelectrics. Nat. Mater. 20, 341–345 (2021).

    Article 

    Google Scholar
     

  • Prosandeev, S. & Bellaiche, L. Characteristics and signatures of dipole vortices in ferroelectric nanodots: first-principles-based simulations and analytical expressions. Phys. Rev. B 75, 094102 (2007).

    Article 
    ADS 

    Google Scholar
     

  • Nahas, Y. et al. Inverse transition of labyrinthine domain patterns in ferroelectric thin films. Nature 577, 47–51 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Tong, P. et al. Thermal triggering for multi-state switching of polar topologies. Nat. Phys. 21, 464–470 (2025).

    Article 

    Google Scholar
     

  • Wang, Y. J., Tang, Y. L., Zhu, Y. L. & Ma, X. L. Meron-antimeron annihilation induced by the electric field in a polar meron lattice. J. Appl. Phys. 131, 224102 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Prokhorenko, S. et al. Motion and teleportation of polar bubbles in low-dimensional ferroelectrics. Nat. Commun. 15, 412 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Hu, L., Wu, Y., Huang, Y., Tian, H. & Hong, Z. Dynamic motion of polar skyrmions in oxide heterostructures. Nano Lett. 23, 11353–11359 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Margaret, M. R. et al. Structural chirality of polar skyrmions probed by resonant elastic X-ray scattering. Phys. Rev. Lett. 129, 247601 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Shafer, P. et al. Emergent chirality in the electric polarization texture of titanate superlattices. Proc. Natl Acad. Sci. USA 115, 915–920 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Li, W. et al. Terahertz excitation of collective dynamics of polar skyrmions over a broad temperature range. Nat. Phys. 21, 1965–1972 (2025).

    Article 

    Google Scholar
     

  • Behera, P. et al. Electric field control of chirality. Sci. Adv. 8, eabj8030 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Salahuddin, S. & Datta, S. Use of negative capacitance to provide voltage amplification for low power nanoscale devices. Nano Lett. 8, 405–410 (2008).

    Article 
    ADS 

    Google Scholar
     

  • Wang, Y. et al. Polar Bloch points in strained ferroelectric films. Nat. Commun. 15, 3949 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Huang, P. et al. Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase. Nat. Nanotechnol. 15, 761–767 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Cugliandolo, L. F. Coarsening phenomena. C. R. Phys. 16, 257–266 (2015).

    Article 
    ADS 

    Google Scholar
     

  • Kibble, T. W. B. Some implications of a cosmological phase transition. Phys. Rep. 67, 183–199 (1980).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Biroli, G., Cugliandolo, L. F. & Sicilia, A. Kibble-Zurek mechanism and infinitely slow annealing through critical points. Phys. Rev. E 81, 050101 (2010).

    Article 
    ADS 

    Google Scholar
     

  • Roychowdhury, K. et al. Dynamics and correlations at a quantum phase transition beyond Kibble-Zurek. Phys. Rev. B 104, 014406 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Du, G. et al. Design of polar skyrmion-based nanoelectronic prototype devices with phase-field simulations. Adv. Funct. Mater. 34, 2405594 (2024).

    Article 

    Google Scholar
     

  • Chazal, F. & Michel, B. An introduction to topological data analysis: fundamental and practical aspects for data scientists. Front. Artif. Intell. 4, 667963 (2021).

    Article 

    Google Scholar
     

  • Du, G. et al. Topological data analysis assisted machine learning for polar topological structures in oxide superlattices. Acta Mater. 282, 120467 (2025).

    Article 

    Google Scholar
     

  • Kalinin, S. V. et al. Machine learning for automated experimentation in scanning transmission electron microscopy. npj Comput. Mater. 9, 227 (2023).

    Article 
    MathSciNet 

    Google Scholar
     

  • Liu, Y. et al. Ferroelectric-based neuromorphic memory devices for bio-inspired computing. Nat. Rev. Electr. Eng. 2, 773–787 (2025).

    Article 

    Google Scholar
     

  • Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).

    Article 
    ADS 
    MathSciNet 

    Google Scholar