Skyrme, T. H. R. A non-linear field theory. Proc. R. Soc. A 260, 127–138 (1961).
Skyrme, T. H. R. A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962).
Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).
Shen, Y. et al. Optical skyrmions and other topological quasiparticles of light. Nat. Photonics 18, 15–25 (2024).
Chong, A., Wan, C., Chen, J. & Zhan, Q. Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum. Nat. Photonics 14, 350–354 (2020).
Wan, C., Cao, Q., Chen, J., Chong, A. & Zhan, Q. Toroidal vortices of light. Nat. Photonics 16, 519–522 (2022).
Song, K. M. et al. Skyrmion-based artificial synapses for neuromorphic computing. Nat. Electron. 3, 148–155 (2020).
Göbel, B., Mertig, I. & Tretiakov, O. A. Beyond skyrmions: review and perspectives of alternative magnetic quasiparticles. Phys. Rep. 895, 1–28 (2021).
Seki, S. & Mochizuki, M. Skyrmions in Magnetic Materials (Springer, 2016).
Tokura, Y. & Kanazawa, N. Magnetic skyrmion materials. Chem. Rev. 121, 2857–2897 (2020).
Kent, N. et al. Creation and observation of hopfions in magnetic multilayer systems. Nat. Commun. 12, 1562 (2021).
Zheng, F. et al. Hopfion rings in a cubic chiral magnet. Nature 623, 718–723 (2023).
Romming, N. et al. Writing and deleting single magnetic skyrmions. Science 341, 636–639 (2013).
Yu, G. et al. Room-temperature skyrmion shift device for memory application. Nano Lett. 17, 261–268 (2017).
Maccariello, D. et al. Electrical detection of single magnetic skyrmions in metallic multilayers at room temperature. Nat. Nanotechnol. 13, 233–237 (2018).
Raftrey, D. & Fischer, P. Field-driven dynamics of magnetic hopfions. Phys. Rev. Lett. 127, 257201 (2021).
Jiang, W. et al. Direct observation of the skyrmion Hall effect. Nat. Phys. 13, 162–169 (2017).
Litzius, K. et al. Skyrmion Hall effect revealed by direct time-resolved X-ray microscopy. Nat. Phys. 13, 170–175 (2017).
Yang, S. et al. Reversible conversion between skyrmions and skyrmioniums. Nat. Commun. 14, 3406 (2023).
Zheng, F. et al. Skyrmion–antiskyrmion pair creation and annihilation in a cubic chiral magnet. Nat. Phys. 18, 863–868 (2022).
Fert, A., Reyren, N. & Cros, V. Magnetic skyrmions: advances in physics and potential applications. Nat. Rev. Mater. 2, 1–15 (2017).
Han, L. et al. High-density switchable skyrmion-like polar nanodomains integrated on silicon. Nature 603, 63–67 (2022).
Shen, Y. et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci. Appl. 8, 90 (2019).
Forbes, A., De Oliveira, M. & Dennis, M. R. Structured light. Nat. Photonics 15, 253–262 (2021).
Gao, S. et al. Paraxial skyrmionic beams. Phys. Rev. A 102, 053513 (2020).
Ni, J. et al. Multidimensional phase singularities in nanophotonics. Science 374, eabj0039 (2021).
Wang, H., Shi, L., Lukyanchuk, B., Sheppard, C. & Chong, C. T. Creation of a needle of longitudinally polarized light in vacuum using binary optics. Nat. Photonics 2, 501–505 (2008).
Tsesses, S. et al. Optical skyrmion lattice in evanescent electromagnetic fields. Science 361, 993–996 (2018).
Davis, T. J. et al. Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution. Science 368, eaba6415 (2020).
Bai, C., Chen, J., Zhang, Y., Zhang, D. & Zhan, Q. Dynamic tailoring of an optical skyrmion lattice in surface plasmon polaritons. Opt. Express 28, 10320–10328 (2020).
Dai, Y. et al. Plasmonic topological quasiparticle on the nanometre and femtosecond scales. Nature 588, 616–619 (2020).
Du, L., Yang, A., Zayats, A. V. & Yuan, X. Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum. Nat. Phys. 15, 650–654 (2019).
Yang, A. et al. Spin-manipulated photonic skyrmion-pair for pico-metric displacement sensing. Adv. Sci. 10, 2205249 (2023).
Lei, X. et al. Photonic spin lattices: symmetry constraints for skyrmion and meron topologies. Phys. Rev. Lett. 127, 237403 (2021).
Karnieli, A., Tsesses, S., Bartal, G. & Arie, A. Emulating spin transport with nonlinear optics, from high-order skyrmions to the topological Hall effect. Nat. Commun. 12, 1092 (2021).
Wang, H. & Fan, S. Photonic spin hopfions and monopole loops. Phys. Rev. Lett. 131, 263801 (2023).
Shen, Y., Hou, Y., Papasimakis, N. & Zheludev, N. I. Supertoroidal light pulses as electromagnetic skyrmions propagating in free space. Nat. Commun. 12, 5891 (2021).
Shen, Y. et al. Topologically controlled multiskyrmions in photonic gradient-index lenses. Phys. Rev. Appl. 21, 024025 (2024).
Wang, S. et al. Topological structures of energy flow: Poynting vector skyrmions. Phys. Rev. Lett. 133, 073802 (2024).
Lin, Y. J., Compton, R. L., Jiménez-García, K., Porto, J. V. & Spielman, I. B. Synthetic magnetic fields for ultracold neutral atoms. Nature 462, 628–632 (2009).
Zhang, Y., Tan, Y. W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015).
Yale, C. G. et al. Optical manipulation of the Berry phase in a solid-state spin qubit. Nat. Photonics 10, 184–189 (2016).
Wang, J. et al. Experimental observation of Berry phases in optical Möbius-strip microcavities. Nat. Photonics 17, 120–125 (2023).
Karnieli, A., Li, Y. & Arie, A. The geometric phase in nonlinear frequency conversion. Front. Phys. 17, 12301 (2022).
Cohen, E. et al. Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nat. Rev. Phys. 1, 437–449 (2019).
Karnieli, A. & Arie, A. Fully controllable adiabatic geometric phase in nonlinear optics. Opt. Express 26, 4920–4932 (2018).
Scully, M. O., Lamb, W. E. Jr. & Barut, A. On the theory of the Stern–Gerlach apparatus. Found. Phys. 17, 575–583 (1987).
Chen, G. et al. Advances in lithium niobate photonics: development status and perspectives. Adv. Photonics 4, 034003 (2022).
Fu, S. et al. Spin-orbit optical Hall effect. Phys. Rev. Lett. 123, 243904 (2019).
Zhang, X. et al. Photonic spin-orbit coupling induced by deep-subwavelength structured light. Phys. Rev. A 109, 023522 (2024).
Feynman, R. P., Vernon, F. L. Jr. & Hellwarth, R. W. Geometrical representation of the Schrödinger equation for solving Maser problems. J. Appl. Phys. 28, 49–52 (1957).
Hahn, E. L. Nuclear induction due to free Larmor precession. Phys. Rev. 77, 297–298 (1950).
Berry, M. V. The adiabatic phase and Pancharatnam’s phase for polarized light. J. Mod. Opt. 34, 1401–1407 (1987).
Karnieli, A., Trajtenberg-Mills, S., Di Domenico, G. & Arie, A. Experimental observation of the geometric phase in nonlinear frequency conversion. Optica 6, 1401–1405 (2019).
Suchowski, H., Porat, G. & Arie, A. Adiabatic processes in frequency conversion. Laser Photonics Rev. 8, 333–367 (2014).
Yesharim, O. et al. Observation of the all-optical Stern–Gerlach effect in nonlinear optics. Nat. Photonics 16, 582–587 (2022).
Karnieli, A. & Arie, A. All-optical Stern-Gerlach Effect. Phys. Rev. Lett. 120, 053901 (2018).
Voloch-Bloch, N., Lereah, Y., Lilach, Y., Gover, A. & Arie, A. Generation of electron Airy beams. Nature 494, 331–335 (2013).
Ornelas, P., Nape, I., de Koch, M. R. & Forbes, A. Non-local skyrmions as topologically resilient quantum entangled states of light. Nat. Photonics 18, 258–266 (2024).
Gottesman, D. & Chuang, I. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402, 390–393 (1999).
Shen, Y., Martínez, E. C. & Rosales-Guzmán, C. Generation of optical skyrmions with tunable topological textures. ACS Photonics 9, 296–303 (2022).
Shen, Y. et al. Topological transformation and free-space transport of photonic hopfions. Adv. Photonics 5, 015001 (2023).
Sugic, D. et al. Particle-like topologies in light. Nat. Commun. 12, 6785 (2021).
Lee, W. H. Binary computer-generated holograms. Appl. Opt. 18, 3661–3669 (1979).
Hu, X. B. & Rosales-Guzmn, C. Generation and characterization of complex vector modes with digital micromirror devices: a tutorial. J. Opt. 24, 034001 (2022).
Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications. Adv. Opt. Photonics 1, 1–57 (2009).
Berry, H. G., Gabrielse, G. & Livingston, A. E. Measurement of the Stokes parameters of light. Appl. Opt. 16, 3200–3205 (1977).