Baskaran, G. & Anderson, P. W. Gauge theory of high-temperature superconductors and strongly correlated Fermi systems. Phys. Rev. B 37, 580–583 (1988).
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).
Nagaosa, N. & Tokura, Y. Emergent electromagnetism in solids. Phys. Scr. 2012, 014020 (2012).
Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).
Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).
Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).
Neubauer, A. et al. Topological Hall effect in the A phase of MnSi. Phys. Rev. Lett. 102, 186602 (2009).
Zang, J., Mostovoy, M., Han, J. H. & Nagaosa, N. Dynamics of skyrmion crystals in metallic thin films. Phys. Rev. Lett. 107, 136804 (2011).
Schulz, T. et al. Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys. 8, 301–304 (2012).
Kurumaji, T. et al. Skyrmion lattice with a giant topological Hall effect in a frustrated triangular-lattice magnet. Science 365, 914–918 (2019).
Hirschberger, M. et al. High-field depinned phase and planar Hall effect in the skyrmion host Gd2PdSi3. Phys. Rev. B 101, 220401 (2020).
Cohen, E. et al. Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nat. Rev. Phys. 1, 437–449 (2019).
Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959).
Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A Lond. Math. Phys. Sci. 392, 45–57 (1997).
Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Ohgushi, K., Murakami, S. & Nagaosa, N. Spin anisotropy and quantum Hall effect in the kagomé lattice: Chiral spin state based on a ferromagnet. Phys. Rev. B 62, R6065–R6068 (2000).
Taguchi, Y., Oohara, Y., Yoshizawa, H., Nagaosa, N. & Tokura, Y. Spin chirality, Berry phase, and anomalous Hall effect in a frustrated ferromagnet. Science 291, 2573–2576 (2001).
Nagaosa, N. Emergent inductor by spiral magnets. Jpn. J. Appl. Phys. 58, 120909 (2019).
Yokouchi, T. et al. Emergent electromagnetic induction in a helical-spin magnet. Nature 586, 232–236 (2020).
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).
Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006).
Heinze, S. et al. Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions. Nat. Phys. 7, 713–718 (2011).
Moreau-Luchaire, C. et al. Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature. Nat. Nanotechnol. 11, 444–448 (2016).
Matsui, A., Nomoto, T. & Arita, R. Skyrmion-size dependence of the topological Hall effect: A real-space calculation. Phys. Rev. B 104, 174432 (2021).
Kimbell, G., Kim, C., Wu, W., Cuoco, M. & Robinson, J. W. A. Challenges in identifying chiral spin textures via the topological Hall effect. Commun. Mater. 3, 19 (2022).
Woo, S. et al. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater. 15, 501–506 (2016).
Juge, R. et al. Current-driven skyrmion dynamics and drive-dependent skyrmion Hall effect in an ultrathin film. Phys. Rev. Appl. 12, 044007 (2019).
Peng, L. et al. Dynamic transition of current-driven single-skyrmion motion in a room-temperature chiral-lattice magnet. Nat. Commun. 12, 6797 (2021).
Hirschberger, M. et al. Skyrmion phase and competing magnetic orders on a breathing kagomé lattice. Nat. Commun. 10, 5831 (2019).
Khanh, N. D. et al. Nanometric square skyrmion lattice in a centrosymmetric tetragonal magnet. Nat. Nanotechnol. 15, 444–449 (2020).
Takagi, R. et al. Square and rhombic lattices of magnetic skyrmions in a centrosymmetric binary compound. Nat. Commun. 13, 1472 (2022).
Okubo, T., Chung, S. & Kawamura, H. Multiple-q states and the skyrmion lattice of the triangular-lattice Heisenberg antiferromagnet under magnetic fields. Phys. Rev. Lett. 108, 017206 (2012).
Leonov, A. O. & Mostovoy, M. Multiply periodic states and isolated skyrmions in an anisotropic frustrated magnet. Nat. Commun. 6, 8275 (2015).
Inosov, D. S. et al. Electronic structure and nesting-driven enhancement of the RKKY interaction at the magnetic ordering propagation vector in Gd2PdSi3 and Tb2PdSi3. Phys. Rev. Lett. 102, 046401 (2009).
Hayami, S. & Motome, Y. Multiple-Q instability by (d − 2)-dimensional connections of Fermi surfaces. Phys. Rev. B 90, 060402 (2014).
Wang, Z., Barros, K., Chern, G.-W., Maslov, D. L. & Batista, C. D. Resistivity minimum in highly frustrated itinerant magnets. Phys. Rev. Lett. 117, 206601 (2016).
Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal current-velocity relation of skyrmion motion in chiral magnets. Nat. Commun. 4, 1463 (2013).
Schütte, C., Iwasaki, J., Rosch, A. & Nagaosa, N. Inertia, diffusion, and dynamics of a driven skyrmion. Phys. Rev. B 90, 174434 (2014).
Grüner, G. The dynamics of charge-density waves. Rev. Mod. Phys. 60, 1129–1181 (1988).
Metaxas, P. J. et al. Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy. Phys. Rev. Lett. 99, 217208 (2007).
Anderson, P. W. & Kim, Y. B. Hard superconductivity: theory of the motion of Abrikosov flux lines. Rev. Mod. Phys. 36, 39–43 (1964).
Fröhlich, H. On the theory of superconductivity: the one-dimensional case. Proc. R. Soc. A Lond. Ser. Math. Phys. Eng. Sci. 223, 296–305 (1954).
Lee, P. A., Rice, T. M. & Anderson, P. W. Conductivity from charge or spin density waves. Solid State Commun. 14, 703–709 (1974).
Psaroudaki, C., Hoffman, S., Klinovaja, J. & Loss, D. Quantum dynamics of skyrmions in chiral magnets. Phys. Rev. X 7, 041045 (2017).
Iwasaki, J., Mochizuki, M. & Nagaosa, N. Current-induced skyrmion dynamics in constricted geometries. Nat. Nanotechnol. 8, 742–747 (2013).
Büttner, F. et al. Dynamics and inertia of skyrmionic spin structures. Nat. Phys. 11, 225–228 (2015).
Birch, M. T. Dataset for: Dynamic transition and Galilean relativity of current-driven skyrmions. Zenodo https://doi.org/10.5281/zenodo.11408317 (2024).