Spatial topology endows topological solitons, such as skyrmions and hopfions, with fascinating dynamics. However, the temporal dimension has so far provided a passive stage on which topological solitons evolve. Here we construct spacetime magnetic hopfions: magnetic textures in two spatial dimensions that when excited by a time-periodic drive develop spacetime topology. We uncover two complementary construction routes using skyrmions by braiding their center of mass position and by controlling their internal low-energy excitations. Spacetime magnetic hopfions can be realized in nanopatterned grids to braid skyrmions and in frustrated magnets under an applied AC electric field. Their topological invariant, the spacetime Hopf index, can be tuned by the applied electric field as demonstrated by our collective coordinate modeling and micromagnetic simulations. The principles we have introduced to actively control spacetime topology are not limited to magnetic solitons, opening avenues to explore spacetime topology of general order parameters and fields.
Electric-current-assisted nucleation of zero-field hopfion rings
South China University of Technology（华南理工大学）；Anhui University（安徽大学）；Chinese Academy of Sciences（中国科学院）；Uppsala University（乌普萨拉大学）；Forschungszentrum Jülich（于利希研究中心）；University of Science and Technology of China（中国科学技术大学）
Nonlinear Dynamics of Hopfion for Frequency Multiplication
School of Physics and State Key Laboratory of Electronics Thin Films and Integrated Devices, University of Electronics and Science and Technology of China（电子科技大学物理学院及电子薄膜与集成器件国家重点实验室）
Stable skyrmion bundles at room temperature and zero magnetic field in a chiral magnet
University of Science and Technology of China（中国科学技术大学）；Anhui Province Key Laboratory of Low-Energy Quantum Materials and Devices, High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院高磁场实验室低能量子材料与器件安徽省重点实验室）；School of Physics and Optoelectronic Engineering, Anhui University（安徽大学物理与光电工程学院）；School of Physics and Materials Engineering, Hefei Normal University（合肥师范学院物理与材料工程学院）；Anhui Key Laboratory of Magnetic Functional Materials and Devices, School of Materials Science and Engineering, Anhui University（安徽大学材料科学与工程学院磁性功能材料与器件安徽省重点实验室）
3D Imaging of Complex Skyrmion and Hopf Topologies in an Extended Sample
STROBE NSF Science & Technology Center and JILA, University of Colorado & NIST（科罗拉多大学与NIST的STROBE国家科学基金会科学与技术中心及JILA）；Department of Physics, University of California, Berkeley（加州大学伯克利分校物理系）；Advanced Light Source, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室先进光源）；Renewable and Sustainable Energy Institute, University of Colorado（科罗拉多大学可再生与可持续能源研究所）；Physics Department, King Fahd University of Petroleum and Minerals（国王法赫德国油矿物大学物理系）；Quantum Center, KFUPM（KFUPM量子中心）；STROBE NSF Science & Technology Center and Department of Physics & Astronomy and California NanoSystems Institute, University of California, Los Angeles（加州大学洛杉矶分校的STROBE国家科学基金会科学与技术中心、物理与天文学系及加州纳米系统研究所）；Center for Memory and Recording Research, University of California San Diego（加州大学圣地亚哥分校记忆与记录研究中心）；Kapteyn-Murnane Laboratories, Inc.（卡普特因-莫恩实验室有限公司）
Spatial Antiferromagnetic Spin Texture as a Nano-Oscillator
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”（乌克兰国立技术大学（伊戈尔·西科尔斯基基辅理工大学））；Institute of Magnetism, National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine（乌克兰国家科学院磁学研究所）
