This work explores the topological properties of altermagnets, a novel class of collinear magnetic materials. We employ equivariant K-theory of magnetic groups and Hamiltonian models to formulate a robust $C^z_4 \mathbb{T}$ topological invariant to classify 2D and 3D altermagnetic systems. Our findings demonstrate that the spin Chern number serves as a robust topological index, corresponding to the half-quantized Chern number of the divided Brillouin zone. This indicator enables the prediction of a topologically protected 2D altermagnetic insulators and 3D Weyl altermagnetic semimetals, highlighting the relationship between altermagnetism and topological phases. Furthermore, our results provide a pathway to the exploration of topological applications in $d$-wave altermagnetic materials.
Mirror Chern insulators in two-dimensional altermagnetic Tc$_2$Cl$_2$O and Tc$_2$Br$_2$O
School of Physics, Northwest University（西北大学物理学院）；Shaanxi Key Laboratory for Theoretical Physics Frontiers（陕西理论物理前沿重点实验室）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论中心）；Fundamental Discipline Research Center for Quantum Science and Technology of Shaanxi Province（陕西省量子科学与技术基础学科研究中心）
Center for Functional Nanomaterials, Brookhaven National Laboratory（布鲁海文国家实验室功能纳米材料中心）；Stony Brook University（石溪大学）；University of Pennsylvania（宾夕法尼亚大学）；Flatiron Institute（平顿研究所）
Ferroelectric Altermagnetic Chern Insulator in magnetic field: electrical control of the Chern number
International Research Centre Magtop, Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所磁顶国际研究中心）；Department of Physics, University of Guilan（吉兰大学物理系）
Spin Hall and Edelstein Effects in Novel Chiral Noncollinear Altermagnets
Leibniz Institute for Solid State and Materials Research, IFW Dresden（德累斯顿莱布尼茨固体与材料研究所）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体的化学物理学研究所）；Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-Saclay（里昂·布里渊实验室，法国原子能委员会，法国国家科学研究中心，巴黎萨克雷大学）；Würzburg-Dresden Cluster of Excellence Ct.qmat, Technische Universitat Dresden（维尔茨堡-德累斯顿卓越集群Ct.qmat，德累斯顿工业大学）；Département de Physique et Institut Quantique, Université de Sherbrooke（舍布鲁克大学物理系与量子研究所）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）
Model Hamiltonian for Altermagnetic Topological Insulators
Universidad del Norte（北大学）；Departamento de Física y Geociencias, Universidad del Norte（北大学物理与地球科学系）；Departamento de Matemáticas y Estadística, Universidad del Norte（北大学数学与统计系）
Two-Dimensional Spin-Antiferroelectric Altermagnets with Giant Spin Splitting: From Model to Material Realization
School of Physics and Technology, Xinjiang University（新疆大学物理科学与技术学院）；School of Physics, Central South University（中南大学物理学院）；Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology（北京理工大学物理学院纳米光子学与超精细光电子系统北京市重点实验室）
Nonrelativistic and Relativistic Contributions to Spin-Momentum Locking and Spin Photogalvanic Effect in an Altermagnetic Multiferroic
University of Milan-Bicocca（米兰比可卡大学）；Consiglio Nazionale delle Ricerche (CNR-SPIN)（意大利国家研究委员会）；Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所）；Ca’ Foscari University of Venice（威尼斯大学）
Symmetry-Selective Strain Control of Spin-Momentum Locking and Spin Transport in Two-Dimensional Pentagonal Altermagnets
Inner Mongolia University（内蒙古大学）；Research Center for Quantum Physics and Technologies, School of Physical Science and Technology, Inner Mongolia University（内蒙古大学物理科学与技术学院量子物理与技术研究中心）；Key Laboratory of Semiconductor Photovoltaic Technology and Energy Materials at Universities of Inner Mongolia Autonomous Region, Inner Mongolia University（内蒙古自治区高校半导体光伏技术与能源材料重点实验室（内蒙古大学））；Inner Mongolia Key Laboratory of Microscale Physics and Atom Innovation, Inner Mongolia University（内蒙古微观物理与原子创新重点实验室（内蒙古大学））
Zeeman-type spin splittings in strained d-wave altermagnets
Key Laboratory of Material Simulation Methods and Software of Ministry of Education, College of Physics, Jilin University（教育部材料模拟方法与软件重点实验室，吉林大学物理学院）；Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), College of Physics, Jilin University（教育部先进电池物理与技术重点实验室，吉林大学物理学院）；International Center of Future Science, Jilin University（吉林大学未来科学国际中心）
A Route to Nonrelativistic Altermagnetic Spin Splitting via Ultrafast Light
State Key Laboratory of Semiconductor Physics and Chip Technologies, Institute of Semiconductors, Chinese Academy of Sciences（中国科学院半导体研究所半导体物理与芯片技术国家重点实验室）；Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences（中国科学院大学材料科学与光电工程技术中心）；Department of Materials Science and Engineering, Northwestern University（西北大学材料科学与工程系）
Southern University of Science and Technology (SUSTech)（南方科技大学）；Quantum Science Center of Guangdong–Hong Kong–Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心）
Direct observation of quadruple spin-texture locking in a 2D d-wave altermagnet
Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University（上海交大李政道研究所与物理天文学院）；Institute for Quantum Science and Technology, Shanghai University（上海大学量子科学与技术研究院）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Department of Physics, The Hong Kong University of Science and Technology（香港科技大学物理系）；Department of Physics, Southern University of Science and Technology（南方科技大学物理系）；State Key Laboratory of Micro-nano Engineering Science, Shanghai Jiao Tong University（上海交通大学微纳尺度工程科学国家重点实验室）；Zhangjiang Institute for Advanced Study, Shanghai Jiao Tong University（上海交通大学张江高等研究院）；Hefei National Laboratory（合肥国家实验室）；New Cornerstone Science Laboratory, Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大李政道研究所新基石科学实验室）
University of Jinan（济南大学）；Institute for Superconducting and Electronic Materials, Faculty of Engineering and Information Sciences, University of Wollongong（伍伦贡大学工程与信息科学学院超导与电子材料研究所）
Chiral Phonons Coupled to Spin-Split Bands in Altermagnetic CrSb and MnTe
CNR - Istituto Officina dei Materiali (IOM)（意大利国家研究委员会材料研究所）；Universität Würzburg（维尔茨堡大学）；Department of Physics and Astronomy, Alma Mater Studiorum, University of Bologna（博洛尼亚大学物理与天文学系）
Atomic-Scale Observation of Symmetry Breaking in Altermagnetic MnTe
University of Washington（华盛顿大学）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；Uppsala University（乌普萨拉大学）；Rice University（莱斯大学）；Czech Academy of Sciences（捷克科学院）；Brno University of Technology（布尔诺理工大学）；University of Nottingham（诺丁汉大学）；Tohoku University（东北大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Metallic d-wave altermagnetism in WFeB: a platform for electrically switchable perpendicular spin-splitter response
Iowa State University（爱荷华州立大学）；Ames National Laboratory, U.S. Department of Energy（美国能源部阿姆斯国家实验室）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；Nebraska Center for Materials and Nanoscience（内布拉斯加材料纳米科学中心）；Oak Ridge National Laboratory（橡树岭国家实验室）；Middle Tennessee State University（中田纳西州立大学）；University of Tennessee Space Institute（田纳西大学太空研究所）
Valley- and Spin-Dependent Electronic and Transport Properties of Two-Dimensional Altermagnetic Titanium-Based Chalcogenide Halides
Northwest University（西北大学）；Shaanxi Key Laboratory for Theoretical Physics Frontiers（陕西省理论物理前沿重点实验室）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论研究中心）；Fundamental Discipline Research Center for Quantum Science and Technology of Shaanxi Province（陕西省量子科学与技术基础学科研究中心）
Generating ferro-spinetic polarizations in altermagnetic insulators
Institute for Theoretical Solid State Physics, IFW Dresden（德累斯顿工业大学理论固态物理研究所）；Würzburg-Dresden Cluster of Excellence ct.qmat（维尔茨堡-德累斯顿卓越集群 ct.qmat）；Centro Atómico Bariloche, Instituto de Nanociencia y Nanotecnología and Instituto Balseiro（巴里洛切原子中心、纳米科学与纳米技术研究所和巴尔塞罗研究所）；CNR-SPIN, Department of Physical and Chemical Sciences, University of L’Aquila（拉奎拉大学物理与化学科学系意大利国家研究委员会自旋物理研究所）；Institut für Theoretische Physik und Astrophysik, Universität Würzburg（维尔茨堡大学理论物理与天体物理研究所）
Catalog of Altermagnetism in Magnetic Wallpaper/Space Groups and Nonsymmorphic Altermagnets
Hefei National Laboratory（合肥国家实验室）；School of Emerging Technology, University of Science and Technology of China（中国科学技术大学先进技术学院）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））；Department of Physics and Center for Theoretical Physics, National Taiwan University（国立台湾大学物理系暨理论物理中心）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）
Altermagnetic spin textures: Emergent electrodynamics, quantum geometry, and probes
Hearne Institute of Theoretical Physics, Department of Physics & Astronomy, Louisiana State University（路易斯安那州立大学物理与天文学系希恩理论物理研究所）；Institute for Theoretical Physics III, University of Stuttgart（斯图加特大学理论物理第三研究所）
Band topology and symmetry-driven magneto-optical response in two-dimensional d-wave altermagnets with staggered spin-orbit coupling
International Research Centre Magtop, Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所）；Department of Physics, University of Guilan（吉兰大学物理系）；Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）
