The recent discovery of altermagnetism was in part motivated by the research of compensated magnets towards highly scalable spintronic technologies. Simultaneously, altermagnetism shares the anisotropic higher-partial-wave nature of ordering with unconventional superfluid phases, which have been at the forefront of research for the past several decades. These examples illustrate the interest in altermagnetism from a broad range of science and technology perspectives. Here we review the symmetry, microscopy and spectroscopy signatures of altermagnetism. We describe the spontaneously broken and retained symmetries that delineate altermagnetism as a distinct phase of matter with d-, g- or i-wave compensated collinear spin ordering. In materials ranging from weakly interacting metals to strongly correlated insulators, the microscopic crystal-structure realizations of the altermagnetic symmetries feature a characteristic ferroic order of anisotropic higher-partial-wave components of atomic-scale spin densities. These symmetry and microscopy signatures of altermagnetism are directly reflected in spin-dependent electronic spectra and responses. We review salient band-structure features originating from the altermagnetic ordering, and from its interplay with spin-orbit coupling and topological phenomena. Throughout, we compare altermagnetism with traditional ferromagnetism and Néel antiferromagnetism, and with magnetic phases with symmetry-protected compensated non-collinear spin orders. We accompany the theoretical discussions with references to relevant experiments.

Anthony J Leggett Institute for Condensed Matter Theory The Grainger College of Engineering University of Illinois Urbana Champaign Urbana IL USA

Center for Emergent Matter Science RIKEN Saitama Japan

Department of Applied Physics University of Tokyo Tokyo Japan

Department of Physics State key laboratory of quantum functional materials and Guangdong Basic Research Center of Excellence for Quantum Science Southern University of Science and Technology Shenzhen China

Department of Physics The Grainger College of Engineering University of Illinois Urbana Champaign Urbana IL USA

Department of Physics University of Tokyo Tokyo Japan

Institut für Physik Johannes Gutenberg Universität Mainz Mainz Germany

Institute for Quantum Matter and Department of Physics and Astronomy Johns Hopkins University Baltimore MD USA

Institute for Solid State Physics University of Tokyo Kashiwa Japan

Institute of Physics Czech Academy of Sciences Prague Czech Republic

International Institute for Sustainability with Knotted Chiral Meta Matter Hiroshima University Higashi hiroshima Japan

Max Planck Institute for Chemical Physics of Solids Dresden Germany

Max Planck Institute for the Physics of Complex Systems Dresden Germany

School of Physics and Astronomy University of Nottingham Nottingham UK

Trans scale Quantum Science Institute University of Tokyo Tokyo Japan

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MAGNDATA: A Collection of Magnetic Structures with Portable cif-type Files. https://www.cryst.ehu.es/magndata/ .