Symmetry, microscopy and spectroscopy signatures of altermagnetism

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.