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Magnetism is one of the largest, most fundamental, and technologically most relevant fields of condensed-matter physics. Traditionally, two basic magnetic phases have been distinguished ferromagnetism and antiferromagnetism. The spin polarization in the electronic band structure reflecting the magnetization in ferromagnetic crystals underpins the broad range of time-reversal symmetry-breaking responses in this extensively explored and exploited type of magnets. By comparison, antiferromagnets have vanishing net magnetization. Recently, there have been observations of materials in which strong time-reversal symmetry-breaking responses and spin-polarization phenomena, typical of ferromagnets, are accompanied by antiparallel magnetic crystal order with vanishing net magnetization, typical of antiferromagnets. A classification and description based on spin-symmetry principles offers a resolution of this apparent contradiction by establishing a third distinct magnetic phase, dubbed altermagnetism. Our perspective starts with an overview of the still emerging unique phenomenology of this unconventionald-wave (or higher even-parity wave) magnetic phase, and of the wide array of altermagnetic material candidates. We illustrate how altermagnetism can enrich our understanding of overarching condensed-matter physics concepts and how it can have impact on prominent condensed-matter research areas.

Recent series of theoretical and experimental reports have driven attention to time-reversal symmetry-breaking spintronic and spin-splitting phenomena in materials with collinear-compensated magnetic order incompatible with conventional ferromagnetism or antiferromagnetism. Here we employ an approach based on nonrelativistic spin-symmetry groups that resolves the conflicting notions of unconventional ferromagnetism or antiferromagnetism by delimiting a third basic collinear magnetic phase. We derive that all materials hosting this collinear-compensated magnetic phase are characterized by crystal-rotation symmetries connecting opposite-spin sublattices separated in the real space and opposite-spin electronic states separated in the momentum space. We describe prominent extraordinary characteristics of the phase, including the alternating spin-splitting sign and broken time-reversal symmetry in the nonrelativistic band structure, the planar or bulkd-,g-, ori-wave symmetry of the spin-dependent Fermi surfaces, spin-degenerate nodal lines and surfaces, band anisotropy of individual spin channels, and spin-split general, as well as time-reversal invariant momenta. Guided by the spin-symmetry principles, we discover in ab initio calculations outlier materials with an extraordinary nonrelativistic spin splitting, whose eV-scale and momentum dependence are determined by the crystal potential of the nonmagnetic phase. This spin-splitting mechanism is distinct from conventional relativistic spin-orbit coupling and ferromagnetic exchange, as well as from the previously considered anisotropic exchange mechanism in compensated magnets. Our results, combined with our identification of material candidates for the phase ranging from insulators and metals to a parent crystal of cuprate superconductors, underpin research of novel quantum phenomena and spintronic functionalities in high-temperature magnets with light elements, vanishing net magnetization, and strong spin coherence. In the discussion, we argue that the conflicting notions of unconventional ferromagnetism or antiferromagnetism, on the one hand, and our symmetry-based delimitation of the third phase, on the other hand, favor a distinct term referring to the phase. The alternating spin polarizations in both the real-space crystal structure and the momentum-space band structure characteristic of this unconventional magnetic phase suggest a term altermagnetism. We point out thatd-wave altermagnetism represents a realization of the long-sought-after counterpart in magnetism of the unconventionald-wave superconductivity.

SignificanceFor many decades, it has been commonly believed that all electronic states of a collinear antiferromagnet (AF) are spin-degenerate, unless the underlying crystal structure lacks centrosymmetry and has spin–orbit coupling. This has been essentially definitional for antiferromagnetism and is widely used experimentally to distinguish ferromagnets from AFs. Recently, it was demonstrated that a new class of magnets, possessing antiferromagnetic order and without net magnetization but showing a typical ferromagnetic response in many aspects, is possible. We predict that FeSb2, which is well known but poorly understood magnetically, is an incipient unconventional magnet of this type and can be pushed to become one by Co or Cr doping. Moreover, the calculated magnetic anisotropy is favorable for exhibiting various anomalous properties. It is commonly believed that the energy bands of typical collinear antiferromagnets (AFs), which have zero net magnetization, are Kramers spin-degenerate. Kramers nondegeneracy is usually associated with a global time-reversal symmetry breaking (e.g., via ferromagnetism) or with a combination of spin–orbit interaction and broken spatial inversion symmetry. Recently, another type of spin splitting was demonstrated to emerge in some collinear magnets that are fully spin compensated by symmetry, nonrelativistic, and not even necessarily noncentrosymmetric. These materials feature nonzero spin density staggered in real space as seen in traditional AFs but also spin splitting in momentum space, generally seen only in ferromagnets. This results in a combination of materials characteristics typical of both ferromagnets and AFs. Here, we discuss this recently discovered class with application to a well-known semiconductor, FeSb2, and predict that with certain alloying, it becomes magnetic and metallic and features the aforementioned magnetic dualism. The calculated energy bands split antisymmetrically with respect to spin-degenerate nodal surfaces rather than nodal points, as in the case of spin–orbit splitting. The combination of a large (0.2-eV) spin splitting, compensated net magnetization with metallic ground state, and a specific magnetic easy axis generates a large anomalous Hall conductivity (∼150 S/cm) and a sizable magnetooptical Kerr effect, all deemed to be hallmarks of nonzero net magnetization. We identify a large contribution to the anomalous response originating from the spin–orbit interaction gapped anti-Kramers nodal surfaces, a mechanism distinct from the nodal lines and Weyl points in ferromagnets.

The recent demonstrations of electrical manipulation and detection of antiferromagnetic spins have opened up a new chapter in the story of spintronics. Here, we review the emerging research field that is exploring the links between antiferromagnetic spintronics and topological structures in real and momentum space. Active topics include proposals to realize Majorana fermions in antiferromagnetic topological superconductors, to control topological protection and Dirac points by manipulating antiferromagnetic order parameters, and to exploit the anomalous and topological Hall effects of zero-net-moment antiferromagnets. We explain the basic concepts behind these proposals, and discuss potential applications of topological antiferromagnetic spintronics.

Giant and tunneling magnetoresistance are physical phenomena used for reading information in commercial spintronic devices. The effects rely on a conserved spin current passing between a reference and a sensing ferromagnetic electrode in a multilayer structure. Recently, we have proposed that these fundamental spintronic effects can be realized in unconventional collinear antiferromagnets with nonrelativistic alternating spin-momentum coupling. Here, we elaborate on the proposal by presenting archetype model mechanisms for the giant and tunneling magnetoresistance effects in multilayers composed of these unconventional collinear antiferromagnets. The models are based, respectively, on anisotropic and valley-dependent forms of the alternating spin-momentum coupling. Using first-principles calculations, we link these model mechanisms to real materials and predict an approximately 100% scale for the effects. We point out that, besides the giant or tunneling magnetoresistance detection, the alternating spin-momentum coupling can allow for magnetic excitation by the spin-transfer torque.

The Hall effect, in which a current flows perpendicular to an electrical bias, has been prominent in the history of condensed matter physics. Appearing variously in classical, relativistic and quantum guises, the Hall effect has — among other roles — contributed to the establishment of the band theory of solids, to research on new phases of interacting electrons and to the phenomenology of topological condensed matter. The dissipationless Hall current requires time-reversal symmetry breaking. When this symmetry breaking is due to an externally applied magnetic field, the effect is referred to as the ordinary Hall effect; when it is due to a non-zero internal magnetization (ferromagnetism), it is referred to as the anomalous Hall effect. The Hall effect has not usually been associated with antiferromagnetic order. More recently, however, theoretical predictions and experimental observations have identified large Hall effects in some compensated magnetic crystals, governed by neither of the global magnetic-dipole symmetry-breaking mechanisms mentioned above. The goal of this Review is to systematically organize the present understanding of anomalous antiferromagnetic materials that generate a Hall effect — which we call anomalous Hall antiferromagnets — and to discuss this class of materials in a broader fundamental and applied research context. Our motivation is twofold: first, because Hall effects that are not governed by magnetic-dipole symmetry breaking are at odds with the traditional understanding of the phenomenon, the topic deserves attention on its own. Second, this new incarnation of the Hall effect has placed it again in the middle of an emerging field in physics, at the intersection of multipole magnetism, topological condensed matter and spintronics. Only in recent years has the Hall effect been predicted and observed in materials with antiferromagnetic order. This Review systematically organizes the current understanding of anomalous antiferromagnetic materials that generate a Hall effect, discussing these systems in the broad context of spintronics, topological condensed matter and multipole magnetism.

Altermagnetism represents an emergent collinear magnetic phase with compensated order and an unconventional alternating even-parity wave spin order in the non-relativistic band structure. We investigate directly this unconventional band splitting near the Fermi energy through spin-integrated soft X-ray angular resolved photoemission spectroscopy. The experimentally obtained angle-dependent photoemission intensity, acquired from epitaxial thin films of the predicted altermagnet CrSb, demonstrates robust agreement with the corresponding band structure calculations. In particular, we observe the distinctive splitting of an electronic band on a low-symmetry path in the Brilliouin zone that connects two points featuring symmetry-induced degeneracy. The measured large magnitude of the spin splitting of approximately 0.6 eV and the position of the band just below the Fermi energy underscores the significance of altermagnets for spintronics based on robust broken time reversal symmetry responses arising from exchange energy scales, akin to ferromagnets, while remaining insensitive to external magnetic fields and possessing THz dynamics, akin to antiferromagnets. The fundamental hallmark of altermagnetism lies in the spin splitting of electronic valence bands. Here, the authors observe splitting in metallic CrSb, revealing an exceptionally large value and energetic placement just below the Fermi energy.

The anomalous Hall effect is a time-reversal symmetry-breaking magneto-electronic phenomenon originally discovered in ferromagnets. Recently, ruthenium dioxide (RuO 2 ) with a compensated antiparallel magnetic order has been predicted to generate an anomalous Hall effect of comparable strength to ferromagnets. The phenomenon arises from an altermagnetic phase of RuO 2 with a characteristic alternating spin polarization in both real-space crystal structure and momentum-space band structure. Here we report an anomalous Hall effect in RuO 2 with an anomalous Hall conductivity exceeding 1,000 Ω −1  cm −1 . We combine the vector magnetometry and magneto-transport measurements of epitaxial RuO 2 films of different crystallographic orientations. We show that the anomalous Hall effect dominates over an ordinary Hall contribution, and a contribution due to a weak field-induced magnetization. Our results could lead to the exploration of topological Berry phases and dissipationless quantum transport in crystals of abundant elements and with a compensated antiparallel magnetic order. By combining vector magnetometry and magneto-transport measurements of epitaxial films with different crystallographic orientations, an anomalous Hall effect can be measured in collinear altermagnetic ruthenium dioxide with an anomalous Hall conductivity exceeding 1,000 Ω –1  cm –1 .

Phases with spontaneous time-reversal ( T ) symmetry breaking are sought after for their anomalous physical properties, low-dissipation electronic and spin responses, and information-technology applications. Recently predicted altermagnetic phase features an unconventional and attractive combination of a strong T -symmetry breaking in the electronic structure and a zero or only weak-relativistic magnetization. In this work, we experimentally observe the anomalous Hall effect, a prominent representative of the T -symmetry breaking responses, in the absence of an external magnetic field in epitaxial thin-film Mn 5 Si 3 with a vanishingly small net magnetic moment. By symmetry analysis and first-principles calculations we demonstrate that the unconventional d-wave altermagnetic phase is consistent with the experimental structural and magnetic characterization of the Mn 5 Si 3 epilayers, and that the theoretical anomalous Hall conductivity generated by the phase is sizable, in agreement with experiment. An analogy with unconventional d-wave superconductivity suggests that our identification of a candidate of unconventional d-wave altermagnetism points towards a new chapter of research and applications of magnetic phases. The classification of magnets now includes altermagnets which possess opposite-spin sublattices connected by rotation and share some features with ferro- and antiferromagnets. Here the authors report the anomalous Hall effect in Mn5Si3 and interpret the results in terms of a d-wave altermagnetic phase.

The origin and efficiency of charge-to-spin conversion, known as the Edelstein effect (EE), has been typically linked to spin-orbit coupling mechanisms, which require materials with heavy elements within a non-centrosymmetric environment. Here we demonstrate that the high efficiency of spin-charge conversion can be achieved even without spin-orbit coupling in the recently identified coplanar p -wave magnets. The non-relativistic Edelstein effect (NREE) in these magnets exhibits a distinct phenomenology compared to the relativistic EE, characterized by a strongly anisotropic response and an out-of-plane polarized spin density resulting from the spin symmetries. We illustrate the NREE through minimal tight-binding models, allowing a direct comparison to different systems. Through first-principles calculations, we further identify the nodal p -wave candidate material CeNiAsO as a high-efficiency NREE material, revealing a  ~ 25 times larger response than the maximally achieved relativistic EE and other reported NREE in non-collinear magnetic systems with broken time-reversal symmetry. This highlights the potential for efficient spin-charge conversion in p -wave magnetic systems. Charge-to-spin conversion, where a charge current generates a spin-current, is critical for spintronic devices. Usually efficient charge-to-spin conversion relies on heavy metals with large spin-orbit interactions, but here, Chakraborty et al show that high efficiency charge-to-spin conversion can be achieved without spin-orbit coupling using recently identified p-wave magnets.