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Altermagnets constitute a new class of magnetic materials that combine properties previously thought to be exclusive to either antiferromagnets or ferromagnets, and have unique properties of their own. In particular, a combination of symmetries connecting magnetic sublattices gives rise to a band spin splitting exhibiting unconventional d, g, or i-wave character. Their unique electronic properties have already led to new spin-dependent transport effects. Here, we consider their spin and charge dynamics on ultrafast timescales. We use a minimal tight binding model that captures the main features of the altermagnetic candidate material KRu$_4$O$_8$. In the framework of this model, we compute the spin-dependent electronic scattering dynamics after ultrashort-pulse excitation and show through these microscopic calculations how electron-electron and electron-phonon scattering processes redistribute optically excited carriers in a 2D slice of the Brillouin zone. We find that the optically excited spin polarization is long lived (~1ps) compared to the electron-electron momentum scattering lifetime of roughly 10fs. This contrasts remarkably with the much shorter spin lifetimes observed in typical ultrafast electronic spin dynamics in conventional ferromagnets and antiferromagnets, making these pulse-driven spin excitation experiments a key probe of altermagnetism.

RuO$_2$ is one of the most highlighted candidates for altermagnetism. However, the most recent muon spin spectroscopy and neutron studies demonstrated the absence of magnetic order in this system. The electronic structure of RuO$_2$ hints at a possibility of realizing a magnetically ordered state upon hole doping, and such a possibility was explored experimentally in Cr-doped RuO$_2$, where it was suggested that this system exhibits the anomalous Hall effect (AHE) due to altermagnetism. In this manuscript, based on our density functional calculations, we revise the results obtained for this system and propose a different interpretation of experimental results. Our calculations suggest that extra holes are bound to Cr impurity and do not dope Ru bands, which remain nonmagnetic. Thus, the observed AHE is not due to the altermagnetism but stems entirely from magnetic Cr ions.

The altermagnet candidate Mn5Si3 has attracted wide attention in the context of non-relativistic spin physics, due to its composition of light elements. However, the presumed structure of the altermagnetic phase had yet to be demonstrated. In this study, we demonstrate a hallmark of altermagnetism in Mn5Si3 thin films, namely the three options, or variants, for the checkerboard distribution of the magnetic Mn atoms. The magnetic symmetries were altered by field-rotation of the N\'eel vector along relevant crystal directions, resulting in anomalous Hall effect anisotropy. The experimental results in nanoscale devices were corroborated by a theoretical model involving atomic site dependent anisotropy and bulk Dzyaloshinskii-Moriya interaction for a single variant.

The recent discovery of altermagnets has opened new perspectives in the field of ordered phases in condensed matter. In strongly-correlated superfluids, the nodal p-wave and d-wave ordered phases of $^{3}$He and cuprates play a prominent role in physics for their rich phenomenology of the symmetry-breaking order parameters. While the p-wave and d-wave superfluids have been extensively studied over the past half a century, material realizations of their magnetic counterparts have remained elusive for many decades. This is resolved in altermagnets, whose recent discovery was driven by research in the field of spintronics towards highly scalable information technologies. Altermagnets feature d, g or i-wave magnetic ordering, with a characteristic alternation of spin polarization and spin-degenerate nodes. Here we review how altermagnetism can be identified from symmetries of collinear spin densities in crystal lattices, and can be realized at normal conditions in a broad family of insulating and conducting materials. We highlight salient electronic-structure signatures of the altermagnetic ordering, discuss extraordinary relativistic and topological phenomena that emerge in their band structures, and comment on strong-correlation effects. We then extend the discussion to non-collinear spin densities in crystals, including the prediction of p-wave magnets, and conclude with a brief summary of the reviewed physical properties of the nodal magnetically-ordered phases., Comment: 17 pages, 6 figures

The recently discovered altermagnets exhibit collinear magnetic order with zero net magnetization but with unconventional spin-polarized d/g/i-wave band structures, expanding the known paradigms of ferromagnets and antiferromagnets. In addition to novel current-driven electronic transport effects, the unconventional time-reversal symmetry breaking in these systems also makes it possible to obtain a spin response to linearly polarized fields in the optical frequency domain. We show through ab-initio calculations of the prototypical d-wave altermagnet RuO$_2$, with $[C_2\|C_{4z}]$ symmetry combining twofold spin rotation with fourfold lattice rotation, that there is an optical analogue of a spin splitter effect, as the coupling to a linearly polarized exciting laser field makes the d-wave character of the altermagnet directly visible. By magneto-optical measurements on RuO$_2$ films of a few nanometer thickness, we demonstrate the predicted connection between the polarization of an ultrashort pump pulse and the sign and magnitude of a persistent optically excited electronic spin polarization. Our results point to the possibility of exciting and controlling the electronic spin polarization in altermagnets by such ultrashort optical pulses. In addition, the possibility of exciting an electronic spin polarization by linearly polarized optical fields in a compensated system is a unique consequence of the altermagnetic material properties, and our experimental results therefore present an indication for the existence of an altermagnetic phase in ultrathin RuO$_2$ films.

Recent developments have introduced a groundbreaking form of collinear magnetism known as "altermagnetism". This emerging magnetic phase is characterized by robust time-reversal symmetry breaking, antiparallel magnetic order, and alternating spin-splitting band structures, yet it exhibits vanishing net magnetization constrained by symmetry. Altermagnetism uniquely integrates traits previously considered mutually exclusive to conventional collinear ferromagnetism and antiferromagnetism, thereby facilitating phenomena and functionalities previously not achievable within these traditional categories of magnetism. Initially proposed theoretically, the existence of the altermagnetic phase has since been corroborated by a range of experimental studies, which have confirmed its unique properties and potential for applications. This review explores the rapidly expanding research on altermagnets, emphasizing the novel physical phenomena they manifest, methodologies for inducing altermagnetism, and promising altermagnetic materials. The goal of this review is to furnish readers with a comprehensive overview of altermagnetism and to inspire further innovative studies on altermagnetic materials which could potentially revolutionize applications in technology and materials science., Comment: Submitted to Adv. Funct. Mater

We show that magnetic insulators with a collinear and compensated order can exhibit a thermal Hall effect even at zero magnetic field if they have altermagnetic symmetries. We predict a finite thermal Hall conductivity vector $\boldsymbol{\kappa}_\text{H}$ for a rutile-inspired effective spin model with Dzyaloshinskii-Moriya interaction. Within the linear spin-wave theory, we identify two magnon branches that carry identical Berry curvature and give rise to a finite $\boldsymbol{\kappa}_\text{H}$, which can be controlled by the N\'eel vector orientation and by strain. The thermal Hall response is further complemented with a spin Nernst response to contrast spin and heat transport in altermagnetic insulators with those in ferromagnets and antiferromagnets. Our results establish the crystal thermal Hall effect of magnons and we discuss material candidates for experimental realization, such as MnF$_2$, CoF$_2$, and NiF$_2$., Comment: 23 pages

Investigating the off-diagonal components of the conductivity and thermoelectric tensor of materials hosting complex antiferromagnetic structures has become a viable method to reveal the effects of topology and chirality on the electronic transport in these systems. In this respect, Mn$_5$Si$_3$ is an interesting metallic compound that exhibits several antiferromagnetic phases below 100 K with different collinear and noncollinear arrangements of Mn magnetic moments. Previous investigations have shown that the transitions between the various phases give rise to large changes of the anomalous Hall effect. Here, we report measurements of the anomalous Nernst effect of Mn$_5$Si$_3$ single crystals. Below 25 K we observe a sign change of the zero-field Nernst signal with a concomitant decrease of the Hall signal and a gradual reduction of the remanent magnetization which we attribute to a subtle rearrangement of the magnetic moment configuration at low temperatures., Comment: 9 pages, 6 figures

The anomalous Nernst effect generates transverse voltage to the applied thermal gradient in magnetically ordered systems. The effect was previously considered excluded in compensated magnetic materials with collinear ordering. However, in the recently identified class of compensated magnetic materials, dubbed altermagnets, time-reversal symmetry breaking in the electronic band structure makes the presence of the anomalous Nernst effect possible despite the collinear spin arrangement. In this work, we investigate epitaxial Mn5Si3 thin films known to be an altermagnetic candidate. We show that the material manifests a sizable anomalous Nernst coefficient despite the small net magnetization of the films. The measured magnitudes of the anomalous Nernst coefficient reach a scale of microVolts per Kelvin. We support our magneto-thermoelectric measurements by density-functional theory calculations of the material's spin-split electronic structure, which allows for the finite Berry curvature in the reciprocal space. Furthermore, we present our calculations of the intrinsic Berry-curvature Nernst conductivity, which agree with our experimental observations.

The newly discovered altermagnets are unconventional collinear compensated magnetic systems, exhibiting even (d, g, or i-wave) spin-polarization order in the band structure, setting them apart from conventional collinear ferromagnets and antiferromagnets. Altermagnets offer advantages of spin polarized current akin to ferromagnets, and THz functionalities similar to antifferomagnets, while introducing new novel effects like spin-splitter currents. A key challenge for future applications and functionalization of altermagnets, is to demonstrate controlled transitioning to the altermagnetic phase from other conventional phases in a single material. Here we prove a viable path towards overcoming this challenge through a strain-induced transition from an antiferromagnetic to an altermagnetic phase in ReO$_2$. Combining spin group symmetry analysis and \textit{ab-initio} calculations, we demonstrate that under compressive strain ReO$_2$ undergoes such transition, lifting the Kramer's degeneracy of the band structure of the antiferromagnetic phase in the non-relativistic regime. In addition, we show that this magnetic transition is accompanied by a metal insulator transition, and calculate the distinct spin polarized spectral functions of the two phases, which can be detected in angle resolved photo-emission spectroscopy experiments., Comment: 6 Figures

Altermagnets (AM) is a recently discovered class of collinear magnets that share some properties (anomalous transport, etc) with ferromagnets, some (zero net magnetization) with antiferromagnets, while also exhibiting unique properties (spin-splitting of electronic bands and resulting spin-splitter current). Since the moment compensation in AM is driven by symmetry, it must be possible to identify them by analyzing the crystal structure directly, without computing the electronic structure. Given the significant potential of AM for spintronics, it is very useful to have a tool for such an analysis. This work presents an open-access code implementing such a direct check., Comment: Submission to SciPost

Altermagnets are compensated magnets belonging to spin symmetry groups that allow alternating spin polarizations both in the coordinate space of the crystal and in the momentum space of the electronic structure. In these materials the anisotropic local crystal environment of the different sublattices lowers the symmetry of the system so that the opposite-spin sublattices are connected only by rotations, which results in an unconventional spin-polarized band structure in the momentum space. This low symmetry of the crystal structure is expected to be reflected in the anisotropy of the anomalous Hall effect. In this work, we study the anisotropy of the anomalous Hall effect in epitaxial thin films of Mn$_5$Si$_3$, an altermagnetic candidate material. We first demonstrate a change in the relative N\'eel vector orientation when rotating the external field orientation through systematic changes in both the anomalous Hall effect and the anisotropic longitudinal magnetoresistance. We then show that the anomalous Hall effect in this material is anisotropic with the N\'eel vector orientation relative to the crystal structure and that this anisotropy requires high crystal quality and unlikely correlates with the magnetocrystalline anisotropy. Our results provide further systematic support to the case for considering epitaxial thin films of Mn$_5$Si$_3$ as an altermagnetic candidate material.

Altermagnetism has emerged as a third type of collinear magnetism. In contrast to standard ferromagnets and antiferromagnets, altermagnets exhibit extra even-parity wave spin order parameters resulting in a spin-splitting of electronic bands in momentum space. In real space, sublattices of opposite spin polarization are anisotropic and related by rotational symmetry. In the hitherto identified altermagnetic candidate materials the anisotropies arise from the local crystallographic symmetry. Here, we show that altermagnetism can also form as an interaction-induced electronic instability in a lattice without the crystallographic sublattice anisotropy. We provide a microscopic example of a two-orbital model showing that the coexistence of staggered antiferromagnetic and orbital order can realize robust altermagnetism. We quantify the spin-splitter conductivity as a key experimental observable and discuss material candidates for the interaction-induced realization of altermagnetism.

The detection of a voltage transverse to both an applied current and a magnetic field is one of the most common characterization techniques in solid-state physics. The corresponding component of the resistivity tensor $\rho_{ij}$ can be separated into odd and even parts with respect to the applied magnetic field. The former contains information, for example, about the ordinary or anomalous Hall effect. The latter is typically ascribed to experimental artefacts and ignored. We here show that upon suppressing these artefacts in carefully controlled experiments, useful information remains. We first investigate the well-explored ferromagnet CoFeB, where the even part of $\rho_{yx}$ contains a contribution from the anisotropic magnetoresistance, which we confirm by Stoner-Wohlfarth modelling. We then apply our approach to magnetotransport measurements in $\rm Mn_5Si_3$ thin films with a complex compensated magnetic order. In this material, the even part of the transverse signal is sizable only in the low-spin-symmetry phase below $\approx 80$ K and thus offers a simple and readily available probe of the magnetic order., Comment: 18 pages, 5 figures

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 spinintegrated 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 signifcance 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., Comment: 13 pages, 9 figures (including supplementary information)

Altermagnets (AM) are a recently discovered third class of collinear magnets, distinctly different from conventional ferromagnets (FM) and antiferromagnets (AF). AM have been actively researched in the last few years, but two aspects so far remain unaddressed: (1) Are there realistic 2D single-layer altermagnets? And (2) is it possible to functionalize a conventional AF into AM by external stimuli? In this paper we address both issues by demonstrating how a well-known 2D AF, MnP(S,Se)$_3$ can be functionalized into strong AM by applying out-of-plane electric field. Of particular interest is that the induced altermagnetism is of a higher even-parity wave symmetry than expected in 3D AM with similar crystal symmetries. We confirm our finding by first-principles calculations of the electronic structure and magnetooptical response. We also propose that recent observations of the time-reversal symmetry breaking in the famous Fe-based superconducting chalchogenides, either in monolayer form or in the surface layer, may be related not to an FM, as previously assumed, but to the induced 2D AM order. Finally, we show that monolayer FeSe can simultaneously exhibit unconventional altermagnetic time-reversal symmetry breaking and quantized spin Hall conductivity indicating possibility to research an intriquing interplay of 2D altermagnetism with topological and superconducting states within a common crystal-potential environment., Comment: 11 pages, 7 figures

The p-wave Cooper-pairing instability in superfluid $^{3}$He, characterized by a parity-breaking excitation gap, is regarded as one of the most rich and complex phenomena in physics. The possibility of a counterpart unconventional p-wave ordering of interacting fermions, in which a Fermi surface spontaneously breaks the parity symmetry, has been an open problem for many decades. Here we identify the realization of the counterpart of p-wave superfluidity in magnetism. We demonstrate a strong parity-breaking and anisotropic symmetry lowering of spin-polarized and time-reversal symmetric Fermi surfaces in a representative p-wave magnet CeNiAsO. As a direct experimental signature we predict a large spontaneous anisotropy of the resistivity. Abundant and robust realizations of the unconventional p-wave magnetism can be identified from suitable non-relativistic crystal-lattice and spin symmetries, without requiring strong correlations and extreme external conditions. This opens new prospects in fields ranging from topological phenomena to spintronics., Comment: 6 pages, 3 figures

Observations of the anomalous Hall effect in RuO$_2$ and MnTe have demonstrated unconventional time-reversal symmetry breaking in the electronic structure of a recently identified new class of compensated collinear magnets, dubbed altermagnets. While in MnTe the unconventional anomalous Hall signal accompanied by a vanishing magnetization is observable at remanence, the anomalous Hall effect in RuO$_2$ is excluded by symmetry for the N\'eel vector pointing along the zero-field [001] easy-axis. Guided by a symmetry analysis and ab initio calculations, a field-induced reorientation of the N\'eel vector from the easy-axis towards the [110] hard-axis was used to demonstrate the anomalous Hall signal in this altermagnet. We confirm the existence of an anomalous Hall effect in our RuO$_2$ thin-film samples whose set of magnetic and magneto-transport characteristics is consistent with the earlier report. By performing our measurements at extreme magnetic fields up to 68 T, we reach saturation of the anomalous Hall signal at a field $H_{\rm c} \simeq$ 55 T that was inaccessible in earlier studies, but is consistent with the expected N\'eel-vector reorientation field., Comment: 4 figures

Altermagnetism is a newly identified fundamental class of magnetism with vanishing net magnetization and time-reversal symmetry broken electronic structure. Probing the unusual electronic structure with nonrelativistic spin splitting would be a direct experimental verification of altermagnetic phase. By combining high-quality film growth and $in~situ$ angle-resolved photoemission spectroscopy, we report the electronic structure of an altermagnetic candidate, $\alpha$-MnTe. Temperature dependent study reveals the lifting of Kramers{\textquoteright} degeneracy accompanied by a magnetic phase transition at $T_N=267\text{ K}$ with spin splitting of up to $370\text{ meV}$, providing direct spectroscopic evidence for altermagnetism in MnTe.

Motivated by the wealth of proposals and realizations of nontrivial topological phases in EuCd$_2$As$_2$, such as a Weyl semimetallic state and the recently discussed semimetallic versus semiconductor behavior in this system, we analyze in this work the role of the delicate interplay of Eu magnetism, strain and pressure on the realization of such phases. For that we invoke a combination of a group theoretical analysis with ab initio density functional theory calculations and uncover a rich phase diagram with various non-trivial topological phases beyond a Weyl semimetallic state, such as axion and topological crystalline insulating phases, and discuss their realization., Comment: v3 fixed typo in Fig. 3

Progress in magnetoelectric materials is hindered by apparently contradictory requirements for time-reversal symmetry broken and polar ferroelectric electronic structure in common ferromagnets and antiferromagnets. Alternative routes could be provided by recent discoveries of a time-reversal symmetry breaking anomalous Hall effect in noncollinear magnets and altermagnets, but hitherto reported bulk materials are not polar. Here, we report the observation of a spontaneous anomalous Hall effect in doped AgCrSe$_2$, a layered polar semiconductor with an antiferromagnetic coupling between Cr spins in adjacent layers. The anomalous Hall resistivity 3 $\mu\Omega$ cm is comparable to the largest observed in compensated magnetic systems to date, and is rapidly switched off when the angle of an applied magnetic field is rotated to $\sim 80^{\circ}$ from the crystalline $c$-axis. Our ionic gating experiments show that the anomalous Hall conductivity magnitude can be enhanced by modulating the $p$-type carrier density. We also present theoretical results that suggest the anomalous Hall effect is driven by Berry curvature due to noncollinear antiferromagnetic correlations among Cr spins, which are consistent with the previously suggested magnetic ordering in AgCrSe$_2$. Our results open the possibility to study the interplay of magnetic and ferroelectric-like responses in this fascinating class of materials., Comment: 8 pages, 5 figures

Resistivity measurements are widely exploited to uncover electronic excitations and phase transitions in metallic solids. While single crystals are preferably studied to explore crystalline anisotropies, these usually cancel out in polycrystalline materials. Here we show that in polycrystalline Mn3Zn0.5Ge0.5N with non-collinear antiferromagnetic order, changes in the diagonal and, rather unexpected, off-diagonal components of the resistivity tensor occur at low temperatures indicating subtle transitions between magnetic phases of different symmetry. This is supported by neutron scattering and explained within a phenomenological model which suggests that the phase transitions in magnetic field are associated with field induced topological orbital momenta. The fact that we observe transitions between spin phases in a polycrystal, where effects of crystalline anisotropy are cancelled suggests that they are only controlled by exchange interactions. The observation of an off-diagonal resistivity extends the possibilities for realising antiferromagnetic spintronics with polycrystalline materials., Comment: 4 figures, 1 table

We demonstrate the emergence of a pronounced thermal transport in the recently discovered class of magnetic materials-altermagnets. From symmetry arguments and first-principles calculations performed for the showcase altermagnet, RuO2, we uncover that crystal Nernst and crystal thermal Hall effects in this material are very large and strongly anisotropic with respect to the Neel vector. We find the large crystal thermal transport to originate from three sources of Berry's curvature in momentum space: the Weyl fermions due to crossings between well-separated bands, the strong spin-flip pseudonodal surfaces, and the weak spin-flip ladder transitions, defined by transitions among very weakly spin-split states of similar dispersion crossing the Fermi surface. Moreover, we reveal that the anomalous thermal and electrical transport coefficients in RuO2 are linked by an extended Wiedemann-Franz law in a temperature range much wider than expected for conventional magnets. Our results suggest that altermagnets may assume a leading role in realizing concepts in spin caloritronics not achievable with ferromagnets or antiferromagnets., Comment: 6 pages

Topological antiferromagnetic systems, which exhibit anisotropic band structures combined with complex relativistic spin structures in momentum space, have shown strong magnetoresistance effects driven by Dirac fermion characteristics. While these new antiferromagnets have been studied in transport experiments, very little is known about their spin-dependent electronic dynamics on ultrafast timescales and far-from-equilibrium behavior. This paper investigates theoretically the spin-dependent electronic dynamics due to electron-phonon scattering in a model electronic band structure that corresponds to a Dirac semimetal antiferromagnet. Following a spin conserving instantaneous excitation we obtain a change of the antiferromagnetic spin polarization due to the scattering dynamics for the site-resolved spin expectation values. This allows us to identify fingerprints of the anisotropic band structure in the carrier dynamics on ultrashort timescales which should be observable in present experimental set-ups.

Magnons in ferromagnets have one chirality, and typically are in the GHz range and have a quadratic dispersion near the zero wavevector. In contrast, magnons in antiferromagnets are commonly considered to have bands with both chiralities that are degenerate across the entire Brillouin zone, and to be in the THz range and to have a linear dispersion near the center of the Brillouin zone. Here we theoretically demonstrate a new class of magnons on a prototypical $d$-wave altermagnet RuO$_2$ with the compensated antiparallel magnetic order in the ground state. Based on density-functional-theory calculations we observe that the THz-range magnon bands in RuO$_2$ have an alternating chirality splitting, similar to the alternating spin splitting of the electronic bands, and a linear magnon dispersion near the zero wavevector. We also show that, overall, the Landau damping of this metallic altermagnet is suppressed due to the spin-split electronic structure, as compared to an artificial antiferromagnetic phase of the same RuO$_2$ crystal with spin-degenerate electronic bands and chirality-degenerate magnon bands., Comment: 6 pages, 4 figures

Tetragonal CuMnAs is a room temperature antiferromagnet with an electrically reorientable N\'eel vector and a Dirac semimetal candidate. Direct measurements of the electronic structure of single-crystalline thin films of tetragonal CuMnAs using angle-resolved photoemission spectroscopy (ARPES) are reported, including Fermi surfaces (FS) and energy-wavevector dispersions. After correcting for a chemical potential shift of $\approx-390$ meV (hole doping), there is excellent agreement of FS, orbital character of bands, and Fermi velocities between the experiment and density functional theory calculations. Additionally, 2x1 surface reconstructions are found in the low energy electron diffraction (LEED) and ARPES. This work underscores the need to control the chemical potential in tetragonal CuMnAs to enable the exploration and exploitation of the Dirac fermions with tunable masses, which are predicted to be above the chemical potential in the present samples., Comment: Submitted to Physical Review X. 20 pages. 9 figures

Magnetism is one of the largest, most fundamental, and technologically most relevant fields of condensed-matter physics. Traditionally, two basic magnetic phases have been considered -- ferromagnetism and antiferromagnetism. The breaking of the time-reversal symmetry and spin splitting of the electronic states by the magnetization in ferromagnets underpins a range of macroscopic responses in this extensively explored and exploited type of magnets. By comparison, antiferromagnets have vanishing net magnetization. This Perspective reflects on recent observations of materials hosting an intriguing ferromagnetic-antiferromagnetic dichotomy, in which spin-split spectra and macroscopic observables, akin to ferromagnets, are accompanied by antiparallel magnetic order with vanishing magnetization, typical of antiferromagnets. An unconventional non-relativistic symmetry-group formalism offers a resolution of this apparent contradiction by delimiting a third basic magnetic phase, dubbed altermagnetism. Our Perspective starts with an overview of the still emerging unique phenomenology of the phase, and of the wide array of altermagnetic material candidates. In the main part of the article, we illustrate how altermagnetism can enrich our understanding of overarching condensed-matter physics concepts, and have impact on prominent condensed-matter research areas., Comment: 25 pages, 13 figures

Parity symmetric photoemission spectra are ubiquitous in solid state research, being prevalent in many highly active areas, such as unconventional superconductors, nonmagnetic and antiferromagnetic topological insulators, and weakly relativistic collinear magnets, among others. The direct observation of parity-violating metallic Kramers degenerate bands has remained hitherto experimentally elusive. Here we observe the antiferromagnetic parity violation (APV) in the bandstructure of Mn$_2$Au thin films by using momentum microscopy with sub-$mu$m spatial resolution, allowing momentum resolved photoemission on single antiferromagnetic domains. The APV arises from breaking the P symmetry of the underlying crystal structure by the collinear antiferromagnetism, while preserving the joint space-time inverison PT -symmetry and in combination with large spin-orbit coupling. In addition, our work also demonstrates a novel tool to directly image the Neel vector direction by combining spatially resolved momentum microscopy with ab-initio calculations.

The Hall effect, in which current flows perpendicular to an applied electrical bias, has played a prominent role in modern condensed matter physics over much of the subject's history. Appearing variously in classical, relativistic and quantum guises, it has among other roles contributed to establishing the band theory of solids, to research on new phases of strongly interacting electrons, and to the phenomenology of topological condensed matter. The dissipationless Hall current requires time-reversal symmetry breaking. For over a century it has either been ascribed to externally applied magnetic field and referred to as the ordinary Hall effect, or ascribed to spontaneous non-zero total internal magnetization (ferromagnetism) and referred to as the anomalous Hall effect. It has not commonly 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 goals of this article are to systematically organize the present understanding of anomalous antiferromagnetic materials that generate a non-zero Hall effect, which we will call anomalous Hall antiferromagnets, and to discuss this class of materials in a broader fundamental and applied research context. Our motivation in drawing attention to anomalous Hall antiferromagnets is two-fold. First, since 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 reincarnation has again placed the Hall effect in the middle of an emerging field of physics at the intersection of multipole magnetism, topological condensed matter, and spintronics., Comment: 21 pages, 6 figures, 1 table

It is commonly believed that in typical collinear antiferromagnets, with no net magnetization, the energy bands are spin-(Kramers-degenerate. The opposite case is usually associated with a global time-reversal symmetry breaking (e.g., via ferro(i)magnetism), or with the spin-orbit interaction is combined with the broken spatial inversion symmetry. Recently, another type of spin splitting was demonstrated to emerge in some fully compensated by symmetry, nonrelativistic, collinear magnets, and not even necessarily non-centrosymmetric. These materials feature non-zero spin density staggered not only in real, but also in momentum space. This duality results in a combination of characteristics typical of ferro- and antiferromagnets. Here we discuss this novel concept in application to a well-known semiconductor, FeSb2, and predict that upon certain alloying it becomes magnetic, and features such magnetic duality. The calculated energy bands split antisymmetrically with respect to spin degenerate nodal surfaces (and not nodal points, as in the case of spin-orbit splitting. This combination of a large (0.2 eV) spin splitting, compensated net magnetization and metallic ground-state, and a particular magnetic easy axis generate a large anomalous Hall conductivity (~150 S/cm) and a sizable magneto-optical 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 {\it points} in ferromagnets., Comment: minor corrections to the references list

The search for novel magnetic quantum phases, phenomena and functional materials has been guided by relativistic magnetic-symmetry groups in coupled spin and real space from the dawn of the field in 1950s to the modern era of topological matter. However, the magnetic groups cannot disentangle non-relativistic phases and effects, such as the recently reported unconventional spin physics in collinear antiferromagnets from the typically weak relativistic spin-orbit coupling phenomena. Here we discover that more general spin symmetries in decoupled spin and crystal space categorize non-relativistic collinear magnetism in three phases: conventional ferromagnets and antiferromagnets, and a third distinct phase combining zero net magnetization with an alternating spin-momentum locking in energy bands, which we dub "altermagnetic". For this third basic magnetic phase, which is omitted by the relativistic magnetic groups, we develop a spin-group theory describing six characteristic types of the altermagnetic spin-momentum locking. We demonstrate an extraordinary spin-splitting mechanism in altermagnetic bands originating from a local electric crystal field, which contrasts with the conventional magnetic or relativistic splitting by global magnetization or inversion asymmetry. Based on first-principles calculations, we identify altermagnetic candidates ranging from insulators and metals to a parent crystal of cuprate superconductor. Our results underpin emerging research of quantum phases and spintronics in high-temperature magnets with light elements, vanishing net magnetization, and strong spin-coherence., Comment: 35 pages including Supplementary information, 8 figures, 1 table

Giant or 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 collinear antiferromagnets with staggered spin-momentum exchange interaction, which generates conserved spin currents in the absence of a net equilibrium magnetization. Here we elaborate on the proposal by presenting archetype model mechanisms for the antiferromagnetic giant and tunneling magnetoresistance effects. The models are based, respectively, on anisotropic and valley-dependent forms of the non-relativistic staggered spin-momentum interaction. Using first principles calculations we link these model mechanisms to real antiferromagnetic materials and predict a $\sim$100\% scale for the effects. We point out that besides the GMR/TMR detection, our models directly imply the possibility of spin-transfer-torques excitation of the antiferromagnets., Comment: 6 pages, 4 figures

It has been suggested that certain antiferromagnetic topological insulators contain axion quasiparticles (AQs), and that such materials could be used to detect axion dark matter (DM). The AQ is a longitudinal antiferromagnetic spin fluctuation coupled to the electromagnetic Chern-Simons term, which, in the presence of an applied magnetic field, leads to mass mixing between the AQ and the electric field. The electromagnetic boundary conditions and transmission and reflection coefficients are computed. A model for including losses into this system is presented, and the resulting linewidth is computed. It is shown how transmission spectroscopy can be used to measure the resonant frequencies and damping coefficients of the material, and demonstrate conclusively the existence of the AQ. The dispersion relation and boundary conditions permit resonant conversion of axion DM into THz photons in a material volume that is independent of the resonant frequency, which is tuneable via an applied magnetic field. A parameter study for axion DM detection is performed, computing boost amplitudes and bandwidths using realistic material properties including loss. The proposal could allow for detection of axion DM in the mass range between 1 and 10 meV using current and near future technology., Comment: 78 pages + appendices, v2: reference list extended, added one more case to fig 23

Time-reversal (T) symmetry breaking is a fundamental physics concept underpinning a broad science and technology area, including topological magnets, axion physics, dissipationless Hall currents, or spintronic memories. A best known conventional model of macroscopic T-symmetry breaking is a ferromagnetic order of itinerant Bloch electrons with an isotropic spin interaction in momentum space. Anisotropic electron interactions, on the other hand, have been a domain of correlated quantum phases, such as the T-invariant nematics or unconventional superconductors. Here we report discovery of a broken-T phase of itinerant Bloch electrons with an unconventional anisotropic spin-momentum interaction, whose staggered nature leads to the formation of two ferromagnetic-like valleys in the momentum space with opposite spin splittings. We describe qualitatively the effect by deriving a non-relativistic single-particle Hamiltonian model. Next, we identify the unconventional staggered spin-momentum interaction by first-principles electronic structure calculations in a four-sublattice antiferromagnet Mn5Si3 with a collinear checkerboard magnetic order. We show that the staggered spin-momentum interaction is set by nonrelativistic spin-symmetries which were previously omitted in relativistic physics classifications of spin interactions and topological quasiparticles. Our measurements of a spontaneous Hall effect in epilayers of antiferromagnetic Mn5Si3 with vanishing magnetization are consistent with our theory predictions. Bloch electrons with the unconventional staggered spin interaction, compatible with abundant low atomic-number materials, strong spin-coherence, and collinear antiferromagnetic order open unparalleled possibilities for realizing T-symmetry broken spin and topological quantum phases., Comment: 26 pages, 4 figures

Time-reversal symmetry breaking is the basic physics concept underpinning many magnetic topological phenomena such as the anomalous Hall effect (AHE) and its quantized variant. The AHE has been primarily accompanied by a ferromagnetic dipole moment, which hinders the topological quantum states and limits data density in memory devices, or by a delicate noncollinear magnetic order with strong spin decoherence, both limiting their applicability. A potential breakthrough is the recent theoretical prediction of the AHE arising from collinear antiferromagnetism in an anisotropic crystal environment. This new mechanism does not require magnetic dipolar or noncollinear fields. However, it has not been experimentally observed to date. Here we demonstrate this unconventional mechanism by measuring the AHE in an epilayer of a rutile collinear antiferromagnet RuO$_2$. The observed anomalous Hall conductivity is large, exceeding 300 S/cm, and is in agreement with the Berry phase topological transport contribution. Our results open a new unexplored chapter of time-reversal symmetry breaking phenomena in the abundant class of collinear antiferromagnetic materials., Comment: 33 pages, 14 figures, 2 tables

Electrical spin-current generation is among the core phenomena driving the field of spintronics. Using {\em ab initio} calculations we show that a room-temperature metallic collinear antiferromagnet RuO$_2$ allows for highly efficient spin-current generation, arising from anisotropically-split bands with conserved up and down spins along the N\'eel vector axis. The zero net moment antiferromagnet acts as an electrical spin-splitter with a 34$^\circ$ propagation angle between spin-up and spin-down currents. Correspondingly, the spin-conductivity is a factor of three larger than the record value from a survey of 20,000 non-magnetic spin-Hall materials. We propose a versatile spin-splitter-torque concept utilizing antiferromagnetic RuO$_2$ films interfaced with a ferromagnet., Comment: 6 pages, 4 figures

The electronic anomalous Hall effect (AHE), where charge carriers acquire a velocity component orthogonal to an applied electric field, is one of the most fundamental and widely studied phenomena in physics. There are several different AHE mechanisms known, and material examples are highly sought after, however in the highly conductive (skew scattering) regime the focus has centered around ferromagnetic metals. Here we report the observation of a giant extrinsic AHE in KV$_3$Sb$_5$, an exfoliable, Dirac semimetal with a Kagome layer of Vanadium atoms. Although there has been no reports of magnetic ordering down to 0.25 K, the anomalous Hall conductivity (AHC) reaches $\approx$ 15,507 $\Omega^{-1}$cm$^{-1}$ with an anomalous Hall ratio (AHR) of $\approx$ 1.8$ \%$; an order of magnitude larger than Fe. Defying expectations from skew scattering theory, KV$_3$Sb$_5$ shows an enhanced skew scattering effect that scales quadratically, not linearly, with the longitudinal conductivity ($\sigma_{xx}$), opening the possibility of reaching an anomalous Hall angle (AHA) of 90$^{\circ}$ in metals; an effect thought reserved for quantum anomalous Hall insulators. This observation raises fundamental questions about the AHE and opens a new frontier for AHE (and correspondingly SHE) exploration, stimulating investigation in a new direction of materials, including metallic geometrically frustrated magnets, spin-liquid candidates, and cluster magnets.

Electrons, commonly moving along the applied electric field, acquire in certain magnets a dissipationless transverse velocity. This spontaneous Hall effect, discovered more than a century ago, has been understood in terms of the time-reversal symmetry breaking by the internal spin-structure of a ferromagnetic, noncolinear antiferromagnetic or skyrmionic form. Here we identify previously overlooked robust Hall effect mechanism arising from collinear antiferromagnetism combined with nonmagnetic atoms at non-centrosymmetric positions. We predict a large magnitude of this crystal Hall effect in a room-temperature collinear antiferromagnet RuO$_2$ and catalogue, based on our symmetry rules, extensive families of material candidates. We show that the crystal Hall effect is accompanied by the possibility to control its sign by the crystal chirality. We illustrate that accounting for the full magnetization density distribution instead of the simplified spin-structure sheds new light on symmetry breaking phenomena in complex magnets and opens an alternative avenue towards quantum materials engineering for low-dissipation nanoelectronics., Comment: 21 pages, 5 figures

Antiferromagnetically doped topological insulators (A-TI) are among the candidates to host dynamical axion fields and axion-polaritons; weakly interacting quasiparticles that are analogous to the dark axion, a long sought after candidate dark matter particle. Here we demonstrate that using the axion quasiparticle antiferromagnetic resonance in A-TI's in conjunction with low-noise methods of detecting THz photons presents a viable route to detect axion dark matter with mass 0.7 to 3.5 meV, a range currently inaccessible to other dark matter detection experiments and proposals. The benefits of this method at high frequency are the tunability of the resonance with applied magnetic field, and the use of A-TI samples with volumes much larger than 1 mm$^3$., Comment: 6 pages, 4 figures. v2 accepted for publication in Physical Review Letters. Many points clarified, some parameter estimates revised

Antiferromagnetic spintronics focuses on investigating and using antiferromagnets as active elements in spintronics structures. Last decade advances in relativistic spintronics led to the discovery of the staggered, current-induced field in antiferromagnets. The corresponding N\'{e}el spin-orbit torque allowed for efficient electrical switching of antiferromagnetic moments and, in combination with electrical readout, for the demonstration of experimental antiferromagnetic memory devices. In parallel, the anomalous Hall effect was predicted and subsequently observed in antiferromagnets. A new field of spintronics based on antiferromagnets has emerged. We will focus here on the introduction into the most significant discoveries which shaped the field together with a more recent spin-off focusing on combining antiferromagnetic spintronics with topological effects, such as antiferromagnetic topological semimetals and insulators, and the interplay of antiferromagnetism, topology, and superconductivity in heterostructures., Comment: Book chapter

The recent demonstrations of electrical manipulation and detection of antiferromagnetic spins have opened up a chapter in the spintronics story. In this article, we review the emerging research field that is exploring synergies between antiferromagnetic spintronics and topological structures in real and momentum space. Active topics include proposals to realize Majorana fermions in an antiferromagnetic topological superconductors, to control topological protection of 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 physics concepts behind these proposals, and discuss potential applications of topological antiferromagnetic spintronics., Comment: Part of a collection of reviews on antiferromagnetic spintronics, 13 pages, 5 Figures