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Altermagnetic spintronics

T. Jungwirth,1 , 2 J. Sinova,3 , 4 P. Wadley,2 D. Kriegner,1 H. Reichlov´a,1 F. Krizek,1 H. Ohno,5 , 6, 7, 8 and L. ˇSmejkal9 , 10, 1

1 Institute of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00, Praha 6, Czech Republic

arXiv:2508.09748v1 [cond-mat.mtrl-sci] 13 Aug 2025

I nstitute of Physics, Czech Academy of Sciences, Cukrovarnick´a 10, 162 00, Praha 6, Czech Republic
2
S chool of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
3
I nstitut fu¨r Physik, Johannes Gutenberg Universit¨at Mainz, D-55099 Mainz, Germany
4
D epartment of Physics, Texas A & M University, College Station, Texas 77843-4242, USA
5
C enter for Science and Innovation in Spintronics, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
6
L aboratory for Nanoelectronics and Spintronics, Research Institute of Electrical
Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
7
A dvanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
8
C enter for Innovative Integrated Electronic Systems, Tohoku
University, 468-1 Aoba, Aramaki, Aoba-ku, Sendai 980-8572, Japan
9
M ax Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany
10
M ax Planck Institute for Chemical Physics of Solids, N¨othnitzer Str. 40, 01187 Dresden, Germany
The research landscape of magnetism has been recently enriched by the discovery of
altermagnetism. It is an unconventional phase of matter characterized by a d-wave (or higher even-
parity-wave) collinear compensated spin ordering, which enables strongly spin-polarized currents
in the absence of magnetization, and features fast spin dynamics. Simultaneously, on the applied
magnetism front, spintronic memories based on conventional ferromagnets are currently turning from
a niche to a mass produced integrated-circuit technology as they start to complement semiconductors
on advanced-node microprocessor chips. Our review connects these two rapidly developing science
and technology fields by discussing how the unique signatures of altermagnetism can impact the
functionality and scalability of future spintronic devices. As a reference, we first briefly recall the
merits and physical limitations of the present ferromagnetic spintronic technology, and of proof-
of-concept spintronic devices based on conventional collinear antiferromagnets and non-collinear
compensated magnets. The main part of the review then focuses on physical concepts of the
altermagnetic spintronics, and its potential interplay with ferroelectricity or superconductivity. We
conclude with an outlook on the nascent experimental research of altermagnetic spintronics, and on
the role of relativistic phenomena.

A. Introduction Altermagnetism1 ,2 has recently attracted broad attention because it enriches the landscape of magnetic phases by a distinct unconventional symmetry class, and because it opens prospects of research and technology advances beyond the realm of conventional magnets. This was highlighted by including the discovery of altermagnetism on the list of the Science breakthroughs of the year 20243 . Altermagnetism is classified and described by spin symmetries which establish that, apart from the conventional ferromagnetism and antiferromagnetism, there exists a third distinct collinear magnetic phase1 ,2 . The s-wave ordering in the conventional ferromagnets (left panel of Fig. 1), which spontaneously breaks the spin-space rotation symmetry and preserves the real-space rotation symmetry, leads to a spin polarized electronic structure with a net magnetization. The conventional antiferromagnetic ordering generates electronic bands which also preserve the rotation symmetries of the crystal lattice, but remain spin unpolarized as in the non-magnetic phase (middle panel of Fig. 1). The altermagnetic ordering1 is unconventional in the sense that it spontaneously breaks both the spin-space and the real-space rotation symmetries. Simultaneously, it preserves a symmetry combining the spin-space and real-space rotations. As such, the ordering in both the position and momentum space has a d-wave (g or i-wave) symmetry with 2 (4 or 6) spin-degenerate nodal surfaces. The electronic structure features a strong non-relativistic spin polarization whose sign alternates from one to the other side of each nodal surface, breaks the time-reversal (T ) symmetry, and preserves zero net magnetization (right panel of Fig. 1)1 ,2,4–6 . Recently, the unconventional T -symmetry breaking spin polarization, generated by a collinear compensated magnetic order, was experimentally confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements in g-wave semiconducting and metallic altermagnets MnTe and CrSb7 –16 , and in d-wave metallic altermagnets KV2 S e2 O or RbV2 T e2 O 1 7,18 . All these altermagnets order above room temperature. In general, robust altermagnetism is predicted in hundreds of materials, including 3D1 ,2,19–22 and 2D2 ,23–30 inorganic, as well as organic3 1,32 crystals. Altermagnetic conduction types can range from insulators and semiconductors to metals and superconductors, and altermagnetism can occur in crystals with common light elements and strong spin-coherence2 . The above characteristics of altermagnetism, together with the THz-range spin dynamics typical of compensated magnets, are prerequisites for future highly

2 scalable spintronic technologies2 . Spintronics based on conventional ferromagnets is currently transitioning from a niche to a mass-production information technology (IT). This is thanks to the magnetic random access memories (MRAMs) which, among other applications, are starting to complement semiconductors on processor chips3 3–36 . For the first time, magnets are breaking the monopoly of semiconductors at the very heart of the computer. Simultaneously, big data and artificial intelligence are boosting demand for high- capacity, fast and energy-efficient memory technologies. Altermagnetism can help future spintronics to meet these challenges. In the following sections we first briefly summarize, as a reference, the present ferromagnetic spintronic technology, and proof-of-concept spintronic devices based on conventional antiferromagnets and non- collinear compensated magnets. We then describe basic physical concepts of the altermagnetic spintronics, and outline their possible extensions exploiting the compatibility of altermagnetism with ferroelectricity or superconductivity. In the discussion we highlight the potential of altermagnetism for the functionality and scalability of future spintronic devices. We conclude with a brief discussion of initial experiments and the role of relativistic and topological effects in altermagnetic spintronics. We note that references given in this article are chosen to illustrate the discussed spintronic concepts, without an ambition to provide an exhaustive overview of the extensive recent literature on altermagnets. For comprehensive reviews on altermagnetism we point, e.g., to Refs. 2, 6, 22, 37–39. Ferromagnetic spintronics The invention of the semiconductor transistor and integrated circuits in the mid of the 20th century, and the subsequent development coordinated for decades by the International Technology Roadmap for Semiconductors (ITRS), led to the astonishingly fast and impactful transition into today’s information society4 0 . However, as the elementary feature size reached the ∼ 1 − 10 nm scale, this development era of semiconductor IT came to an end, with the last update of the ITRS published in 20164 1 . From 2017, the ITRS has been replaced by the International Roadmap for Devices and Systems (IRDS), in which spintronic MRAMs have taken a prominent role among the new emerging technologies complementing semiconductors3 6 . For example, the role of the semiconductor flash memory as the traditionally leading technology for embedded non-volatile memories is nearing its end because scaling it below the ∼ 20 nm node becomes prohibitively costly. Similarly, battery- backed semiconductor static random access memory is becoming unsuitable because of high leakage currents at the advanced nodes. Embedded non-volatile MRAM is emerging as a scalable, low-cost, leakage-free, power- efficient and radiation-hard alternative3 3–36 . This has led major foundries to introduce embedded MRAMs into their advanced-node processor chips, particularly for mobile, automotive and AI applications. The spintronic MRAM represents a revolutionary magnetic IT. The traditional role of magnets in IT has been in data storage on ferromagnetic hard-disk drives (HDDs)4 2 . Similar to magnetic-relay bits of the early computing machines, HDDs employ magnetic fields and mechanically moving parts for their operation. The spintronic MRAM does not use magnetic fields and is implemented on a solid-state integrated circuit. Instead of magnetic fields, MRAMs use the strong spin polarization of the electrical current, provided by the ferromagnetic order, for both reading and writing. In an MRAM bit, passing the spin-polarized current between a reference and a recording ferromagnetic electrode, magnetized anti-parallel or parallel, generates the tunneling magnetoresistance (TMR) readout signal which can reach hundreds of per cent4 3–45 . At a higher amplitude of the spin-polarized current, writing is facilitated by the spin-transfer torque (STT) which switches the magnetization of the recording electrode back and forth, depending on the direction of the writing current4 5,46 . An alternative three-terminal MRAM geometry is based on writing by the relativistic spin- orbit torque (SOT)4 7,48 . Here switching of the recording ferromagnetic electrode is controlled by an in-plane electrical current converted via the spin-Hall effect (SHE) into a transverse spin current in an adjacent relativistic charge-to-spin conversion electrode4 9,50 . The sensitivity of ferromagnets to magnetic fields and the stray fields generated by ferromagnets, which enabled the traditional magnetic technologies, are thus not needed for either reading or writing the spintronic MRAMs. They remain merely a limiting scalability factor which is suppressed at the expense of elaborate multilayer-stack designs of the present ferromagnetic MRAM bits. The non-volatile bi-stability of ferromagnets in principle enables to approach the fundamental physical limit of the minimum necessary energy to operate (erase) a bit, associated with the irreversible nature of the common computing schemes. According to the Landauer’s thermodynamic principle5 1,52 , this minimum dissipated energy is given by kB T ln 2, where T is the temperature and kB is the Boltzmann constant. At room temperature, the Landauer limit is thus in the meV range. On one hand, the least dissipative operation at the Landauer limit was so far demonstrated5 3,54 , using an elaborate quasi-equilibrium switching protocol5 1,52 , in exploratory ferromagnetic bits at switching times above µs, i.e., at switching speeds orders of magnitude below the GHz speed of commercial MRAM chips. On the other hand, the switching energy of the MRAM bits, consumed as Joule heating ∼ RI 2 tp , is many orders of magnitude above the Landauer limit4 5 . Here R is the resistance of the bit, tp is the writing pulse-time, and I is the

3 required writing current whose amplitude is independent of tp as long as the pulse-time is comparable or larger than the coherent spin-precession time-scale4 7,55,56 . In ferromagnets, this threshold time-scale is typically in the ns-range, corresponding to the GHz-range of the ferromagnetic resonance frequency governed by the weak relativistic magnetic-anisotropy energy. Instead of further linearly decreasing with decreasing tp , the switching energy increases by three orders of magnitude when moving from ns to ps writing pulse-times, because below the ns threshold, the required amplitude of the writing current becomes inverse-proportional to the pulse time4 7,55,56 . The quantum uncertainty principle imposes an additional fundamental constraint on the downscaling relationship between the operation time and energy. A single-photon pulse in a form of a narrow wave-packet peaked at the meV energy of the Landauer limit, i.e. peaked at the THz frequency, can have its pulse-length scaled down at most to the ps range. The simultaneously least-dissipative and fastest control of a bistable memory bit is thus in the ps (THz) range5 1,52,57,58 , which is beyond the reach of the present ferromagnetic MRAMs. Spintronics with conventional antiferromagnets and non-collinear compensated magnets Compensated magnets can share with ferromagnets the non-volatile bi-stability. In contrast to ferromagnets, however, the resonance frequencies are in the THz range due to the strong inter-spin-sublattice exchange energy. Moreover, compensated magnets with their vanishing net magnetization generate negligible stray fields and are insensitive to common magnetic fields. Over the past two decades, these favorable features have motivated research in spintronic devices whose functionality is governed by the compensated magnets. The field gained particular momentum after the experimental demonstration of electrical writing and reading at room temperature in memory bits fabricated from a collinear antiferromagnet CuMnAs5 9 on chips that can be controlled from a PC via a standard USB interface6 0 . Subsequent THz experiments6 1 showed that the switching pulse-length can be reduced to ps and, by using THz photoconductive switches, the ultra-fast switching was simultaneously spatially controlled down to nano-scale6 2 . Unlike ferromagnets, the switching energy did not show the unfavorable upturn for the pulse length reduced from ns to ps. However, the amplitude of the switching energy remained on the scale of 10 meV per atom which is still far from the Landauer limit of meV per bit. In the experimental memory bits realized in CuMnAs or other conventional collinear antiferromagnets, reading and writing is typically realized via T -symmetric responses. These can include, e.g., a combination of anisotropic magnetoresistance and SOT, or ordinary resistivity changes induced by switching into magnetic nano-textured multiple-metastable states5 9,63 . The weak relativistic origin of the responses in the former case, and heating close to the N´eel temperature and absence of bi-stability in the latter case hinder the energy efficiency. The utility of these relatively inefficient T -symmetric responses is dictated by the nature of the magnetic ordering in the conventional collinear antiferromagnets, with opposite spins in the crystal related by either translation or inversion. Such ordering does not generate the strong T -symmetry breaking and spin polarization in the non-relativistic electronic spectra typical of ferromagnets. Only weak T -symmetry breaking transport effects were detected in the collinear antiferromagnets, since in these magnets they require the relativistic spin-orbit coupling and non-linear response to the applied electric bias6 4 . (For reviews on antiferromagnetic spintronics see, e.g., Refs. 65–70.) Electrical readout by T -symmetry breaking linear responses, including the relativistic anomalous Hall effect (AHE) or the non-relativistic TMR, was demonstrated in the family of Mn3 X (X=Sn,Pt,...) non-collinear compensated magnets, in combination with electrical switching7 1–76 . Despite the vanishing net magnetization, the compensated non-collinear magnetic order can generate T -symmetry breaking electronic spectra and spin-dependent transport phenomena, reminiscent of ferromagnets7 7–85 . However, due to the non-collinearity of the magnetic order, spin-up and spin-down electronic states are strongly mixed which is reminiscent of the relativistic spin-orbit coupling. (For reviews on the non- collinear compensated magnets see, e.g., Refs. 6, 86–88.) Altermagnets, discussed in the following sections, share the favorable THz-range resonance frequency, lack of magnetic stray fields, and T -symmetry breaking linear responses. In addition, the collinear magnetic order in altermagnets can generate well separated and conserved spin-up and spin-down transport channels, which in ferromagnetic MRAMs underpin the strong TMR and STT responses for reading and writing. Altermagnetic spintronics and its interplay with ferroelectricity or superconductivity The right panel of Fig. 1 shows position-space and momentum-space cartoons of the T -symmetry breaking spin-dependent currents in a model d-wave altermagnet. For the electric bias E applied along the x (kx ) axis, the linear-response electrical current is dominated by electrons form the spin-up crystal sublattice (spin-up Fermi surface). Following the d-wave symmetry of the altermagnetic order, in which opposite-spin electronic states are related by a crystal-rotation symmetry, the spin polarization of the electrical current reverses from up to down when E is rotated to the y (ky ) direction8 9 . In contrast, the model ferromagnet in the left panel of Fig. 1 has only one (spin-up) magnetic

4 atom in the unit cell which determines the (spin-up) polarization of the electrical current, independent of the direction of the applied bias. In the model of a conventional antiferromagnet in the middle panel of Fig. 1, the translation symmetry relating the opposite- spin sublattices implies that both sublattices contribute equally to the electrical current, rendering the current spin unpolarized for any direction of the applied bias. In the model d-wave altermagnet, the longitudinal electrical current for E applied along the in-plane diagonal direction is spin unpolarized, reminiscent of the conventional antiferromagnet. However, the electrical currents of the spin-up and spin-down electrons are deflected by opposite angles from the diagonal direction of the applied bias8 9 , resulting in a transverse pure spin current3 1,89 . This non-relativistic phenomenon, dubbed spin-splitter effect (SSE), is reminiscent of the relativistic SHE4 9 . However the transverse spin currents in the two cases have a principally distinct phenomenology8 9 . The SSE breaks the T -symmetry and the polarization axis of the spin current is determined by the spin- polarization axis of the altermagnetic order. In contrast, the SHE is a T -symmetric response with the polarization of the spin current determined by the direction of the applied electric bias. In terms of the effects’ amplitudes, the SSE is predicted to outperform the SHE by two orders of magnitude because of its strong non-relativistic magnetic-ordering origin8 9 . Finally, for E applied along the out-of-plane direction of the model altermagnet in Fig. 1, both spin-up and spin-down electrical currents propagate along E, resulting in no transverse spin current and an unpolarized longitudinal current. The spin-polarized electrical currents were predicted to enable the giant magnetoresistance (GMR) effect in a stack comprising two altermagnetic layers separated by a metallic non-magnetic spacer2 3 . In analogy to ferromagnets, the altermagnetic GMR in the current-in- plane (CIP) geometry, illustrated in the left column of Fig. 2, is determined by the ratio of the spin-up and spin- down electrical conductivities for the given direction of the applied bias2 3 . When the altermagnetic order is orientated such that a spin-polarized electrical current is allowed in the out- of-plane direction, as shown in the middle column of Fig. 2, altermagnetism also enables the counterparts of the ferromagnetic current-perpendicular-to-plane (CPP) GMR for the metallic spacer, and the TMR for an insulating spacer2 3 . Remarkably, altermagnetic TMR effects of comparable amplitudes to ferromagnets are also predicted in the geometry where the total out-of-plane current is spin unpolarized2 3,90 . This is because even in this so called spin-neutral-current geometry, there are transport channels at individual momenta which are spin polarized, as illustrated in the right column of Fig. 2. We also emphasize that, while the GMR determined by the spin-dependent momentum-integrated conductivities is symmetry-restricted to d-wave altermagnets, the TMR is allowed by symmetry for any crystal orientation and in all d, g or i-wave altermagnets1 ,23 . Besides the above analogies to spintronics based on metallic ferromagnets, altermagnetism enables to readily extend the concepts to insulators. Similar to conventional antiferromagnets, and unlike ferromagnets, the altermagnetic ordering is common in insulators1 ,2 . In Fig. 3 we show, next to the stacks with metallic altermagnets, examples of TMR multilayers with insulating altermagnets. Here an unpolarized electrical current injected from a non-magnetic metallic electrode is spin-filtered through the thin-film altermagnetic insulator. The spin-filtering effect depends on whether the magnetic orders in the two altermagnetic layers are arranged parallel or antiparallel, resulting in the TMR effect9 1 . Insulating altermagnets open a possibility to eliminate the parasitic energy dissipation associated with Joule heating. Instead of the dissipative electrical currents, the altermagnetic insulators can be switched by electric fields. A particularly efficient and non- volatile switching mechanism is predicted in insulating multiferroic materials with coexisting altermagnetic and ferrolelectric orders (Fig. 3)9 2,93 . Note here that ferromagnetic ferroelectrics are elusive, again because of the general incompatibility of the ferromagnetic ordering with insulating electronic structures9 4,95 . If in a TMR stack, at least one of the altermagnetic electrodes is multiferroic, it can be switched by an electric field instead of current via the so-called altermagneto- electric effect9 3 , illustrated in Fig. 3. To explain the phenomenon, we first highlight that the local atomic spin density in an altermagnet can be decomposed into an isotropic dipole contribution and an anisotropic higher-partial-wave component (d-wave component for the model d-wave altermagnet in Fig. 3)4 ,96 . While the dipoles are ordered antiparallel between the neighboring magnetic atoms, i.e. have the N´eel-like order, the higher- partial-wave components are identical, i.e., are ordered ferroically. This latter non-N´eel ferroic component is one of the key signature of the altermagnetic ordering4 ,96 . Its sign is determined in the model altermagnet by the sense of the uniaxial anisotropic distortion of the local crystal environment of the magnetic atoms. When the altermagnet is ferroelectric, and the sense of the uniaxial distortion reverses between the two magnetic atoms upon reversing the electric polarization P , the sign of the higher-partial-wave component of the local spin density flips (Fig. 3). Since the sign of the higher-partial-wave spin-density component in the position space determines the sign of the alternating spin polarization of the energy iso-surfaces in the momentum space, the latter also flips when reversing P . In the context of the insulating altermagnets, we next recall a recent study of TmFeO3 5 7 . The experiment in this altermagnet demonstrated a reorientation of the axis of the ordered spins by a ps-long THz-field pulse at energy of ∼meV per atom. The study was performed at low temperatures, and only a transient reorientation

5 was demonstrated without accomplishing a stable non- volatile switching. Nevertheless, it showed that reaching the Landauer limit is becoming a legitimate, albeit still challenging goal in the research of ultra-fast altermagnetic spintronics. The predicted altermagnetic order in the insulating parent cuprate La2 C uO4 of high-Tc superconductors pointed towards yet another research avenue, this time exploring and exploiting the interplay of altermagnetism with superconductivity1 ,2,97 . The research direction offers alternative routes to suppress Joule heating, to generate strong spin-dependent responses and, in a broad sense, to merge the fields of spintronics and superconducting quantum technologies. An example is the infinite magnetoresistance (IMR) effect, predicted in a stack comprising two metallic altermagnetic layers separated by a spacer made of a conventional s-wave superconductor (Figs. 2 and 3)9 8 . Here the altermagnet suppresses the superconducting transition temperature via the inverse proximity effect. The strength of the suppression depends on whether the mutual magnetic configuration of the two altermagnetic electrodes is parallel or antiparallel, rendering the spacer superconducting in one configuration and, at the same temperature, non-superconducting in the other configuration. In the IMR effect, the altermagnetic order plays merely a role of suppressing the s-wave order parameter in the adjacent conventional superconductor. Other proposed device schemes employ the unconventional d-wave (or higher even-parity-wave) altermagnetic symmetry which can support unconventional Cooper pairing. The superconducting spin-splitter effect9 9 , schematically shown in the left column of Fig. 4, is an example of such device concepts. The phenomenon is based on the predicted unconventional spin-triplet Cooper-pairing correlations enabled by the spin-split altermagnetic electronic structure. As a result, an applied supercurrent from a conventional superconducting electrode adjacent to the altermagnet in the geometry analogous to the dissipative SSE induces an edge magnetization with opposite signs at opposite edges of the altermagnet. Also in analogy to the dissipative SSE, the sign of the edge magnetization reverses when reversing the supercurrent. In the right column of Fig. 4, we show two examples of superconducting quantum phenomena based on unconventional spin-singlet Cooper pairing with a finite center-of-mass momentum, which can be also promoted by the altermagnetic symmetry of the spin- split Fermi surfaces. A superconducting order parameter corresponding to the fine-momentum Cooper pairing1 00 is spatially inhomogeneous and contains nodes at which the order parameter changes the phase by π. As a result, a Josephson junction (JJ) comprising two conventional superconducting electrodes separated by an altermagnet shows 0 − π oscillations as a function of the length of the altermagnet1 01–104 . Here the JJ ground state oscillates between having the same phase of the order parameters in the two superconducting electrodes, or having a phase difference of π. Compared to ferromagnetic JJs1 00 , with a typically exponential decay superimposed on the 0 − π oscillations, the altermagnetic JJs can show undamped 0 − π oscillation over a large range of JJ lengths1 01 . In altermagnets, the center-of-mass momentum of the Cooper pair locks to discrete crystal axes which eliminates the rotational Goldstone mode and preserves a sizable superconducting gap1 05 . Moreover, besides proximity induced, it was also predicted that the unconventional spin-singlet finite- momentum superconductivity (Fulde-Ferrell-Larkin- Ovchinnikov phase) can coexist with altermagnetism in the same crystal1 06–110 . Both the proximity- induced and the intrinsic cases are predicted to lead to non-reciprocal transport thanks to the broken time-reversal and inversion symmetries by the combined order parameters1 05,109,111,112 . In the corresponding superconducting diode effect, a dissipationless supercurrent flows in one direction, while a dissipative normal current flows in the opposite direction (Fig. 4). Remarkably, the superconducting diode efficiency in altermagnets can reach 100% (in the presence of an external magnetic field), corresponding to the superconducting critical current going to zero for one of the two opposite current directions1 12 . The altermagnetic 0 − π JJs and superconducting diodes attract a broad interest as testbeds of the proximity-induced or intrinsic unconventional Cooper pairing in altermagnets, as well as for their potential applications in cryogenic ultra-low-dissipation electronics and spintronics, and quantum sensing and computing technologies. Additional tuneability of these devices can be achieved via gating or doping when employing altermagnetic semiconductors. Since altermagnetism is readily compatible with insulating/semiconducting band structures, it circumvents the limitations of ferromagnetic semiconductors in which enhancing the Curie temperature towards room temperature tends to diminish the semiconducting properties1 13,114 . MnTe is a prototypical room-temperature altermagnetic semiconductor whose intrinsic band structure features a band-gap and an alternating (g-wave) spin splitting, both on the favorable ∼ 0.1 − 1eV scale1 . Outlook: Experiments and relativistic effects In the previous section, we focused on phenomena and device functionalities based on strong non- relativistic physics of the altermagnetic ordering. This originates from the interplay of the single-particle crystal potential of a desired symmetry and many- body electron-electron interactions1 ,2,38,39 . The crystal- potential and the electron-electron interaction terms in the non-relativistic Hamiltonian preserve the spin- space rotation and crystal rotation symmetries, and the T -symmetry. The altermagnetic ground-state breaks

6 these symmetries spontaneously. In contrast, the single- particle spin-orbit coupling term in the 1/c2 expansion of the Dirac equation breaks the spin-space rotation symmetry already on the level of the (relativistic) Hamiltonian. Similarly, the single-particle Zeeman term in the Hamiltonian, describing the dipolar coupling of the spin and the magnetic field, breaks the spin- space rotation symmetry, as well as the T -symmetry. Therefore, on one hand, care must be taken to properly disentangle in experimental data contributions of the spontaneous symmetry breaking by the altermagnetic ordering, from contributions of the symmetry-breaking relativistic Hamiltonian terms. On the other hand, the altermagnetic ordering can constructively interplay with relativistic (and topological) physics, giving rise to a range of additional spintronic phenomena. In the following paragraphs we give examples of both of these types of the relativistic effects. The disentanglement of the non-relativistic and relativistic effects have been of concern since the initial experiments exploring the predicted spin-dependent transport phenomena in altermagnets. In measurements searching for the SSE in the d-wave altermagnetic candidate RuO2 1 ,4,89,115–117 , the competing SHE induced by the relativistic spin-orbit coupling was found to play a major role in the detected signals1 18–120 . Similar care has to be taken in measurements of the anomalous Hall effect (AHE) in the presence of an external magnetic field due to the competing contribution from the Zeeman term, as was the case of the initial AHE measurements performed again in RuO2 1 21 . We note that the research of this altermagnetic candidate is further complicated by an ongoing debate on the intrinsic vs. extrinsic origins of the measured magnetic signals in some of the studied samples, or by the absence of any detected magnetic response in other samples1 15–119,121–139 . The seeming controversy in the reports has been attributed to a fragile magnetic ordering in the correlated ruthenates, including RuO2 , which may be affected by the film-thickness, strain, impurities, or disorder1 29 . Zero-field remanent AHE signals, excluding the competing Zeeman effect, were detected in an altermagnetic candidate thin-film Mn5 S i3 1 40 , and altermagnetic MnTe, VNb3 S 6 or FeS whose robust collinear compensated magnetic orders were confirmed by neutron diffraction measurements1 41–143 . The AHE in collinear magnets, including altermagnets, vanishes by symmetry in the absence of the relativistic spin-orbit coupling6 . This makes the AHE an example of a constructive interplay of the T -symmetry breaking by the altermagnetic ordering and the relativistic spin-orbit coupling. Here we emphasize that the Hall vector requires the same broken symmetries as the magnetization vector. In other words, when the relativistic AHE is allowed by symmetry, the weak relativistic magnetization is also allowed in the altermagnet. However, the remanent AHE in altermagnets is not generated by T -symmetry breaking due to the weak magnetization, but rather by the strong T -symmetry breaking induced by the compensated altermagnetic ordering, as shown theoretically for the intrinsic Berry-curvature mechanism of the AHE in a number of altermagnets4 –6,140,141,143 . The ac T -symmetry breaking linear response, such as the X-ray magnetic circular dichroisms (XMCD), exhibits different spectral dependences of the signal generated by the altermagnetic ordering and by the magnetization. The two contributions can thus be qualitatively disentangled. This was theoretically and experimentally demonstrated in MnTe1 44 , showing that the altermagnetic contribution dominates in MnTe at zero field, while the magnetization contribution takes over at applied fields of a few T. In combination with the photoemission electron microscopy, the XMCD was used to monitor the controlled preparation of single-domain altermagnetic states in MnTe, as well as predesigned domain walls and topological vortices1 45 . The demonstrated control of the altermagnetic domains and textures is a key prerequisite for the future systematic experimental exploration of the proposed altermagnetic-spintronic device concepts. Another example of the constructive interplay of the altermagnetic ordering and the spin-orbit coupling are the spin-dependent conductivity and the related SSE in g and i-wave altermagnets, where these effects vanish by symmetry in the non-relativistic limit. In the g- wave altermagnet MnTe with the magnetic easy-axis in the c-plane, the relativistic conductivity tensor in the c-plane is spin dependent for the spin component along the c-axis1 41 . This remarkable relativistic spin- polarization orthogonal to the in-plane magnetic easy- axis can be understood by exploring the effect of the spin-orbit coupling in the momentum-space nodal plane parallel to the c-plane of this g-wave altermagnet. The spin-orbit coupling lifts the spin degeneracy in this nodal plane7 , and generates a d-wave like alternating spin polarization along the c-axis, enabling the corresponding spin-dependent conductivity and the SSE. The collinear relativistic spin-polarization in the nodal plane, i.e. the absence of a spin texture with a momentum dependent spin-axis, is symmetry protected in MnTe. Simultaneously, the amplitude of the relativistic spin splitting is large, reaching ∼ 100 meV thanks to the heavy Te atoms. The collinearity and the large spin- splitting amplitude strengthen the resulting relativistic spin-dependent transport effects. We note that until the discovery of altermagnetism, the experimental realization of the strong and collinear spin polarization induced by the spin-orbit coupling has been a long-sought elusive goal in the research of relativistic spintronics. Besides the diffusive spin-dependent conductivity mentioned above, it opens the possibility in altermagnetic 2D topological insulators to realize robust non-dissipative spin-transport phenomena, including the quantum spin (or anomalous) Hall effect2 9 . We expect

7 that these, as well as the other altermagnetic phenomena and device concepts outlined in this review, will attract broad experimental research interest, eventually leading to advances ranging from highly-scalable spintronic IT to viable superconducting and topological quantum technologies. Acknowledgments TJ acknowledges support by the Ministry of Education of the Czech Republic Grant No. CZ.02.01.01/00/22008/0004594 and ERC Advanced Grant No. 101095925. JS and LˇS acknowledge support by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - DFG (Project 452301518) and TRR 288 – 422213477 (project A09). LˇS acknowledges support by the ERC Starting Grant No. 101165122. DK acknowledges support by the Czech Science Foundation Grant No. 22-22000M, and the Lumina Quaeruntur fellowship LQ100102201 of the Czech Academy of Sciences. HR acknowledges Max Planck Dioscuri Program (LV23025). 1 ˇSmejkal, L., Sinova, J. & Jungwirth, T. Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry. Physical Review X 12, 031042 (2022). URL 9 Osumi, T. et al. Observation of a giant band splitting in altermagnetic MnTe. Physical Review B 109, 115102 (2024). URL https://arxiv.org/abs/2308. 10117v1http://arxiv.org/abs/2308.10117https:

https://link.aps.org/doi/10.1103/PhysRevX.12. 031042. 2105.05820. 2 ˇSmejkal, L., Sinova, J. & Jungwirth, T. Emerging Research Landscape of Altermagnetism. Physical Review X 12, 040501 (2022). URL http:

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abs/2405.024
2405.02409.

13

Ferromagnet Altermagnet Antiferromagnet !, # !, # !, #↓ !, #↑ !, #↑ !, #↑ !, #↑ !, #↓ !, # !, # ! " #! #" #↓ #↑ ## FIG. 1. Left: Cartoons of the position-space crystal structure and the momentum-space Fermi-surfaces showing the generation of the spin-polarized current in a model ferromagnet. Middle: Same for a model antiferromagnet with opposite-spin sublattices related by translation, rendering the electrical current spin unpolarized. Right: Model altermagnet with the longitudinal current for the applied bias along the x-axis dominated by electrons form the spin-up sublattice, and for the bias along the y-axis by electrons from the spin-down sublattice. Corresponding spin-polarizations of the longitudinal currents are also shown in the momentum space8 9 . For the bias applied along the in-plane diagonal, spin-up and spin-down electrical currents are deflected by an opposite angle form the bias direction. In the corresponding spin-splitter effect8 9 , the longitudinal current is spin unpolarized while a pure spin-current flows in the transverse direction. For the bias applied in the out-of-plane direction, the current is spin unpolarized.

CIP-GMR, IMR CPP-GMR, TMR

P a r a l l e l A n t i - p a r a l l e l #↓ # # # # # # #↑ #↑ #↑ #↓ #↓ #↑ #↓ $↓ $↑ FIG. 2. Left column: Schematics of the altermagnetic counterparts of the giant magnetoresistance effect in the current-in-plane geometry (CIP-GMR)2 3 in a stack comprising metallic altermagnetic electrodes separated by a metallic spacer (cf. middle-left panel in Fig. 3). The altermagnetic infinite magnetoresistance (IMR) effect9 8 can be realized when the spacer layer is formed by a conventional superconductor (cf. top-left panel in Fig. 3). Middle column: Altermagnetic counterparts of the current- perpendicular-to-plane (CPP) GMR and the tunneling magnetoresistance TMR effect2 3 with metallic and insulating spacers, respectively (cf. middle-left and bottom-left panels in Fig. 3). Right column: TMR in the spin-neutral-current geometry2 3,90 .

14 AMM SC AMM AMM NM AMM AMM NI AMM NM AMI NI AMI NM

−$ AMFE + + +$ + +

NM AMFE NI AMFE NM

2 3,90,98
FIG. 3. Left column: Trilayer stacks corresponding to the IMR, GMR, and TMR effects (cf. Fig. 2) . AMM stands
for altermagnetic metal, SC for superconductor, NM for normal metal, and NI for normal insulator. Top-middle: TMR stack
9 1
with altermagnetic-insulating (AMI) layers . Bottom-middle: TMR stack with altermagnetic-ferroelectric (AMFE) layers.
Left: Schematics of the altermagneto-electric effect providing a strong non-relativistic coupling, mediated by lattice distortions,
9 3
between the sign of the ferroelectric polarization (P ) and the sign of the alternating spin polarization . Only representative
examples of multilayers are shown in the left and middle column; other multilayers can be also considered, e.g., combining
different types of altermagnets in the stack.

15

Spin-singlet ! = 0 pairing Spin-singlet ! ≠ 0 pairing Spin-triplet ! = 0 pairing 0 − & Josephson junctions

' ' ' '+& )! " =0

Supercurent spin-splitter ' '+& )! " = 0 $↓

Superconducting diode ) !"

AM-FFLO AM-FFLO )! " )! " j $↑ $↑

) j $↓ FIG. 4. Left column: Comparison of spin-singlet (top) and spin-triplet Cooper pairing (middle) in altermagnetic Fermi surfaces, and schematics of the supercurrent spin-splitter effect9 9 due to the spin-triplet pairing correlations in the altermagnet separating two conventional superconducting electrodes (bottom). Opposite magnetizations are generated at opposite edges of the altermagnet upon injecting the supercurrent. Right column: Finite-momentum spin-singlet pairing (top), and corresponding 0 − π Josephson-junction1 01–104 (middle) and superconducting-diode1 05,109,111,112 (bottom) quantum devices. In the Josephson junctions, the ground state with zero supercurrent corresponds to the same phase of the order parameters in the two superconducting electrodes, or to a phase difference of π, depending on the length of the altermagnet separating the two superconductors. In the superconducting diode, a supercurrent flows in one direction and a normal dissipative current in the opposite direction. AM-FFLO stands for a coexisting altermagnetic phase with a superconducting Fulde-Ferrell-Larkin- Ovchinnikov phase.